2Why Can’t Students Do _____? Let’s look at the steps from elementary school through Calculus, to see where the problem comes from. We’ll follow one thread, which includes:Proportional ReasoningSlopesDerivativesCaveat: What we do here this morning is not sufficient, but it includes the right steps.
4Mr. Big & Mr. SmallUse the paper clips and buttons to measure Mr. Small.Give me the paper clipsUse the buttons to measure Mr. BigQuestion: How many paper clips tall is Mr. Big?How do you know?The most common high school answer is eight!Why?
5Proportional Reasoning Most students reason additively rather than proportionally unless they are explicitly told to do the latter.Students must learn how to reason proportionally AND when to do which type of reasoningthe way they learn to do it is to have experiences to guide them:Measure lots of stuffPredict other stuffWhat changes students is discourse, and NOTHING ELSE!
6Adding CableA 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?You have the tools to do this problem without me giving you any other numbers…so don’t ask me!
7My way C = circumference, d = diameter: Why didn’t you do it that way? (C + ΔC) = π (d + Δd)But, C = πd, so…ΔC = π Δd ≈ 100 feetWhy didn’t you do it that way?
8Change, Rate & SlopesMost students think of change as involving the difference between two points. They DO NOT think of change as an entity in itself.Then, we go to rates and slopes, and so they DO NOT ever get the “per unit” thing…This is where the graphing calculator explorations of y=mx+b come in handy:Adjust m and lookcreate tables from graphs(!)find slopes from graphsPlay green globs
9Another problemYou 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.)
10The Frog PuzzleProfessor 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.First trip to the pond55 frogs caught and bandedSecond trip to the pond72 frogs caught; of those 72 frogs, 12 were found to be banded.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.
12CalculusNow, going to dy/dx without any one (or more) or these* understandings is, quite literally, impossible:Proportional ReasoningChange as an entityPer unit*There’s really more, like thinking about f(x) as an entity rather than a collection of points, etc.
13How do we develop these understandings? Practice alone won’t do itInstead, we develop the understanding by using these types of problems this wayStart with something concrete (and manipulible) – e.g., measureForce students into a cognitive conflict (e.g., 8 buttons vs. 9) or other surprise (don’t need lots of numbers)Students must socially construct an understanding (e.g., justify to one another)Students must reflect on the situation – this is very important!Students must bridge the idea to something else