So when we look over the horizon of the calendar in the hope of some sunshine, it’s hard not to feel troubled and apprehensive. Yet we go on. There are syllabi to be completed and children to be coached, special days to be planned and meetings to be attended. In fact, it is the details of our daily lives that save us from a sense of complete hopelessness.

The cover story theme of this issue comes at an opportune time, perhaps, although we didn’t really plan it that way. The authors in each of the articles dealing with the theme focus on how we can bring the heroic to the humdrum. We do live in extraordinary times, times that call for a different perspective, times when an aggregate of small acts can make a big difference – to combating the huge challenges that face us, whether they are political, cultural, environmental, or economic. The person who decides to stop and lend a hand to an older person struggling to cross the street, the one who helps someone fill out a form, the one who calls out the harasser on the bus, the one who carries a cloth bag to the market or the one who chooses the inconvenience of separating trash. These are all small acts of decisiveness – and while they may not add up to our notion of the heroic, they add up to a big change.

So we can all choose to be heroes of our own space and time, in our own space and time. By making the choice that is – individually – a little harder, a little less convenient, a little more time consuming, a little less comfortable. But so much more better for the whole lot of us.

**Happy 2017!**

Over the last few years there has been a noticeable decline in the reading habits of students. Among the various reasons for this, video games and other gadgets are thought to top the list. Research has shown that excessive visual stimulation can be detrimental to concentration and can take away interest from the printed word affecting language skills of students. Difficulties with English as Second language (ESL) also play a part for some children. Further, in the language building progression of listening, speaking, reading, and writing, our school systems tend to focus more on writing. The other three aspects, which are more important for language building, get sidelined. Lastly, language demands of the various Boards are quite high and often the gap between this and the child’s language level is large. When the medium of instruction is itself not comprehensible, it affects the attention, interest, and behaviour of children in the class, thus impacting their academic performance.

There are homes where children are both exposed to English and also encouraged to read and listen to stories early on. Children in these homes not only grow up with a lot of books around them but also get to play language games. Language games (like word building, word ending, etc.) have the dual effect of encouraging bonding among family members and inculcating an interest in language. On the other hand, there are children who have little or no exposure to English at home. This could limit their pronunciation skills, vocabulary, comprehension and expression, and therefore leads to a struggle with the language.

**What is language?**

As we all know the components of language are:

• phonology (system of sounds) – that helps in accurate pronunciation and spellings,

• morphology (word parts) – that helps to understand the meaning of words better,

• semantics – vocabulary,

• syntax – which is the sentence structure and

• pragmatics – which is the social conventions of language.

There could be deficits in each of these areas affecting the receptive and expressive language of children.

- Many children are unable to read as reading instruction is not explicit in primary classes and children are expected to start reading on their own. The textbooks too are often not graded for readability.
- There are some children who can read but are unable to understand the vocabulary or use them appropriately.
- Some others can understand but are unable to express themselves and
- A last group who have oral language but struggle with written expression.

So it is important that each of the language areas be built consciously for the whole class by the English as well as subject teachers.

The author is a special educator who has worked for many years with children with dyslexia and also in training school teachers with remedial methodology for the whole class. Her current focus is scholastic backwardness and language. She can be reached at subhav05@gmail.com.

Everybody loves a hero. You see, even if he gets left behind in Mars all by himself (I am referring to the movie ‘Martian’), he is unfazed, undeterred, unaffected, and unbeatable. He, to somebody ‘old’ like me, looks so unreal and un-human.

But, everybody loves a hero. So, let me keep my skepticism aside, and see how we can convert this popularity into what I would call educational or teaching advantage.

**Advantage self**

Everybody loves a hero, because he has so much heroism inside him.

The most obvious thing to learn is the inspiring strength a hero seems to embody as well as exhibit. Take perseverance for example – obstacles come in all shapes and sizes, but our guy just goes on.

How would it be if we were to ask children to re-read a heroic story or re-watch a hero movie with the express objective of finding where and how many times our hero persists? Further, follow up this analysis with a mapping of the same traits in our own lives and maybe actually conclude that we too have our moments of perseverance.

We can hop from one characteristic to another, see what we love in our hero and continuously dig for the occurrence of the same in our lives. Characteristics that I would love to explore would include courage, sense of humour, initiative, responsibility, righteousness, … and many more.

The authors run an open unschool called Aarohi and invite all readers to visit and see how open learning can be an amazing way to work with children. They also conduct training retreats and online training for teachers and parents. Visit www.aarohilife.org.

In this concluding part of Nest watch, I have an assorted collection of nests that are quite different from what one is accustomed to seeing or hearing about. I am describing these nests in the fond hope that when teachers take students on eco-tours into protected sanctuaries, they can look out for these nests.

**Wild pigs**

We know them as wild boars. Recently they have been rechristened by biologists as wild pigs. These are animals that are easy to observe in the forests although the loss of habitats is fast posing a threat to survival of these creatures. Once upon a time there were at least six to seven species of wild boars roaming the forests of India. Today there are just two species left – the wild pig and an even more endangered pygmy hog found in the north-eastern regions of our country. Generally nocturnal, they possess a strong sense of smell but poor eyesight. They are not scared of humans, are known to destroy farm crops and come looking for left over foods around human habitation. They roam about in groups-described as sounders – which consist of the mature sow, sub-adult sows, and young ones. The males are generally solitary and enter the sounder during the mating period. The wild pigs are quite brave and will charge at you at the slightest hint of insecurity; they charge even at a tiger if they sense danger. Offence it seems is their defence, so be careful when you see them.

There are two kinds of nests that the wild pigs make. The farrowing (for giving birth) and the resting nests. The pre-parturition nest or the farrow nest is made by the sow 24 hours before giving birth. The nest is made of plant materials available close to where the nests would be built. The nest is a length wise structure generally oval to oblong in shape. These are made in select places chosen by the female. The selected place is generally first dug out by the females before the construction starts. The depressions are lined with leaves and are generally built next to trees or wooden logs for added protection. Depending on the weather sometimes they are built in open grasslands or next to water. The nest often resembles a long bed of woven basket. The top of the nest is like a dome to cover and keep the babies warm. A bigger and larger sow often builds a bigger nest than the smaller sows.

The farrowing nests are used for only a week after which they are shifted to a resting nest. The nests, besides providing warmth to the newborns are believed to also protect it from predators such as leopards, when the sow is away foraging. She never moves far away from the young ones, but the danger of a predator is high. Thus the farrowing nests are constructed to serve as a camouflage too.

The resting nest is not as complex as the farrowing nest. However it is not sometimes possible to distinguish the two as noted by researchers who examined several farrow and resting nests.

In general, these are used, as the name suggests, by resting females and hence do not posses all the characteristics of a farrowing nest. Sometimes the excavations were shallow, often linings were scarce with just the bedding materials arranged sufficiently to allow the pigs to rest.

The author is a consultant for science and environment education. She can be reached at scopsowl@gmail.com.

Idioms are an integral part of any language; additionally, they lend the language its character and richness. Idioms make the speaker sound more fluent, natural, and of course more interesting. Idioms are also able to voice a thought much more succinctly and therefore cut down on verbiage. For instance, instead of explaining at length that you think what someone else has is much more appealing than what you have, but you are fully aware deep inside that you are actually downplaying all the positives on your side, you can express the thought in a nutshell (!) by saying: The grass is always greener on the other side of the fence; or if you want to be more crisp: The grass is always greener.

Our school textbooks are generously strewn with idioms so that students are introduced to them in print, savour them, and then internalize them so that they become a part of their own language system. Unfortunately, most students continue to graduate to the next few classes without the slightest clue of what an idiom is since they only have a blurred idea of the meaning and fumble through the idiom exercises, emerging none the wiser!

How do we warrant that students pause to reflect on the meaning of idioms so that they can learn to use them in their speech and writing in order to make the writing and their conversation richer and more engaging?

http://www.myenglishteacher.eu/blog/colour-idioms-list-and-their-meanings/ is a very educative site that lists 90 colour idioms, explaining them and also using them in sentences so that the reader has a clearer idea of the meaning and the usage. Teachers could use the site to help students expand their vocabulary, and students too could actually do this work independently since the site is self-explanatory. But then, if wishes were horses….and the rulers of teachers were magic wands! As teachers, we know that we need to step in to design a method to encourage and assist students master the idioms and use them actively in everyday speech and writing.

Let’s have a look at some of the colour idioms listed on the website. A student who reads the sentence, *The thief was caught red handed* may figure out that the thief indeed did not escape, but the student may not necessarily understand that the thief was caught while committing the crime. Another sentence like *Charlie is grandma’s blue-eyed boy* may be interpreted to be a grandson with blue eyes (and that could also be true as far as the story or the lesson is concerned!). Similarly, does the sentence *I was in the dark about the surprise party* mean that I was in a place where there was no sunlight or light? What does the child decipher from such sentences? Does the child just stumble over them without understanding the full import and in the process ignore what could be an opportunity to enrich his/her language? Regrettably, this is what happens in most cases. Let us see if it is possible to take the students through idioms in a guided manner.

Have a look at the following 13 pictures and think of what colour idioms they could be associated with. They may not all be easy, but if the students have already been introduced to the idioms, it should not prove to be a cumbersome or an impossible task.

The author is a teacher educator and language trainer based in Hyderabad. She can be reached at manaswinisridhar@gmail.com.

At Shishuvan, integrated learning forms a vital part of the school’s pedagogy. Around nine years ago I joined Shishuvan. Even then, encouraging hands-on activities and project-based learning was a part of Shishuvan’s methodology. On a sunny afternoon as we sat in the staff room, a meeting was called by the leadership team and we were oriented towards the concept of integrated learning. Frankly, I was absolutely confused and wondered if I would ever be able to think in that direction. Today, it’s so ingrained that some of us at Shishuvan have been able to break the barriers of subjects and are able to look at any concept/event/phenomenon from a multi-disciplinary approach.

As a teacher, I strongly recommend this approach. It offers a multi-faceted learning of concepts to the students. It challenges teachers to go beyond their scope of knowledge about the topic. Whenever I create links and explore any topic, I feel that I have learned much more than what I knew before. Over a period, teaching the same topic may become monotonous but this approach gets us to work with a team and new ideas keep coming which once again excites the teacher to take on the activities.

Sitting together and brainstorming on how to link concepts is a pre-requisite to make this approach effective. Looking into the curriculum of the class, previous knowledge of the students and an awareness of contemporary issues/developments is definitely the key to planning the topics to be covered.

Geography is not a subject that can be effectively taught in the classroom. From a subject known for rote memorization, it is becoming a subject which is now being explored in many ways. Adding a multi-disciplinary approach will further ensure that students have a deep understanding of geographical concepts and also find the learning meaningful and interesting. One such topic is ‘Rocks’. Let us see how to make this topic really interesting and comprehensive.

Typically, the learning objectives in the topic of rocks in geography include the following: students should know what rocks are, understand their uses and how different types of rocks are formed. Understanding about rocks also provides a base to study the topic of formation of soil and the concept of weathering.

Before beginning, the teacher can give an assignment to the students as homework. Students can be asked to list the objects in their house/neighbourhood/on their way to school/in school where stones or rocks have been used. The students can then be divided into groups and each member of the group shares the information. They then generate a common list. Each group is given a chance to read out five objects at a time and all those who have listed the same can put a cross mark against that so that they do not repeat the same. The list can be then displayed. The teacher can also add some important uses if they have been missed out.

Apart from showing videos on the formation of each type of rock, you can make models showing the formation of igneous and sedimentary rocks. Demonstrate the eruption of volcano and how lava flows. There are many websites which give information about the material and procedure for demonstrating this. Some of the sites that can be visited are:

http://www.volcanolive.com/model.html

https://sciencebob.com/make-your-own-volcano/

The discussion about how igneous rocks are formed by cooling of lava released during an eruption of a volcano can be done after the demonstration.

A model of landforms made using clay soil can be made to show how surface runoff always carries sediments along with it. A beaker with water can be taken and every day for a week, the students can put a handful of soil collected randomly from their surroundings. This beaker should be placed in a sunny spot. They can analyse how the particles settle. Over a week, once water starts evaporating, they will be able to see how the sediments start forming layers. This can continue till the water evaporates leaving just the sediments in the beaker. This will give a thorough understanding of the concept of sedimentary rocks.

The author teaches Geography at Shishuvan High school, Mumbai. She can be reached at prachirandr@gmail.com.

A young couple, Rohit Dhankar, working towards his Ph.D in Mathematics and Reena Das getting ready to write her Ph.D thesis in Sociology were approached by a friend who wanted a couple to run a new school for him. The friend wanted the couple to be trained by the visionary educator David Horsburgh, whose village school Neel Bagh, 100 kilometers from Bangalore, was the Mecca for alternative schooling in India. For Rohit and Reena, who took up the offer, the time they spent with David at Neel Bagh was transformational. Coming from a background of mainstream schooling, David’s class-less, teacher-less, curriculum-less school with its emphasis on learning by doing, was a real eye opener. They came back to Jaipur inspired, and without waiting to finish their doctoral degrees, they started Digantar with seven children in 1978. Reena told me that her father, who was an army doctor, never forgave her for not becoming the college professor he thought she had almost become. Telling him that she wanted to spend all her time with children and didn’t need a Ph.D for that was no good and he never understood how she could so casually throw away all the hard work that she had put in.

The elementary school stayed with some 25 students for ten years or so before Rohit and Reena decided to expand their circle of influence to rural children. Digantar Shiksha evam Khel Kud Samiti was registered in 1987 and today they run three schools catering to 600 children on the rural outskirts of Jaipur. To give you an idea of how different these schools are, here is a partial list from their website:

- Children have the freedom to learn at their own pace. Children learn from each other in collaboration. That is the reason they are not distributed according to classes; each learning group has mixed age groups and multiple levels of learning.
- No fee is charged from children in any form.
- The NCF 2005 and RtE highly recommend Continuous and Comprehensive Evaluation (CCE); Digantar schools have been using CCE since 1978.
- This school has established itself as a source of help to other organizations and schools all over India in the past 35 years.

I have been to some gatherings that Rohit has been a participant in and every time I have been awed by his encyclopedic knowledge about all aspects of education, especially about the philosophy of education. (I have walked through the library at Digantar and know that it is not only philosophy he is interested in). I also noticed that only the brave, or people who know him very well, dare to take him up on an argument. He is extremely incisive and whatever side I start on, I notice that most times when his opponent finally gives in, I for one am definitely seeing Rohit’s point of view. I think this is a sign of great authenticity and not of great persuasiveness or scholarship. Rohit commands great respect in the world of school education in India.

He was not there when I recently went to Jaipur and took my autowallah down the 200 metre kutcha road leading to the gate of their beautiful five-acre office complex. (The schools are a further six km away from here). But a pleasant, smiling, slightly – built, and very fit Reena was there and I took an instant liking to her. Finding out that I had come by an overnight bus from Delhi, and before that by a 48-hour train journey from Kerala, and that I had come to Jaipur only to meet her (of course, without an appointment), she gave me a lot of her precious time. She showed me around the office and other buildings that house the various programmes that Digantar runs. These include:

- The Academic Research Unit (TARU), the resource support wing that works on teacher training, and curriculum, and material development.
- The spaces from where the foundation course, ‘Certificate program in foundations of education’ runs. The participants stay for eight weeks spread over the year and all their stay and study happens here.
- The office of Shiksha Vimarsh, a unique bi-monthly Hindi magazine on the theory and practice of education.

I hope that gives an idea of the uniqueness and comprehensiveness of the vision of Digantar’s work in education. Let me conclude with some appropriate words from their website.

‘Digantar, from the Sanskrit “Dik” and “Antar”, means a change in direction. In our case, it is a well thought out change in direction towards a more meaningful, appropriate, and complete education for rural children.

**Been around for:** 36 years

**Number of teachers/staff:** Digantar runs three village schools (Digantar Vidyalaya) but the organization is larger than its schools. It runs a teacher training and curriculum development division and publishes a bi-monthly Hindi magazine on education.

**Number of children:** 600 children study in the three Digantar Vidyalayas

**Classes handled:** 20 learning groups (12 primary, 5 upper primary, 2 secondary and 1 senior secondary levels)

**Approximate fees per child:** The children pay no fees

**USP:** Digantar Vidyalayas serve the rural community and run along democratic lines

**Location:** Outskirts of Jaipur

**Website:** http://www.digantar.org

The author got his degree from IIT Kharagpur in 1988. This article has been written under a Wipro education fellowship. He can be reached at arunelassery@hotmail.com.

]]>In the first part of this series (Separation Techniques – Part I – November 2016), we discussed some techniques that happen on a routine basis. In this article we will explore and learn about other techniques of separation of various substances by using easily available and local resources.

**Can you think of an idea or method to separate water and oil?**

Before going to the school, I worked on designing an alternate model/apparatus for separating water and oil. This reminded me of my college days where we separated two liquids using a ** separating funnel. **After a discussion with colleagues, I finally designed a model using a water bottle, a straw, some clay (for

When I arrived at the school the next day, the school teacher said, *“I was also thinking about the separation of oil and water but I didn’t get anything.”* What the teacher said motivated me since I realized that she was also trying to find out the process or method of separation. This showed that she was eager and curious to learn. Without losing time I showed her my apparatus and then both of us entered the classroom. The students were then asked, ** “Can we separate oil and water?”** They responded that it is impossible to separate the two. Then I asked them to list out other substances that cannot be separated. They responded,

In an earlier lesson on separation techniques, the students had reflected on how it was not possible to separate chalk powder and soil. Taking that further, I asked the students if they could separate

• Soil and salt

• Sand and salt

• Salt from salty water (mixture of salt in water)

• Sugar from sugar water (sugar solution in water)

• Salt and Naphthalene ball

Almost in unison they said, ** “We can’t separate them.”** At this point, the class teacher referred to the process of salt formation from sea water by asking,

The author is with the Azim Premji Foundation. He can be reached at avneesh.shukla@azimpremjifoundation.org.

Robots have always been a subject of fascination for young minds. Be it in a TV series (Vicki, the robot in Small Wonder), cartoons and comics (Irona in Richie Rich) or movies (Terminator Series), robots are an attraction for children. In this article, we reflect upon the experiences we had with students (over a broad age-range) during informal interactions such as science popularization events. We hoped to get some insight into students’ ideas about robots, their interest in the subject, and the possibility of using robotics as a starting point to introduce them to design and technology. Robotics happens to be a field with the potential to combine different disciplines, such as electronics, biology, and design. Given its popular status, it can be a helpful tool in classrooms as it provides a fun context for children to start exploring and playing around with technology.

The word ROBOT comes from the Czech word “robota” meaning forced labour. Colloquially people understand a robot as ‘one that mimics a human being’(Koren, 1985).

**Why bring robotics into education?**

Over the last couple of decades, a growing body of research in the area of early childhood education has emphasized the value of using constructivist methodologies in learning. Some researchers advocate the subject of robotics as part of early education since it can support the integration of constructivist practice and philosophy by engaging children and teachers in the active design of meaningful projects (Bers et al., 2002, p.124). Researchers have also suggested that robotics could be used to address the tenets of constructionism – learning by designing, manipulating (computational) objects to think with, exploring of powerful ideas, and being reflexive.

**Informal interactions**

The observations discussed below are mainly from four independent science popularization events with special focus on one that took place in a college in Wadala, Mumbai. In this college, we conducted a basic robotics workshop aimed at creating interest in and increasing awareness about robotics. We were also interested in eliciting students’ ideas about robots. A total of 159 secondary and higher secondary science students participated in this workshop (74 girls, 80 boys, 5 did not indicate gender). We had sessions that involved completing a short questionnaire focused on students’ ideas about robots and their uses and demonstrations of a squiggle bot (http://www.instructables.com/id/Easy-Squiggle-Bot/) with instructions on how to make one, and a session addressing student queries regarding career options in Science, Technology, Engineering and Mathematics (STEM) disciplines specifically in the area of robotics. The students chose the order of sessions randomly, based on availability of the resource persons for the sessions.

A squiggle bot is a rudimentary beginner’s robot that can be used to scribble using markers or sketch pens.

While this workshop did not have many opportunities for students to work with their hands owing to time, space, and resource constraints, it did provide some insights into students’ ideas about robots and how robotics could serve as a potential starting point for design and technology education.

Disha Gupta, Adithi Muralidhar and Sugra Chunawala work at the Homi Bhabha Centre for Science Education, TIFR, Mumbai. They can be reached at disha/adithi/sugrac@hbcse.tifr.res.in.

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.

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