Some years ago I started a note in my journal with a rather bold title, Explanatory Terms Matter. Below it I wrote five pairs of words and nothing else.
Plus one or positive one.
Area inside or area of.
Doubts or questions.
Cancel or compare.
Solve or simplify.
The note stayed unfinished for a long time, and I think I know why. Each pair looked too small to deserve a page.
It took me a while to see that the smallness was the whole point. A student meets a definition once. They meet our everyday words a hundred times a week. Whatever those words carry, correct or careless, slowly settles into how the student thinks.
Words Are Not Packaging
We often treat the language of a lesson as packaging. The mathematics is the gift; the words are only the paper around it. Geneviève Boulet, who teaches mathematics education in Canada, argues the opposite in a paper I return to often. She describes a group of teachers who were asked what a polygon is. They had used the word for years. After much probing, it became clear that they did not know exactly what a polygon is, and that their knowledge of the shape and their words for it were tied together. Where the words were vague, so was the idea (Boulet, 2007).
1. Plus One or Positive One?
Her sharpest example is a fraction. When we read 2/3 as “two over three”, we describe how the symbol sits on the page: a two, a bar, a three. When we read it as “two out of three”, we describe the quantity. One reading is about ink. The other is about mathematics. Thomasenia Lott Adams makes a similar point about reading mathematics in general, summing it up in a neat line:
Words tell. Numerals listen. Symbols show.” (Adams, 2003).
If the words tell the wrong story, the symbols will show it faithfully. With that in mind, here are the five pairs from my note.
On the board I write −3 − (−5). A student reads it aloud: “minus three minus minus five.” Every word in that sentence is the same word, and yet the three minus signs are doing two different jobs. The first and the third tell us that a number is negative. The middle one tells us to subtract.
Read it as “negative three, subtract negative five”, and the structure appears by itself. One is the sign of a number; the other is an operation between two numbers. The same holds for the plus sign. In 2 + 1 we add one. In +1 standing alone we have the number positive one.
This is not a primary-school issue that disappears in Class 11. It returns, in disguise, at the very first page of modulus. We write |x| = −x when x < 0, and a good student objects: “Sir, a modulus cannot be negative.” The student has read −x as “negative x” and has concluded that it must be a negative number. It is not. It is the opposite of x, and when x is negative, its opposite is positive. A student who has heard “the opposite of x” a few hundred times does not raise that objection at all.
2. Area Inside or Area Of?
We say “the area of a circle is πr²” so often that nobody blinks. But a circle, strictly, is a curve: the set of points at a fixed distance from the centre. A curve has length, not area. What has area is the region the circle encloses, the disc.
Is this pedantry? In Class 6, perhaps it is harmless. In Class 12 it is not. The chapter on application of integrals asks for “the area under a curve”, and students quietly learn that area always means the region between a graph and the x-axis. Then a question asks for the area bounded by y² = 4x, the y-axis and the line y = 4, or between two curves that cross each other, and the student does not know where to begin. The phrase they memorised does not name the region; it only names a habit.
“Area inside the curve” or “area of the region bounded by” are longer phrases, I admit. But they make the student ask the one question every area problem begins with: which region? Once the region is drawn, the integral is usually the easy part.
3. Doubts or Questions?
In Indian classrooms, “doubt” has come to mean question. “Any doubts?” we ask at the end of a lesson. Coaching institutes run “doubt-clearing sessions” and set up “doubt counters”. The word is so common that we no longer hear what it says.
Listen to it once. A doubt is something to be cleared, like a debt or a blockage. It suggests that something has gone wrong with the student, and that the teacher’s task is to remove it. No wonder the class goes silent after “Any doubts?” Nobody wants to be the one who admits that something has gone wrong.
A question is a different thing. A question is how mathematics moves forward. “Why does this work?” “Does it work for negative numbers too?” “What if the triangle is obtuse?” None of these is a deficiency to be cleared. Each is the beginning of some more mathematics.
A doubt is something to be cleared. A question is something to be explored.
So I have changed my last line. Instead of “Any doubts?” I ask “What questions do you have?”, and on good days, “What would you like to ask about this?” The difference in the number of hands is small, but it is real.
There is a lovely irony here too. In its true sense, doubt is precious in mathematics. “I doubt this step is valid” is exactly the attitude of a careful mathematician, and that is the doubt we should welcome. It is the other kind, the label for “I didn’t get it”, that we can retire.
4. Cancel or Compare?
“Cancel” may be the most overworked word in school algebra. We cancel common factors in fractions, we cancel terms on both sides of an equation, and students soon begin to cancel whatever looks alike.
So let us see what “cancel” is hiding. When we go from 3x = 3y to x = y, we have not cancelled anything. We have divided both sides by 3, which is allowed only because 3 is not zero. Now look at x² = x. “Cancel x” gives x = 1, and the root x = 0 is lost without any warning. The word “cancel” made the division by x invisible, and with it the condition x ≠ 0.
Sometimes “cancel” hides something else entirely. From 2ˣ = 2³ we “cancel the 2” and write x = 3. From log a = log b we “cancel log”. What we are really doing here is comparing: the function is one-one, so equal outputs force equal inputs. And that is why “cancelling sin” in sin x = sin α is wrong. Sine is not one-one, and the honest answer is x = nπ + (−1)ⁿα, not x = α.
Even a JEE favourite like the limit of (x² − 1)/(x − 1) as x → 1 deserves this care. We do remove the factor (x − 1), and rightly so, but the reason is that x is close to 1 and not equal to it, so x − 1 is not zero. That reason is the whole lesson. “Cancel” throws it away.
If we say “divide both sides by x, provided x ≠ 0” or “compare the exponents, because 2ˣ is one-one”, the condition travels with the step. In my experience, most of the wrong answers in this topic come from that one missing condition.
5. Solve or Simplify?
The last pair sounds the most obvious, and yet I see it mixed up every week. We simplify an expression and we solve an equation. They are different tasks with different answers.
To simplify (x² − 9)/(x − 3) is to rewrite it in an equivalent form, x + 3 with x ≠ 3. The answer is another expression. To solve x² − 5x + 6 = 0 is to find the values of x that make the statement true. The answer is a set of numbers, here {2, 3}.
When the words get mixed up, so does the writing. A student asked to “solve” an expression sets it equal to zero on their own. Another writes a chain like x² − 5x + 6 = (x − 2)(x − 3) = x = 2, 3, in which the equals sign means three different things in one line. The first equals sign says the two expressions are identical. The next ones are trying to say “and therefore”. Simplifying is a chain of equal expressions; solving is a chain of equivalent statements. Keeping the two verbs apart keeps the two kinds of chains apart.
What This Asks of Us
None of these five needs a new syllabus or a new textbook. They need only our attention, because the habits are ours before they become our students’. Lung (2005), quoted by Boulet, puts it well: technical vocabulary should be seen as a point of departure for exploring concepts, not as the end point of learning. Careful words are not the destination. They are the road.
If you want to try this, start with just one pair this week. Pick the one you say most often, and notice yourself saying it. Say “negative five” when you mean a number and “subtract five” when you mean an operation. Say “the region bounded by” before you set up an integral. Replace “Any doubts?” with “What questions do you have?” for five classes, and count the hands. Say “divide, provided it is not zero” every time you are about to cancel. Ask your students, before they start writing, whether they are simplifying or solving.
Our students will forget many formulas after their examinations. They will not forget the way we spoke about mathematics. Let it be a way of speaking that leaves them asking questions.
Stay Mathactive.
— Ashutosh Kumar
References
- Boulet, G. (2007). How does language impact the learning of mathematics? Let me count the ways. Journal of Teaching and Learning, 5(1). https://doi.org/10.22329/jtl.v5i1.125
- Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher, 56(8), 786–795.
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