"Tools for the monoglossic are especially powerful in environments structured with significant positioning distinctions. Mathematics classrooms are just such places."
"The power is in the subtlety."
Summary:
By examining a broad set of mathematics classroom transcripts from multiple teachers, Wagner and Herbel-Eisenmann find that the word just played a big part in the mathematics classroom and could be used to suppress and invite dialogue. They focus attention on 'just problem' as central to mathematics classroom conversation. Students had the feeling that teachers should not use just in the class and thought just was kind of an aggressive word. In addition, the word just was the 27th most common word used in mathematics classroom discourse, more common than multiply, why and because.
Wagner and Herbel-Eisenmann point out that just serves as an adverb that seems to be synonymous with simply in most cases (28%) where the teacher is positioned as one who authorizes processes or procedures. The second most common usage is relatively synonymous with only (21%), and this usage can be accepted by students. The third but most powerful usage of just includes situations that represented varying degrees of frustration (22%). The authors invite mathematics teachers and educators to pay attention to subtle words as this kind of subtlety is powerful.
Three Stops:
1. 'Exclusive" verbs v.s. 'Inclusive' verbs
I am interested in the classifications of 'exclusive' verbs and 'inclusive verbs'. Rotman (1988) distinguishes between 'exclusive' verbs, which describe an action that can be done independently from others (e.g., write, calculate, copy), and 'inclusive' ones, which include an action that requires dialogue (e.g., describe, explain, prove). I recall my mathematics class and can think of some exclusive and inclusive verbs that I apply in my teaching. I use a lot of describing, explaining, and gestures in my teaching process. For example, I have been using “Can you tell me more?” and “What do you notice?” in our dialogues. I tried to focus on students’ understanding by encouraging them to share their ideas and thoughts. I found that ‘inclusive’ is a perfect word to describe the role of teachers and students during the interaction. We must allow each other to play a role in the mathematics dialogue. Sometimes, when I was introducing a new concept or theory to students, however, I may use ‘exclusive’ verbs such as repeat, copy or do without knowing it.
2. Suppressing dialogue, like any suppression, is an act of power (p. 10).
3. "What is my discourse like?"
The authors encourage us to note what our discourse is like among the mathematics class and to prompt reflective awareness. When I was in my first year of teaching, I invited some experienced teachers to come to my math class. Because when I was giving a class, I can hardly notice my problems in the dialogue. I learned a lot through their feedback. I realized that I often repeated students' questions or answers that made our conversations less succinct. In the beginning, from my point of view, why I did that was to make sure students get my point or gave them responses. But in most cases, repeating what students already said is not an efficient or positive response. If I didn't invite my colleagues to my class, I would have never known it was not good behaviour. As mathematics educators, we are supposed to keep asking ourselves, "What is my discourse like?" and "How might I change it to reflect my intentions?" (p. 13).
Question:
What do you think of the application of 'exclusive" verbs and 'inclusive' verbs in your mathematics class?
References:
Aijmer, K. (2002). English discourse particles: Evidence from a corpus. Philadelphia: John Benjamins.
Rotman, B. (1988). Towards a semiotics of mathematics. Semiotica, 72(1/2), 1–35.
Wagner, D., & Herbel-Eisenmann, B. (2008). “Just don’t”: The suppression and invitation of dialogue in the mathematics classroom. Educational Studies in Mathematics
Hi Erica,
回复删除Thank you so much for your in depth summary regarding this article. I never realized how crucial this word 'Just' is to use in math classes. I appreciate your thinking regarding exclusive and inclusive verbs to use in classes. I would like to share my feelings as a student when I usually saw exclusive verbs like 'to calculate', 'to find', 'to solve' in the math questions when I was in grade 8, 9 and 10. I really felt so relaxed and straightforward, uncomplicated when I had to apply a direct formula to get an answer in any math problems. And I felt so complicated, complex, vague when I saw such inclusive words as 'describe', 'explain', 'prove' in my math questions at that time. I think there were many reasons behind it, one of those was, my math teaching was not appropriate to make me think that math could be creative, exploratory in nature; there was no place to do mistakes or sufficient time in my class to think math in a versatile sense. I really intrigued to the idea to involve the inclusive verbs in our math classes with sufficient time to think and to do work over it.
A very interesting discussion! The exclusive and inclusive verbs identified by Rotman are especially interesting to me -- and also Sukhwinder's associations with them. It's interesting that the exclusive verbs can sometimes seem calming because they are so individual and straightforward, and sometimes seem threatening because they are 'commands'. I think there is space for both straightforward, procedural moments and (lots more) exploratory, communicative moments in our math classes, and as teachers, we can make decisions about the mood and needs of the class to judge where to go with things.
回复删除Jianying, very interesting to hear about the insights you got from having fellow teachers visit and observe your classes! I've had this same experience, and it can be really enlightening, especially if you and your observers understand teaching and learning in similar ways and have a good collegial relationship.
PS: Madison phoned to let me know that she will be catching up her post and responses this week -- she had a major car breakdown on the weekend that took up way too much of her time, and she apologizes for the delay!
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