MATH JOURNALS: FROM MATH TALK TO THE HOME-SCHOOL CONNECTION A HANDBOOK FOR TEACHERS OF GRADES 1-6 by Jennifer Tippett B.Sc. Hons., Trent University, 2000 B.Ed., Queens University, 2001 PROJECT SUBMmED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF EDUCATION IN MULTIDISCIPLINARY LEADERSHIP UNIVERSITY OF NORTHERN BRITISH COLUMBIA February 2013 © Jennifer Tippett, 2013 UNNERSITYofNORTHERN BRmSH COLUMBIA UBRARY Prince George, B.C. 11 Abstract Mathematical education has evolved over the past 15 years resulting in significant changes. New math standards and a variety of mathematical communication skills are tolls which are now required to be incorporated into teachers' daily practices. The central aim of this handbook is to provide teachers with specific strategies, tips, and examples of how to effectively incorporate oral and written mathematical communication skills into today' s classroom. This handbook is designed to lead teachers through the natural progression of mathematical learning and through the development of a home-school connection. Incorporating math journals into the classroom involves establishing a positive learning environment where students feel comfortable sharing ideas and asking questions of others. After establishing a positive learning environment, teachers will then have the opportunity to develop student discussion and to create various group structures. Through this process, teachers will be able to determine what students understand which will drive instruction as well as providing students with the vocabulary and oral skills necessary for sharing and understanding their mathematical learning. Written math communication is most successful when it is done through the use of a problem solving math journal where students are challenged to put their mathematical ideas on paper in an effort to solve word problems and demonstrate their understanding. Written communication also offers students the opportunity to reflect upon their written material and orally share solutions. Once math journals have been established in the classroom, teachers can then proceed to involve parents through the use of a Home-Math Journal which gives lll students an authentic audience and allows parents to become involved and informed in how math is being taught today. Following and building upon the above progression of skills will increase students mathematical learning as they become skilled at expressing their mathematical thinking. This increased ability to express ones ideas will ultimately increase parents and teachers ability to understand what students comprehend and how they can best support their mathematical learning. IV TABLE OF CONTENTS Abstract 11 Table of Contents 111 List of Tables v List of Figures Vl Acknowledgements Vll Chapter 1 Introduction Project Overview Rationale Importance of the Project 1 1 2 4 Literature Review Oral Communication Why oral communication is important How to incorporate mathematical communication in the classroom How to support oral communication in the mathematics classroom Written Communication Why written communication is important How to incorporate written communication in the math classroom How to support students ' mathematical writing Home-School Connection Why involve parents in mathematics education How to involve parents to create a home-school connection How to support parents and the importance of math journals Conclusion 34 35 36 39 43 48 Chapter 3 Research Methodology 50 Chapter 4 Handbook Introduction Table of Contents Introduction Math Talk - Oral Communication Why math talk? How to promote good math talk. 52 54 56 Chapter 2 6 7 7 12 21 22 23 26 59 61 64 v Chapter 5 References How to support math talk. Math Journals- Written Communication Why write? How to promote good math journal writing. How to support math journal writing. Home Math Journals- the Home-School Connection Why create a home-school connection with math journals? How to create a home-school connection with math journals. How to support an effective Home-Math Journal. Summary Appendices Appendix A- Initial Letter Home Appendix B- Helpful Hints Appendix C -Reflection Sentence Starters Appendix D- Sample Problems 79 87 88 92 106 113 114 117 135 145 151 152 153 154 155 Conclusion 158 162 Vl List of Tables Table 1. Bloom's Taxonomy 69 Table 2. Math Talk Strategies 72 Table 3. Books for Teaching Math Concepts 93 Table 4. Math Writing Strategies 98 Table 5. Sample Questions and Prompts to get Students Thinking 137 Vll List of Figures Figure 1. Math Journal Sample. This is a sample math journal entry completed by a Grade 2 student. 101 Figure 2. Home-Math Journal entry. This figure illustrates the student solution to a Home-Math Journal problem. 120 Figure 3. Pizza Pieces Problem. Student solution to the pizza pieces problem. 123 Figure 4. Pizza Pieces Problem. Home Partner solution to the pizza pieces problem. 124 Figure 5. Pizza Pieces Problem. Student reflection and sample feedback to the Home-Math Journal entry. 124 Figure 6. Chocolate Problem. Student solution to the chocolate problem. 128 Figure 7. Chocolate Problem. Home Partner solution to the chocolate problem. 129 Figure 8. Chocolate Problem. Reflection and sample feedback to the Home Math Journal entry. 129 Vlll Acknowledgements I wish to acknowledge the ongoing support and encouragement of my supervisor, Dr. Andrew Kitchenham, who helped to guide me on this journey. I would also like to thank my committee members, Dr. Henry Harder and Dr. Colin Chasteauneuf for their support and feedback in the creation of this project. Most importantly, I would like to take this opportunity to sincerely express my love and thanks to my amazing husband, whose endless love and continuous support has helped me to complete this project. During the course of this endeavour, our family has grown with the addition of three wonderful children, and I would like to thank them for their unwitting support and co-operation. 1 Chapter 1: Introduction Project Overview The teaching of mathematics has changed significantly in the last 15 years. In fact, mathematical communication has been one of the major focal points of recent calls for reform in mathematics education (National Council of Teachers of Mathematics [NCTM] , 2000). The new math standards require the use of a variety of mathematical communication skills to be incorporated into teachers ' daily practices which include speaking, writing, demonstrating, and depicting ideas visually. Many educators of mathematics also believe that communication is an important part of mathematics. Communication allows students to share and clarify understanding while also asking questions, reflecting, and discussing learning with others. Communication allows students to develop rich mathematical understandings while also developing their oral and written communication skills as they learn to be clear and convincing. Teaching math communication is very similar and has many ties to the way teachers teach students to write. We know that individuals learn to speak well before they learn to write; therefore, developing oral communication skills is a natural place to start teaching mathematics communication. There are handbooks that address oral communication (e.g. , Chapin, O' Connor, & Anderson, 2009; Parrish, 2010) and which discuss the variety ofways to use math journals (e.g., Bums, 1995; Countryman, 1992); however, there is not one resource that seems to address the natural progression of how we should be teaching students to effectively develop their mathematical communication skills. In addition to developing students ' mathematical communication skills, it is important to involve parents in the learning of mathematics. Parents are an invaluable 2 resource as they are able to nurture children physically, emotionally, and intellectually (Fan & Williams, 2010). While parents can usually find time to read a story to their children, thus instilling a love for literature, they are often unsure as to how to instil an appreciation or love of mathematics. Not only is mathematics a subject that is necessary for students to attain in order to function adequately in society, mathematics is also a subject which should be seen as being enjoyable and fun. Parents ' attitudes towards mathematics have an impact on children's attitudes, thus it is important for parents to show interest and enthusiasm towards their children ' s mathematical learning (Fan & Williams, 2010). Parents need support from classroom teachers so that they can actively become involved in their children' s mathematical learning. Parents need to be shown how to best support their children' s learning as the methods used to teach mathematics have changed significantly in the past 15 years (de Abru & Cline, 2005). This project provides teachers with information and ideas to begin to create mathematical communities within their classrooms that are built upon math communication in both oral and written forms, while also bridging the home-school gap through the use of a Home-Math Journal. Rationale This project involved the creation of a handbook which is designed to help teachers develop students' oral and written communication skills by first focussing on oral communication prior to developing written communication skills (see Chapter 4). This resource will help teachers to fmd ways to reach students and change the way they are teaching mathematics. Furthermore, this handbook will show teachers how to direct and guide students as they develop deeper mathematical understandings. In addition to addressing oral and written communication with students at school, it will assist teachers in developing a 3 home-school connection by getting parents involved in the learning of mathematics. Parent involvement is critical in helping to develop positive relationships and open the lines of communication between parents and the school. This handbook will address the natural progression from oral to written communication followed by how to develop a home-school math connection to inform and assist parents in helping their children with the learning, understanding, and communicating of mathematical ideas. In addition to addressing the oral and written communication of mathematical ideas, the handbook also demonstrates how to develop an effective home-school connection. There is a large body of research that has shown a positive connection between parent involvement and a student' s academic achievement (see Fan & Williams, 2010 for a full discussion) . Developing a positive relationship between the home and school seems to have a significant effect on student achievement, while also helping parents to understand the ways that math is being taught in schools (Fan & Williams, 2010). This project introduces a homework task that builds upon the use of math journals where students and parents are able to explore mathematical problem solving and written math communication. The Home-Math Journal helps to open the lines of communication between the parent and child as they compare, discuss, and reflect upon their solutions. Not only does the Home-Math journal help parents become involved in students' mathematical learning, but it also helps parents understand how to help their children at home when they are having difficulties. In my experience as an educator, these resources on their own are in high demand as every year I have parents who are unsure of the way students are learning math today and also unsure of how they can best support their children at home. Many parents feel that since they did not perform well in 4 school themselves, that they will not be able to support their children ' s mathematical learning (de Abreu & Cline, 2005). The handbook helps to address parents' concerns while offering a positive homework activity for parents (or another home partner) and students to engage in as it will help to further develop students ' mathematical understandings. Parents and students will be able to see similarities and differences in their solutions and reflect on these solutions while also fmding misconceptions and how these misconceptions may have occurred. Thus, in addition to addressing how teachers should introduce oral and written communication skills into the classroom, this resource also provides teachers with a resource to develop a positive homeschool mathematics connection through the use of a Home Math Journal. Importance of the Project The creation of the handbook, Math Journals: From Math Talk to the Home-School Connection, is a much-needed resource for three main reasons. First, the handbook provides important information on how to successfully introduce oral and written communication into a mathematics classroom. Currently, teachers need to go to a variety of sources in order to locate strategies and information regarding the use of math communication. Furthermore, there are many strategies that teachers can use to help promote good math talk in their classrooms, to which this resource will provide teachers easy access. The resource also explores how teachers can become comfortable promoting and supporting math talk and math journal writing in their classrooms and schools. The handbook breaks down these skills and shows teachers how to develop their roles as mediators of math communication (Brown & Hirst, 2007). 5 Secondly, this resource will help teachers to see the link between oral and written communication and the importance of developing these skills in our students to develop deeper mathematical understandings and overall comfort levels. Very few of the resources available explain the importance of developing students ' oral mathematics communication skills prior to attempting student written communication skills. The development of these skills is very important because students need a lot of practice with math talk before they can progress to writing their math ideas on paper (Turner, Styers, & Daggs, 1997). Finally, the handbook also provides a positive home-school relationship in mathematics while offering an important and relevant homework activity which requires and promotes the interaction of parent and child as each learns from the other. The Home Math Journal helps to inform parents of how math is being taught and how their children understand different math concepts. Furthermore, the handbook provides parents with strategies and suggestions as to how they can best support their child ' s learning at home, especially if their children are unsure how to begin solving or communicating their ideas on paper. 6 Chapter 2: Literature Review Given that mathematical communication has become a major focus for reform in mathematical education (NCTM, 2000), there is a need to examine how we as teachers can guide students to become effective communicators of mathematics. Students need to become proficient in both oral and written mathematical communication. While oral and written communication sound simple on paper, these concepts involve a variety of different skills as students need to be able to work collaboratively, share, question, reflect, clarify, and represent their ideas in a variety of forms. Structuring the mathematics classroom around mathematical communication helps to create rich learning experiences for students as they explore and develop deeper mathematical understandings. In creating a handbook on the effective use of math journals, it was important to take a close look at what research and resources were available on this topic in order to build on the information that has already been presented and to create a resource that would be easy for teachers to understand. When teaching students to communicate effectively, it is first important to have them engage in oral discussions before embarking on written mathematical tasks. I have structured the literature review below into three sections: oral communication, written communication, and the home-school connection as these sections correspond to the divisions within my handbook. Many of the resources and studies examined below help to demonstrate that the most important skill that we can teach students today is how to make sense of information, as students today are often bombarded with information everyday from different sources (Rakow, 1999). In order to help students make sense of information in mathematics, we need to examine how students develop a deeper understanding of mathematical concepts and the roles that oral and written communication play in this 7 learning. Furthermore, it is important to develop a positive home-school connection in the learning of math to both educate parents on how math is being taught while also providing rich tasks that parents and students can engage in together to further develop these deep mathematical understandings. Oral Communication At the beginning of the school year, the first initiative teachers should be focusing on is creating a safe, "communication friendly (p.31)" (Jarman, 2008) environment. Students should feel safe participating and sharing ideas in a variety of different situations and in different groupings. In order to help develop a safe learning environment, teachers need to ensure that rules are set from the beginning of the year that address acceptable and unacceptable behaviours. Behaviours that may hinder mathematical communication should be addressed immediately and not tolerated (Cobb, Wood & Yackel, 1994). Once a safe and communication-friendly environment has been established, teachers can begin to see the benefits of oral communication in their classrooms. The safe space created in a classroom will allow for students' individual confidences to develop and grow. The need for oral communication in math is increasing and research has demonstrated that oral communication is central to students developing their own understandings of mathematic concepts. The following explores why oral communication is important, how to incorporate oral communication into the mathematics classroom, and how to provide support to students as they become more proficient at oral communication in math class. Why oral communication is important. Historically, mathematics has been a subject where teacher talk often dominated class discussions and students came to rely on teachers to be the experts with the answers as opposed to learning that they could work out 8 their own solutions and develop deeper mathematical understandings. Since mathematics is often conveyed using symbols, oral and written communication in mathematics is not often recognized as being as an important part of a mathematics program. However, it is through communication that ideas become permanent fixtures that can be reflected upon, refmed, discussed or amended. The communication process also makes ideas public which challenges students to think and reason about mathematics and how the results of their thinking relate to the thinking of others (NCTM, 2000). Students do not learn to communicate about mathematics naturally, thus teachers need to guide students and teach them the skills necessary to allow them to share and communicate their mathematical ideas effectively (Cobb, Wood & Yackel, 1994). Brown and Hirst (2007) examined the need for mathematics reform and the importance for sociocultural practices (i.e. , questioning, reasoning, and justifying). They used an extension ofVygotsky' s experimental-developmental method to examine the influence of social and cultural processes on learning and development in the mathematics classroom. The study was conducted over the course of one year and involved eight elementary teachers who were selected from Grades 1, 2, 4, 5, and 7. The eight teachers were given a professional development session by the first author to become familiar with the sociocultural approach to teaching and learning. These sessions involved the first author co-planning and co-teaching lessons with these teachers to help the teachers become familiar with the investigative processes and ways of interacting within a mathematics community. Teachers were shown how to structure lessons so that the teacher and students would work in small groups to represent a problem, and then compare their representations with other group members and fmally with other groups. 9 Brown and Hirst (2007) emphasized that the role of the teacher should be as a mediator in math talk. With the teacher acting as a mediator, students were found to be able to construct their own understandings. Each teacher' s class was videotaped twice during the year while they were doing mathematics. Anecdotal notes were taken regularly throughout the school year with regards to teacher-student and student-student interactions and students and teachers were asked to keep a reflective journal throughout the year. Closer to the end of the study, each teacher was interviewed individually about his or her feelings about teaching mathematics. During these interviews, a video of each teacher' s classroom teaching was played to stimulate recall and allow for reflection and discussion. Through an analysis of these classroom interactions, teacher interviews, and student reflective journals, Brown and Hirst (2007) found that when teachers shared their expertise and mediated the students ' learning, students were able to build upon the teachers ' modeling and make sense of math concepts. Furthermore, when teachers acted as reflective and critical users of math talk and were able to act effectively as mediators, the quality of student learning was vastly improved. The authors demonstrated that math talk is crucial in the development of mathematical understanding and proficiency. Furthermore, Brown and Hirst (2007) showed the importance of teacher modelling and mediating in getting students to express their ideas orally. In order to teach students to talk and reason together effectively, Mercer and Sams (2006) designed, implemented, and evaluated an experimental intervention program called Thinking Togeth er. The research included 406 children and 14 teachers, and was carried out from 2002 to 2004. Teachers were selected for this study based on their willingness to participate in research on children 's talk. The main goals of the program created for this 10 study were to increase student' s awareness of the use of spoken language as a tool to think together, to develop children' s abilities to use language both collectively and on their own, and to enable students to use language in their study of math and science. Teachers in this study were given one half-day training in-service about the program with follow-up meetings and discussions in their own schools during the 23-week intervention. A matched set of control classes was also used to compare findings . Teachers were provided with 12 detailed lesson plans which began with a teacher-led whole class introduction, followed by a group discussion, and a follow-up whole-class discussion and reflection on the learning. Each class also developed ground rules for class discussions which were posted in each teacher' s room and reviewed often in class (Mercer & Sams, 2006). Mercer and Sams (2006) gathered the following data: pre- and post-intervention video recordings of a focal group in each class, video recordings of other groups of children in the same classes engaged in the Thinking Togeth er lessons, video recordings of teacher-led whole class sessions, post-intervention audio recordings of interviews with teachers and students, and pre- and post-intervention tests of children ' s knowledge and understanding in maths and science. In an analysis of teachers' interactions with children, Mercer and Sams (2006) looked at the extent to which teachers: used "why" questions, used reasoning words (because, so, if), offered personal reasons to back up statements, checked that everyone who had an idea had been heard, and sought agreement amongst the class at the end of a debate. In an analysis of students talk in groups, Mercer and Sams (2006) examined how talk had changed by examining how children in all classes used talk to solve problems and to look for changes in the pre-intervention and post-intervention talk of students in these classes. 11 The results of this study showed that the individual teacher approach differed and mattered in the overall findings , therefore training teachers for more than half a day may have made more of a difference, as some teachers did not follow the ideals of the program in the same ways. Furthermore, Mercer and Sams (2006) found that students needed to be taught how to interact with one another in order to improve the quality of students ' language and overall understanding of the underlying mathematical concepts. Thus, the research showed that students can be taught to use talk as a tool to improve their reasoning skills, and group talk activities can also help the development of individual students' math reasoning, understanding and problem solving skills. This research study supports the notion that talk is an important factor in improving student understanding and in empowering students to become comfortable with math concepts and with the sharing of their strategies and solutions as they learn. Furthermore, Mercer and Sams (2006) demonstrated that a teacher' s approach has a large impact on how well students are able to effectively develop their communication skills. In shifting mathematics teaching from a classroom that is dominated by teacher talk to one that focuses mainly on student talk, teachers will be empowering students to create their own mathematics understandings. Through oral communication in cooperative group situations, students are able to make their thinking permanent and accessible to others. Students are then challenged to question, reason, and justify their thinking which leads to greater understanding and use of higher level thinking skills. Once students have been taught how to interact with one another in the mathematics classroom, teachers will take on a less direct role where they become responsible for modelling talk behaviours and mediating class discussions. Having the opportunity to discuss mathematics concepts cooperatively with 12 peers allows students the opportunity to justify their thinking, learn from others ideas, and ultimately increase their own personal mathematics understanding. How to incorporate mathematical communication in the classroom. The information above gives support as to why oral communication is and should be an important aspect of mathematics teaching and learning. It is then up to teachers to find ways to incorporate mathematics communication in the classroom. For traditional teachers who are used to disseminating facts, algorithms, and knowledge to students, a shift to oral communication where students create their own knowledge often takes time and forces traditionalist teachers out of their comfort zones into an area that is foreign to them. Students are thinkers who should be given the opportunity to work in different groupings to discuss, critique, and create mathematical understanding, and not just factual regurgitation. In transforming teachers from traditionalist mentalities, Cobb, Wood, and Yackel (1994) purported that teachers need to establish social norms within the classroom where student thinking is valued even more than correct answers, where students are expected to express thinking, explain, and justify solutions as well as listen to and ask questions about the ideas of others. Students should also be prepared to share their solutions with the class at all times because class discussions typically followed pair or small-group collaboration (Cobb et al. , 1994). Once social norms have been established within the classroom, Cobb et al. (1994) and Stein, Engle, Smith, and Hughes (2008) advocate the need for a problem solving approach to learning mathematics. The problem solving approach helps teachers shift from being simply the dispensers of knowledge to orchestrating student learning (Stein et al., 2008). Furthermore, teachers should realize that the greatest learning comes out of situations 13 involving conflict, confusion, surprise, and during social interactions (Cobb et al. , 1994). Students need challenging problems, collaborative group work, and class discussions about student solutions which will allow students to express their thinking and create learning opportunities. Cobb et al. (1994) have found that once norms have been set in the classroom, teachers can present a problem, have students work on the problem in pairs and follow up with a class discussion of student solutions with no need for external rewards. The class norms and challenging math tasks allow students to become engaged in the task as opposed to ego-oriented as they develop autonomy, both social and intellectual (Cobb et al. , 1994). Students begin to use one another as resources and become skilled at sharing solutions with the whole-class . Meanwhile, teachers learn how to orchestrate discussions in order to build on personal and collective sense-making as they guide student thinking in mathematically sound directions (Stein et al. , 2008) . Teachers need to ensure that class-discussions are not just a show and tell of correct answers; instead teachers need to select students to present solutions in a way that scaffolds the learning. Teachers need to help draw connections between solutions or link ideas to strategies that were the most helpful or efficient (Stein, et al. , 2008). Soter, et al. (2008) explored and evaluated nine approaches to classroom discussions to better understand quality classroom discourse. The nine approaches were found to span three different stances towards text: an expressive stance where students explore meaning in a text through conversations similar to the conversations adults would engage in about texts; an efferent stance where students derive information from texts; and a critical-analytic stance, where students focus on inquiry that allows for reflection and reasoning and the listening of others ideas. The nine approaches were: Grand Conversations, Book Clubs and 14 Literature Circles (Expressive Stance); Instructional Conversations, Questioning the Author and Junior Great Books (Efferent Stance); Collaborative Reasoning, Philosophy for Children and Paedia Seminar (Critical-Analytic Stance). The authors collected and coded transcripts from students in Grades 3 to 9 which ranged in length from five to thirty minutes. Through the coding of transcripts, Soter et al. (2008) found that students contributed most to conversations in the expressive stance, teachers contributed most to conversations using an efferent stance and students and teachers shared control of conversations in the critical-analytic stance with students showing more elaborated explanations. The results of this study also demonstrated that the expressive stance led to reasoning words which indicated taking a position; the efferent stance and expressive stance both demonstrated a use of reasoning words which indicated generalizations; and the critical-analytic stance showed reasoning words which demonstrated speculation and overall led to more higher-level thinking and reasoning (Soter et al., 2008). The authors demonstrated the need for a shift to a critical-analytic stance where students are more engaged, involved, and challenged to think critically about mathematical concepts. Turner, Styers, and Daggs (1997) explored ways to encourage mathematical thinking in students. The authors believed that learning mathematics helped students make sense of their world as they learned to reason, solve problems, and explore. The authors further believed that teachers lacked examples of how to change their teaching from rote; dull math to math that was challenging and would ultimately increase student engagement. Thus, Turner et al. (1997) set out to make math instruction more challenging and less rote. A challenging activity was defined as one that pushed students to create new understandings, 15 allowed students to struggle with their thinking and reasoning, and provided opportunities for students to select appropriate problem-solving strategies. Turner et al. (1997) designed activities and created math units that were challenging, supported students ' autonomy, and involved collaboration. Two lessons were shared in this article and sample interactions between a teacher and his/her students in the class were discussed. There was no indication of who the teacher was or how he or she was selected. The researchers found from these two sample lessons, that student engagement increased as students questioned, explained, and talked through their solutions. Furthermore, when teachers promoted student autonomy by giving students control of their learning, the researchers reported that it demonstrated more motivation for students to think on their own. Turner, et al. (1997) also believed that collaboration was linked to students thinking through and justifying their strategies. Collaboration made students question, critique, and rethink, which were skills that students were familiar with, though not in math class. Collaboration activities also helped students to become a community of learners. Researchers found that students enjoyed the discussions and took pride in being able to see strengths and weaknesses in others work. By having students discuss their mathematical ideas, allowing them to create their own understanding and explanations through math talk and journal writing, teachers were able to make a difference in student autonomy and overall mathematical understanding. This research study demonstrated the importance of communication as a way to increase student motivation and participation. By fostering student autonomy and collaboration, Turner, et al. (1997) showed that students can build their overall mathematics communication skills and understanding. 16 Si1bey (2003) also discussed the importance of oral communication as she shared the importance of having students write in math class from her own experiences as a math teacher and leader. It is explained that writing in math class actually helped students to work their way through problems, gave them opportunities to listen to others solutions and allowed them to organize their own thinking. Silbey (2003) gave some strategies for getting students to discuss information in the classroom as a first step, followed by a process for getting students to write which used a combination of individual think time, small-group discussion, reporting back to the class as a whole, and finally, individual student written drafts of their solutions. Following these key strategies, Silbey (2003) demonstrated how teachers should assess and reflect on the written information submitted by their students, which included an examination of student understanding, use of math terminology, ability to articulate their thoughts in written form and any other patterns or common errors that might become evident. Silbey (2003) also made reference to the importance of effective questioning in the development of students critical thinking skills. This research study provided further support for the need for oral communication and collaborative practices in math prior to students being comfortable formulating their mathematical thinking in writing. Oral communication is a powerful tool that can be used to assess student understanding while also informing instruction. In their book, Chapin, O' Connor and Anderson (2009) explored the use of math talk to improve student learning. The researchers used data that were collected during Project Challenge, an intervention project which ran from 1998 to 2002 in one low-income urban school district in the United States. The authors hoped that combining a solid curriculum 17 with instruction based on mathematical understanding, and a heavy emphasis on talk and communication about math, would allow these learners to be able to develop a deeper understanding of mathematic concepts, and produce learners who would persevere through challenging problems while gaining individual mathematical confidence. It took the first seven to eight months before the authors began to see an improvement in students' reasoning and before student responses became more complex and sophisticated. Chapin, et al. (2009) believed that talk in the mathematics classroom caused misconceptions to surface, which allowed these misconceptions to be addressed and student understandings to be examined. By using a variety of discourse formats (whole-class, small group, partner), students were able to improve their ability to reason logically. These discussion formats allowed students to hear alternative viewpoints, begin to give justification to their own reasoning, and demystified mathematical concepts, making everyone's thinking public. Allowing students to talk about their thinking and problem solving gave students the chance to observe, listen to, and participate in mathematical thinking, thus pushing them out of their comfort zones. Whole-class discussions provided motivation to students as they were excited to share and showed a genuine interest in the ideas and claims of others. Overall, talk in the mathematics classroom made student thinking public, helped students elaborate, model, build on and add to complex ideas (Chapin et al., 2009). In addition to different discourse formats, Chapin et al. (2009) introduced and explained five different "talk moves" that teachers could implement in their classrooms for different purposes. The first talk move was revoicing which could be used to deal with unclear answers that students may give. In this move, the teacher tried to repeat some or all of what the student had said, and then asked the student if the teacher's revoicing was 18 correct. This move allowed students to hear their own ideas expressed orally and were then able to add to, or correct any aspects of their answers that no longer made sense to the student. The second talk move was rep eating which could be used to understand what other students understood of another's reasoning. In this move, the teacher asked one student to repeat or rephrase what another student had said and then immediately followed up with the first student to ensure that the repeating of the second student was accurate. This move helped students to realize that other individuals were listening to their ideas and thus, the students needed to make their thinking and explanations explicit. The third talk move was reasoning which could be used to have students make their reasoning explicit. In this move, the teacher had a student make a claim, and ensured that all students had heard and understood the idea presented. The teacher then asked if another student agreed or disagreed with the claim presented, and asked that student to explain why or why not. This move allowed students to explain their reasoning and justify their thinking as they explained why they did or did not agree with the claim presented. The fourth talk move was adding on which could be used to increase participation in the discussion. In this move, the teacher began by revoicing two positions that had emerged, and modeled how to respectfully check with the originators of the two positions that her revoicing was accurate. The teacher then asked others to contribute by asking them to either agree, disagree, or to add to the other students ' comments. The final talk move discussed by the researchers was waiting which should be used regularly in the classroom following a posed question. In this move, the teacher posed a question to the class and then waited for at least five seconds for students to think before 19 calling on someone to answer. Wait time is also important to remember after calling on a student to answer. Wait time allows students to organize their thoughts before they start to answer. Wait time is also important to remember when dealing with students whose first language is not English. The remainder of the authors' book discussed more in depth how these talk moves and formats could be used to help students reason, solve problems, and discuss mathematical computations and concepts. The authors also provided some troubleshooting ideas for teachers who may be dealing with students who do not talk, students that monopolize the talk, off-task behaviour in small group discussions, and many other issues that may arise when introducing math talk into the classroom. This resource has provided one example of what types of handbooks are available for teachers and again this resource only focused on oral discourse, while not extending to written communication or parent involvement. Similar to the ideas of Chapin et al. (2009), Stein et al. (2008) advocated five steps to help teachers be successful orchestrators of whole-class discussions: Anticipate, Monitor, Purposefully Select Responses, Sequence, and Connect. In the first step, teachers need to anticipate how students may interpret the problem, any strategies that students may use to tackle the problem that is presented, and how these strategies might relate to the math concepts being presented. By anticipating student thinking, teachers are able to prepare possible questions or activities that may be able to support possible student misconceptions. The second step in being a successful leader of class discussions is monitoring. During this step teachers need to pay very close attention to mathematical thinking. When students are working in pairs or small groups on the problem that has been presented, teachers need to be actively circulating through the classroom with the goal of identifying 20 mathematical learning/strategies that are being used and which of these ideas should be shared with the whole class. Teachers need to make sense of student thinking, even when this thinking may be incorrect. Stein et al. (2008) suggest listening in, asking questions and taking notes to be able to reference student thoughts later on and help with the next step: Purposefully Selecting Responses. During this third step, teachers need to use the information they gathered during the student work sessions to select students to present math ideas that are important to be shared, air out common misconceptions and offer the greatest learning for the class. Teachers may wish to ask for volunteers to present; however, it is important to have an idea of what each student can contribute to the discussion in order to scaffold the learning, help make connections, and guide student thinking. The fourth step presented by Stein et al. (2008) is sequencing which involves teachers deciding which groups of students should present first and how the different solutions may be related or contrasted. By sequencing student presentations of solutions in a particular way, the resulting discussion will be more predictable and coherent (Stein et al. , 2008). Once student solutions have been sequenced, teachers move to the fifth and fmal step: Connecting. During this step teachers need to guide students to make judgements about approaches and their effectiveness. Students will be comparing responses, reflecting, evaluating, and revising their ideas as they examine different strategies and develop powerful mathematical thinking (Stein et al. , 2008). These five steps presented by Stein et al. (2008) follow a launch-explorediscuss lesson structure similar to that presented by Chapin et al. (2009) and Cobb et al. (1994) . To begin oral communication in the classroom teachers should begin by establishing social norms and specifically teaching students how to communicate about mathematics. 21 Teachers should ensure that students are aware of the expectations for math discussions and foster student autonomy and collaboration. By focusing on collaboration and problem solving, students will begin to use self expression as a way to create their own learning and will begin to view their peers as resources. Through collaboration in partner, small-group, and whole-class scenarios, student thinking will come alive and students will be able to justify the strategies they used to solve problems as they become critical thinkers of mathematics. Finally, teachers will begin to design their mathematical lessons in a way that promotes student thinking and understanding through a launch-explore-discuss lesson structure (Stein et al. , 2008). The launch-explore-discuss lesson structure will allow teachers to anticipate and prepare for possible student answers. Teachers will also become skilled at monitoring, selecting, and sequencing student solutions to be shared with the whole-class as they focus on making connections and guiding student learning (Stein et al. , 2008). How to support oral communication in the mathematics classroom. Once oral communication has been established and students are familiar with the five talk moves that Chapin et al. (2009) presented, and the launch-explore-discuss lesson structure has been established as presented by Stein et al. (2008), teachers need to fmd ways to support student math talk that takes place in the classroom. Some of the main issues that teachers will be facing when incorporating math talk into the classroom are dealing with unclear answers, allowing enough think time, determining what students really understand, ensuring that student thinking is clear to others, dealing with talk monopolizers and students who refuse to talk, dealing with off-task behaviours, students who don ' t listen and of course dealing with students who are at different levels (Chapin et al., 2009). Cobb et al. (1994) demonstrated that establishing clear classroom norms and focussing on problem solving would help 22 alleviate many of the issues that may arise during classroom discussion sessions because students are clear of the expectations and are actively engaged in the solving of mathematical problems. Chapin et al. (2009) offer some suggestions to deal with many of the issues above. Teachers are encouraged to ask questions and help guide student thinking without monopolizing classroom discussions. Teachers should try to call on different students to participate to ensure that one student or group of students do not begin to take over the discussion. It is important for students to understand that their views, opinions or uncertainties are valuable and necessary for rich classroom discussions (Chapin et al. , 2009). The beginning of the school year will require more modelling and teacher intervention, but as the year progresses and students engage in and learn from one another' s ideas, students will become more skilled at explaining their mathematical thinking in a way that others will understand. Furthermore, students will also become skilled at asking questions to ensure that they understand the ideas of others, making the teachers role more of a guide than as a director. Ensuring that teachers are presenting challenging problems, providing support to students, monitoring student learning, and guiding student discussions will also help to address many of these issues (Stein et al. , 2009). Written Communication Once students have become comfortable discussing their mathematical ideas orally in pairs, small-groups and with the whole-class, teachers will be able to begin to introduce written mathematical communication. The following explores the research as to why written communication is important in the mathematics classroom, ways to incorporate written communication into the mathematics classroom, and how to provide ongoing support to students as they engage in math journal writing. 23 Why written communication is important. When ideas are written on paper, those ideas are made permanent and become available for sharing, revising, and revisiting. Putting ideas on paper help students to develop clarity and understanding of mathematical concepts. A math journal is an important tool for students to explore how to express their problem solving solutions and strategies with others. Bums (1995) identified the need for getting students to talk about their ideas to help them clarify their thinking and develop deeper understandings. She gave many examples of ways that she encouraged students to talk about their written math ideas; however, she did not really expand on the need to focus on math talk prior to writing. Since writing provides insights into student's thoughts and ideas, Bums (1995) demonstrated that written communication should be used to assess student understanding. Furthermore, Bums (1995) reminded teachers that partial understanding and confusion are a natural part of the learning process. Math writing not only provided teachers with insights into student thinking, but also provided an excellent vehicle for communicating with parents about what their children were learning and the progress that they were making (Bums, 1995). Albert and Antos (2000) argued that students need to see how math is relevant to themselves, thus, they developed the Daily Math Journal Project to engage fifth graders in exploring and explaining mathematical concepts. Albert and Antos (2000) had at least two students each day take home a daily math journal to help students see connections between their daily lives and the math concepts being taught. Selected students for that day took home a math journal and had to find a problem to solve in their daily life that involved math. Students then had to write what the problem was and how they had solved it. Then they had to pose a similar problem for the class to solve. The process was started using a teacher 24 modelled example, as well as providing a writing tips sheet for each step of the process which students could refer to at any time. Each math class then began with students ' sharing their journal entry from the night before with the class. Following the students ' sharing of the problem and solution, the class then worked in groups to solve the problems that were posed by their peers. In engaging in the sharing of solutions, students were able to talk and learn from one another as well as learning different ways to solve a problem. In this way the students were active participants in their learning, using a collaborative learning situation and deepening their overall understanding of mathematic concepts (Albert & Antos, 2000). The researchers found that the journal helped inform parents about how their child understood math beyond rote math assignments, while also giving a voice to every student. The process promoted social interaction as students shared their solutions and then worked in groups to solve the similar question posed by their peers. The act of putting ideas on paper helped students to create new thinking and an awareness of math concepts. This article demonstrated the link between oral and written communication, while also demonstrating the importance of making math relevant to students. The writing of mathematics was seen as a way for students to document their thinking and share their strategies and process with the class at a later time, while also allowing for self-reflection. Furthermore, this article supported the link between writing and parents, in that written math helped bridge the gap between math teaching and keeping parents involved and informed. Jurdak and Abu Zein (1998) examined the effect of journal writing on achievement and attitudes towards mathematics. The subjects for this study consisted of 104 students between the ages of eleven and thirteen in the middle school at the International College in 25 Beirut. Jurdak and Abu Zein (1998) sorted students into two groups: journal writing (JW) and non-journal writing (NJW) . Students in the JW group were given 7-10 minutes at the end of the period, three times a week, to write a diary-like entry to a journal prompt given by the teacher related to that days ' work. This process was continued for twelve weeks before both groups completed the Mathematics Evaluation Test (MET), which consisted of multiple choice questions, to determine achievement, and an Attitudes Towards Math ematics Questionnaire to determine students attitudes towards learning mathematics (Jurdak & Abu Zein, 1998). The results showed that the JW group demonstrated better concept understanding, procedural knowledge and mathematics communication that the NJW group. The two groups did not differ significantly on their overall attitudes towards math or overall math achievement; however, the writing was done in only 7-10 minutes in class, and no written or verbal feedback or modelling occurred. As journal entries that were not completed were essentially homework, parents were found to request more of this activity as it was enjoyable for students, gave students a vehicle for self-expression and learning, and provided a window into their thoughts and feelings about mathematics. These fmdings are similar to much of the research on math journals, in that it made student thinking and understanding accessible to others. Writing in math class is an important part of any mathematics program as it makes student thinking public which allows others to access, review, and offer feedback to the individual. By making ones thinking public, the students ' ideas are accessible and available for students to come back to at a later time and revise or review (Bums, 1996). Mathematical writing allows students to express ideas or feelings towards mathematics, work out their 26 thinking, and share their thought process with others. Furthermore, students are able to see their own misconceptions and become clearer in expressing their ideas so that others are able to understand the steps that they took to solve a problem. Math journal writing allows students to build upon and utilize their oral communication skills to assist them in getting their ideas on paper. Math journal writing is a tool that allows students to organize and sort through their thoughts as they solve problems and build upon their mathematical understandings and is a skill that should be used along with oral math communication. By having students mathematical thinking on paper, teachers are able to assess student understanding, uncover any misconceptions and ultimately use this information to help guide student thinking and drive instruction (Countryman, 1992). How to incorporate written communication into the math classroom. In the book Writing in Math Class ?: A Resource for Grades 2-8, Burns (1995) provided myriad strategies and suggestions of how to begin to get students writing in math class. Through many samples of student writing, Bums ( 1995) demonstrated how she responded to specific content, and how she encouraged students to write more and add details to their written explanations. Bums (1995) encouraged teachers to be upfront with students about the purposes for writing in math class so that the students will be able to understand how writing will help them learn math. Writing is a tool which can be used to help students think about ideas as it allows students to gather, organize, and clarify their thoughts. Far too often, students learn to compute before they really understand why the computation procedures make sense, thus Bums (1995) encourages teachers to allow students to develop their mathematical thinking and number sense through reflecting on their thinking, justifying their reasoning, and judging the reasonableness of their solutions. 27 Bums (1995) presented and described the use of a variety of different types of written assignments including: math journals or logs, math problem solving, explaining mathematical ideas, creative writing, and other general writing assignments. Bums (1995) demonstrated in each of these sections how she used written prompts to help students get started with their writing. Prompts did not need to be used; however, they were often found to help students who were unsure how to begin explaining their ideas. Bums (1995) also explained the importance of student writing as well as offering examples of student writing across the grades. She demonstrated the importance of incorporating writing into the math that happens in the classroom. Bums (1995) also suggested that student math writing should explain: what he/she did, what he/she learned, and what he/she is not sure about or still wondering about. Student responses should explain students reasoning and convince the reader that their solution is correct while also showing how they got to the solution. Bums (1995) provided further support to the notion that writing is a necessary part of a mathematics program, as it provided a window into each individual students thinking and understanding. The information gathered from students' written work could then be used to drive instruction while also being revisited for self-assessment purposes. Furthermore, Bums (1995) suggested how to help students who are struggling by prompting them with questions. She also shared the importance of feedback to student writing and explained that feedback should be encouraging, substantive, honest, and specific. Teachers need to demonstrate through feedback that they value and are interested in student thinking. This resource was an example of a handbook that is available to teachers on written math communication. 28 Crespo (2003) offered a unique way to get students writing in math and explained how she had used letter writing in her teacher-education classrooms as well as with collaborating elementary teachers as a way to help students to explain their mathematical thinking. This research article discussed how the use of pen-pals in math class could get students to write and explain their ideas in order to promote the deepening understanding of mathematical concepts. The first step in setting up math pen-pals presented by Crespo (2003) was to set up pairs (or trios) of students so that younger students were paired with an adult student in the teacher-education classroom. The adults usually wrote to one or two youngsters once a week for the duration of the course (approximately twelve weeks) . Once the pairs had been established, the younger student usually initiated the exchange with a Dear Pen Pal letter, which tended to be more of a personal introduction where the student described what they liked and disliked about math, asked their adult pen pal a question, listed the topics they have been studying and posed a math problem for their adult pen-pal to solve. By having students pose a problem to their pen-pal, it allowed them to think about the known and unknown aspects within a problem and explore and experiment with the wording of the problem (Crespo, 2003). Writing a problem out challenged students to think about what to include and what to leave out. The process of documenting and sharing students learning in writing helped the students to make their thinking clear and often showed the importance of explaining their ideas in more than one way. Furthermore, students were able to let their pen-pal know what did and did not make sense to them, and also allowed the teacher candidates to model how they solved problems in different ways. 29 Students were found to be motivated as they had an authentic audience and learned through the "tutoring" of their pen-pals. Since there was time between mailings, students were able to reflect, think and discuss strategies with their peers before responding to their pen-pal. Similar to Burns (1995), Crespo (2003) also suggested giving students phrases or sentence starters to assist students in getting their ideas down on paper. The author also found that the interactions between pen-pals helped students to learn and understand the importance of justifying their answers in order to avoid questions from their pen-pal. Crespo (2003) also suggested having all of the older pre-service teachers ask the same question to the students, which allowed for whole class discussion and sharing within the elementary classroom, thus allowing for brainstorming and class discussions either before or after the younger students had responded. Pen-pal letters provided a rich context for mathematical communication as students constructed and worded problems, as well as solved, justified and explained their own solutions to a question posed to them. Crespo (2003) supported the importance of writing as a way to document one' s thinking and also improve student written communication skills. The more practice students have with different forms of communication, the clearer their explanations and communication became. Furthermore, Crespo (2003) has demonstrated how to use an authentic audience to increase student motivation and increase learning for both older and younger students. This resource is valuable as it demonstrated how written math communication can provide a rich context for sharing strategies and explaining and justifying solutions to problems. Due to the call for students to be able to communicate math through speaking, writing, demonstrating and depicting visually (NCTM, 2000), the need for journal writing is 30 becoming more evident as it encompasses all of these aspects of mathematical communication. Williams and Wynne (2000) provide an account of their approach to incorporating journal writing into the mathematics classroom. These two teachers decided to have only one class participate at a time. Each of these teachers taught in the high school setting and provided a great deal of insight into how to approach the task of getting students to write in math class which could be transferred to the elementary setting. In each teacher' s classroom, the students responded to either affective or mathematical writing prompts in order to assess student knowledge and feelings towards math concepts. Similar prompts were used in each class and assessment rubrics were also similar in order to allow for class compansons. The teachers began by using the journal twice a week; however, they switched to only once a week as the teachers found that they needed more time to respond to student journals. Students were expected to complete their journal outside of class time and hand in their journals on Friday at the beginning of class. Students were told to try to limit their entries to one page of explanation and were told that less than a page would not be sufficient. Student e~tries were graded on a rubric, which was included in the syllabus for the course, thus available to students for self-assessment prior to their journals being handed in. At the beginning of this project, Williams and Wynne (2000) were somewhat discouraged in that some students claimed that they did not know how to write what they wanted to say and many students did not enjoy journal writing. By the midpoint of the project, the teachers had reduced the number of entries to only once a week and students appeared to be relieved at the reduced workload. Toward the end of the project, the researchers found that students had enjoyed the journal writing and requested that they 31 continue as they felt it helped them to express their ideas better and gave them immediate feedback on their understanding. These journals also provided excellent feedback on what students really understood of the curriculum and where students were struggling. Although in a high school setting, this research further demonstrated how journal writing could assist students in their self-assessment and growth as mathematic learners, while also providing valuable information to teachers on student understanding. It also demonstrated that student engagement with writing is a process that develops over time and not something that students will necessarily enjoy immediately. It takes time for students to see the benefits of written math communication, especially if writing in math class is a new concept for them. Flores and Brittain (2003) discussed how they used writing with preservice teachers to help them understand the importance of using writing in the math classroom and to demonstrate how writing could help students to understand concepts more deeply. Writing made one ' s thinking permanent as one could revisit and analyze the writing later on. Every week, the instructor in the preservice teachers' methods course introduced a new mathematical concept. At the end of each session, preservice teachers were asked to write about their impressions of the lesson, thus providing valuable reflections which could be revisited later on. In a sense, these students were acting like pupils in an elementary class might. In this way, the preservice teachers were able to understand and experience the value of this activity for elementary students. It was shown through the responses of these student teachers that writing helped with the thinking, confidence, and misconceptions of students. It allowed the preservice teachers to look back on their thoughts and reflect on their growth. 32 By providing different types of writing, the preservice teachers were able to choose how to best express themselves. The writer was able to personalize his or her thinking and attempt to "make sense" of problems or concepts as well as providing a document of their learning for assessment or sharing purposes. Furthermore, this article showed that cooperative learning was a positive approach to learning mathematics and also helped students to fully understand concepts as teachers developed the critical thinking skills of their students. Flores and Brittain (2003) also explained that writing could help to support a positive learning environment as it took some of the pressure off of the student to come up with the correct answer right away. It also allowed students to use strategies and methods that worked and made sense for them as they solved problems at their own pace. Thus, the writing in math was seen as a way to help relieve anxiety. Many students found the ability to write about what they really felt to be cathartic. It also helped the preservice teachers to see through their experiences with writing and how writing in math could produce critical thinkers. In sharing and discussing their writing with others, these students were also able to learn from one another and expand their own mathematic understandings. Flores and Brittain (2003) did caution that writing was a tool and like any other tool, it may not be as effective for some as it was for others. This resource further demonstrated the importance of written communication as a way to document one ' s feelings and thinking while also showing how the sharing of these ideas can help to expand individuals' repertoires of mathematical strategies and understanding. Countryman's (1992) book provided proof to the concept that writing in math class increased students' ability to understand mathematical concepts. She gave many 33 examples of different techniques that she used with middle and high school students to encourage written work in math, including the use of journals, freewrites, learning logs, written descriptions of problems, autobiographies, and formal papers. The author believed that using writing as a technique in math, actually opened up the mathematical field to many students who at one time may have felt left out as students began to feel that they could talk about and access math more freely. Countryman (1992) believed strongly in students and teachers keeping journals in order for teachers/students to reflect on their own experiences that were taking place in their classrooms. Writing allowed students to develop skills of planning, inferring, construction, symbolizing, interpreting, and reflecting. Furthermore, by adding writing to the math classroom, everyone was active, collaborative, and was participating in the learning of mathematics, while also constructing their individual understandings of mathematical concepts. Countryman (1992) believed that "knowing mathematics is doing mathematics"(p. 2), and that this could only happen in classrooms where students were encouraged to explore, justify, represent, discuss, and ultimately be an active part in their mathematical learning. This resource acts as another example of a handbook that is currently available to teachers on the use of written math communication. This resource does not discuss how to begin with oral communication or how to develop a positive parental involvement. Establishing mathematical writing in the classroom can be done using many different strategies; however, using a problem solving approach to a math journal is one of the most effective ways to get students thinking, solving, and sharing their mathematical strategies with one another. It is important for teachers to remember that math writing should not be taught in isolation from oral communication, but instead be integrated with one another to 34 allow students to express ideas orally with peers, ask questions, construct understandings, and at last document this learning in their math journals. How to support students' mathematical writing. Once students have had a chance to explore mathematical writing, teachers can begin to incorporate math journal writing as an important part of their math program. Using a math journal that centres around problem solving allows students to engage in challenging mathematical concepts, use problem solving skills, discuss ideas with their peers, and then solve and document problems independently. As with any initiative, students will require support from teachers in order to ensure students are reaching their mathematical potential. One of the techniques that Bums (1995) and Whitin and Whitin (2002) strongly advocated is the use of writing prompts. By providing students with a list of prompts for different aspects of their mathematical writing, teachers are helping students to begin getting their ideas on paper. Often when students are just starting to write about their mathematical thinking, being able to formulate sentences which help them to explain their thinking is a very challenging task. With the use of prompts, students have somewhere to go to help them put their ideas on paper. As students become more skilled at putting their ideas on paper, they will likely rely less on the prompts as they are more comfortable and skilled at explaining their thinking both orally and in written form (Bums, 1995). Most struggling students will have difficulty putting their ideas into words when first starting math writing, so having prompts would be a great asset. In addition, having students work cooperatively in pairs or small groups to discuss and solve the problem prior to having individual students document their thinking will allow for the flow of ideas, sharing of different points of view, and allow students to learn from one another. Cooperative learning 35 is a large part of the mathematics classroom which centres on mathematics communication (Mercer & Sams, 2006). Collaboration allows students to question, critique, and rethink (Turner, et al. , 1997). Students need to be comfortable sharing ideas (correct or incorrect), challenging and questioning one another, and ultimately constructing their own understandings. Allowing students to talk through their ideas before putting ideas on paper is another strategy that will help students transition from math talk to math writing (Mercer & Sams, 2006). Finally it is up to teachers to respond to students ' written mathematical ideas in a prompt and informative way. Simply giving empty feedback like "good work" will not help to guide student learning and understanding (Silbey, 2003), instead teachers need to be able to give constructive feedback to students by fmding areas of their mathematical writing that could use improvement or clarity and offering suggestions for students to try to incorporate into their next journal entry (Bums, 1995). If teachers notice that many students are making the same errors, or need help with the same aspects of their writing, a mini-lesson specifically designed to teach that skill may be useful to be done with the whole-class. By assessing mathematical writing, teachers will be able to develop a clear understanding of the needs of individual students and the class as a whole. Math writing provides teachers a window into student thinking which allows teaches to prepare and use strategies to help move student thinking forward through a combination of collaborative practices and individual support and feedback. Home-School Connection There is a large body of evidence to demonstrate the importance of involving parents in their children' s education and more specifically in their mathematical learning (Bernhard, 36 2010; Carey, 1998; Cavanagh, 2009; Litton, 1998; Menendez, 2008; Mistretta, 2004; Zanger, 1998). Many parents struggle with how to best assist their children at home mainly due to the fact that they are unsure of the way that math is being taught today (Cavanagh, 2009). Parents are often more familiar with the use of algorithms and procedural knowledge than the more conceptual knowledge approach that students are presented with today (Whiteford, 1998). Thus, it is imperative that teachers try to find ways to involve parents in the mathematical education of their children to help educate parents on the different instructional approaches being used and how parents can best assist their children at home. The following examines why parental involvement is so important, ways to involve parents in positive ways, and how to provide support to parents once they become engaged in home math learning activities. Why involve parents in mathematics education. When parents see their children struggling with their mathematical learning, they often instinctively want to cheer them up by comparing their own past struggles with math. Parents really need to remain positive for their children and try to forget their own past experiences with mathematics and focus on how to show them the importance of math in the real world (Cavanagh, 2009). By maintaining a positive attitude towards mathematics, students ' anxieties towards the subject are often decreased, which helps lead to increased confidence and achievement (Cavanagh, 2009). Hong, Y oo, You, and Wu (20 10) also examined parental involvement as it related to students' mathematical achievement. Student survey data from their study demonstrated that value and behaviour were influenced by parents, and that parents' values do make a difference in the achievement of their children (Hong et al. , 2010). Once again, this research has demonstrated that parents need to find ways to value and support their children in 37 positive and effective ways. Furthermore, survey data collected by Fan and Williams (2010) proved that communication with the school should be positive in order to avoid negative associations. If students believed that their parents valued education, they were more likely to achieve and feel confident towards their achievements. Thus, parental involvement with the school should remain positive in order to strengthen the bond between home and school and demonstrate that parents value their children's education (Fan & Williams, 2010). Maintaining a positive outlook on math is an important contributing factor for student success; however, this is difficult if parents are unclear or unsure of how math is being taught in their children' s classrooms. It is important for teachers to fmd ways to involve and inform parents of the similarities and differences of how math is being taught today compared to when they were in school. Most parents focus on the procedural knowledge that they had learned when they were in elementary school, and are often confused by the conceptual approach and relational understanding of the math their children face today (Whiteford, 1998). Furthermore, parents who have immigrated to Canada are often even more disadvantaged at providing assistance to their children as they are unsure of the mathematical terminology being used and often do not feel comfortable approaching the teacher/school with questions or concerns (Bernhard, 2010). It is important for teachers to take into consideration the variety of needs that parents in their classrooms may have and find ways to approach and meet these needs. Many parents who had been surveyed (de Abru & Cline, 2005) about their perceptions of how math is taught today stated that the changes in the teaching of mathematics seemed to make math more enjoyable for their children; however, many worried that there would be disadvantages to this new way of teaching. Specifically, parents 38 felt that students lacked basic math skills (i.e., memorized math facts) . These parents who had been surveyed indicated that they wanted more information on how math was being taught and how they could best assist their children at home (de Abreu & Cline, 2005; Menendez & Civil, 2008). Cavanagh (2009) reported that parents' attending workshops run by the schools and districts from the Prince William County in the suburbs ofWashington, wanted to be able to assist their children by building on what the schools were teaching while helping their children to enjoy math. Parent workshops should be designed to empower parents to assist their children at home in meaningful ways that align with the focus of the school. Parents need to be able to experience the use of manipulatives and today' s teaching strategies in order to better understand how their children are learning (Menendez & Civil, 2008). Parents also indicated that they needed more knowledge around math standards, assessment, and practical ways to help motivate their children to do math at home. Furthermore, many parents surveyed have indicated that problem solving, conceptual understanding, and applications of mathematics have been the most challenging aspects of helping their children at home (Mistretta, 2004). By involving parents in the learning process and building ways for parents to interact with their children ' s mathematical learning, parents will begin to value meaningful and fun ways to learn with their children (Kliman, 2006). Involving parents in the mathematical learning process would allow parents and students to explore concepts and problems together, which would help to build a jointlearning experience where parents were working in partnership with their children and the teacher to help increase student achievement while also making math learning fun and enjoyable. Thus, there is an obvious need for teachers to take steps to communicate with 39 parents about mathematical concepts, skills, and problem solving. As well, parents need to understand assessment tools and have sample home activities which are geared towards the mathematical concepts being learned in the classroom so that they can best support their children ' s learning at home. By getting parents involved teachers will be opening the lines of communication between the classroom and the home. How to involve parents to create a home-school connection. The information above provides a significant body of evidence which demonstrates that positive parental involvement and attitudes are important for student achievement. It is up to teachers, administrators and school districts to understand the needs of parents and fmd ways to bridge the home-school gap in the teaching of mathematics and open the lines of communication between the school and the home. There are many different ways that parents can be encouraged to become more active in their children ' s mathematical learning. Bernhard (2010) examined three different intervention programs which were geared towards getting Latin American parents in Canadian school systems more educated and involved in their children's education. The first intervention involved 12 parents who met each month to discuss aspects of their children's experience with Canadian schools. Discussions took place in a Toronto community centre. Facilitators began each session by initiating a discussion before asking parents for their input on other issues that they wished to discuss. These sessions helped parents to become more familiar with terminology and these parents were shown to be more likely to attend parent meetings following the intervention program as they were more confident in their role and what was expected of them (Bernhard, 2010). 40 The second intervention program built upon the first and involved a 10-week project which later became known as the Canadian Parenting Workshops (Bernhard, 2010). These workshops were designed to teach Latin American parents about school report cards, contacting school superintendents and members of parliament, learning about school committee structures, and the process of designating children for special needs. Results from this intervention showed that parents were more empowered once they understood that their values and views had validity. Furthermore, parents better understood what was expected of them, how to collaborate with teachers, express concerns, and also build upon their ethnocultural differences (Bernhard, 201 0). The third intervention was the Authors in the Classroom Program (ACP) which had parents and children work together to self-author books about themselves, their families , and their goals. This intervention helped to strengthen the links between and among children and their families . These family activities helped these immigrant parents to realize the value of transmitting heritage to their children. These three different intervention programs demonstrated the need to involve and educate immigrant parents about the education system. These parents, like other parents, required information on how math was being taught and how they could build a partnership with their child to positively help them at home. Providing parents with necessary information about their child's mathematical learning could be done through parent workshops, parent information nights, or through the use of authentic mathematical tasks for parents and their children to complete together. Parents who had been surveyed had even requested their desire to attend courses for parents (de Abru & Cline, 2005). These different ways of involving parents in the education system will be examined more closely. 41 Another aspect of teaching mathematics today with which parents are often unfamiliar involves the use ofmanipulatives. Carey (1998) demonstrated that giving students mathematical manipulatives to be used at home along with math tasks and reinforcement worksheets, not only helped parents to get some experience with a hands-on approach to learning math but also gave them an authentic way to enjoy spending time helping their children with mathematical concepts that were being presented in the classroom. This approach of sending home manipulatives as a type of homework assignment helped to create a working partnership with the parents while also reinforcing topics being taught in the classroom. The use of manipulatives as a homework task also helped parents to better assist their children at home and see first hand how manipulatives could be used to support the learning of mathematical concepts. Family math nights were yet another way that parents could get involved in mathematical learning with their children. These evening events usually involved math games or activities that were designed to get parents and their children engaged in mathematical tasks that were fun and encouraged students to explain their thinking while they completed the different math games/activities (Cavanagh, 2009). Parents were given the opportunity to play or engage in mathematical tasks with their children in the school or classroom setting, which allowed teachers to also interact and field questions/concerns of parents as they arose. One of the downfalls of family math night tended to be in the attendance as busy family schedules often did not allow for all parents to attend; however, it did provide yet another way of informing and educating parents on fun ways to engage in mathematical tasks with their children at home (Cavanagh, 2009). These events often end 42 with a variety of ideas that parents can then take with them and use with their children at home. When students are able to see real-world connections of their learning, they are more likely to demonstrate an interest in mathematics. One way that parents could try to get involved in their children ' s mathematics program which incorporates these real-world connections is through community organizations that offer mathematical challenges to families. Fawcett and Shannon-Smith (2009) revealed that activities which took place in partnership with community organizations such as the zoo, park, minor league baseball parks, or theatres were received with positive feedback from students and parents. These types of programs are not very common, therefore, it is important for parents to look for as many opportunities to "do" math with their children as they can. Finding ways to do math at home will help students to see the math that lies around them; for example, cooking, measuring, making change and other related activities (Loftus, 2009). The use of math games is another strategy to involve parents in a fun and productive way (Kliman, 2006). The idea is to get parents and students playing games that foster, support and build upon mathematical learning, which could be done in a variety of ways. Kliman (2006) created games that incorporated geography and math in order to attempt to foster family activities as part of the family culture. Providing math games that were geared towards aspects of the mathematical learning that was occurring within the classroom may have provided more promising results for Kliman (2006); however, most parents reported that they enjoyed playing the games. Parents have shown that they were eager to spend quality time learning alongside their children (Zanger, 1998) and math games would provide the opportunity to do this in a way that was fun while also being productive for both parents 43 and students. Math games could be taught in class and then taken home for students to teach and play with their parents, or they may be introduced at a family math night event where teachers were able to field questions and assist families. Having parents participate in the writing of story problems with their children is another way to help educate and involve parents about the problem solving approach that students are learning today. Creating story problems allows students and parents to fully understand the necessary and extra information that these problems contain. Furthermore, having home partners (e.g. , parents, aunts, uncles, siblings) create these story problems together increased their sense of pride and ownership while also motivating students to try to solve problems written by other home partnerships (Zanger, 1998). This initiative was made more successful by Zanger (1998) by having the stories published and distributed to all families within the school. Once parents and their children saw their work in a published form, motivation to produce and solve story problems that others had created increased (Zanger, 1998). There are many ways to educate and involve parents in their children ' s mathematical learning. Depending on the needs of a parent population, teachers/schools may choose to offer workshops, information sessions, family math nights or other effective strategies to educate and involve parents in their children ' s mathematical learning journey. The key to success appears to be that parents need to experience and become educated in how math is being taught to their children so that they can better support their children' s learning at home in a positive way. How to support parents and the importance of math journals. As demonstrated above, parents need to be well informed of the way that math is being taught in school as 44 well as being provided with a variety of different ways to become involved in their children ' s mathematical learning. Once parents have been given information on how math is being taught in school, whether that be through a family math night, newsletters, workshops or any other method that meets the needs of both the teacher and parents, home math initiatives can be successfully established. Parents should approach learning mathematics, like learning or developing mastery in a sport. In order to get better, students need to practise and have many experiences. Furthermore, parents need to focus on student understanding and not so much on the grades that they are receiving in math. By focusing less on student grades, parents will begin to see the aspects of mathematical problems their children are actually understanding, and where their misconceptions are coming from, in order to best support their learning at home (Haury & Milbourne, 1999). For many parents, this mindset will be a big shift and one that will likely need to be revisited in order for this to become the new way of thinking about success and support of mathematical learning. By getting involved in their children ' s learning, parents will be able to challenge their child to fmd other solutions to different problems, encourage them, and help them to explain and justify their answers/thinking. When explaining ones thinking in math, a math journal is a great tool. Math journals not only help teachers understand and assess student understanding, but they are a great vehicle for communicating with parents about what their child is learning and the progress that they are making (Albert & Antos, 2000). These journals offer a great opportunity for parents to see what their children are thinking and where their misconceptions/mistakes are occurring and perhaps allowing a way to offer suggestions to parents of how they can best support their child at home. Math journals are also a great way to point out to parents that 45 partial understanding and confusion are natural to the process of learning and that students need many experiences to deepen and expand their learning in order to become more confident in their abilities as a problem solver (Bums, 1995). Teachers may also choose to use journals as a home connection with parents/guardians to establish the home as another resource for student learning. These journals may be completed by students and their parents together as they solve problems and describe their solutions in writing (Whitin & Whitin, 2002). These home journals could help bridge the gap between home and school, by involving the family as well as demonstrating some of the strategies and questions that are being focussed on in the classroom, and how journals are being used as a way to keep track of mathematical reasoning. Joumaling also provides a way to show parents what to do if their child is unsure how to begin or solve a problem. Providing parents with prompts to use with their children when they are unsure of how to proceed with a math problem is one way that teachers can support parents (Bums, 1995). Phrases such as: ' Explain the problem to me', 'What do you have to do in the assignment?', ' Tell me what you're supposed to write about' , 'Do you have any idea about what the answer is? ', ' What ideas do you have?' , ' Tell me something you know', 'See if you can say those words again in your head and then write them down on your paper', provide parents with the words they can use when helping their children at home. These prompts are a great place to start when introducing writing and problem solving at home because one of the big problems that parents come across is how they should best help their child when they continually say that "they don' t get it", or "don't know where to start". 46 Providing a math journal opportunity that parents and students could work on together would allow students to work on their problem solving and written communication skills in a safe environment while also allowing them to share their learning and thinking with their home partner. Students would also be able to compare their own solution with the solution of their home partner' s as they begin to understand the thinking of others. We know that students learn more when their thinking is challenged and when they can vocalize their thoughts, strategies and ideas with others (Countryman, 1992). Engaging in discourse is sure to enrich the learning experience for students and parents and provide a great way for students to put what they say into words through written communication. Being able to take this joumaling experience home and experience it alongside their parents/caregivers, would make for rich discussions and focussed learning for both the parents and students. Parents would be able to get a better understanding of what was being expected of their children, and how problem solving could help to demonstrate student misconceptions and deepen their overall level of understanding. Students are thinkers who are exploring and experiencing the world around them. Each individual brings with them new perspectives and theories to the learning environment, which allows students and parents to learn many new ideas and strategies from one another. By bringing students ' own ideas and prior knowledge to new situations, students and parents would be able to plan, organize, and consolidate their learning through different forms of communication. Through a combination of oral and written communication strategies, students would be learning how to explain themselves mathematically while keeping their thinking clear and coherent. Furthermore, by sharing their thinking and ideas with a family 47 member who has solved the same problem, they would be able to make comparisons and understand that there are many ways to solve, or represent solutions to the same problem. Students need to understand why they are talking and writing about their mathematical reasoning. These discussions and written documentation offer parents and the teacher a view of what the student may be thinking and how they are processing problems. Uncovering what students truly understand is critical for teachers to plan effective future lessons, as well as for students whom are able to go back and reflect on prior learning and bring new information to their problem solving approaches. Once this rich opportunity for oral and written communication has been introduced and used regularly with students, one can ' t help but be excited to observe the impact that these strategies could have as a homeschool connection and the many ways that this approach could help keep parents informed of what is being taught and how they could best support their children ' s mathematical learning. Not only would a home math journal improve student understanding, but also it will inevitably help to prepare students for explaining their thinking on standardized assessments. Furthermore, journaling could be combined with math games or math manipulative activities (Bums, 1995) where activities were sent home followed by students writing about their learning or what they noticed while playing/completing these activities. These joumaling activities could be further supported by newsletters sent home to address concerns/issues that teachers notice through the journals. Furthermore, teachers may offer support by sending home sample sentence stems, information on how to reflect, prompts that parents could use to help a struggling student, as well as any other information that parents may need to better support their children's learning and help get student ideas written down on paper. 48 Conclusion The research shown here demonstrates that teachers and schools need to invest time at the beginning of the school year to determine the needs of their parent population and then find ways to educate parents on the changing ways that math is being taught today before introducing ways for parents to become more actively involved in their children ' s mathematical learning. If districts and schools become united in this effort to engage and educate parents, some of the onus will be taken off teachers as workshops could be offered within the area as opposed to at the school level. In any case, investment in the education and involvement of parents in their children ' s mathematical learning has been shown to lead to increased student achievement over time. Families need to understand that parents need to remain positive to increase student achievement and teachers need to provide a way for parents and students to interact in a positive way with mathematical concepts. Furthermore, parents need to understand that it is important for them to find ways to "do" math with their children both through the school and on their own. Students need to see the math that is around them for them to better appreciate the importance and fun of the subject. Once parents are engaged in mathematical activities at home with their children, it is imperative that teachers provide ongoing support to assist this newly established homeschool connection. Teachers can provide this support through communication in a Home Math Journal, through handouts on prompts or sentence stems to assist with written communication, or through newsletters or parent meetings. Mathjournaling is a great way to get parents involved in aspects of the mathematical program that parents are often less familiar with, such as problem solving and written 49 communication. If teachers incorporate a Home-Math Journal with their students they should see an increase in parent involvement as well as positive feedback as parents are given a task to complete with their children. Providing a homework task that parents and students complete together can eliminate the problem of students not wanting to ask for help with homework tasks, and instead allows parents and students to learn together and explore mathematical concepts in a fun and safe environment. Positive parental involvement of this kind will allow for increased student achievement over time. An examination of the above research and resources on math communication demonstrates that there does not appear to be any information on the use of a problem solving math journal with parents. Much of this research demonstrates the importance of parental involvement in education and how parental involvement can actually improve student motivation and achievement, but not on the specific use of a math journal as a collaborative tool to build parent understanding of how math is being taught, while also giving students a chance to learn from their parents in a meaningful way. Furthermore, while many articles discuss the importance of math talk prior to writing, there does not seem to be a resource that demonstrates how to teach math talk leading into the teaching of math journals, and ultimately into the Home-Math journal. 50 Chapter 3: Research Methodology I developed the handbook presented in the next chapter using information collected from the variety of sources which I discussed in the previous chapter. In particular, I have incorporated Cobb, Wood, and Yackel's (1994) research on using math talk with students as it demonstrated that students needed to be challenged, given opportunities to work collaboratively, and encouraged to share and express their thinking with peers in order to create rich learning opportunities. In examining math writing, I have incorporated Flores and Britain's (2003) research on using math writing with preservice teachers to help them understand the value in using math writing to help students develop clearer understandings of math concepts. Flores and Britain (2003) verified that mathematical writing helped to make student ideas permanent while supporting the learning and encouraging students to use different strategies and methods. Countryman's (1992) work was also incorporated as it demonstrated that writing allows students to organize, interpret, symbolize, communicate, plan, infer, and reflect about their learning. By writing in math class, Countryman found that all students were able to be involved and share their thinking. Finally, I have utilized research by Cavanagh (2009) which confirmed that sharing negative experiences with mathematics was detrimental to student success in math and that parents who were surveyed wanted guidance as to how they could assist their children with math at home. The work of Whitin and Whitin (2002) was also included as they discussed the possibility of using math journals to bring parents into the learning experience which ultimately inspired the creation of the Home Math Journal which is presented in the handbook. Through this research as well as my own experiences in developing math talk and math journal writing in my own classroom, I have created this handbook to allow other 51 teachers to begin developing effective math communication within their classrooms. My experiences in creating a Home-Math Journal, which I used with Grade 2 and 3 students and parents along with the research that was addressed above, helped me to demonstrate the importance of parental involvement in the teaching and learning of mathematics. In this handbook, I have demonstrated first hand how successful the use of a Home-Math Journal has been for me and how it could be adapted to meet the needs of students, depending on their age or learning style. The handbook begins with an introduction to the handbook followed by sections on Math Talk, Math Journals, and Home-Math Journals. The handbook follows the natural progression of mathematical learning as the research has shown that students need to begin by talking about mathematical concepts prior to being comfortable putting their ideas on paper. Once oral and written mathematical communication has been established in the classroom, parents can become involved in the process by establishing a Home-Math Journal for students and parents to complete and learn from together. The handbook concludes with a summary of the main points of the book and an appendix of forms and samples that could be used or adapted to begin developing oral and written communication in the mathematics classroom. 52 Chapter 4: Math Journals Handbook The following handbook was created to assist teachers in understanding how to effectively develop math journal writing with students in their classrooms. A review of the literature demonstrated that there did not appear to be one resource that fully explained the importance of developing oral communication with students prior to introducing written mathematical communication. The following handbook guides teachers through this process while explaining the importance of math talk in the classroom and how math talk should be established in the classroom prior to introducing mathematical writing. The handbook also establishes the interconnectedness of oral and written communication as the two skills are closely linked and should not be taught in isolation; instead, teachers should find ways to use oral communication to help bridge the gap to written communication. This handbook also takes teachers past simply involving students in math journal writing, and demonstrates how math journals can be used as a technique to involve and educate parents on the way that math is being taught today. Hopefully, this handbook can inspire teachers to effectively use math talk and math journals both in their classrooms and as a tool to keep the lines of communication open between home and school. 53 54 Table of Contents Table of Contents 54 Introduction 56 Math Talk- Oral Communication 59 Why Math Talk? How to Promote Good Math Talk? Role of the Teacher. Establishing Social Norms. Asking Good Questions. Math Talk Formats. Math Talk Strategies. Planning and Implementing Talk in the Classroom. What to Talk About? Math Talk from Grades 1-6. How to Support Math Talk? Giving Students Time to Think. Determining what students understand and dealing with unclear answers. Helping Students Make their Thinking Clear. Dealing with Talk Monopolizers. Dealing with Students Who Won't Talk. Dealing with Off-Task Behaviour and Students Who are Not Listening. Dealing with Students Who are at Different Levels. Math Journals- Written Communication Why Write? How to Promote Good Math Writing? Asking Good Questions. Math Writing Formats. Math Writing Strategies. Math Journals- What Do We Write About? Planning and Implementing the use of Math Journals in the classroom. Math Writing from Grades 1-6 How to Support Math Journal Writing? Using Prompts. Cooperative Learning Environment. How to Respond Effectively. Helping Struggling Students. Helping Students Make Writing Clear. 61 64 65 66 67 70 71 74 77 78 79 80 80 81 82 83 84 84 87 88 92 94 94 97 100 102 104 106 106 107 107 109 110 55 Working with Students at Different Levels. Home- Math Journals- The Home-School Connection Why Create a Home-School Connection? Creating a Home-School Connection with Math Journals Creating a Home-Math Journal and Sample Journal Entries. Tips for Involving Parents in Mathematical Learning. Parent Feedback after Journaling. Strategies to Communicate with Parents. Home Math Journals from Grades 1-6. Supporting an Effective Home-Math Journal Letters Home to Parents. Using Sentence Stems. Helping Parents Ask Good Questions. How to Encourage Reflection in our Students? How to Help Children Who are Stuck? Dealing with Journals that Don't Come Back. Keep Parents Informed. Looking For Math Around Us. 111 113 114 117 119 130 131 133 135 135 135 136 139 140 141 142 143 143 Summary 145 Appendices 151 Appendix A- Initial Letter Home Appendix B- Helpful Hints Appendix C - Reflection Sentence Starters Appendix D- Sample Problems References 152 153 154 155 162 56 Introduction The first step to improving students writing starts with the first days of school where teachers work with their students to build a sense of community; a place where all students feel safe and one where they know that their ideas will be respected. To assist with this safe environment, it is important to have structures in place right from the first day of school that will help to support oral discussion in all areas of the curriculum, but particularly in mathematics. In order to develop a classroom environment that is "communication friendly (Jarman, 2008, p. 31 )" students need to feel secure. Thus, teachers need to create a classroom ethos which fosters respect for others ideas and opinions and encourages risk-taking (Capacity Building Series, 201 0). A positive classroom environment allows all ability levels in the classroom to feel involvement in the learning process, as they are given time to think, play, and explore with mathematical concepts (Education For All, 2005). To create a positive learning environment, norms need to be set where students understand the rules for discussions and learning in the classroom. Students should be aware that there is only one speaker at a time; students need to understand the importance of giving others a chance to share their ideas. Also, if a student does not agree with a statement, he or she needs to be able to disagree in a polite way. Furthermore, students need to listen carefully to the ideas of others and ensure that the speaker is finished before adding on their own ideas or comments. Students need to ask questions, build upon the ideas of others, and be willing to clarify their ideas for the class. Once these classroom norms are set, teachers will also need to establish effective classroom management techniques for having students work in a variety of different groupings. Students should be made aware of what a discussion should look and sound like. For example, students should be expected to be seated knee-to-knee when 57 speaking in small groups or pairs with eyes on the speaker. Teachers need to make students aware early on of their expectations for the discussion/group work and have effective ways of bringing the groups back on task and getting the attention of the class quickly in order to provide important feedback/information to guide or mediate the discussions. Teachers also need to be familiar with different cooperative learning techniques in order to provide effective oral and written tasks for students (Education For All, 2005). Teachers must also spend a great deal of time at the beginning of the school year developing a learning environment where students can work cooperatively with other students and where they are explicitly taught how to respect the ideas and opinions of others. The classroom environment should be active and filled with enthusiasm for learning, from both the teacher and student (Early Math Strategy, 2003). Furthermore, Jarman (2008) stated that it is important to structure the physical space to support talk and have spaces for students to talk with each other. Right from the beginning, the teacher should be having students work in a variety of group settings as the class gets to know one another as this helps demonstrate that they can all learn from the ideas and strategies of others, as well as from their mistakes. The role of the teacher includes orchestrating discourse by posing questions to challenge student thinking, listening carefully and monitoring for understanding, and encouraging each student to participate in some way, even if that means having students repeat the ideas they have heard (NCTM, 2000). The student' s role includes listening and responding to the teacher and to one another, using a variety of tools to reason, make connections and solve problems, as well as communicating and making convincing arguments of representations, procedures, and solutions. Furthermore, students should feel comfortable with free exploration as they build confidence in their abilities as learners. 58 To develop inquiry in students, Jarman (2008) also suggested that resources be carefully selected that will help to spark interest and curiosity. When students are encouraged to think creatively, follow their own lines of inquiry, make connections, and solve problems, they are developing skills of life-long learning while also developing their own selfconfidence. J annan (2008) stated that mathematics communication occurs when students begin to explain their thinking, discuss concepts, estimate, predict, and generalize based on their individual experiences. Once a safe and communication-friendly environment has been established, teachers can begin to see the benefits of oral communication in their classrooms. Creating a safe space within the classroom will allow for students ' individual confidences to develop and grow while also providing a solid foundation from which written communication can follow. Teachers can support their students' development through the modelling of appropriate group behaviour as well as using a combination of Shared, Guided, and Independent practice opportunities. 59 60 Math Talk- Oral Communication Oral language is important in that it is the foundation for more complex literary skills which are important for student success in today's knowledge society. Students today need to be literate in their logical-reasoning, problem solving and decision making skills, in order to be successful in our technology driven society. Thus, the most important skill that we can teach students today is how to ' make sense of information' as they are often bombarded with information everyday from many different sources (Rakow, 1999). In order to help students make sense of information in math, we need to examine how students develop a deep understanding of math concepts. We know that students need to be given time to think, play, and explore with the "big ideas" in math (Education For All, 2005), and that the classroom environment should be active and filled with enthusiasm for learning, from both the teacher and student (Early Math Strategy, 2003). In addition to an enthusiastic learning environment, effective math talk requires a safe and inclusive classroom where students feel comfortable expressing their ideas and opinions freely and working collectively to construct their own meaning of mathematic concepts. Students should feel comfortable with free exploration as they build confidence in their abilities as learners. Research has shown that in order to fully understand a concept, students should be able to see, hear and feel math, as they explore through a more problem solving approach to learning (Early Math Strategy, 2003). In order to foster interdependence and responsibility for their learning as they work with their peers and talk through their strategies, students need to have many opportunities to verbalize, and reformulate ideas, discuss and compare as they begin to think mathematically (Education For All, 2005). 61 Math talk is an essential process for learning math because through communication, students are able to reflect upon, clarify, and expand their ideas and understandings of mathematical concepts and relationships (Education for All, 2005). Furthermore, math talk in early years has shown to significantly lead to children ' s growth in understanding math concepts (Kilbanoff, 2006). Orchestrating and leading math talk based on student responses in a way that advances the learning for all is a difficult task and requires a shift from the teacher as simply a dispenser of knowledge to an engineer of a learning environment where students work through problems and construct their own understandings (Stein, 2008). The role of the teacher in a whole-class discussion is to develop and build on individual and collective sense making as teachers learn to guide student thinking in mathematically sound directions. Before embarking on implementing math talk in the classroom, it is first important to establish the classroom as a safe learning environment, where students are comfortable taking risks and supporting the learning of others. Once a safe learning environment has been established where each student listens to every individual in a nonjudgmental, inquisitive, and attentive way, teachers can focus on developing students oral communication skills. The NCTM (2000) urges teachers to increase communication in the mathematics classroom, both through talk and writing. Since we know that students learn to speak before learning to read and write, it makes sense that oral communication should be introduced before written communication. As students begin the school year, teachers ' should focus on creating a collaborative work environment as this collaboration makes students critique and rethink, which are critical skills in mathematics (Turner, Styers, & Daggs, 1997). Collaboration 62 between students also helps students to become a community of learners where students can begin to take pride in being able to see the strengths and weaknesses in others work, while also helping to increase student thinking and reasoning skills (Soter, Wilkinson, Murphy, Rudge, Reininger & Edwards, 2008). Through this type of collaboration, students will begin to use each other as resources and share solutions in whole-class discussions which are orchestrated by the teacher (Stein, 2008). Turner, et al. (1997) also found that teachers were able to make a difference in students' autonomy and overall mathematical understanding as students questioned, explained, and talked through their solutions with one another. Students felt that they had more control over their learning and student engagement increased. Communication was shown to increase student motivation and participation while also building students mathematics communication skills and understanding (Turner, et al. , 1997). Another important reason for focussing on oral math communication within the classroom prior to written mathematics is that math talk allows students to become comfortable formulating their mathematical understanding prior to being able to document these ideas on paper. Students' should be given many opportunities to share and hear ideas from others and learn to make their thinking clear and concise (Silbey, 2003). Furthermore, oral communication is a powerful tool that can be used to assess student understanding while also informing instruction. In using discourse in learning math, student thinking becomes public and gives opportunities for students to negotiate meaning from others ' ideas. These discussions help to teach students to make their ideas clear and concise while also allowing them to learn from their mistakes or the mistakes of others (Silbey, 2003). Mercer and Sams (2006) found that students could be taught to use talk as a tool to improve their reasoning skills, and group talk activities could also help the development of 63 individual students' math reasoning, understanding, and problem solving skills. Thus supporting the notion that talk is an important factor in improving student understanding and in empowering students to become comfortable with math concepts and with the sharing of their strategies and solutions as they learn. Math talk requires teachers to be keen observers of students work and mathematical talk in order to best support student learning. Furthermore, the use of higher-order thinking helps lead to deep learning (Capacity Building Series, 2012). As teachers have students work with each other to develop a sense of community, the students should be talking about their learning with one another. In this environment, the teacher takes on the role of a coach, where they are responsible for expanding upon student solutions and ensuring that all students have a voice and are heard. Research has shown that in order to fully understand a concept, students should be able to see, hear, and feel math, as they explore through a more problem solving approach to learning (Early Math Strategy, 2003). A problem-solving approach allows students to use math manipulatives and truly explore the ideas in their own ways, making the learning personal to themselves. The learners are given opportunities to foster interdependence and responsibility for their learning as they work with their peers and talk through their strategies. Each student is involved and accountable for their own learning while they work on their abilities to verbalize, reformulate ideas, discuss, and compare (Education For All, 2005). These opportunities to talk out their thinking with others not only builds appropriate social behaviour in our students and their ability to provide constructive feedback, but it also builds their understanding of the concepts and encourages them to think mathematically. Through these effective math talk sessions, 64 students begin to realize that their ideas and opinions are valued and essential to the learning of the class as a whole (Capacity Building Series, 2010). Oral communication helps students to express their observations and the patterns that they find in mathematics. Talk allows them to rehearse their ideas and use each other and the teacher as sounding boards for their thinking (Whitin & Whitin, 2002). Students engage in discourse to share, shape, and improve their understanding of a concept, which inevitably moves their own thinking forward. Discourse helps students learn to be clear and convincing, provides more chances for equal participation, and more opportunities to observe and listen to the ideas of others. Students are able to practice working in groups, and math talk also allows time for students to process their thinking while making ones thinking public. Finally, math talk helps misconceptions surface so that the teacher can address these areas, while also giving students the opportunity to think, discuss, extend, elaborate, write, verbalize, listen, and read (Education for All, 2005). In order to promote effective math talk in the classroom, teachers need to begin modelling "math talk" by using think-alouds to support student talk. When first starting math talk in the classroom, it is important for teachers to understand that while social beings, students need to be specifically taught the skills and behaviours needed to enable the sharing and considering of others ideas in addition to how to defend their own ideas and opinions, and build on and question the opinions contributed by others (Cobb, Wood & Yackel, 1994). Initially teachers will need to have a more hands-on role in the classroom and engage in a great deal of modelling until these skills have been effectively taught. Eventually teachers will begin to act as facilitators to their students' discourse by asking questions and mediating 65 opposing ideas. Students often have difficulty hearing the ideas of others and linking those ideas with their own, therefore, the teacher needs to be able to help students make connections between student ideas and thinking. With the teacher acting as a mediator, students were found to be able to construct their own understandings of math concepts (Brown & Hirst, 2007) . Furthermore, when teachers acted as reflective and critical users of math talk and were able to act effectively as mediators, the quality of student learning was vastly improved and students were able to gain new perspectives through the sharing of ideas (Brown & Hirst, 2007). In starting to use math talk in the classroom, teachers should try to create problem solving questions that are challenging, support students ' autonomy, and involve collaboration (Turner et al. , 1997). The use and encouragement of collaboration allows students the opportunity to listen to others solutions and allows students to organize their own thinking (Capacity Building Series, 201 0) . Silbey (2003) suggested having students discuss information in the classroom using a combination of individual think time, small-group discussions, and reporting back to the class as a whole before having students create written drafts of their solutions. The use of math talk prior to writing helps students develop their ideas and ensure that their thinking is clear and concise. Creating an inquiry-based classroom where students are allowed to develop their own understanding fosters the development of more engaged and self-directed learners (Capacity Building Series, 2011). When establishing math talk in the classroom, the role of the teacher is often quite different from the previous show and tell of correct answers or the basic math fact focused classroom that many of us have had experience with (Stein, 2008). Instead, teachers need to 66 become skilled at filtering student ideas to help demonstrate how each math strategy helped to illustrate different math ideas. During whole-class sharing sessions, teachers need to move beyond the sharing of only correct responses, which gives students no reason to listen, and help students to draw connections between different solutions or how these ideas link to strategies that were most useful, efficient, accurate etc.(Stein, 2008). Teachers should not feel that they should avoid telling students anything, which was a common misconception when math talk was first introduced (Stein, 2008), instead teachers should actively shape student ideas to increase student mathematical thinking while effectively guiding whole-group discussions. As would be expected, this requires a lot of improvisation on the teacher' s part during the discussions, because the more solutions there are to the same problem, the harder it is to orchestrate and link ideas for students (Stein, 2008). Creating an environment where social interaction is the focus for mathematical learning requires teachers to establish social norms where student thinking is valued even more than correct answers, where teachers expect students to express thinking, explain and justify solutions, and listen to the ideas of others. During pair collaboration, teachers should expect students to solve the problem, agree on an answer, and respect each others ' ideas. If norms are set from the beginning of the school year, there will be no need for external rewards; instead students will become task-oriented as they develop both social and intellectual autonomy (Cobb et al. , 1994). Stein (2008) suggests having two main norms established prior to implementing inquiry-based, student-centred instructional tasks. The first is student authority whereby students are the authors of their own mathematical ideas. The second is student 67 accountability whereby students are expected to be able to account for how their ideas connect to the ideas of others. These norms require students to work together and listen and ask questions of others to fully understand the ideas of other individuals ultimately leading to an increased understanding of mathematical concepts, strategies and solutions. One of the most important roles of the teacher in promoting student discourse is asking good questions (Silbey, 2003). Questions that teachers ask can completely influence the thinking of their students, along with the overall outcome of the discussions that happen in the classroom. Asking good questions is an important aspect of encouraging student thinking. Questioning is also a skill that does not come easily, even to experienced teachers. In asking good questions, it is first important to remind yourself never to say anything a student can say. Instead, teachers and parents should be asking questions and allowing students to come up with the answers. Asking questions instead of stating information that students could be providing helps send a message to the students that their participation is essential. Furthermore, it helps to support a constructivist classroom approach where students are creating their own knowledge while investigating together. When learning to ask good questions, it is important to remember to question-listenquestion (NCTM, 2000). The idea is to pose a question to promote deeper mathematical thinking. Teachers need to listen actively to students ' thoughts and ideas in order to develop and ask questions that will help to build student understanding. By recording yourself teaching, you will be able to hear whether your questions actually do help students to develop deeper understanding or to get them to a desired answer. As teachers, we want to ensure that 68 the student is learning from the questions we are asking. Teachers should listen to students ' responses and guide them based on what they are thinking. When teachers are preparing lessons, it is important for them to anticipate student thinking and create questions in advance that will link to the goals for that lesson. Also, teachers and parents need to pose open-ended questions which will help to build student selfconfidence. By asking questions, we are helping students to become more engaged, mathematical thinkers. It is very common for teachers to ask leading questions that have only one answer, especially when it comes to the field of mathematics. While this can act as a good starting point for asking questions, teachers need to quickly move from this direct approach, to a less-direct approach in order for students to develop their skills and understanding. In order to improve the talk in the classroom using a less-direct approach, teachers need to ask non-leading questions. For example, teachers may ask their students: "What are you doing?", "Why are you doing it?", and "How does this strategy help you?" (Bums, 1995). These questions help teachers to understand what students are thinking and how they are going through the problem solving scenario. In addition to posing open-ended questions the use of verbs from Bloom' s Taxonomy is also a helpful tool. The Bloom ' s continuum contains six categories that range from more concrete to more abstract, from lower order thinking to higher order thinking skills. These revised verbs are presented in Table 1. Creating these open-ended questions often takes more time, thus it is strongly suggested that these questions be prepared as much as possible before class. Many of the closed questions we instinctively ask in math class can be turned into open ended questions with a little bit of thought as we try to tap into students higher order thinking 69 verb Table 1 'Bloom's Taxonomy Description Remembering Retrieving, recognizing, and recalling relevant knowledge from long-term memory. Understanding Constructing meaning from oral, written, and graphic messages through interpreting, exemplifying, classifying, summarizing, inferring, comparing, and explaining. Applying Carrying out or using a procedure through executing or implementing. Analyzing Breaking material into parts, determining how the parts relate to one another and to an overall structure or purpose through differentiating, organizing, and attributing. Evaluating Making judgments based on criteria and standards through checking and critiquing. Creating Putting elements together to form a coherent or functional whole; reorganizing elements into a new pattern or structure through generating, planning, or producing. (Anderson & Krathwohl, 2001 , pp. 67-68) skills through the use of the verbs from Bloom ' s Taxonomy. One way of knowing that you have asked a great question, is that math questions worth asking often promote discussion and dialogue. A question may be posed to the class and students may work in small groups, student pairs or mixed-ability groups until they can agree on one or more good answers before they share their reasoning with the whole class. Another indication that you have asked a good question is that good questions will lead to more good questions. Students who explore open-ended questions become great inquirers and will begin to pose questions worth 70 asking and exploring themselves. By asking good questions, we are helping students to become more engaged, mathematical thinkers. Chapin e. al. (2009) discussed three different types of talk that could be used in the classroom. The first was whole-class discussions where the teacher acted as a facilitator of the talk and guided the discussion. Whole-class discussions did not focus on giving correct answers, but instead focused on student thinking. Teachers and students were exploring the steps taken in students' reasoning together as a learning team. The second type of talk was small-group discussions which usually involved three-six students. During small group talk the teacher circulated and didn't control the discussion, but instead observed and interjected if needed or appropriate. Once students had experienced some teacher direction in the whole-class discussion, students were be able to direct more of the talk on their own as they learned how to ask questions, challenge, and share ideas with their peers in socially acceptable and polite ways. The third type of talk that Chapin et al. (2009) advocated was the use of partner talk. Partner talk was the same idea as a think-pair-share as it allowed students to practice and rehearse their reasoning with another student before contributing to the class discussion. Partner talk also allowed students to bring up questions in a non-threatening/safe environment as they began to formulate how to ask the whole group. This type of talk was used a lot when first introducing talk in the classroom as it allowed students to practice talking with just one other student before feeling comfortable sharing their thinking with the class as a whole. Furthermore, the use of partner talk was also a great way to introduce a new 71 topic as it once again allowed students to feel comfortable as they delve into new content (Chapin et al., 2009). There are many different high-yield strategies that teachers can use to help develop students' oral communication skills with mathematical concepts. Many of them are the same as the strategies teachers would use in other disciplines when having students work on their discourse. The overall goal is to get students talking about different aspects of their mathematical thinking and learning. By making these tasks enjoyable and fun, students are more likely to become engaged in the task allowing teachers to get a better understanding of what students understand and how they are processing mathematical concepts. Table 2 lists some possible strategies that could be used to encourage effective discourse and mathematical thinking. Having a variety of different ways to encourage math talk with students helps to keep students interested and involved while also giving teachers a tool kit of ideas to meet the learning goals for that day. All of these strategies can help give diversity to the mathematical environment and offer different ways to get students excited and talking about their learning, while deepening their overall understanding of the concepts being examined. Many of the strategies listed in Table 2 involve students writing their ideas on paper and then sharing these ideas with peers. Students often need to be able to solve problems and work out their solutions on paper prior to feeling comfortable sharing their thinking with others depending on the individual learning styles of the students. Without strong discourse, writing down ones mathematical thinking • would be far more difficult for students, since the rich discussions offer students a chance to rehearse their ideas before writing them down. 72 Strategy Table 2 Math Talk Strategies Description Solve and Discuss A group of students go to the board and solve the same problem while the class solves that problem at their desks. Students at the board share their thinking, solutions, and strategies with the class, while students compare and discuss the different examples. Mental Math Students are encouraged to put down paper and pencils and attempt to answer simple math problems/scenarios in their heads. Students are then encouraged to share aloud how they got to their answers which demonstrated different thinking and strategies to get to the same solution. Step-By-Step Each child performs a different step towards the solution of the problem. In this scenario, all group members have a role and are involved in the solution. Think-Pair-Share (or Student Pairs) Students work to solve problems in pairs before group discussion/sharing sessions. Partner work gives students a chance to verbalize their thinking and use their partner as a sounding board to ensure their solutions make sense and are clear. Whole Class Practice and Student Leaders The class works independently on a solution to a problem and then selected students help to lead the discussion as students share their solutions to come to a class consensus. Scenarios Students act out or role play a mathematical relationship in a visually memorable way. Involves creativity in the demonstration of student understanding. Small Groups Students work in groups of 3-5 on a problem. Group solutions are then shared with the class. 73 Sp eakers and Listeners Students take turns being the speaker and the ' active listener' who has to paraphrase, question, and make sense of their partner' s ideas. Numbered Heads A strategy used to get students into different groupings within the classroom. Students are given a number (i.e. 1-5) and then all of the number 1' s get together, all of the number 2' s, and so on. Choral Answers Students answer as a class. This approach works well for quick facts that are being shared or fmal answers, not for the process being used as this will vary between groups or individuals. 1, 2, 3 Flash Students have the opportunity to write their answer on a piece of paper or use their fmgers to show their final answer. Students then show their answers when asked. The 1,2,3 Flash is a good way to do a really quick check to see if students are on the right track (Silbey, 2003). Math Vocabulary & Math Word Wall Students are introduced to different math vocabulary and are encouraged to use math vocabulary effectively. The use of a math word wall helps to promote math vocabulary while also helping to remind students of vocabulary that they have learned, as well as how to spell then, which is helpful as students start writing. Math Strategy Wall Similar to a word wall, however a strategy wall lists different mathematical strategies which students have used and how each strategy could be used. For example: Guess and Check - think about the question, think of a possible answer and check to see if it is correct. Catch the Mistake and Fix Students are given a problem and a solution that contains an error. Students take on the role of the teacher to find that error and fix the solution. (Education For All, 2005) 74 Parallel Tasks Students are given different numbers to use in solving similar questions. Parallel tasks allow students who are more comfortable with smaller numbers to still apply the same skills in a differentiated way allowing for differentiated instruction. When mediating discussions in the classroom, Stein (2008) advocated a LaunchExplore-Discuss lesson structure with student ideas as the launching point. During the launch phase, a problem was presented to the class for students to discuss and solve in small groups or pairs. The second phase involved the exploration of mathematical ideas as students worked in pairs/small groups to solve or work through the problem. The third phase was the whole-class discussion and summary phase where teachers orchestrated the class discussion by purposefully selecting pairs/groups to present their solutions and then helping students to link ideas before being able to summarize key points from the discussion. When implementing inquiry-based, student-centred instructional tasks that centre on mathematical discussions which follow the launch-explore-discuss lesson structure, there are five practices that will help teachers to plan and implement talk effectively in the classroom: Anticipating, Monitoring, Selecting, Sequencing, and Making Connections (Stein, 2008). The first practice involved anticipating likely student responses. This required teachers to think about how students may interpret a problem, what strategies (correct or incorrect) students may choose to tackle the problem, and how these strategies may relate to the math concepts being presented. By anticipating possible student responses prior to a problem being presented to a class, teachers had the opportunity to have possible questions or 75 activities planned to help students recognize where they went wrong. Anticipation required teachers to at a minimum, have completed the tasks themselves and solved them in as many ways as possible (Stein, 2008). The second practice involved monitoring student responses during the explore phase. During this phase, teachers needed to pay very close attention to students' mathematical thinking and circulate throughout the room with the goal of identifying mathematical learning/strategies being used and deciding which of these should be shared with the whole class. Taking notes during the explore phase was a strategy suggested to help teachers to keep track of the ideas/strategies being explored by different groups in order to help teachers to effectively guide the whole-class discussion later on. It was important for teachers to make sense of student thinking during their circulation through the room, even when this thinking was incorrect (Stein, 2008). The third practice involved selecting students/groups to present during the discussion and summary phase. By purposefully selecting student responses, teachers were more likely to discuss important mathematical ideas, which allowed teachers to air out common misconceptions (Stein, 2008). While teachers may have asked for volunteers to present their ideas, it was important to have selected students who have useful ideas to share with the class for discussion. The fourth practice involved purposefully sequencing student responses. At this stage, teachers were responsible for organizing student responses which have had similar misunderstandings or which used related or contrasting strategies in the solutions. Once teachers have selected and sequenced student responses they will be able to help students to make connections between strategies and allow teachers to build on student ideas. By 76 purposefully sequencing the responses, the discussion will be more predictable and easier to follow (Stein, 2008). The fifth practice involved making connections between student responses and key ideas. Teachers and students were making judgements about approaches and the effectiveness of these approaches. Teachers needed to understand the math that was being presented in each students' answers and find ways to make connections between the solutions. Furthermore, teachers needed to ensure that all students had the opportunity to participate meaningfully to each class discussion. Teachers were responsible for planning and asking questions that would help move student thinking forward, thus they should have carefully scaffolded student learning and used problems that had meaning for students. When problems related to real life, students were much more eager to get involved in the solution and were also able to see a purpose for mathematics. Teachers were responsible for helping to lead students to compare responses, reflect, evaluate, and revise ideas, discover different strategies and how they could be used to solve different problems. By making connections, students will be able to build on ideas and strategies as they develop powerful mathematical ideas and understandings (Stein, 2008). During a Think-Pair-Share teachers need to change the way they ask questions from questions like "Do we all get it?" or, "Does anyone have any questions?", to asking questions that give the learners an opportunity to communicate their reasoning as to why they chose a particular method and how their choices made sense. Thus, teachers should transform their closed-questions, those that can be answered with one word, to open-questions, those that require explanations. If students are not familiar with this type of questioning, it is important to progress to these types of questions slowly so that students will become more comfortable 77 and confident that they know what constitutes an appropriate response. The think-pair-share is a wonderful way to introduce these types of questions as it allows students to think on their own and then share and talk through their solution with a partner before sharing with another pair to students and finally facilitating a whole-class discussion. When first beginning math talks with students, teachers need to plan what their discussion will be, the types of questions they will use to promote discussion as well as possible misconceptions that may arise based on the problem being presented. Although having a solid plan is important, it is equally important to be able to improvise and respond to situations that come up in the discussions that teachers did not anticipate (Chapin et al., 2009). Teachers need to follow student thinking and continue to ask open-ended questions to help students come to their own understandings of math concepts. Once a plan is in place, it is important to think about what types of things you are going to get students to talk about. There are many different directions a math talk can go depending on the type of understanding you want students to develop. Teachers may choose to have students discuss math concepts, building relationships between concepts, vocabulary, computation, reasoning, problem solving, and problem solving strategies (Chapin et al., 2009). It is important to note that some of the best discussion will come out of problem solving situations or questions that are presented to the students; however, it is important to have students discuss all aspects of their mathematical learning. 78 Math talk may look a little bit different across the grades ; however, the main thing that will change is the depth of conversation and the variety of different types of math talks that students will be able to handle. In Grades 1 and 2, students will be spending the majority of their math talk time in a full class discussion where the teacher is able to model math talk and monitor and support student ideas while helping to make connections between the ideas of the students. Students will of course be introduced to partner and small-group talk; however, these groupings will be used a lot less and for shorter periods of time as students in these grades often need more focused tasks and are only starting to build upon their math talk strategies and skills. In Grades 3 and 4, students will begin to use more partner and small group discussions as they explore mathematical concepts with their peers. Students who have had math talk experience in the earlier grades, will be able to build upon their knowledge and self-confidence, and really begin to investigate the mathematical concepts via oral discourse. Students at this age will be better able to defend their thoughts and ask critical questions of their peers to help them understand math concepts and the thinking of their peers. In Grades 5 and 6 students will be relatively proficient at clearly stating their ideas, defending their ideas and asking critical questions. At this age group, math talk can take many different formats as students are better apt to mediate, question, and expand on the ideas of others. With practice, students will be able to become confident in their oral math communication skills and in turn, will develop deeper understandings of the math around them. 79 Introducing math talk to students can be challenging as many students are unfamiliar and thus uncomfortable discussing their mathematical thinking. Historically math teaching has focussed on the rote teaching of math facts and math was thought to have only one correct answer. However, as students engage in discourse about mathematics, it quickly becomes evident that students create their own understandings of math concepts and often approach problem solving situations in very diverse ways. Teachers need to create an environment within their classrooms that allows students to learn mathematics through social interaction (Cobb, Wood & Yackel, 1994). Students need challenging problems, collaborative group work, and class discussions about student solutions in order to increase their mathematical understanding. When students express their mathematical thinking, teachers are able to create learning opportunities as students work together, cooperate, listen, and share (Cobb et al. , 1994). Math talk requires teachers to take on a facilitator role where they are helping to keep the discussion going without controlling the conversation. Taking on a role that is less directive is not always an easy task for teachers, especially if math talk is a new initiative for them (Mercer & Sams, 2006); however, with some practice teachers will start to see the growth in student understanding as students begin to make connections and construct meaning with less support. There are many different issues that can arise when conducting math talks in a classroom. The following offers some suggestions of how to approach these concerns, should they arise in your classroom. 80 The most important talk move that should be used regularly is waiting (Bums, 2005; Chapin et al. , 2009). Teachers should wait a minimum of five seconds after posing a question before calling on someone to answer. If many students do not have their hand up after this time, it may also be effective to restate the question or explain it in a different way as this may help clarify the question as well as providing more time for students who hadn 't quite formulated their ideas (Capacity Building Series, 2011). Wait time is also important to remember after calling on a student to answer. Sometimes students will raise their hand as they have an idea; however, they may need a few extra seconds to find a way to put their thinking into words. The use of wait time allows students to organize their thoughts before they start to answer. Wait time is also important to remember when dealing with students whose first language is not English and also helps to draw out quieter students as they are shown that everyone' s ideas are valued and think time is respected (Capacity Building Series, 2010). In order to understand what other students understood of another students ' reasoning, Chapin et al. (2009) suggested the use of repeating. In this move, the teacher asked one student to repeat or rephrase what another student had said and then immediately followed up with the first student to ensure that the repeating of the second student was accurate. Repeating helped students to realize that other individuals were listening to their ideas and thus, the students needed to make their thinking and explanations explicit. Through the use of questioning and having students restate another student' s ideas, teachers can begin to gage 81 how much a student truly understands about a mathematical concept or solution. Once again, teachers should be using open-ended questions to attain a clearer picture of what students understand as this type of questioning requires students to explain their thinking as opposed to simply sharing a final answer. Once teachers have a clearer understanding of what students understand, teachers will need to help students whose answers are unclear. Chapin, et al. (2009) suggested using a revoicing technique where the teacher tries to repeat some or all of what the student had said, and then asks the student if the teacher' s revoicing was correct. Having students paraphrase allows students to hear their own ideas expressed orally and are then able to add to, or correct any aspects of their answers that no longer make sense to that student. Teachers and students also need to remember to listen with an open mind and if the answer still does not make sense, teachers or students should ask for clarification. Teachers could clarify student thinking by saying "Tell me more ... " or "Explain your thinking/answer another way ... ". Another way to deal with unclear answers is through the use of questioning; however, paraphrasing is a great first step to ensure that what was heard was correct as it allows the student to determine if what was heard was accurate, as well as giving the student an opportunity to correct or edit their explanation (Capacity Building Series, 201 0). To help students ensure that their thinking is being stated clearly, teachers could have students work in pairs with the partner acting as a sounding board. One student would explain their thinking aloud and their partner would be responsible for paraphrasing what they heard, checking to ensure that they heard correctly, and then asking questions until they too fully understood that students' thinking. Teachers may also choose to pair with an older 82 'buddy' class, similar to a reading buddy, but instead using math in a mentor-like role (Crespo, 2003). The younger student would then explain their idea aloud and the older student would be responsible for asking questions and helping the younger student adapt or modify their answer to ensure that their thinking was clear and concise. These types of oral discussions would also greatly help students to eventually put their ideas on paper later on. Chapin, et al. (2009) suggested using reasoning to help make student thinking explicit. In this move, the teacher had a student make a claim, and ensured that all students had heard and understood the idea presented. The teacher then asked if another student agreed or disagreed with the claim presented, and asked that student to explain why or why not. Reasoning allowed students to explain their reasoning and justify their thinking as they explained why they did or did not agree with the claim presented. Having students share their solutions and thinking aloud with others helps students to hear their ideas and use their peers as resources to make their thinking more concise. In every classroom there are the students who always have their hands up and tend to monopolize the discussion. To help deal with talk monopolizers it is important to use wait time and explain to the class that you are giving students time to think and waiting for some more hands to show that students are ready. Another strategy to help encourage participation by others is repeating the question. Repeating often helps students who are almost there, but just needed to hear the question again. Also, having students talk to a partner first will help to encourage ideas and allow for the rehearsal of students reasoning giving students more confidence to share with the class as a whole. 83 Teachers need to find ways to invite new speakers which could be done by simply asking for more ideas or explanations. Some teachers also use the 'gender rule' where a boy is asked followed by a girl and then a different boy then girl. The 'gender rule' strategy further helps teachers to choose a variety of participators while keeping the floor open for discussion. Finally, teachers may have to call on students who do not have their hands up. Before calling on these hesitant students, it is usually a good idea to have done a partner talk first so that students have had a chance to begin to formulate a response before having to do so with the class. Teachers may also want to pull the avid participators aside and explain why they aren 't being selected each time even though their hands are up (Chapin et al., 2009). For students who are extremely hesitant to participate, teachers should ensure that the students understand the problem by using revoicing and checking for understanding through the use of questioning. Teachers should also use partner talk first to allow the less confident or quiet students a safe and comfortable environment to share their thinking. Finally, it is important to remember that it will take time for hesitant and quiet students to feel completely comfortable with oral discussion in math, especially if this is something that is new to them (Chapin et al, 2009). As a way to help encourage new points of view to a discussion, Chapin et al. (2009) suggested using adding on. In this move, the teacher began by revoicing two positions that had emerged, and modelled how to respectfully check with the originators of the two positions that her revoicing was accurate. The teacher then asked others to contribute by asking them to either agree, disagree, or to add to the other students ' comments. Another idea is to have hesitant students illustrate their thinking on paper. Some students are able to 84 demonstrate their thinking better on paper and having a visual aid may help students to explain their thought processes (Bums, 2005). When using small-group discussion, it is important that groups are given challenging problems that will help to involve all individuals. Humans are social beings by nature, therefore it is understandable that discussions will occur when placed in a group. The role of the teacher is to redirect the talk to the task at hand in order to keep talk productive (Capacity Building Series, 201 0). If a problem is too simple, one student or the group will have solved the problem too quickly which doesn' t allow for the discussion and contribution of all members that small-group discussions require. Teachers need to be monitoring discussions within the classroom and refocusing conversations as necessary. In every class teachers struggle with how to deal with students who are not listening to the discussions, either in a whole-class or small-group situation. To help with students who are not listening, teachers should use the ' repeat' move which makes students respond to the ideas of others. Also, calling on students randomly after having them share their thinking in pairs also encourages students to pay attention as they begin to realize that they need to be ready to answer at all times and that their ideas and contributions are an important and respected part of the whole-class discussion. In every classroom there are students working at a variety of levels, especially in mathematics. While pairing students in a variety of ability levels is often common place for group work situations, this method is not as effective with math discussions as the higher 85 level students will solve the problem, share their thinking, and monopolize the learning of the others who are struggling. Thus, pairing students in similar ability levels and personalities will help to encourage better talk and help to keep the discussions going. Similar ability grouping also allows the teacher to support more needy groups easily and teachers know that these individuals are getting a chance to develop their understanding of the mathematical concept (Chapin et al. , 2009). It is important for students to have some variety in their learning so that they can learn from one another and learn to help guide/orchestrate talk from their peers in a respectful way. All of the above strategies for supporting math talk in the classroom are important to help encourage and promote equal access to participation for all students. It is necessary that the norms for respectful discourse are well established in the classroom from the beginning of the school year so that students understand standard tum-taking, how to respectfully disagree with another' s' point of view or answer, and how to ask questions. Students need to feel safe participating and sharing their thinking with others in the classroom and teachers need to use the strategies above to help encourage participation from everyone. In using discourse in learning math, student thinking becomes public and gives opportunities for students to negotiate meaning from others ideas. These discussions help to teach students to make their ideas clear and concise while also allowing them to learn from their mistakes or the mistakes of others. Students and teachers alike become skilled at active listening in both whole-class, small group, or one-on-one discussions. Through active listening, students ask questions of others as they engage in sense-making. When first introducing math talk into the classroom, students may have difficulty putting their ideas into words. Teachers need to mediate ordinary and mathematical language to help students 86 communicate clearly. Furthermore, providing rich learning activities can also help to encourage effective discourse from students (Bums, 1995). 87 88 Math Journals -Written Communication We know that students learn to speak before learning to read and write, therefore it makes sense that verbal communication should be introduced before written communication. In order to effectively communicate in writing, students should be very comfortable discussing and sharing their ideas verbally. Once oral communication has been established, teachers can introduce written math communication with their students. Similar to oral communication, it is important for teachers to be upfront with students about the purposes for writing in math class so that the students will be able to understand how writing will help them learn math (Bums, 1995). Writing is a tool that can be used to help students think about ideas as it allows students to gather, organize, and clarify their thoughts. Far too often students learn to compute before they really understand why the computation procedures make sense, thus Bums ( 1995) encourages teachers to allow students to develop their mathematical thinking and number sense through reflecting on their thinking, justifying their reasoning and judging the reasonableness of their solutions. Writing is a complex process, but one that can help to clarify students ' understanding of a concept/problem, as well as providing a record of the thinking and processes behind student solutions (Van de Walle, 2005). Furthermore, writing allows students to combine their personal experiences with mathematical language, which increases students understanding and communication skills. Students should be encouraged to write about their conceptual thinking and problem solving processes on a regular basis as it allows them to ask questions, share their feelings, and build their confidence towards different aspects of their understanding. Countryman (1992) believes that writing is an essential part of a mathematics 89 program because students need to organize, interpret, explain, construct, symbolize, communicate, plan, infer, and reflect about their learning through a variety of writing situations. Writing provides a window into each individual students thinking and understanding (Bums, 1995). This window provides teachers with ample information which can then be used to drive instruction while also being revisited by students for selfassessment purposes. Since writing is such an important, yet complex task for students, it is important that teachers use discourse prior to having students write down their ideas. Teachers may choose to use a paired, small-group, or whole-class discussion prior to students writing their thoughts, reflections, and learning on paper. Students also need to have a clear understanding as to why they are writing in math class. Thus teachers may say something like: "I need to know what you do and do not understand, and when I read your math writing it helps me to know what you do and don ' t understand.", or "Your math writing shows me what you understand and how I can best help you learn.". To help students progress to writing clearly, teachers should begin by giving students open-ended prompts to help spark their thinking. Furthermore, teachers are encouraged to introduce writing into their classrooms with more familiar math concepts before building up to the more complex concepts (Van de Walle, 2005). Starting with simple mathematic concepts will allow students to increase their confidence and self-esteem as they embark on documenting their metacognition. Once students become more confident in their written math skills, teachers can begin to see the internal dialogue of their students. Writing helps to open a window into the minds of students (Flores & Brittain, 2003) while giving students time to reflect, think, and discuss strategies before embarking on the task of writing their ideas on paper. 90 When student thinking is written down, teachers are able to attain insights into student's thoughts and ideas. Burns (1995) demonstrated that written communication should be used to assess student understanding. Often in mathematics it is difficult to ascertain whether students fully understand a concept; however, oral and written communication can help to make student understandings, or misunderstandings, evident. Teachers can then use the written discourse as a teaching tool to help address student misconceptions and ensure that all students are understanding a concept before moving on to the next concept. In addition to being a useful assessment tool, math writing also provides an excellent vehicle for communicating with parents about what their children are learning and the progress that they are making (Burns, 1995). The use of a math journal by Albert and Antos (2000) not only helped to inform parents about how their children understood math, but it also promoted social interaction as students shared their thinking and awareness of math concepts. The writing of mathematics was seen as a way for students to document their thinking and share their strategies and processes with the class at a later time, while also allowing for self-reflection. Furthermore, Albert and Antos (2000) purported that written math helped bridge the gap between math teaching and keeping parents involved and informed. Flores and Brittain (2003) also found that writing helped with the thinking, confidence and misconceptions of students. Student teachers were able to look back on their written math thoughts and reflect on their growth. Finally, Flores and Brittain (2003) found that written math communication helped to support a positive learning environment as it took some of the pressure off of the student to come up with the correct answer right away. It also allowed students to use strategies and methods that worked and made sense for them as they solved problems. Thus, the writing in 91 math was seen as a way to help relieve anxiety while producing critical thinkers. In sharing and discussing writing with others, students were able to learn from one another and expand their own mathematic understandings (Flores & Brittain, 2003). Countryman (1992) believed strongly in students and teachers keeping journals in order for teachers/students to reflect on their own experiences that were taking place in their classrooms. Writing allowed students to develop skills of planning, inferring, constructing, symbolizing, interpreting, and reflecting. Furthermore, by adding writing to the math classroom, everyone was active, collaborative, and were participating in the learning of mathematics, while also constructing their own individual understandings of mathematical concepts. Countryman (1992) believed that "knowing mathematics is doing mathematics" (p. 2), and that this could only happen in classrooms where students were encouraged to explore, justify, represent, discuss, and ultimately be an active part of their mathematical learning. The research presented above demonstrates that mathematical writing is important in helping teachers to understand what students know as the writing acts like a window into student thinking and also allows teachers to assess and address misconceptions. Furthermore, mathematical writing allows students a way to organize their thinking and share their ideas with others. Writing in math has also shown to increase confidence and self esteem in students while relieving student anxieties, as solutions can be thought through and demonstrated in several ways. Writing allows students the opportunity to reflect on their solutions and make amendments to misconceptions that may become clear once students share their ideas with one another. In addition to allowing for reflection, writing is also a wonderful tool for teachers to assess where gaps, confusions, or misconceptions lie, and what connections students are making between new and prior knowledge. Teachers can also use 92 math writing as an effective way to communicate with parents about student progress, as it makes student thinking accessible. Writing is a very powerful tool for students and teachers alike, as it can freeze ideas so that we can return to them later for reflection, refmement, discussion, and amendment (Whitin & Whitin, 2002). There are many different types of writing activities that teachers can use to get students writing in math class. If this is the students' first experience with math writing, a great way to get them inspired and thinking is through the use of picture books to introduce concepts. Table 3 below lists just some of the books that could be used to introduce math concepts and spark math discussions which can lead to a variety of written tasks. The following website contains lists of other picture books which could also be used for a variety of different mathematical concepts: http ://www.the-best-childrens-books.org/math-forkids.html. In addition to using picture books, having students write about how math talk helped them solve a problem is another way to ease students into the writing process. Students may write about how talking with a partner helped them, or what they learned by sharing their ideas. Another way to ease students into writing tasks in math class would be having students play a math game and then having them write about the strategy(s) they used, or how to play/win the game. Another creative way to introduce math writing is through the use of creative writing activities. Students may write a short story or Haiku poem about a math concept. Having students embark on creative writing tasks allows 93 Table3 Booiought 25 in November f.VId 13 more m De<::embet·. r f s l\e gove: 6 TO MiS$ Muller. how many post-it Home-Math Journal entry. This figure illustrates the student solution to a Home-Math Journal notes does Mr s. Tippett have now? Show your w?rh and explai'' how you ~ r j ~ ~ j r::..· jL E-.:·... .Ju t:. ~ c:.. l!ep··-" vfflt;(' ~i ~ ~ t i The first journal entry that was completed with the students was done completely in class and I was able to monitor and model what was expected when students took the journal home. Students had already had experience with our in-class math journals and knew that they needed to show how they got the answer and to explain their thinking clearly. For the initial journal entry, I gave students 20 minutes in class to solve the first problem independently and explain their thinking (Figure 2). Students used the blank area of the notebook to show their work and solve the problem using any strategies/methods they chose. 121 Students then proceeded to explain their thinking in words on the lined area below. After students had completed their math journals ensuring that their ideas were clearly explained, I had students tum to the next page in their math journals. Here I had glued in my own solution to the same problem. I explained to the class that this was what their parents would be doing at home, only they would be writing out their solutions on the page following the student' s solution. Students would get to answer the problem first on their own and then have their ' home partner' answer the same question using their own strategies and explaining their thought process. I explained to students that their home partner could be a parent, grandparent, Aunt, Uncle, older sibling, or anyone at home who could help to answer the problem. The reason I made the response partner so open was that I knew many parents in the school that I was teaching at worked and may not have always been available to work with their child, thus allowing for a bit of flexibility on weeks that were hectic in the home, and also allowing students to see different approaches to problem solving from different individuals. After the student and their ' home partner' had each solved the problem individually, it was explained that they would then talk about how they had each solved the problem and explained their thinking by using the pictures, numbers, and words. Once partners had shared their solutions with one another, the student would then complete a ' reflection ' about the solving and sharing of that problem. In class for the first example, I shared how I solved the problem with the class and had a few students in the class share their solutions with the class. Each student then completed a reflection so that they could practice reflecting and understand how this process would work at home. I had them use different reflection starters, which 122 were glued into the back of their journals (Appendix C) to help students to frame their reflections. Following the in-class example, I began to send the Home-Math Journal home on Tuesday and Thursday nights. Each Tuesday and Thursday night, the students were given a different journal prompt/problem (Appendix D) which they were to complete for the following day. After a week of using this twice a week schedule, I found that students and parents/home partners did not have enough time to complete the journals, especially given the one night tum over. So, the following week I began to send the journal home on Monday nights and had students return the completed journal on Friday mornings. Giving families the whole week to fit the journal into their busy schedules, allowed families flexibility and ensured that students would have time to complete the journal without rushing through this very important learning task. Students were responsible for answering the problem and explaining their thinking, having their ' home partner' solve the same problem, and then ' reflecting' about the question, the solutions, and the strategies that they had used to solve the problems. The students were excited at the idea that their parents had ' homework' ! (Bratina, 1996). I reminded students that they were to answer the problem on their own using any strategy that would work for them. Even though I rehearsed the process many times with my students, I still had some students who failed to follow the clear instructions. Two students had their ' home partners' solve the problem, but did not solve the problem on their own first, which made reflecting more difficult for these students. I then began to label each page for the students with the headings ' Your Solution, 'Home Partner Solution' and 'Reflection '. These headings seemed to help parents and students know what they should be demonstrating on each page as they 123 progressed through the steps in the correct order. Labelling the pages for students also seemed to help many of the students, as few others forgot to do the 'reflect' page after completing the problem on their first journal entry. figure 3. Plna P•!'Pl'r• !:!ricin! pal pJeces of r~ and !!"'""' ~~s an the ~ She put 19 Vf!lCM ••• oltogetlt<:rc 6 •f t~ ~r Pizza Pieces Problem. Student solution to the pizza pieces problem. rehe s , t ~~~ £!; orAd meres 01 ..-\VeT ..CC!tfii.it::: 124 figure 'f. Pizza Pieces Problem. Home Partner solution to the pizza pieces problem. put II r r e~ cl !}qc;unct pt 1 , , "' , baau 5t !Cf i~ I g of 19 u tltet< l<.>tre e f r~ tv {inc;{ c td P?Y ·\, ~ a( ICf, 1 ..!: ' b1rC1c figures. ~ -\o 1 'ou:.\- '\OI.A i ~ ~ ~ ~ ~ ~ ~~ ~ ~ \0ltt.re... e ~~ . -\\\v,.\- Q.,(ir; t Pizza Pieces Problem. Student reflection and sample feedback to the Home-Math Journal entry. ~. h)l.".: n .cdc. f'(\y :fer. jl"il£jj 1 \ or , o£1 t\ e fl21ita. v;?,,c,1 ,.:. ~ c!o'rJ» "' ~ v.rt. c,-v1 The above example demonstrates the use of the three sections of the home math journal. The first image, Figure 3, shows the student's solution to the Pizza Pieces problem, 125 the second (Figure 4) shows the home partner' s solution and the third page (Figure 5) shows the student's reflection after having shared and discussed solutions with their home partner. On the top half of the reflection page I also provided my written feedback to the student. In this example you can see how the student was trying to use what she knew about pizza and fractions to help her solve this problem; however, she also used what she knew about subtraction to find how many were left. After sharing with her home partner, this student realized that her use of fractions was inaccurate ("I drew the picture of the pizza which didn ' t make sense.") even though she did get the correct answer. Figure 3, 4 and 5 demonstrate the different strategies that students and parents may use and the different ways that students and home partners might explain their strategies and thinking. Following each journal, I also responded to the home partner team about something that worked really well and a suggestion to the child for the improvement of either their solution, or their reflection. My feedback on student and home partner journal entries was very important as it allowed students to get individual feedback about their solutions and how they could make their solutions clear and easy to understand. I provided this feedback on the reflection page, usually at the top (blank) of the page as students usually didn't use this space for their reflections. Post-It-Notes could also be used to provide feedback in this journal similar to the in-class math journal presented in the previous chapter. Once again, it is important to provide feedback on the same page if possible so that students can easily make connections between the feedback and their solutions. At the end of the term, parents were asked to provide feedback of how the Home-Math Journal process was going during parent/teacher interviews in order to get an idea of how parents were receiving the initiative and whether they required any further assistance or resources to help the process run more 126 smoothly. Parent teacher interviews offered a perfect venue for further opening the lines of communication, educating parents on the mathematical learning of their children as well as troubleshooting areas of concern that parents may have had (Peressini, 1998). It was extremely interesting to see how the parents solved the problems and the types of reflection statements that the students came up with. Some parents showed how they knew, where others just told what they did and what the answer was. I had some students who worked with an older sibling at home instead of a parent, which was also an interesting partnership. The students were highly motivated to take the math journal home and were beginning to use different reflection starters as they became more comfortable expressing themselves in written form. One Grade 3 student noticed that their ' home partner' did not really explain their thinking. His reflection for this journal entry was as follows: "The difference of my moms and I is that mom wasn 't trying so mine is better then my moms but we got the same answer. I wonder why mom doesn 't explain what she does. " Another student noticed that their 'home partner' had found a different solution to the problem, but that it also worked. The following is her reflection: "In this case my mom instead switched the cat and the frog. But she is still right. I liked the part where she explained how they were in line. The similarities are that me and my mom both put the bird at the end of the line. " I really enjoyed reading the parent solutions, and was pleased to see that many of the parents had made a special effort to find a different way to solve the problem than their child. I had some parents use multiplication to show the answer when their child drew a picture and added. Seeing home partners solve problems differently really helped to get students thinking 127 about the connections between multiplication and addition and how they could use these two different methods to get to the same answer: "You did the journal differently than me because you used multiplication instead of adding. I could have made my journal better by adding more detail. I was surprised that you came to the same conclusion as me. " Students were also able to reflect about their misconceptions, as seen in the following reflection about the mother goose and her goslings problem: "I noticed that our answers were different. My answer was 14 and my mom was 49. Now I understand that I should have added 7 seven times. I was surprised because I thought I was right. Sharing with my home partner helped me understand. Next time I will try to be better. " This reflection is very interesting because the student realized that they did not add enough times, but also because the parent also showed a misconception by multiplying 7x7 to get 49, instead of 7x3 to get 21. The parent and child seemed to have missed the information that was provided in the picture. The picture showed the mother goose with three goslings and asked students to fmd how many goslings there would be if there were seven mothers. This home partner team just read the question and when they read ' the same number of goslings' assumed that if there were 7 moms, there must be 7 goslings for each mom. I responded back to the student and home partner to praise the depth of the reflection while also clarifying the misconception. It is important for students to see that everyone makes mistakes and sometimes information could easily be missed in a question. It is also a great opportunity for parents to talk about re-reading questions and using picture cues to assess all of the given information and going back to double check their answers. In the following reflection, the 128 student realizes that she missed the last part of the word problem where 13 more post-it notes where bought: "We both had a lot in common but one thing, !forgot the 13 post-it notes." figure 6. l hr +,1 fr ·"""' \l ~~~ ~ ~ 1 Chocolate Problem. Student solution to the chocolate problem. SA f#J btrn;;.:._ ·'I :rrro:ml ,_ fre{l 'ras co 01e. anJ .::.D ~ 129 figure 7. Chocolate Problem. Home Partner solution to the chocolate problem. figure a. Chocolate Problem. Reflection and sample feedback to the HomeMath J oumal entry. In the above chocolate problem, the student used numbers, pictures and words to explain their thinking and show how they came up with 10-5=5 (Figure 6). The home partner 130 also used pictures but showed the math in a step-by-step manner as opposed to putting 3 and 2 together to get 5 (Figure 7). By demonstrating the same solution in a different way the student was able to see similarities and differences between the solutions. Acting out these types of problems with students also helps them to make connections. This student still seemed confused in their reflection as to how their home partner got 8-3=5 and they did 105=5 (Figure 8). In this situation the teacher may choose to meet briefly with this student to go over both solutions and ensure that they make these connections prior to sending home the next problem. The more questions that the students had, the better their explanations of what they did became. The parents that took the extra time to explain their steps in words, really inspired their children to do the same. Parent entries became a very powerful modeling tool as the students took cues from past journals with their ' home partner' and used them to improve the quality and quantity of their own work. The dialogue that parents were having with their children was fantastic and was demonstrated in the great reflections that students were writing. Parents were allowing their children to attempt problems on their own before giving hints and allowing them to follow through on their thinking and explain their strategies and solutions in their own ways. Students were then found to be going back and determining where their thinking went wrong and why. Most importantly, students were excited and motivated to complete these journal entries, and were very upset when I kept them for the week of parent-interviews. Bratina (1996) offered six tips for involving parents in mathematical learning: 131 1. Ensure that parents understand the task. Parents should be approached early in the year and the Home-Math Journal process should be explained well in advance of the Home-Math Journal being sent home. 2. Provide clear and easy directions. By sending home a letter which clearly outlines the Home-Math Journal process prior to the start of the initiative, as well as having the steps of the process pasted in the front of the math journal for easy reference will help parents and students know what to do. 3. Keep the activities simple. The idea is to have students use their oral and written communication skills to solve problems. The problems selected for the HomeMath Journal do not need to be complex, but rather should promote student thinking and understanding. 4. Use frequently for increased impact. By sending the Home-Math Journal home once each week, parents and students get into a routine and build upon their skills over time with weekly feedback from the teacher. 5. Get feedback from parents and students. At the end of the term, teachers should use the Home-Math Journal as a discussion point during parent-teacher interviews and use any feedback received from parents to help improve the process or plan for the year ahead. I wasn't sure how parents would take the idea of a Home-Math Journal when I first introduced this initiative in my classroom and therefore decided to speak to this assignment in person during parent/teacher interviews. I was happily surprised at the amount of very positive feedback that I received from parents. Parents commented on their children's 132 engagement, and on an increased willingness of their children to receive help from their parents at home. I was also able to explain the relevance to standardized testing, which was fabulous because each year I was always asked how I was preparing my students for the testing and what types of questions students could be asked to answer. The Home-Math Journal was a fantastic resource for explaining the importance of students justifying and explaining their thinking during math problems. Parents were excited with the journal because it allowed them to work with their children on specific and valuable tasks. One parent commented that she "really enjoys the Home-Math Journal and has really liked the idea of working together on the same problem". Another parent stated that they "weren 't sure how to help when their child didn 't know how to begin the problem" . They went on to explain that they gave hints as to how to solve, or they solved the problem and showed their child how to solve the problem. I then explained that the goal of the problem solving process was to have the students decide how to solve the problem. I referred them back to my first hand-out which had a list of questions they could ask their child, as well as telling them to try offering a strategy to try. For example, "Would drawing a picture help you understand the problem?" or "What information do you need to help you?" . The parent conference was a great time for me to explain and show parents how to help get their children started on a problem using questioning, without giving students the answers. Another parent also stated that they " loved the Home-Math Journal because it really got the parents involved" and "helps them to get their child to understand things in different ways" . It was great to receive such great feedback because most parents had mentioned at the beginning of the year that they wanted more math drill practice. I think parents were really starting to see how students needed to be able to apply their skills and really think about what 133 was happening in the problem. Some students used manipulatives like Lego to help them solve the problem, where others drew pictures. I did not receive any negative feedback from any of my parents about the Home-Math Journal. One parent joked about the idea of homework being given to them, but went on to explain the value that she was getting from the experience. It was fascinating and empowering to hear the specific and positive feedback that parents were able to share about their home partnerships with their children. Some parents were asking if ' they got the right answer' and were eager to see the feedback from the last problem. Many parents said it was the only time that their children really listened to their suggestions and were accepting of ' help' with their homework. Apparently I had many students in my room who did not want to do any extra practice or revision with their parents at home and the Home-Math Journal time was a saving grace as it required dialogue between parent and child. The parents then went on to say that they wished more of the homework could come home like this, because they were really able to work with their child like a team. When starting a home math initiative, it is always important to ensure that parents are well informed about what your expectations are for them, why this initiative is being introduced, and how this initiative will help improve student achievement and understanding. Once parents have a solid understanding of the background behind the initiative and what their role will be, it is also important to keep the lines of communication open. I have found that the use of a daily planner (or agenda) helped students and parents keep track of due dates and other important information that was occurring in the classroom. The daily planner has acted as an effective communication tool when parents had quick questions to ask of the 134 teacher, as they could just write in their child ' s planner and have a response from me that night, as I checked planners each afternoon before they went home. In addition to a daily planner, I also tried to keep parents informed through the use of a monthly newsletter. In the newsletter I was able to let parents know the educational focus in each area of the curriculum that I had planned for that month. It also gave me a forum for introducing new topics or initiatives being introduced in the classroom. If the initiative was very new, it may have required distinct letters sent home that further explained the process, as in the case of the Home-Math Journal. Finally, the best way to communicate with parents is obviously in person. While this is often difficult to arrange, taking advantage of parent-teacher interviews or meet the teacher nights are two venues that lend themselves to these types of open discussions. I have found the meet the teacher night to be a great time to explain my math teaching to parents, have them experiment with math manipulatives and view their children ' s in-class math journals. I then used this opportunity to begin to explain that students will be progressing to a HomeMath Journal which would be coming home once a week. Outlining this project at this time early in the year gives parents a bit of a heads-up of the upcoming initiative and also allows parents to ask questions at that time or prior to the start of the Home-Math Journal initiative. The parent-teacher interviews typically fell around the end of the first term and offered a wonderful venue to reflect on how the Home-Math Journal process, how it was going at home, and also allowed teachers to field any questions that may be surfacing from parents. It was also a way for the teacher to get feedback from families on how the journal was being used at home, as well as a way for teachers to go back through the journal entries and show the growth in student understanding and written communication. Often parents 135 don't see this growth on their own, but once it is pointed out, they began to see more clearly what the expectations were and how they could help their child to improve their written communication and reflection, as well as working on their problem solving skills. The Home-Math Journal may look relatively similar across the grades as the basic steps in completing the journal are the same. The students will solve and show their solution to a problem on their own page, then a home partner will solve and show how they solved the same problem on a second page. The third page will then be a reflection completed by the student following the sharing of the two solutions by the student and their home partner. The one thing that will change across the grades is the amount and sophistication of the written work. In Grades One and Two, students will likely be using more pictures to demonstrate their knowledge, while in Grades Three to Six students will become more comfortable and confident in explaining their thinking in words. Students will also be solving problems that reflect expectations in their respective grade levels, thus becoming more complex as students progress. The more complex problems will also lead to an increase in explaining and justifying answers. Introducing the Home-Math Journal should begin with an initial letter home to parents which clearly outlines the process and reasoning for the initiative. Teachers will have already planted a seed at the beginning of year at the meet the teacher night, so parents should already have been given a heads-up that this Home-Math Journal initiative would be 136 starting soon. Following the initial letter home, you may want to draft another letter later in the year to address an area of the Home-Math Journal that you have noticed to be occurring in many students journals that needs to be addressed or corrected. The second letter may outline further information for parents, or provide specific examples to help address the area that many students or parents are struggling with or that need to be improved. Using a letter for this purpose allows the teacher to provide support in a friendly way while also addressing an area that many individuals are struggling to improve upon at one time. One of the most helpful strategies for students and parents is the use of sentence stems. For the Home-Math Journal I have put together a list of sentence stems to help students explain their thinking (Appendix B) as well as separate sentence stems to help students reflect with their home partner (Appendix C). To make these helpful tools handy, I made sure that they were glued right into the Home-Math Journal so that parents and students always knew where to find them and always had them accessible to assist them if they were unsure how to start explaining their ideas. Table 5 on pagel37 also contains sentence stems which could be used with students to get them writing about mathematics. This table also lists some questions that could be asked help students who are stuck on a problem or who are having difficulty putting their thinking down on paper. The table is broken up into different areas of mathematical problem solving including mathematical thinking, sharing representations, retelling, making connections, sharing feelings towards math, and predicting. Table 5 could be used by parents or teachers to help support the expression of both oral and written mathematics in today' s classrooms. 137 Table 5 Satnple Questions and Prompts to get Students Thinking Questions to AsK Student Prompts to Assist Student Writing Help get StudentS thinKing. How else could you have ... ? How are these - - the same? How are these different? About how long .. .? (many, tall, wide, heavy, big, more, less, etc.) What would you do if. .. ? What else could you have done? If I do this, what will happen? Is there any other way you could ...? Why did you ... ? How did you ... ? How do you know? What does (this)_ _ represent? How did you know where ... ? How did you know which ... ? How did you know when ... ? Could you use some other materials to ... ? How could you record your work? How could you record your discovery? How could you share your discover? How did you estimate what the answer could be? How did you prove your estimate? Help studentS share their representations. How have you shown your thinking (e.g., picture, model, number, sentence)? Which way (e.g., picture, model, number, sentence) best shows what you know? How have you used math words to describe your experience? How did you show it? How would you explain _ _ to a student in Grade _ _ ? (a grade lower than what student is currently in) I decided to use a... A graph (table, T-chart, picture) shows this the best because ... I could make this clearer by using a ... The math words that help someone understand what I did are .. . 138 Help students reAeet on their worK. What mathematics were you investigating? What questions arose as you worked? What were you thinking when you made decisions or selected strategies to solve the problem? What changes did you have to make to solve the problem? What was the most challenging part of the task? And why? How do you know? How does knowing _ _ help you to answer the questions _ ? A question I had was ... I was feeling really .. . I decided to , I was thinking .. . I found _ _ challenging because ... The most important thing I learned in math today was ... Help maKe connections. What does this make you think of? What other math can you connect with this? When do you use this math at home? At school? In other places? Where do you see _ _ at school? At home? Outside? How is this like something you have done before? This new math idea is like ... I thought of.. I did something like this before when... We do this at home when we ... I remember when we ... Help share feelings, attitudes or beliefs about mathematics. What else would you like to fmd out about ? ---How do you feel about mathematics? How do you feel about - - - - - - - -? What does the math remind you of? How can you describe math? The think I like best about mathematics is ... The hardest part of this unit on _ _ is .. . I need help with because ... . Write to tell a friend how you feel about what we are doing in mathematics. Mathematics is like - - because ... Today, I felt .... Help studentS to retell. How did you solve the problem? What did you do? What strategy did you use? I solved the problem by ... The math words I used were ... The steps I followed were .. . 139 What math words did you use or learn? What were the steps involved? What did you learn today? What do(es) mean to you? Help predict, invent or problemso1ve. What would happen if. .. ? What decisions can you make from the pattern that you discovered? How else might you have solved the problem? Will it be the same if we used different numbers? What things in the classroom have these same shapes? How is this pattern like addition? What would you measure it with? Why? How are adding and multiplying the same? My strategy was successful because ... Explain to a young child or someone that wasn ' t involved ... Draw a picture to show how you solved the problem. Prove that there is only one possible answer to this problem. Convince me! Tell me what is the same? What is different? How do you know? (Capacity Building Series, 2011a) Helping parents ask effective questions will help parents understand how math is being taught, as it demonstrates the inquiry approach to learning that is being used today. Furthermore, asking effective questions will help parents to maintain consistency in how mathematics is being approached in the classroom while also assisting parents in understanding what their child knows, and where their misconceptions lie. Providing parents with prompts and questions that could be used in a variety of situations will give parents the tools necessary to assist their children to create their own understandings and reach solutions with more independence. As teachers, we want to empower parents to probe the thinking of their children and encourage understanding without giving them the answers. The following is a table that could be sent home along with a letter 140 to provide parents with the vocabulary and questions that may assist them in working with their children (Table 5), while also acting as a valuable resource for classroom teachers. By asking questions of our students, we will not only promote student understanding and learning, but we will also uncover what students really understand and where their misunderstandings are occurring. We are not doing students any service by giving them the answers or helping them too much. Instead, parents and teachers alike should be asking questions and helping students to create their own mathematical understandings. Having students engage in reflection is yet another way to get students to gain full understanding of mathematical ideas. Without metacognition, students will not be able to fully understand a concept as they need to be able to understand what they know and then make connections between different ideas. To assist students in becoming reflective thinkers, teachers and parents need to provide many opportunities for discourse. Furthermore, teachers will need to provide support to parents, as reflection is likely not something that they themselves engaged in with regards to mathematics. Thus, parents and teachers should encourage the oral sharing of solutions with a partner whom is responsible for actively listening and asking questions to ensure understanding. Students should be encouraged to share what they learned, what they did, why they did it, and anything that they didn 't, or still don't understand. Teachers and parents need to model reflection to students so that they can see and hear the internal dialogue that goes on when one shares their thought processes. Again, this is not something that will come easily to students, as they have likely not had a lot of experience reflecting. However, students will pick up the skill a lot faster than an adult, 141 especially if parents and teachers are frequently modeling effective reflection. Teachers and parents should also model how to share solutions by using any numbers, or pictures to help explain and demonstrate their solution to others. It is important to note that teachers should be using this reflection/sharing technique during the in-class math journal. So, students should have had a significant amount of practice with the skill prior to being expected to reflect at home with their home partner. When students become stuck on a problem, teachers and parents should first try to ask questions to assess what students understand. Some questions and prompts to use may be: • "What do you have to do on this assignment?" • "Tell me what you ' re supposed to write about." • "Do you have any idea about what the answer is?" • "Why do you think that?" • "What ideas do you have?" • "Tell me something you know." • " What could help you get to the answer?" • "See if you can say those words again in your head and then write them down on your paper." Also, referring to the table on pages 135-137 will help to provide parents and teachers with the tools necessary to assist students in moving forward with their understanding. Providing a list of questions and prompts like the ones in table 5 on pages 135-137 in a handout or letter home to parents would allow them to be utilized throughout the school year and possibly even be pasted to a fridge or kept in a safe place for use during the Home-Math Journal entries. 142 As with any homework assignment, it is important for teachers to be prepared for the Home-Math Journal to not come back on time despite constant reminders in the daily planner. In my own personal experience with the Home-Math Journal, missing journals on Friday mornings was more common at the beginning of the initiative before parents and students got into the routine of solving the problem and handing it in each week. For students who did not hand in the journal on Friday morning, I made a note in the students daily agenda to the family to remind them that the journal needed to be at school on Monday. If the journal did not show up at school on Monday either, I then phoned home to touch base and see if the book had gone missing, if they had received my note on Friday, and just to see how things were going. Making this phone call seemed to end the issue in most cases. In the odd case students had actually lost the Home-Math Journal in which case we created a new one and sent it home the following week. From my experience, students took pretty good care of this homework initiative and were pretty conscientious about bringing it back because they wanted to share what they had done at home and get my feedback about their responses. In any case, it is important for teachers to have some extra Home-Math Journals set up in case one does go missing. Another suggestion would be to have some pages stapled together for that week so that students could still bring the new task for the week home, and these pages could always be stapled into the Home-Math Journal once it returned to school. Keeping the lines of communication open with home through a short phone call does seem to help keep parents informed and often clears up the problem of missing Home-Math Journals quickly. 143 It is very important to continue to keep parents informed, which could be done in many ways. Personally, I have found that using a monthly newsletter to inform parents of the expectations that will be covered that month in each of the subject areas really helped parents to know what we were covering that month and what was corning up next. The monthly newsletter helped parents know what we were studying, and where we were headed for that month. I also tried to add in important dates that parents needed to be aware of for that month. In addition to the monthly newsletter, I also used the daily planner/agenda as a communication tool with parents. The planner provided a quick and easy way to make notes back and forth with families if they had any questions or if I needed to inform them of something that was going on that day. More formally, I used the meet the teacher night as an initial way to introduce the way that math was being taught and explained the use of the Home-Math Journal to parents, since I typically began the Home-Math Journal shortly after this event. At the end of term, I used parent-teacher interviews as a venue to discuss the journal further, problem solve, answer questions, and guide parents in how to help their child be more successful. By keeping the lines of communication open, parents became an important part of the learning process and were more aware of what their child was learning and how they were progressing mathematical information. When encouraging mathematical understanding in our students/children, it is important for students to experience and see that math is all around them (Kliman, 2006). 144 Involving children in simple tasks like measuring while cooking, making change, measuring distances for a new fence, etc. , helps them to see why math is so important. Thus, parents need to help students to make these connections and involve them in these experiences. Every day we can fmd ways to involve children of all ages in the math around us. By making mathematics learning fun and engaging, students are more likely to want to be involved and explore the mathematical concepts further. One way to spark this idea with parents could be embedded in the monthly newsletter in a section at the end entitled 'Math Around Us', where teachers could provide a strategy or idea for exploring mathematics in the world with students. This monthly tip would allow teachers to give parents one idea each month and hopefully encourage parents to continue to find ways to explore mathematical concepts with their children more often. The use of a Home-Math Journal has proven to be an effective way to promote communication, problem solving, and mathematical writing while also involving and keeping parents informed. The Home-Math Journal helps to build students' self-confidence which in tum helps to improve student achievement. Parents also become educated on how math is being taught and how to assist their children at home. The teamwork of parents and students working at home allows parents to unveil what their children understand while promoting student metacognition, and their oral and written communication skills. The Home-Math Journal provides teachers with a valuable homework task as opposed to worksheets which don 't allow students to develop their personal understandings. The Home-Math Journal is a great way for teachers to encourage students to accept ' help ' from home supports while bridging the gap between school and home and building a team of learners. 145 146 Summary Whitin and Whitin (2002) discussed the possibility of using a math journal to bring parents/caregivers into the learning experience and inspired the creation of a Home-Math Journal with my Grades 2 and 3 students. At first my students were a bit apprehensive and unsure if their parents would be willing to help them with this task every week, but once they had taken the journal home the first week, they were eager and excited to continue this process. The students were able to work on their problem solving and written communication skills in a safe environment while also being able to share their learning and thinking with their ' home partner' . They were also able to compare the two solutions as they began to understand the thinking of others and fmd different ways to solve/explain ones ' thinking. We know that students learn more when their thinking is challenged and when they can vocalize their thoughts, strategies, and ideas with others. The use of discourse has really enriched the learning experience for students (and parents) and has provided a great way for students to put what they 'say' into words through written communication. Being able to take this joumaling experience home and experience it alongside their parents/caregivers, made for rich discussions and focussed learning for both the parents (or home partners) and students. Parents were able to get a better understanding of what was being expected of their child, and how problem solving could help to demonstrate student misconceptions and deepen their overall level of understanding. Students are thinkers whom are exploring and experiencing the world around them. Each individual brings with them new perspectives and theories to the learning environment, which allows students (and parents) to learn many new ideas and strategies from one another. By bringing their own ideas and prior knowledge to new situations, students (and parents) are 147 able to plan, organize, and consolidate their learning through different forms of communication. Through a combination of oral and written communication strategies, students were learning how to explain themselves mathematically while keeping their thinking clear and coherent. Furthermore, by sharing their thinking and ideas with a family member who had solved the same problem, they were able to make comparisons and understand that there were many ways to solve, or represent solutions to the same problem. Students need to understand why they are talking and writing about their mathematical reasoning. These discussions and written documentations offer the parents (and teachers) a view of what students are thinking and how they are processing problems. The type of information provided by mathjoumaling is critical for teachers to plan effective future lessons, as well as for students whom are able to go back and reflect on prior learning and bring new information to their problem solving approaches. Now that this rich opportunity for oral and written communication has been introduced and used regularly with students and parents, one can' t help but be excited to observe the impact that these strategies will continue to have on student learning. Not only has this approach to teaching mathematics improved student understanding, but it will inevitably help to prepare students for explaining their thinking on standardized tests at the end of the year. By the end of the school year when testing occurs, students should be 'experts ' at writing clear and concise explanations to a variety of problem solving questions and should have more confidence in their abilities as mathematicians and problem solvers. I look forward to continuing to observe the results of this Home-Math Journal initiative with my students in years to come. The Home-Math Journal has definitely opened the lines of communication with home and shown parents the importance of problem solving and writing about ones thinking. It is 148 through this process that we (parents and teachers) are able to see where children ' s misunderstandings occurred. It is from here that we could find ways to best help and guide student understanding. Parents were working on explaining problems in different ways to help demonstrate for their child that sometimes there is more than one way to get to the answer of a problem. The Home-Math Journal also helps teachers to see the misconceptions that some parents have when reading problems, which also helps students to see that we can all learn from our mistakes and use that knowledge as we progress to new problems. The learning that was taking place was phenomenal and the reflections were demonstrating that students were really starting to think about their own thinking and the thinking of others. In regards to my students ' written communication, the Home-Math Journal was definitely inspiring them to explain themselves more clearly as they had a real audience, which made this form of written communication authentic and inspiring to my young writers . I looked forward to seeing the dialogue documented in the Home-Math Journals every time they were returned, and will continue to treasure these responses from my students in years to come. It was interesting to look back and reflect on each student' s first few journal entries at the end of the year, as their writing skills continued to improve and become clearer and easier to follow. I am very happy with the results of my personal inquiry and will definitely continue to use math journals and Home-Math Journals in my teaching. Being able to discuss the Home-Math Journal with parents at parent/teacher interviews was very powerful and helped to clarify the information for parents, while also allowing me to offer suggestions to help parents lead their child to the answer on their own. Parents and students were both excited about this new assignment which allowed them to work cooperatively together to complete a specific task, thus building the home-school connection. 149 Before embarking on a Home-Math Journal, it is important to remember the natural progression of steps that needs to be firmly established before this initiative can be a success. First, teachers need to set the stage and ensure that the classroom environment is one where students feel safe and secure offering their opinions and receiving feedback from their peers. Once a collaborative environment has been established, teachers need to have students working on expressing their mathematical ideas orally. Again, teacher modelling is a very effective method for helping students to understand what your expectations are for students' math talk. After students become more comfortable expressing their ideas aloud, they will be ready to start writing and documenting those mathematical thoughts and ideas on paper. Once again, the use of teacher modelling and student samples will help students to grasp what the expectations are for students ' math writing and how they could go about explaining their thoughts on paper. Finally, after an in-class math journal has been used in the classroom, students will be ready to take home the Home-Math Journal where they can work on their problem solving, oral sharing, and metacognition as they learn from a home-partner and practice reflecting about their problem solving experiences. If the above progression of skills is followed, teachers and parents are bound to see an increase in student understanding because teachers and parents will have a better understanding of what students know and thus how they can best support their learning or inquiry. Also, students will be able to build upon their oral and written math communication skills as they develop deeper mathematical understanding through the rich discussions and focused learning. If used correctly, this handbook will help teachers to create a community of learners who are interested in the claims of others, who are able to learn in a safe environment, and keep parents informed and involved through the use of the Home-Math 150 Journal. Oral and written communication help students to organize and share their thoughts with others while making their thinking public. These public documents help teachers to foster student autonomy, collaboration, build on their mathematical communication skills and ultimately drive instruction forward. 151 152 Appendix A- Initial Letter Home Horne-Math Journals Dear Parents, Coming home on Monday nights for the next few weeks, will be a Home-Math Journal. This is a journal where your child (and you) will be solving math problems and writing about their mathematical thinking, learning and sharing. This journal will be handed in to me by Friday each week so that I can provide feedback before the next math journal. Below are some tips to help at home: There are 4 Guidelines for Home-Math Journals: Students must explain in their journal: 1. What he/she did. 2. What he/she learned. 3. What he/she is not sure about or are still wondering about. 4. Did your 'home partner' solve the problem similar to you?, and What did you learn from sharing with your home partner? What Do I Do If My Child is st ~ Try asking the following questions: "What do you have to do on this assignment?" "Tell me what you're supposed to write about?" "Do you have any idea about what the answer is?" "Why do you think that?" "What ideas do you have?" "Tell me something you know." * "See if you can say those words again in your head and then write them down on you paper." * * * * * * Remember: * * Math journals are designed for students to explain their thinking, therefore the focus is not on getting the correct answer right away, but on demonstrating how they got their answer. This journal helps parents and teachers to see where students' misconceptions lie in order to help focus student learning. Talking about solutions with others helps students to write their ideas on paper. Encourage talking and questioning with your child at home. Have fun! Sincerely, 153 Appendix B- Helpful Hints (glued into front cover of Home-Math Journal) Home-Ma1:h Journal HelpfUl Hin1:S Page 1- Your Solution Explain and Show what you did to get the answer. "I think the answer is ..." "I think this because ... " "I decided to use ... " "I solved the problem by ... " "I figured it out by ... " " "I knew I had to because "The steps I followed were ..." Tell what you learned Tell what you are not sure or wondering about. Use a picture or equation to show how you got your answer. Page 2- Your Home Partner's Solution (same as above) Page 3 - Reflection • Share your solutions with each other • What did you learn by sharing your solution with your home partner? • What surprised you? • What questions do you still have? • Did you solve the problem the same way? • What was the most challenging part of the problem? Why? • **use the sentence starters at the back of your book to help you** 154 Appendix C- Reflection Sentence Starters (glued into back of Home-Math Journal) RefleCtion Sentence Starters • I noticed that ... • I had trouble with ... because ... • I was surprised that ... because ... • I could have made my journal better by ... • Now I understand that ... • I wonder why ... • Sharing with my home partner helped me to ... because ... • This question helped me to ... • I did something like this before when ... • The most important thing I learned from this problem was ... • Something that was different about our answers was ... • A question I had was ... • We both ... • Next time I will try to ... 155 Appendix D - Sample Problems Below are some sample problems that I used with my Grade 2 and 3 students in their home math journals. I found that using students names in the word problems helped to motivate students to solve problems as they enjoyed seeing their name or the name of a peer in the problem. These types of questions could be adapted for other grades or other math problems could be found in textbooks, on the internet, through other teachers, or created by teachers based on the expectations outlined in the curriculum documents and on teachers goals for that lesson. Sample Problems 1. In long jump, Samiee jumed 65 em. Janelle jumped 25 em farther than Sarniee. How far did Janelle jump? Show your work. Explain how you know. 2. Frog, cat, bird, and two rabbits are lined up to buy tickets for the Animal Express. A rabbit is first in line. Frog is between a rabbit and cat. Bird is not next to cat. Cat is not last in line. How could the animals be lined up? Show your work. Explain how you know. 3. Mrs. Tippett loves to collect post-it notes. She bought 25 in November and 13 more in December. If she gave 8 to Miss Muller, how many post-it notes does Mrs. Tippett have now? Show your work and explain how you know. 4. The mother geese and their goslings are out for a walk. There are 7 mothers (M), each with the same number of goslings (G). How many goslings (G) will there be altogether? Show your work and explain how you know. (Accompanied with a picture of a mother goose with 3 goslings behind her). 156 5. Word is getting around that From Cracks to Crackers is a perfect book for young ants. Aunt Bee' s bookstore sold 3 copies on Monday, 6 copies on Tuesday, 12 copies on Wednesday and 24 copies on Thursday. If the pattern continues, how many copies will the bookstore sell on Saturday? 6. Michai put pieces of red and green peppers on the pizza. One half of the pieces were red. He put 6 red pieces on the pizza. How many green pieces did he use, and how many pieces were there altogether? Show your work. Explain how you know. 7. Patrick, Scott, Natalie and Marina like to go to the park on Saturday. One likes to swim, one likes to play tennis, another likes to play soccer, and one likes to ride a bike. Patrick likes the water. Scott doesn' t like to chase balls. Natalie doesn 't like to use a racquet. What does each one like to do at the park? Show your work. Explain how you know. 8. It is Bargain Bonanza Day at the Petfood Land. Ebhoni pays for 8 bags of Healthy Hamster food and gets 16. Jonelle buys 5 boxes of Canary Crunch and gets 10. Quinn buys 9 bags of Nibble Kibble food, how many bags will he get? Explain how you know. 9. Wings are flapping and bird seed is flying off in all directions! Jeremy looks at the bird feeder. He counts 16 birds in all. There are three more red birds than blue birds. There are two fewer yellow birds than blue birds. How many red birds are at the feeder? How many yellow birds? How many blue birds? Show your work. Explain how you know. 157 10. Teanna has a juice container that holds 2 L of juice. Darren has a water bottle that holds 500 mL of water. How many water bottles would Darren have to pour into Teanna' s juice container to fill her container? Explain how you know. 158 Chapter 5: Conclusions Mathjournaling is a great way to get parents involved in aspects of the mathematical program that parents are often less familiar with, such as problem solving and written communication. If teachers incorporate a Home-Math Journal with their students, they should experience an increase in parent involvement as well as positive feedback from parents who will likely be appreciative of the homework task that they get to complete with their children. A homework task like the Home-Math Journal often eliminates the problem of students not wanting to ask for help with homework tasks, and instead allows parents and students to learn together and explore mathematical concepts in a fun and safe environment. Also, the Home-Math Journal helps build a mathematical community by involving parents in a team approach to experiencing mathematics. This type of positive parental involvement will allow for increased student achievement over time. I believe that the handbook addresses many concerns that teachers are currently faced with when teaching mathematics to develop student understanding. I believe that the handbook presented here is a valuable resource to teachers in the elementary grades in three ways. First, the proposed handbook provides a clear and comprehensive understanding of why oral communication is important for students within a mathematics classroom and how teachers can foster and support this valuable discourse within their own classrooms. The resource also helps provide teachers with suggestions on how to create a communication friendly environment within their classrooms so that discourse is valued between all individuals and a community of mathematicians can be fostered. Secondly, the resource addresses written communication and how math writing should follow and build upon the oral discourse that was addressed in the first section of this 159 text. The resource provides teachers with an understanding of why written communication is important, how it can help teachers to assess student understanding, as well as offering suggestions and strategies to help promote good math journal writing in the classroom. The handbook provides teachers with strategies to create a community of learners where student autonomy is fostered and where students engage in collaboration to build their mathematical communication and understanding. Teachers are shown how to prepare math lessons through a launch-explore-discuss lesson structure, listen to student discussion, select and sequence student responses to be shared with the class, and how to help make connections between student ideas. The resource also provides teachers with suggestions of how to support students who are struggling with written communication and ultimately help them to make their thinking explicit and concise. Finally, the above resource introduces, explains and demonstrates the power of building a positive home-school relationship in the learning of mathematics. Through the use of a Home-Math Journal, parents and students will be solving, sharing, discussing, and reflecting upon each others solutions as they continue to build mathematics understanding. The Home-Math Journal experience will help students to demonstrate how they are learning math to their parents, while also demonstrating where their misconceptions lie and how parents could best help struggling students at home. The handbook provides an authentic and rich homework task that will actually help to open the lines of communication with parents while also getting parents involved in a positive way in their children' s learning of mathematical concepts. The handbook not only explains why the home-school connection is so important, but it also lays out how to set up and manage a Home-Math Journal, as well as how to help parents to support this learning at home. 160 I believe that the handbook presented here has a great deal to offer teachers, as it provides three very important and interconnected aspects of math communication in one resource. Having oral, written and home math communication accessible in one text makes introducing effective mathematical communication much easier for teachers as teachers would currently have to access many different resources to cover the breadth of information found in this handbook. Having the necessary information available in one resource will not only make teachers lives easier, but will also make the teaching and learning of mathematics a much more enjoyable experience as all of the pieces of the puzzle are available together and are presented in a developmentally appropriate order. The motivation for this project stemmed out of having completed my own personal inquiry into math journal writing in my own Grade 2 and Grade 3 classrooms, where I had the opportunity to develop a problem solving math journal with my students. During the course of this personal inquiry, I had to access many texts in order to better understand how to get my students talking and then writing about their mathematical experiences. I then extended this inquiry in an attempted to get parents involved in the process by developing a Home-Math Journal with my students. The success I experienced with this personal inquiry was so rich that I wanted to create one resource that encompassed all aspects of mathematical communication. My ultimate goal was to assist teachers in developing math journals and Home-Math Journals in their own classrooms. The main limitation to this project is that it is created for teachers in Grades 1-6; however, the initiatives presented have really only been tested on Grade 2 and Grade 3 students. In order to help with this limitation, I have tried to provide suggestions in each of the three sections of the handbook as to how oral communication, written communication, 161 and home connections may change or adapt across the grades. These sections may require more information once the strategies have been tried with different grades. The recommendations that I have for this handbook are that it be shared with teachers as a model of how math journaling can meet the new standards set out by the NCTM (2000) and school districts everywhere. There has been a large shift in how math is being taught today, and this handbook helps to address many of these areas while providing useful and important learning tasks. I believe this resource should be published in order to be more accessible to teachers. Reflecting back on this project, I have realized that this was a labour of love. Mathematics is an area of my teaching that I knew I had to improve as the way that I had learned math was so drastically different from the constructivist view that I was to be teaching. Based on my own personal inquiry and the huge success that I have had with the creation of the Home-Math Journal, I knew that there was a need for a resource that would explain the progression from math talk, to math writing, and eventually to building the homeschool connection. The above resource answers the math problem of using discourse and involving parents in the learning experience as outlined by the NCTM (2000) . While all change involves uncertainty (Fullan, 2007), the above resource provides clear and easily accessible information on how to begin to implement math talk, math journaling, and a Home-Math Journal which should help teachers to understand the need for mathematical change. 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