How To Raise A Child Prodigy In Mathematics

“The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.” ~ Charles Caleb Colton

Every child is a gifted child because he or she is naturally inquisitive and inquisitiveness is all there is needed to start a journey in mathematics. Hence, every child has a potential to become a mathematical prodigy. If every child doesn’t pursue mathematics as his/her interest, it is not the failing of the child. It is the failing of an educator, if not the least of an education system.

In my experience, I have seen scores of children with promising aptitude in mathematics and yet very few of them pursued their interest. This is mostly because the pupil had to go through dry as dust routines and rules and processes in order to ‘learn’ mathematics. Mathematics is one of the most misunderstood subjects. If you ask someone about what is mathematics. It is very likely that they would talk about learning numbers, arithmetic and/or algebra. They might throw in some geometry there too. In reality these are just a narrow subsets of what maths really is.

The Mathematical Process

Defining Mathematics is not as easy as defining biology, chemistry, Sociology or any other field of inquiry or study. In fact, it is still a philosophical debate as to what is Mathematics? But we can focus on Mathematical process rather than defining mathematics as a field of study. According to Professor Robert H. Lewis “Mathematics is not about answers, it’s about processes…The real ‘building’ in the mathematics sense is the true mathematical understanding, the true ability to think, perceive, and analyze mathematically.” Hence, mathematics is a system of robust thinking. More generally, mathematical process deals with statements/facts and using a logical grammar to arrive at certain results.

Is It Important For A Child To Do Well In Mathematics?

Why should children learn heaps of facts and results that they might not apply in their lives?(Here, I will not get into philosophical debate, I will only concern myself with practical issues). Although, mathematical facts and results have their huge importance (From Music to Programming and Medicine, I apply these facts and results frequently), but what is of utmost importance is the logical and critical thinking children develop by combining these facts into results. There is a completely different language, a grammar of logic that gives mathematics its power. By Learning mathematics a child is etching that rational capability onto his/her brain. And the importance of rational thinking in routine life is … well.. HUGE!

Rules Of The Game

When training a young mathematician, a parent or a teacher should keep following points in mind. Strict adherence to these is a key to success in teaching mathematics.

• The stress should be placed on the system of proof rather than on axioms and results.

By this I mean that stress should not be placed on memorizing. This is especially true at the initial stage of learning. Because axioms and results would change as a child delves further in the study of mathematics. From number theory to topology from sets to categories etc. axioms change but the logical grammar stays of consistent. And this grammar is the true gist of mathematics. 

• Teach logical grammar by examples only and never by rote memorization of a set of rules (at-least before age 12)

Classically mathematicians have learned this logical grammar through reading proofs and retaining the logical grammar used in those proofs. A child should employ the same technique. He or she should not specifically learn that logical grammar as rules. Apart from the fact that children abhor learning a set of dry rules there are two main reasons for this approach: Firstly, proof in mathematics is used for both i) establishing the truth of a statement ii) to explain why  it is true. Formal logic theory often blurs the later purpose by overly complicating the proof for a young mind. Secondly, to understand logic theory, one needs a fair bit of abstract concepts and children rarely develop capability of abstraction before age 12.

• Rely on visual explanations rather than on words.

The most important thing is to minimize the use of words to describe things and processes. Rather use visual examples because heavily relying on words hinder concept building. In early 90s, Dr. Matthew Peterson et al. of University of California tested this idea of teaching mathematics with this visual approach. This simple idea became so phenomenally successful in a multi state study that In 1998, they branched off to create an independent non-profit organization, now known as MIND Research Institute. you can learn more about their work and approach in the following Ted Talk video.

•  Never burden a child. Make it a source of enjoyment rather than a duty.

A child should and would enjoy the process of learning mathematics if it will be taught as described below. If the lesson becomes a burden, drop it. Because if a child drops a lesson he or she will not learn something new. But if a child is made to go through a lesson he or she detests, the child will not only not learn something new, s/he will also develop a distaste for the subject and might even stop finding mathematics beautiful. This is the plight of our modern education system bthw.

The Curriculum 

Following are some resources that can be used to develop keen interest and capability of mathematical thinking. They can be used alongside a standard school course in Mathematics. With time students will develop connections with their course and their excursions in mathematics and will not only develop a keen enthusiasm for the subject but also necessary logical competence (which schools don’t teach) to pursue any field of inquiry in life, including higher mathematics.

The classification of the following material is according to my own theory and experience in mathematics and a little understanding of developmental psychology.  A teacher or parent may try any permutation of the following program they deem fit according to their child’s needs or interests. Almost all of these resources are available for free online. Links are provided.

Up to Age 3

Two things; music and visual patterns. If you really want to bring up the genius in children, start introducing them to patterns as early as possible. Before birth is not early enough. Music is a great example of patterns. Focus more on the breadth of musical patterns covered. Anything from Flamenco to Classical western music. Indian folk music to African syncopated percussion. Everything goes. Remember, the stranger the better. It will be a great idea to introduce them to micro-tonal music as well.  For visual stimuli, I would suggest Images of fractals and other complex visual patterns. At this stage the learning would be rather passive a child will shift interest in a fraction of a second but a perception of a second would be enough to ingrain pattern recognition in the child’s brain.

Ages 3 to 7

Children between these ages have a surge of expansion in their cognitive capabilities. A three year old can understand almost 3000 words. Can compare objects based on size and colours etc. As they age they can tell stories and form their own rudiemntary concepts. This is an age of language development. And what other time would be a better to introduce logical grammar. Hence, at this stage a child should be introduced to computer programming.

• Scratch Programming Language

I would recommend MIT’s Scratch. Scratch is a programming language and an online community where children can program through interactive games, and animations. It is very easy and as children create with Scratch, they learn to think creatively, work collaboratively, and reason systematically. This reasoning is would build the sound foundations for higher mathematics in future. Scratch is designed and maintained by the Lifelong Kindergarten group at the MIT Media Lab. Hence, the platform is very well researched. Following is a Ted Talk by Mitch Resnick that explains the platform and philosophy in detail.

Online video tutorials of Scratch can be accessed from here.

Ages 7 to 12

At this stage proper mathematics can be introduced because now is the time when children are gradually separating from objective thinking to and developing subjective thinking. However, complete faculty of abstract thinking is not yet developed and they can not yet comprehend hypothetical statements. But this is the most important time in a gifted Mathematician’s life. Because only 25% of children would grow out of this stage and develop the proper abstract thinking. Hence, according to Piaget’s theory of cognitive development, for 75% of population, the development of cognition stops here.

• The Elements by Euclid
  

  A page from Byrne’s Illustrated version of ‘The Elements’. Each proposition is proved as such Followed by Q.E.D(Quod Erat Deomnstrandum) Lit.”Which is what had to be proven”

  I have spent a great deal of time reviewing maths courses for children of this age and the best resource that I could find, was written 2800 years ago. ‘The Elements’ by the Greek Mathematician, Euclid is still the best source out there to introduce both mathematical substance and thinking. The best of the best is the illustrated version of the first 6 books by Oliver Byrne published in 1847. This version is a remarkable feat of genius because Byrne attempts to present Euclid’s proofs in terms of pictures, using as little text – and in particular as few labels – as possible. What makes the book especially striking is his use of colour. Children would fall in love with the book and I cannot praise this book enough. Byrne only illustrated first 6 out of 13 books. These books deal with geometry. For Number theory, I would recommend, books 7 to 9 but as they are not illustrated, a parent or teacher would have to put in a little effort to explain the proof in Byrne’s method. Or one can also delay number theory until, after all 6 books are covered and till then a child would have developed a decent capability to understand a written proof as well. Byrne’s fully illustrated version can be accessed here for free.

Ages 12 And Onwards

Transition to abstract thinking is now complete. At this age a rigorous course in Pure Mathematics can be started. Unlike the previous category, there are a number of candidates in this one. I will, however, discuss only 2 of the top and an optional resource on logic theory.

• A Course of Pure Mathematics By G.H Hardy

My first choice is G.H Hardy’s Course in Pure Mathematics is
because Hardy was one of the greatest mathematicians of last century. And he has written the book in a language that
would be understood by a high schooler if the book is followed properly in its entirety. One critic has rightly described his style; “Hardy combines the enthusiasm of the missionary with the rigor of the purist in his exposition..”. The book is freely availabe online and can be downloaded from here.

• Basic Concepts of Mathematics by Elias Zakon

This book helps the student complete his or her transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later higher mathematics courses and there are many advanced topics in it that a student can leave and they are listed as optional and a student may omit them at this age and later in his/her life may comeback to them. The
book is freely availabe online and can be downloaded from here.

• 100% Mathematical Proof by Rowan Garnier & John Taylor (OPTIONAL) 

This is an optional resource. And it beautifully yet simply explains the rules of logic as they are applied to mathematics. The book is full of proof examples and has a considerably flat learning curve which makes learning enjoyable.