Show Kids What’s “Under the Hood”, Before They Drive the Car-Unlocking the Mental Machinery of Mathematics
Unlock the mental machinery of mathematics.
My experience of school mathematics was just like many other peoples’. I remember drill and worksheets. I have an automatic recall of times tables, but no mental imagery associated with them- the number just appears. It’s not visual, just verbal- you say 7 x 8, and “56” pops up. I remember in later years doing “every odd numbered questions”. Hearing others talk about their mental imagery or strategies for dealing with multiplication, I was briefly jealous. What was I missing, when I just have a blank spot in my head where times tables appear?
I remember lots of days sitting through the “taking up of the homework”, which often took most of the class. Assign new work, do some worked examples, repeat, day in, day out. I will say this for that method of teaching: you had to be prepared, homework done, because you might be the one called on to put the solutions up on the board.
I did fine, because I could learn that way. I didn’t question, I just did. Day in, day out, I, to borrow, @MrSoClassroom’s phrase, just “did the math”. I wouldn’t particularly say I was an active “doer of mathematics”.
Looking back, it would have been nice if we got to discuss the ideas and concepts which I now know are incredibly interesting. I am mostly self-taught throughout my 30s, with regards to the big ideas of math, through relentless reading and Internet study.
There was one course though, which gave me glimmers and glimpses of the true power and beauty of mathematics. It was Calculus, grade 13 at the time, which finally showed me how powerful the machinery of mathematics was. I know for many, Calculus is the “wall” they crash into to, the limit of mathematics for them, or something they just get through. This is because calculus is often taught as a series of procedures-differentiate, integrate, repeat.
My teacher showed us “under the hood”- we had to develop the tools we were to use from first principles. On the tests we had to show how the formulae were developed. We had to develop them, and then use them. To give a grade 4 example, this would be like being given a test where kids have to show how the standard algorithm works, and then use it to multiply. That would be a great assessment. I would love to see teachers do that in grade 4. I often say kids should have to get an “algorithm” license before they use them. Learn how it works, show me, and you can use it ever after.
We don’t often open up and show kids what is “under the hood”, but we can. We have that power I think number talks in K-6 are a good example: if algorithms are the machinery, how do they work? What’s under the hood?
When we speak of teaching conceptually, that’s the only way my teacher knew how to do it. It’s how he understood calculus. After we worked with the concepts, we applied them to interesting problems. That same year, I had the opposite experience with chemistry. It was taught as a series of rules and procedures. I deployed formulae on the test in the right places, and got a good grade. I had not idea what a mole was, or why it mattered, and I failed university chemistry. Guess what my best grade in that first year was, though? Calculus.
So what’s the big idea behind calculus? I read once that the big idea of calculus could be seen as, “up close, even the curviest lines are straight”. It’s about rates of change, and is a powerful problem-solving tool if seen as such. We won’t all need to deal with differential equations, but rates of change do matter in all our lives.
I say, let’s show kids “what’s under the hood” more often. They won’t hit the “conceptual wall” so hard, if at all. For some, that wall is fractions. Let’s show them how fractions work, and why they matter. For some, the wall is algebra. Let’s spend lots of time on reasoning algebraically before we leap to the “language of x”. For some, it’s integers. Give kids interesting contexts like temperature, and work with number lines often.
Nobody is saying kids will develop rigorous proofs, or understand the mathematical world “all the way to the bottom” (Jordan Ellenberg’s phrase), but they understand the ideas of the K-12 curriculum much better.