Mathematics Teaching: “From Concrete to Abstract” Approach
Problematic Teaching Method Leads to Mathematics Phobia
Mathematics is universal. It is considered as one of the most important subjects in school level curriculum across space and time, primarily owing to its importance in day to day function. Mathematics continues to enjoy an unabated status as without it the most sophisticated scientific and technological problems could never be solved. But the very utterance of the word ‘Mathematics’ seems to arouse two diametrically contradictory feelings and reactions. Some feel at home with mathematics, while a huge mass of students develop an aversion towards Mathematics. Their score in mathematics remains very low. As a consequence, students develop serious anxiety. Students start escaping mathematics. The incessant avoidance of mathematics entails what many have termed as Mathematics Phobia. This fear of mathematics is universal. One of the causes of Mathematics Phobia results from weak teaching method and mathematics pedagogy. Students are instructed to follow (almost like a ritual) a particular method while solving any mathematical problem. The traditional method of teaching mathematics compels students to abstract certain mathematical rules out of nothing. The concrete - pictorial part remains absent from the scope of mathematics teaching.
A Different Approach: “From Concrete to Abstract”
The problem of fear of mathematics among students could be solved if we take a different approach in teaching from elementary level onward. ‘From Concrete to Abstract’ approach in teaching mathematics could be a point in case. Abstract ideas in mathematics could be formed by drawing on concrete examples. It is a system of learning that uses physical/concrete/pictorial resources to build a child’s understanding of abstract mathematical topics without any difficulty. We believe that conceptualization of the mathematical concept must remain fundamental in our approach of teaching.
Case Study of Multiplication Algorithm
What exactly do we mean by ‘From Concrete to Abstract’ approach? How does it influence the pedagogy of mathematics teaching? To address the questions, let us consider the traditional way/method (also known as multiplication algorithm) of solving 2 digit long multiplications (for instance 37×23). It has become some kind of a ritual that we instruct students to follow the step by step method:
1. Begin the multiplication from the right and continue towards left. But we never make it very clear as to why we begin from the right. This is sometimes awfully perplexing to the young learners as they learn to write anything from the left.
2. Complete the ordinary multiplication of the single digits, but put zeros usually in the second line of computation. Here the students have to imagine the existence of a zero.
3. Some leading digits are to be written up on top. It is again not very clear to the students as to why they would take some digits on the top.
4. Finally students are required to complete the multiplication by performing addition.
| - | 1 2 | - |
|---|---|---|
| - | 3 | 7 |
| X | 2 | 3 |
| 1 | 1 | 1 |
| 7 | 4 | 0 |
| 8 | 5 | 1 |
The obvious question is: Why does the multiplication algorithm work? Most of the times, students learn the method without realizing the rationale behind it.
But there are many ways with which one could clarify the abstraction involved in multiplication algorithm. One such way is geometrical interpretation of multiplication. The geometric model of multiplication is area of a rectangle. Hence, computing the product of two numbers (let’s say 37×23) is a computation of the area of a rectangle with side lengths 37 and 23 units. Thus, here 37×23 corresponds to the area of a 37-by-23 rectangle. The student has to divide this rectangle into convenient parts (usually in terms of 10s, 20s, 30s etc.), compute the area of each piece, and add up individual areas (as shown in the picture). There is no doubt that by drawing on such technique, teachers could easily explain the traditional algorithm.
This is how we must strengthen the mathematical ability of the students. We have no doubt that teaching mathematics at this point becomes fun and students would be able to comprehend the most abstract mathematical concept from the concrete examples, as shown above.
