Last week in Lausanne, I was watching a talk that started with some standard material on finite groups of Lie type. There is this typical construction where you start with a connected reductive group over the field with elements (where is a power of a prime ) and the Frobenius endomorphism . The fixed points form a finite reductive group and this construction creates a link between algebraic groups and algebraic geometry.
Presumably, most of my readers have no idea what these things are, but don’t worry, they aren’t the point. Here is my problem: to understand algebraic groups, I need to understand enough algebraic geometry to utilize the above construction, and so I have some motivation to learn algebraic geometry. On the other hand, to motivate certain necessary things in algebraic geometry, it would be awfully nice to understand some of the theory of algebraic groups. Can you see my problem? I want to learn both and to use the knowledge of one to motivate the other. But I know neither!
Shawn discussed the use of analogies in teaching yesterday and I whole heartedly agree with him. What do you do, however, when you are trying to motivate a subject by comparing it to a subject the students will only understand once they comprehend the subject that you’re trying to teach them?
There is an easy solution to this problem: use a different analogy!
Sure. But I’m teaching precalculus and I’m pretty sure that the material is best motivated by calculus, the precise subject I’m supposed to be preparing the students for. So, I’m thinking right now that we should abolish precalculus and make calculus a two-year course. Come to think of it, let’s make calculus an -year course (for ) and start showing our students calculus when they are 7 or 8. Don Cohen does it! At least then we could introduce most of the mathematics we teach in context instead of pulling out of the void definitions and results whose purpose is to build up yet more definitions and results, most of which the students consider to be as enjoyable as a rattlesnake bite, but without the fun of asking your friends to suck out the poison. Yes, this is the way to do it.
Ah, but I don’t get to teach it like that. I have to teach precalculus and I only get one semester (six weeks, actually, as it is summer school). Fine.
Stream of consciousness (with apologies to James Joyce)
Maybe all is not lost what if I just start teaching it like it is calculus what if I start with the problem of describing the average velocity of an object and then build up the formulas and interpretation and then work towards instantaneous velocity then I could talk about how this motivates the need to understand lines really well or at least the point-slope form of a line I then need to transition to giving lots of examples of functions that occur naturally maybe even getting the students to provide me with examples I think I definitely need to get hyperbolic functions in they always seem to get the shaft but I love those cool things and the way they connect to exponentials is really awesome did you ever notice that awesome might mean inspiring some awe so that if you wanted something to be full of awe it should be awful hah I like that I’ll have to use that when some student does really well okay back to what I was doing so yeah hyperbolic functions they are connected to exponentials and so are trig functions I wonder if I can convince them of that and if they will believe that the exponential function is pretty important I don’t think so it is all a bit too much catnip for mathematicians we need to get back to throwing balls into garbage cans and other things that suck students in and get them invested in the answer I still like the structure here though start with the problem to be determined later and add layers that require the standards but get the students to request the layers and the standards for the second part of the course on sequences and series I’ll introduce the area problem and use that at the end of the course could it be that these students will be ready for calculus that is probably too much to ask but I can dream right
The Mind Map
I don’t understand how the mind map thing really works; I should probably read one of the books on the subject. Here, anyway, is what I did in about 20 minutes. It feels like a mess and doesn’t seem to capture what I’m trying to do.
In fact, the problem is that there is just way too much in there. For instance. when covering linear equations, do I really need to do more than slope and point-slope form at the beginning? Sure all the rest is useful, but only when you get to a specific problem. And why do we need all this garbage about quadratics? Other than root finding, we certainly won’t use it in calculus. On the other side of the fence, I think introducing the binomial theorem might be a waste of time. When I taught calculus, I used it just once: to prove the power rule for arbitrary natural number powers. Just a guess, but I don’t think anyone followed and only one student (hi, Ben!) probably remembers that we used it. The whole sum of n squares, cubes, etc. has to go. I better not play this game too long or there will be nothing left. It is just that I feel the need to be overly cautious: don’t include material that I find appealing; include material that is suited to the task ahead.
The Standards List
I’ve put it off long enough. Here is my first draft of a standards list. I’ve left off some things that are in the mind map because I’m trying suppress stuff they should have seen (like the distance formula) and which I’ll thus probably talk about but not assess (oh that already sounds dangerous to me). Reading this list a couple of times makes me feel rather nauseous. If you’d rather see a copy of the spreadsheet I made for this, click here.
- Compute the average rate of change of a function
- Interpret the average rate of change of a function
- Interpret the instantaneous rate of change of a function
- Describe the connection between the average rate of change and instantaneous rate of change of a function
- Slope of a line
- Point-slope form
- Function identification by graph: linear
- Function identification by graph: quadratic
- Function identification by graph: rational function
- Function identification by graph: exponential function
- Function identification by graph: logarithmic function
- Function identification by graph: trigonometric functions
- Sketch graph by type: linear
- Sketch graph by type: quadratic
- Sketch graph by type: exponential function
- Sketch graph by type: logarithmic
- Sketch graph by type: trigonometric
- Sketch piecewise defined functions
- Explain (intuitively) continuity
- Test continuity
- Find the effect on a function of a horizontal shift to the graph of the function
- Find the effect on a function of a vertical shift to the graph of the function
- Determine the equation of a circle given the radius and center
- Determine the equation of an ellipse given the center, vertex and co-vertex
- Describe radian measure in terms of the circumference of the unit circle
- Convert radians to degrees
- Convert degrees to radians
- Use the coordinates of the unit circle to find sine and cosine
- Use the unit circle to determine the sign of sine and cosine
- Use the unit circle to show that sin^2(x) + cos^2(x) = 1
- Use the unit circle to show sine is odd and cosine is even
- Compute sine and cosine for the 16 standard angles
- Convert rectangular coordinates to polar coordinates
- Convert polar coordinates to rectangular coordinates
- Use the law of sines
- Use the law of cosines
- Use sum formulas for sine and cosine
- Use product formulas for sine and cosine
- Use half-angle formulas for sine and cosine
- Solve equations with trig functions
- Write periodic solutions to equations with trig functions
- Graph sine, cosine and tangent (base functions)
- Graph horizontal and vertical shifts of trig functions
- Graph horizontal and vertical reflections of trig functions
- Graph horizontal and vertical stretches of trig functions
- Describe the effect of reflections and stretches on an arbitrary function
- Find asymptotes of a rational function
- Find the sign graph of a rational function
- Graph a rational function
- Distinguish between a sequence and a series
- Compute geometric series
- Describe a Riemann sum as a series (intuitive)
- Describe exponential growth in terms of exponential functions
- Describe the rate of change of a polynomial as x increases versus the rate of change of an exponential as x increases
- Define hyperbolic trig functions in terms of exponentials
- Graph hyperbolic trig functions
- Determine if a function has an inverse
- Compute inverse trig functions on standard values
- Compute logarithms
- Use logarithm properties to compute logarithms
- Perform arithmetic on complex numbers
- Explain connection between trig functions and exponential functions via de Moivre’s formula