Parallels between Language and Math
Research has overwhelmingly shown that language is an innate ability which manifests itself into an individual language using a minimal language input. Similarly, researchers have found that babies are born with rudimentary counting skills (https://today.duke.edu/2013/10/babymath). If such mathematical ability is nurtured in a right direction, anybody could solve any real world problem with exceptional mastery. Yes, it is the direction that matters not the number of formulas to remember, not even the amount of practice.
Before we move on to understand the direction of math teaching, let us analyse how we learn languages. Natural spoken languages as mother tongues are always learnt primarily through listening. Error correction, teaching grammar rules, or motherese (pampered speech used by mothers) has proven to be of little use when it comes to first language acquisition. In some communities, adults do not speak to babies at all. Yet, babies learn their mother tongues at the right age through input they receive when adults speak to each other. Another riveting example will be the way a deaf and mute person acquires the structure of a language, say Tamil, through signs and lip movement and tend to imbibe that in his own vocabulary of signals. He learns the language without error correction or the explicit grammar rules.
However, error correction and grammar rules are useful for conscious editing purposes (Krashen, 1979). In particular, it helps in error-free writing in one’s second language (L2). It helps minimally when it comes to speaking (which is a subconscious process). Speaker might implement a grammar rule that is taught to him but the language will not become natural until he focusses on the meaning rather than structure. So, grammar rule must always be followed by examples with a focus on meaning rather than the structure. However, teaching language through grammar is an endless endeavour. In fact, Noam Chomsky found that language is generative and innate only when he was asked to formulate a comprehensive grammar book. The book went on to hundreds of thousands of rules with no sight of a finish line.
Alright, error correction or grammar seldom helps in speaking. Then what makes people speak a language naturally? It is the quality (not even the quantity) of listening and reading (together called language input). Language learning is a stress-free process. Listeners do not need to focus on the structure of the language. When the meaning is acquired, the structure is also subconsciously acquired. Only such subconsciously acquired language helps in natural speaking. The listening involves setting of certain parameters internally when an ideal listener understands the meaning of an utterance (Chomsky, 1981). The input so provided should be one level higher than the present ability in case of second language acquisition (Krashen, 1979)). The parameter setting process is less complex compared to the huge innate language rules (as evidenced by the similarities in the structures of all world languages). Such minimal input is enough to form an infinite different set of language structures. One could learn a language in a short span of 6 months given a conducive environment (language immersion).
The above facts about language acquisition in the previous paragraph are quite ideal. We have spoken about an ideal listener. An ideal listener is the one who is stress-free (relaxed and void of any communication apprehensions) and a creative thinker. Being stress-free, confident, and cheerful is a natural requirement for any creative process (John Cleese on creativity). Language is a highly creative process. We tend to create infinite different structures of language through limited input. For example, the sentences that appear in this paper would have never been written or spoken by anybody in the history of the universe nor will appear in future of the universe.
Let us discuss the importance of thinking when it comes to language. My former colleague, Ms Kanchana Menon always says it is the lack of thinking ability rather than language skills that is responsible for L2 learners’ poor writing / speaking when we speak about quite a few Indian children learning English. The only place I have seen people improving their language is when they were made to think in a thinkers’ club named ‘Your Voice’, initiated by Mr Aravind, the Manager, Soft Skills Academy with my previous employer, KG Educational Institutions, Coimbatore, India. Language interacts with various cognitive centres of the brain for its development. One may find it harder to comprehend philosophy because the philosophers use extremely complex words/sentences. Higher order thinking always requires complex structures to express it. Also, if we apply Darwinian philosophy here, higher order thinking ability naturally selects complex sentences to express it. It creates such structures naturally without being exposed to such complex reading. It is definitely not a superficial effort to make the language complex. Therefore, thinking forces a person to use a better language to express it. So, avenues to think deeper, good language input and a natural, stress-free communication environment is all that an English teacher might provide to achieve results.
Now, let us turn towards mathematics. It has many parallels with language. It is also infinitely extendable. All that is required to solve problems is the underlying logic learnt through the problems we learn at school. Formulas are of limited use just like grammar. Application of formulas is limited to solving problems of similar type. It works quite well for a short term, examination based problem solving. It does not help a person in real life problem solving and inductive reasoning. Sometimes formulas and shortcuts are detrimental to inductive reasoning.
For example, questions such as: “Find the units place of 232583+…” have certain shortcut methods prescribed in books on quantitative aptitude. However, such shortcuts critically undermine students’ logical thinking ability. Students fail to understand an extremely simple logic that powers of certain numbers follows a pattern in their unit places. It consumes less time as well. However, many quantitative aptitude books give scant regard to such logical thinking using the available patterns and trends related to the problem. Arriving at different patterns and analysing them are critical skills in inductive reasoning. As bank managers or as marketing managers people have to observe several such patterns and arrive at decisions. The purpose of such questions in competitive exams becomes futile by such approach. Such an approach has been rigorously used by exam coaching centres. However, such high score producing centres are detrimental to the nation’s skill development.
Interest rate related problems are something that we encounter both in personal and professional lives. We are all taught a magical formula for compound interest which many remember only when we prepare for exams. But compound interest can be calculated using the much basic knowledge of percentages. For example, a person invests 10000 in a bank which provides him 10% interest rate compounded annually. How much will he get at the end of the third year?
A person accustomed to the logic will apply the same in one single step as: First year 10000+1000 (10% of 10000)=11000; Second year 11000+1100=12100, Third year 12100+1210 = 13310.
This method is more logical and even quicker than the application of formulas.
The formula for compound interest, P (1+R/100)N is an intimidating one. It must be taught only after teaching the below logic explained through the above example.
In the first year, amount grows to 110% (10% added to 100% amount) which is equal to 1.1.
So, 10000*1.1 is first year’s output (11000)
In the second year, 10000*1.1 becomes the input which will again grow to 110% or 1.1
So, the amount after second year will be, (10000*1.1)1.1 (=12100)
Third year, (10000*1.13) or 10000(1+10/100)3. This can be generalized to P*(1+R/100)N
However, many of us have learnt the formula first and implement them in competitive exams.
Work related problems are even more important. It helps officers determine the number of employees or material needed for a project. Many teachers tend to teach their students in a narrow fashion aiming at different problems that could encounter in the exams. For example, problems of the following type are very common: “A finishes a work in 2 days, B in 3 days. How long will it take for both of them working together?” Some teachers prescribe ab/(a+b) is the instant solution for such problems (a(=2) and b(=3) are the time taken for A and B respectively to complete the job.
Some teachers suggest a formula be applied. If ‘c’ is the time taken for both a and b, 1/c=1/a+1/b. This works great for all such problems. But, the practice of logic is missing here.
This problem could be taught / learnt by introducing the concept of speed of work or productivity (which is a real world term) of employees which is work done divided by the time taken (p=w/t). Here work is 100% done. So work done is 100/100 =1. Productivity for A will be 1/a, B will be 1/b. Combined productivity is 1/a+1/b = ½+1/3 = 5/6. From our formula, p=w/t, t will be w/p. So, the time taken for 100% work to be completed by A and B will be 1/(5/6) = 6/5 or 1 1/5 days.
The above logic will help solve even problems with varying amounts of work done which is usually the case in the real world. For example, “if a painter paints one-fourth of a room in 2 hours, another painter paints ½ the similar sized room in 1.5 hours. How long will it take for both to complete a similar sized room?” This method is equally quick as the application formulas, it is stress-free (no burden of new formulas) and it improves the accuracy and extendibility. A person who has only learnt the formula by rote could never solve this simple extension of the earlier type of problems. Such a person cannot apply his logic in the real world. This could be the reason why in spite of our math books being heavy and having complex problems, we fare poorly in math compared to other countries. Such gaps in education, if bridged at its roots, will help in the nation building.
In addition to forcing formulas, some math teachers ask students to practise similar problems again and again. My parents too used to ask me to work out math problems instead of going through it. I have never worked out a single problem for my exams. I only used to have a look at the logic used. Many people tend to learn in this way. Repetitive drilling (as done in some private schools in India to achieve better results in math) does not work both in language learning and mathematics.
Another common teacher trait is the failure to provide options for the questions. This action unintentionally deprives students of vital time management skill and logically arriving at answers.
Making students memorize formulas or providing dangling (rote learnt) shortcuts will only increase students’ dependence on memorised knowledge instead of expanding their thinking ability. Just like languages, math works based on exposure to different kinds of logic. The logic gained from mathematics pervades the real life as well. It helps improve one’s creativity in the real world. As mentioned in the initial stages of discussion, creativity can be nurtured only in a relaxed, confident and cheerful environment. So, pack all the formulas behind and look at the inner beauty of mathematics through the lens of logic and spread the enthusiasm that
you receive.
To sum up, in both language teaching and math sessions, all that a teacher might have to do is to nurture thinking and creativity by creating the required ambience. Conventional teaching (based on structure, formulas and procedural steps) undermines (sometimes kills) thinking and creativity. Therefore less of teacher self and more of learner self must be encouraged to multiply thought showers in the classrooms.