Conceptual Knowledge

Conceptual & factual knowledge can be considered 2 levels at which mathematical ideas can be understood. Factual is understanding at a shallow level, often without much understanding. Conceptual knowledge is understanding at a deeper level.

Richard Feynman calls these 2 levels as "knowing" and "understanding".

Concepts are abstract mental constructions. Physical objects & events can be experienced through our 5 senses. Concepts cannot be experienced through our senses. They cannot be directly pointed out. They need to be understood by a mental process which we may call as a “sixth sense”.

A concept is a model, theory, general principle – an idea, an inference that is used to explain and connect ideas & information. Mathematics contains many such concepts: congruence, equality, linear or non-linear relationship, function, derivative, imaginary number, internal consistency etc.

Let us take the same 6 statements quoted as examples of factual knowledge, in the previous chapter, and relook at them from the perspective of conceptual knowledge.

They would not be about “what these statements are?” but about understanding why these statements are considered true.

1. 2, 3, 5, 7 are prime numbers less than 10.

a. What is a prime number?

b. Why 2, 3, 5 & 7 are the only prime numbers less than 10?

c. Why 4, 6, 8 & 9 are not prime numbers?

2. 8 X 7 = 56

a. What is the meaning of this statement?

b. Can this statement be represented visually in multiple ways?

c. Can this statement by derived from first principles?

d. What are other statements which follow from this statement?

e. Given this statement can you work out 6 X 9?

3. Odd + Odd = Even

a. What is Odd?

b. What is Even?

c. Can this statement be derived from first principles?

d. From the above statement, can we derive what is Odd + Even?

e. Are there other relations between Odd & Even?

4. A Square is a quadrilateral with all its sides and angles equal.

a. What is a quadrilateral?

b. What is the difference between a quadrilateral & a square?

c. Are there other quadrilaterals with all sides equal?

d. If all angles of a square are equal, what is its measure? Why?

e. Is the square a rectangle?

5. A Right Angle equals 90 degrees

a. What is a right angle?

b. Why is it 90 degrees?

c. What is a degree? How is it measured?

d. Why is it called a right angle?

e. What is the importance of a right angle?

6. While evaluating an expression involving arithmetic operations, the multiplication & division operations have to be done before the addition & subtraction operations.

a. Why is this called a convention? What is a convention?

b. What is the need for this convention?

The fact that 5 is a prime is not visible by looking at various representations of 5. Properties of 5 need to be probed to understand the property of being prime and what is it that makes 5 a prime number.

Understanding concepts involves “mental effort” apart from the use of the 5 senses.

The habit of “thinking about thinking” (metathinking) and introspection would be very helpful for understanding concepts.

Concepts are difficult to learn as they are products of mental processed. Math has a preponderance of concepts. This is what makes for most of the difficulties in learning & teaching it.

Math concepts are also built on other concepts like a pyramid built with playing cards. Each concept is related in many ways to concepts below it in hierarchy and many others which are at the same level. So a weakness in understanding a particular concept will create difficulties in understanding concepts related to it.

A unique aspect of mathematics is its reliance on a sequence of dependent concepts. Unlike subjects such as history, where concepts are broader and less interdependent, math involves a deeper chain of connected ideas. This makes the learning process fragile; missing a single concept can disrupt comprehension due to the interlinked nature of mathematical ideas.

Because they are mental objects, concepts learnt in one topic may be applicable in another totally different context. E.g the number of diagonals of a polygon can be related to the total number of handshakes when two cricket teams shake one another’s hands at the end of the match. Here each vertex of a polygon can be visualised as a player and the line joining the vertices can be imagined as a handshake. Between 2 vertices or players, there can only be one line or a handshake.

Since concepts are understood through a mental process, learners may understand concepts in unique lays, depending on their “previous knowledge” & life experience. E.g One person can understand an even number as something which can be paired in twos. Another can visualise it as a weighing balance with both pans at the same level. Hence, we say that learners “construct” concepts in their mind in unique ways. We can think of concepts as “unique mental visual images” formed in our brain.

Since they are mental objects, concepts cannot be taught only through “direct instruction” since that mode may capture only certain perspectives of the concept. A particular perspective given by a teacher may not make sense to some children, who do not have the experiences necessary to understand the teacher’s perspective.

A teacher may tell that an odd number is one which leaves a remainder of 1 when divided by 2. A child still struggling to understand division may not understand this concept. Hence, she may end up memorising that statement.

Difficulty of teaching concepts may be making teachers to convert the concepts into rules which can be memorised by students. Memorising is an easier skill compared to understanding. This is a classic example of a concept to be understood made into a set of rules to be memorized.

It is this mental effort of understanding concepts that makes learning math an unique opportunity to develop thinking skills like logical & structured thinking. The corollary is that converting concepts into rules does not develop such thinking skills.

Mental effort is not visible. One of the options available to a teacher is to make this thinking visible. This can be done by questioning, discussing & listening to students’ thought processes.

Math & Visualization

Research published by cognitive neuroscientist Vinod Menon and his colleagues in Stanford Medical School in 2015 found that when a child works on a math problem, five different brain areas are active, and two of them are related to visual pathways. If math lessons involved more visualizations and other multisensory components, they would tap into these math-learning networks.

According to math educator Jo Boler, “When they (students) build or model with maths or move with maths, all of these help develop these different parts of the brain"