Mathematics as a discipline has many characteristics, but none may be as central to the subject as problem solving; none may be as evocative and memorable; and, perhaps, no other activity holds so much scope for the thrill of discovery. If we as teachers are to convey and instill a lasting love of mathematics in our students, then we must exploit to the fullest extent the possibilities offered by this activity.
One may view problem solving as a playful game of tease – a game between the teacher and the student, or between the math enthusiast and the subject itself. The element of play is essential. Here we use the word ‘play’ in its original sense and not the sense it has acquired in modern times: allowing free movement and exploration, with no compulsion, no sense of “I must prove myself” or “I must show that I am better than you.”
The reader may wonder whether we are making a big deal out of this; after all, most people would imagine that school mathematics is all about solving problems. That certainly is the impression that children and parents would carry with them from their experience at school. However, these ‘problems’ are merely back-of-the-chapter exercises; they cannot properly be called problems at all – they are merely drill exercises. To be taught, say, the quadratic formula for finding the solutions of a quadratic equation, and then to apply the formula to 10 different quadratic equations, differing only in small ways from each other: that is not problem solving at all! That is merely doing a set of exercises to reinforce the algorithm, and ‘drill’ is a more appropriate word for such an activity, not ‘problem solving.’ No, what we are referring to in this article is not this but the culture of solving non-routine problems.
No article about problem solving can leave out the incisive observations made by the mathematician George Pólya, who has written with such eloquence and clarity on this topic. Here is one quote of his that is absolutely on the dot: It is better to solve one problem [in] five different ways, than to solve five problems [in] one way.
We will have occasion to share more quotes from Pólya later in the article.
The title of this article refers to the ‘joy’ of problem solving, but that does not mean that one does not feel pain during the process! Quite often, the dominant feeling during problem solving is one of frustration, of feeling rather stupid. Solving a problem in geometry, you stare at the figure, and then you stare more, and still more, all the while feeling that you have run into a wall. And then, all of a sudden, a glimmer of an idea strikes you, maybe while out on a walk, maybe while you are in bed, trying to fall asleep, or when you wake up in the morning, and the idea stays with you. You continue chewing on it, off and on, and, bit by bit, the pieces start to fall in place, like a jigsaw puzzle. Then, one day, the picture is complete, and you gaze at it in amazement, marvelling at its beauty, and wondering why you never got the central idea in the first place. This, more or less, is the experience of problem solving. Would not it be wonderful if every student could have a taste of this in school? Is it not part of our work that every student experiences this taste?
A peculiar syndrome that confounds Indian education is that of “completing the syllabus.” We must, absolutely must complete the syllabus – and woe to the teacher who does not! This phenomenon brings out a dichotomy we see sharply highlighted in education systems all around the world and particularly so in India: focusing on what to think, rather than how to think. We tend to be quite obsessed with the ‘what’ and on ensuring that every little detail in the syllabus is attended to, down to the last comma and full stop, but curiously neglect the ‘how’, which surely is the most essential part of education, for the ‘how’ is about learning how to think. What better way is there to learn the ‘how’ than to actively plunge into problem solving? We quote Pólya again:
Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet … and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.
One can learn how to think only by actually doing so and finding what it is all about; there is no technique for it….
However, it is important that the teacher not be carried away in his enthusiasm when it comes to posing challenges for the student. He must assess for himself as accurately as possible the level that is most appropriate to the student. If he poses challenges that are too easy, that are not challenges at all, he runs the risk of creating a dismissive attitude in the student, a lack of engagement, a lack of creative tension. If he poses challenges that are too challenging and much beyond the ability of the student, he runs the risk of creating a mental block in the child, a lasting feeling that “this subject is not for me,” or a permanent impression of inferiority. As the mathematician David Hilbert noted,
A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts.
There is a fine line to tread here. We need to find that line by careful experimentation and by getting to know the child as well as possible – and that can happen only through close contact between teacher and child.
At the primary and upper primary level, problems involving numbers are available in plenty. Children love such problems! For example, there is the whole family of problems called cryptarithms, which are simply coded forms of arithmetic problems with obvious restrictions such as: (a) two different letters cannot represent the same digit; (b) numbers are not permitted to start with the digit 0.
A particularly simple instance of such a problem is the following. Find the digits O, N, G if
ON + ON + ON + ON = GO.
It turns out that this has a unique solution, but one must not say this in advance; children need to discover it on their own. However, a tiny change gives rise to interesting effects. The following problem has multiple solutions (and children should find them on their own):
ON + ON + ON = GO.
In contrast, the following has no solutions at all:
ON + ON + ON + ON + ON = GO.
These are interesting discoveries for children to make for themselves – that some problems have multiple solutions, while others have no solution. All these phenomena have parallels in real mathematical applications. Here is a well-known example: it is not possible to find positive integers a and b such that a2=2b2 (this is just another way of saying that the square root of 2 is an irrational number). A more spectacular and famous example is Fermat’s Last Theorem (‘FLT’), which is the statement that there do not exist positive integers a, b, c and n>2 such that an+bn=cn.
Here are two cryptarithms that I have tried out with many batches of students.
- Find the digits A, B, C, D given that ABCD x 4 = DCBA. (This means that when we multiply the four-digit number ABCD by 4, its digits are reversed.) A related cryptarithm is ABCD x 9 = DCBA.
- Find the digits T, W, O, H , R, E given that TWO x TWO = THREE.
- Given a square ABCD with side midpoints E,F,C,H, we draw lines as in Figure 1. Find the ratio of the area of shaded region HIFJ to that of square ABCD.
(I got this problem from the Twitter account https://twitter.com/Cshearer41)
- Look at the configuration of the four circles (Figure 2), all touching each other. If the largest circle has radius 2 units and the two circles shaded blue have radius 1 unit, what is the radius of the circle shaded red?
- In Figure 3, we see a configuration of three squares in a row. Three angles x, y, z have been marked. Find the angle sum x + y + z.
I have experienced greater class engagement from the first one (the digit reversal problem) than from any other non-routine problem! Perhaps this is because the code can be unraveled using pure mathematical reasoning. There is something extremely appealing about being asked to find a four-digit number whose digits are reversed under multiplication by four. The utter simplicity of the statement is part of its appeal.
Another rich source of non-routine problems is magic squares. We can start with questions such as, “Why do we not have 2 x 2 magic squares?” Then we move to 3 x 3 magic squares. It turns out that using the numbers from 1 to 9, just one magic square can be made; every other magic square using these numbers is simply the original one looked at in another way. However, to show this requires a bit of work. This too is a wonderful topic for middle school children. Closely related to this is the subject of magic triangles.
It is important to dip into Euclidean geometry to as great a depth as possible, simply because geometry is such a rich source of beautiful problems. It also happens that Euclidean geometry is a highly neglected area in school mathematics. Let me share here a few problems that have been excellent sources of class discussion.
The last problem is very famous and it allows for some very elegant solutions.
In this short write-up, I have focused on problems at the middle school level. However, the beauty of this subject is that we can find problems that are both enjoyable and accessible to the student and the teacher alike, at every level. We can also find problems that even students and teachers who are highly capable will find challenging.
The beauty of problem solving is that it is nearly impossible to have an encounter with it that does not enrich and empower us in some way. Each time, one learns fresh the truth of Pólya’s dictum, that it is better to find five different ways to solve a problem than to solve five problems in the same manner. Is it not time that we allow this dictum to come alive in our classes? Is it not time that we work towards nurturing a culture of non-routine problem solving at every level of the school? If our intention as mathematics teachers is to instill in the child a lasting love for the subject, then it is imperative for us to educate ourselves in this matter and to take forward the spirit of problem solving to as great an extent as possible.
The author is the Director of Sahyadri School KFI. He has been in the field of mathematics education for many decades. He is the author of several mathematics books for students and teachers and serves as Chief Editor of the magazine At Right Angles. He has a deep interest in nurturing inquiry into fundamental issues among teachers and students at the school level. He can be reached at [email protected].