Understanding Mathematics vs. "Doing Mathematics"

This morning, as I try to get back into a routine ahead of Monday's return to work from March Break, I started reading Kathy Richardson's How Children Learn Number Concepts - A Guide to the Critical Learning Phases.

This was a book that was given to me earlier this year after my role was expanded from working with grade 7-12 mathematics teachers to a full K-12 mathematics co-ordinator role. With only a background in intermediate & senior math, I've learned so much from primary and junior math teachers this year, and I'm eager to learn more about how students acquire concepts of number, relation, and computation.

In her introduction, Richards describes what she calls Critical Learning Phases - "crucial mathematical ideas that students must understand if they are to find meaning in the mathematics they are expected to learn." These crucial ideas are "insights, rather than facts or procedures," meant to be carried forward as students engage in higher levels of mathematics. 

What are the big ideas of mathematics we want students to be able to understand, as opposed to just chugging through a calculation and ending up with a correct answer?

I am still early in the book (and learning about how we learn to count... something I definitely take for granted!), but the reading has got me thinking about what we teach in high school math.

What are the mathematics in grade 9 we want students to be able to understand?
How is this different than the mathematics we want students in grade 9 to be able to do?
And how does this distinction change my teaching practice?

A change in perspective?

For instance, one of the skills we teach in grade 9 math is how to find the rate of change from a linear graph. Locate two points on the line, find the change in the "rise" and the change in the "run," and use those differences to calculate the rate of change. That is the mathematics we want students to be able to do.

But what is the big idea behind that skill, the crucial understanding? That certain relationships have a constant rate of change? What a rate of change is? That by using ratios and proportions we can make predictions when the rate of change is constant?

A variety of questions (worksheet?) might be a good way to practice determining the rate of change given two points on a line, but that's not the way I would choose if I wanted students to walk away with a deeper understanding of what a rate of change even is.

In the classroom, too often, I got sucked into just teaching the skills I felt students needed to know in order to succeed in the course (i.e. do well on the tests and exam). 

Perhaps I should have spent more time on figuring out what the crucial ideas were in the curriculum - they are not immediately obvious to me! - and tailoring the learning to develop those deep understandings. I wonder about the permanence of such ideas, rather than the impermanence of memorized procedures.

NB - I am not against practicing computations for getting the correct answer... many, MANY feats of engineering around us, as well as getting the recipe for the cookies I baked this afternoon right, depend on performing calculations and getting the correct answer! But perhaps it should not be the sole focus?


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