Improving Mathematical Discourse and Understanding in Classroom | TE 855, Study Guides, Projects, Research of Teaching method

Download Improving Mathematical Discourse and Understanding in Classroom | TE 855 and more Study Guides, Projects, Research Teaching method in PDF only on Docsity! Improving Mathematical Discourse & Understanding in the Classroom Melissa M. Goodrich TE 855 Fall 2005 A21468836 THE PROBLEM: Imagine stepping into a mathematics classroom where students are engaged in a high- quality math discussion. The students are working on various math problems, approaching them from different perspectives, and are communicating their solutions in a variety of ways. The arguments they are presenting support their thinking, and they have the ability to tell others about their work. The classroom discussion that is taking place is allowing students to not only evaluate their own ideas, but also the ideas of others. This is a classroom where mathematical discourse is taking place, and students are using mathematical reasoning to help them understand mathematics more. This situation may not seem difficult to accomplish for some, but for a majority of math teachers, they realize that the traditional models of teaching and learning mathematics do not help them create mathematically powerful environments like the one I described above (Ittigson, 2002). I am one of those teachers. As a high school mathematics teacher, I don’t feel that many of our math classrooms are using methods of teaching that is encouraging this mathematical discourse. In many classrooms, even my own at times, few interactions are taking place. The teacher is dominating the conversations by asking all the questions and if students do not know, the teacher will give the answer. As a result of this, “teaching by telling” is the most common pedagogy (Leonard, 2000). For this reason, I have chosen to focus on this problem in my research project. NCTM’s Standards recommends teachers to move away from teacher-centered classroom to one that is centered on student thinking and reasoning. They feel students should be resourceful problem solvers, and be able to communicate their ideas both orally and in writing. Principles and Standards emphasizes that all of us that are involved in teaching children mathematics, should be able to produce this discourse in our classroom 2 intellectual authority in the teachers’ hands, therefore taking any responsibility for thinking and reasoning away from the students. She too feels that there are few examples and guidelines out there to help teachers get away from this traditional style of teaching. One comment that Stein made that really caught my attention is the idea that many students know that their answer is correct because the teacher stops asking the question. Therefore, because of this, students often ask “Why?” when the teacher asks them to explain their reasoning. They feel that they don’t have to explain their reasoning because they already know that their answer is correct. “Helping Students Become Mathematically Powerful” is an article written by R. Ittigson who also mentions that the NCTM Principles and Standards wants us teachers to construct a classroom where students are problem solvers, using a variety of approaches to a problem, and can use arguments to support their thinking, and can also explain their reasoning to others. She feels that the only way that teachers can be successful creating a classroom scenario like this, they would have to be provided models, resources, and other suggestions that will spark students enthusiasm and interest in problem solving, help them communicate their ideas effectively, and help them grow to become mathematically powerful learners. The third author who backed up my thoughts regarding this problem was J. Leonard, who wrote “Let’s Talk about the Weather: Lessons Learned in Facilitating Mathematical Discourse”. She believes that mathematical discourse encourages the students’ interpretations of math problems and increases student-to-student interactions and she feels that teachers need to be in charge of facilitating this classroom discourse. She states, “Studies show that children learn and retain more information when they are able 5 to articulate what they know to others” (Leonard, 2000). I was very impressed with a comment that she made that relates to the way that students learn. She explains that students do not come to school as empty disks waiting to receive new information. Instead they already have a wealth of stored knowledge that they can access and updated. By having teachers facilitate this mathematical discourse, students can construct new knowledge by connecting it with the previous experience and knowledge they have stored up. Leonard suggests to teachers that they need to learn ways to facilitate discourse in the classroom, try various approaches, make observations, collect data, and then analyze the results. Just like anything else, you have to see what works with the children in your classroom, and what doesn’t. After reading this, I knew that this is what I would need to do in my own classroom and would need to encourage my co-workers to do the same. To start the process, I knew I would need to first do some research to find some approaches to improve the discussions that took place in my Algebra classes, which would also achieve the second goal I had set when starting my review of the research. I was very excited to begin researching some ways that I could help my students understand the mathematics they were learning in my math classes, by facilitating more mathematical discussions in my own classroom. I was pleased to find an abundant amount of information that I could use to try different approaches in my classroom. Not only did I find a list of examples and suggestions to try, I was also able to read scenarios from other teachers as they tried these approaches in their own classrooms. One person who did an excellent job suggesting ways to improve discourse was M. Lampert in her book, “Teaching Problems and the Problems of Teaching”. I felt that her book related to my research topic in many ways and provided me with many suggestions to increase 6 discussions and reasoning in my classroom. Her use of real situations that occurred in her classroom helped me see how these suggestions could really help, no matter what age the students are. I knew how difficult it was to get high school age students to open up and communicate their thoughts and reasoning. However, Lampert shows us that by making a few simply changes in the classroom, any age students can begin communicating their thoughts and analyzing the thoughts of others. After reading various chapters of her book, there were a few that I thought would be extremely helpful in my own situation. Chapter 7 entitled, “Teaching While Leading a Whole-Class Discussion” was particularly helpful to me during this research project. Lampert offers many suggestions for interacting with the whole class at once, maintaining overall coherence amongst the group, while including many different individuals into the experience. She feels that it is very important to get all the students engaged in the conversation, especially those who are not normally verbal participants. As I read through this chapter, the idea of “teaching in the moment” caught my attention right away. I knew that in my own classroom, I often lead discussions in the whole class environment. I knew that to create this learning environment I wanted that was student driven, I would have to learn how to react to students questions and reactions in the moment. Lampert uses many interactions in the whole class environment that are constructed with the observations that she is witnessing from the students studying on their own, in groups, or as a whole group. She has a way of guiding the conversations and questions so that they strongly link the concept that is being learned in the math classroom. Lampert states how important it is to focus on the details of the speech acts because “particular words and intonations are some of the most 7 back to the idea I mentioned before about students coming into the classrooms with previous knowledge that can help them engage in these conversations. Some of the research I read stated that getting students to talk in the classroom is sometimes the easy part, because students come in with the previous knowledge of some of these topics and they are eager to share their thoughts. However, the more challenging part may be encouraging students to interpret and comment on the ideas of their classmates. As I stated earlier in my paper, NCTM Standards feels that this discussion amongst classmates to analyze and communicate their ideas is very crucial to the learning process and to understand mathematics. In one of the articles I read, they stated two important goals that would help improve the discourse between the students when it comes to discussing their ideas. The first goal is that students need to respond to other students comments rather then just state their own ideas, and the second goal is that the students need to use one another’s ideas as the basis for thinking and learning the mathematics (Gamoran, Louis, & Mendez, 2000). I felt that these were great goals, and I could see how they would help in my classroom. When I thought back to the current communication that was taking place in my classroom, I realized that students were discussing on their own ideas, and would rarely respond to their classmates’ solutions. When I thought about why that was, I realized I wasn’t encouraging them to do that. So I was very eager to see what methods the authors of this article, “Students’ Building on One Another’s Mathematical Ideas” used to meet the two goals I described above. As I continued to read this article, the authors described three ways to help students move away from simply stating their own ideas during whole class discussions. These three solutions not only move students away from doing this, but it also helps them build on 10 other students’ ideas. The three ways include (1) having students state whether they agree or disagree with an idea that has been brought up in the classroom, (2) have students provide new evidence for someone else’s idea, and (3) have students draw on a classmate’s idea in creating their own conjectures (Gamoran, Louis, & Mendez, 2000). To finish up my discussion on the literature that I reviewed, I wanted to discuss a topic that came up in quite a few of the articles I read, and that is the idea of the types of questioning used to improve mathematical discourse. Lampert touched on this idea a little when she mentioned the way you begin a conversation can make a huge difference. Many authors feel the same way and offer up some suggestions to teachers on how to improve their types of questions. Teachers need to get away from the traditional questions like “What did you get for your answer?”, and “How did you get that answer?” to help promote their students’ mathematical thinking. When discussing this idea with my co-workers, we all decided that this was an area that we all needed help with. In the article that we read in class by Herbel-Eisenmann & Breyfogle, they feel that even though focusing on the questions asked is important, examining the interaction pattern is important too. In their article they explain the idea of using “focusing” questions once as student has stated their solution. In this situation, the teacher will pick out certain features from their solution, ask them to explain what they mean, and then restate what the student had said. This pattern helps students articulate their own thinking and encourage other students to make sense of one another’s strategies and reasoning (Eisenmann & Breyfogle, 2005). As you can see, there was a lot of literature out there that helped me reach my two goals: to find research to support my thoughts that there was a lack of discourse to help 11 with mathematical understanding in classrooms, & also to find suggestions and ideas to use in my own classroom to improve this problem. Once I was done gathering the information, it was time to put a plan into place that I could use in my own classroom. MY APPROACH TO THE PROBLEM: Once I finished reviewing the literature, I knew that I would have to sort through all of the suggestions that were offered to me, and decide which ones would work best in my classroom. I had to think about the types of classes I have, the subjects I teach, what forms of discussions & styles of teaching I use in my classroom, and much more when deciding what approaches I should use. The first thing that I needed to determine was the class that I was going to implement this plan in and how I was going to record and evaluate how the strategies worked. I felt that I should start with just one of my classes, so that I could really focus on it and see the results. What I ended up deciding was to implement this plan in my Algebra class rather than in my Geometry class. In the past years, I have taught both subjects, and I have always found it more difficult to get students to communicate their thoughts and reasoning in the Algebra classes. Next, I had to decide how I would keep track of what was taking place in the classroom, what was different about the previous ways I had taught, and how I was going to record the results. The first thing that I did, was make a list of the types of questions I asked students in the past, and the types of responses I got from them. Many of these comments and questions are similar to the responses and questions that I have mentioned in this project so far. This is what I would use to compare my results with, of the new approaches I would use in my classroom. To gather my information, I decided that the best way would be to take lots of notes after I implemented the strategies. I was also lucky enough to have a co- 12 up. One of the students who went about completing the problem this way was “Sierra”. Even though I noticed over half the class doing it this way, I also realized that not all of them were getting the correct answer. I also expected this due to the fact that using this method, students may forget a couple of the patterns. Even though the majority of the class was using this process, I also saw some other procedures being done. I witnessed one of my Algebra students, I will call him “Josh”, drawing circles in his notebook. The circles were in pairs, and lined up in groups of four. I was anxious to hear what his explanation of his solution would be. Finally, one of my students by the name of “Jon”, had the expression 2 x 2 x 2 x 2 written down with the solution with the answer of 16. After giving the students about 5 – 10 minutes to work on this problem and find a solution, I knew it was time to begin the conversation and see what would result. I began the discussion by taking Lampert suggestion and asked the class, “Does anyone have anything to say about this problem?” I didn’t want my students to feel like they could only respond if they had the correct answer. This is very important to me because I find the same students always answering questions in my classroom. And it always seems to be the students who get the answer right. I know that it was my fault that is was happening though, because I would ask students something like, “Who has the correct answer?” or something that would make the students feel that they would need to have the answer correct to participate in the conversation. Then once the correct answer was given, I would agree and then move on. I knew this would need to be changed during this particular conversation. “Sierra” was the first to raise her hand. She stated that she ended up with 14 possible ways to answer the question, and that is all she said. I was prepared that this would happen; students would just say the answer and not explain their 15 reasoning because that is what has been happening so much in the mathematics classrooms. I did not want to tell Sierra that her answer was wrong. I wanted to make sure that I was keeping the student’s self-esteem up and did not want anyone to feel bad about giving an incorrect answer. Plus, I knew that Sierra chose a great method to use to solve the problem (remember that she used the common method of just making a list), she just missed a couple. I wanted her to explain to the class what she did, so I asked her, “Can you explain your reasoning and procedure that you used, Sierra?” She explained that she had made a list of all the ways to answer the quiz. Once she said this, another student, “Lanie”, chimed in that she also went about it that way but received 16 possibilities. To keep the conversation going, I thought about using some “focusing” questions to get the students to think about the strategies they used, and also to restate what the students had said. After discovering that both students made the lists, I decided to ask them to explain the way they went about making the list. I soon found out that Sierra just kept making a list until she felt she covered all the possibilities, while Lanie tried to use a pattern when making the list (TTTT, TTTF, TTFF, TFFF, …..). After both of them did a nice job explaining their procedure, I asked the others students if anyone else had another way of completing the problem. In the past I would have found myself just telling the class that Lanie had the correct answer, and would have moved on. This time I knew that I would have get the rest of the class involved to decide which one was correct. When no one raised his or her hand, I asked Josh if he would like to share his procedure. I wanted to make sure Josh knew I was interested in his response. He began describing the reasoning behind why he made the circles. In then end, I realized that he was drawing the picture to resemble a bubble sheet, one column being the true responses 16 and one being the false responses. I ran into a little conflict at this point that I wasn’t prepared for. Some students were making a comment about Josh’s approach, and that it was too much work and he didn’t need to do it that way. However, I knew the importance of complementing students work and making them feel good about their ideas. What I did is continued to guide Josh into answering some questions about why he chose this method, and after some time, students began to analyze his plan, and understood that this in fact did lead to finding the correct answer. To end the discussion, I had Jon explain the reasoning for his expression of 2 x 2 x 2 x 2. After discussing this way, students were able to see how this could help them solve problems that involved a lot more true or false questions, which is what I wanted them to understand. Overall, I thought the discussion involved a lot more discourse then some of the others I had previously. CONCLUSIONS: Obviously, this was the first time that I was implementing the strategies that I had researched to help with the problem of lack of discourse and understanding in the mathematics classrooms. I knew I had to try this with a lot more story problems and activities throughout the semester. Over the past month, I have included a lot more story problems as warm-ups in my Algebra class, and have also tried to have students work on some of them in groups. Some days the dialogue and conversations were so impressive in my class that I had to just stand back and smile. I was so impressed with the fact that more of my students have felt more comfortable answering questions and contributing to class discussions. I think that my students feel that they can contribute more now and explain their reasoning even if they have the wrong answer. I see more and more of them 17