The other day my kid asks me to present math problems about area. Early middle school: simple multiplications and divisions. He got taught a formula for areas of trapezoids: A = 1/2h * (b1 + b2).
I decided to show him how to #unittest his #math solution, by giving him a different approach: A = a + 2b, where a = area of the square, and b = area of each triangle on the sides of that square.
He threw a fit and refused to accept my approach, because it wasn’t the same as he had learned.
@aeveltstra
But they're not given a nor b, therefore there's no such formula as A=a+2b. They are only given h, b1, and b2, so rewriting your formula into the information they are actually given (the first line of working out is writing the formula, then substituting the given values into it) it would actually be A=hb1+h(b2-b1)/2=h(b1+(b2-b1)/2), so it's either memorise that formula or A=h(b1+b2)/2. Which would you choose to memorise?
@SmartmanApps I recognize learning materials are malleable. The choice you present assumes they aren’t. Maybe you had hoped readers wouldn’t recognize that? I would teach both your simplest formula, which itself is a derivative, and the formula I presented. The fact that your learning materials don’t include that is caused by the prior choice to teach your simplest formula. It shouldn’t be used in turn to teach only that formula: that is circular reasoning.
@aeveltstra
"my kid’s response proves" why we don't try and teach them 3 things at once.
"testing the pupil’s application of it to the problem" - the teacher does that when they mark it.
"The choice you present assumes they aren’t" - does nothing of the kind. Only does what every teacher does and teach 1 thing at a time.
"prior choice to teach your simplest formula" - we don't teach only the simplest formula, but we only teach 1 thing at a time. Composite shapes is a high school topic.
@SmartmanApps You yourself said teachers choose to teach the simplest formula. Do you wish you retract that statement?
I also am not trying to teach 3 things at once. We are in agreement on that.
The fact that something currently is a high-school topic, shouldn’t stop us from reviewing whether it SHOULD be. Breaking apart a complex shape into simpler ones and adding them should be a middle if not a primary school topic. As soon as you learn addition and multiplication.
@SmartmanApps You’re the one who claimed the trapezoid function is the simplest. You’re the one who claimed that’s why it’s being taught. I used that as a reference. Then you say it isn’t. So, you are contradicting yourself. Do better.
@SmartmanApps Ah. But I didn’t try to teach 3 things. The kid already knows the middle-school, grade 7 way of calculating the area of a trapezoid. What I’m teaching, is just 1 thing: how to check whether the calculation was performed correctly. Wouldn’t teachers want pupils who can double-check their work? Out here in my line of work, that ability is very valuable.
@SmartmanApps Yes, one thing at a time. And instead of teaching a formula to memorize, without any understanding of how it came to be, what should be taught is that understanding. This is as easy as teaching kids addition: here’s a shape, there’s a shape, oh, and look, a third shape that is the exact same size as the first. And each of those shapes have an area calculated by multiplication, which they already know. Build upon prior knowledge.
@aeveltstra My high school geometry teacher hated my alternate proofs and more than once I had to argue it out and demonstrate it to her to get full points on my work.
@gooba42 Teachers are as much constrained by their learning materials as the people they teach. They got taught by the same educational machine which probably has taught people in mostly the same way for decades, making change really hard. Any variety brought by pupils and from the outside (like me, a parent), is met with great resistance as a result, for allowing it means a teacher has to expend more labor than they deem acceptable.
Educators should avoid making belief that the approach they teach is the only possible one.
You cannot verify your approach by repeating it. That only tests whether it leads to the same outcome, but doesn’t verify. Instead, you need to find a different way to get to the same conclusion, and then compare.
@aeveltstra
"Educators should avoid making belief that the approach they teach is the only possible one" - we don't teach that. We do teach that this is the simplest formula.
"You cannot verify your approach by repeating it" - the approach has already been verified in the proof of the derivation of the formula, which is shown at the time we teach the formula.
@SmartmanApps a) Then some of you do a bad job conveying that to their pupils, as my kid’s response proves. Maybe you want to encourage each other to do better.
b) We aren’t unit-testing the formula itself - we are unit-testing the pupil’s application of it to the problem. If they can approach the same problem with a different formula that should result in the same outcome, they can compare outcomes and either affirm or reject their own work.
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