Sundaram is an ex-principal and Math educator, who has recently completed a book explaining the logic underlying concepts and procedures in the K-8 math curriculum and suggesting effective ways of teaching them. The free book is available to all here.
“Love or Hate” Dichotomy
No other subject possibly evokes such strong and opposite emotional reactions as Mathematics does. This is the reason why in mathematics it is usually either “I understand it completely and love it” or “I cannot comprehend it at all and hate it.” There is rarely a middle ground. In many cases this “hate” develops into “math avoidance”, “math anxiety” or even “math phobia”.
Why does this happen? Mathematics is actually an easy subject to learn.
Nature of Mathematics
The nature of Mathematics is like a double-edged sword.
The content of all subjects learnt in school can be classified as three kinds of knowledge – Conceptual, Procedural & Factual. The table below summarizes them.
Mathematics is quite different from other subjects. It is roughly 50% concepts, 45% procedures and only 5% facts. In other subjects usually the conceptual aspect is smaller, bulk of the content being factual knowledge. In these subjects, memorization becomes essential, at least to pass examinations.
But in mathematics the need for memorization is very less.
In mathematics, even procedures are based on underlying concepts. For example a student cannot master all varieties of additions, with & without carry over, unless her understanding of place value is strong.
The concepts are also hierarchically arranged like a pyramidal house of cards, with the higher-level concepts based on understanding of the lower-level concepts. The concepts are also linked horizontally among various topics of Mathematics itself. Concepts also undergo subtle modifications as they become more complex and more widely applicable.
This closely interlinked nature of Mathematics concepts is both an opportunity and a threat. When a concept is understood, the related concepts can be easily understood. But when a concept is not understood, all the related concepts become difficult to understand.
Mathematics Is Easy
Human brains have evolved to deal with abstract concepts. It is because of this edge that we have over other living beings, that we are at the top of the heap. Human brains are also good in remembering information that is relevant to us. Information related to emotions are also more easily stored and recalled. But the human brain is not good at memorizing vast amounts of information which may not be or appear not to be relevant to us.
Mathematics is a concept-laden subject. In addition, mathematical thinking springs from within us. Even before arriving at school, children already know a lot of mathematics. For all other subjects we need external tutors.
Children understand 'fair share' before learning fractions
The concepts in K-8 Mathematics, except for integers, can all be related to our life experiences and patterns in the environment. Hence, if taught appropriately, children would be able to understand mathematics easily.
But children in K-8 stage cannot understand math concepts taught through lectures & mere number work. Teaching should be through relating math concepts to our life experiences and patterns in the environment. Where life experiences may not be feasible, they have to be mirrored through activities with or without materials.
So Mathematics can be understood easily in K-8 grades. A strong foundation in K-8 Mathematics would make Mathematics learning in higher classes also easy. Hence primary school is very crucial in developing good learners of mathematics.
Mathematics is also a subject with enormous number of explorations, puzzles & games. Hence Mathematics can be learnt with joy and happiness.
Why is Mathematics learning difficult?
Pedagogy (teaching methods)
Mathematics in our schools is not taught effectively. The current pedagogy of lectures, textbook, homework, practice drills and examinations is not appropriate for mathematics. Very few schools have practical work & activity-based teaching in mathematics.
Teacher training courses do not train on teaching methods (pedagogy) appropriate for a concept-laden subject like mathematics. They are taught pedagogies which are suitable for fact-laden subjects.
Overload of “rules”
Most math concepts are converted to rules in the classroom. Procedures are also converted to rules. Students have to memorize all these rules.
For reasons explained under “Nature of Mathematics”, as concepts transform gradually, the related rules also change. So the load & variety of the rules to be memorized keeps on increasing. Like the proverbial straw on the camel’s back, in many students, at some point during their schooling, there is a mental breakdown resulting in math anxiety, avoidance and phobia.
Consider an example of two “rules” on the addition of fractions that are found in textbooks.
To Convert a mixed number to a fraction, apply this rule:
Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the common denominator as in the fractional part of the mixed number.
To Add Fractions with different denominators, apply this rule:
First find a common denominator by taking the least common multiple of the denominators. Then convert all the addends to have this common denominator. Then add using the rule number 1.
Remembering such rules is intimidating, even to an adult.
Many children and adults can add fractions perfectly without having to memorize any rules...as long as they understand what fractions represent & how they behave. Whether it is Equivalence or LCM (Least Common Multiple), it makes logical sense once they are able to visualise what is going on.
Ineffectiveness of “drills”
Procedures in math have to be practiced with understanding of the underlying concepts. The mindless practice known as “drill” only produces boredom and disinterest in mathematics.
Math & Intelligence
Unfortunately understanding mathematics has been identified by society with intelligence. Children who are not good in mathematics are at the receiving end of comments from many teachers and develop low self-esteem. This quickly leads to lack of motivation.
Curriculum developers seem to be unaware of these issues. The students in K-5 stage are mostly in the concrete-operations stage. They need a lot of time to learn concepts of mathematics. But the curriculum is so overloaded with unnecessary content that there is no time for effective learning.
Textbooks are very unimaginative, focussing mostly on numerical work. The problems at the end of chapters are far removed from real life experiences. Hence students get convinced that mathematics has no relevance to their lives.
The Examination System
The examination system also robs students of substantial learning time. All the assessments are mostly paper-pencil tests. The system has no means to locate & nurture students with a “mathematical eye”.
Many parents carry the scars of the math anxiety they had during their schooling. They invariably pass on their anxieties to their children further demotivating the children.
Any change in math education is a very long drawn affair which may take decades. But the world around us is changing rapidly. Understanding & application of mathematics is becoming a necessary skill in the knowledge society. Would our students be ready for the future?
Note: GenWise offers a series of short online courses titled 'Math Unmuddled'. This series takes up topics that pose common difficulties for children and clarifies these to build strong conceptual foundations. Sundaram has played a key role in the design of these courses. Click on the links below for more details.
Early Algebra starts on April 26, 2021
Integers starts on April 30, 2021
Fractions starts on May 10, 2021
Triangles starts on May 11, 2021
Ratios and Proportions starts on May 24, 2021
Percentages starts on May 25, 2021