July 5, 2010

The Calculus Carrot

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 \mathbf{G} over the field with q elements (where q is a power of a prime p) and the Frobenius endomorphism F: \mathbf{G} \to \mathbf{G}. The fixed points G = \mathbf{G}^F 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 n-year course (for n \approx 10) 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.

  1. Compute the average rate of change of a function
  2. Interpret the average rate of change of a function
  3. Interpret the instantaneous rate of change of a function
  4. Describe the connection between the average rate of change and instantaneous rate of change of a function
  5. Slope of a line
  6. Point-slope form
  7. Function identification by graph: linear
  8. Function identification by graph: quadratic
  9. Function identification by graph: rational function
  10. Function identification by graph: exponential function
  11. Function identification by graph: logarithmic function
  12. Function identification by graph: trigonometric functions
  13. Sketch graph by type: linear
  14. Sketch graph by type: quadratic
  15. Sketch graph by type: exponential function
  16. Sketch graph by type: logarithmic
  17. Sketch graph by type: trigonometric
  18. Sketch piecewise defined functions
  19. Explain (intuitively) continuity
  20. Test continuity
  21. Find the effect on a function of a horizontal shift to the graph of the function
  22. Find the effect on a function of a vertical shift to the graph of the function
  23. Determine the equation of a circle given the radius and center
  24. Determine the equation of an ellipse given the center, vertex and co-vertex
  25. Describe radian measure in terms of the circumference of the unit circle
  26. Convert radians to degrees
  27. Convert degrees to radians
  28. Use the coordinates of the unit circle to find sine and cosine
  29. Use the unit circle to determine the sign of sine and cosine
  30. Use the unit circle to show that sin^2(x) + cos^2(x) = 1
  31. Use the unit circle to show sine is odd and cosine is even
  32. Compute sine and cosine for the 16 standard angles
  33. Convert rectangular coordinates to polar coordinates
  34. Convert polar coordinates to rectangular coordinates
  35. Use the law of sines
  36. Use the law of cosines
  37. Use sum formulas for sine and cosine
  38. Use product formulas for sine and cosine
  39. Use half-angle formulas for sine and cosine
  40. Solve equations with trig functions
  41. Write periodic solutions to equations with trig functions
  42. Graph sine, cosine and tangent (base functions)
  43. Graph horizontal and vertical shifts of trig functions
  44. Graph horizontal and vertical reflections of trig functions
  45. Graph horizontal and vertical stretches of trig functions
  46. Describe the effect of reflections and stretches on an arbitrary function
  47. Find asymptotes of a rational function
  48. Find the sign graph of a rational function
  49. Graph a rational function
  50. Distinguish between a sequence and a series
  51. Compute geometric series
  52. Describe a Riemann sum as a series (intuitive)
  53. Describe exponential growth in terms of exponential functions
  54. Describe the rate of change of a polynomial as x increases versus the rate of change of an exponential as x increases
  55. Define hyperbolic trig functions in terms of exponentials
  56. Graph hyperbolic trig functions
  57. Determine if a function has an inverse
  58. Compute inverse trig functions on standard values
  59. Compute logarithms
  60. Use logarithm properties to compute logarithms
  61. Perform arithmetic on complex numbers
  62. Explain connection between trig functions and exponential functions via de Moivre’s formula




  1. I’d skip the hyperbolic functions and do more stuff with summations, not just arithmetic and geometric series. Put the logarithms in a bit earlier, along with the exponentials.

    It is possible to start earlier with calculus. My son started when he was in 5th grade, with me helping him derive the formula for how high a ball bounces, based on the assumption that the speed out of the bounce was a constant times the the speed into the bounce (the “coefficient of restitution”). We took time lapse photos of balls bouncing, and he measured the heights of the bounces and fitted the formulas to it to estimate the coefficient for different balls. It didn’t win any prizes at the County Science Fair (I think the judges assumed more parental involvement than was the case, despite explicit mentor statements), but he learned a lot of math and physics.

    I didn’t tell him that what he was doing was calculus until after he’d done it, to avoid scaring him off. Afterwards, he was so excited that he bought himself “Calculus for Dummies”, which he read through several times.

    Comment by gasstationwithoutpumps — July 5, 2010 @ 4:05 pm | Reply

  2. I agree logarithms need to go much earlier. The topics at the end of the semester are often the ones the students least remember. And hyperbolic can certainly be cut.

    I’ve taught polar in Pre-Calculus but it could go if you’re tight.

    Comment by Jason Dyer — July 7, 2010 @ 7:12 pm | Reply

  3. Thanks for the advice (and the time it took you to read and come up with said advice). I’ve been teaching the course for two days now (5.5 hours of class time) and have already realized I had missed a few standards that I definitely want them to know (domain and range for instance). Let me explain my reasoning for putting things where I did, though, and then maybe you can help unmuddle my thinking.

    Historically, at Suffolk, the topic that gets the shaft in precalculus is trig. Our students are coming in woefully underprepared. I decided that this was going to get a higher priority in my class. Also, exponential functions and logarithms are more natural (says me) in the context of calculus (especially logarithms). To me, this means that precalculus isn’t a great place to teach it. Okay, I need to expose them to it, but what I want to do is actually teach them enough calculus (intuitive version) in this class that, by the end, I can introduce these ideas more organically. Now, I have already introduced the graphs of exponential functions and logs, and given examples of their use, so at least their aware.

    I don’t want to cut hyperbolic functions. I’m being stubborn here. The professor next semester will almost certainly skip them (apparently, I’m the only one here who doesn’t) and as none of the students are physics majors (four forensic science students and two CS) they are unlikely to use them in their applications. But, darn it, this video about cosh rocks. And…it fits into my whole scheme of showing that all the new functions we talk about come (somehow) from the exponential function. I know, I know. You’re right. It is just that I hate admitting it.

    Comment by Adam Glesser — July 8, 2010 @ 8:12 am | Reply

    • Dunno about the situation where you are, but in Calculus they assumed we knew logarithms already and I got zero reinforcement.

      Which was bad because in Pre-Calculus they assumed we knew them already from Algebra II, and in Algebra II they didn’t do that good a job of teaching it. I never really learned everything about them until I started teaching them myself!

      Other issue: non-Calculus students — especially those on a Statistics track — may need the logarithms.

      Comment by Jason Dyer — July 8, 2010 @ 9:31 am | Reply

      • You make a good point about assuming they learned something in the past. I mentioned the slope-intercept form of a line as something they knew and I wanted just to quickly refresh them on and they looked at me as if I was speaking in Hungarian. I showed them what I meant and they almost all agreed they had never seen it before.

        I am lucky in this class as I know that all of the students are moving on to calculus. We have two precalculus tracks and you only take the one I’m teaching if you need to take calculus.

        What I might do is try to keep the material in class from becoming too compartmentalized and throw some log stuff (as well as the other stuff) in all over the place. It might be a bit messy (which is antithetical to my preferred style) but this is an experimental course for me.

        Comment by Adam Glesser — July 8, 2010 @ 9:38 am

  4. […] This post was Twitted by samjshah […]

    Pingback by Twitted by samjshah — July 26, 2010 @ 9:21 pm | Reply

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