**variable**- we can

*work with*quantities, and learn things about them and draw conclusions from them, even when we don't know what they are.

**inverse operations**- operations can be undone (and sometimes they can't, at least not uniquely) and this is useful for solving all kinds of problems.

**functions**- Different kinds of rules that map a set of numbers to another set of numbers follow certain patterns.

**transformations**- rules can be changed in systematic and useful ways.

**equivalence**- How do we know when things are the same? How do we know when they are not the same?

**proof**- Usually a big feature of a geometry class, but I'd argue at least as important in algebra. How do we know for sure that something is true?

I've been following this discussion too and my super clumsy input on a pretty big idea for algebra is: "how can we represent numerical relationships symbolically in a way such that the notation helps us express and derive general rules for the behavior of number?"

ReplyDeleteSuper wordy, I know. But I think that algebra is just a symbolic system imposed on number and representing numbers using the notation of algebra allows us to make simple, powerful and beautiful leaps of logic.

It's all just about representing a number with a letter, then letting the value attached to that letter rove and observing what happens.

Gah... I don't know how to write it out nicely, but it's about the symbols and how powerful they can be for generalizing knowledge.

Kate

ReplyDeleteFrom this list I'd say that the idea of inverses is the big idea that usually gets shortchanged. This came up recently when Sam Shah posted about inverse trig functions. I think that dragging inverses into the light and pointing out that so many of our curricular habits in the math sequence rely on the process of learning how to do something and then learning how to undo that process would help our students make sense of what we are asking them to do. Especially by the time they get to Algebra II and are struggling with radicals and logarithms.

Quantity and equivalence are "huge" ideas for students to develop in preparation for upper level math. In fact, combine those with number sense components and I believe you'll nurture strong math students.

ReplyDeleteLike it. At first glance, would want to add some probability, statistical reasoning, something about rate of change, something about structure, proof, and modeling. I am very interested in such a list--without getting too detailed. I wonder how some modern mathematics needs to be valued, like apportionment, scheduling, bin packing, graph theory, ...

ReplyDelete