Mathematics, its teaching/learning has evoked wide discussion ever so often. Aptly so because everyone has encountered Mathematics (and language) in his/her life at some point of time or other and consequently has a definite opinion about it. Hence the need to reflect upon the images that arise in us about what constitutes 'Mathematical Intelligence', and how it may be fostered, if at all.
Popularly, mathematics is treated as being synonymous with numbers and arithmetic. Subsequently, dexterity with numbers and efficiency in arithmetic is merited as mathematical intelligence. But maths is numbers no more than music is just notes. Mathematics is like a language, a language to communicate the sciences.
It is difficult to precisely define what mathematical intelligence is. What will be attempted is to broadly characterise the components that might constitute mathematical intelligence.
Recalling, in a lighter vein, a cartoon of Dennis the Menace: Margaret, Dennis' friend, expresses difficulty in trying to sculpt a horse. Dennis retorts: 'Simple! Chip off what don't belong to the horse!' In a similar manner, let us begin by looking at what mathematical intelligence may not be.
Research findings have this to report : 'Many high school students can't interpret graphs, don't understand statistical notions, are unable to model situations mathematically, seldom estimate or compare magnitudes, never prove conjectures, and most distressing of all, hardly ever develop a critical, skeptical attitude towards numeric, spatial and quantitative data or conclusions.' (John Allen Paulos, Beyond Numeracy)
So, taking a cue from that, perhaps mathematical intelligence may not be confined to the ability to:
- 'work out' a dozen problems efficiently
- master techniques
- remember formulae
- show familiarity with numbers
- manipulate algebraic expressions
- prove theorems /riders axiomatically
- work with pedantic logical formalism
The scope of mathematical intelligence is much more than the above. A language is not just the alphabet, grammar, vocabulary, neat handwriting and punctuation. These should be like the infrastructural support for communication and literature. Likewise, our perception of Mathematics will direct our characterisation of what it means to have a flair for it.
'Mathematics as an expression of the human mind reflects the active will, contemplative reason and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness and supreme value of mathematical science.' (Courant and Robbins, What is Mathematics?)
Accordingly, one might say the essence of Mathematics lies in :
- intuition / common sense
- creativity / originality
- visualisation / imagination
- ability for generalisation and abstraction
- ability to see connections / differences
- ability to apply
- process of thought formulation
- estimation / approximation
- analysing for errors / absurdities
- looking at a problem from different dimensions; a variety of solutions; open-ended problems.
So goes a story about a great mathematician, Karl Friedrich Gauss, when he was in Class 2. In an attempt to quieten a noisy class, the teacher is said to have instructed the class to add numbers all the way from 1 to 100 and settled to take a nap only to be woken up minutes later by a lad who claimed the answer to be 5050. On being questioned how he arrived at the answer so soon, Gauss is said to have offered the following explanation:
First he wrote the numbers in this way:
1 + 2 + 3 + ...+ 98 + 99 + 100 = N
Then he wrote them backwards under that:
100 + 99 + 98+ ... + 3 + 2 + 1 = N
The sum of each pair was equal to 101 :
101 + 101 + 101 101+101 + 101=2N
So 1+ 2+ 3 + +100 = (l00xl0l)/2 = 5050
To generalise would be to say:
1+ 2 + 3 + + (n-2) + (n-l) + n = n (n+l) /2
This is what we understand by seeing something intuitively'. To visually understand the same thing, the following figure will be helpful.
Area of shaded portion = 1 + 2 + 3 + n
But area of the whole rectangle is n (n+l)
So area if shaded portion is also = n (n+ 1) / 2
The blossoming of mathematical intelligence in a person and the scope that exists in an educational ambience to foster it seem to be largely influenced by the myths that exist in our minds about math aptitude and math learning. So it becomes necessary to reflect upon those myths critically. I outline a few which occurred to me and invite comments.
- 'Practice makes for understanding.'
Experience of generations proves the contrary. Practice, if resorted to, after a thorough comprehension of the concept, brings about speed and efficiency. But 'practice' is neither a means for comprehension nor a substitute for comprehension. More often than not, practice (which invariably manifests as 'doing sums' mechanically by rote) dulls an inquiring mind and is counter productive to math intelligence.
- 'Emphasis on understanding is time consuming; techniques make for expediency and efficiency.'
On the face of it, this might seem true if priorities are short-term (a test tomorrow, complete the home-work in 1 hour). But if any learning of lasting value has to take place, the prescription of a technique summarily defeats the purpose.
- 'Answer is what is important; the answer is an end in itself.'
A mathematician terms it 'the tyranny of the right answer'. If the vision is restricted by blinkers to viewing only the answer, vital aspects in the thought process of the student will fail to get attended to.
- 'All wrong answers are equivalent.'
Answers can be wrong for a variety of reasons. Every wrong answer has its own dimension and seriousness. The wrong answer yields insights when it is indicative of an incorrect thought process.
- 'Students either can do Maths or can't.' (like being tone-deaf for music)
Everyone has an innate ability for Maths like an ability for speech/language. An individual's talent for poetry or literature is dependent on his/her predeliction.
- 'Math learning is sequential/hierarchial'
Numbera Arithmetica Algebraa Calculus
This is the popularly practised sequence in the teaching of Maths based on the assumptions about a student's ability to perceive abstraction and relationships and patterns. Actually, a child's mind is capable of phenomenal imagination and visualisation which can be exploited from early years. Math learning is lateral.
- 'There is one standard procedure to learn Maths.' (The famous maxim of 2 + 2 = 4 )
The consistency, periodicity and predictability of some elementary tenets should not be wrongly understood to stand for strait-jacket procedures. An individual learns at his/her own pace, in his/her own style.
- 'In order not to forget Maths, you have to keep in touch by constant practice.'
What one can perhaps 'forget' in Maths are formulae and other such relationships/information. That is not a big loss because such information is always available in resource books or can be retrieved by working from first principles. Problem solving ability once acquired is not possible to lose. Is there anything like 'forgetting' to ride a bicycle because you've not ridden one for long?
- 'Exam score is the index if mathematical intelligence.'
This myth has come to exist by an erroneous presumption of a logical converse. What is true is that if Maths has been learnt well, the student can score well in any exam. How does that guarantee the converse? In addition to these myths, there are a few functional constraints in large-group learning situations that serve to colour the image of math intelligence:
- standardized norms / expectations of the group
- group expected to learn at the same pace / same style
- maths as an 'academic' exercise' out there' - very detached from the self - very divorced from common sense
- wrongly timed emphasis
- teaching objectives having to match exam / assessment objectives and suitable interpretation of performance.
Given the stigma. attached to mathematical intelligence, within the boundaries of myths and constraints that exist, one way of our responding to the challenge may be to attempt to sharpen the skills of students by providing appropriate contexts of work.
The follow:ing skills are widely recognised as very essential for Maths learning:
- Pattern Recognition
- Mental Arithmetic
- Reading scales / graphs / maps
- Estimation & Approximation
- Scale Drawing
- Problem Solving
- Error Analysis
Each of these skills is a tool for an aspect of mathematical intelligence. Pattern recognition is involved in generalisation and formulation of a problem. Visualization is an important component of Geometry, Trigonometry and Mensuration. To sum up, a quotation from Confucius states our philosophy of teaching Maths:
'Give a hungry man a fish and you've fed him for the day.
Teach the hungry man how to fish and you've fed him for a life-time.'
Math intelligence is knowing how to fish in whatever waters and our attempt may be to provide hook, line and sinker.