This page is a sub-page of the page on our Mathematical Explainatorium.
There is a huge bureaucratic structure in education that could be called “the difficultification industry,” which makes a living from organizing the mathematical curriculum in a way that emphasizes the difficulties – in effect telling learners that most mathematical concepts are too difficult to address at an early age, because before you start with concept/subject C you have to master concept/subject B and before you begin with B you have to master concept/subject A. This linear view on curricularized education in effect kills the highly non-linear sparks of curiosity and interest that are the most powerful drivers of generative learning.
A good example of this is given by the so called Rubrik’s cube, a “mathematical bomb” which exploded on the youth community back in the 1980s. Little kids, driven by a glowing interest, were busy inventing their own algorithms, creating and exploring mathematical structures. And how did the school system respond to this golden mathematical opportunity? In Sweden, the “difficultification industry” outlawed it, and came up with a rule that the kids were not allowed to play with the cube in school. Imagine what could have happened if instead they had exploited this wave of mathematical interest and brought in people with mathematical knowledge that could have started to explain some of the mathematics behind the cube. But that type of disruptive event has no place within the linear curriculum system.
Children have no problems learning language abstractions, neither do they fail to see the advantage of using such abstractions. No language teacher would dream of telling a pupil that you cannot yet use the word “vehicle” because it’s too abstract at this stage, you must talk about “meaningful” things such as “car,” “boat,” and “bicycle.” Yet, this happens all the time within the difficultified math education process.
Neither do children have problems with learning abstract and meaningless games – in fact they love to do so. Chess is a good example. No one outside of the difficultification industry would ever dream of telling a kid that “chess is too abstract for you, you should not play it until you are 15 years old.” Also, no chess-playing kid ever runs into problems like: “I don’t understand chess because I don’t understand why the bishop is moving diagonally.” Yet, within the difficultified math education process, questions like this are being asked all the time, because, inspired by the anti-abstract and “meaningful” approach of “week-day mathematics,” meaning is being desperately sought for in the wrong places.
Sadly, people with real knowledge of mathematics are increasingly rare within primary and secondary schools. In fact, many people who did not understand (or were not even exposed to) the mathematical material presented during the later stages of math education are recruited to teach in the early stages, i.e., they teach mathematics for young kids. This has catastrophic consequences for maintaining and stimulating the interest in mathematics among young people, on which the future of our high-tech society ultimately rests.
Alan Kay – one of the early pioneers of the graphics work station computer, and a long-term pedagogical innovator in the field of children and mathematics – has said that “The kids can literally smell the fear of the math teacher in front of them.” This fear stems from the traditional teacher role as a “knowledge filter”: I’m going to teach you what I know. In practice, this knowledge filtering strategy is almost invariably translated into carrying out a large number of very similar calculations from a book. This didactical strategy could be termed the Goebbels’ principle of math education: If you carry out a calculation a large enough number of times, then you will learn something from it.
In fact, it is the exact opposite. The will to understand what is going on is gradually turned into mindless mechanical repetition. An excellent example of this was given by our daughter Ylva, who came home from school one day (when she was in fifth grade) and said:
It is interesting to observe that many “math-damaged” people can remember exactly the point in time and the situation that made them give up on understanding math. A common “weapon of math destruction” used by scared knowledge-filtering teachers is to ridicule pupils that ask questions they do not know how to answer. The classical example of such a question is the following: “Why is minus times minus equal to plus?” The typical (fearful) response of such a teacher makes the pupil feel that s/he has asked a stupid question, and that s/he is the only one in class that “hasn’t gotten it.” In fact, s/he may be the only one in class that realizes that this question actually needs a sensible answer.
How can we change this destructive pattern? One way would be to bring in early math teachers that really know the subject. However, this is not a realistic alternative for several reasons, the main one being the low status and low salary that is associated with early teaching. People that know mathematics and know how to communicate that knowledge are eagerly sought after and much more highly paid elsewhere.
We must transform the destructive knowledge-filtering role of the early math teacher into more of a knowledge coach: “I will help you to find out more about the tings that you are interested in.” This implies the introduction of some form of support system (mathematical helpdesk), where questions can be routed to knowledgeable people that can provide live answers and stimulating conversation around the subject.
During the early stages of math education, we focus only on meaning. In Sweden there is a concept called “weekday-mathematics” (“vardagsmatematik”) (German: “Alltagsmatematik”), which tries to discover the structures.
We desperately need to take a complementary approach. We need “week-end mathematics”, the kind of math that fascinates people and engages them in conversation – the kind of math that you can talk about in the pub on Saturday night – the kind of math that can get you laid. Is there such a kind of math? Yes there is. It’s the math of many dimensions, of curved spaces, the math that Einstein used and that the actress Cameron Diaz dreamed about understanding, something that resulted in the book \, E = m c^2 . Of course there is such a kind of math. That’s why people become mathematicians. It’s just that we are so terribly bad at bringing it to kids.
And why is that so? Basically because we suffer from the tradition that we have to be able to prove everything that we show. And mathematical proof has become so hard to teach that it has been mostly taken out of the curriculum. So we need to bring in “mathematical show” as early as possible, something that can be done by making use of modern computer simulations and interactions.