This page is a sub-page of our page on The Impossibility of Teaching Mathematics.
The patterns underlying traditional math education
The Problem/Solution/Elimination pattern is concerned with the difference between solving and eliminating problems. Solving problems has the effect of making their symptoms dissolve, which means to ‘disappear in solution’ – just like salt-crystals in water. However, from this solution the symptoms of the problem can be easily crystallized in various ways. This tends to crystallize (= institutionalize) any organization that has been formed in order to solve problems.
Eliminating problems means dealing with causes instead of symptoms – which tends to transcend the problems as well as the organization . In fact, eliminating the problems dissolves the organization – instead of crystallizing it. Instead of becoming institutionalized, the organization ‘dissolves into solution’ – from which it can be conveniently re-crystallized if the problem should ever show up again.
The judicial system is seen as a way to solve the problems of crime by confining the criminals in jail. This activity is supported by the legal system, and the re-crystallization of convicted criminals is assured by various forms of economic dysfunctionality, most of all the natural reluctance to employ an ex-convict. Many criminals live their lives within this loop, while others organize and set up some kind of legal front in order to break the cycle. In this way the organized criminal is seen as a form of emergent evolutionary outgrowth – supported by the problem/solution pattern.
The pattern describes how conceptual difficulties (= concept-formation-problems) in mathematics education are ‘solved’ by promoting algorithmic ability, i.e. by teaching various forms of ‘arithmetic schemes’ in a more or less fundamentalist fashion. This dissolution process is modeled as driven by a pedagogical system that operates according to the principle of “behave or degrade”, i.e. the traditional math-test-metric. This leads to severe forms of understanding dysfunctionality, which in turn drives the crystallization of the conceptual difficulties. By nurturing the difficulties, the loop creates anticipated conceptual difficulties, which are summarized in the biggest mental block of all – “mathematics is difficult.” This conclusion is here modeled as an effect of the (be)cause: “I never understood it when I was at school.” This is a natural form of defense reaction that ties in with the Discriminator Questions pattern, which describes the filtering of the educational system and expresses the view that “when you do not understand during the later stages of your education, you often end up teaching in the earlier stages“.
Another variation on the problem-solution pattern appears in the later (often called higher) parts of mathematical education and research. This pattern shows the conceptual difficulties being dissolved into academic status, driven by an assessment system that operates according to the well-known principle of publish or perish.
This leads to another kind of understanding dysfunctionality – where research articles are written, not in order to be genuinely understood, but rather in order to “pee in an academic habitat,” which means to “fend of intruders” and stake an intellectual claim which is as large as possible. Moreover, this leads to the nurturing of difficulties for yet another reason – namely in order to maintain the status of the professional mathematician: Mathematics is difficult – because – I understand it, and I am smarter than you.
The Discriminator Questions
Interpretation: This pattern shows the filtering influence of a series of discriminator-questions that confront people that participate in the later (= higher) part of the educational process. The first discriminator is given by the question: “Do you understand?“ If the answer is ‘yes’, the next discriminator-question is “Do you want to make money?“ If you answer this question by ‘no’, then you probably go into some form of Academia, where you are confronted with the third discriminator-question: “Are you creative?“. If your answer to this question is ‘yes’, you probably become a researcher, otherwise you probably become a (academic) teacher. This is the pattern that operates behind the well-known academic contempt that the successful researcher has for the successful teacher. Within academia, if you are a good teacher, you are considered by default to be a bad researcher – until you have delivered evidence to the contrary.
Going back to the second question, if you did understand your later education, but you do want to make money, then you go into business. In this case, if you are creative, you become a consultant and market your skills in various ways, whereas if you are not, you become an employee. In both cases you work as a specialist (= dealing with special problems, i.e. trouble-shooting) – the difference is mainly the form (= security) of employment: F-skatt for the consultant, A-skatt for the employed specialist.
Going back to the first question, if you didn’t understand your later education, but you do want to make money, then you also go into business. If you are not creative, you get employed as a “routine-ist” (≈ clerk). This is a person that takes part in operating the everyday schedule of things – as opposed to a specialist, that deals with special situations.
If, on the other hand you are creative, you become an artist, and often a commercially successful one. Going back one level, if you didn’t want to make money, but you are still creative, then you also become an artist, but in this case you are not so commercially successful. Instead you tend to “realize your inner artistic potential“ in various ways, often enduring substantial forms of economic hardships on the way.
Finally, if you didn’t understand the material of your later education, didn’t want to make money and weren’t creative, then you are subjected to the fourth discriminator-question: “Are you socially conscious?“ If your answer is ‘yes’, then chances are that you become a teacher in the earlier parts of the educational process.
The Discriminator-Questions pattern has a bearing on why early teaching has become so dominated by women. The social form of intelligence displayed in the traditional female role-model is very lowly valued by the economic society, which is reflected in the low salaries associated with early teaching.
An important aspect of the Discriminator-Questions pattern is the order in which the questions are asked. In this pattern the question concerning social intelligence is the one that is asked last, which reflects its lesser degree of estimated importance. Since there are four different questions involved, there are 24 different ways to order them. Each one gives a different filter of discrimination. It is an interesting exercise to play around with the order of these questions and reflect a bit on the corresponding labeling of categories.
The Discriminator-Questions pattern has a profound impact on the educational situation in general, but its effects are probably most pronounced in mathematics. The aspect of the pattern that concerns this issue is the following: Many people who did not understand the material presented during their later education in mathematics (= higher courses) are recruited to teach in the earlier parts of the education process.