EdCamp Questions on Math Teaching and Learning

I’m heading to EdCamp Houston this morning (my first EdCamp!) and I’m pondering questions I’d like to dig into.

My two major interests (today) are: Debunking the Right Answer Myth and Teaching Algebra with Arithmetic.

Here are my initial questions for each:

Debunking the Right Answer Myth

(and it’s cousin, the Failure is Bad Myth)

What makes people think that all math problems have a right answer? Can we change this?
How do we inspire teachers to see (and promote) that math problems don’t have only one right answer? (And some have no answers or it’s impossible to find any.)
How do we convince students to embrace failure?
How do we get teachers to encourage failure?
Do teachers and students really know the value of failure?

Teaching Algebra with Arithmetic

Can you teach algebra with arithmetic?
Where do you see the properties of math in everyday activities? (Like the distributive, commutative and associative properties)
What’s up with zero and one, anyhow? What are they good for, how do we use them, and why?
Why is algebraic manipulation so hard for students to understand?

What do you think?

Are there other questions in these areas I should consider? What am I missing?

Leave your thoughts in the comments, please. And share with your PLN!

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I remember my group theory class (abstract algebra) from university covered 1 and 0 in a round-about way. The reals are a group under addition with the identity 0 (meaning 0 doesn’t change a number under addition and that it is the “answer” to a number plus it’s inverse). Groups have to have an identity and all elements not the identity have to have an inverse. So addition works fine. Multiplication is the problem because the reals are NOT a group under multiplication unless you remove 0. The identity is 1 and the inverse of any number a, is 1/a. Conventions had to figured out for what to do when multiplying by zero and when looking at the inverse of zero (1/0). Making 0 times anything 0 matches the additive definition of multiplication and that just leaves dividing by zero, which doesn’t really match up with anything.

It would be interesting if someone tried to make some math where you could divide by zero in a way similar to how we have i (and complex numbers) for when you end up with the square root of a negative number, but that’s a big digression and may not get anyone anywhere but confused.

The first part was kinda technical (and it’s been years since I studied it so maybe rusty too), but it kind of highlights why and when 0 and 1 are special numbers (it depends on the operation you are using). And that they are connected to the idea of inverses (which is how you “get” to the identity with the operation). What they do is kind of make the set of numbers cycle through and connect to each other (be a closed set under the operation).

Perhaps an example: Adding three to a number is pretty easy to do and we can also take three away (this is the inverse, we are adding (-3)). So what are we doing if we both added three and took three away? Without zero to connect these things we have a big problem.

3 Responses to EdCamp Questions on Math Teaching and Learning

Matt says:

April 6, 2013 at 11:43 am

I remember my group theory class (abstract algebra) from university covered 1 and 0 in a round-about way. The reals are a group under addition with the identity 0 (meaning 0 doesn’t change a number under addition and that it is the “answer” to a number plus it’s inverse). Groups have to have an identity and all elements not the identity have to have an inverse. So addition works fine. Multiplication is the problem because the reals are NOT a group under multiplication unless you remove 0. The identity is 1 and the inverse of any number a, is 1/a. Conventions had to figured out for what to do when multiplying by zero and when looking at the inverse of zero (1/0). Making 0 times anything 0 matches the additive definition of multiplication and that just leaves dividing by zero, which doesn’t really match up with anything.

It would be interesting if someone tried to make some math where you could divide by zero in a way similar to how we have i (and complex numbers) for when you end up with the square root of a negative number, but that’s a big digression and may not get anyone anywhere but confused.

The first part was kinda technical (and it’s been years since I studied it so maybe rusty too), but it kind of highlights why and when 0 and 1 are special numbers (it depends on the operation you are using). And that they are connected to the idea of inverses (which is how you “get” to the identity with the operation). What they do is kind of make the set of numbers cycle through and connect to each other (be a closed set under the operation).

Perhaps an example: Adding three to a number is pretty easy to do and we can also take three away (this is the inverse, we are adding (-3)). So what are we doing if we both added three and took three away? Without zero to connect these things we have a big problem.

David Wees says:

April 6, 2013 at 12:05 pm

Those sound like pretty good questions. I hope you find some people to discuss them with!

Trever Reeh says:

April 7, 2013 at 12:00 pm

Those are great questions to start out with. Have fun at EdCamp you will have a blast.