I’ve written before about refining my “why math” elevator speech and being more deliberate in how I frame why math is worth learning. Both of those happen at specific times — my framing attempts to start each course with a clear vision of why math is worth learning, and my elevator speech is deliberately responsive to student protests about “when am I ever going to use this”. I’m working to send that message on a more regular basis in my classes, and one big opportunity is how I introduce a new topic or unit.

We started optimization today in my Calc A class. I framed it the way I see it. I said something like:

Optimization problems pretend that they’re problems you might need to solve in the real world, though they’re usually hilariously contrived questions that ask you to be unnecessarily specific about something that doesn’t matter very much. But they involve drawing fun pictures, finding creative ways to model things with algebra, and paying carefully attention to detail, so they’re worth solving.

I don’t know if that’s particularly inspiring. It probably sells optimization short. I hope someday I’ll have a repertoire of great, engaging three-act tasks and modeling scenarios that make optimization the most thrilling part of the differential calculus curriculum. But right now I don’t, and I want to be as forthright as I can — to avoid digging myself into a whole in the future, to build some element of trust that I see the world in a similar way to my students, and to be honest about the value of optimization. And if I’m being honest with myself about the value of optimization, I believe that it’s worth learning because it offers a great visual perspective into calculus, it requires students to build algebraic skills that aren’t necessary in many other areas of calculus, and often trips students up who lose the forest for the trees. I believe that’s worth learning.

This doesn’t mean saying, “I’m sorry this isn’t the most exciting topic.” I’m not apologizing for anything. I love teaching math, and I really believe it’s worth learning. But I don’t believe it’s worth learning because students will use it in the “real world”, and I enjoy poking some fun at that perspective when I get the chance. Instead, this is about framing the true value in math — reasoning, modeling, generalizing, questioning, arguing — and unapologetically teaching students through that perspective.

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howardat58Ok, try this:

Put a sequence of y values at equall spaced x values on a piece of graph paper (how old fashioned!). Ask them to sketch a curve trough the points (easy if well chosen). Then a maximum point or a minimum point wi;; probably appear. If it doesn’t, start again. With luck they will all come out with different estimates. Now, suggest some form of curve fitting to the three or four data points surrounding the maximum. If three they’ll get a quadratic. If four a cubic. Calculus can then be used to get an estimate of the maximum point, and it can be automated.

If they go for a cubic they have gone some way to creating a cubic spline curve, so send them to the library or the internet.

Real life RARELY generates nice functions that apply to the whole range of interest. It comes as sequences of data points.

Optimization is a powerful thing, and used widely in engineering and other areas. When you get to partial differentiation it gets more interesting.

dkane47Post authorThat’s a cool idea — and sounds like it could create some perplexity for students. You’ve mentioned cubic spline curves before — I still haven’t learned enough about them to feel confident teaching them. I agree about the power of optimization, but I’m making an argument about teaching student optimization. I don’t think that power is present in the types of problems that are most useful in teaching students to be good at optimization.

howardat58I’ll get back to you on this, cubic splines are not difficult !

I guess you have tried the “minimise the cardboard in a box” thing.

They could also see functions that can be differentiated, but the equatioin f'(x) = 0 cannot be solved algebraically —-> numerical methods

howardat58Construction of cubic spline, descriptively as far as possible!

Just read the words first time!

Start with two points P1 = (x1,y1) and P2 = (x2,y2)

Need the line joining the two points, call it L12, throw away text book methods.

Simply, for any k the point (1-k)P1 + kP2 is a point on the line L12

k = 0 gives P1 and k = 1 gives P2

If in doubt check with k=1/2.

To get x and y into this (ie to get the equation of the line) we need to use the fact that as k goes from 0 to 1, x goes from x1 to x2 , and y goes from y1 to y2

So k = (x – x1)/(x2 – x1) and 1 – k = (x2 – x)/(x2 – x1) … I love the symmetry here

Then y = (1 – k)*y1 + k*y2

Substitute and simplify to get y = (x2*y1 – x1*y2)/(x2 – x1) + (y2 – y1)/(x2 – x1)*x, or just imagine it done !………………………………………………..intercept………………………slope……………

At which point one may well think “Why didn’t I do this with numbers”

Now just imagine that this is done for P2 and a new point P3 to get line L23

Then repeat the whole process with the two lines, using a new k going from x1 to x3

This gives a quadratic function which passes through all three points. Call it Q123

We now need a new point P4 to get line L34

Then repeat the whole process with the 2 lines L23 and L34, using a new k going from x2 to x4

This gives a second quadratic function which passes through the three points P2, P3 and P4. call it Q234

Now do the process again, with a new k going from P1 to P4, using the two quadratics Q123 and Q234

You will get a cubic function passing through all 4 points.

Only the middle bit is used, from P2 to P3 as we no move one step along, with points P2 to P5, and do it all again, well, not so much, since we have already done a lot of it.

The cubic spline curve is the collection of all the “middle” bits, and you should be able to see that the bits join up nicely (same slope at the join).

If you have a programming whizz kid in your class they can code this up fairly quickly. it is much simpler if the x values are equally spaced, and even better if you use 1, 2, 3, 4, … for the x’s.

dkane47Post authorThis is some good stuff to chew on! Thank you.

howardat58Remember Polya, draw some pics, try some not too simple numbers. (avoid zero!)

Let me know how it goes.

It is such a structured thing that the computer program is shorter than the explanation.