Computer-based maths education
Content
Summary: Why we need to base maths education on computers
Conrad Wolfram
Maths education in schools lags behind maths outside in a crucial, crippling way: it isn't based on computers doing all the calculating. We continue to insist that our entire population spends years trying to learn how to calculate by hand— solving equations, multiplying matrices or sketching graphs—instead of focus on how to conceptualise and solve problems mathematically while getting computers to do the procedural calculating. Almost no-one outside education does anything above basic arithmetic by hand any more. When was the last time a bridge was designed by hand-calculating the differential equations? Or someone calculated the distribution in an epidemiological study by hand? When did you last add up your bills at home manually or need to know how to solve a quadratic equation? In the real world, mathematical techniques are used more than ever but increasingly without those involved knowing the associated manual methods. Instead their intellect is applied to formulating evermore complex problems mathematically, thinking through what questions need to be answered, what calculation to instruct the computer to perform, and how to interpret the computed result back to reallife. Before computers were applied, the limiting step was almost always the individual’s capacity to calculate; now the issue is usually our ability to formulate an idea or interpret a result appropriately. Using computers for calculating is liberating everyone outside education to employ high-powered mathematics even if they don't know the details of how it's being worked out. But propose this complete computer-based maths approach for education and hear cries of “dumbingdown”: that even if what’s taught is far more relevant, using computers in this way is mindless and somehow not educational. Do those espousing this view really believe maths in engineering, finance, science or every day living has become less conceptual since computers? I'd argue quite the opposite— that technical professionals need greater conceptual understanding of their subject and application of mathematics for the complex modelling and greater optimization underlying today's highly technological existence. It's worth remembering that successful technology is often about increasing separation of the “how something is done” from the “what you're trying to achieve”, whether it's “engine setting” from “driving”, “image formation” from “photography” or as here “calculating techniques” from “working things out”. Technology often enables automation—initially for simple uses, then if it works for advanced or professional uses too until the technology is ubiquitous. And in the last few years, that's the state that computer technology for calculating has reached: simply surpassing what humans can do at all levels, with automation that encodes calculating expertise for everyone, which collectively no single human would have. So why has this chasm opened up between maths education and maths outside? Central is a confusion and disagreement over what maths constitutes, what's needed and therefore what's taught. Many politicians (and if it's not communicated otherwise, much of the Public) believe
2 Summary: Why we need to base maths education on computers. Conrad Wolfram, 2009. that maths = calculating or even just “numeracy”, and therefore that computers doing the calculating either destroys maths or denudes it of intellectual content. Yet when those same politicians talk about the need for people to study mathematical subjects for the “knowledge economy”, they presumably mean maths that is appropriate outside education—the much wider and now computer-based variety that isn't just maths for its own sake. Practically, these two definitions are increasingly at odds because if hand-calculations are required, they dominate maths curricula at the expense of the practical maths needed outside education. Failure to recognise this trade-off is delaying fundamental, difficult and vital reform. I sometimes hear that we are now incorporating computers in maths education step-by-step; progress is being made towards computer-based maths. Yes and no. I'd argue “the system's mindset” is stuck with an outdated desire for hand calculations. Computers are present but often supplement or, amazingly, are used to try to teach rather than replace hand-calculating. Certainly there are enlightened and innovative teachers who have made the most of the available technology. But in the end the system—in particular exams without computers—requires that everyone knows advanced hand-calculating procedures and therefore forces all teachers into teaching the traditional subject, best described as “the history of mathematical calculating”. Until there's fundamental change in this line of thought, our curricula will necessarily be stuffed with calculating techniques, taking away the vast majority of valuable time in maths education from conceptualisation. There's no lack of pressure for change: after all maths is more central to our lives than ever, but almost no-one believe maths education is working well. Governments see too few who are mathematically trained and even then not sufficiently; students typically find maths boring and hard; teachers find maths challenging to teach; technical professionals think it's dumbed down and bifurcate into wanting the subject to become more “rigorous”, or finding it irrelevant for practical applications. Few believe that radical improvement isn't important. And yet, correctly applied, computer-based maths could offer that radical change—within education just as it has outside over the last couple of decades. It is not a choice between the vocational and intellectual; it's both more conceptual and more relevant maths, maths that takes you so much further than you could ever go by hand. For example, why not teach calculus to 10-year olds? Traditionally, because computing integrals is hard and requires years of practice in formula manipulation beforehand. But conceptually, “finding areas” is quite straightforward, as well as important in a huge range of practical applications. Stripping out hand calculations means realistic, fuzzily defined, computationally hard problems can be investigated (just as in real life). There’s much more scope for open-ended exploration—setting up problems and interpreting results, trying many different examples quickly, testing the limits. These steps require intellectual insight, while getting students so much further in seeing what they can do with maths and how. I'm aware I've left many issues unaddressed in this quick summary. There’s the relationship between concepts of calculating and wider mathematical understanding; and cases—mostly in primary education and for estimating—where I do believe learning hand-calculating and especially mental arithmetic remain worthwhile. There’s how practically to transition from hand to computer-based maths, while improving inclusiveness. And let’s not forget changing maths' image into something exciting and fun!
Conrad Wolfram
Maths education in schools lags behind maths outside in a crucial, crippling way: it isn't based on computers doing all the calculating. We continue to insist that our entire population spends years trying to learn how to calculate by hand— solving equations, multiplying matrices or sketching graphs—instead of focus on how to conceptualise and solve problems mathematically while getting computers to do the procedural calculating. Almost no-one outside education does anything above basic arithmetic by hand any more. When was the last time a bridge was designed by hand-calculating the differential equations? Or someone calculated the distribution in an epidemiological study by hand? When did you last add up your bills at home manually or need to know how to solve a quadratic equation? In the real world, mathematical techniques are used more than ever but increasingly without those involved knowing the associated manual methods. Instead their intellect is applied to formulating evermore complex problems mathematically, thinking through what questions need to be answered, what calculation to instruct the computer to perform, and how to interpret the computed result back to reallife. Before computers were applied, the limiting step was almost always the individual’s capacity to calculate; now the issue is usually our ability to formulate an idea or interpret a result appropriately. Using computers for calculating is liberating everyone outside education to employ high-powered mathematics even if they don't know the details of how it's being worked out. But propose this complete computer-based maths approach for education and hear cries of “dumbingdown”: that even if what’s taught is far more relevant, using computers in this way is mindless and somehow not educational. Do those espousing this view really believe maths in engineering, finance, science or every day living has become less conceptual since computers? I'd argue quite the opposite— that technical professionals need greater conceptual understanding of their subject and application of mathematics for the complex modelling and greater optimization underlying today's highly technological existence. It's worth remembering that successful technology is often about increasing separation of the “how something is done” from the “what you're trying to achieve”, whether it's “engine setting” from “driving”, “image formation” from “photography” or as here “calculating techniques” from “working things out”. Technology often enables automation—initially for simple uses, then if it works for advanced or professional uses too until the technology is ubiquitous. And in the last few years, that's the state that computer technology for calculating has reached: simply surpassing what humans can do at all levels, with automation that encodes calculating expertise for everyone, which collectively no single human would have. So why has this chasm opened up between maths education and maths outside? Central is a confusion and disagreement over what maths constitutes, what's needed and therefore what's taught. Many politicians (and if it's not communicated otherwise, much of the Public) believe
2 Summary: Why we need to base maths education on computers. Conrad Wolfram, 2009. that maths = calculating or even just “numeracy”, and therefore that computers doing the calculating either destroys maths or denudes it of intellectual content. Yet when those same politicians talk about the need for people to study mathematical subjects for the “knowledge economy”, they presumably mean maths that is appropriate outside education—the much wider and now computer-based variety that isn't just maths for its own sake. Practically, these two definitions are increasingly at odds because if hand-calculations are required, they dominate maths curricula at the expense of the practical maths needed outside education. Failure to recognise this trade-off is delaying fundamental, difficult and vital reform. I sometimes hear that we are now incorporating computers in maths education step-by-step; progress is being made towards computer-based maths. Yes and no. I'd argue “the system's mindset” is stuck with an outdated desire for hand calculations. Computers are present but often supplement or, amazingly, are used to try to teach rather than replace hand-calculating. Certainly there are enlightened and innovative teachers who have made the most of the available technology. But in the end the system—in particular exams without computers—requires that everyone knows advanced hand-calculating procedures and therefore forces all teachers into teaching the traditional subject, best described as “the history of mathematical calculating”. Until there's fundamental change in this line of thought, our curricula will necessarily be stuffed with calculating techniques, taking away the vast majority of valuable time in maths education from conceptualisation. There's no lack of pressure for change: after all maths is more central to our lives than ever, but almost no-one believe maths education is working well. Governments see too few who are mathematically trained and even then not sufficiently; students typically find maths boring and hard; teachers find maths challenging to teach; technical professionals think it's dumbed down and bifurcate into wanting the subject to become more “rigorous”, or finding it irrelevant for practical applications. Few believe that radical improvement isn't important. And yet, correctly applied, computer-based maths could offer that radical change—within education just as it has outside over the last couple of decades. It is not a choice between the vocational and intellectual; it's both more conceptual and more relevant maths, maths that takes you so much further than you could ever go by hand. For example, why not teach calculus to 10-year olds? Traditionally, because computing integrals is hard and requires years of practice in formula manipulation beforehand. But conceptually, “finding areas” is quite straightforward, as well as important in a huge range of practical applications. Stripping out hand calculations means realistic, fuzzily defined, computationally hard problems can be investigated (just as in real life). There’s much more scope for open-ended exploration—setting up problems and interpreting results, trying many different examples quickly, testing the limits. These steps require intellectual insight, while getting students so much further in seeing what they can do with maths and how. I'm aware I've left many issues unaddressed in this quick summary. There’s the relationship between concepts of calculating and wider mathematical understanding; and cases—mostly in primary education and for estimating—where I do believe learning hand-calculating and especially mental arithmetic remain worthwhile. There’s how practically to transition from hand to computer-based maths, while improving inclusiveness. And let’s not forget changing maths' image into something exciting and fun!
