**Shweta Shripad Naik**

Often, educators as part of their teaching, attempt to bring familiar/known examples from students’ lives into the classroom. When teaching mathematics in schools, this happens on several occasions and in various degrees. Teachers use real life contexts when introducing a concept or conveying a math problem or even while correcting a solution. However, there are certain portions of mathematics teaching that remain unaffected by what real-life mathematics would entail for that situation. In this article, I report how teaching that emphasizes learning of mathematical procedures, limits the use of real-world contexts. Specifically, I discuss how problems that seem obviously solvable from the students’ real life point of view become inaccessible to them, when constrained by the *“steps”* that school mathematics require.

I describe here a lesson from a 7^{th} standard classroom where students are learning to solve problems of profit and loss. This topic is considered part of commercial mathematics. Commercial mathematics is defined as the math that is used in the practical world of commerce and real-life. Among all the topics in mathematics, in this one, the use of real life contexts is inevitable. In the lesson narrated here we see how real-life contexts help students solve the problems, and at the same time cause difficulty for teachers to bring every student to focus on the *“steps”*. The *“steps”* are an integral part of teaching mathematics in school and here I describe how giving priority to steps in teaching, that too without understanding, create a disconnected mathematics. I discuss an example from research where educators bring to light alternative strategies that are intrinsic to real-life contexts.

**Don’t solve the problem, do the “steps”!**

Punit^{1} Sir, is a middle school mathematics teacher and teaches students from diverse backgrounds in a low socio-economic locality. He was teaching profit and loss to the students on a particular day. After a rhetorical explanation on how to decide profit or loss in a transaction, he ventured into solving “application” problems. The phrase “application” is commonly used for those type of problems, where one uses the mathematics learned to solve real life problems. He wrote the following problem on the board.

“Shaila bought something for Rs. 40 and sold it for Rs. 60, then what happened?”

The students immediately responded in chorus, “Profit of Rs. 20”. I got the impression that the students understood the concept of profit. However, Punit Sir was not happy about the chorus answer, he asked everyone to speak one by one. He asked Raima to respond. As soon as she got up, he asked her, what is given in the problem. She was confused for a moment, as I think she thought she had to solve the problem. Finally to answer, “what is given?” Raima read out aloud the question that was written on the board. May be because she thought the “given” is nothing but the problem. The teacher accepted it as an answer and re-phrased it as, “yes, what is given to us is that the purchase price is Rs. 40 and selling price, Rs. 60”. Then he called on Shahid, and asked him what the next step would be. Shahid couldn’t respond at that moment. What Punit Sir expected from the students was, to write down the formula for profit. The work expected to have specific labels – the *“steps”*, a routine to arrive at the answer. This routine includes, writing what is given, what is to be found, writing down the relevant formula and then actually substituting and solving it. We have all been exposed to this at least once in our lives. But for the students, the problem was already solved. Shaila gained a profit of Rs. 20. And hereafter the whole confusion started.

Students at this moment were quiet and listening, maybe they were not sure what was expected since they already gave the answer. The teacher then wrote the formula and “step-by-step” [formula – substitution – calculation – answer] arrived at the answer as “Shaila earned a profit of Rs. 20”. Students were happy that their answer matched with their teacher’s answer. The teacher continued with the next problem that he narrated orally.

The students seem to be fond of their teacher; they listened to him patiently. He had a certain way of speaking which sounded as if he was telling a story. In the next problem, while he was writing on the board, he was also narrating the problem loudly with pauses and change in pitch of his voice, making the narration of the problem much more interesting.

“I bought a basket of oranges for Rs. 160. The basket contained 2 dozen oranges. I went to the market to sell these oranges. To the first customer I told that the oranges are for Rs. 100 a dozen. The customer started bargaining and I sold the oranges for Rs. 65 a dozen. Did I earn profit or loss?”

This problem was not from the textbook and the teacher made it up at that moment. The teacher called Aisha to the board. This time she knew that she was expected to produce the steps similar to the one written on the other side of the board. She wrote the formula for profit, and then added 65 twice at the corner of the board (see picture) and then subtracted the sum from the original purchase price Rs. 160. The teacher did not interfere. He let her finish.

After Aisha finished writing, Punit Sir asked her to go back and asked if anyone else wanted to “correct her answer”. I was not sure whether the students understood that the solution written on the board was wrong. The teacher also did not point out that what she actually calculated was the loss, though the formula she wrote was for profit. It seemed to me that the first step of her solution was wrong and therefore there was no discussion on the steps that followed.

Even though the teacher did not discuss Aisha’s solution, he explained the context of the problem again. This time, making the story juicier. He used a lot of gestures to narrate how hot the day was when he went to the market to sell the oranges, and how he decided to go home early and therefore, he sold all the oranges to the very first customer. And, then he asked again, “so what happened?” To which the students responded, “it is actually a loss”. After spending a minute working in their notebooks, most of them started saying that it was a loss of Rs. 30. But the story doesn’t end here. Punit Sir insisted on writing the solution of the problem in *“steps”* and this time the students were also needed to notice that they had to use a different formula than what was written on the board. Students were quiet again.

Now what could be the difficulty – the students understood that it was a loss-problem, they even calculated the loss, and they were also convinced by the context of the problem. Still, they were unsure and a little bit confused about writing down the ‘steps’. There were two more problems and the scenario remained unchanged. Students were able to understand whether the situation presented led to loss or profit, but as soon as they were asked to present the solution in ‘steps’ there was uncertainty and fear to attempt the problem. Some examples that they solved together are given below.

Salim bhai bought one bicycle for Rs. 2200. After a year, he sold his bicycle at the cost of Rs. 1800. Find out whether he made profit or loss, and how much?

Reena Auntie bought a TV for Rs. 15400/-. Soon after, she decided to move to another city. So she sold it to her neighbour friend for 13000/-. Find out whether she made profit or loss, and how much?

These contexts, due to their familiarity, made sense to the students. There was even a discussion of how a used TV will have reduced costs and therefore Reena Auntie’s deal made sense. Some students raised a concern about the repairing expenses that Salim Bhai might have incurred on his cycle, but are not part of the problem. The situations in the context were active in the students’ thinking about the problems, but the procedural part of the solution remained unaffected by this understanding. What were the students’ ways of reaching the answer, and why were they not part of the classroom discourse? And where is the space in school mathematics to account for the students’ ways of doing mathematics?

While many of you must have been tempted to know how these students arrived at the answers, Punit Sir always asked for a solution of a solved problem. “How did you arrive at the answers?” this one question would have given us insight into the *students’ steps* of solving problems. Punit Sir might have his own agenda to push for the *steps*, but it appeared to me that the students’ understanding of profit and loss in each problem was personalized. What I mean by ‘personalized’ is students’ identifying with stories of Shaila, Salim Bhai, and Reena Auntie, as it was Shaila’s profit for them, and Salim Bhai’s loss but not as an application of the general profit and loss formula.

If context made the students understand the problem easily, it might have also played a role in how they solved the problem mentally. Now you must be thinking what could be the role of context in mathematical calculations. Nunes^{2} and her colleagues, in their work on coconut sellers brought forward some of these alternate strategies. The coconut seller in the study did not use the school-like strategy, which is 35 ×10, instead he used his knowledge of the price of three coconuts, i.e., 105 to find the price of nine coconuts and then added the last 35 for the tenth one. Here is a transcript of their dialogue.

Customer: How much is one coconut?

M: 35

Customer: I’d like 10. How much is that?

M: (Pause) Three will be 105; with three more, that will be 210. (Pause) I need four more. That is…(pause) 315…I think it is 350.

The mathematical work that entails the work done by coconut seller is as follows:

35×3 (which he might already know)

105+105

210+105

315+35

And also 3+3+3+1

May be in this case 35×10 would have been the efficient way of solving the problem, however the logical flow in the seller’s response somehow reveals much more about his understanding of the mathematics. There are *steps* in this solution as well, but they are specific to this problem. The *steps* are derived from the inherent nature of the problem. In Punit Sir’s class, something of a similar nature could have been observed, but an emphasis on a specific format was a loss for us.

It is not difficult to be convinced that early emphasis on generalized steps is going to be harmful. In Punit Sir’s class with his persistent instruction he made students follow the *steps*, and on the very next day, the students stumbled upon the following problem that needed a slightly different method from the general method.

Anthony bought eight dozens of banana at the cost of Rs. 30. He then sold five dozens for Rs. 45 a dozen and three dozens for Rs. 35 a dozen. Find out whether he gained profit or loss, and how much?

I feel that so much energy of teachers is invested into arriving at a general format and then to teach variations of the generalized form, that there is very little energy and space remaining to manage *students’ mathematical* ideas. I would hope that we facilitate all possible ways of doing problems and then abstract the general part of it. Moreover, real-life contexts have the potential to derive different strategies of calculation as part of students’ shared knowledge. I believe, as teachers, we still need to explore this more!

- All names used in this article are not real, they are pseudonyms.
- Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools.
*British journal of developmental psychology*, 3(1), 21-29.

The author is a Scientific Officer in the mathematics education research group at HBCSE, Mumbai. She holds a bachelor and masters degree in mathematics. She can be reached at shwetnaik@gmail.com.