Making Progress, Arithmetically
Today is 11/12/13, a rather pleasant-sounding date because the numbers form an arithmetic sequence, albeit a trivial one. It’s not the only date in 2013 for which the month, date, and year form an arithmetic sequence. How many others are there?
Several nights ago, my sons asked if they could do bedtime math, but Eli asked if we could do problems other than those on the Bedtime Math website, because “they’re a little too easy.” So instead, I navigated to the MathCounts website and opened the 2013-14 MathCounts School Handbook. We scrolled to page 9 and attacked the problems in Warm-Up 1.
Things were going well until we reached Problem 8 in the set, which read:
The angles of a triangle form an arithmetic progression, and the smallest angle is 42°. What is the degree measure of the largest angle of the triangle?
Eli asked, “Daddy, what’s an arithmetic progression?” pronouncing arithmetic as “uh-rith-ma-tick” instead of “air-ith-met-ick.”
I could have just answered Eli’s question by stating the definition:
An arithmetic progression is a sequence of numbers for which there is a common difference between terms.
But such a definition isn’t very helpful, since I’m not sure that either Eli or Alex know what sequence, common difference, or term mean. It would have led to even more questions.
Plus, I’ve always believed that kids understand (and retain) more when they discover things on their own. Call it “discovery learning” or “inquiry-based instruction” or any of myriad other names from educational jargon, it just means that giving kids the answer is not the most effective way for them to learn.
So instead, I said, “Let me give you some examples.” And then I wrote:
1, 2, 3
3, 5, 7
Alex said, “Oh, I get it! An arithmetic progression is a nice pattern of numbers.”
So I said, “Well, let me give you some patterns that aren’t arithmetic progressions.” And then I wrote:
2, 4, 8
“That’s a nice pattern, isn’t it?” I asked. “But it’s not an arithmetic progression.”
“Oh,” said Alex. He thought for a second, then revised. “You have to add the same amount every time.”
And there you have it. Three examples, and my sons were able to define arithmetic progression. It’s not as sophisticated as “a common difference between terms,” but “add the same amount every time” is a sufficient definition for a six-year-old.
So they generated an arithmetic progression with 42 as the smallest term:
42, 45, 48
Eli said, “I don’t fink vat’s enough.” When asked to explain, he said he thought that the angles in a triangle add up to 180 degrees.
“Are you sure?” I asked. He wasn’t. Nor was Alex. So I asked if they could convince themselves that the sum of the angles is 180°.
Alex said, “Well, the angles in a square add up to 360°, and you could cut it in half.” So we did:
They then reasoned that each triangle would have a sum of 180°. “But maybe that only works for a square,” I said. “How do you know it’ll work for other shapes?”
Eli suggested that we could cut a rectangle in half, too:
Understand, this is NOT a proof of the triangle sum formula. When they get to high school and need to demonstrate the rigor that the Common Core State Standards are demanding, well, then we’ll worry about formal proof. But for now, I’m okay with six-year-olds who can demonstrate that kind of reasoning.
They then took another guess, but this time they chose three numbers that added to 180:
42, 59, 79
Realizing that the difference between the first and second terms was 17 and the difference between the second and third terms was 20, they revised:
42, 60, 78
They concluded that the largest angle had a measure of 78°. And all was right with the world.
So why am I telling you all this?
Partially, it’s because I’m a proud father.
But more importantly, it’s because this vignette demonstrates that teaching is an art, and successful teaching doesn’t happen by accident. It’s not easy, as many people believe. What’s easy is the perpetuation of bad teaching, a la Charlie Brown’s teacher, or textbooks that simply present information with the belief that students will absorb it by osmosis. Good teaching, however, requires content knowledge and pedagogical knowledge, and it demands teachers who can handle unexpected classroom twists and turns and have the ability to adjust on the fly.
A student is convinced that a right triangle isn’t a right triangle because the right angle isn’t in the lower left corner? You better find an effective way to clarify that misconception. (Hint: Don’t use a traditional textbook where every picture of a right triangle shows the right angle in the lower left corner.)
Students think that 16/64 = 1/4 because you can “cancel the 6’s”? Uh-oh. Better find some counterexamples pronto, and help them understand why 16/64 can be reduced to 1/4.
Your students don’t know the definition of arithmetic progression? Then you better figure out a way to help them define it, and just writing your definition on the chalkboard isn’t gonna cut it.
Someday soon, I hope to add my project at Discovery Education to the list of examples of good teaching. Until then, I’ll just keep blathering about my sons.