The key: Math by understanding pedagogical problem of math universalisation can be addressed if every child learns math by understanding and only by understanding. This requires a solution of problems at both the conceptual level as well as at what we may call the’ math linguistic’ level. The Math Linguistic Problem: Maths is not only alphanumerics. Math has many languages: thing language, the language of actions, individual and group activity, the language of shape and size, picture language, sound language… and of course, the language of pencil and paper, slate board and chalk, of numerals and symbols. We may term this latter language as the alphanumeric language of math. Alphanumerics are not mathematics. It is only one of the languages of math. For children, the alphanumeric language is a new and unfamiliar language. The difficulty that children have with primary math is mainly at the math linguistic level with the alphanumeric language, rather than at the conceptual level. A Two-Stage ProcessTherefore the learning process must be broken up into two stages. First, the stage of conceptual understanding, where the child learns and understands the concept in a familiar language. The second stage is the stage of translation of this understanding from the familiar language into the unfamiliar alphanumeric language. Over the four years of primary school, the child must steadily develop knowledge of and comfort with the alphanumeric language (representation). To that extent, towards the end of primary school, one can increasingly work directly with alphanumerics as the child develops familiarity with this new math language. For the first stage of understanding a concept, the child must encounter the concept in a familiar math language. Since doing and understanding are closely related, this first encounter must be in the language of doing. Things language, actions language, the language of shape and size, are universal and familiar math languages for children. These are the languages in which the child must first encounter a new concept. This is illustrated with three examples.
1. The addition is Joining
Almost all of primary mathematics can be built on a single concept which is: Addition is joining. Things can be joined. Shapes can be joined. With jodo blocks (a math manipulative) the child learns how to represent 3+1 and 2+2. With jodo blocks, the child also learns to use the symbols >, < and = which represent the bigger, smaller and the same size. With this, the child can discover that 3 + 1 = 4, in the language of things. After achieving complete familiarity and understanding with these operations of making and comparing, the understanding can be translated into alphanumerics. The addition is thus discovered by a process of making and comparing things. All concepts of primary math can be traversed by a process of performing activities, working with things, and problem-solving in a carefully designed sequence.
2. Multiplications mean rectanglesMultiplication of integers is repeated addition of the same number to itself. In the things language of addition as joining of jodo blocks, joining the same number to itself repeatedly generates a rectangular shape. Making rectangles is another way of understanding multiplication tables. The three times table on the mathemat is all the rectangles with one side made with three plugs. The student constructs, counts and writes the multiplication table.
3. Fractions is not a new subject, but a part of the division.
Fractions can be made and understood by dividing length, or area, or volume, using a length of cord, or tiles/paper, or bottles of water respectively. The addition of fractions is encountered in the same way: Addition is joining. Once understood with things, the next problem, the more difficult one, of translation into alphanumeric language can be addressed separately. If we collapse the two problems of understanding and of alphanumeric representation into a single problem, understanding fractions becomes difficult and almost impossible. Separating the two problems, and addressing each separately, with understanding in things language coming first opens up the possibility of universal proficiency with fractions.
Universal Active Math
The above three examples are only specific illustrations of a comprehensive and general, two-stage method: Learning a new concept by a structured sequence of solving problems with things, shapes and sizes. After conceptual understanding is gained in this familiar, universal, things language, the second stage of translating into pen and paper representations of pictures and alphanumerics is achieved by another structured sequence of problem-solving.
In the first stage the children construct material structures in a manner which facilitates the formation of appropriate mental structures. In the second stage they are helped to translate these mental structures into the pencil-paper symbols of the alphanumeric language.
With this two-stage approach, it is possible to make every mainstream student understand and be comfortable with all the competencies of elementary school math including fractions, decimals and negative numbers. Thus we can propose and construct a comprehensive pedagogy for universalisation of primary math, which we call Universal Active Math.
Three connotations of ‘Universal’
The term ‘Universal’ here has three different connotations.
Firstly, a pedagogy for achieving universalisation under existing real conditions.
Secondly, the use of a universal language- the language of things, for the child’s first encounter with a new concept.
Thirdly, math as a universal language of the natural and social sciences.
It must be emphasized here that the above pedagogy does not require a change of syllabus. However, it does change the manner in which the syllabus is transacted.
Pedagogy by itself will not result in universalisation. Universalization needs something more than subject enrichment in the classroom. It necessitates systems for mass implementation and rigorous methodology.