**The Formation of Mathematics Education**

**How Mathematics Education became a Ritual and what this Ritual does in Modern Society**

**Sverker Lundin, University of Gothenburg**

sverker.lundin@gmail.com

sverker.lundin@gmail.com

**The practical arithmetic**

Sociological mechanisms connected primarily to social elites

In Sweden, most of these books were organized similarly, starting with a presentation of the number system, and leading to practical "recipies" for how to handle specific problems.

To master the practical arithmetic, you needed to memorize rules and tables, together with various "shortcuts", to be able to produce the right answer quickly and without mistakes.

Shortcuts and practicalities played an important role in the practical arithmetic, for making computations manageable.

A second origin of mathematics education is a discourse promoting mathematics as a high status science by connecting it to truth, logic, rationality, God, etc.

High expectations

Persistent self-critique

Other functions

When an increasing number of children, stayed increasingly longer in school,

classroom management

emerged as an urgent topic of discussion:

One solution, which was to become highly successful, was exercises, presented in textbooks, designed to keep children busy and thus silent. In Sweden this was called "silent practice" ("tyst övning")

In the Swedish discussion of these exercises, two kinds of arguments reinforced each other:

The philosophical idea of (inner) self-activity, was reinterpreted as meaning that children work by themselves with their textbook.

But ever since this form emerged it has been claimed to malfunction:

... despite some 12 or so years of compulsory mathematical education, most children in the developed world leave school with only a limited access to mathematical ideas. (Noss & Hoyles,

1996

)

Mathematics, then, has degenerated from the most intellectually educative subject of all into a mechanical cramming of rules actually wrecking to the intellect. (Petrini,

1905

)

The teaching of arithmetic amounts to the most stupid bondage of the intelligence of the Swedish people, which the learned pedantery since centuries has managed to impose on reasonable beings; for something unreasonably weirder and harder than arithmetic, as it is generally presented in textbooks and by teachers according to them, has never in any subject been created, if not in astronomy before Copernicus. ... very few of those, who learned arithmetic in school, are able, beside the habits of work, to handle and answer the often most simple of arithmetical questions. (Otterström,

1849

)

The expectations of what will come out of this highly specific and stable arrangement are extreme:

Proponents of education, business and society, powerfully and unanimously express that good meaningful knowledge of mathematics is a precondition for self-confidence, democracy, economic growth and life-long learning. (

The Swedish National Center of Mathematics Education, 2001

)

If it seems doubtful that mathematics education actually manages to provide people with "the power of mathematics", it is well known that it manages very well to perform other "functions" in society:

It takes care of children, so that their parents can work

It distributes children over stations in society, in a way that appears to be objective and (relatively) legitimate.

The aim of learning was

displaced

by the idea of the power of mathematical knowledge and the idea of pure rational thinking.

In these recepies, difficulty was caused by units of measurements (for money, volume, weight, length, time, etc.) having other bases than ten, necessitating work with fractions.

**I start here**

ca. 1800

**how it became...**

**what it does...**

Specific

Easy to recognize

Archaic

Dated technologies & practices

2010

1980

1920

Regulated & Monitored

Fabricated / Produced / Made

- an institution, constantly

kept

in order

Stable

Obligatory &

Time Consuming

performed in great numbers

Geographically invariant

Special time and place

Follows formalized schedule with specific rythm of training and assessment

**Mathematics in the Education System**

**A definition of the problem**

These ideas of what it meant to "know mathematics" were reflected in more or less local "education systems" where assessments were used as a means of

establishing hierarchical differences between persons

. In Sweden this kind of schooling emerge in the course of the 19th century, first in military schools, then in the public "Läroverk".

In these settings

mathematics as a school subject

took shape as something you could spend several years

doing

. It became stablilized through textbooks structured according to examination requirements. At the same time it became increasingly different from contemporary mathematical science and from various kinds of contemporary professional use of mathematical techniques.

Fantastic ideas connected to Mathematics

ca. 1650

Fantastic ideas connected to Education

ca. 1850

ca. 1850

ca. 1500

Sociological mechanisms connected primarily to "the people"

A first origin of elementary mathematics education is the practical arithmetic: a system of techniques for answering questions pertaining primarily to trade and economy.

It was presented in books consisting primarily of rules, tables, explanations and exercises.

In the practical arithmetic we find the object of the persistent critique of mathematics education, directed against its own core - namely its "traditional past", which it constantly tries to lay behind.

In Sweden, these visions influenced how the practical arithmetic was presented:

Shortcuts and practicalities were replaced by "realistic" exercises illustrating the power of mathematics.

Around 1800 arithmetic got intertwined with the emerging educational theory.

The metaphors of growth and development

Taboo of "knowledge transmission". The teacher should not explain, but somehow induce self-movement.

The idea that development must be driven by "self-movement" (Selbsttätigkeit). The children must desire to learn and they must be active because the purpose is to learn to be "self-moved", i.e. to be free.

The idea of the importance of the senses, in particular seeing (Anschauung).

The idea of the not-yet-human child and its potential for development (Bildsamkeit).

The idea of the perfect human that was to be

built

by education.

Here we have a second origin of "high expectations".

The importance of starting with the very simple.

The idea of continuous progression, without any gaps.

The importance of always adjusting teaching to the "standpoint" of the child.

Mit Grube (1816-1884) hat die Hauptentwicklung der Rechenunterrichts ihren Abschluss gefunden. Denn indem derselbe ... das Rechnen in den Dienst der sittlichen Bildung stellte, hat er demselben den Zweck gegeben, der weitere Steigerungen ausschliest. Danach hat es heute auch keinen Sinn mehr, nach dem obersten Zweck des Rechnens erst noch forschen zu wollen. Der Rechenunterricht soll in seiner Weise die Idee des erziehenden Unterrichts mit verwirklichen helfen. Wie der gesamte Unterricht ein hauptmittel zur

entwickelung des sittlich-religiösen Charakters

sein soll, so nun auch der Rechenunterricht an seinem Teile.

- Berthold Hartmann, 1893

- How could a single teacher manage a group of children of different ages and different abilities?

On the one hand the exercises were deemed

necessary

for practical reasons.

On the other hand they were deemed

beneficial

for knowledge development.

In my interpretation, the establishment of silent practice as the preferred way of managing the classroom, helped determine what ideas of learning that "made sense" in mathematics education. Rules and reading were excluded for "double" reasons: practical and theoretical.

The "fantastic ideas" of educational theory, together with the "fantastic ideas" of mathematical knowledge, made it possible for silent practice to appear as (partially) meaningful.

Unconscious ambivalence

Superficiality

Unconscious ambivalence

Superficiality

Social norms induce a pressure to conform.

To evade this pressure we perform superficial acts of conformance. We act "as if" we followed the norm.

Fascinatincly, these superficial acts of norm-evasion "works", in the sense that they release us from the pressure.

Because of how they perform a double function for us, working both for the norm and against it, they become very important. Personally, we can be somewhat compulsive. Culturally, the correct performance of these acts can be safeguarded by informal and formal sanctions.

When we are forced into taking part in these superficial activities, we learn both what norms we should respect, and how to evade them.

Acts of norm-evasion keeps norms in place, but at a certain distance from our personal life. They establish running-room for us as individuals.

Politeness

Tourism

Religion

What mathematics education does, is to demonstrate, for us moderns, that we understand how important it is with mathematics and science...

...while it at the same time makes it possible for most of us, most of the time, to not care at all about mathematics.

We have delegated the work of worshipping mathematics to children. We (i.e most citizens in modern society) take pleasure in not having to take part in mathematics education.

Mathematics education amounts to a "show", a superficial display, of how we officially understand modern life to be: filled with mathematical problems, only manageable with the help of mathematical knowledge.

Reflection about authenticity and superficiality can function as a means of emancipation from obsessive behavior, and open up for discussion of what norms to follow and how.

But it can also lead to contempt for the superficial and a displaced and less pleasurable - and perhaps even stronger - obsessiveness, aimed at "reform".

**Sketch of an explanatory framework**

Applied to mathematics education

A concluding question concerning authenticity, reasonableness and obsession

The acts protect us from the total imposition of the norm

On the one hand we "know very well" that mathematics education does not work, but at the same time we do not really understand it.

The superficial enactment of the norm suggests the possibility of authenticity - in our case an actually well-functioning mathematics education:

Authentic original

Degeneration/Failure

Present day superficiality

These acts can be seen as cultural resources, not necessarily unproblematic

Conducted in a similar way in many different places

It is as such an institution that mathematics education will be discussed here. Other settings, such as mathematical research, engineering and "everyday life", falls outside the scope of this presentation.

Mathematics education is thus defined as

a highly specific social arrangement

expected to endow youth with "the power of mathematical knowledge"

well known to persistently fail

well known to perform other, less illustrious functions in society

They make it possible to sustain a self-image that is partially false

Perhaps one can say that what was formed, in these places, was various kinds of "

rites of passage

", where persons destined to take more or less elite positions in society were forced through

ordeals

consisting of "mathematical" training and assessment.

The often very difficult problems that students learned to solve in these settings were usually not scientifically relevant, nor practically useful. (In Sweden this is very clear as regards algebra and geometry.)

The problems the students learned to master,

signified usefulness and rationality

through the associations connected to mathematics...

The number of exercises became an important selling point for textbooks, and in Sweden they increased approximately exponentially in the second half of the 19th century.

Note that this interpretation of the history of mathematics education is based mainly on work with Swedish material.

Here we see a first origin of the extreme expectations put on mathematics education.

... mathematics and in particular geometry was here presented as a means to fundamentally transform people and society.

The "learning activities" coming out of these visions, circled around what

could be called

the fundamentals of arithmetic, but with a completely different aim than that of mastering the practical arithmetic..

Passing the test

indicated what kind of person you were

- rational, useful, disciplined, of good character.

Authentic politeness?

While the practical arithmetic was a (low status) tool that anybody could learn how to use...

Authentic tourism?

This past is associated with

mechanical computation

,

memorization

, superficial manipulation of

digits

,

rule-following

and generally strange and difficult ways of going about.

But this is an evaluation resulting from the specific perspective of mathematics education itself - its fantastic ideas of Mathematics and Education.

In fact the practical arithmetic most probably worked very well, for the particular problems it was designed to solve, and it could be mastered by anyone willing to put in the necessary effort.

The "mechanical" rules were criticized and replaced by algebraic principles and proof, focused on "understanding why" instead of "knowing what to do".

But as critics have pointed out and historians have shown, it is much more difficult to apply abstract mathematical knowledge than the discourse about its power suggests.

This has generally not lead to doubt in mathematics itself. Instead, failure is explained with reference to

mechanical

and superficial

"tradition"

, that must still somehow be present,

disturbing

the pure and beneficial power of mathematics.

Here we see a first origin of the "other functions" of mathematics education.

Here we have a second origin of the "other functions" of mathematics education.

Med Grube har räkneundervisningens huvudutveckling hittat sitt slut. I och med att han ställer räkneundervisningen i den sedliga bildningens tjänst, har han gett den ett mål som utesluter ytterligare förbättring. Därför är det idag inte längre meningsfullt att forska kring räkneundervisningens mål. Räkneundervisningen skall på sitt sätt hjälpa till att förverkliga undervisningens bildande mål. Precis som undervisningen som helhet bör vara ett huvudmedel för att utveckla elevens sedliga och religiösa karaktär, skall också räkneundervisningen på sitt sätt bidra till detta.

With Grube, the main development of the teaching of arithmetic has been completed. Because, as the teaching of arithmetic has been put in the service of moral education, it has been given a goal that cannot be further improved on. Therefore, it no longer makes any sense, to want to inquire into the highest goal of arithmetic. The teaching of arithmetic, should in its particular way, seek to contribute to the realization of the idea of a developing education. As education is a major instrument for the development of the moral-religious character of children, so is also the teaching of arithemtic for its part.