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"How do I involve my child in challenging mathematics? He gets good marks in school tests but I think he is smarter than school curriculum."

"My daughter is in 4th grade. What competitions in mathematics and science can she participate in? How do I help her to perform well in those competitions?"

"I have a 6 years old kid. He hates math. How do I change that?"

We often get queries and requests like these from parents around the world. Literally. In fact, the first one came from Oregon, United States, the second one from Cochin, India, and the last one from Singapore.

We have created this article to answer these kinds of questions and help you to help your children.

Fifth grade is the first time a student is exposed to algebra and geometry simultaneously (in most schools). Note that the arithmetic of exact computation (example: what is the exact value of 31 + 57?) is fundamentally different from the arithmetic of approximation or geometric reasoning. They are processed by different parts of the brain (that is neurologically they are distinct).

Unfortunately, the school arithmetic (or olympiads organized by some private organizations) are mostly focused on the exact computation. This approach is counterproductive for the development of a young mind. It is prudent to keep the student away from these harmful practices.

An alternative practice is to expose the student to "approximation" (**example:** is the sum of 31+243 larger than 300?)

Imagination plays a big role in mathematical science. Vernacular literature can be the greatest source of childhood imagination. Greater focus on a second language (like English in India or Spanish in the U.S.) may be an impediment to the child's mental development. Let your child read a lot of storybooks in vernacular literature.

Dyscalculia is a mental state (like dyslexia) where the child has difficulty in processing numbers especially to associate symbols with magnitude. For example, most of us when looking at the symbols 5 and 500, associate distinct spatial magnitudes to each of them (5 is small, 500 is large). If your son finds it difficult to do so, he or she may perform terribly in school arithmetic test.

Please note that it might be counterproductive to force him to learn tables and do well in those tests (in fact learning tables is a bad thing to do under any circumstance). It may drive him away from mathematical science altogether. If he has dyscalculia he may do well in visualization (say geometry) instead of exact arithmetic. Some branches of mathematics are free of 'magnitude' in a sense (some forms of geometry are examples). Let your son try those things if he fairs badly in computational mathematics.

It is not at all uncommon to have dyscalculia (remember jt is less of a disease and more of a mental state so do not panic). About 2 to 3% of any population has this state of mind in all countries. Unfortunately, most schools are ill-equipped to handle such situations.

Though math olympiads can be an excellent exposure to nonstandard problems, not all mathematicians favor them.

"Many of the winners in mathematical Olympiads that I know have, unfortunately, not been very successful as mathematicians when they grew up unless they continued to study like hell.... there are people who, though slow-witted at Olympiads, are good at solving problems that may take years to solve, and at inventing new theorems or even new theories." - Dimitry Leites

So it is prudent to use competitions and olympiads as motivators and focus on "learning the subject for its sake".

Computation techniques like Abacus or Vedic mathematics are not only useless from a mathematician's point of view, they may have a negative pact on a child's brain. Mathematics has very little to do with computation. Most of mathematics is pure logical reasoning. Do not waste your child's time and mental ability for the sake of some swift multiplication.

Some argue that apart from computation these techniques have another purpose- improving concentration and focus. Whether or not these claims are true, it is better to use other methods (like meditation, team sports, chess) for improving concentration and focus.

A dry lab with simple equipment for physics, chemistry, and astronomy experiments is useful. Books like Quest Vol. 1 and 2 are sources of simple experiments. You may also buy a simple telescope and a microscope. City dwellers may visit the rural countryside in weekends for clearer skies and less distraction.

Astronomy can be a great motivator for mathematics and physics. Consider the Orion constellation. This is the most prominent constellation in the night sky. It has one red star located at the crest. Investigations like "why is it red" or "why the Orion is the brightest" or "how far the stars are from each other" are great motivators for discussions in physics, astronomy, and mathematics.

**Instruments **

- A moderately powered telescope
- Calipers, thermometers, simple volt-, ohm- and ammeters, potentiometers, diodes, transistors, simple optical devices
- Double-beam oscilloscope, counter, ratemeter, signal and function generators
- Analog-to-digital converter connected to a computer, amplifier, integrator, differentiator, power supply.

Students in 13 to 18 year age group from all around the world may participate in this contest. A fifth-grader may begin preparing for it at home using the lab. Intel science fair is another similar contest. These links are useful:

https://www.googlesciencefair.com/

These two contests are available for American students but the problems presented here are useful for all students of 5th and 6th grade.

Computers are a necessary evil. You may use software like Mathematica for illustrating math concepts with visuals. But it is generally more useful to allow the visuals to form in the mind first. In fact, locus questions are most critical in the formation of mathematical reasoning in the young mind. But note that if the brain of the child lacks proficiency in visualization one should not force them to do so. A child may be capable of handling discrete mathematics (say counting, combinatorics) better than visuals (geometry). Great Romanian mathematician Louise Posa is one example. Computers may aid you to identify this trait. Elementary computer programming especially in C may help in honing this aptitude in a child.

Exposure to the internet and television is generally counterproductive due to a lack of restraining and similar factors. Especially replacing indoor and outdoor sports by video games has been statistically proven to be counterproductive to a child's physical and mental development.

A great book for locus problems (geometry in motion) - Vasiliyev's Lines and Curve (it is available in our library).

Last but not the least, books play a very important role in the academic development of a child. Here is a great list of books. Some of them are out of print but available in our library. Not all of them can be used at the age of 10 but gradual exposure to these resources is priceless.

**Miscellaneous**

- Mathematical Circles: Russian Experience; Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg
- Challenges and Thrills of Pre-College Mathematics; V Krishnamurthy, C R Pranesachar
- Excursion in Mathematics
- Mathematical Olympiad Challenges; Titu Andreescu, Razvan Gelca
- Mathematical Gems Vol. 1, 2, 3; Dolciani Series
- Intuitive topology by V.V. Prasolov

**Geometry**

- Geometric Transformations; Yaglom
- Lines and Curves; Vasilyev
- Problems in Plane Geometry; Sharygyn
- Geometry Revisited; Coxeter Greitzer
- Geometrical Etudes in Combinatorics; Alexander Soifer

**Number Theory**

- Elementary Number Theory by David Burton
- Elements of Number Theory by Sierpinsky
- 104 Problems in Number Theory by Titu Andreescu

**Algebra**

- Elementary Algebra and Higher Algebra; Hall and Knight
- Inequalities through Problems; Venkatchala
- Inequalities; Korovkin (Little Math Library)
- Functional Equation; Venkatchala
- Complex Numbers from A to Z; Titu Andreescu

**Combinatorics**

- Principles and Techniques in Combinatorics; Chen Chuan-Chong, Koh Khee-Meng
- Introduction to Combinatorics; Brualdi

**Problem Books**

- Problem-Solving Strategies; Arthur Engel
- The IMO Compendium by Dusan Djuki, Vladimir Jankovi,

**Calculus**

- Pre-Calculus; Tarasov
- Calculus Volume 1 and 2; Apostle

Please note that though a teacher is indispensable for learning, a great deal can be achieved by self-help with sympathetic and flexible facilitation from parents. It is unwise to occupy a child's free time by coaching and tutorials at an early age. Use the above-mentioned methods with care and we hope you will be able to help your child to do what he or she is most comfortable.

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