Universalisation of primary math

Dr Vivek Monteiro

Universalisation1 In the context of India, the problem of universalisation of elementary education has two important aspects. First, the magnitude of the problem: It concerns about 300 million children between the ages of 0 and 14 years, at any one time. Second – the legal mandate for universalisation. We begin by looking at the second aspect.

The constitutional mandate
Under the Indian Constitution, for 43 years, universal access to education was a Directive Principle: Article 45 provided for free and compulsory education for all children until the age of 14. However, the Directive Principles of the Indian constitution are not legally enforceable.

That position changed in 1993, with a historic judgment of the Supreme Court. Thereafter, in 2002, the 86th amendment introduced to include the right to free and compulsory education explicitly as a fundamental right of children between the ages of 6 and 14.

As a consequence of these developments universal access to education for young Indian citizens is a right in the same sense that universal suffrage is a right of every citizen above the age of 18, irrespective of gender, class or caste.

Quality of education
What is meant by free and compulsory education? What kind of education? Though the Indian Constitution does not explicitly mention the question of the quality of education to be provided, the issue does arise implicitly through the commitment in its Preamble to equality of opportunity for all Indian citizens.

Quality for equality
Indian society is highly unequal characterised by inequality based on economic class, caste, gender, community and region. Education must therefore be an instrument for promoting and building equality of opportunity, in a manner which counters and compensates for the widespread inequalities in other areas. Under the Indian Constitution, education of the poor and disadvantaged cannot be inferior to that available to the socially and economically privileged sections. On the contrary, it must be of such a high quality that it can compensate for the endemic inequality faced by the poor and disadvantaged.

It is evident that the requirement of “quality for equality” cannot be met in a market framework, where education is a commodity to be purchased. The low purchasing power of the poor will necessarily mean lower quality education. Privatisation of education may provide better quality of education for those who can afford it, but it cannot address the constitutional mandate of education for equality. It is therefore the responsibility of the State to provide education of high quality to all its citizens outside of and independent of the market. This cannot happen without strengthening the government school system.

The fundamental flaw of the new legislation drafted in the form of the Right to Education Bill is that it fails to address the problem of declining standards in government schools and tries to substitute them with private schools.

Having stated this basic requirement of the Indian Constitution of “Quality education for equality”, we now adopt a limited ‘preliminary’ level of “universalisation” for the purpose of this article. This preliminary level must be seen only as a stepping stone towards the universalisation mandated by the Indian Constitution. We focus our discussion on the subject of universalisation of elementary mathematics.

What is universalisation?
By universalisation we mean that all children, except a few who have special learning difficulties, must be proficient in math. By proficiency in math at a given class level we mean that the student should be comfortable with math concepts, should feel confident that he or she has understood each concept, should be able to correctly perform math skills and operations and should be able to represent and transform simple real life problems into math problems at that level. The student should have a two-way translation skill, representing real life problems with things and in numbers and narrating a real life situation for a given math expression. Basically, by the end of primary school, the child should be comfortable and able to count numbers of three and more digits, perform the basic operations of addition, subtraction, multiplication and division with these numbers, and should understand and be able to work with fractions and decimals. She or he should understand simple measurements in space and time and should be introduced to basic sizes and shapes for geometry.

Quantitative measure of universalisation
We can adopt the official quantitative thumb rule of 85×85. Eighty five per cent of children of a given class level should be proficient in at least 85 per cent of the math competencies prescribed in the national/state curriculum for that class level. Let us therefore define ‘proficiency’ as achieving a score of 85% or better in a comprehensive assessment designed for testing understanding and competency. In India, achieving universalisation of elementary math would require approximately 19 million children in each class to score 85% or better in well-designed math tests.

Universalisation as a scientific problem
Universalisation must be seen as a scientific problem. This means that it must be taken up with the same seriousness with which we send humans into space, or rockets to the moon, or with which we eliminate diseases like polio and small pox.

Achieving universalisation would necessitate that each year 22.2 million children enroll and remain in primary school and that 85% or more of these children in each class achieve proficiency as already defined. This in turn would require that all (100%) elementary school teachers in the country be comfortable with elementary math. Is this possible? Based on our experience with pilot projects in Maharashtra and Goa in realistic situations we believe that it is feasible to make systematic incremental improvements to approximate the targets in a span of about four to five years. But it will require the solution of problems at two levels: pedagogical and logistical. We begin with a discussion of the pedagogical problem.

The key: Math by understanding
The 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
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.

For children, the alphanumeric language is new and unfamiliar. 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 process
Therefore, 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 is that 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.

Three examples:
1. 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 bigger, smaller and 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. 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 means rectangles
Multiplication 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 the 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 division
Fractions can be made and understood by dividing length, or area, or volume, using a length of cord, or tiles/paper, or a bottle of water. 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 representing 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.

Universalisation2 Universal active math
The three examples given above are only specific illustrations of a comprehensive and general, two-stage method: Learning a new concept by a structured sequence of problem solving 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.

This two-stage approach should make it possible for every student to 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.
First, a pedagogy for achieving universalisation under existing real conditions.
Second, the use of a universal language- the language of things, for the child’s first encounter with a new concept.
Third, which we do not discuss here, math as a universal language of the natural and social sciences.
It must be emphasised here that the above pedagogy does not require change of syllabus. However it does change the manner in which the syllabus is transacted.
Pedagogy by itself will not result in universalisation. Universalisation needs something more than subject enrichment in the classroom. It necessitates systems for mass implementation and rigorous methodology.

Mass implementation: Standard Operating Practices
For universalisation the comprehensive pedagogy must be implementable on a mass scale. This means that it must be converted into standard operating practices (SOPs). The important question here is: Can learning math by understanding be converted into SOPs? Or is it something that will always be the privilege of only the few students who are fortunate to be taught by gifted and inspired teachers? Without SOPs there can be no mass programme, encompassing tens of thousands of schools.

A carefully designed activity kit in every classroom, teachers trained in the use of activity for learning math, workbooks for helping children to translate hands-on understanding into pencil and paper language, comprehensive assessment modules for continuous diagnostic assessment, and good manuals for both teacher and students separately, constitute the five elements of an SOP package.

Can we measure universalisation?
Assessment plays an essential and defining role in achieving universalisation. In mathematics it is possible to assess very accurately, by properly and scientifically designed questions, students’ understanding of the subject. This can be done at an individual level, classroom level, or at a mass level. Whether at the district level, state level or national level, progress, or lack of it, towards universalisation, has to be continuously and objectively assessed. In math this can be done quantitatively, accurately and rigorously. This fact makes it possible to make incremental progress towards universalisation at a school level, cluster level, block level and district level. The method of benchmarking, widely used in industry, and the method of successive approximations, well known in science, are of direct relevance for taking systematic incremental steps towards universalisation.

What kind of test?
The assessment test must consist of all three components: practical, oral and mental math, and written. Giving teachers well-designed sample tests can help shape both desired classroom practice and enhance non-rote, activity-based learning. The unit tests and semester exams can become critical diagnostic tools for every teacher and school to proceed systematically forward, identifying and remedying weaknesses in a student’s understanding. To perform this function the assessments must be developed with the complete participation and confidence of the teachers and be specifically designed as a diagnostic tool, testing for understanding.

Neglect of a basic necessity
It is unfortunate that continuous and comprehensive assessment is not receiving the importance that it deserves, at all levels, from school to national. It is difficult to understand this lacuna in national and state educational policy and practice. The NCERT performs tests only biannually and that too only at the class three and class five level in some sample districts. Both the NCERT as well as the ASER tests do not meet the standards of ‘comprehensive testing’. In fact, scientific assessment must be done at all levels, in all classes, in a transparent manner. This must be done both by the teachers as well as by independent bodies. The NCERT tests for mathematics at the class five level show wide disparity between districts and states. With well designed universal tests, the best performing classes, schools, panchayats, blocks, and districts can serve as engines to pull the rest of the nation towards universalisation by benchmarking.

In mathematics it is possible to assess understanding competency wise, without complicated tests, and arrive at a very detailed understanding of the status of the class, school, block, district, or state. It is also possible to design a carefully weighted test such that the average score for the constituency which is being assessed by itself will give one a good idea of the status of the constituency. Independent end of year testing is necessary, and can reliably establish the progress towards, or distance from, the universalisation norm. Our experience in a number of different situations shows that achieving a progress of 10% each year in the average score, over the previous year, is a target which is entirely feasible.

Universalisation3

SOPs and Systems for UAM
The logistical aspect is no less important than the pedagogical aspect. For building a mass programme, after converting the pedagogy into standard operating practices (SOPs), incorporating the best practices available locally, nationally and internationally, a number of critical questions remain. Some of these questions are:

How many days of actual instruction do the students receive in an academic year? How many days are subtracted on account of non-academic burdens imposed on the teacher?
How many hours of quality instruction does the student receive in a year in each subject?
How much time from the school day is subtracted for serving and consuming the mid-day meal (khichdi)?
What is the average down time in a school day due to various factors?
We have observed that in many government schools a great deal of down time takes place for entirely avoidable reasons like delay in provision of the mid-day meal every day, or delay in provision of textbooks each year.
It is encouraging that some of these questions are explicitly addressed in the new legislation on the “Right to education” that is on the national agenda. The legislation however is not specific on the systems of accountability. This is a crucial lacuna which must be filled.

Accountability at all levels
Though teachers must be accountable for the achievements of their students if universalisation is to be attained, it is equally important that other stakeholders in the system are also held accountable for their function. The role of the administration to ensure that all the necessary inputs are available at the proper time is crucial. Administration must be held accountable along with teachers for instructional down time.

Mass teacher participation
Mass participation of the teaching community is, of course, critical to achieving universalisation. Teacher training is an important component but is obviously, not the only important factor. Given the commitment and accountability of all major stakeholders, it is possible to conceive a nationwide systematic programme which will over a period of four years cover all four classes of primary school. How can tens of thousands of teachers be trained for a programme on this scale?

Teacher training
The teacher training necessary for this programme consists of four days of intensive training followed by one-day sessions every six weeks. These interactions should be linked to the unit and semester tests and be held immediately after each test. This can and should be done within the Sarva Shiksha Abhiyan parameters for it to become completely replicable. The strength of the learning by doing approach, which is the cornerstone of the universal active math, is that teacher skills continuously improve as they implement the programme. Our experience in implementing this programme in many hundreds of schools in different states has emphasised the need for continuous school and classroom visits by a math facilitator to help the teacher. The block resource persons and cluster resource persons can be trained to perform this function. Where the programme has been rigorously implemented average achievement levels have improved from the first year itself. In classrooms where rigorous implementation has taken place the norms defining universalisation have been reached and exceeded. A copy of the tests administered, and the results for some ongoing programmes such as SSA in three talukas of the state of Goa, can be obtained by sending an email request to the author.

In summary
From the above discussion, which is based on experience in implementing programmes in realistic situations, we can assert optimistically that the necessary and sufficient conditions for achieving universalisation are in the realm of feasibility. Universalisation of primary math in the limited sense that we have defined, is attainable, within the available resource parameters.

The author is a trade unionist with a background in mathematical physics. He is the founder advisor of Navnirmiti, which is a self reliant organisation working for universalisation of quality education. He can be contacted at [email protected] .

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