Approach to problem solving in mathematics

While mathematics falls under science, problem solving is an art. Just like in art, you get better and better as you do more; you get better at problem solving as you solve more and more problems. However, you need a structured way to solve problems to make it more effective and internalize the lessons learnt. Here I will list few steps and how students can apply these steps in problem solving for their board, and JEE main & advanced. This is again a very mathematics oriented approach. If this helps in Physics and chemistry, it is well and good.

Draw a diagram whenever possible

Chinese saw it long back when they said, “A picture is worth a thousand words”. Is it the reason why mathematics Olympiad is dominated by Chinese students?

This becomes extremely important in coordinate geometry. I have seen reluctance from students from drawing the picture. They find it difficult to draw diagrams. This shouldn’t be so. Take for example, drawing 3 normal lines on a parabola. Students dread this because the normal lines do not look like normal at all. But have you seen the real drawing in your book. Just check. The point from where 3 normal lines are drawn on the parabola is little farther from the vertex. Moreover, out of 3 normal lines, only one is in the same side of the point while 2 are on the other side. If you look carefully and keep these things in mind, drawing is not at all a problem. In fact it will make things quite easy for you. Similar pattern you can find in all the difficult drawings especially in conic section of coordinate geometry.

The same thing can be done in 3D even though drawing 3D is difficult. Draw only the relevant things that are asked. For example, if you have to draw a line on a plane, simply draw two perpendiculars line and assume that you are looking at it from the side view so that the plane looks like a line. This will help you draw.

Identify the purpose of the problem

The purpose of question is to find or prove something. It may be find the value of an expression, value of variable, prove or disprove a hypothesis. While this sounds quite obvious, it doesn’t look so when you see interpretation of problem by many students. It is not that the language is difficult but the span of concentration is low in today’s times where instant gratification is the most desired thing in the world. Read the question carefully and clearly note down what the question is asking.

Note down any information given about the quantities, both known and unknown

Information about any variable helps you set the boundary of your problem solving. If it says that the variable whose value is to be found is positive. You can remove all negative real numbers including 0 out of your options.

Note down the information and inferences that you can draw

Sometimes you have to draw the information by looking at the equations and problems. For example, if an equation of odd degree is given, you can be very sure that all its roots are not imaginary. If there is an equation of even degree with leading positive coefficient and negative last term, you can be sure that there are at least 2 real roots, one positive and one negative.

Give a name to all variables and constants in the problem

Once you have written down the unknown and known quantities, give those names (x, y etc.) which can be readily used in framing the equations and working through formulae.

Then use all related knowledge till you find sufficient equations for unknowns

Finally solve. Use all your knowledge to solve the question. Your goal is to solve the problem and get the right answer. While solving, keep your mind open for any new insight that you can get into the problem. Remember that solving mathematical problems is also a skill that you build by using formulae, processes, methods, and finally insight or intuition. Most of the students confuse this insight with tips and tricks. This insight or intuition comes from solving lot of problems of different types. Keep solving and discovering.

~ Pankaj Priyadarshi

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