Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the category “Flashback Friday”

Flashback Friday: “Sharing” in Sums

When my own kids were VERY young, we used to play “finger math.” Hold up some fingers on each hand and ask “How much is 2 and 3?” Of course, in time it changed, bit by bit so that “2 and 3” became “2 + 3,” holding up the fingers became their job, and then there were no fingers at all. The specific memory involves asking my daughter “How much is 3 and 5?” Right away she said “Eight!” Since we had been playing for a little while, I teased her, saying “No, it can’t be eight. You told me 4 and 4 was eight!” She replies with “Look, Mama, you give one from the five over to the three and then it’s four and four!”

Why don’t students automatically think that way more often?

I know, some do, but it seems that addition is so often accomplished one of three ways:
-It is a “known fact” they are able to recall immediately.
-They “stack the numbers” (either on paper or mentally) and add using the algorithm.
-Ummm, I’m too lazy so I’ll use a calculator.

Now, occasionally students will use alternative mental strategies, but often only IF the numbers are “compatible” to begin with, such as 25 + 75. Why don’t we encourage them to MAKE the numbers “more compatible,” if they can? Who wouldn’t rather add 40 + 63 than 39 + 64? What about 50 + 75 instead of 48 + 77? How can we improve our students’ “computational FLEXibility” instead of just focusing on their “computational ABility?”

Enduring Mathematical Understanding doesn’t come from knowing how to apply an algorithm, it comes from looking for alternatives, strategies, and shortcuts that arise from a deeper sense of number!


Flashback Friday: “It’s Like the Diving Board”

Watching diving during the Olympics always brings back a memory from a number of years ago. I taught high school math in my previous district, and this particular story involves a young lady I was fortunate to have in class for three years during a stretch of time where we were phasing in Contemporary Mathematics in Context. She was in the original “pilot” group at the Junior High and then moved to the high school where I first implemented the next three years of the curriculum with that particular group while the rest of the school phased in each course a year later.

We had been working on analyzing the effects of a, b, and c on the graph of y = ax^2 + bx + c. Nothing as fancy as axis of symmetry, vertex, and specific roots, just “How does changing the value of a affect the graph?” “What about b and c?” She had been sick for a few days so she came in at lunch to spend some time working on what she had missed. We started with looking at y = ax^2 and changing the value of a – no problem. Next we investigated y = ax^2 + c, for different values of c, and again the effects were quickly realized. We also had a successful discussion about “why” changing those values affects the graph the way it does. Then it was time to look at b. In the past students usually “discovered” things like “If b is positive it actually moves the graph to the left (and down) but if b is negative it moves to the right (and down.)” “That’s if a is positive, but if a is negative it does the opposite.” Of course the whole “positive – left” and “negative – right” idea is something they have to “remember.” There’s not a lot to grab onto until you start looking at things more algebraically.

With this student we started by looking at a few graphs where b was not zero, and then she said, quite unexpectedly, “Oh, it’s like the diving board.” “Huh?” “It’s like the diving board problems.” It takes me a bit, but then I see the light!

One of the main contexts in which quadratics were first introduced was using diving platform/diving board scenario. Initially it is a 10m platform. Students are told that, due to gravity, any object will fall such that d = 4.9t^2 where t is time in seconds and d is distance in meters. Using that (and the assumption that divers are basically “falling” with negligible effect from the “jump”) students develop the rule h = 10 – 4.9t^2. The graph and table are analyzed, other “free fall” situations are discussed and then (finally the relevant part) it returns to diving. First students are asked to explore what would happen if there was no gravity, and a diver leaves a 3 meter board with a velocity of 5 meters per second. Of course, it is humorous to imagine a diver continuing to fly though the air at a speed of 5 meters per second. Students are well-versed in linear modeling and come up with h = 3 + 5t, and similar rules for different initial height and velocity situations. Then, we come back to reality where gravity DOES exist, and work towards h = 3 + 5t – 4.9t^2. This student went back and grabbed the connection that b is the “speed the diver is traveling when they leave the board,” which translates to the slope or rate of change of the graph right as it crosses the y-axis!

Since that day I have had a new perspective on the effect of b on the graph of y = ax^2 + bx + c. (Maybe it is more useful to write it as y = c + bx + ax^2.) The graph crosses the y-axis at c. If b is positive, it is increasing as it crosses, but if b is negative it is decreasing. It a is positive, it begins to curve so that the “U shape” is right side up, but if a is negative, it begins to curve so that it is upside down. Based on the magnitude of b, you can draw the steepness of the tangent line at the y-intercept (because, of course, that’s what it is!) Based on the magnitude of a, you can draw a “skinny” or “wide” parabola. Try it! No quadratic formula, no factoring, no axis of symmetry – no “math” – and you can sketch a fairly accurate graph. Sure, you won’t have the exact x-intercepts or vertex, but all of the features of the graph are based on understanding and reasoning, not just “remembering.” How had I never seen this before?

Why do students resist mathematics in “story” situations? When concepts are learned within a context they are more likely to develop Enduring Mathematical Understanding!

Flashback Friday: Half and Double

When my son was about six years old we would “play math” when we were in the car, or getting dinner ready, or just “playing.” I remember asking him what 7 x 8 was, and hearing a reply that went something like this: “Well, 7 x 8 is the same as 14 x 4, and 14 x 4 is the same as 28 x 2, so. . . 56!” It was not quite what I expected, but then I recalled a brief conversation we had had some time earlier involving the number 12.

Son: “3 x 4 is 12, and so is 6 x 2.”
Me: “Yes. . .?”
Son: “Does that always work?”
Me: “What?”
Son: “That if you double 3 you get 6, and half of 4 is 2.”
Me: “Well, what do you think?”

I could see the gears turning as he thought about it in his mind – picturing the four groups of three changing into the two groups of six. I don’t recall the resolution of the discussion, but apparently it was enough to satisfy him so that it became a strategy that he began to use on a regular basis. I am certain that I did not encourage it as a “method,” just an interesting occurrence.

A few weeks after the “7 x 8” question, I asked it again, expecting that he might remember “56” from his previous “derivation.” Here’s what I got: “Well, 7 x 8 is the same as 14 x 4, and 14 x 4 is the same as 28 x 2, so. . . 56!” He showed exactly the same enthusiasm for “solving the puzzle” that he had a few weeks earlier, not recollecting that he had, in fact, solved it before.

Contrast that with the child who has memorized 7 x 8 = 56, and would be able to repeat it back in the blink of an eye. Who has more Enduring Mathematical Understanding?

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