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Topic: Enrichment
Replies: 38   Last Post: Aug 24, 2012 1:33 PM

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Robert Hansen

Posts: 11,345
From: Florida
Registered: 6/22/09
Re: Enrichment
Posted: Aug 17, 2012 6:50 AM
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First off, what you describe is not "some calculus". Some calculus would be (for example) starting with the tangent/slope discussion, constructing a very simple notion of limit, deriving the difference quotient, using that to derive the power rule and then the student can go off and apply that to the polynomials they are familiar with in algebra. That would be "some calculus". "Showing" a student some calculus is not same as "teaching" a student some calculus. I have a strict requirement that teaching involves the student getting the material. I am not saying that you are a bad teacher if the student doesn't get the material, only that it must be perfectly clear that your goal is that the student get the material.

I don't think graphing a straight line or a parabola is an indicator of anything. A monkey could almost do these things. Surely you meant to say more than this? Calculus is not just the geometric notion of a tangent to a line or the area under a curve. Calculus is

This is what I would consider to be "teaching some calculus"...

1. I talk about tangents, using a circle at first because they should be familiar, and then more generic curves. My goal here is to ensure that they understand that a tangent intersects a curve at one and only one point. It is uniquely defined at each and every point. Using visual aids that "zoom" in, I want them to realize that if this line "tips" in either direction it will cross the curve again.

2. Can we determine, analytically, the slope of this tangent? So now we start building up the difference quotient argument. God was kind and gave us parabolas for this purpose. After we setup our difference quotient (using y=x^2 as the curve) the limit argument is pretty straight forward.

3. We look at our result. We plot the slope and see if it makes sense. We look at critical points. We solve for the maximum. We talk about a parabolic arch with a ladder leaning against it.

You can't get through step 2 without fluency in algebra and analytical geometry. You can't recognize the situation or follow the path of reasoning. The intuitive notion of a tangent to a curve (step 1) is not some calculus. Analytically determining that tangent (step 2) is some calculus.

You said "enrichment". To me the enrichment occurs during step 2. That is the "wow factor". When I look at what you wrote you seem to be saying that we can fake the "wow factor". That we can make the student think they are doing calculus (raise their self esteem) even though they are clearly not doing calculus (or math). And you further suppose that this will make them more interested in math, even though they still have yet to experience math.

Have you ever seen this strategy work? I have looked far and wide and I have never seen this work. I have seen classes that raise the self esteem of students by faking the "wow factor". That works because it is so simple to fool a child. But it is a dastardly act. Later, they will fail miserably when they are called upon to actually do math.

Let me ask you this. If your son or daughter was in this state regarding math and for whatever reason, you wanted them to succeed at math, would you ask them about the slope of a tangent to a curve or would you ask them "What is 1/2 + 1/4?" Me personally, I would start with "What is 1/2 + 1/4?"

Bob Hansen

On Aug 16, 2012, at 6:23 PM, Peter Duveen <> wrote:

> Bob, Chandry, this is what I am getting at.
> It seems to me that calculus is placed on a pedestal as something unachievable by the masses. But looking at it, I see a lot of material in the calculus that could be approached by students on a sixth or seventh grade level. If they can graph a straight line and a parabola, they are game for calculus, at least some calculus.
> The second point is teaching to the test versus enrichment. The question may be how can one jump start or help to kindle a person's intellectual curiosity. This will depend partly on the students interests, partly on his innate intelligence, and partly on the assets of the teacher/instructor/tutor, and the assets of the school (computers, laboratory equipment, etc.).
> Each case is different, and each teacher is different. There is a lot of trial and error involved. I doubt very much that, in spite of all the "research" being done, one will ever systematize a method for teaching that is 100 percent effective and efficient. There are too many variables involved.

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