“The ideas come first, and then the words.” This popped out of a recent e-mail from a frequent reader – and former colleague at MSU. The discussion had started elsewhere, but quickly led into how math and science (and others) are taught TO and how they are learned BY our students.
The statement was attributed to an unnamed science educator who, according to my correspondent, advocated ‘active interaction with phenomena.’ The statement really resonated with me, as I have often felt the same thing, and preached it to future, as well as in-service, teachers. I’d always say ‘concepts before definitions’, but I think I like the ideas-first phrasing better.
I’d much prefer having a 5th or 6th grader walking around talking about ‘the distance around a circle’ [or ‘the top number on a fraction’], rather than attaching fancy labels to these notions before they’re ready. Even then, I’d be inclined to wink and say to students, “you know, high schoolers call that ‘circumference’ – if you want to feel smart, you may use that, too.”
Speaking of the distance around a circle – circumference, if you prefer – here’s one of my favorite classroom activities that illustrates these discussions.
Gather a collection of circular objects of all sizes. Things like a ring, a jar lid, a pizza pan, a hula hoop, etc. The more the better. Ask the various groups to choose (at least) five of these objects and carefully measure the distance around and the distance across each object and compare them. A string works well for gaining more accuracy here. Here’s where the fun starts.
It is an absolute joy to watch the astonishment and curiosity begin to form. “Omigosh! The distance around is about 3 times the distance across each time, no matter the size! Is that always true? Did you guys notice that?”
Further discussion (and perhaps more accurate measurements) usually results in discovering and conjecturing that ‘the distance around a circle’ is always a ‘little more’ than 3 times ‘the distance across the circle’. Maybe around 3.1 or so.
At this point, one can go several directions. My former colleague, in classes with future science teachers, would go on to discuss the measurement discrepancies and lead into the concept of ‘experimental error’, and – more importantly – how that isn’t the same thing as ‘mistake’. I love that.
In a class of upper elementary students, another direction is to confirm that this relationship (ratio) is, in fact, ALWAYS true! (How cool is that?!) Indeed, since it’s constant, we’ve given it a name (rather than having to forever repeat that long phrase). Specific reasons aside, math types long ago named that constant ratio with the 16th letter of the Greek alphabet. Or PI, as we know it.
So now you know: Yes, PI is roughly 3.14 or 22/7 (a little over 3), as most of us can instantly spout. But it’s more. It’s actually a ratio of the circumference to the diameter of any circle (which happens to be a constant close to 3). And you discovered it yourself! Perhaps you’ll never look at pi the same way again.
But, back to the beginning. Maybe this column has been too ‘mathy’ for you. But it seems that this idea of ‘interaction with phenomena’ or ‘ideas before words’ is so much more crucial and effective in math/science teaching. If needed, you can probably make your own transfers to other fields.
Perhaps you’ll wish to go enjoy a cup of coffee and a piece of ‘pi’ to rejoice in your new discoveries.
