Conrad Wolfram, a British technologist and businessman, and founder of Computer-Based Math (www.computerbasedmath.org), defines math as consisting of 4 steps:
- Posing the right questions.
- Translating a problem from the real world into a math formulation.
- Computation.
- Translating the results back into the real world.
The problem with the current methods of Math education, he asserts, is that we spend 80% of the time teaching students computation (step 3), and we teach them how to do it by hand. Thus we have little time left teaching them steps 1,2 and 4.
I think Step 1 is very important and should be stressed above all the others. Posing the right questions means that the students must first understand the problem. Without this understanding it would be almost impossible to solve the problem.
I have seen too many test papers filled with all sorts of calculations but it was obvious that the student didn’t understand why he was doing those calculations. There would be an answer but it would be so obviously impractical or unrealistic and the student did not even bother reviewing his solution or asking why that was so. This is what happens when there is too much focus on computation and getting at the answer, but not enough focus on understanding and translating the results to reality.
It is like producing students who are experts at changing tires, but they will change the tires even if the problem is an oil leak or an overheated engine. They don’t know how to ask the right questions. They don’t know what the real problem is.
Wolfram proposes that we should begin teaching students at an earlier age to use computers for calculations, which can do them so much faster, more accurately, and at several more orders of difficulty. If we do that, then we have more time to focus on the other steps.
The obsession with making students calculate by hand is eating up a lot of time, and actually kills the interest of the general populace. No wonder a lot of people say “I hate Math” or “Math hates me.” They have the mistaken notion that math equals calculation instead of it being an approach to understand and solve real-world problems like “where do I invest my money so that it gives the best returns” or “which life insurance policy is most advantageous for me and my family?” or even “how do I win this poker game?”
Incidentally, I have a friend who is very good at math and also very good at poker. He was able to build his house from his winnings in poker-playing. Now, isn’t that an interesting and successful application of Mathematics?
It is understandable that in the development of math education, there was a huge focus on computation — because there weren’t any computers back then and the only way you could get to the results was to compute by hand. But that is not the case today.
A complex algebraic equation that may take several minutes or even an hour to compute by hand can be done in seconds by software like Mathematica (invented by Conrad’s brother, Stephen). Instead of teaching students the how of solving such an equation, teachers can instead focus on the why — on what it means in real world, and why it is important, and why it matters.
Wolfram also discusses one of the most common objections to this approach, which is that students must “learn the basics” first and that is why there is so much focus on computation. But what exactly do we mean by learning the basics? He comes up with this analogy.
Do people need to understand the mechanics of a car in order to learn how to drive it? Well, maybe in the early days of cars, it was necessary to have some knowledge of how an engine works and so on, because there was less automation and you had to do a lot of things manually just to even start the car.
These days, there is so much automation that you don’t even need a key to start the car, or learn manual transmission. Just push a button and step on the pedal and away you go. So now we have millions of people who can drive cars without really understanding how they work, but they know how to get from point A to point B, which is really what driving a car is all about.
With computers, we have the ability to teach our kids to handle complex mathematical equations without really doing the nitty-gritty work of solving them by hand. Instead of being disinterested or intimidated because of the long calculations, they will instead be more focused on the implications of the problem and how the results matter in real life, which is really what mathematics is all about.
Here is a video of Conrad Wolfram’s original talk back in 2010:
Originally published in Sunstar Davao.
Email me at andy@freethinking.me. View previous articles at www.freethinking.me.
The problem with “solving problems” is it requires a pulling of the self away from the mystery, it’s the tree in the garden of eden, the tree of the knowledge of greater than and less than; doing math involves abstracting from the divine ocean we swim in, ignoring the qualities to see quantity;
In the movie “the exorcism of emily rose”, whenever the people encountered the mysteries of spirit, they had trained themselves to “count to 5”, to pull themselves back;
A psychiatrist listened to a 45 minute rambling monologue branching into manifold directions of unreality; at the end of it he asked the patient to subtract 7 from a hundred, and in the manic and febrile, quasi-hallucinatory state, the patient thought for a moment ad replied “a hundred and nine”, and waited confidently for the next question, until gradually the doctor began to shake his head no, and the patient apologized for the obviously wrong answer, thought again, and answered ‘a hundred and seven’.
The patient walked away amazed by the doctor’s magic trick, a student who had followed along with math class through high school and into college like lots of other people, and learned that day that there was a competition in the brain, between the ability to do math and the ability to think within the molten sea of madness and vision.
From then on, the patient avoided almost all math in life, (which in the IT field is surprisingly easy to get away with), never really knowing how much money he had, only that it was ‘enough’, and realized that this modification left him able to hallucinate and ideate far more floridly than his peers.
In a recent book about online dating and the patterns revealed by the “dataclysm”, the abundance of data, the writers concluded “no compelling evidence supports matching sites’ claims that mathematical algorithms work” (excerpted in a New York Times article 1/23/2015). “That is because what creates a relationship can’t be expressed in data or photographs…they have to stop thinking in individual terms and start thinking in rapport terms. Basically, they have to take the ‘enchantment leap’. This is when something dry and utilitarian erupts into something passionate, inescapable, and devotional. (These days) we live in a culture and an online world that encourages a very different mind-set; in a technical culture in which humanism, religion and the humanities, which are the great instructors of enchantment, are not automatically central to life.”
Humanity, most of them, they have gone down into the rabbit hole, the test tube of human knowledge. And down there they can tell time and manipulate chemical reactions and build impressive machinery and even their silicon computer space age stuff, but they imagined, in the 50’s, when my parents were taking it all in, that they would eventually understand everything, in just another ten or twenty years, and then be able to solve all the problems and be infinitely magical.
I like to say one day humanity will grow up, put away their “toys” of science and mathematics, in the toy box in the attic to be glanced at wistfully once in a while but left behind as neoteny mostly, and come up and out and join the rest of the species, and the children, as we pursue our inner nature and our destiny, and search for mystery and wonder.
We get the inspiration early on to go out into the universe of mystery and wonder and really open our minds there, and you can’t let the academy professors or the priests and teachers and scared little rabbits in the forest – the worldly minded camarilla – distract you from what’s really happening, happening for all the other species, and the plants, the spirits, the two year olds, the shamans, and the schizophrenics, and the Daoists for that matter, yeah, we just wanna pursue inner nature and destiny, and search for mystery and wonder, and as for that nonsense you all tell yourselves about that invisible force in the ground that’s always sucking you in, makes everything suck for you, what is it? That mystical force that Newton came up with in 1666 when the demon threw Eve’s apple at his head? What was that particle your scientists can’t seem to find, oh yeah, the graviton, you guys got the gravity stuff, yeah, well it’s ideas like that that cause us to walk away from your world, sorry, nothing personal, go do whatever it is you guys do in your gravity, but hey, why do you feel the need to pull everybody down into the test tube of human knowledge?
I have a quote on my wall here, taken from the NY Times, from a guy who won the Nobel Prize for physics; this guy admits that science is a test tube of human knowledge that can’t really know about things past the “Waterloo” of psychiatry, like schizophrenia and shamanism:
“Steven Weinberg, of the University of Texas at Austin, who won his Nobel in ‘79 for using Higgs theory to unify two of the forces of nature, declared (mournfully) in the New York review of books: “physical science has historically progressed not only by finding precise explanations of natural phenomena, but also by discovering what sorts or things CAN be precisely explained. These may be fewer than we had thought”.”