Enquiring into the origins of any phenomenon is the most difficult task. An answer to the question “chicken came first or the egg?” is still eluding many a philosopher. Thus, to set any process in motion requires a lot of talent and resourcefulness. Once the initial momentum is achieved external forces can come into play to overcome the inertia and add to the initial achievement.
In this context it is essential to recall the famous adage “first impression is the best impression”. Some say it is the last impression.
There is an aura created around this subject and people have come to look at it with awe. Mathematicians have been elevated to a pedestal and a mathematical phobia is prevalent, in general, among the masses.
Hence, the primary teacher who initiates a child to this subject has the prime task of removing this phobia and creating a first impression that is best for them, ensuring that they are perfectly at ease while continuing to learn the subject at higher levels. As every new bit of information adds only if consistent with the existing, it is essential to provide adequate foundation for the years to come.
The intention is not to lecture on the normal teaching methods or aids which all teachers are well equipped with. Rather, the focus here would be on some of the essential competencies that are missing in students reaching secondary levels.
The question therefore is “what primary methods to meet secondary needs?” At this juncture I should justify how I qualify to speak on this subject.
I have been teaching mathematics at all levels from class 4 to 12 in schools. Also, as the mother of two children, I have taught them from standard 1. Thus, to a certain extent know the mental calibre of children at all these levels.
Many a time I have heard parents complain that their children have been doing very well scoring 80% to 90% till primary classes but have shown an abrupt decline in performance at the secondary level.
We all know that mathematics learning is a continuous process and with a concentric syllabus vertically there should not be such complaints. Why should not there be a smooth transition from a lower level to a higher level?
For instance, from class 6 to 8 all theorems prescribed for classes 9 to 10 are covered but the difference is in the treatment. Whereas in the middle classes it is experimental, from class 9 upward it is totally logical. This transition leaves the student disoriented till they could get acclimatized to the new analytical approach. What was required of the middle school teacher was an inductive correlation to logical constructs in the experimental approach.
Hence, here the emphasis will be on those teaching strategies that have relevance in the contexts of gaps in learning in primary from a secondary perspective.
Need-based Mathematics
The belief that mathematics is all numbers, that it is only for those who are intelligent with numbers and computation should be corrected. Children should be taught that mathematics is after all man-made. Only where there has been a need mathematics had found an expression. Thus, no concept should be taught unless a need is created for the same in the learners’ minds.
Teaching Number System (Class 1 - 6) from Natural Numbers up to Integers
When we teach natural numbers in class 1, before introducing the numerical symbols the need for the numbers themselves should be generated. Before introducing fractions in class 3, the need for fractions should be created by illustrating how natural numbers cannot explain certain life situations, their inadequacy leading to the need for new numbers, the fractions.
When introducing the concept of integers in class 6, students should be apprised how the numbers so far studied cannot explain situations involving directions. Thus, a need is generated for new numbers.
Gong back to the natural numbers, at class 1 level, the need for these numbers is generated by providing situations consisting of one and many (one moon and many stars), identifying and isolating one from many, recognizing the unit for the physical entity which would eventually end up in working a continuum in terms of the discrete.
Incidentally this opens up a new angle in mathematics teaching. People normally opine that mathematics teachers are taciturn. They believe only in doing and not talking.
Talk Mathematics
Mathematics has its own language which has been developed from our day-to-day language, in this context, through the medium of instruction, English.
One and many has an analogue in English, singular and plural. It is important that the student can articulate mathematical rules by expertly transiting between the technical and common languages. Only the language ingrained in the mind could help to transfer the concept to figures and then to computation.
For instance, let us take the concept of equivalent fractions. The definition in a class 3 text book reads: “two fractional numbers are said to be equivalent if and only if they represent the same part of the whole”.
It may sound a bit high for the mental ability of the students. But we do teach the concept of a fraction as some part of the whole. The first step is to see “how many parts?” The second step is to see the equal parts of the same whole.
Same unit divided into 8 equal parts and 4 equal parts respectively will show 2 parts of eight and one part of four as the same part of the whole and hence equivalent.
Each time handling a concept may involve different sensory experiences yet should culminate in articulating in the common language since the human mind processes information only through primary language tools most effectively.
If by some chance a child fails to comprehend the full import of a concept through its experiences at a particular level, the language component will remain residual in the brain and will emerge with new understanding with evolved maturity of age at a later stage.
This once again leads to the next point I wish to emphasize.
Reinforcement
For an indefinite number of times across the grades the same concept figures with perhaps a progressive increase in information. It is vital to go back every time to the origins and relate to the increased levels. Some teachers feel that as certain topics are to be dealt with in a higher class they need not be taken at a lower level and some others feel the reverse.
At a lower class the introduction will take longer than in a higher class. But the level of appreciation in a higher class of the same concept is much more.
You should see the eyes sparkle when I explain the fundamentals of decimal system of numeration in a higher class. Almost every time the topic figures a few questions are important irrespective of the levels such as
What are digits?
What is decimal system of numeration?
Are there other systems?
Which number enabled decimal system possible?
What is area? perimeter?
Invariably children, and sometimes even teachers, have been unable to articulate. These ideas remain as a gut feeling and refuse to find shape in words.
There is a mix up between a digit and a number. All these once again emphasize the importance of language correlation and training. There is further a necessity for reinforcement irrespective of its relevance, from the examination point of view.
I feel that mathematics should be taught as even poetry is taught in languages. It should be critically analyzed and appreciated. Why do we say "a chord", "the centre of a circle"? "A radius", if it is a segment, and "the radius", when it is the fixed distance of any point on the circle.
By comparing the usage of articles in English in the mathematical concept, more than one chord and a unique centre could be understood with a better clarity.
If by such training a child has grasped, learnt and stored in their memory, the expressions and rules that apply to problem solving, applications come automatically.
After a certain time, transition between language, rules and their transference to problem- situations will be simultaneous.
When we are laying elaborate foundations to understanding the concepts, we should also not sacrifice computational speed.
Modern vs Traditional
There is a misinterpretation of modern math. There is nothing modern about it. It is just an elaborate effort to coordinate the language of mathematics with the ordinary language, in an endeavour to explain the how and why of the traditional and fast computational techniques. We accepted without asking why, that negative times negative is positive. The modern practitioner is expected to explain why even if not questioned. But it does in no way prevent the teacher in providing tips for memory and shortcuts to computation once conceptual foundation is laid.
Author’s note: This was meant for addressing primary teachers during a workshop organized in an in-service course, a regular feature of Kendriya Vidyalaya Sangathan. It never materialized due to paucity of time. However, my then-Principal Mr. Dhushyanthan encouraged me to publish it in KVS magazine “sangam” and CBSE publication “CENBOSEC” volume 30, No3., September 1992 edition.
- Chithralekha Gurumurthy
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