COMMUNICATION:

WHAT DIFFERENCE DOES IT MAKE?

 

 

 

 

 

 

by

 

Kristen A. Shanefelter

 

 

 

 

 

 

 

 

 

 

 

Department of Curriculum & Instruction

College of Education

University of Maryland

College Park, MD

 

May, 2004

 

 

 

 

 

ABSTRACT

This paper details one teacher’s account of her action research project.  It reports on a study that investigated the effects of a focus on communication in mathematics on second graders’ abilities to explain how they solved problems.  Data were collected through student artifacts, informal interviews, audiotapes, and a reflective journal.  Information was analyzed in regard to student performance on oral and written responses, questioning, explaining mathematical thinking, sources of mathematical ideas, and responsibility for learning.  The results showed that focusing on communication in mathematics positively influences students’ abilities to explain how they solve problems.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BACKGROUND/RATIONALE

When I first decided to become a teacher, I had grand ideas of changing everything I disliked about school.  I thought I’d be able to prevent students from experiencing some of the struggles that I had to overcome as a learner.  Based on my success as a swimming instructor, I imagined teaching to be a relatively easy profession and I assumed that I would instinctively adopt the best practices to enable everybody to learn. Then, midway through my college career, I started taking methods classes.  That is when my perspective began to change.

The course that influenced me the greatest was a mathematics methods course.  Throughout the semester, I had to assume the role of both a primary student and an elementary school teacher.  Suddenly, I realized how difficult it was to break down and explain a concept or strategy that had become so ingrained in my cognitive processes that I didn’t even realize I used it.  The process in question had often become automatic to the point that it seemed I just knew the answer.  I had no clue how to explain what steps I took to arrive at my solution, much less any inkling of how to teach my methodology to others.   I also became aware of how a lack of experiences and a limited knowledge of vocabulary could negatively impact a primary student’s understanding.  

Later, when I became a second grade teacher in a middle-class suburban area, I noticed that while many of my students were able to quickly arrive at an answer, they were unable to prove or explain why their solution made sense.  For several years, I discussed this dilemma with my colleagues and I frequently explained my concerns during parent-teacher conferences.  I also took some professional development courses.  As a result, I implemented an increase in the use of manipulatives during mathematics instruction and I insisted that my students use them during assessments to explain their reasoning.  Over time, I noticed an improvement in my students’ abilities to show me how they solved problems, but I still felt like something was lacking.

Recently, I began a masters program at the University of Maryland.  One of the classes I took was a course about misconceptions in mathematics.  I didn’t see how it would relate to my concerns about my students’ abilities to communicate the steps they took to solve problems, but I thought it sounded interesting and it had the added bonus of fulfilling one of my program requirements.

As my classmates and I began to delve into the field of mathematics education, I became aware of the multitude of influences that can affect a child’s understanding of any particular concept.  I read about the importance of allowing children to explain their perspectives and to communicate their thought processes, for it is only when students are given an opportunity to voice their thinking that teachers truly comprehend what is happening in the minds of their pupils.  

This summer, I took another mathematics course.  This class further enlightened my understanding of the value of communication.  I realized that, historically, children in the United States have not been taught to express their thought processes in mathematics.  Instead, they have been encouraged to memorize facts and formulas. In essence, many American students were unintentionally given the message that the route was less important than the final destination.   Consequently, studies have found that American students tend to be less successful than students in other countries when it comes to applying their mathematical knowledge to a variety of situations.  This realization both concerned and inspired me to learn what I could do to reverse this trend.

            I studied information from the TIMSS Report, and I strove to learn more about improvements in reform curriculum.  I pored over material from the National Council of Teachers of Mathematics, and I began to realize that although I had always valued the need to explain processes and solutions, I had never truly provided a classroom atmosphere that would nurture the development of these skills.  

            As a new school year approached, I began to wonder what changes I could make to help reach my goal of enabling students to not only solve mathematical challenges, but also be able to articulate how they arrived at their answers.  I couldn’t help but wonder how a focus on communication skills in mathematics would affect students’ abilities to explain how they solved problems.  I allowed this question to mill about in the back of my mind, but decided not to act until I actually met and had the opportunity to assess the strengths and needs of my second grade students.

            Throughout the first few weeks of school, I paid careful attention to my twenty-eight second graders.  I watched as these children, of varying ethnic but similar economic backgrounds, interacted with each other and learned the class routines.  I posed a variety of math scenarios.  I was thrilled to see how quickly some of my students were able to reach a solution, but the number of blank stares that instantly replaced the eager hands raised in the air when students were asked to explain how they reached their solutions troubled me.   A few children volunteered to share their ideas, but more often than not the explanations were, “I just knew it” or, “I did it in my head.”  On occasion, I would hear, “I added” or, “I drew a picture,” but I still didn’t get the in-depth explanations I had hoped to elicit.  I didn’t know what was added or drawn, and I certainly didn’t know the students’ reasoning.

            I asked my colleagues, who taught many of these same children as first graders, to comment on their perceptions of the students’ mathematical abilities.  The teachers explained that there was a “wide range within the grade.”  This description matched the grades I had seen on the students’ report cards from first grade.  I then asked if the teachers thought the students were capable of solving and explaining grade appropriate tasks.  Since all but one of the students in my class were either on or above grade level in math, the teachers assured me that most of the students would be able to solve grade appropriate tasks.  Again, I inquired as to whether or not these students would be able to explain their thinking or their solutions.  When responding to this question, the teachers hesitated and then admitted that they weren’t as confident about the students’ potential for success in this area.

            Midway through the last week of August, I administered a pretest for our first county-mandated mathematics unit.  The results varied dramatically.  Some children were proficient in solving a variety of problems, while others exhibited only a limited understanding of the skills needed to pass the pretest.  The one thing that nearly all of the students had in common was a deficit in communication skills.  One task directed the students to first “circle the correct number sentence” and then “use what you know about place value to explain your choice.”  Eighty-six percent of the students were able to accurately accomplish the first part of the problem but none of them were able to correctly answer the second portion of the task.  In fact, only thirty-six percent of the children even attempted to write an explanation.  On a different task, the students were supposed to list some odd numbers that were less than ten.  They were then instructed to “use what you know about odd and even numbers to explain your thinking.”  This time, eighty-two percent of the students tried to draw or describe their thinking; however, only eleven percent of the students were successful.  I found these results to be appalling yet they solidified my desire to investigate how a focus on communication skills in mathematics would affect students’ abilities to explain how they solve problems.

RESEARCH METHODOLOGY 

Once I determined my area of focus, I decided to formulate an action research plan based on the information cited in the book Action Research: A Guide for the Teacher Researcher (2003) by Geoffrey Mills.  This book explains both the concept of and the theory behind action research.  It describes each step of the action research process and provides guidelines for performing one’s own action research project.   In addition, the book depicts teachers’ anecdotal experiences with implementing teacher research projects.

            According to Mills, an area of focus for action research should be about an issue that “involves teaching and learning, is within your focus of control, you feel passionate about,” and “you would like to change or improve” (2003, p. 26).  After I confirmed that my topic met all of these criteria, I submitted a proposal for my action research project.  I posed the question, “How will a focus on communication skills in mathematics affect students’ abilities to explain how they solve problems?”

 Determining my research question made me feel as if a huge weight had been lifted off my chest because I now had an area of focus.  At the same time, I knew that I still had a long way to go.  Immediately, I began to ponder the directions that this project could take and I fretted about how I was going to actually complete the action research.  After working myself into a frenzy, I decided to take the project one step at a time, starting with acquiring authorization.

My principal instantly approved of my topic and she offered her full support.  I explained my proposal to the parents of my second graders and I opened myself up to questions.  Amazingly, very few people had concerns.  Two parents asked if I would be able to devote my full attention to their children while undertaking such a project.  I assured them that I perceive myself to be primarily a teacher and then a teacher researcher, and that I truly believed that this endeavor would only strengthen my skills as an educator.  After that, the only other comments I received were offers of help.  Although one parent eventually declined to give written permission for her child to participate in the project, I respected her point of view and, in general, felt extremely fortunate to be working in such a supportive community. 

Subsequently, I began to immerse myself in the research project.  Without delay, I made a concerted effort to better record and reflect upon my teaching practices.  On several occasions I attempted to take field notes during my math block.  Unfortunately, I struggled with taking consistent and accurate notes while still facilitating the discussion.  As a result, I often ended up missing important pieces of information or inadvertently allowing the children to get off topic or have gaps in the discussion.  I became frustrated and it seemed that my shift in demeanor did not go unnoticed by the children.  I observed that the more frustrated I became, the less eager the students were to participate and share their thoughts.  In fact, during a session in early October, eleven of the fourteen children who had their hands raised high, lowered their hands so that their fingertips barely extended past their shoulders.  They also averted their eyes from my gaze.  I felt horrible.

I remembered that Geoffrey Mills’ book described the action research process as a model that “invariably ‘spirals’ the researcher back into the process repeatedly” (2003, p. 18).  It also stated that “good teachers have always reflected critically upon their practices” (2003, p. 20).  I decided that the book made sense, and that I should view my mishaps with taking field notes as an opportunity to reevaluate my data collection methods and action research plan. 

I debated creating a survey to determine the attitudes of my pupils toward problem solving but then realized that I was straying from what I really wanted to learn.  I remembered a piece of advice that was offered to me during a research festival that I attended last year.  The gist of the advice was that while it is worthwhile to branch out and consider all of your options, it is important to not get too carried away.  I remembered hearing the personal tale of a woman who kept thinking of new ideas and collecting more and more data until she was so overwhelmed that she lost all direction and nearly gave up.  Fortunately, she managed to narrow her thinking and she produced an exceptional project.  Nonetheless, I wanted to try to avoid getting myself into her predicament. 

I decided that for the time being, I would limit my field notes to occasional notations during class discussions and instead concentrate on collecting the majority of my information through written reflections and journaling, student writing samples, and informal interviews.  When the time came for me to submit my official action research plan, I knew that I also needed to start organizing the data I was collecting, procuring resources to supplement my knowledge base, and developing a rubric to provide for consistency of grading.  However, I was unclear about how I should proceed.  I came to the conclusion that I should revisit Action Research: A Guide for the Teacher Researcher (2003) and see what advice Geoffrey Mills provided.  In chapter two, he suggested making an “initial foray into the professional literature…to try to better understand the problem on which you are focusing” (p. 28).  He went on to explain that “the literature may suggest other ways of looking at your problem and help you to identify potential promising practices that you may use in your classroom to correct the problem” (p. 28).

LITERATURE REVIEW AND DATA COLLECTION

I started my literature review by re-examining some of the texts that had so greatly influenced me over the summer. I first turned to Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000).  The National Council of Teachers of Mathematics published this book, and it describes a vision for pre-kindergarten through twelfth grade mathematics education.  It also provides guidelines to help evaluate the effectiveness of mathematics programs.  The book consists of six principles that highlight the basic characteristics of a high-quality mathematics program.  In addition, it contains five standards that describe the mathematical content that students should learn to be successful, and five standards that depict the mathematical processes that students draw upon to acquire and use their content knowledge.

Next, I reread Making Sense (1997) by James Hiebert, Thomas Carpenter, Elizabeth Fennema, Karen Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piety Human.  This book explains the ramifications of, and the need for, teaching with understanding.  It provides a framework, consisting of five critical dimensions, for helping teachers examine and analyze classrooms to ensure teaching for, and learning with, understanding.  Finally, this book accounts experiences from four classrooms that share several core features within each dimension and illustrates how the dimensions play out in real settings.

            Reading these books served to re-ignite my enthusiasm for focusing on communication within the mathematics classroom.  According to Making Sense, “Teachers must ensure that all children learn to communicate about mathematics” (Hiebert et al., 2003, p. 72) because “communication increases the likelihood that students will think again about their own method, and hear about other methods that may work just as well or better” ( p. 20).  Principles and Standards for School Mathematics further elaborated by stating that “students, like adults, exchange thoughts and ideas in many ways – orally; with gestures; and with pictures, objects, and symbols.  By listening carefully to others, students can become aware of alternative perspectives and strategies. By writing and talking with others, they learn to use precise mathematical language and, gradually, conventional symbols to express their mathematical ideas” (National Council of Teachers of Mathematics, 2000, p. 127).  Both sources agree that students need to have frequent opportunities to investigate and communicate about their mathematical thinking. 

            I decided that the information I was reading matched with my personal beliefs and my professional goals, so I started to provide problematic tasks and opportunities for students to reflect upon and discuss their thinking more regularly.  I resolved to incorporate these types of activities into my mathematics sessions at least three times per week.  To learn more about how to approach the implementation of this teaching method, I went to my school library.  There I found two resources that proved to be invaluable. 

            One resource was Connect to NCTM Standards 2000 (2000) by Francis Fennell, Honi Bamberger, Thomas Rowan, Kay Sammons, and Anna Suarez.  It contains a description of the ten curriculum standards that make up the Principles and Standards for School Mathematics 2000, as well as four lessons for each standard to model ways the process standards can be used to provide meaningful mathematics lessons.  This book enabled me to better understand the process standards and it served as a guide to help me develop lessons that integrated these standards into my personal teaching style. 

            The other resource I uncovered was entitled Problem Solving Strategies: Crossing the River With Dogs and Other Mathematical Adventures (Herr & Johnson, 1994).   It is “organized into chapters, with each chapter introducing a new problem-solving strategy.  Each chapter presents several problems” (p. 3) and sample solutions.  I found that I could easily adapt the problems to meet the needs of my second graders thereby ensuring that I presented tasks that could be solved using a variety of strategies and that lent themselves to collaboration and communication.  The book also provided insight into some of the approaches children use to solve problems.  This knowledge enabled me to prepare myself for differing techniques and possible shifts in classroom discourse. 

In addition to utilizing the professional resources that were available at my school, I joined the National Council of Teachers of Mathematics.  As an incentive for joining the organization, I was given free subscriptions to the Journal for Research in Mathematics Education and Teaching Children Mathematics.  The membership also granted me access to the member website, professional publications, conference opportunities, and other resources to supplement my knowledge base and provide ideas for student tasks.

Later in the semester, I needed to attend a professional meeting so I was unable to teach my mathematics class.  I asked our staff development teacher to cover for me and I explained that I wanted her to have the students begin the period by solving and discussing the following question: “Marty decided to start a rock collection.  On Monday, he collected two rocks.   On Tuesday, he collected three rocks.  On Wednesday, he collected four rocks.  If Marty continues to collect rocks at this rate, how many total rocks will he have by Saturday?”  I told her that the children were allowed to discuss their ideas with each other and I asked her to please allow at least two or three students to share their strategies and solutions with the entire class.  Afterward, the staff development teacher commented that she had really enjoyed working with my students and listening to their conversations.  She was impressed that so many of the students had used different methods to reach the same conclusion.  I briefly explained the action research project in which I was engaged and I summarized some of the research I had read.  She told me that what I was saying reminded her of some of the things she had recently read in the book Helping Children Learn Mathematics (2002), which was used as a basis for our county curriculum guides.  I asked to borrow the book and she kindly lent it to me.

            Helping Children Learn Mathematics details a study conducted by the National Research Council on “what research says about successful mathematics learning from the preschool years through eighth grade” (Kilpatrick & Swafford, 2002, p. 7).  When I read the book, I noticed that it mimicked much of the information I had previously read in the TIMSS Report and it reiterated many of the conversations my peers and I had held throughout our summer course.  The book stated that “despite the dramatically increased role of mathematics in our society, mathematics classrooms in the United States today too often resemble their counterparts of a century ago” ( p. 3).

            Helping Children Learn Mathematics recommends “new goals for mathematics learning” (Kilpatrick & Swafford, 2002, p. 1) and “a course of action for achieving those goals” (p. 1).  It claims that “mathematical proficiency involves five intertwined strands: (1) understanding mathematics; (2) computing fluently; (3) applying concepts to solve problems; (4) reasoning logically; and (5) engaging with mathematics, seeing it as sensible, useful, and doable” (p. 1).  In the section about reasoning, I read that “one of the best ways for students to improve their reasoning is to explain or justify their solutions to others” (p. 14) and that “as students reason about a problem, they can build their understanding, carry out the needed computations, apply their knowledge, explain their reasoning to others, and come to see mathematics as sensible and doable” (p. 14).  Again, my attention was drawn to the role of communication in developing mathematical proficiency.

            Later in Helping Children Learn Mathematics (Kilpatrick & Swafford, 2002), I read suggestions for what teachers can do to help students become mathematically proficient.  Many of the suggestions seemed commonsensical such as, “Be committed to the idea that all children can become proficient in math,” (p. 37) or, “Use an instructional program and materials that, based on the best available scientific evidence, support the development of math proficiency” (p. 37).  However, I realized that this book would have been incomplete without any mention of the teacher’s role in the learning process. 

            Making Sense (Hiebert et al., 2003) contains a section that specifically describes the role of the teacher.  It states that a truly professional “teacher knows each of his or her children well, understands the mathematics that should be learned, selects tasks that enables each student to engage in problematic mathematics, and orchestrates the complex world of the classroom so that the children reflect about their thinking and participate in mathematical discussions” (p. 72).  Principles and Standards for School Mathematics took this idea one step further when it said that teachers need to “extend students’ mathematical reasoning by posing new questions and asking for arguments to support their answers” (National Council of Teachers of Mathematics, 2000, p. 122). 

            At the mention of questioning, I remembered that we had dedicated a substantial amount of time during my summer course to asking appropriate follow up questions.  I reread my personal notes from class and, in the process, came across a handout entitled “Asking Questions: The Teacher’s Role in Discourse.” I wondered if I was successful at asking effective questions, and I decided to find out by focusing my attention on the words I spoke during class discussions. 

            After reviewing my journal entries and reflecting on my feelings about the class discussions, I realized that I was often cutting my students short and taking over the explanation.  I was rephrasing their words so that what they said “made sense” and so that the other kids could quickly and easily “see” what strategies or logic the student used to solve the problem.  During the actual discussion, I justified my comments by telling myself that it was my job to make things clear to my students.  Upon further reflection; however, I knew that I was not allowing my students to truly engage in effective discourse.  Instead of being a co-teacher and a co-learner and encouraging the students to take most of the responsibility for understanding, questioning, and explaining the mathematical understandings of others, I was still playing a major role in influencing the direction of the discussion.

            To remedy this situation, I decided to concentrate on refraining from commentary and instead practice asking questions to elicit student explanations.  As a memory aid, I decided to keep a copy of “Asking Questions: The Teacher’s Role in Discourse” close at hand during future math conversations.  I also came to the conclusion that I should once again follow the advice of Geoffrey Mills and review the related literature, as I agreed with his notion that “taking time to immerse yourself in the literature allows you to reflect on your own problems through someone else’s lens” (Mills, 2003, p. 29).

            With the help of the librarian at the university, I came across two articles that directly related to my questions about the teacher’s role in creating a classroom atmosphere that encourages and supports communication.  The first article was located in Mathematics Teaching in the Middle School (1998).  It was written by Laura Van Zoest and Ann Enyart, and it is entitled “Discourse of Course: Encouraging Genuine Mathematical Conversations.”  This article reports on the need for, and difficulty with, initiating student discourse in mathematics classrooms.  It describes methods for teachers to analyze and reflect on their teaching practices in order to provide for better classroom conversations.  Finally, the article describes one teacher’s experience with initiating meaningful discourse in her middle school mathematics class.

The second article was located in the Journal for Research in Mathematics Education (1999).  The name of the article is “Creating a Context for Argument in Mathematics,” and the author is Terry Wood.  This article describes an investigation of a second grade math class in which students were expected to solve disagreements through argumentation.  It focuses on the role of the teacher in creating a classroom environment that promotes learning through sharing, listening, evaluating, questioning, challenging, and discussing.  Finally, it explains research citing the need for discussion in constructing knowledge of mathematical concepts. 

            As I perused these writings, I was struck by how similar my experiences were to those of the teachers in the articles.  I could really relate to Ann in “Discourse of Course: Encouraging Genuine Mathematical Conversations” when she said, “Another thing I see myself doing a lot is interpreting for the students.  When they do an inadequate job of explaining their reasoning, I feel honor bound to explain it to the rest of the class.  When a student has a good idea from which the rest of the class could benefit, I find myself explaining for him or her and interpreting” (Van Zoest & Enyart, 1998, p. 151).  I could also sympathize with her when she said, “I give away the ‘right answer’ by acknowledging a definite ‘yes!’ that the student is correct” (p. 151).  Before reading the article, I hadn’t realized that I also tend to confirm correct answers, but afterward, I knew that I needed to work on allowing the students to determine which approaches and solutions were correct.  I was encouraged when I read Ann’s remark that once she adapted her teaching style and allowed the students to be the decision makers, “one pleasant surprise was finding that if I let a wrong answer stand, it usually didn’t last very long without some comment” (p. 152).  This inspired me to do everything in my power to suppress my natural tendency to comment on the correctness of an answer and start allowing the students to have control.           

            Terry Wood’s article stated that the children in her study “as listeners, were to do more than pay attention and listen politely, they were to take an active role and to take responsibility for assisting others in making sense of mathematics” (Wood, 1999, p. 181).  This description explained exactly what I was hoping to achieve in my classroom.  In order to reach that goal, I decided to model my conversation with the students after the teacher in the study.  Before having a student explain his or her thinking, she “created a context not only for inquiry… but also for argument” by focusing the other “children’s thinking on listening and considering questions to ask” (p. 181).   The teacher explicitly stated her expectations by saying, “Your job is to be listening to what he’s saying and trying to decide if you have a question, so that you can ask it.  You might think, ‘I’m not sure of what you’re saying?’ or ‘I’m not sure how you did it’ or ‘You didn’t count the way I thought you should,’ or ‘Could you tell me one more time?’ Okay?” (p. 181)  Although I had often told the kids to listen carefully to decide if what the speaker is saying makes sense, I had never thought to model sample questions.  I decided that this was an excellent strategy and I immediately incorporated it into my teaching.

            “Discourse of Course: Encouraging Genuine Mathematical Conversations”  directs teachers to videotape or audiotape “for a minimum of fifteen minutes a segment of your classroom that involves talk” (Van Zoest & Enyart, 1998, p. 152) as a means for focusing on discourse.  I had originally considered videotaping my class but I didn’t know how I could effectively do that without excluding the one child who was not granted permission to participate in my study.  After reading this article, I brought up my concerns during a discussion with my action research study group.  They agreed that videotaping was probably not the best method of data collection for my project. We discussed the fact that “it is generally accepted in action research circles that researchers should not rely on any single source of data, interview, observation, or instrument” (Mills, 2003, p. 52).  From previous experiences, I knew that transcribing audiotapes can be extremely time consuming but we decided that, in order to provide for triangulation, or the use of multiple sources of data, I should probably attempt to record several discussions per week.  Our rationale was that as long as I didn’t exclusively audiotape the girl who did not have permission to participate in my study, this would be an acceptable form of tracking the development of our class discussions.  In order to save time, I decided to not transcribe every tape.  Instead, I opted to jot down comments or gestures that struck me as significant during class and then later listen to each tape and take notes or reflect on the discussions as appropriate.  In addition, I resolved to transcribe only the portions of tapes that I deemed to be valuable to my research.

I started taping the next day and was proud to discover that the students were capable of coming to a consensus about the correctness of an answer.  I posed the question, “The red, blue and green dinosaurs lined up to go to the big party.  They lined up in this sequence, red, blue, blue, green, red, blue, blue, green.  If there were thirty dinosaurs in line, how many of the dinosaurs were green?”  After the students spent about ten minutes talking with each other and working out their solutions using the tools of their choosing, I invited some volunteers to share their strategies.

An eight-year old Caucasian boy named Mike came up to the white board to share his solution. Before he started explaining his answer, I told the students, “As Mike is speaking, I want you to listen to what he is saying and decide if you have any questions about his explanation.  For example, you might think, ‘I don’t understand what you’re saying’ or ‘Why did you do that?’ or ‘Your answer doesn’t make sense to me’ or ‘Can you show me that again?’  Remember, we need to listen to him first and then we can talk about his strategy.”  Then Mike proceeded to explain his thinking.  He used a black marker to write, “RBBGRBBGRBBGRBBGRBBGRBBGRBBGRB” on the board.  He explained that R stood for the red dinosaurs, G stood for the green dinosaurs, and B stood for the blue dinosaurs.  Next, Mike went back and circled all of the Gs.  When I asked him why he circled the Gs, he told me, “I had to find out how many green dinosaurs were in line so I circled all of the Gs and counted them.”  After that, he wrote “6 green dinosaurs” on the board.  Many of the students in the class looked at him, looked at their papers, and looked back at him again.  Six children pointed at the board and moved their hands through the air as they counted each of the Gs that were circled on the board.  I bit my tongue and waited to see what would happen.  Finally, three children raised their hands.  I said, “Mike, it looks like some of your classmates have questions.”  He called on Jessica who said, “That’s not right.” Mike replied, “Uh-huh.”  For what seemed like thirty seconds or more, there was silence.  Then, Alexander joined the discussion by saying, “No, it’s not right.  See, the pattern was red, blue, blue, green and you did it right but you circled seven Gs.”  Mike looked at the board again and then pointed to each G that he circled while he counted aloud.  He came to the conclusion that there were seven green dinosaurs in line so he wiped out the “6” and wrote “7.”  Then he looked the paper where he did his original work and said, “Huh?”  I asked Mike what was wrong and he said, “But I got six here.”  I asked him which answer he thought was right.  He said, “The one up here” as he pointed to the board.  I asked him what he could do to find out why the answer on the paper didn’t match.  He said, “Check my work.”  Mike looked down at his paper and said, “Red, blue, blue, green, red, blue, blue, green, red, blue, blue, green, red, blue, blue, green, red, blue, blue, green, red, blue, blue, green, red, blue, blue, red, blue.  Whoops, I skipped that green.  Now, I know what happened. It is seven.”  Since no one else had his or her hand raised, I asked, “Does Mike’s solution make sense?”  The students nodded and/or verbalized their agreement.

As the school year progressed, I was pleased to see that despite the fact that I had strayed from the county mandated lesson sequence, my students were achieving success on their unit assessments.  On the unit one assessment, twenty-nine percent of the students demonstrated a complete understanding and seventy-one percent demonstrated a developing understanding of the second grade concepts. In addition, seventy-five percent of my students demonstrated a developing or complete understanding of the above grade level indicators.  On the unit two assessment, all of my students demonstrated a complete understanding of the second grade objectives and eighty-two percent demonstrated a complete or developing understanding of the third grade concepts.  The unit three assessment revealed that eighty-six percent of the students demonstrated a complete understanding and eleven percent demonstrated a developing understanding of the second grade concepts. One hundred percent of the students demonstrated a developing or complete understanding of the above grade level indicators. What impressed me the most; however, was that on all three tests, every single student attempted to answer the questions that required a written explanation.  Some answers were incorrect because the students were still unclear about the concepts and some answers were incorrect because they didn’t match the specific requirements stated in the rubric, but I was thrilled to see that most of the explanations were logical and they matched the strategy the students chose to utilize.

In early January, I introduced the “Math Sentence Starters” (see Appendix A), an outline of sample sentences students can use to guide their written responses.  I modeled how the “Math Sentence Starters” can be used to elaborate upon written explanations by expanding some of the children’s original answers from earlier in the year.  Then, we did some sample problems and explanations together.  Finally, the students were given individual copies to keep in their math journals. 

The following week, I showed the students the “Math Rubric” (see Appendix B) that my team of  second grade teachers had developed collaboratively after reviewing rubrics that had previously been created and modifying them to meet the needs of our instructional program.  I explained that I would use the rubric to score their written responses and that it was imperative that they understood how to use the rubric themselves.  To facilitate their understanding, I hung a large poster of the rubric in the front of the classroom.  I went over my interpretation of the scoring gradations and then gave each child a copy of the rubric to keep in his or her math journal.  Afterward, we scored a series of teacher-created  “student samples” and then discussed the scores until a consensus was reached. 

I periodically asked the students to self-score their written responses.  Independently, I would also score their responses.  If there was a discrepancy between our scores, we would talk about our reasoning for choosing our scores and then see if we could reach an agreement.  I felt that this was an integral part of the rubric implementation, because it served to ensure that we had a joint understanding of how to use the rubric to determine what was expected from a well-written response. 

Before long, I saw an improvement in my students’ explanations of the approaches they took to solve problematic tasks.  The written responses became more detailed and tended to list specific strategies, rather than containing only numbers, a picture, or a brief statement such as, “I added.”   An example of this improvement can be seen in the work of a seven-year old Indian girl named Nayantara.  During the first week of January, she solved a problematic task, which stated that, “same shapes are same numbers.” The task was as follows:

                        +          = 10                     +         = 8

           1.  What number is         ?

            2.  What number is             ?

3.     Explain how you found the numbers. 

Although Nayantara was able to correctly determine that, “The triangle is five” and “the square is three,” her written explanation of, “I used my number grid to help me” did not provide much insight into how she arrived at those answers.  Toward the end of January, I posed the problem, “Yoshio buys food for the monkeys at Swing City Zoo.  It costs $3.00 to buy the food to feed 2 monkeys for one day.  How much does Yoshio have to pay for food to feed all 18 monkeys on Monday?  Please explain in writing how you determined your answer.”  Nayantara responded, “$27.00.  First I drew a picture of 18 monkeys.  Second I crossed out two at a time because Yoshio feeds two monkeys at a time.  For each pair I crossed out I counted $3.00.  For nine pairs I got $27.00 because 9x3=27.”  When compared to the first response, Nayantara’s second solution demonstrates a more complete explanation of the process she used to solve the problem. It provides insight into the strategy she used and it clearly states how she determined her answer.  The more I compared the student work samples, the more encouraged I became by the visible progress I was observing.

When the January issue of Teaching Children Mathematics arrived, I was excited to find an article that was closely related to my action research project.  It was written by Susan Scharton and is entitled, “I Did It My Way: Providing Opportunities for Students to Create, Explain, and Analyze Computation Procedures” (2004).  The article accounts one teacher’s experiences and reflections on teaching using traditional methods versus teaching in a way where students are encouraged to develop and communicate their own techniques for solving problems.  Scharton claims that “confusion often occurs when students use memorized procedures without understanding how and why they work” (Scharton, 2004, p. 279) and that “giving students experience with solving problems and allowing them to communicate their problem-solving strategies to others are essential components to developing understanding and benefit students in a variety of ways” (p. 278). Her statements corroborated the information I had learned in my classes and through my research.  Furthermore, they  served to convince me that my teaching methods were justifiable and educationally sound.  She claimed that “listening to the methods that others had used broadened students’ procedural ‘repertoire’; they began to ‘try on’ other students’ ways of computing” (p. 179).  Scharton went on to say “students often discard confusing methods that their classmates demonstrate; therefore, hearing about the strategies of their peers does not harm them” (p. 179).  As I thought about these words, I wondered if her statements held true in my classroom so I decided to see what my students had to say about the benefits or lack thereof of listening to other children’s methods.  During an informal survey, twenty-three of my students agreed that it was helpful to learn how other people solved the same problem.  The remaining five students raised their hands to indicate that they didn’t know if it was beneficial to their learning or not.  No one indicated that communicating about problem solving was not helpful.  When I shared these results with the class, a seven-year old Asian girl named Katie explained why she liked having students share their thinking.  She said, “because now I don’t feel like I’m stupid if I do it a different way than everybody else.  Plus, I can find out some other ways to do it and I can try them later.” A six-year old Caucasian boy named Jake agreed with Katie.  He said, “Yeah, so we can use those strategies next time.”  

Unfortunately, soon after this discussion, I came down with a stomach flu that lasted nearly a week.  Since I was unable to secure the same substitute teacher each day and I was too sick to properly explain my approach to teaching mathematics, I resorted to providing plans that were much more traditional in nature.  Instead of solving problematic tasks and having rich discussions, my students listened to lectures and completed written assignments that required little more than regurgitating facts.  Following that setback, inclement weather forced the school system to delay or cancel school multiple times.  Consequently, my math time was eliminated or severely restricted.  During this same period, my administration strongly encouraged that the second grade teachers begin to review previously taught concepts and have the students practice filling in bubbles in preparation for the Cognitive Test of  Basic Skills (CTBS).  Although I do not agree with the degree of importance assigned to the CTBS, I do understand the consequences of my students not showing growth in comparison to the scores of the second graders from last year. Therefore, I begrudgingly succumbed to the influences from above and began to teach via drill and practice.

  Just as my frustration at having to teach traditionally and my stress at not being able to collect very much new data were coming to a head, the February issue of Teaching Children Mathematics arrived.  It contained an article by Larry Buschman entitled “Teaching Problem Solving in Mathematics” (2004).  At first I almost ignored the article because I feared it would only make me feel more discouraged.  Fortunately, I decided that if nothing else, it would serve as more information for my action research project. 

In “Teaching Problem Solving in Mathematics” (2004), Buschman explains why many schools have not implemented the reforms called for by the National Council of Teachers of Mathematics, and he describes the challenges teachers encounter when attempting to teach mathematics through problem solving.  Buschman claims that many teachers feel that “problem solving can be difficult to teach and even more difficult to learn” (Buschman, 2004, p. 302).  To be successful at teaching problem solving, “teachers must change not only what they teach but also how they teach” (p. 305), they “must model for students problem-solving abilities that they neither possess or have seen demonstrated by others” (p. 306), and “they may need to change some of their most basic beliefs about what constitutes mathematical literacy” (p. 307).  In addition, teachers need to have administrators who will become involved in the learning process and who will provide “staff development activities over an extended period of time” (p. 309). 

I was amazed at how relevant this article was not only to my action research project, but also to the challenges I was facing at work on a daily basis.  In the article Buschman states that “moving mathematics instruction away from drill and practice and toward teaching mathematics through problem solving” (Buschman, 2004, p. 302) is “central to the reform movement” (p. 302) and that “instead of teaching children how to solve certain types of problems using traditional problem-solving strategies, teachers can encourage children to solve problems in ways that make sense to them” (p. 304).   While I was encouraged to hear that he agreed with the methods in which I believed, I was frustrated that so much of what I was reading touted the benefits of teaching problem solving, yet my county curriculum emphasized drill and practice.  I was also discouraged at my inability to incorporate problem solving into my current, test-preparation-oriented mathematics routine.  Then I read that “some of the best problems for children to solve exist in the day-to-day activities that occur in the classroom” (p. 304).  I suddenly realized that just because I was being pressured to teach using drill and practice during my math block, that didn’t mean that I couldn’t incorporate problem solving into the other areas of the school day.  I was relieved to not have to completely compromise my beliefs and my research, and I was motivated to integrate mathematical problem solving into other subject areas.   

When the CTBS was finally over, I was excited to resume my original teaching practices. At the same time, I was fearful that so much time had passed that even with my efforts to take advantage of opportunities to do problem solving throughout the school day, my children would not be as adept at thinking and talking mathematically as they previously were.  To my relief, my students shared their strategies and debated solutions as if we had never taken a break at all.  In fact, a parent volunteer joked, “You have it made.  The kids do all the work and you get the paycheck.” 

 

 

ANALYTICAL PROCEDURES

            Before long, it was time for me to start analyzing my data to determine my outcomes.  I had read about this portion of the action research process, and we had held extensive discussions in my action research class, but I still felt anxious and overwhelmed.  My materials were fairly organized and I had continuously examined the information I had gathered and made adjustments accordingly throughout the data collection process, but I just couldn’t fathom where to begin. For some reason, I was intimidated by the prospect of scrutinizing my research.  In addition, I was irritated that many of the work samples were located in the students’ math journals.  The journals were not only bulky and difficult to transport, but they were also challenging to compare because I had to spend time locating the entries in each journal.  Furthermore, I couldn’t write my observations and reflections directly on the pages because I intend to send the journals home at the end of the school year.   At the same time, I did not want to copy each journal entry.  I just couldn’t justify consuming such a large quantity of paper.  So, instead of grinning and bearing it, I procrastinated. 

            During this time, the March 2004 issue of the Journal for Research in Mathematics Education arrived.  Since the journal contains collections of case studies and reports of research, among other manuscripts, I decided to peruse the text to see if I could get some ideas about how to conduct my analysis and how to organize my final report.

            While looking through the Journal for Research in Mathematics Education (2004), I came across a case study of one teacher’s experiences with implementing whole-class discourse in her third grade class of primarily Latino students.  Kimberly Hufferd-Ackles, Karen Fuson, and Miriam Gamoran Sherin wrote this article entitled  “Describing Levels and Components of a Math-Talk Learning Community.”  Since the topic of this study closely resembled my topic, and since according to the authors, “The goal of this article is to introduce a framework that can help to guide teachers’ work in this area and to facilitate researcher and teacher educator understanding of this process” (Hufferd-Ackles, Fuson, & Sherin, 2004, p. 81), I thought it would be the perfect resource to guide me through this phase of my action research.

            The segment of the article that discussed the coding system and “the four distinct, but related components that captured the growth of the math-talk community over time” (Hufferd-Ackles, Fuson, & Sherin, 2004, p. 87) caught my attention.  The researchers created a table to organize and explain the developmental progression of four areas: “(a) Questioning, (b) Explaining math thinking, (c) Source of mathematical ideas, and  (d) Responsibility for learning” (p. 87).  I immediately recognized that I could use the same table to assist me in analyzing my data.

            I decided to put ten of the audio recordings of class sessions, spanning the course of my action research project, in chronological order.  Afterward, I listened to the sessions and used the table from “Describing Levels and Components of a Math-Talk Learning Community” (Hufferd-Ackles, Fuson, & Sherin, 2004, p. 88-90) to determine the levels of the students’ actions and my actions during each session.  I noted the levels on a table (See Table 1) and later compared them to see if there were any signs of growth. 

 

Table 1.

 

 Levels of Questioning, Explaining Mathematical Thinking, Source of Mathematical Ideas, and Responsibility for Learning

 

 

Tape 1

Tape 2

Tape 3

Tape 4

Tape 5

Tape 6

Tape 7

Tape 8

Tape 9

Tape 10

 

 

 

 

 

 

 

 

 

 

 

Questioning

1

1

2

2

2

3

2

2

3

3

 

 

 

 

 

 

 

 

 

 

 

Explaining Mathematical Thinking

0

1

1

1

1

2

2

3

3

3

 

 

 

 

 

 

 

 

 

 

 

Source of Mathematical Ideas

1

1

1

2

2

2

2

2

2

2

 

 

 

 

 

 

 

 

 

 

 

Responsibility for Learning

0

0

1

1

2

2

2

2

3

3

 

 


            According to Table 1, at the start of the project my class was at a level one for “Questioning.” Toward the end of the project, my class progressed to a level three.  For “Explaining Mathematical Thinking,” my class started at zero and ended at three.  In the category titled “Source of Mathematical Ideas,” we went from a one to and ended at two. In the last category, “Responsibility for Learning,” my class evolved from a zero to a three.  These changes indicate improvement in all four categories.  The information on Table 1 shows that early in the project, I took the primary responsibility for questioning, explaining solutions, providing mathematical ideas, and evaluating while the students took a more passive role in the learning process.  Toward the latter part of the study, the students became more involved.  They asked each other questions and tried to explain their strategies and solutions.  The students volunteered to share their ideas even if they were different than someone else’s ideas and they also helped to decide if an answer or explanation made sense.  Table 1 also indicated that while the students did become a greater source of mathematical ideas, there was still some room for growth. 

            I decided to review my journal entries and reflections to see if they matched the information I had gleaned from Table 1.  My notes suggested a similar trend in the roles and responsibilities of the students.  In fact, several times I had remarked that I was excited about the students being so involved, and that I was shocked at the variety of strategies that they employed.  I was thrilled that the children seemed to be developing more skills to discuss and write about the ways in which they solved problems. I mentioned that I was surprised that they were so capable and that I was impressed with their abilities to ask questions, make comments, share their techniques and thought processes, and generally help each other understand the material we were studying.  My entries also showed that I struggled with giving up control and allowing the students to determine the direction of the lesson.  I did get better at permitting the students to interrupt my explanations to share their ideas and opinions, but sometimes I felt as though I didn’t have time to get off track so I would limit student discussion and forge on with my intended lesson.  Therefore, I agreed that there was still room for growth in regard to letting the students take a more active role in developing mathematical ideas.

            Although my notes clearly showed that I was under the impression that the children had improved in their ability to explain their thinking, I knew that I needed to analyze the students’ written responses before I could honestly state a conclusion.  Since some of their work was loose, some was in journals, and some had been transcribed onto another paper so that I could send the originals home to the students’ families, I decided that the easiest way to compare the information was to record it on a chart.  Because I had an abundance of data and I wanted to avoid feeling overwhelmed, I elected to limit my selection from which to study.  I selected thirteen samples, in chronological order, that spanned the length of the research project.  Next, I used my “Math Rubric” to score each of the thirteen samples for every student.  I recorded the scores in the appropriate row and column on the chart.  I color-coded the chart so that each scoring value was a different color.  For each writing sample, I also compiled the number of students who earned each score and converted that information into a line graph (See Appendix C).  After that, I calculated the mean scores for each writing sample and created a column graph to show those results (See Appendix D).  Finally, I figured out the mode score for each sample and recorded that information on a column graph (See Appendix E). 

            While I had hoped to see the scores dramatically increase to where everyone was consistently earning a score of three, this was not the outcome.  Instead, according to the chart, the line graph, and the column graph of the mean scores, the student scores tended to fluctuate.  Sometimes the scores even decreased from one work sample to the next.  Over time; however, there was a general trend where the number of responses earning a score of zero or one decreased and the number of scores earning a two or a three increased.   This interpretation can be clearly seen by the increased presence of the blue column, or score of three, in Appendix F.  On the first writing sample, only four percent of the students earned a score of a two or a three, and on the second sample, thirty-nine students earned a score of a two or a three.  By the twelfth and thirteenth sample, the percentage of students earning a two or a three rose to eighty-eight point five and eighty-seven respectively.  The mode scores in Appendix E indicate that at the beginning, the majority of students were earning a score of one and in the middle, the majority of students were earning a score of two.  Only toward the end of the study did the mode score improve to three. 

As I analyzed the chart and graphs, I wondered why the scored writing samples did not reflect the degree of progress that I had anticipated.  All of my data sources pointed toward an improvement in students’ abilities to explain how they solved problems, but I noticed that the writing samples indicated less growth than the table or the reflective journal.  Upon reflection, I realized that a number of factors could have contributed to this lack of correspondence.  First, the scores only represent a sample of the written responses that the children completed.  Perhaps, if I had chosen other responses, or a greater number of responses, I would have found different results.  Second, each scoring value on the “Math Rubric” encapsulated a range of responses.  For example, in response to the question, “The principal of Heart Elementary School wants to give all of her 2nd grade children valentines.  There are 26 students in Mr. Cupid’s 2nd grade class.  There are 27 students in Ms. Chocolate’s 2nd grade class, and 28 students in Mrs. Friend’s 2nd grade class.  Approximately how many cards will the principal need to buy?”  Corrieanne’s response of, “90.  I rounded.” would be given a score of two because her strategy worked for the problem and she tried to explain her answer.  Jake’s response of, “I rounded 26 to 30.  I rounded 27 to 30.  I rounded 28 to 30 and I got 90.” would also be given a score of two because his strategy worked and he tried to explain his answer.  It would not be assigned a score of three because it does not explain how he arrived at the answer ninety.  Although both Corrieanne and Jake received the same score, Jake’s response is clearly more developed since it contained specific information regarding how he rounded each number.

Another reason why the scores of the writing samples did not match my expectation could be that the written scores were derived from a variety of problematic tasks covering a plethora of mathematical concepts.  Since they were not all about the same topic, and they were not all written in the same format, it is difficult to accurately compare them.  Familiarity also plays a role in students’ abilities to solve problems.  When a teacher first introduces a topic, students usually tend to draw on their background knowledge to help make connections and better understand the material.  If a concept is new or different, the child will most likely be less successful than if he or she was presented with a problem covering the same material later in the unit.  An additional contributor could be that many of the problems were geared toward more advanced thinking.  I did not focus solely on the concepts and approaches that were designated in my second grade curriculum guide.  In fact, I frequently posed problems that could have been appropriate for third and fourth grade students.  Maybe if I had concentrated on simpler tasks, the students would have shown improvement more rapidly.  Finally, I have observed during my eight years of teaching second grade that the ability to verbalize thoughts and ideas typically develops before the ability to write descriptions of the same thoughts and ideas.  If this developmental progression is typical in math, it would help to explain why my notes and reflections after discussions reflected a belief that the students were dramatically improving in their abilities to explain how they solved problems. 

OUTCOMES AND REFLECTIONS

After reviewing all of my data and reaching the point where I wanted to state a conclusion, I knew that I needed to take a number of things into consideration.  One thing I needed to consider was the fact that any pretest versus posttest improvement did not necessarily provide support that an emphasis on communication was the sole contributor to increased success.  I realized that most students naturally show improvement after being exposed to the materials and the format of the questions, not to mention instruction on the specific concepts.  Maturity plays an important role in both the development of concepts and the ability to explain them.  These same factors had to be considered when examining any improvement in oral and written responses throughout the year.   According to Helping Children Learn Mathematics, “Developing strands of proficiency individually is much harder than learning them together.  In fact, it is almost impossible to master any one of the strands in isolation” (Kilpatrick & Swafford, 2002, p. 17).  Therefore, I needed to take into account that my teaching style changed in more ways than simply putting an emphasis on communication skills.  

With all of these factors in mind, including the bias that is inherent in a teacher reporting on his or her own action research, I still believe that focusing on communication skills positively impacted my students’ abilities to explain how they solved problems.  When I compare my students from this year to my memories of students from years past, I am amazed at the difference.  In the past the majority of the children tended to replicate the strategy I had demonstrated in class.  Very few children were willing to branch out and try something different.  Although my prior students were usually able to determine the correct answer, they rarely verbalized or clearly explained their thinking in writing.  This year, many of my students learned to take greater responsibility for their learning. They asked questions, shared their responses, evaluated each other’s ideas, and generated more of their own problems, strategies, and solutions.  Over the past few months, I also observed some of my second graders using strategies that were shared by their classmates during discussions.  A few of my students even tried more than one approach to solve the same problem.  My journal entries, student artifacts, tables, charts, and graphs help validate these claims.  In addition, they demonstrate an overall improvement in my students’ abilities to explain their mathematical thinking in writing.

Based on these findings, I will definitely apply many of the strategies I learned and employed while conducting my action research project to my teaching next year.  However, now that I am more familiar with the research, available resources, and teaching practices, I feel that I will probably make some minor changes. 

            The first thing I will change is that I will start using problematic tasks and modeling effective questioning from the beginning of the year.  I want my students to know right from the start that I value their opinions and that it is okay to try and to share different strategies.  I want the children to know that it is important and acceptable to respectfully challenge other people’s solutions and to try to make sense of what they’re hearing.   If I employ these practices from the start, I feel that it will take less time for the students to take an active role in their education and act as co-teachers and co-learners rather than just passive participants.

            I want to work with my principal, my teammates, and my staff development teacher to find a way to balance my preferred style of teaching with the expectations listed in the county curriculum guide.  This year, I felt frustrated with trying to mesh teaching through problem solving with teaching in a more traditional method, particularly during the CTBS test-preparation period.  In the future, I would like to do everything in my power to provide my students with a more consistent learning environment.  Even if I occasionally need to teach using traditional methods, I would still like to dedicate a greater portion of time to discussing and writing about mathematics.

Beginning in September, I will periodically use the data from this year to help me reflect upon my teaching practices.  If I continue to monitor my progress, I believe that I will better ensure that I provide an atmosphere that promotes open communication and the sharing of learning responsibilities. I tend to be reflective by nature, but this year I struggled with finding enough time to record all of my thoughts and ideas.  It seemed that every time I sat down to write, someone would stop by my room or another task would distract me.  Therefore, for next year, I plan to buy a hand-held tape recorder and verbalize my reflections during my commute home.  If this method is not feasible, I will set aside a specific time to write in a secluded environment.

I will introduce the “Math Sentence Starters” earlier in the year, too.  Although odds are very good that I will not write another in-depth paper about my action research next year, I do plan to track my students’ progress over the course of the year.  I really felt like this made a difference in my ability to understand the thought processes of my students.  In order to better monitor their improvement in written responses, I will work with my team to modify the “Math Rubric.”  In my opinion, the one I used for this project was not capable of accurately depicting the extent of student progress because each score encapsulated too broad a range of responses.

I am contemplating not only having the students self-score their responses, but also having them record their progress on a chart or a graph.  I feel like the visual representation may further motivate them to do well.  The responses I choose to score and track will probably be more consistent with grade-level indicators, and I may compare scores within each unit rather than trying to analyze them across an entire year.  That way, the problems I evaluate will be more similar in content and format.  Of course, I will still pose more challenging tasks on a variety of mathematical topics throughout the year. 

CONCLUSION

I have recently been informed that I may be moving to third grade next year.  Although I am terrified at the prospect of learning a whole new curriculum, I am somewhat relieved to know that I can apply much of what I’ve learned from my action research project to the objectives for third grade.  I now know that focusing on communication in mathematics positively affects students’ abilities to explain how they solve problems, and I feel strongly that communication should be an integral part of mathematics instruction.  Next year, I will still belong to the National Council of Teachers of Mathematics, and I will have access to all of the resources inherent in that membership.  I will also be able to reference the wealth of materials I located during my research.  Therefore, I will have a solid foundation for expanding my knowledge and implementing communication in mathematics regardless of which grade I teach.  I am looking forward to observing the impact I hope to make by utilizing my newfound knowledge and skills.

 

 

 

 

 

 

REFERENCES

Buschman, L. (2004).  Teaching problem solving in mathematics. Teaching Children

Mathematics, 10 (6), 302-309.

Fennell, F., Bamberger, H., Rowan, T., Sammons, K., & Suarez, A. (2000).   Connect to

NCTM standards 2000, Chicago, IL: Creative Publications.

Herr, T., & Johnson, K. (1994).  Problem solving strategies: crossing the river with dogs

and other mathematical adventures.  Berkeley, CA: Key Curriculum Press.

Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., Murray, H., et al. (1997). 

Making sense.  New Hampshire: Heinemann.

Hufferd-Ackles, K., Fuson, K., & Sherin, M. (2004).  Describing levels and components

of a math-talk learning community. Journal For Research in Mathematics Education, 35, 81-116.

Kilpatrick, J. & Swafford, J. (Eds.). (2002).  Helping children learn mathematics. 

Washington, DC:  National Academy Press.

Mills, G. E. (2003).  Action research: a guide for the teacher researcher. Upper Saddle

River, NJ: Pearson Education, Inc. 

NCTM (2000). Principles and standards for school mathematics.  Reston, VA:  National

Council of Teachers of Mathematics.

Scharton, S. (2004). I did it my way: providing opportunities for students to create,

explain, and analyze computation procedures.  Teaching Children Mathematics,

10 (5), 278-283.

Van Zoest, L. R., & Enyart, A. (1998).  Discourse, of Course: Encouraging Genuine

Mathematical Conversations.  Mathematics Teaching In The Middle School, 4,

150-158.

Wood, T. (1999).  Creating A Context For Argument In Mathematics Class.  Journal For

Research In Mathematics Education,30, 171-192.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

APPENDIX A

 

 

MATH SENTENCE STARTERS

 

Topic Sentence/Sentence 1:

                                                Use words from the problem and give your answer.

 

Sentence 2:  (choose one)

                                    To solve the problem, I …

                                                I solved the problem by …

 

Sentence 3:

                                                This is what I did to solve the problem.

 

Sentence 4:  (choose as many as you need)

                                                First, I ….

                                                Next, I …

                                                Then, I …

                                                Following that, I …

                                                After that, I …

                                                I compared … to …

                                                I used the operation …

                                                Finally, I …

                                                Last of all, I …

 

Closing Sentence:

                                                This is how I solved the problem.

 

 

 

 

 

 

 

APPENDIX B

 

 

 

 

                     Math Rubric  

 

3    My answer matches the question.

I clearly explained the strategy or strategies I used to solve the problem.

 

2      My strategy worked for most or all of the problem.

 I tried to explain my strategy.

 

1      I attempted to solve the problem.

I did not try to explain my strategy or my strategy is unclear.

 

    0  Incorrect answer

 No explanation

 No response

       Unreadable

 

 

 

APPENDIX C

 

APPENDIX D

 

APPENDIX E