From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: My weak student can beat your weak student!
Date: 24 Jul 2000 18:18:44 GMT

A student just left from an hour of private office help. Since I
really oughn't go out for a drink at mid-day, I hope venting some steam
in public will have a purgative effect instead. This is for real.


[Student has come in with the weak background that leaves calculus
instructors scratching their heads. We talk about computing derivatives
as limits of difference quotients -- student had flubbed the derivative of
3x+(4/x) by virtue of failing miserably to subtract the necessary fractions
using common denominators. We discuss the need for algebra skills, then 
move on. That, perhaps, was my fatal error...]


Student: "I'm having some real problems in this class [Calculus 1]. 
Can you show me how to do problems like this in case we have some on
the next test?" [Student points to a problem of the form, "Sketch a
function which satisfies f'(x)>0 for x in [0,1], etc."]

Me [somewhat puzzled, since there's nothing to 'do' on those problems]: 
"OK, let's try one to see if you were with me in class today. Sketch a
function which has f' positive everywhere but f''(x) > 0 for x < 2 and
f''(x) < 0 for x > 2."

S: "So I need  f' to be increasing on the first part."
[Sketches something like y=log(x) ] "Like this?"

M: "You were right -- you needed  f'  to be increasing. Now, f' measures 
the slope of the tangent line, right? So you mean to say the slope of the
tangent line is getting greater as you move from left to right?"

[S is silent, appears to have thought so.]

M: "What would you say the slope is here?" [Points to left edge of the graph,
slope is around  4. Hard to tell -- this is freehand, no grid.]

S: "Around 1?"

M [puzzled]: "Hm, well the picture's a little unclear but OK. And here?"
[Points to rightmost point, where the slope is really a little less than 1.]

S: "Maybe 4?"

M [pause]: "You're saying the slope at this second point is four times as
great as the slope at the first point?" [Effort to mask incredulity is
probably wasted.] "What does 'slope' mean?"

S: "See, I can't describe it so well. I know the formula..."

M: "If you understand it, you should be able to describe it in half a dozen
words, tops.  Look, let's do this accurately" [Produce graph paper with 
half-inch grid, recreate general shape, mark two interior points on graph 
near left and right edges.] "OK, here are the two points on the graph. 
Show me the tangent lines" [Provides a ruler.]
	[Student draws the line segments well.] "OK, now what's the slope
of this first line?" [The points (1,2) and (3,6) are conveniently near the
endpoints of student's line segment.]

S: "I'd have to write it down and I..."

M: "Sure go ahead. Write anything you need to."

S: [Carefully writes, correctly,  "(y1-y0)/(x1-x0)=". Counts coordinates
1, 2, 3, 4, 5, 6;  1, 2;  1, 2, 3;  1. Writes "(6-2)/(3-1)=4/2=2".]
"The slope is 2".

M: "Right. But you made it much too hard for yourself." [Thinking ahead to
the other point, whose coordinates are around  (10,18)...] "See, all you
needed to do is to count the _difference_ between the y-coordinates and
then between the x-coordinates" [Draws the little triangle]. "Most students
just remember 'rise over run'. See, that's what the  '6-2' measures is
the _difference_ between the y-coordinates -- what we usually just call
the 'rise'." [Small lecture followed.] "So now what's the slope of the
second line?"

S: [Draws the small triangle this time. Coordinates are not lattice points.]
"Well, the rise is..." [counts off] "It's more than two; could it be
two and a half?"

M: [Surprised that this could be a question, though probably 2+1/3 is closer.]
"Yes, sure, that's close enough. And the run is...?"

S: [Counts carefully.] "3" [Looks for confirmation] "So the slope is
two-and-a-half over 3".

M: [Sensing that we have yet to hit bottom] "Right. Two-and-a-half thirds.
As a simple fraction that would be..."

S: [Unsure] "You mean this?" [Writes "{2.5}\over{3}"]

M: "Yes, but you can write that as a simple fraction, you know, a ratio of
two whole numbers. Here, you've written that numerator as a decimal, which
is fine but you can write it as a fraction, too. What's two and a half as
a fraction?"

S: [Writes  "2 {1\over 2}"] "You mean like this?"

M: "Well, that's a mixed number. You can write it as a simple fraction.
What is it -- eleven ninths? seven fourths? What is two and a half as a
fraction?"

S: [Draws parentheses around the "2" and the "1/2"] "One?"

M: [Losing patience now] "No! If I give you two and a half bucks, have I
just given you one dollar? What's two and a half? It's two AND a half.
That means two PLUS a half"

S: "Oh so it's back to common denominators". [Starts to fumble with halves.
Perhaps recognizing this is supposed to be the kind of thing one should
be able to do mentally, announces:] "Three halves."

M: [Barely resisting the impulse to be insulting, reaches for some coins.]
"Look, I haven't got any half-dollars, so pretend these other coins are all
half-dollars. [Throw in pairs] Here's one dollar, here's another, here's
an extra half-dollar. I've just given you two and a half dollars, and it's
what? Five half-dollars. That's five halves of a dollar. Two and a half
is five halves." [Small diatribe about the need for students to actually
_learn_ the material they deal with in math classes. Small concession
thrown in for student's emotional well-being, recognizing that the student
probably got shafted by lousy teachers early on. I can't change that now...]

M: "OK, great. So this numerator is five halves. Now that slope is
five halves over 3. Now that's a compound fraction. Do you remember how
to simplify those?"

S: [tired and embarassed now] "No."

M: "Here, let me remind you". [Big fraction bars used to show the
invert-and-multiply routine.] "So you get five-sixths for this slope.
Now, is that bigger than the slope we had over here? Is it even bigger
than one?"

S: "I don't know."

M: [Well-practiced skills of teachers' patience clearly wearing thin]
"What is five sixths? Can you describe what that is in some other way?
What is that number?"

S: "Is it, um, negative..."



I didn't let the student finish describing the number.

Meet you in the bar in five minutes.