Sociomathematical Norms, Argumentation, and Autonomy in Mathematics (Erna Yackel & Paul Cobb)
When I first saw the title of the article, I thought that it would be very boring with many theoretical ideas related to social issues. However, after I read it, I really felt that this is a very enlightening article, reminding me of a fact that we need to place more emphasis on the discussion and interaction between the teacher and students in the classroom disregarding the subject areas. I have to admit that in my previous teaching, sometimes, I didn’t take my students’ mathematics voice seriously. Because of the time limitation and the main purpose which aims at how to let my students achieve high scores in all types of examinations, I took more teacher-centered way to treat my teaching, especially when I was teaching grade 12 students who have an urgent need to get good scores in the National Entrance Examination, at that period of time, I had less communication and interaction with my students in the classroom. What I usually did during that period was to convey the knowledge throughout the whole class and asked my students to copy down the important notes and solutions step by step. I always thought that the solutions I provided to my students were more elegant than what they got and the teacher-centered teaching could save their time. Also, I was also very satisfied with seeing that my students could follow the skills or solving techniques that I taught them to solve some similar problems. What I did was to view my students as passive recipients or empty vessels waiting to be filled and kill their interests and motivation to construct the knowledge themselves. I admit that if students can construct the knowledge themselves, it will be very beneficial for developing their conceptual understanding of mathematics. But, what I talked above doesn’t necessarily mean that I always used the teacher-centered way in my teaching.
The authors define the term of ‘sociomathematical norms’ and clarify the subtle difference between general social norms and sociamathematical norms in the paper. They stated: “normative understandings of what counts as mathematically different, mathematically sophisticated, mathematically efficient, and mathematically elegant in a classroom are sociomathematical norms. Similarly, what counts as an acceptable mathematical explanation and justification is a sociomathematical norm.” It means that mathematical difference, mathematical sophistication or efficiency, mathematical elegance, and an acceptable mathematical explanation and justification form the sociamathematical norms in the math classroom. Also, in my view, they also form an ascending chain in which each norm is developed one by one. These norms remind me of the fact that if students can make mathematic difference in their response or solutions, they are able to develop a sophisticated or efficient solution among different solutions themselves. If they can see or compare the similarities and difference in solutions through argumentation, it’s also nature to see that they are able to develop conceptual understanding of the problem and present an acceptable explanation and justification accordingly. I recognize that these norms are so essential and important that they can be seen often in the math classroom in which there is sufficient interaction between the teacher and students. In my former teaching, if I had enough instruction time when I taught in grade 10 and 11, I also encouraged my students to engage in math learning with group discussions, whole-class discussions, and the communication with me. So, I can say that I’m quite familiar with the episodes that the authors described in the examples although I’m devoid of the academic ability to advance these episodes to a theoretical level. In my math classes, as long as there is no conflict between the teaching pace and my teaching strategies, I preferred to use the way with many inquiries. I totally agree to the author’s opinion that mathematical difference is interactively constituted by the teacher and the students. To give the students enough time to present their solutions fully should be greatly emphasized by teachers. Students are different individuals, having different mathematical thinking. So, they are the source of generating different solutions although not of them are correct. Just like what the authors said that the accuracy is not an important issue in developing math understanding, what more important is whether students can reflect their understanding through identifying similarities and differences among various solutions. Such reflective activity has the potential to contribute significantly to children’s mathematical learning. (Yackel & Cobb, pp464) Students can truly establish the conceptual understanding when they protect their own solutions or learn from others’ argumentation. In addition, students’ different thinking or solutions also help the teacher broaden his/her thinking. For example, in my teaching, if I got quite stuck when solving a complicated problem, I would turn to ask help from my students rather than from my colleagues. I usually posed the question first, and then asked students to raise their opinions of how they thought about it. If students raised some ideas that I didn’t know, I definitely would ask the students to give me an explanation, and then I saw students discussing, arguing between the similarities and differences among various ideas. Students are always thrilled to see that they can solve a problem that the teacher is not able to, at that time, they are very willing to present their explanations and justify their explanations. It’s true that their explanation and justification also provide me with an opportunity to develop my better conceptual understanding.
The teacher’s reactions to a child’s solution can be interpreted as an implicit indicator of how a solution is valued mathematically, such as “Yeah”, “Perfect”, “Well done!”, “really?”, etc. I’m really impressed with reading the episode in example 4 because I always did the same way to see if my students could persist in their solutions or explanations in my teaching. The questioning, like “Is it right?”, “Really?” can help me identify whether my students have got a strong conceptual understanding. If they haven’t, they would swing in front of such questionings. I have ever used the same example when I saw my students protesting my questioning and told them what they needed is to justify their reasoning. “If you can justify the accuracy mathematically, why do you swing?” This was a question I would ask when seeing my students’ protest. I admit that these soicomathematical norms are helpful for students to foster the development of intellectual autonomy which is a major of mathematical education.
To sum up, this is a good paper, setting forth a way of analyzing and taking about mathematical interaction in the classroom from the perspective of sociomathematical norms.
When I first saw the title of the article, I thought that it would be very boring with many theoretical ideas related to social issues. However, after I read it, I really felt that this is a very enlightening article, reminding me of a fact that we need to place more emphasis on the discussion and interaction between the teacher and students in the classroom disregarding the subject areas. I have to admit that in my previous teaching, sometimes, I didn’t take my students’ mathematics voice seriously. Because of the time limitation and the main purpose which aims at how to let my students achieve high scores in all types of examinations, I took more teacher-centered way to treat my teaching, especially when I was teaching grade 12 students who have an urgent need to get good scores in the National Entrance Examination, at that period of time, I had less communication and interaction with my students in the classroom. What I usually did during that period was to convey the knowledge throughout the whole class and asked my students to copy down the important notes and solutions step by step. I always thought that the solutions I provided to my students were more elegant than what they got and the teacher-centered teaching could save their time. Also, I was also very satisfied with seeing that my students could follow the skills or solving techniques that I taught them to solve some similar problems. What I did was to view my students as passive recipients or empty vessels waiting to be filled and kill their interests and motivation to construct the knowledge themselves. I admit that if students can construct the knowledge themselves, it will be very beneficial for developing their conceptual understanding of mathematics. But, what I talked above doesn’t necessarily mean that I always used the teacher-centered way in my teaching.
The authors define the term of ‘sociomathematical norms’ and clarify the subtle difference between general social norms and sociamathematical norms in the paper. They stated: “normative understandings of what counts as mathematically different, mathematically sophisticated, mathematically efficient, and mathematically elegant in a classroom are sociomathematical norms. Similarly, what counts as an acceptable mathematical explanation and justification is a sociomathematical norm.” It means that mathematical difference, mathematical sophistication or efficiency, mathematical elegance, and an acceptable mathematical explanation and justification form the sociamathematical norms in the math classroom. Also, in my view, they also form an ascending chain in which each norm is developed one by one. These norms remind me of the fact that if students can make mathematic difference in their response or solutions, they are able to develop a sophisticated or efficient solution among different solutions themselves. If they can see or compare the similarities and difference in solutions through argumentation, it’s also nature to see that they are able to develop conceptual understanding of the problem and present an acceptable explanation and justification accordingly. I recognize that these norms are so essential and important that they can be seen often in the math classroom in which there is sufficient interaction between the teacher and students. In my former teaching, if I had enough instruction time when I taught in grade 10 and 11, I also encouraged my students to engage in math learning with group discussions, whole-class discussions, and the communication with me. So, I can say that I’m quite familiar with the episodes that the authors described in the examples although I’m devoid of the academic ability to advance these episodes to a theoretical level. In my math classes, as long as there is no conflict between the teaching pace and my teaching strategies, I preferred to use the way with many inquiries. I totally agree to the author’s opinion that mathematical difference is interactively constituted by the teacher and the students. To give the students enough time to present their solutions fully should be greatly emphasized by teachers. Students are different individuals, having different mathematical thinking. So, they are the source of generating different solutions although not of them are correct. Just like what the authors said that the accuracy is not an important issue in developing math understanding, what more important is whether students can reflect their understanding through identifying similarities and differences among various solutions. Such reflective activity has the potential to contribute significantly to children’s mathematical learning. (Yackel & Cobb, pp464) Students can truly establish the conceptual understanding when they protect their own solutions or learn from others’ argumentation. In addition, students’ different thinking or solutions also help the teacher broaden his/her thinking. For example, in my teaching, if I got quite stuck when solving a complicated problem, I would turn to ask help from my students rather than from my colleagues. I usually posed the question first, and then asked students to raise their opinions of how they thought about it. If students raised some ideas that I didn’t know, I definitely would ask the students to give me an explanation, and then I saw students discussing, arguing between the similarities and differences among various ideas. Students are always thrilled to see that they can solve a problem that the teacher is not able to, at that time, they are very willing to present their explanations and justify their explanations. It’s true that their explanation and justification also provide me with an opportunity to develop my better conceptual understanding.
The teacher’s reactions to a child’s solution can be interpreted as an implicit indicator of how a solution is valued mathematically, such as “Yeah”, “Perfect”, “Well done!”, “really?”, etc. I’m really impressed with reading the episode in example 4 because I always did the same way to see if my students could persist in their solutions or explanations in my teaching. The questioning, like “Is it right?”, “Really?” can help me identify whether my students have got a strong conceptual understanding. If they haven’t, they would swing in front of such questionings. I have ever used the same example when I saw my students protesting my questioning and told them what they needed is to justify their reasoning. “If you can justify the accuracy mathematically, why do you swing?” This was a question I would ask when seeing my students’ protest. I admit that these soicomathematical norms are helpful for students to foster the development of intellectual autonomy which is a major of mathematical education.
To sum up, this is a good paper, setting forth a way of analyzing and taking about mathematical interaction in the classroom from the perspective of sociomathematical norms.
1 comment:
Julia --
Thanks for an interesting blog.
I can see that you are confronting differences between the traditional lecture-based, teacher-centred ways of teaching math that you are most familiar with, and the constructivist, discourse-based, learner-centred teaching ideas you are encountering here at UBC.
Do you think that you will be able to use some of these new ideas when you teach in China? Will the culture of the school (and the parents) allow it? Do you feel competent and confident enough to try a different way of configuring the classroom experience for your students? And do you think it would work well with your background and personality?
It's important to think about these kinds of issues when trying to bring about educational reforms. The things you are thinking and feeling about these new ways may well be shared by other teachers who are also successful at teaching in traditional ways. Reflecting on what would make it possible for you to shift your teaching can give insight into what's needed in general to support teachers.
Thanks!
Susan
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