What makes Mathematics hard to learn?

What makes Mathematics hard to learn?

Feb 16, 2008. An essay from Marvin Minsky

Why do some children find Math hard to learn? I suspect that this is often caused by starting with the practice and drill of a bunch of skills called Arithmetic—and instead of promoting inventiveness, we focus on preventing mistakes. I suspect that this negative emphasis leads many children not only to dislike Arithmetic, but also later to become averse to everything else that smells of technology. It might even lead to a long-term distaste for the use of symbolic representations.

Anecdote: A parent once asked me to tutor a student who was failing to learn the multiplication table. When the child complained that this was hard, I tried to explain that this was not a big job because there are less than 50 facts to learn (because AxB = BxA).

	2	3	4	5	6	7	8	9
2	4	6	8	10	12	14	16	18
3	6	9	12	15	18	21	24	27
4	8	12	16	20	24	28	32	36
5	10	15	20	25	30	35	40	45
6	12	18	24	30	36	42	48	54
7	14	21	28	35	42	49	56	64
8	16	24	32	40	48	56	64	72
9	18	27	36	45	54	63	72	81

	2	3	4	5	6	7	8	9
2	4
3	6	9
4	8	12	16
5	10	15	20	25
6	12	18	24	30	36
7	14	21	28	35	42	49
8	16	24	32	40	48	56	64
9	18	27	36	45	54	63	72	81

However, that child had a larger-scale complaint: “Last year I had to learn the addition table and it was really boring. This year I have to learn another, harder one, and I figure if I learn it then next year there will be another one and there’ll never be any end to this stupid nonsense. " This child imagined ‘Math’ to be a continuous string of mechanical tasks—an unending prospect of practice and drill. It was hard to convince him that there would not be any more tables in subsequent years.

To deal with the immediate problem, I made a deck of “flash cards,” each of which showed two digits on the front and their product on the back. The process was to guess each answer and, if it was correct, then to remove that card from the deck. This made the task seem more like a game in which one can literally feel one’s progress as the size and weight of the deck diminishes. Shortly the child excitedly said, “This deck is a really smart teaching machine! It remembers which products I’ve learned, and then only asks for the ones I don’t know, so it saves me from wasting a lot of time!”

However, a more serious problem was that this child had no good image or “cognitive map” of what might result from learning this subject. What function might Math serve in later years? What goals and ambitions might it help to achieve?

Anecdote: I asked a younger child “how much is 15 and 15” and she quickly answered, “I think it’s 30.” I asked how she figured that out so fast and she replied, “Well, everyone knows that 16 and 16 is 32, and then I subtracted the extra 1’s.”

Traditional teacher: “Your answer is right but your method was wrong: you should add the two 5’s to make a 10; then write down a 0 and carry a 1, and then add it to the other two 1’s.” The traditional emphasis on accuracy leads to weakness of ability to make order-of-magnitude estimates—whereas this particular child already knew and could use enough powers of 2 to make approximations that rivaled some adult’s abilities. Why should children learn only “fixed-point” arithmetic, when “floating point” thinking is usually better for problems of everyday life! More generally, we need to find out more about how each child regards each subject. How might it answer questions like “What am I doing here, and why? ”What can I expect to happen next?” “Where and when am I likely to use this?

Students need Cognitive Maps of their Subjects

Until the 20th century, mathematics was mainly composed of Arithmetic, Geometry, Algebra, and Calculus; then Logic and Topology started to rapidly grow. Then the 1950s saw a great explosion of new ideas about the nature of computation—ideas that are now so indispensable that our primary school curriculum is out of date by a century.

Today, our new computational concepts have become so useful and powerful that we should start teaching them in earlier years. We usually think of Arithmetic as a subject in itself. But we can also think of it, instead, as just a certain bunch of algorithms—which suggests that we could begin with simpler and more interesting ones!

For example, if you look at my book on Computation, you’ll see many concepts that apply to very wide ranges of phenomena—yet in the first hundred pages of that book, you’ll find virtually no Arithmetic. For example, we could engage our children’s early minds with some theories about Finite State Machines, and this would provide them with useful ways to think about the programs that they create with the low-cost computers that they now possess. Languages like Logo and Scratch can help children to experiment with geometry, physics, math, and linguistics—and then go on to make practical systems that their communities can develop and share.

For Geometry, we can make interactive graphical programs that can lead young children to observe and explore various sorts of symmetries—and thus begin to grasp the higher-level ideas that mathematicians call “The Theory of Groups”—which can be seen as a basis for Arithmetic, but could also help to understand many aspects of other subjects.

For Physics, children can have access to programs that simulate the dynamics of structures, and thus become familiar with such important concepts as stress and strain, acceleration, momentum, energy—and vibration, damping, and dimensional scaling.

In any case, we need to provide our children with better cognitive maps of the subjects we want them to learn. I asked several grade-school teachers how often they actually used long division. One of them said, "I use it each year to compute the average grade.” Another claimed to have used it for keeping track of financial transactions (but couldn’t recall a specific example) and none seemed to have adequate maps of mathematics as a potential lifetime activity. Here is a simple but striking example of a case in which a child lacked a cognitive map:

A child was sent to me because he was failing a geometry class, and he gave this excuse: " I must have been absent on the day when they explained how to prove a theorem."

No wonder this child was confused—and seemed both amazed and relieved when I explained that there was no standard way to make proofs—and that “you have to figure it out for yourself”. One could say that this child simply wasn’t told the rules of the game he was asked to play. However, this is a very peculiar case in which the ‘rule’ is that there are no rules! (In fact, automatic theorem- provers do exist, but I would not recommend their use.)

Bringing Mathematics to Life

What is mathematics, anyway? I once was in a classroom where some children were writing LOGO programs. One program was making colored flowers grow on the screen, and someone asked if the program was using mathematics. The child replied, “Oh, mathematics isn’t anything special: it’s just the smart way to understand things.” Here are a few kinds of questions that pupils should ask about the mathematical concepts we ask them to learn:

Geometry: Why are 2 triangles congruent, when their corresponding sides are equal? Answer: a triangle has only 3 degrees of freedom—which means that there is no way to change its shapes once those lengths are constrained. And that’s why structures built with triangles are so strong! We all live in a 3-D world, but few people learn good ways to think about 3-D objects. Should this be seen as a handicap? How many different ways can you paint 6 colors on the faces of a cube? Can you envision how to make a cube using 3 identical 5-sided objects? We know that gloves come in left- and right-hand forms—but why are there only two such versions of things?

Arithmetic: Why is prime factorization unique? Few high-school teachers are able to prove this? How does recursion lead to exponentiation? How do populations grow? How do animal organs grow? Why does “compound interest’ tend to add more digits at constant rates?

Logic: If most A’s are B’s, and most B’s are C’s, does this imply that some of the A’s must also be C’s? (Most people say Yes, but the answer is No.) We all try to use logical arguments, but we should also learn the most common mistakes. Is it possible that when Mr. Smith moved from Company A to Company B, this raised the average IQ of both institutions?

Statistics: Even a small background in mathematics can help to approach many other fields and areas—and few mathematical subjects rival Statistics in the range of its everyday applications. How do effects accumulate? What kinds of knowledge and experience could help children to make better generalizations? How should one evaluate evidence? What’s the difference between correlation and cause? What are the most common forms of biases—and why one needs to be skeptical of anecdotes. What are the most common kinds of mistakes, such as Post Hoc vs. Propter Hoc?

In particular, it seems to me, that we should try to get children to learn use the “T-test” method, which is a simple statistical test, yet which handles huge ranges of situations. Also they should understand using square roots to assess variations. Example: Basketball scores are frequently not statistically significant!

Combinatorics: Consider that, when we teach about democracy, few pupils ever recognize that, in an electoral-college voting system, a 26% minority can win an election—and if there are 2 tiers of this, then a mere 7% minority could win! How do cultural memes manage to propagate? How does economics work? At what point should we try to teach at least the simplest aspects of the modern Theory of Games?

Abstract Algebra and Topology: These are considered to be very advanced, even postgraduate. Yet there are many phenomena that are hard to describe if one lacks access to those ideas—such as fixed-points, symmetries, singularities, and other features of dynamic trajectories, all of which appear in many real-world phenomena. Every large society is a complex organization that can only be well described by using representations at many different levels of abstraction—e.g., in terms of person, family, village, town, city, country, and whole-world economy—and “higher mathematics” has many concepts that could help to better understand such structures.

How can we encourage children to invent and carry out more elaborate processes in their heads? Teachers often insist that pupils “show their work”—which means to make them “write down every step.” This is convenient for making grades, as well as for diagnosing mistakes, but I suspect that this focus on ‘writing things down’ could lead to mental slowness and awkwardness, by discouraging pupils from trying to learn to perform those processes inside their heads—so that they can use mathematical thinking in ‘real time’. It isn’t merely a matter of speed, but of being able to keep in mind an adequate set of alternative goals and being able to quickly switch among different strategies and representations. This suggests that OLPC should promote the development of programs that help pupils to improve their working memories, and to refine the ways that they represent things in their minds:

The Impoverished Language of School-Mathematics.

There’s something peculiar about how we teach math. If you look at each subject in elementary school—History, English, Social Studies, etc.— you'll see that each pupil learns hundreds of new words in every term. You learn the names of many countries and organizations, the names of leaders and battles and wars, the names of many authors and books—thousands of new words every year.

However, in the case of school-mathematics, the vocabulary is remarkably small. The children do learn the names of various objects and processes—such as addition, multiplication, fraction, quotient, divisor, rectangle, parallelogram, and cylinder, equation, variable, function, and graph. However, they learn only a few such terms per year—which means that, in mathematics, our children are mentally starved, by having to live in a “linguistic desert.” It really is hard to think about something until one learns enough terms to express the ideas in that this subject. Specifically, it isn’t enough just to learn nouns; one also needs adequate adjectives! What's the word for when you should use addition? It’s when a phenomenon is linear. What's the word for when you should use multiplication? That’s when something is quadratic or bilinear. How does one describe processes that change suddenly or gradually: one needs terms like discrete and continuous. To talk about similarities, one needs terms like isomorphic and homotopic. Our children all need better ways to talk about, not only Arithmetic and Geometry, but also vocabularies for the ideas one needs to think about statistics, logic, and topology. This suggests an opportunity for the OLPC children’s community: to try to set up discussion groups that encourage the everyday use of mathematical terms—communities in which a child can say “nonlinear” and have others admire, and not discourage her.

Mentors and Communities:

If one tries to learn a substantial skill without a good conceptual map, one is likely to end up with several collections of scripts and facts, without good ways to know which of them to use, and when—or how to find good alternatives when what you tried has failed to work. But how can our children acquire such maps? In the times before our modern schools, most young children mainly learned by being forced to work on particular jobs, and ended up without very much ‘general’ competence. However, there always were children who somehow absorbed their supervisors’ knowledge and skills—and there always were people who knew how to teach the children who were apprenticed to them.

I’ll come back to this in another Memo about the disadvantages of modern age-based classes. Today most education is broader, but apprenticeship itself now is rare, because few teachers ever have enough time to interact very much with each of their students: a modern teacher can only do so much. The result is that no one has time to deal thoroughly questions like “What am I doing here, and why? ”What can I expect to happen next?” or “Where and when am I likely to use this?” However, now we can open new networks through which every child can communicate. This means that we can begin to envision, for each of our children, a competent adult with enough “spare time” to serve as a mentor or friend to help them develop their projects and skills. From where will all those new mentors come? Perhaps that problem will solve itself, because our lifespans are rapidly growing. The current rate of increasing longevity today is one more year for every four, so soon we may have more retired persons than active ones!

Of course, each child will be especially good at learning particular ways to think—so we’ll also need to develop ways to match up good “apprenticeship pairs.” In effect we’ll need to develop “intellectual dating services” for finding the right persons to emulate!

In any case, no small school or community can teach all subjects, or serve the needs of individuals whose abilities are atypical. If a child develops a specialized interest, it is unlikely that any local person can be of much help in developing that child’s special talents and abilities. (Nor can any small community offer the range of resources to serve children with limited abilities.) However, with more global connections, it will be easier to reach others with similar interests, so that each child can join (or help form) an interactive community that offers good opportunities.

(Some existing communities will find this hard to accept, because most cultures have evolved to reward those who think about the same subjects in the same ways as do the rest! This will pose difficult problems for children who want to acquire new ways to think and do things that their neighbors and companions don't do—and thus escape or break out of the cultures in which they were born. To deal with this, OLPC will need to develop great new skills of diplomacy.)

Emphasizing Novelty rather than Drudgery?

Actually, I loved arithmetic in school. You had to add up a column of numbers and this was fun because there were so many different ways to do it. You could look here and there and notice three 3's and think, “that's almost a 10 so I'll take a 1 off that 7 and make it a 6 and make that 9 into a 10." But how do you keep from counting some numbers twice? In this case, you could think: “Now I won’t count any more 3's.” How many children did these things exactly as they were told to do? Surely not those who became engineers or mathematicians! For when you use the same procedure again, there’s little chance to learn anything new—whereas each new method that you invent will leave you with some new mental skill (—such as a new way to use your memory). For example, how do you remember when you “carry” the number of 10s of a sum? Sometimes I’d mentally put it on my shoulder. How do you remember a telephone number? Most people don’t have too much trouble with remembering a 7-digit ‘local’ number, but reach for a pen when there’s also an area code. However, you can easily learn to mentally put those three other digits into your pocket—or in your left ear, if you don’t have a pocket!

Why are so many people averse to Math? Perhaps this often happens because our usual ways to teach arithmetic only insist on using certain rigid skills, while discouraging each child from trying to invent new ways to do those things. Indeed, perhaps we should study this subject when we want to discover ways to teach aversions to things!

Negative Expertise

There is a popular idea that, in order to understand something well, it is best to get things right from the start—because then you'll never make any mistakes. We tend to think of knowledge in positive terms—and of experts as people who know exactly what to do. But one could argue that much of an expert’s competence stems from having learned to avoid the most common bugs. How much of what each person learns has this negative character? It would be hard for scientists to measure this, because a person’s knowledge about what not to do doesn’t overtly show in that individual’s behavior.

This issue is important because it is possible that our mental accumulations of counterexamples are larger and more powerful than our collections of instances and examples. If so, then it is also possible that people might learn more from negative rather than from positive reinforcement? Many educators have been taught that learning works best when it seems pleasant and enjoyable—but that discounts the value of experiencing frustrations, failures and disappointments. Besides, many feelings that we regard as positive (such as beauty, humor, pleasure, and decisiveness) may result from the censorship of other ideas, inhibition of competing activities, and the suppression of more ambitious goals (so that, instead of being positive, those feelings actually may reflect the workings of unconscious double negatives). See the longer discussions of this in Sections 1-1 and 9-4 of The Emotion Machine.

See also “Introduction to LogoWorks” at [1].

Nitpicker's maxim: Anything worth doing is worth doing badly. Raskin's addendum: At first.