Problem-solving: What Is It And How Do We Do It Better?

In mathematics in general (as well as reflected in school criteria sheets) there are two main areas of which both need mastery in order to be good at maths: technical processes and problem-solving. Technical processes involve the learning of algorithms and methods commonly used to solve problems, adding to and maintaining your collection of these over time. A level of mastery in this field comes from remembering and improving on skills learned in previous terms of work and an ability to recall them when necessary in a test setting. Problem-solving on the other hand is the method of adaption when faced with a problem whose solution you have previously not seen which usually involves the combination of multiple technical processes whose identity and order are not immediately obvious.

As an inherently creative process then, problem-solving is difficult to discuss in concrete terms. However, I will proceed with this article talking about my own personal experiences with problem-solving and the techniques which work well to help achieve the best results for me in it.

To be an excellent problem-solver, one must have an efficient and meaningful way of getting thoughts out of the mind and down onto paper. Intense thought about complex mathematical processes is hampered by our minds trying to juggle the task of holding on to any piece of information not relevant to the current thought. Hence, to clear your mind and allow efficient thought I recommend the following tips:

. Write down the key information in the problem straight away at the top of the page. . Draw diagrams to fully conceptualise the problem -as big and as accurately as possible without obsessing over details. Any diagram that is smaller than one third of the page is practically useless. . Always keep a pen in your hand and your hands on the desk poised ready to write. It is a small mental trick that avoids the mathematical paralysis that is similar to writers block. . Brainstorm possible methods that you think may help get to the solution of the problem quickly without too much in-depth thought and then start exploring more thoroughly than one if you believe is most likely to yield an answer.

Now that we are collating and keeping track of our ideas effectively, it remains for us to improve our knack for deducing the correct method to the solution for each problem. The most successful method for this is to spend time breaking down the problem into its simplest elements and then relating it to something that you are seeing the solution to before. This process typically involves changing the original question into a different question that is easier to understand and conceptualise.

For example, today I was doing a problem which involved a sedan overtaking a car involving accelerations and this was difficult for me to get my head around. I replaced the van and car which have a fixed area to non-point objects of zero size and adding distance between those two points which were the lengths of the car and van. This simplified the problem and allowed me to complete the question based on previous problems that I had seen analysing the speeds of objects with no size.

In this way, there is no easy fix for difficulty in problem-solving. Experience is usually the only answer to develop the "knack". However, employing the services of a tutor or other professional who has prior knowledge in how to do the questions which challenge you is an easy way to add to your collection of solutions.

Finally, based on my experience as a tutor, common things which students do not want to do when doing problem-solving and should get used to are: defining new variables not given in the question, adding lines to geometric diagrams and solving equations with little to no numbers in them.

Also, never be above a long and complicated solution. If you can see two different methods of solving the problem and one is shorter, by all means use that. However, if you find a solution which you know will work eventually but will take lots of time, do not be above doing a long algebraic solution.

The biggest and most useful piece of advice that I can give to any new problem solver is too literally just try anything! Think of something to do which you think will help you get closer to a solution and do it without thinking about it more. I can almost guarantee that your inaction in trying something to solve the problem and find the full solution in your head will invariably take longer than if you had just found something to do and busied yourself with doing it.

Best of luck in all of your mathematical problem-solving adventures and I hope that this article contains some helpful tips to get you started on your path of becoming an excellent mathematician.