How can I help my child with problem solving ?
I often talk to parents about helping their children with problem solving skills.
In mathematics, children are first taught how to count, make mental blocks of 10, recognise simple 2 dimensional shapes and once these fundamentals are ‘mastered’ the various systems (MYP / IGCSE / IBDP / Common core) will start teaching algebra. As students progress we eventually get them to learn about curves and ‘functions’, teaching them some useful formulae along the way and preparing them for engineering or financial degrees.
Most of the knowledge and application of these formulae are what many mathematicians would call ‘pattern copying’ type work. You learn the steps, you understand a formula, you practice 2-3 variations of the problem and once the test comes, the question is exactly the same. Since mathematics is a language and we are learning to speak ‘Mathanese’, it makes sense to help everyone on the journey to calculus or statistics achieve every milestone with a ‘copying’ mindset; to encourage all students to replicate work which really is simply the language, the basics, the abc so that one day perhaps they could go further.
Creative thinking isn’t easily achieved
The issue is, once postgraduates are required to think up of a decent project, or students need to write up an IA (Internal Assessment) for the IB exam, they have very little to go with in terms of creativity or problem solving since they’ve never in fact been motivated to ‘create’ or ‘invent’ anything at all.
You could ask many a postgraduate simple problem solving questions and they may have issues with the easiest questions. So how can we train ourselves to solve mathematical problems ?
In the very early stages, extracting information is often all that is required. Understanding sum = adding, difference = subtracting, product = multiplying, quotient = dividing. Seeing words like consecutive = in a row (2,3,4)
At a more intermediate level we can identify number patterns. Primes, squares, cubes, fibonacci numbers. Even numbers are 2n numbers, odd numbers are 2n-1 numbers.
Olympiad questions are much trickier
For the real questions, the olympiads, the UKMT, Gauss, Putnam and Math Counts type problems. Number theory is a must. A student will need a solid grasp of number theory, divisibility and multiplicity rules. An astute ‘Mathlete’ will be able to divide and multiply larger numbers mentally and have shortcuts for each question type.
With this ‘equipment’ in mind, here are 10 key points to effective problem solving:
- Orientation: Details in the question, reading it carefully.
- Defining: Is it a ‘to find’ or ‘to prove’ type problem
- Recognising: Is it similar to another problem you’ve done before ?
- Identifying: Indentify the hypothesis and the conclusion
- Intuition: Can you guess a possible solution ?
- Keywords: Can you identify keywords which are important (perfect squares, prime numbers, infinite sequences)
- Reading it out loud: Sometimes this helps !
- Trial and error: Experiment, plug real numbers in, ‘get your hands dirty’
- Pre ending: Think of the ‘penultimate step’ which is often the epiphanical part of a satisfying mathematical problem
- Try it out: even if it seems impossible at first.
Great rewards await for those who put the work in
True problem solvers think laterally, they are able to attain peripheral vision or see ‘through’ questions. A problem solver can bring elements from other parts of Mathematics into application to serve as a proof (for example finding a visual or geometrical proof to a problem)
Don’t give up, just like every mental milestone, the journey is ever so important. We are suffering today from instant gratification and immediate answers. Time spent contemplating an issue will inevitably lead to ideas being generated. Some of the greatest human discoveries happened by accident !
Problem solvers get jobs at google. Problem solvers ace interviews. Olympiad winners have a golden ticket to universities.
Give it a try 🙂 it’s a beautiful journey. Encourage the kids to learn to think critically, creatively and for themselves.
You could think of this maxim while you’re facing a complete mind melting problem:
‘Time thinking about a problem is always worth spending, even if you appear to make no progress at all’
About the author: Karim Arditi is the program director for Mathemagic, Higher Math and Science Studies. Karim is simply passionate about mathematics and its intricacies.
References: Art and Craft of Problem Solving, Paul Zeitz, 2007. Copyright John Wiley and Sons, inc
Find more challenging problems to test your wits with here