**This PERC talk was from Anna Sfard about how we construct meaning socially**

How we talk about things, says Sfard, matters. How we talk about things changes what we see, and also what we do.

### How do we talk about math or physics? How do we talk about* learning *math and physics?

We need more than one way of talking.

For example, she showed a clip of a teacher discussing negative numbers with her class. She had students re-invent the rules for adding and subtracting signed numbers. SHe expected that students would have a hard time with “minus times minus” but instead the first problem was about “plus plus minus”. What is 2 times (-5)? One student gave the answer that 2 times -5 is -10 because “5 is the bigger number”. The class followed this student, citing that rule for the answer for several other problems. When the teacher presented her argument, the children were skeptical and openly disbelieved her.

What went wrong?

We can see learning as acquisitionist and participationist.

### If learning is change, what is is that changes when a person learns math/physics?

**Acquisitionist**. Piaget would say “one’s concepts/mental schemas change. One constructs and acquires those new constructs.” Learning depends on our cognitive makeup.

P**articipationist.** Vygotsky wuld say one’s participation in an activity, form or practice changes. The form of activity or way of performing a task changes. Learning depends on human agency.

Acquisitionist and participationist are not incompatible. Both are useful. You can choose which one to use depending on what you’re looking at.

What about school learning?

### School learning means a change in thinking. But what is *thinking?*

People develop skills by doing things together. These activities are communicated by communication. So “commognition=communication + cognition”. Learning, says, Sfard, is a form of communication.

### What is math or physics?

Math is a way of thinking. It’s a discourse. We communicate about math through keywords, symbols, descriptions, and processes (like proofs). But is communication all there is to math (or physics)? What is number? It’s an idea we made up to communicate something about the world. Apart from people, number (or force) doesn’t really exist. We can’t describe “force” without using the word force!?

Let’s go back to the example of the math class and negative numbers. That class created new rules together, though not the ones the teacher wanted them to arrive at. What they arrived at wasn’t consistent or contingent on other things that they knew (or would eventually know). The teacher has knowledge that the students didn’t have. She was an expert and needed to have an active role in that conversation. However, she was likely to use words in a different way, with different rules, than the students would. So, the class needed to agree about who gets the final say if people don’t agree. Invention isn’t enough! The teacher didn’t make it explicit how she goes about talking about ideas, and she renounced her leadership. So, the students invented their own rule, a rule that wasn’t that great, and the teacher’s leadership was difficult to regain because they hadn’t formed an agreement on the role of teachers and students.

{ 3 comments }

I once tutored a girl in first year High school Algebra on negative numbers and asked her why she thought a negative times a negative was a positive. She said the most brilliant thing: ‘It’s kind of like saying “I am not not going to do such and such”, which means I am going to do it.’

Great series of liveblogs – thanks for sharing these.

I like that story, Sean, but why would that then infer a rule for multiplication of negative times negative, rather than addition of negative and negative (which gives a positive)? It’s a useful mnemonic if you remember it’s for multiplication, but I can imagine some kids getting confused if they weren’t the ones to come up with it.

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