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Knowledge Profile of Math
Before we look at effective ways of learning math, we need to understand the nature of math as a subject to be learnt.
Like any other subject learnt in school, the knowledge in math can be classified into 4 kinds.
1. Factual Knowledge – that needs to be “remembered & recalled”
2. Procedural Knowledge – that needs to be “practiced”
3. Conceptual Knowledge – that needs to be “understood”
4. Habits & Attitudes to be developed – that needs to be “internalised”.
Deep & Shallow Knowledge
Concepts are abstract ideas which cannot be sensed by our 5 senses. They have to be grasped by a 6th sense using our thinking skills, a process which we call understanding. An example is the idea of a prime number. We cannot observe this just by seeing a number.
Understanding itself is a difficult idea. There is no clear understanding of what is understanding. Math is a subject which contains a preponderance of concepts. Children in primary school do not have the mental capacity to directly "understand" concepts. Hence the pedagogy has to be planned accordingly.
Factual knowledge consists of statements, definitions & conventions. The statements may be mathematically & factually correct or incorrect. Their correctness however can be verified.
Factual knowledge is about communicating any math content. The communication can be done even without understanding what is being communicated. Hence a student can memorize a statement like " 7 is a prime number" without understanding what is a prime number and why 7 is a prime number.
Hence we can think of factual knowledge is communicating conceptual knowledge at a shallow level.
Understanding the deeper concept makes it easier for a learner to "remember" the associated factual knowledge easily and precisely using the correct vocabulary,
Necessary & Arbitrary Knowledge
Another perspective with which we can view these two levels of knowledge is Necessary & Arbitrary knowledge.
Conceptual knowledge is "necessary" to understand math. Factual knowledge could be called "arbitrary".
Factual knowledge are the vocabulary with which we communicate conceptual knowledge. The vocabulary can be different in different languages and contexts. We can describe prime numbers either as "line numbers" or as numbers which have "only 1 and the number itself as factors".
Math also has an equal amount of procedures & computations to be mastered. Many procedures themselves are based on understanding of certain concepts. For example, the procedure for adding two 2-digit numbers is based on a concept called "regrouping" or "carry over".
Procedures need to be practiced many times, with an understanding of the underlying concept, to attain mastery. If addition of two 2-digit numbers is practiced many times, with an understanding of whether a "carry over" procedure needs to be performed and if yes, how it should be performed, then we will master that procedure. Mastery means we would be able to do all varieties of adding two 2-digit numbers, quickly and accurately.
Habits & beliefs about math and learning math cannot really be taught. An example is the realisation that mistakes in math are stepping stones to learning correct math. These cannot be taught directly by a teacher. A student has to internalise these by observing teachers, peers, parents & the society at large, over a period of time.
You can see that the four kinds of knowledge have been arranged in ascending order of difficulty – both in learning & teaching.
The difference between these 4 kinds of knowledge would become clear if we understand the difference in the four common school subjects - Language, Math, Social Studies & Science.
We will represent in a table, the proportion of the four kinds of knowledge in these four school subjects.
An Analysis of Subjects Learnt in School
In terms of concept, language & social studies have the least, science some more and math has the maximum. We are talking about social studies & science as taught in schools, not what they are as disciplines. Concepts in social studies are too difficult for school students to understand. Science, as taught in schools, is mostly history of science.
In language the concepts are grammar & phonic rules which are a minor aspect of language. That is why children can pick up a language without learning formal grammar & phonic rules.
Math has a lot of concepts, even in pre-school. These concepts are both hierarchical and connected across different levels.
In terms of facts & information, Social Studies has the maximum, followed by language & science. Almost the entire vocabulary in a language needs to be learnt from an external source. Science, as studied in school, has a lot of information related to history of science. This is the reason social studies is seen as a "memorising" subject.
Math has very little of facts & information. Mostly it is about numerals and other symbols used in operations & equations. For a student who has understood math, almost no preparation is needed before any assessment.
In terms of Procedural knowledge, language has the maximum skills in terms of LSRW. Math has almost the same in terms of computations & algorithms (procedures). Skills in science are lesser and are related to conducting experiments and sketching. Skills in social studies are even less and are about drawing and using maps.
This analysis reveals that each subject is made of varying amounts of the four kinds of knowledge. The implication, which we will see in a later chapter, is that each of these subjects need to be taught differently.
We will look at each of the four knowledges in detail in subsequent articles.
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