Math(s), the real world, and computers

Nov 16 2010 Published by under Uncategorized

Over at her Guardian blog, Grrlscientist just posted a video where Conrad Wolfram suggests that instead of teaching hand calculation, it would be better to teach children mathematics through computer programming. I think Wolfram is more than simply wrong - he's dangerously wrong.

In the video, Wolfram argues that in the real world, mathematics is essentially a four-step process:

1: Pose the right question

2: Figure out the mathematical formulation

3: Calculate

4: Apply answer back to real world.

Wolfram argues that education is too focused on how to do step three by hand, and insufficiently focused on the other three steps, which are clearly more important in the real world. He suggests that using computers to do the calculations in school would be a "silver bullet" (his words) that would go a long way toward fixing math education.

I think Wolfram is perhaps confusing his world with the entirety of the real world. I'm not sure that computer calculation is always as immediately available or convenient in the real world as his suggested teaching method would require.

For example:

That's a partial view of one of the swimming pools I supervise. The two people in the picture are lifeguards, engaged in the process of painstakingly scrubbing the walls and floor of the pool in an attempt to knock off as much of the mustard algae as possible. While they're doing this, I want to figure out how much algaecide will be needed to address the problem.

Unfortunately, I don't know the pool volume, and that's kind of important to know if you want to know how much of a chemical to add to reach a given concentration. I've got that information - and a computer - in my office, which is not at the pool. I don't need an exact volume - as long as I'm within 5% or so it will be good enough. That's hand calculation territory, provided I can remember enough geometry to do the job. (The pool is essentially an irregular hexagon in shape at the surface.)

My world is a touch more blue-collar than Wolfram's. That's the case for a lot of people, many of whom need to use math on a regular basis, and need to do so in places where computers are not normally available, and not normally necessary. Computer-based math simply adds unnecessary complication.

And it does so with potentially devastating costs to society.

We already - at least here in the United States - live in an environment where the gap between the rich and the poor continues to increase as an accelerating rate. Rearranging mathematics education to depend more heavily on computers works great if and only if you have easy and regular access to a computer.

This may be another area where Wolfram is confusing his own world with the real world.

In the real world, many children and many schools still do not have adequate access to computers for the sort of education that they're doing now, much less a new math curriculum along the lines Wolfram proposes. His new math curriculum would simply provide another way to leave those students farther behind.

5 responses so far

  • fizzchick says:

    There are much simpler questions (not that this is a bad example). You're at the grocery store. Which brand of pasta, A or B, will feed your kids more cheaply? How many boxes/bags of pasta, jars of sauce, and gallons of milk can you buy without overdrawing your account when you get to the register? Are you going to need to stop for gas on this 50 mile drive, or can you make it home and buy gas tomorrow? These and a ton more questions are deeply relevant, and computers are generally unavailable when you need to know the answer, particularly if you are poor and thus smartphoneless. But getting the answers wrong can result in problems ranging from embarrassing to devastating to the family budget.

  • Peter says:

    Even though I am not a huge fan of Conrad Wolfram, I must defend him a little -- so get ready to flame me 😉

    I think there's only a slight misunderstanding. He does want people to be able to calculate (watch ca 6:14+). All he says (in the TED-usual, exaggerated way) that right now calculations are the only thing happening at school and that's in part a waste of time. I would add: and we're doing it badly; check out http://blog.mrmeyer.com for examples.

    Wolfram does not say that everybody should always use a computer. He claims a) that to program a solution can help understand problems better and b) that computers are good tools to introduce complicated concepts at earlier stages than we do today. I must admit that I agree with him on both accounts.

    For example: at some point in your school career you might have learned how to solve a system with two equations and two variables. That's nice. Unfortunately, it is a huge waste to teach just the 2x2 case. Unless you take a linear algebra course at university you might not find out that it is conceptually the same to solve a system with 100 equations and variables (or in fact, the billion equations that google needs to check for its pagerank) -- the idea to solve it is the same, you just have to do more steps in you calculation.

    Now, the same effort that taught you the 2x2 case could be spend to teach you how to write a small program that solves all cases. In turn you'd gain a much deeper understanding that you'll probably remember much longer (and you might remember that this actually can mean something whenever you google yourself).

    To use another example: at the moment, teaching math in school is like teaching soccer by telling people to run in a circle, a lot. Sure, if you don't have some basic stamina, it does not make much sense to go out on the pitch. But running in a circle is not soccer... What I'm trying to say is that we need both, the workout -- train mental calculations just like we train stamina -- and time to actually play the real game.

    Don't get me wrong, I totally see your point about people not being able to afford computers and having hands on problems. But then again it's "only" a vision. Besides, many people cannot afford books about history or the time to read them -- yet we teach history in schools.

    • Sean says:

      "Unless you take a linear algebra course at university you might not find out that it is conceptually the same to solve a system with 100 equations and variables (or in fact, the billion equations that google needs to check for its pagerank) — the idea to solve it is the same, you just have to do more steps in you calculation."

      This is not really true. Solving large systems of linear equations is an active field of research, and the methods go far beyond the traditional method you would learn in school to solve a 2 x 2 system (Gaussian elimination). See for example:

      http://news.slashdot.org/story/10/10/22/1236215/Astonishing-Speedup-In-Solving-Linear-SDD-Systems

  • captainahags says:

    Although I'm no expert by any means, I do tutor calculus at my college, along with precalc and trig, and I have to agree somewhat with wolfram and some with the author. 90% of the kids I tutor have the mathematical ability to crunch the numbers and find the answer once I show them how to set it up, but can't set up a problem for themselves. This pops up most often with optimization problems- I have 1000 cm^2 of material to build a box with a square base and open top, what are the dimensions of the box with the largest volume and what is that volume? Most of them, if I set it up for them (and there's not a whole lot to set up, obviously) have almost zero trouble after this. On the flip side, however, I do get a decent number of kids who need to go to their calculators for EVERYTHING- even if the answer only requires figuring out if a quadratic is positive or negative for some small value.

  • gardener says:

    Most people, even those with moderately low income have cell phones with them at all times. They have calculator functions. So the argument that we all have to be good a t computation in order to succeed "in the real world" is less convincing than it was in the past. In school, we really do spend a great deal of time on computation - to the exclusion of more interesting, challenging and practial math instruction.