Slippery SlopesBarney Ricca

AMTRA 2012

Why Can’t Students Do _____?

0Let’s look at the steps from elementary school through Calculus, to see where the problem comes from. We’ll follow one thread, which includes:0 Proportional Reasoning0 Slopes0 Derivatives

0Caveat: What we do here this morning is not sufficient, but it includes the right steps.

Some pictures

Mr. Big & Mr. Small

0Use the paper clips and buttons to measure Mr. Small. 0Give me the paper clips0Use the buttons to measure Mr. Big0Question: How many paper clips tall is Mr. Big?

0 How do you know?0 The most common high school answer is eight!

0Why?

Proportional Reasoning0Most students reason additively rather than

proportionally unless they are explicitly told to do the latter.0 Students must learn how to reason proportionally AND

when to do which type of reasoning0 the way they learn to do it is to have experiences to

guide them:0 Measure lots of stuff0 Predict other stuff0 What changes students is discourse, and NOTHING

ELSE!

Adding Cable

0A cable is just long enough to go around the equator of the earth. (Assume the earth’s equator is a circle.) I want people to be able to walk under the cable, so I want to raise the cable to stand everywhere 15 feet above the earth’s surface. How much more cable is needed?

0You have the tools to do this problem without me giving you any other numbers…so don’t ask me!

My way

0C = circumference, d = diameter:0 (C + ΔC) = π (d + Δd)0 But, C = πd, so…0ΔC = π Δd ≈ 100 feet

0Why didn’t you do it that way?

Change, Rate & Slopes0Most students think of change as involving the

difference between two points. They DO NOT think of change as an entity in itself.

0Then, we go to rates and slopes, and so they DO NOT ever get the “per unit” thing…0 This is where the graphing calculator explorations of

y=mx+b come in handy:0Adjust m and look0create tables from graphs(!)0 find slopes from graphs0Play green globs

Another problem

0You have a block of wood with a total mass of 540 kg. This type of wood has 0.85 g in each cubic centimeter. Suppose you were to add 38 g of wood to the block. By how much would you increase the volume of the entire block? Do not substitute into a formula; explain the relevant arithmetical reasoning in your own words. (Hint: Be sure to think about change in volume rather than entire volume.)

The Frog Puzzle0 Professor Thistlebush, an ecologist. conducted an experiment to determine the

number of frogs that live in a pond near the field station. Since he could not catch all of the frogs he caught as many as he could, put a white band around their left hind legs, and then put them back in the pond. A week later he returned to the pond and again caught as many frogs as he could. Here are the Professor's data.

0 First trip to the pond0 55 frogs caught and banded

0 Second trip to the pond0 72 frogs caught; of those 72 frogs, 12 were found to be banded.

0 The Professor assumed that the banded frogs had mixed thoroughly with the unbanded frogs, and from his data he was able to approximate the number of frogs that live in the pond. If you can compute this number, please do so. Explain in words how you calculated your results.

The Frog Puzzle – part 2

0Check out the video…0 http://fnoschese.wordpress.com/2012/03/02/disrupt-

this-my-challenge-to-silicon-valley/

Calculus

0Now, going to dy/dx without any one (or more) or these* understandings is, quite literally, impossible:0 Proportional Reasoning0 Change as an entity0 Per unit

*There’s really more, like thinking about f(x) as an entity rather than a collection of points, etc.

How do we develop these understandings?

0 Practice alone won’t do it0 Instead, we develop the understanding by using these

types of problems this way0 Start with something concrete (and manipulible) – e.g.,

measure0 Force students into a cognitive conflict (e.g., 8 buttons vs. 9)

or other surprise (don’t need lots of numbers)0 Students must socially construct an understanding (e.g.,

justify to one another)0 Students must reflect on the situation – this is very important!0 Students must bridge the idea to something else