Rong but Reasonable-Part 1
Ashwin Guha, Math Educator
Have you wondered about the ‘w’ in ‘write’ and ‘wrong’? Why Shakespeare is a playwright, though he did not ‘wright’ plays but ‘write’ them? Going by pronunciation, a beginner would be justified to spell ‘rite’ and ‘rong’. Neither is right, but they are reasonable. This article is about wrong but reasonable answers we often encounter in school math.
As a math educator, I interact with individuals having a wide range of competence in math — from primary school kids to doctoral students. I am curious to understand how they learn, reason and respond — especially to unfamiliar situations — in their daily lives or in their classroom lessons. I am particularly fascinated by the incorrect answers to math questions from students. They are far more interesting than correct answers, since they reveal more about the thinking process that goes inside the student’s head.
As teachers, we are often in a hurry to achieve desired learning outcomes. We look for correct answers from students to assess their learning. Once we receive a satisfactory response, we race ahead to the next topic. I argue that there is some value in exploring incorrect or incomplete answers. An immediate benefit is that it helps teachers find learning gaps among students. There is another subtler benefit to inquiring into wrong answers. It develops the art of reasoning. Because even wrong answers may sometimes be reasonable.
When a student gives a wrong answer in a class, it easy to be dismissive. It takes a bit of effort to show why the answer is wrong. It takes more effort to understand why the student thinks it is right. But it is a worthwhile exercise, even if done occasionally.
Let me illustrate with the following question which I often ask 10 or 11-year-olds.
Divide the equilateral triangle into 3 identical parts
These are the three most common responses I get.
Option (a) is obviously wrong. Option (c) is obviously correct. The middle response looks suspicious. Can you see why Option (b) is not correct?
Yet without fail, I get this response from students. Let us see why this answer looks reasonable.
1. We get three same-shaped pieces, namely, triangles.
2. The triangles appear to be of the same size.
Can you see that Option (b) is a better answer than Option (a)? It shows a basic understanding of shapes, which younger children, say, four-year-olds may not have. Yet it falls short of mathematical rigor. And that is okay.
Usually, the students to whom I ask this question are aware of ideas like isosceles triangle, obtuse triangle etc. although they may not be aware of congruence of triangles or the formula to calculate the area. That makes it all the more interesting, since this question could be the opportunity to introduce the idea of congruence.
At this point, I ask students to relook at Option (b). They begin to realize something is amiss. The triangular pieces on the flanks do not look like the middle triangular piece. When I (or fellow students) bring up this issue, kids serve a brilliant riposte.
‘Sir, you haven’t drawn the figure properly! If you divide the base into three parts accurately, we will get identical parts.’
This highlights an essential and implicit and often overlooked part of maths — abstraction. Our simultaneous emphasis on precision and approximation leaves children confused. In a chapter on construction, we insist that the student draws line segments of length 6.4 cm precisely. But in a chapter on mensuration, we approximate a farm several acres in size to a small rectangle a few centimetres across on a page. To be clear, this is not really approximation as much as abstraction. Children do not realize that we are capturing only the abstract idea in the rough diagram, whereas adults believe they do. For the student, Option (b) is a valid solution, because it is a correct solution to a similar problem below.
Divide the slice of pizza into three identical parts
The slice of pizza is kind of like a triangle, and the solution works for the slice. So, it is reasonable to expect it will work for the triangle as well.
This inductive logic should be appreciated and encouraged among students, even if it fails to provide the correct solution in this instance. Because this is how one develops familiarity and understanding with a new idea.
Going back to our original problem, I claim to skeptical students that no matter how accurately you draw the lines, the three parts will never be identical. I invite you to consider this question for a moment. How do you convince that there is an inherent flaw in Option (b) and you are not just being a stubborn teacher?
One way is to have students cut out a triangle, cut the three pieces and see if they can superimpose the pieces on each other. This approach is time-consuming and often messy in a classroom. But it is worth attempting.
The second way, is to show that the flank pieces are obtuse angled triangles, whereas the middle piece is acute angled triangle. Can an obtuse-triangle ever be identical to an acute-triangle? No. This argument makes use of prior knowledge and a level of abstraction that is just about comprehensible to 11-year-olds. (For younger kids, the previous approach is the best)
Note that we did not use terms like congruence or calculate the area of the pieces.
As an aside, we can in fact see that Option (b) is partially correct, since the area of the pieces is the same. This would be a discussion for advanced learners.
Also, can you find another solution to the problem, other than Option (c)?
‘Why spend so much time on a wrong answer?’ you may ask. After all, we already have a clear-cut answer in Option (c).
First, this exercise illustrates the fact that though there could be many wrong answers to a question, there are different degrees of ‘reasonableness’ used to arrive at a wrong answer. Let us encourage our children to reason. Often that reason could be ‘It looks correct.’ We need not dismiss them as illogical or unscientific. Instead we could say ‘That’s good, but we can do better.’
This is the crux of mathematics. We never encounter a real-life situation where we need to divide a triangle into three identical parts. But we encounter situations where we need to justify our reasons, convince others how they could be wrong, understand why they believe their choices to be correct. This happens ALL THE TIME. It is only by viewing an issue from multiple perspectives, we can foster a meaningful dialogue. Often, that means adopting a perspective that we do not subscribe to. This involves not just mastery over mathematics and logic, but also imagination and empathy, qualities that are unfortunately overlooked in the study of math.
Math is projected as the most objective field of study, and for good reason. Axioms and methods lead to clear, well-defined and unambiguous conclusions. We cannot invoke our right to hold individual opinions and agree to disagree in the mathematical arena. However, this does not mean that mathematical thinking lacks nuance. The proofs in math are not just to stress-test the product (namely, theorems and formulae) but also to stress-test the process (namely, the art of deduction). In order to ascertain the robustness of a process, we must test it on both flawless and defective products. Similarly, to ascertain the validity of our reasoning, we must test in on both right and wrong answers. And the aim of math is to develop this skill in reasoning. Hence, in addition to teaching the rigor of right answers, we must also inquire into the reasonableness of wrong answers.
So, let us actively seek wrong answers in our classrooms. It brings about a lively engaging debate out of every math problem. The classroom experience will be all the richer for it.