Let’s Talk Math:

Investigating the Role of Conversation in a Secondary Mathematics Classroom

Entering into my student teaching experience this semester I knew that I wanted my classroom to be an interactive learning environment.  It was important to me that the students were active participants in the learning process rather than simply passive observers.  I had done some research on this in the past and had a few ideas going in, but I would soon find that there was much more yet to be learned about just what made for a productive learning environment.  Specifically, I had not considered how much talking would affect my classroom dynamic.

This semester I taught two 90-minute classes each of which had a very unique personality.  At 8 am I started the day off with a geometry class.  It was made up mainly of freshman and sophomores, though I did also have two juniors and a senior in the class.  They were a quiet group, partly due to the time of day, but mostly as a result of the mix of personalities.  Given a choice they would have preferred that I do all the talking while they listened quietly.  This made classroom management fairly simple and overall they seemed to be learning the material.  During the last period of the day I taught algebra C, which is the last part of algebra II.  This group was a mix of juniors and seniors and they were a particularly noisy group.  Again this was likely partially caused by the time of day, but the mix of personalities was the main cause.  Most of them were already friends and this was their last required math class.  Practically every time I spoke at least half of the class would turn to their neighbor to comment on something I had said.  Managing this group was a taxing and exhausting experience and often I found myself a little hoarse by the end of the period.  In spite of the constant noise level, it seemed that the students in the class were also on average learning the material.

With these two classes as different as night and day I began to ask myself just how much talking should happen in a classroom, and who should be doing it.  Initially I preferred to teach the geometry class, because they were so easy to manage, but as the semester progressed I found that when appropriately managed I enjoyed the talking in the algebra class because it helped me to be more aware of what students were and were not understanding.  By the time I reached the halfway point in my experience my views of each class had switched.  Geometry, while still easy to manage, was boring, and frustrating in that getting students to answer questions was, on some days, like pulling teeth.  Algebra on the other hand was fun, while we still dealt with noise issues, the back and forth exchange of ideas kept students engaged and me from feeling like I was talking to a brick wall.  Mostly though they asked questions and only those who had looked ahead in the book had enough confidence to answer.

Progress had been made in both classes towards my original goal of an interactive learning environment, yet I had the feeling that neither of my classes had it right yet and I was beginning to understand that it was not just about finding a happy medium in the noise levels.  More important than how much talking was going on, was what they were talking about and just how they were approaching ideas.  I needed to figure out what the role of conversation should be in the classroom and how I could help my students understand that and rise to it.   So with my question in mind I took the phrase teacher-learner to heart and began my research, digging for a solution to my classroom conundrum.

Research

In the process of my research I discovered that conversation in the context of mathematics is a topic that has received a fair amount of attention in recent years.  More commonly referred to as inquiry or discourse I had stumbled upon what is at the forefront of mathematical education reform.  The authors I found wrote about this topic from a variety of perspectives.  Some wrote from a purely theoretical perspective discussing the reasoning behind and potential implications of mathematical discourse while others wrote from a more practical perspective discussing what did and did not work in their particular classrooms.  Still others offered helpful hints for developing a classroom centered on discourse and ways of evaluating the current state of a classroom.  In the end I divided my research into three categories, theoretical insight, practical advice, and case study examples.  Many of the articles or books actually addressed more than one of these categories, but for simplicity sake I have grouped them based on what I found to be most helpful in each.

Making discourse a part of a classroom in a way takes mathematics back to its original form.  As a result, use of discourse can help students to understand the heart of mathematics and what makes it a field worthy of study.  Calvert discusses this in her book, Mathematical Conversations within the Practice of Mathematics.  She explains that mathematics is a complex process of problem solving, reasoning, guessing and checking, and reflection rather than a simple linear path from problem to solution.  Allowing students to participate fully in all of these processes not only grants them insight into the work of a mathematician, but also gives them valuable skills they might not have acquired otherwise (Calvert).  Similarly Moschkozich points out that “learning mathematics is a discursive activity that involves participating in a community” in her article Examining Mathematical Discourse Practices.  Because mathematical practices involve developing meanings with special attention to different perspectives they are not something that can be completed without discourse taking place. In fact discourse is a natural and essential part of practicing mathematics in any setting (Moschkovich).  Boaler explores the communicative approach in part of her book What’s Math got to do with it?.  In this approach mathematics is viewed as a form of communication, not unlike a second language, rather than a set of rules.  This encourages students to talk about what they are learning and share it with their peers in a variety of ways.  Mathematical communication involves not only words but also graphs, tables, equations, and numbers.  But unlike a traditional classroom, in the communicative approach these are all very connected as different ways of sharing the same ideas or information.  When viewed as a form of communicating mathematics is no longer something to be good or bad at, it is instead something that all students are capable of and something that requires a community working together at to be successful (Boaler).  Each of these authors was in the end reinforcing the same idea; mathematics would not exist without discourse or communication.  It would be doing students a disservice to teach mathematics without allowing an exchange of ideas to take place in the classroom just as it did when the mathematics was originally being developed.

The second set of articles that I read outlines other teachers’ attempts to create effective mathematical conversation within their own classrooms.  Martin led a group of seventh graders in a discussion on the nature and existence of cubes.  He asked students to first work with two-dimensional visual representations of a cube and then asked to verbally respond to the connection between their drawings and actual cubes.  Later students also had an opportunity to discuss their ideas in smaller groups.  The main technique used in Martin’s discussion was to give students a variety of opportunities to express their ideas and engage in discourse.  This helped to include as many students as possible in the topic and also brought out a wide variety of ideas on the subject (Martin).  Knuth and Peressini gave examples of what they referred to as univocal and dialogic communication in the classroom.  Univocal communication occurs when the teacher uses discourse the bring student to a specific understanding by directing their thinking and discussion.  Dialogic communication on the other hand is a back and forth exchange of ideas where both student and teacher have input into the understanding that is reached.  Discourse generally falls somewhere between these two extremes, but in this authors perspective, dialogic discourse leads to a deeper understanding of mathematics on the part of the student (Knuth and Pressini).  In the article Math Instruction for Inquiring Minds, Chapko and Buchko discuss that traditional teaching methods, while often easier to implement, are a form of educational neglect.  They are each principals in schools that chose to change their math instruction to a more dialog based system.  Chapko stated that “society, career fields, and technological advances mandate that students possess math knowledge beyond basic computational skills” and she believes that hands on discussion based classes help students to get what they need out of math.  Buchko noted the “fundamental shift in thinking, teaching, and learning” that has occurred since switching over to inquiry based mathematics and notes the particular flexibility required of teachers who choose to teach this way (Chapko and Buchko).  Manouchehri and Enderson looked at a specific classroom example in order to analyze the way a successful discussion based classroom functioned.  The two most important factors were the role of the teacher and classroom structure.  In this case the role of the teacher was to ask questions of students to push their thinking to a deeper level.  She was always encouraging and accepting of multiple correct ideas about a topic.  Students did not look to her for answers, but rather sought out answers for themselves in essence eliminating any position of authority.  Students in this class understood the flow of the learning and when presented with a situation dove in without needing specific guidance.  They worked in groups acknowledging everyone as valuable participant.  Without this willingness to explore and discuss, learning would not have taken place.   So the role of the teacher was two fold in the classroom, establishing a community where students felt motivated to learn and were unafraid to trust their own ideas and to provide encouragement along the way without stepping into a position of authority (Manouchehri and Enderson).  Each of these classrooms had something specific to show about mathematical classroom discourse.  From the importance of a variety of opportunities to share, to the difficulties faced by teachers it is clear that discourse does not always look the same and that there are a variety of challenges to be faced along the way.  If successful, this type of instruction can not only give students a deeper understanding of the subject, but also bring students to actually enjoy math and see it as something more than a set of rules to be memorized.

The final and perhaps most helpful set of articles I read gave specific suggestions as to how to establish a classroom where discourse can take place.  Hutchinson presented what she believes to be the stages of a good discussion in her article When Talking in Class is a Good Thing.  A good discussion must start with a good topic, which is to say a problem that has either more than one way to approach it, or multiple correct solutions.  Secondly it is important that every student has a chance to respond to the topic be it orally or in written form.  When the discussion begins it is important that students not feel intimidated about sharing their ideas.  The role of the teacher is critical at this point in that they must be encouraging and supportive of all strategies and solutions.  In an effective discussion teachers can learn about their students while their students are learning about the topic at hand (Huchinson).  Sherin provided another structure for discussion in her article Facilitating Meaningful Discussion of Mathematics.  The first step in this type of discussion is generating ideas.  At this point the goal is to get a many ideas out as possible without worrying about directing the conversation in a specific direction.  Next the various ideas are compared and evaluated by the students.  A teacher may push certain ideas to the front of discussion, but they should not direct the conversation more than that.  Finally the discussion is focused in on one or two student questions that are chosen by the teacher.  This allows the teacher an opportunity to control the direction of the class without dictating what they should be learning.  Throughout the entire process the role of the teacher is one of facilitator and guide and all of the ideas and learning come directly from the students (Sherin).  Discussion based classes can be noisy, which may in turn lead to discipline issues.  Fay and Funk suggest methods of discipline that will not detract from the learning process in Teaching with Love and Logic.  Some of these include standing closer to the student who is having the behavioral issue, or deferring off topic comments to a later time.  These strategies along with others mentioned in the book allow the classroom to return to its normal flow without having to take out time for discipline (Fay and Funk).  Finally Stein provides a method of evaluating the discourse happening in a classroom in her article Let’s Talk.  Discourse can be evaluated in four categories, questioning, explanation of thinking, source of ideas, and responsibility for learning.  Each category receives a score on a scale of 0 to 3 where 0 represents a teacher directed classroom and 3 represents a community of learners with the teacher as an equal member.  Teachers can use this tool to assess their classroom and identify specific areas for improvement (Stein).

The research I found only touched the surface of what is currently a hot topic in mathematics education, yet it provided several insights into my question of the role of conversation.  Mathematics is at its heart a subject fueled by communication with others.  It is a subject that has been built up by people working together refining ideas and guessing at unknown possibilities.  Then removing conversation would be akin to removing a large portion of the learning process.  Yet creating a community that is accepting of classroom discourse is a difficult and slow process.  It is most effective when several teachers or an entire school will commit to doing it together.  And even though the discussion may be made up entirely of student ideas, the role of the teacher and facilitator and guide is still hugely important.  Helping students to develop their ideas and correct their own misconceptions is part of the role of the teacher.  Finally because discourse is such a complex process it is important to evaluate and refine the teaching process on a regular basis.  So armed with the new set of knowledge I was ready to go back into the classroom and begin to redefine conversation in my classroom.

Action

Each of my classes needed something different in order to create successful classroom discourse.  In geometry I needed to stimulate conversation by encouraging students to voice their ideas and affirming them when they do.  Algebra on the other hand needed no encouragement to start talking, but rather needed direction in order to transform what was happening from simply talk to discourse.  I tried out several different methods in each class knowing that no one method was likely to reach all students.  I used the pair share technique in both classes, but for different reasons.  In geometry it gave students a chance to share their idea under less pressure and in algebra it provided a space to get out the basic observations so that we could get deeper into the subject more quickly.  I also tried whole class discussions with both groups, but mainly used this technique in geometry where students were less prone to interrupting each other.  Both classes participated in lab day, which was a once a week opportunity for them to work through some sort of inquiry based activity in small groups.  Finally I treated proof as a written discourse for students who were struggling with the concept.  By using this variety of activities I was able to engage the majority of my students in some sort of mathematical conversation by the end of the semester.

In pair share I would pose a somewhat general question to the class and then ask student to share their ideas with a neighbor.  Then I would let students talk for a few minutes while I listened in on a few groups.  I was unable to reach every group, but I tried to hit a different set each time.  This gave all students a chance to give their ideas and hear from others before the discussion actually began.  My hope was that this would boost confidence and get more students involved in class and for many students it did seem to help.  In geometry this also woke students up a little more if they were beginning to drift off.  My algebra students it seemed had been using pair share from the beginning whenever they wanted.  By making it an actual activity I was able to communicate that it was ok to talk, but there were specific times and topics that were more appropriate than others.  It also allowed me to put a start and stop time on the individual conversations, which cut down on wasted time waiting for students to quiet down.  Overall this was a very good strategy for getting students talking and establishing talk times during class, but it did not facilitate deeper understanding and the conversations that occurred tended to stay on the surface of the topic.

The whole class discussions that I attempted were not as successful as I had hoped they might be.  Part of the reason for this I believe was the classroom set up.  Students were seated in rows facing forward, an arrangement that I could not change due to the fact that I shared the classroom with an evening class.  Also I had trouble coming up with appropriate topics that did not have a single answer and were still pertinent to the material at hand.  The discussions I had in geometry were more successful when they made connections between geometry and another discipline.  Some of these topics included the similarities and differences between the constitution and the axioms of geometry and the structure of a proof as compared to a well-written essay.  I was not able to come up with this sort of connection for algebra, which is probably partially why discussions were less successful in that class.  Still, even in geometry I did not get the level of participation I hoped for making this a less successful part of my plan.

Lab days were by far the most successful part of my attempt to implement inquiry and discourse into the classroom.  Students responded very positively to this, particularly those in the Algebra class and were always disappointed if for some reason the lab was moved back. Generally I would give students a set of manipulatives or a problem to play with and a hint at the goal of the lab.  Some labs had more structure than others taking students step by step through several ideas and asking them to draw connections between them, while others were more open ended giving students a chance to be creative in their solutions.  These labs were always completed in groups encouraging students to work together and learn from one another.  Students tended to have a better grasp of material covered on lab day than on the other material in the chapter, which is further evidence to support their effectiveness.  These days were also more enjoyable, both for me and for the students because we all got to be involved and active.  This is something I will definitely repeat in the future and continue to refine as a part of my teaching.

Proof made up a large part of my Geometry curriculum and was also a very challenging topic for many students.  For me proof is at the very heart of mathematics so I did not want to let the topic go by the wayside.  I had noticed that students could explain a theorem to me in conversation, but could not write out a formal proof.  This brought me to the idea of proof as a written form of discourse.  I encouraged students to use paragraph proof form and simply think of it as explaining as they would in conversation.  This helped several students, especially when it came to constructing indirect proofs.  I only wish that I had been able to introduce proof this way initially because those students that used this idea tended to make fewer mistakes and have more coherent proofs when they were finished.  Unfortunately students who had mastered two-column proofs were unwilling to change.

I tried several techniques for implementing discourse into the classroom during the latter portion of my student teaching, and while I found some success, my classroom came nowhere near those described in my research.  I believe this is because I did not do enough and I did not have enough time to let students adjust to the changes.  Developing a classroom community that is open to deep mathematical conversations is essential and without that nothing will work.  Developing this though takes time and I do not believe that the few weeks I had in the classroom were enough.  I saw my students make progress and I believe that had this been in place in all math classes they would definitely have been capable of incredible discussions.  As it was though our discussions were less than they could have been.  Based on this I believe that it is important to make meaningful conversation a part of the class from day one if you hope to create an atmosphere that will generate discourse.

Conclusion/Next Steps

My student teaching experience has taught me many things and my investigation into the role of conversation has provided particular insight into who I hope to become as a teacher.  It is important to me that what I teach is as close as possible to actual mathematics and that means that I must incorporate discourse into my classroom.  I believe that this is something that must be established early on in a class if students are to adjust to it.  Discourse can and should take on a variety of forms in any given classroom because not all students access knowledge in the same manner.  It should be used both as a method of discovery and for reflection.  Above all discourse must be a positive experience where students are encouraged and supported and not told their ideas are incorrect.  I have learned that discourse takes patience, especially at first, and I must be willing to wait out my students.  In the end a classroom where deep mathematical conversations take place is more rewarding for both student and teacher because both have the opportunity to teach and learn without fear of failure.

As I move forward in my teaching career I will continue to try and make mathematical conversation a part of my classroom.  I will establish it early and return to it often in order to help students become accustomed to the practice of sharing their ideas.  Stepping out of the role of teacher and into the role of learner I will learn from my students and they will learn from me.  I hope that I can make my classroom a community of learners where everyone is valued for the insight they bring and no one is seen as being good or bad at math.  Through discourse I will strive to bring mathematics back to life for students and allow them to feel like mathematicians in their own right.  None of this will be easy and most of it will take some time to perfect, but I believe that it will be worth the extra time and effort.  Through meaningful conversation I believe students can gain a deeper understanding of mathematics as well as gain valuable thinking and problem solving skills, which they can apply in other classes and in life outside the classroom.  Finally I will choose to incorporate discourse into my classroom because it makes teaching a more meaningful experience for me when I can continue to learn alongside my students.


Annotated Bibliography

Boaler, Jo. What’s Math Got to do With it? New York: Penguin Group, 2008.

This book discussed the problems facing mathematics education today and gave suggestions as to how they might be overcome.  The topics discussed included the nature of mathematics, effective classroom approaches, educative assessment, grouping systems, girls in math and science, and teaching strategies.  I found the portion on classroom approaches to be most useful on this particular project.

Calvert, Lynn M. Gordon. Mathematical Conversations within the Practice of Mathematics. New York: Peter Lang Publishing, 2001.

The main focus of this text was how conversation can be used to foster a deeper understanding of mathematics.  It gave many classroom examples to support the theories presented.  I found the portion on the practice of mathematics where they discussed the work of a mathematician to be beneficial to my inquiry.

Chapko, Mary Ann and Marian Buchko. “Math Instruction for Inquiring Minds.” Principal (2004): 30-33.

This article was written by two principals who had switched their schools over to inquiry and discourse based mathematics.  They explained why they had done it and how it had affected their schools.  This was helpful because it allowed me to see hat might happen if this were applied on a whole school level.

Fay, Jim and David Funk. Teaching with Love and Logic. Goldon: The Love and Logic Press, 1995.

This book focused on provided teachers with a variety of classroom techniques that made the classroom a positive environment.  It discussed discipline from a variety of perspectives both theoretical and practical.  The practical side of this was what I referenced most in my time in the classroom because this style of discipline fits in nicely with discourse-based teaching.

Huchinson, Mary Jo. “When Talking in Class is a Good Thing.” Ed Digest (2010): 46-8.

The author here outlined what she believed to be the process of facilitating a good discussion.  Specific steps were given along with tips for getting all students involved in the activity.  This was helpful to me as I attempted to facilitate my own discussions in my classroom.

Knuth, Eric and Dominic Pressini. “Unpacking the Nature of Discourse in Mathematics Classrooms.” Mathematics Teaching in the Middle School (2001): 320-325.

The goal of this article was to identify different types of discourse and the effects of each.  The authors identified two specific styles of discourse, univocal and dialogic, but noted that most discourse falls somewhere between the two.  This helped me see that a discussion was not always effective discourse depending on who had control over the outcome.

Manouchehri, Azita and Mary C Enderson. “Promoting Mathematical Discourse Learning from Classroom Examples.” Mathematics Teaching in the Middle School (1999): 216-22.

This article was almost more of a case study of a single classroom where discourse was implemented effectively.  After showing a lesson they looked at all of the factors that contributed to this classroom running so smoothly.  What was interesting to me what the role of the teacher in this process and also what she had facilitated as a classroom community.

Martin, Christopher. “Philoposhy in the Mathematics Classroom.” Mathematics Teaching (2008): 16-17.

In this article the author outline a classroom discussion that he had lead in a 7th grade classroom.  His discussion bridged the gap between mathematics and philosophy.  I was interested to see not only his discussion technique, but also how to help students make connections and facilitate interdisciplinary discussions.

Moschkovich, Judit. “Examining Mathematical Discourse Practices.” For the Learning of Mathematics (2007): 24-30.

This article looked at practicing mathematics as participating in a community.  It outlined some of the theory of discourse and discussed the importance of language in the classroom.  I used the theoretical perspective of this article to frame my understandings of these ideas.

Sherin, Miriam Gamuran. “Facilitating Meaningful Discussion of Mathematics.” Mathematics Teaching in the Middle School (2000): 122-125.

This article also outlined a discussion technique for the classroom.  It focused on the role of the teacher in each portion of the discussion and how they could slowly take more and more control, but still led the students be in the lead.  This would be helpful if you had a specific goal for the discussion or a topic that you wanted students to focus on more than other topics.

Stein, Catherine C. “Let’s Talk.” Mathematics Teacher (2007): 285-289.

The purpose of this article was to provide teachers a way of evaluating discourse in their classroom.  It outlined a scale of 1 to 3 for several catergories.  I used this to evauate my classroom before I began to experiment with discourse.