To compute by hand, or not ?

Some comments in response to a video TED talk that Conrad Wolfram gave called “Teaching Kids real math with computers”

I did enjoy Wolframs video, but I think its too tempting to take away computations.. in doing so we risk losing the math understanding behind them.  I just think we need more of both : more understanding and more facility with actual practical problem solving.

Technology can help to explain math better. I just found this video today, which contains a superb intro to Bezier curves, among other things.   I think this would be pretty engaging to high school students, be a motivation for them to learn calculus.

One of the things I wanted with GridMaths is to take some of the pressure off long working out but also have the student solve the problem for themselves. Its easier to line up numbers in long multiplication and you tend to make less typos. By design, GridMaths is _not_ a calculator, there are other apps that do that well.

I’m currently working on a hybrid approach where you can use ASCIIMathML to get good looking math expressions, which should open GridMaths up to wider use at high-school level. Maybe we can make worked problems less of a chore and more about understanding … but still keep the facility and practice with computations at a good level.

I can remember an awful lot of fiddling around with sharpening pencils and erasers at school..  push-pencils solved that, but I must have bought 15 different compasses :]  Once a person can write legibly, maybe the emphasis should segway from paper to electronic tools, but still keep the ability to do the constructions, structure the essay, think critically, type well, organise computations etc.

I recently saw a report from one school where they moved to 1-to-1 tablets, where they saw savings of ~20k/month in stationary costs.. maybe exaggerated and an affluent school, but paper is kind of expensive at volume.

Its certainly a lot quicker to make a GeoGebra construction on a tablet, than to do it with ruler, compass, dividers, protractor and pencil … you can concentrate on the concepts and specifics, rather than paper-management. Then the diagram you construct is malleable, you can drag and interact… that’s a big innovation, and helps understanding.

I wonder what ways software can help take the load off teachers, so the mechanics are easier for them, and they can spend more time teaching face to face.

An easy Putnam problem : areas in a stretched circle

I’ve been thoroughly enjoying Paul Zeitz’s book “The Art and Craft of Problem Solving“.

One of the early problems in the book is from 1994 Putnam math competition, but surprisingly easy once you see that it can be scaled into a much simpler question.

The problem :

Find the positive value of m such that the area enclosed by the ellipse x^2/9 + y^2=1, the x-axis, and the line y=2x/3 is equal to the area in the first quadrant enclosed by the ellipse x^2/9 + y^2=1, the y-axis, and the line y=mx.

Heres a drawing of the areas mentioned using GeoGebra, with each area in separate quadrants for comparison, with an approximate area.  The Qn asks what is the slope of the line through point F [or -ve slope of line through Q here].

Problem as stated, with areas colored

Well its a bit hard to pick the point Q [ the slope -m ] so that these colored regions have the same area…

… but then you realize that the whole problem becomes easy if you simply scale the ellipse back to the unit circle.

To do this, x is scaled back by 3x, so areas become 1/3 of what they were [y remains the same].  So the problem now looks like this :

“]

scaled back to the unit circle : areas are x 1/3, slopes x 3

So the line that gives the same area is y=x/2.  When this is scaled back to the original ellipse, the slope gets divided by 3;  so the line we want is y=x/6.

Quite surprised to see this in a Putnam, but it does show a really common motif in math, namely :

Transform to a simpler domain, solve it there, then transform the solution back.

The astute reader will have noticed that I cheated : the areas of the colored regions in the top picture are of course ~1.65 or 3 x area of regions in the second diagram.

The areas marked were calculated by rough approximation with a polygon and the GeoGebra area function. [  The ellipse itself was drawn to fit using foci – I wasn’t sure how to edit the formula in GeoGebra to make it exact. ]