Sweety Rastogi

Margaret Mead had said, “Children must be taught how to think, not what to think.” However, our teaching of math, particularly in schools, does exactly the opposite. Math classrooms produce a way of thinking that is only about exams and test scores with little preparation for the real world. Students are implicitly encouraged to pursue rote learning and are not given credit for thinking out-of–the-box.

In the math curriculum in schools:
• Prime numbers are not of prime importance as the concept is not extended after the primary level.
• Commercial mathematics (percentage, profit /loss, discount) has no mention after class 8.
• Logarithms are introduced in chemistry (class 11 and 12) without mentioning them in mathematics across levels – primary, middle, secondary, or higher secondary.
• Vectors, differentiation, and integration are introduced in physics (11 and 12) before they are in mathematics though they are core concepts in mathematics.

This is a very small list of the problems in our curriculum. We need a system that rewards curiosity and creativity. Our education system needs to be progressive in its approach by letting a child blossom the way he/she wants to and create his/her own path. This incident that I narrate below is what got me thinking about the objectives of our curriculum.

One day, I received a WhatsApp message from Amrit, a former student. It was a math problem that he wanted me to solve.

The caption read: ‘These question marks need to be replaced with numbers and what is the name of this process?’

The answer was easy, but it took me 10 minutes to figure out why and to which grade this problem must have been given.

Amrit was good at math; I was surprised that he had sent me the problem. I wrote out the answer:

819 ÷ 3

= (600÷3) + (210÷3) + (9÷3)

= 200 + 70 + 3 = 273

I gave the caption to the image – “This is inclusive learning. Students who find it difficult to understand the division process or are still learning to divide can cope easily.”

Amrit who is now the father of a 6-year-old sent me this emoji in reply:

He wrote that we never discussed this way of division in any of the classes during his years in school.

Yes, it is true. In school, concepts have to be taught within the time allotted and both the mentor and mentee must compress their work according to bell schedules and holiday breaks.

Division means sharing; but how many teachers are able to explain the meaning of division?

Division means ÷ (the symbol of division).

This is what we tell our students and we make sure that they remember this symbol so that they know they have to divide when they see it. I have 30 candies and I want to share them with 30 students of my class. This means when I divide the candies among my classmates, each student will get one candy.

Division by zero – We generally skip this concept because its answer is∞ (infinity). But have we ever thought why division by zero is not possible? I have 30 candies and I do not want to share, so no division will happen. Division by zero means that no sharing is happening. 0÷30 means I have nothing to share with 30 people so each one will get nothing, which means zero.

Take another example

1509 ÷ 26

Quotient = 58 and Remainder = 1

Why does the student have to use the traditional method of division in which he has to find the multiple of 26 closest to 150 and then proceed. Why can’t he use some other way? Like this:

Look at the above images and ask yourself.

Can’t we explain the area models along with the traditional methods of teaching division and multiplication?

In the above area model of multiplication, we decompose the factors into tens and ones, use them to represent the side lengths of our rectangle and then multiply the parts. The sum of the parts gives us the area or the product of our equation.

In this example, we multiplied 10×40 to make 400, 10×1 to make 10, 2×40 to make 80, and 2×1 to make 2. Then we added each of these four parts to find the total area of the rectangle or the product. 400+10+80+2=492.

Expose your students to alternative methods as well. Let the student decide what is easy for him to understand.

Students find factors and multiples easy because we as teachers generally test only the basic level of remembering and understanding. And they believe that they are proficient with this concept.

I asked in class 10 while teaching HCF and LCM
6 = 3 X 2
2 is a factor of 6 True / False
6 is a multiple of 2 True / False

And then extended to
-6 is a multiple of -2 True / False
-3 is a factor of -6 True / False

and then I asked
3 = 1.5 x 2, so 2 is a factor of 3 True / False.

Most students were confused.

Factors of a number are always lesser or equal to the number.

If we take 2×3=6, 1×6=6

Here 1,2,3, and 6 are factors of 6. The number 6 does not have a factor greater than 6. Here factors are natural numbers.

If we take -2×3=-6, here 3 is not a factor of -6 because 3 is greater than -6. Factors are not applicable to negative integers.

If we take 2×1.5=3, here also 2 and 1.5 are not factors of 3. Factors are also not applicable to decimal numbers.

So, factors are always positive integers (natural numbers).

This concept of factors and multiples is restricted to positive integers, but the gap in learning which the students were carrying from the last five years was evident from their expressions as the concepts of factors and multiples starts from class 5.

There is an urgent need of a free common platform where both the mentor and mentee can assess themselves to see where they stand and move to the next level at their own pace.

The mentor should know how to frame skill-based questions and questions related to higher order skills (analyzing, evaluating, creating). For example, a number has exactly eight factors including one and itself. Two of its factors are 21 and 35. What is the number?

In this question, the student needs to analyze and then evaluate the final answer.
21 = 3×7 ….… (i) and
35 = 5×7 ……. (ii)

Since 21 and 35 are factors of the required number, the factors of 21 and 35 are also the factors of the required number.

Assuming the number to be X, we have factors of X as 1,3,5,7,21,35, X (the number itself), m (last factor)
The last factor is m=3×5, since that is the remaining pair left without multiplying.

Hence the required number is X = 3x5x7=105.

Therefore all eight factors of X are 1,3,5,7,15,21,35,105

But, the same question has a one line solution as well – in order to find the number, we can find the least common multiple of 21 and 35 which is 105.

In traditional math classrooms – work is often assigned by the teacher with the progression “I do, we do, you do”. It often becomes the default approach to learning and the result of this is students lose their thinking capabilities and are able to learn only through guided practice.

Even after verticalization across grades, learning gaps continue to persist and are hard to rectify in higher grades. Learning gaps are like micro-pollutants, which increase as the student moves to higher grades.

Don’t you think after looking at these learning gaps, the progression for math classrooms should be ‘you do’ and ‘we do’? The ‘I do’ by the teacher needs to go to make it a learning classroom.

Change can’t happen at once whether in the teaching or in the learning pattern. Instead of starting a lesson with direct instruction, give students “thinking tasks” they can do, ideally in groups. Create highly engaging, non-curricular thinking tasks as problem solving activities, mental puzzles, which should motivate students. As the school year progresses and students become more accustomed to this mode of working and thinking, the activities and challenges can be replaced with tasks directly related to the curriculum.

Promote active learning by positioning students to be doers – rather than recipients – of mathematics. It is evident that a revolution in math teaching and learning is requisite now, but how? And when? These need to be addressed on priority.

The author is a teaching practitioner who has been advocating for liberating the education process to accommodate change. She considers teachers as catalysts of change in transforming the education process. She calls herself a co-learner in the journey which she is undertaking with her students. The author has 25 years of experience in the field of education as a teacher, mentor, curriculum planner and moderator. She is presently working as science stream leader (9-12) in Learning Paths School, Mohali. She can be reached at srastogitms@gmail.com.