The Purpose of Mathematics
Let’s play a game before we proceed with reading this article. I will put forward a general question. You will then ask a Math teacher that question. I will predict the answers to the question even before you ask them and hence even before you get your responses. If I predict even one answer correctly I win, if I do not, you win. You get to choose your teachers. The teacher you select could be from any class, any education board and from any era. Ready to play? Here we go. The question is: ‘What are some of the general challenges you encounter in the teaching/learning of Mathematics?’ While you now go around asking the question, I will carry on with this piece predicting the answers and thereafter explaining how I did the magic.
1. Students dislike ‘word problems’
2. Students forget what they learnt in the previous class
3. Students get bored and want to know why they have to ‘do this’
I am very sure you will get the same responses. It could be that all of these responses do not come from each teacher but put together these would be the set of challenges teachers face in the teachinglearning of Mathematics. In my interactions with teachers across the country, it’s been the same rabbit that comes out of the same hat, every time; and I have no accomplice in the audience.
There is no magical mystery to this poser actually. The truth is that over the past several decades we in India have not really changed our approach to Mathematics. When we talk of ‘approach’ what we mean is what do we see as the purpose of Math, what are the salient features about the subject that need to be identified as core and how then do we address these areas as we design the teaching-learning programme.
Up to a few decades ago schools were still a product of the industrial age. Only the more advantaged students attended schools that educated them as future academic, cultural and organisational leaders. That system does not meet the needs of today that requires not merely mathematically literate workers but employees who understand the complexities and technologies of information- age communication, who can work cooperatively in teams and who can solve open-ended problems. Hence, as students engage with a math curriculum in school, problem solving should not be a distinct topic but should permeate the entire programme and provide the context in which concepts and skills can be learned.
That is why in any credible mathematics programme, problem solving forms a central core. For a student to understand that there is a situation that is begging a solution, to even know just that and to start thinking about solutions, sharing their thinking and approaches with other students and teachers and working out strategies is how the mathematical thinking process should evolve. The problem can be very simple or extremely complex. Mathematics then develops as a tool to solve that problem. The problem is at the centre, not the math.
Sadly, our math curricula mostly put problem solving at the end of a concept after all the maths is done. Chapter-ends in many textbooks have a section called problem solving; alienated as it were from the main part of the chapter. Let us look at an example:
Sania eats a fourth of a cake for breakfast. Her brother eats a fourth as well. How much cake do they eat all together?
This is a very simple ‘direct’ word-problem. However many third graders may not be able to solve it. They can write one-fourth, they can shade one-fourth in diagram but they fail to understand how to proceed in applying the concept. This is most often because they have not been taught how to visualise math concepts. The way they have been taught is: This is the symbol of a fourth (some teachers erroneously and mostmisleadingly call it ‘one by four’): ¼,
The top part is called the numerator; the bottom is called the denominator and so on. So then ¼ + ¼ = what? A student brought up in the ‘problem solving is central’ approach would be, on the other hand, able to feel the problem, see the situation, and intuitively know and be allowed to opine that one fourth and one fourth is two fourths just like one chocolate and another chocolate is two chocolates.
The salient features of effective mathematics learning are concept development and skill development. Both go hand in hand while an assessment for learning and an assessment of learning follow closely.
The ability to visualize is essential to mathematics. Mathematics is one of the most concrete subjects, especially so in the foundation years and middle years. That is why activities that grow out of problem situations are valuable. This leads to effective and deep concept development. Students who develop a deep understanding of concepts have a lifelong retention of them and are able to apply them with ease in any relevant situation.
To be able to ‘do’ math mentally or otherwise falls under skill development. The learning outcome of any mathematics session should be a skill no matter how short the session is. To illustrate and explain this further let us look at a case of a teacher taking a 30-minute class on adding fractions. The teacher should be able to walk into the session (and also actually out of it having accomplished the mission) with the desired learning outcome well defined in advance and in writing e.g.
By the end of this session my students will be able to add ‘like fractions’ by demonstration, orally and in written mathematical symbols.
What is also key is frequent, effective and informal formative assessment. Every classroom session should contain at least two to three opportunities to informally assess the students’ understanding of the concepts that are being studied. For example, a task asking students to show by drawing and writing a story how three fourths of the cake might be consumed could be given. The focus is still on the problem and mathematical thinking is required for this. The students’ work will give a very good indication to the teacher on how well the concept has gone down and whether or not the teacher can proceed; if not, the teacher moves towards a plan to ‘re-form’ the students understanding of the concept.
When problem solving becomes an integral part of instruction, and concept and skill development evolve out of that, students gain confidence in doing mathematics, engage with it at a superior level developing persevering and inquiring minds. There is also much deeper retention and hence greater and more permanent enjoyment for teacher and student alike. No mystery, no magic and no ‘problem’! Game over.
Sarita Mathur is a free-lance education consultant offering services to schools, both rural and urban, in India and abroad. An alumnus of St. Stephen’s College, Sarita has a degree in Mathematics, Education and a post-graduate degree in Operations Research. She started her career with the India Today group as Assistant Manager Marketing and then after a period of 12 years went on to join the education sector. The Mathematics background and her well-honed sense of systems and processes had her institutionalize several long lasting and important changes as Principal of The Shri Ram School placing it firmly on the map as a progressive and leading school of India. Sarita has served as a consultant on the International curriculum of the CBSE and also serves as advisor/ consultant to several curriculum companies, schools and start-up ventures. She is currently actively engaged with Scholastic India and the Shiv Nadar Foundation.