“Who has finished?” may be one of the most misleading questions we ask in a Maths classroom. A teacher gives students a question to solve. Within a few seconds, a few hands go up. Some may continue working on the problem. A few are still staring at the question, perhaps drawing something, trying a different approach or simply thinking. Those who finish first often attract a attention. “Very good”, “excellent”, or “you are quick at Maths”. Nobody says it aloud, but the message can be easily understood as finishing first is a sign of being good at Maths.
Not a race
But is it actually true? At some point, without consciously intending to, have we allowed speed to become a measure of mathematical intelligence? Maths certainly involves fluency. Students need to develop familiarity with numbers, basic mathematical operations, formulae and procedures. Being able to recall basic facts quickly can be useful. But mathematical understanding is not always a race. A student may solve a problem in 30 seconds because the learner might have seen a similar question often earlier. Another may take three minutes because the learner is trying to understand what the problem is actually asking.
The first student may be faster, but does that necessarily mean better conceptual understanding? Not always. In fact, some of the most valuable mathematical thinking happens in the silence between the question and the answer. A student may be visualising the problem, recalling something learned earlier, trying to accommodate new knowledge, testing a possibility, rejecting an approach, and trying another. From outside, it may look as though the student is doing nothing and disengaged. Inside the student’s mind, quite a lot may be happening.
Yet, classrooms often leave little room for this invisible work. “Come on, it’s easy”. “You should be able to do this quickly”. “Everyone else has finished” ... Repeated often, such comments teach students that Maths is about speed. The consequences can be significant. A learner who needs more time may begin to believe that Maths is not easy and compare oneself with other classmates who finish fast. Gradually, the thinking changes from “I need more time to solve this” to “I am slow and not good at Maths”. What began as a difference in processing time becomes an identity.
Different ways of seeing
This is particularly troubling because mathematical ability does not develop at a uniform pace. Children bring different experiences, prior knowledge, confidence levels, and ways of thinking into the classroom. Some recognise patterns quickly. Others need to represent a problem visually. Some are comfortable manipulating numbers mentally. Others need to write down their reasoning before they can see the next step. None of these differences, by themselves, tells us who will become the deeper mathematical thinker. Students who ponder over a question often ask challenging doubts.
Consider two students working on the same problem. The teacher explains how a cylindrical road roller re-carpets a particular stretch of a road, then derives the formula for the Curved Surface Area (CSA) of the solid cylinder. The formula was derived and a question was given. The first learner immediately applies the familiar formula and arrives at the correct answer. The second spends several minutes drawing a diagram, trying a simpler example, pondering over, trying to connect and eventually arriving at the same answer through a different route. Who understands Maths better?
We may be tempted to praise the first student because of speed. But, perhaps, the second student has developed something equally important: the ability to make sense of an unfamiliar problem. This distinction becomes even more important when students encounter problems they have never seen before.
After giving a familiar direct question to find out the CSA of the cylindrical roller, the teacher frames the question differently. This time, the question was to find out the area of the road stretch covered by the roller. Adequate information regarding the diameter of the circular side of the roller, length of the roller, and the number of revolutions it takes to cover the road stretch was also provided.
Speed can help when the path is familiar. But unfamiliar problems demand something else: reasoning, flexibility, persistence, and the willingness to sit with uncertainty. A student who has been trained to expect immediate answers may struggle when a problem does not resemble anything in the textbook. The student may ask, “Which formula should I use?” The student is not necessarily lacking neither mathematical ability nor computational skills. Perhaps, the classroom has simply taught the learner to look for a procedure before looking at the problem.
This time, the second student may come up with a reasonable solution, as the learner has taken time to encounter problems logically earlier also. Connecting concepts to the real world and applying them in a context requires faster thinking ability and cognitive skills. The first learner’s conceptual clarity may be good but applying the procedural knowledge in a real-world context needs more in-depth understanding and clarity as demonstrated by the second learner.
This is one reason why Maths education needs to distinguish between fluency and understanding. We would not want a student to spend five minutes calculating something that can eventually be recalled instantly. Fluency matters, but it should support thinking, not replace it.
The teacher pitches in and explains that, when the roller revolves once, its curved surface only touches the ground. When it revolves ‘n’ times to complete the road stretch, the portion covered by it will be the area of the road. The product of the CSA of the roller and the number of times it revolves, i.e., (2πrh × n) square units will give the area of the road stretch that has been re-carpeted.
When a teacher poses a question, the first hand that goes up may not belong to the student with the deepest understanding of the material. It might simply belong to the student who is quickest to respond. The teacher explains the concept of ‘volume’ for three-dimensional solid objects as the amount of space occupied by the object. The students have already learned that the volume of a cuboid is calculated by multiplying its dimensions: Length(L), Breadth(B), and Height(H). To derive the formula for the volume of a cylindrical object, the teacher asks the class, “Who knows the answer?” The first learner responded quickly, saying πr²h. When asked how the learner arrived at that answer, there was only silence. Instead of immediately celebrating the first correct response, the teacher might say, “Take a minute to think about it.” Additionally, rather than simply asking, ‘What is the answer?’ the teacher could instead inquire, “How did you arrive at that answer?” or “Who found a different method?” This encourages deeper understanding and critical thinking among the students.
The second learner took more time but demonstrated a better conceptual understanding by explaining that the volume is simply the product of the area of the cross-section of a solid object and its height. This learner approached the problem from a different perspective and arrived at the correct response. This example shows that a quick response does not necessarily indicate conceptual clarity or deep thinking. For the first learner, the concept of the volume of a cuboid, expressed as Length x Breadth x Height (L x B x H), is understood but the mathematical aspect is not fully grasped. He/she fails to recognise that L x B represents the area of the cross-section of the cuboid, which is obviously a rectangle. When this area is multiplied by the height (H), which lies in the space, it results in the volume of the cuboid.
Thought process matters
The teacher engages in a thoughtful discussion and concludes that the concept of volume is determined by ‘the product of the surface area of the base of the object and the height’ it occupies. For a cylinder, the area of the circular base is calculated using the formula (πr²) square units, where r is the radius. The height of the cylinder is represented by ‘h’. Therefore, the formula for the volume of a cylinder is: Volume = πr²h cubic units.
The focus shifts subtly, from speed to thinking. Students then start internalising the fact that, rather than arriving at a correct answer swiftly, the thinking process matters more. There is another danger in making Maths a race: comparison. When students constantly see who finishes first, Maths becomes competitive even when the lesson was never designed to be. Some of the greatest mathematical discoveries did not emerge from rushing towards an answer, but from people who were willing to stay with a problem, question an assumption, try another possibility, and remain uncertain for a while.
In a Maths classroom, our students too deserve that intellectual space. The moments of pause need to be gauged as moments of deep thoughts, ensuring clarity, critically analysing the situation, and arriving at meaningful conclusions.
Perhaps, instead of asking which student finished first, we should sometimes ask which student can explain the Maths behind the answer. Instead of rewarding speed alone, we should reward clarity of thought. When a child takes more time, the teacher’s response should not be, Haven’t you finished? but Take your time. I want to see how you are thinking.’
If Maths is about making sense of patterns, relationships and ideas, and real-world connections, we should stop the culture of rewarding students for reaching an answer first. That reward becomes significant if the learner can explain the real Maths behind the question. If we want children to become mathematical thinkers rather than swift answer-producers, it is time we stopped using the clock to decide who is good at Maths. If we mean teach children how to think, we should stop measuring how quickly they can finish thinking as well.
The writer is a Maths teaching professional, Central Board of Secondary Education, Hyderabad.