This is a first installment of the Sgt. Pepper’s Lonely Hearts Journal Club, where we will monologue about interesting papers.

This week’s pick is:

Alexandre V. Borovik – Calling a spade a spade: Mathematics in the new pattern of division of labour

# Summary

The paper looks at the socio-economic roots of the crisis of mathematical education. As the economy of developed countries is turning more and more technology- and knowledge-intensive, the demand on mathematical ability becomes bimodal: most of the population doesn’t require even the most rudimentary arithmetical skills, while a very small cognitive/technological elite requires ever-specializing knowledge. This deep knowledge hard to acquire both because of length of training and scarcity of quality teachers.

The dire state of mathematical (and science) education is then both caused by the market economics (it is a very expensive and highly uncertain investment for pupils, demand is low in terms of number of employments) and in the long-term it feeds back into the system as scarcity of deep expertise.

# Quotes

The paper is chock-full of good quotes. Here is a selection, paper is worth reading in entirety.

Position of mathematical education and the source of its crisis

concentrate on mathematics education, as an important and well documented area of interaction of mathematics with the rest of human culture.

[…] forces that drive these changes come from the tension between the ever deepening specialisation of labour and ever increasing length of specialised learning required for jobs at the increasingly sharp cutting edge of technology.

As an example of long-term infeasibility of over-specialization:

There are more mobile phones in the world now than toothbrushes.[…] However, practical necessity forces us to teach a rudimentary MP3/MP4 technol-ogy, in cookbook form, to electronic engineering students; its mathematical content is diluted or even completely erased.

This leads to an interesting contrast between high- and low-skilled labour:

In the clothing industry nowadays, cutters are replaced by laser cutting machines. But a shirt remains essentially the same shirt as two centuries ago; given modern materials, a cutter and a seamstress of yesteryear would still be able to produce a shirt meeting modern standards […]

What a 19th or 20th century cutter would definitely not be able to do is to develop mathematical algorithms which, after being converted into computer code, control a laser cutting machine. Design and optimisation of these algorithms require a much higher level of mathematical skills and are mostly beyond the grasp of the majority of our mathematics graduates.

The driver for this change:

It is this tension between the ever-increasing degree of

specialisation and the ever-increasing length of specialised

education that lies at the heart of the matter.

Disappearance of middle-level skill level from the point of view of economy:

Despite popular perception, the middle is gradually disappearing to create an ‘hourglass economy’.

[…] consequent declining need, among most of the population, regarded as employees or workers, for the kinds of skills (language skills, mathe-matical skills, problem-solving skills etc.) which used to be common in the working class […]

Dumbing-down is a rational—from the capitalist point of view—reaction to these labour-process developments. No executive committee of the ruling class spends cash on a production process (the production of students-with-a-diploma) that, from its point of view, is

providing luxury quality.

And from the from the point of view of the students:

Certain levels of mathematics education are not supported by immediate economic demand and serve only as an intermediate or preparatory step for further study. From an individual’s point of view, the economic return on investment in mathematical competence is both delayed and less certain.

The outcome (emphasis mine):

As a result, the West is losing the ability to produce competitivelyeducated workers for mathematically intensive industries.

Any chance to remedy the status? The path through standard education is hard:

The job market is changing fast and improving education is a slow and difficult process.

Mathematics education has a 15 years long production cycle, which makes supply-side stimuli meaningless.

Many people have high hopes for computerization of mathematical education. Borovik is rightfully skeptical:

[…] when a certain previously “manual” mathematical proce-

dure is replaced by software, the design and coding of this software requires a much higher level of mathematical skills than is needed for the procedure which has been replaced—but from a much smaller group of workers.

Another possible path is deep education via homeschooling, “math circles” (clubs) or via “*Zunft system” – *highly specialized, deep mentorship in an almost family setting. The problem with those solutions is that they do not scale to sufficient number of pupils as the number and time of high-quality mentors is very limited.

Borovik provides several examples of tensions and problems in the current education system (memorization, meaningless repetitive tasks, discussion around long division, the impact of smartphones and universal mathematics-solver software).

The ultimate aim would be to provide “deep mathematics education” – a concept developed by Maria Droujkova:

When I use the word “deep” as applied to mathematics education, I approach it from that natural math angle. It means deep agency and autonomy of all participants, leading to deep personal and communal meaning and significance; as a corollary, deep individualization of every person’s path; and deep psychological and technological tools to support these paths.

This is important because:

The potential for further intellectual metamorphoses is the

most precious gift of “deep mathematics education”.[…]

I lived through sufficiently many changes in technology to become convinced that mathematically educated people are stem cells of a technologically advanced society, they are re-educable, they have a capacity for metamorphosis.

Finally, this presents the Democratic nations a trilemma:

(A) Avoid limiting children’s future choices of profession, teach rich mathematics to every child—and invest serious money into thorough professional education and development of teachers.

(B) Teach proper mathematics, and from an early age, but only to a selected minority of children. This is a much cheaper option, and it still meets the requirements of industry, defence and security sectors, etc.

(C) Do not teach proper mathematics at all and depend on other countries for the supply of technology and military protection.

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