What’s it about math?

S Sundaram

In the last two decades, three phrases have entered the vocabulary of many math educators – math phobia, math anxiety and dyscalculia. Why is phobia only associated with math? Why don’t we hear of history or language phobia? We will try and understand this phenomenon in this article.

There are two inter-related reasons. First is the failure of many educators and teachers to realize that the purpose and content of math are different from that of other school subjects. The second is the inappropriate pedagogy used in math education as a consequence of this.

Math is different
However, most learners intuitively realize that math is different compared to other subjects. But they find it difficult to pinpoint the nature of their difficulties with math. Most can easily describe what they are learning in language, science or social studies, but have difficulty in describing what it is that they are learning in math. Numbers, computations and geometry is what most can come up with.

To understand how math is different, we need to go into the components of any subject that is to be learnt.

What are subjects made of?
The learning of any subject can be classified into the following components – concepts, skills and information.

Concepts are mental abstractions, which need to be constructed in our mind by experience and introspection. Ideas like prime numbers and the operation of subtraction are two examples. They may be mirrored by physical entities and events but cannot be directly “perceived” or “pointed out”. You cannot show what “prime-ness” is. We can only point out a few numbers which have this quality. Concepts are the most difficult to learn.

Skills, whether physical or mental, are to be learnt through practice. Performing a long division or colouring a map without crossing boundaries are examples.

Information is “bytes” of information to be remembered or memorized. The names of types of triangle or the birth year of Akbar are examples. We may not even know if a piece of information is true or false. Different pieces of information that we learn, may not even be related to one another. Information is the easiest to learn, as it means “remember” or “memorize”.

Each of these components also needs a different way of learning. Information can be delivered through lectures and remembered. Skills, however, have to be practiced and mastered. Concepts have to be understood.

Difference between subjects
Let us represent, in a table, the relative proportion of concepts, skills and information in four common subjects taught in school. The numbers are just indicative. The purpose of the table is mainly to understand the difference between the subjects.

Subject Language (%) Math (%) Science (%) Social Studies (%)
Information 40 10 50 70
Skills 50 40 20 20
Concepts 10 50 30 10
Total 100 100 100 100

Information is maximum in social studies followed by science. By science we mean science as taught in schools where the emphasis is on information about definitions, discoveries and inventions. So both become subjects with a lot of memorization. Information is the easiest component of learning.

Skills are maximum in language where without listening, speaking, reading and writing skills, a language cannot be mastered. Though skill of experimentation is a major part of science, in schools, laboratory work is not given real importance. Math also needs a lot of skills to be practiced. “Drilling” is a common word used about math education.

Concepts are the most difficult aspect of learning as they involve a lot of introspective thinking and abstraction. Language has the least amount of concepts which is basically grammar and phonic rules. Social studies can include deep concepts, but these do not form part of the school curriculum. Math has the highest level of concepts.

Math is different and difficult
Math is concept heavy. Concepts are abstract mental ideas whose understanding requires a lot of experience, modelling and introspection. Concepts have to be “caught” by the learner. They cannot be directly “taught”. These elements make it a difficult subject to understand, and there is no clear understanding of what “understanding” is.

Another difficulty is that all math concepts are related across the curriculum and also related hierarchically across grade levels. We can think of the math curriculum as a pyramid built with playing cards, stacked one on top of another. The weakness of even one card at the lower level will weaken the entire structure. It is similar to the saying that a chain is as strong as its weakest link. A weakness in subtraction will affect understanding of division, fractions, algebra, etc.

How math needs to be taught
Basic ideas in math like numbers, shapes and operations were developed by observing patterns in the world around us. Hence in pre-school and primary school, these ideas can be taught to children using the environment around us and our daily life experiences.

But very soon, these basic concepts start developing into complex ones. Continuous layers of abstraction get added on like the layers of skin on an onion. Hence along with teaching of these basic ideas, the very process of thinking about these ideas and the inter-relationships between them should also be developed. If these skills are not developed early on, these concepts will become so abstract that students will not be able to understand them.

Beginnings of math anxiety and phobia
If the teaching of math is not appropriate, the students’ lack of understanding keeps deepening, as the complexity of the concepts keeps increasing. A learned psychologist has compared the mental state of a student who cannot understand what is happening in the class to that of a novice swimmer who struggles just to keep his nose above water!

In addition, negative comments about weaker students reduce the self-confidence of students. Hence, each math class brings a lot of anxiety. Anxiety itself reduces the motivation to learn and hence further reduces the capacity to focus and learn. For many students the level of anxiety keeps building up leading to the psychological condition of a phobia.

What is the way out?
Since math phobia is a condition which develops over a number of years, there is no short-term solution to it. In the long run, there are several interconnected strategies which could be adopted.

Appropriate pedagogy
Concepts cannot be directly pointed out. They have to be demonstrated in a variety of indirect ways so that the student can catch them. My guru used to say “Concepts are not taught. They are caught”. So the concept of an even number should be shown to students by equal sharing which is a daily event they are familiar with. This can be done using simple manipulatives. Just providing a definition (that an even number is divisible by 2) may not produce understanding. It can even mislead as this statement is not true if numbers are written in odd number bases!

A math activity centre with plenty of manipulatives and models is extremely necessary for children to “practice and explore” math.

Some examples of skills in math are computations and constructions in geometry. In general, skills have to be practiced in a variety of ways. But math skills are intimately woven with underlying concepts. They have to be practiced with understanding of the concepts. Mastering addition of two digit numbers needs a clear understanding of the place value concept. Practicing skills mindlessly may lead to mistakes when the procedure needs to be modified to suit the situation. My guru used to say “drilling may produce only holes and not understanding.”

Information in maths is mostly the symbols, names and definitions. Many terms used in math are never heard by students outside the math classroom – obtuse, coprime, etc. Some have different meanings when used in daily life – interest, imaginary, etc. In such cases, students have to be helped in absorbing these terms and their connotations by repeatedly using them in context in class for extended periods of time.

Appropriate age-wise pedagogy
Children do an average of 15 years of schooling, during which they go through several stages of mental and emotional development. Developmental psychology tells us that until the age of 10, their ability to understand abstraction is limited.

Hence as children proceed through different class levels, the method of teaching should change to be in consonance with their mental ability to learn complex concepts, skills and information.

Unfortunately, we see that, in our schools, all subjects at all class levels are taught using the same process – lecture, blackboard, chalk, duster, students cramped in rows, textbooks, classwork, homework, examinations, pass/fail, report cards.

Teacher training
Such changes in pedagogy require a sea change in our teacher development and certification courses and on-the-job training. In our schools, continuous teacher training is almost never heard of!

Primacy of the primary
The strong foundation for developing motivation and competency to learn math has to be laid in primary school. In primary school, all skills are important. Hence in primary school the curriculum has to be reduced and teachers have to ensure that each and every child who leaves primary school attains at least 80% understanding of the entire curriculum. A pass mark of 40 or 50% is inappropriate in math in primary school.

Assessments
Formal examinations, which reduce the number of teaching periods, should be replaced up to primary with class tests and that too of the formative type. Methods of assessment should be broadened to include oral, practical and project assessments. Timed tests should be reduced to the minimum. The textbook should become a reference, rather than the “bible”. Mistakes should be seen as “learning opportunities” and not “shaming opportunities”

Math is critical
The world has changed dramatically in the last five decades. In this digital and knowledge society all children must be given an opportunity to learn and enjoy math which is going to be extremely necessary for them to lead an empowered life.

The author has worked as a principal, teacher trainer and educational consultant in several schools in different parts of India. He retired as the principal of Reliance School in Jamnagar in 2013 and has settled down in Chennai. His areas of interest are primary mathematics, school leadership, quality in education and technology in education. He is currently working on a book on understanding the various concepts underlying all the topics in the K-8 math curriculum. He can be reached at [email protected].

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