Teaching Concepts

Each Kind of Knowledge Needs a Different Pedagogy

We need to realise that each of these 4 kinds of knowledge are qualitatively different and hence need a different kind of teaching & learning process to be effective.

Each of the topics in math, contain a mix of these 4 kinds of knowledge. Hence even within a topic, different sub-topics may need to be taught differently.

One of the reasons why math is universally considered a difficult subject is that, most teachers use the same teaching method - blackboard, chalk, talk, textbook, class work, home work, time-bound tests - for all topics in math. Or for that matter for subjects other than math also.

Concepts Are Difficult for Children

Concepts are mental ideas or objects which cannot be sensed directly by our 5 senses. They have to be understood by a sixth sense in our mind. But we have seen that evolution has equipped our brains to understand abstract concepts.

The difficulty with math is that even preschool & primary school math contains many concepts for which the children are “developmentally”, as per Piaget, are not ready.

Hence all the concepts need to be presented to children in a way that they can make the connection and “construct” them in their minds. So “direct instruction” is not possible in case of concepts.

Buckminister Fuller has captured this idea beautifully. He said, "If you want to teach people a new way of thinking, don't bother trying to teach them. Instead, give them a tool, the use of which will lead to new ways of thinking."

Mathematician John Horton Conway, put it a bit differently. He said "You don't educate people by telling them useful things; you educate them by telling them interesting things"

Children do not understand the purpose of learning. Learning needs to be made interesting to them.

Assimilation & Accommodation

Concepts are learn through a reorganization of existing ideas in the mind. This consists of two processes; assimilation & accommodation as named by Piaget.

Assimilation is how we, see the world with the current knowledge we have.

Accommodation is how we learn and change our lens to reorganize what we know into more advanced thinking. It is a new perspective in which we see the information that we have in the mind.

This task is mostly mental, requiring a lot of introspection, which a teacher can at the most facilitate. That is why teaching concepts is difficult. A teacher has to understand the current knowledge of a student, teach the new knowledge and simultaneously help the student adjust his/her perspective keeping in mind the student's abilities & weaknesses.

Teaching Concepts

Concepts cannot be taught. They can be caught by using an appropriate tool (manipulative)

Let us see several strategies which can be used, to understand conceptual knowledge.

1. Demonstrate with manipulatives

   1. Place value notation of a 2-digit numbers like 23 is usually explained as “2 in the ten’s place and 3 in the ones place”. This stream of words has no meaning to most children who cannot understand what is a “ten’s” place & a “ones” place.

      1. The place value concept can be mirrored to children by representing 23 as a collection of 2 bundles (of ice cream sticks) and 3 sticks.

   2. The idea of a prime number is very difficult to understand even for high students. Telling children that they are numbers which can be divided only by 1 and the number themselves does not help matters.

      1. Representing numbers with tokens in various geometric shapes is a better approach.

      2. Composite numbers are those which can be arranged as rectangles.

      3. Prime numbers can only be arranged in a straight line.

   3. Different kinds of triangles can be easily formed and understood by using long thin broomsticks. This is far more effective and efficient that drawing triangles on a notebook page.

2. Relate to life experiences

   1. Addition & subtraction are processes which can be related to life experiences of children. Children collect toys & other knick knacks. They also exchange them with their friends.

      1. Addition can be related to a case where your toys increase as you receive many toys.

      2. Subtraction can be related to a case when you give away toys to friends.

   2. Division can be related to our normal habit of sharing things in our homes, schools and with our friends.

3. Visual Images & imaginations

   1. Odd & Even numbers can be understood through a story which can be visualised & understood by children

      1. Imagined a party situation where a group of people who are asked to make pairs.

      2. If one person is left without a partner the number of people would be odd.

      3. If no one is left without a partner, the number would be even.

   2. The interconnection between different kinds of quadrilaterals can be best brought out by a family-tree type of diagram.

      1. This can bring out the idea that – a square is a rectangle which is a parallelogram which is a trapezium which is a quadrilateral.

      2. And the idea that – all squares are rectangles, but all rectangles are not squares.

4. Discussions & Arguments

   1. The teacher could ask a question leading to a concept and allow students to respond and others to either agree or disagree.

   2. In this process a student gets exposed to different ways of looking at a problem and ultimately arrive at a solution acceptable to every one.

5. Metacognition

   1. Metacognition or Reflection is thinking about one’s thinking. It is active monitoring & control of the cognitive process.

   2. It is very essential in any deep learning.

6. Connecting to Previous Knowledge

   1. At the conceptual development stage, when students first encounter new ideas and skills, thinking about the relationships between their prior knowledge and new knowledge tends to help students have better understanding.

7. Constructing Knowledge

   1. Concepts are mental ideas that have to be "understood" in individual ways.

   2. Thinking and introspection are necessary processes in "constructing" concepts in the mind.

   3. The time & space for this has to be provided to students.

8. Use of non-examples

   1. Sometimes, use of a non-example will enable an understanding of a concept by creating a counterpoint.

   2. Comparing factors of 7 & 8 may make it easier to understand the difference between prime & composite numbers.

9. Math Activity Centre

   1. Setting up a Math Activity Centre is an excellent way to learn concepts.

   2. The centre can be in a separate space, fully equipped with all kinds of manipulatives and videos. Children can be taken to that centre for a block period so that they can work on different activities under the guidance of a teacher.

10. Practical Examination

    1. A practical examination should be a part of assessment of math learning.

    2. Practical examination can consist of questions which require student to demonstrate their understanding of concepts using manipulatives and sketches.

11. Clearing Misconceptions

    1. Concepts are built both hierarchically and bridging different topics.

    2. We have seen that at the lower levels, some concepts may be taught in a simpler way which may need to be corrected at higher classes.

       1. An example is the idea that multiplication always increases magnitude. This is valid with whole numbers, but not with fractions.

       2. Hence while starting the topic of fractions, the teacher should revisit this old idea and tell the students that the old idea now needs to be revised.

       3. If this is not done, the old misconceptions would continue in the subconscious mind and hinder more complex understandings.

12. Concept Formation is a Continuous Process

    1. We "construct" our conceptual knowledge with the existing experience & understanding that we have. Hence as we grow in experience, the strength & depth of our conceptual understanding would also change.

    2. So we will be permanently in the stage of "instrumental" understanding of a concept.

       1. An example of the idea of a prime number.

       2. We start with a visual idea that prime numbers can only be represented as a line.

       3. Then we understand the fact that a prime number has only two factors, 1 & the number itself.

       4. Then we may understand the idea that prime numbers are the building blocks of whole numbers.

       5. These understandings may keep on getting more deeper and complex.

       6. A mathematician's understanding of prime numbers would be vastly deeper than that of a school student.

There is no "royal way" to understanding concepts. It is difficult and takes a lot of time & effort on the part of the student.