Volume 6 Issue 3: The word ​‘heuristics’ has always left me fumbling for the perfect definition. Rule of thumb seems adequate but too informal, as a teacher trainer, I’ve always preferred to illustrate rather than define this word. And the first article in the November issue does just that- Gaurav Bhatnagar, leads you gently into the Exponential Series- a heuristic definition! The relaxed tone continues with V. G. Tikekar setting the stage to arrive at formulae which are usually given and proved using mathematical induction. He, on the other hand, uses the narrative mode to set the context which explains and then proves the sum of squares and sum of cubes formulae. From story we move to song, V. Ramprashanth’s article Tonic Ambiguity is a delightful depart from our usual articles and it explains very simply, using just the concepts of ratios, how two hearers can place the same tones at different positions on the musical scale. CoMaC and then Shailesh Shirali, wrap up the Features section with a bouquet of proofs.

ClassRoom opens with Prithwijit De’s trademark investigative style – the title simply says A Triangle Problem – read on to find out more. Student Corner is our happy space and this time, we have Vinay Nair’s students talking about Happy Numbers and Bodhideep Joardar Investigating Pi. Next, we have Naveen Yadav combining Set Theory, Geometry and Combinatorics in Scalene Triples and Ujjwal Rane follows up on his previous article on Fagnano’s Theorem with a very visual, very practical alternative proof. A video of this result has also been uploaded by him on YouTube, it is available at https://​youtu​.be/​5​M​r​N​M​-​VxXd8. Low Floor High Ceiling features a modification of an investigation into triangles having the same area and perimeter and conditions for congruency. Learn about G numbers in Hara Gopal’s article. We’re particularly delighted to bring in yet another new sub-section to ClassRoom – Course Correction features vignettes from pedagogical experience, ways in which pitfalls leading to misconceptions can be addressed and used to advantage to strengthen student understanding. This time, we feature the solution of a problem from a traditionally ambivalent topic- Combinatorics.

On to TechSpace, we have not one but two articles this time. Thomas Lingefjard describes the use of technology to introduce concepts in Calculus and Sangeeta Gulati describes changes in the Desmos software and how it can now be used to build activity kits. Problem Corner is a treasure trove of interesting problems in addition to the usual Middle School, Senior School and Adventures in Problem Solving articles.

Review features an old classic – R. Ramanujam takes us through a lovely guided tour of Hardy’s A Mathematician’s Apology. And there is an unexpected bonanza, with this issue, we will feature a mini- review, it may be of a video clip, a game, a story…………..all things mathematical in essence. This time, J. Shashidhar talks about Marcus du Sautoy’s Story of Mathematics. Read the facing page to know more about this fascinating mathematician.

Pullout has graduated to Upper Primary – with this issue, we bring you a compendium of information about teaching Integers- surely the point when students begin to have negative thoughts about mathematics. I’m sure you’ll enjoy reading how to combat those tendrils of fear.

Sample surveys conducted by the National Sample Survey Office (NSSO) are the most widely used sources of household level information about consumption, employment, and other socio-economic indicators in India. The representativeness of samples, the wide range of topics surveyed, and the availability of a long-time series are some of the reasons for the appeal of NSSO data for research and policy. This paper assesses the quality of the NSSO data for Nagaland and Jammu and Kashmir, which lie in India’s politically restive ethno-geographical periphery. It argues that the NSSO data for these states during 1973 – 2014 lack representativeness and inter-temporal comparability due to faulty sampling frames, frame and sample non-coverage, and biased samples. It quantifies the impact of data quality on statistics of interest to policy-makers. The paper shows that the estimates of monthly per capita consumption expenditure (MPCE) are sensitive to non-coverage and argues that the incidence of poverty is underestimated because NSSO surveys failed to capture the complete distribution of consumption expenditure due to non-coverage. In Nagaland, the degree of non-coverage was so high that in most years between 1993 – 94 and 2011-12 the state’s poverty headcount ratio was the lowest in the country despite the possible overestimation of its poverty line. The paper concludes with a discussion of the implications of the use of non-representative survey data. Put together, the unreliability of government statistics in Jammu and Kashmir and Nagaland highlights systemic problems that have wider implications for our understanding of the relationship between state, statistics, and policy-making.

Education policies have resulted in several enduring legacies in keeping with the ever-changing society, as well as its political manifestos. Because of the dynamic nature of society in general, and the speed with which the world changes, bringing with it changes vis-a-vis every aspect of socio-economic-political change, national education policies from 1968 have seen changes in 1986, 1992 and now a draft policy in 2016. It is expected that each policy, when introduced, will take into account the demands of the day while preserving the central core of its goals. National educational policies are, or at least should be, the consequence of a clear understanding of the socio-cultural beliefs of India, and at the same time having clarity in introducing a system that is aligned to the goals enshrined in our Constitution and has the avowed purpose of creating a democratic society of enlightened citizens. In this Issue, we have articles which examine education policies from 1968, which was the starting point chosen for this issue. The draft policy of 2016 has been closely examined, as have the ways in which language and literacy have been addressed. Another article is an examination of the history of educational policy documents. Other articles have been written about specific aspects such as CCE and the Midday Meal Scheme. This is only a sampling and we hope that readers will find this Issue interesting.

Volume 6 Issue 2: It is a pleasure to share with you, our readers, the collection of articles in this, the July 2017 issue. The (un)popular view of mathematics being a terrifying subject takes a completely new twist with the first article in which a mathematician takes on a terrorist threat! The hunt for answers to a mathematical problem is usually an absorbing one, at least to aficionados of the subject but Arun Vaidya’s fascinating story I M Code makes it a matter of life and death.

Following this, we have an article on another application of mathematics: Interpolation by Sankaran Viswanath. You will see again how mathematics is a tool for prediction, and how data can be fitted into mathematical expressions which then provide a mathematical model. From here, we move on to card tricks; yes, fun and mathematics can go together — and At Right Angles shows you how in Suhas Saha’s Ternary Base Magic Trick. A quick peek behind the magic reveals patterns based on the ternary base, it’s not as complex as it sounds, read on to find out. Our Features section ends with Shailesh Shirali’s exposition on Quadrilaterals with Perpendicular Diagonals, a nice bouquet of Arithmetic, Algebra and Geometry for you.

In ClassRoom, we have the second part of the Inequalities series started in the March 2017 issue, again, both Algebra and Geometry are used to first prove the arithmetic mean- geometric mean inequality and then apply it in several situations to illustrate the power of this relationship and also view its implications in graphs, geometric figures, functions􀀿􀀿􀀿.the list of connections seems endless! Moshe Stupel and David Ben-Chaim appear next with their article Three Elegant Proofs, the name says it all, we promise it lives up to its title. CoMaC, as usual, provides an indepth analysis of an often-asked question, now increasingly appearing even in WhatsApp forwards: What’s the next number? Is the answer really unique as the question implies it to be? More on numbers with Swati Sircar and Sneha Titus, writing on the Sums of Consecutive Natural Numbers; mental mathematics becomes visual all of a sudden, and this Low Floor High Ceiling activity is sure to appeal to a variety of learning styles. Vinay Nair takes up the theme of Divisibility by Primes and provides some powerful tests using an ​‘osculator’. Students are sure to be intrigued. ClassRoom concludes with a Proof Without Words on a property of the Orthocentre of a triangle.

For some time now, we have been featuring articles by students and we are particularly happy when they write in with their own discoveries. So much so, that from this issue onwards, we have devoted space to Student Corner in the ClassRoom section. Featured this time are Bodhideep of class 6 and Parthiv of class 11, you are sure to be impressed with their discoveries.

Our cover this time, features Golden Quadrilaterals and the illustrations have been provided by Michael de Villiers, who continues his series on constructive defining. These beautiful quadrilaterals have been defined by investigation and are an interesting activity for students who believe that everything in mathematics is pre-defined and that there is nothing new in mathematics to be discovered.

Problem Corner has seen some changes over the last few issues. In a deliberate attempt to avoid a ​‘camp’ approach to problem solving and to make this section more inclusive, we have a wide variety to interest our readers. Prithwijit De sets the ball rolling with his article on ProblemPosing. This is followed by Middle and Senior Problems addressed to different age groups. CoMaC presents a theorem about a triangle and a problem about a rational number; the titles are deliberately bland but these are as fun as Shailesh Shirali’s Adventures in Problem Solving.

The Review this time will certainly have you leaping to order this book: The Cartoon Guides to Calculus and Algebra, a series whose name says both all and nothing. Can such a serious subject be illustrated with cartoons? With mathematical rigour? Read the Review and I’m sure you’ll be convinced.

Our issue concludes with the PullOut — Padmapriya Shirali focuses on Large Numbers and how students can grasp this concept. I am sure that adults too will enjoy this refresher course and pick up tips on how to make this topic child- friendly and approachable. So it’s over to you now! Happy reading.…

This paper explores the construction of hybrid spaces through the observation of middle school children engaged in short science projects in an informal science learning programme. Hybrid spaces are not just physical structures, but refer to contexts, relationships and knowledges developed by children as their social worlds and identities merge with the normative expectations of school science. Hybrid spaces have been characterised in three different ways: as a convergent space between academic and traditionally marginalized knowledges and discourses; as a navigational space, or a way of crossing and succeeding in different discourse communities; and as a space of cultural, social and epistemological change where competing knowledges and discourses challenge and reshape both academic and everyday knowledges. (Moje et al., 2004; Barton et al., 2008). This paper characterises such hybrid paces by analysing activities of children working on short projects in the broad area of: ​‘Trees, plants and insects’, during a summer camp held at the Azim Premji University. Children seemed to primarily use the ​‘third space’ to navigate between different funds of knowledge and succeed in science. They developed science artefacts such as scrapbooks and a children’s magazine, and negotiated new roles for participating and expressing their developing science identities. They also brought in local knowledge and activities from their home contexts such as gardening, cooking and socialisation with members of their own and wider community. Informal settings help in the negotiation, construction and development of these hybrid spaces, and is particularly meaningful for children who otherwise see science as being alien and outside their everyday lives. Children brought different funds of knowledge into their participation and discussions from both their formal and informal experiences linked to science. Children decided their own trajectory of learning experiences in consultation with the facilitator. This paper also describes various possibilities in informal settings and learning experiences within and outside formal school settings, which help children explore and engage more deeply with their developing interests in science.

This working paper will focus on conceptualising why informal labour continues to persist in the Indian economy. More specifically, it will try to theorise why, despite generating growth, informal labour has been locked in unequal and exploitative terms of contracts. Crucially, it will try to understand why informal labour processes do not evolve mechanisms for providing security and support to labour in the form of benefits and rights. In this paper, I will focus specifically on highly exploitative and unequal informal labour processes. I categorise under this category those labour processes that do not provide a mechanism to reproduce labour power. These processes are predicated on maximising the appropriation of surplus from labour, without providing any provision to support the workers in ill health, facing work related accidents and injuries, or due to retrenchment. While informal labour is defined as labour without legal rights or contracts, I specifically focus on those processes where there do not exist any norms or mechanisms of providing security to labour within the production process. Hence, I refer to these processes as highly exploitative and unequal labour processes. This paper will be structured in the following manner. Firstly, it will introduce the concept of labour process and argue that highly exploitative informal labour processes in India broadly fall into two distinct labour processes namely: informal coercive accumulation process and petty commodity production. It will then derive concepts from the Paris Regulation School to modify the labour process framework. It will then describe concrete case studies conducted by other researchers, to demonstrate that this framework can be utilised to make sense of field work based research on the informal sector. Lastly, I will assess the questions posed in this paper based on the derived framework and will argue that informality is connected to the nature of capitalist development in India.

The quality of education depends largely upon the teacher. Hence building teacher capacity through continuous professional development of teachers is critical to the quality of education. Systems and programs for continuous professional development for school teachers in the formal educational system in India are inadequate and often ill-conceived. Apart from other lacunae, they often do not recognize the professional identity and agency of the teacher. Voluntary Teacher Forums (VTFs) that are facilitated by Azim Premji Foundation in different locations of India, as part of an integrated and multimodal approach to continuous professional development, try to address this central issue. This study shows that VTFs are evolving as very useful platforms for collaboration and peer learning amongst teachers in various locations. It shows that given an environment that allows for easy access to meaningful opportunities, teachers will commit their time, talents and resources to their own professional development. The study further recognizes however, that the real issue is to understand how such forums can be made to happen in different locations in a context as complex as the Indian public education system. While this study highlights certain crucial dimensions of the VTFs – including what happens within these forums and the efforts that go behind it – these continue to be subjects for our future research works.

Economic reforms sought to replace the Indian statedominant economy with a liberal, competitive market economy. However, a plethora of recent scams indicate that collusive rentsharing arrangements between business and policy makers are prevalent. A nexus has developed, linking politicians and business, as well as the bureaucracy. These trends signal a breakdown of competitive markets and the accumulation of wealth via corruption. This paper explains crony capitalism within a framework of interaction between four stakeholder groups whose motives and behaviour have altered in the recent context. These are political executives, political parties, business entities and the bureaucracy. Using the coal mine block allocation controversy as a case study, we examine the modus operandi of crony capitalism as a capture of the policy process. We examine the limitations of the present corrective political and legal processes. 

The paper concludes with a set of recommendations for curbing crony capitalism via reform of : 

(a) the political funding system,
(b) the policy process,
© the audit institution and
(d) the business environment

Volume 6 Issue 1: The Art of the Matter — can a picture be a powerful pedagogical tool? At Right Angles believes it can, and if you read the write up on the facing page, I’m sure you will agree with us. Going to the heart of the art is key to mathematics and there’s more of the same inside this issue. We begin with Ramya Ramalingam, a sixteen-year-old school girl, unravelling the mysteries of Knot Theory for us. And Haneet Gandhi picks up where her series on Tessellations stopped, with a fascinating article on tiling and the pictures we can create with different combinations and permutations of polygons.

In the Classroom section, Khushboo Awasthi opens up the Square Root Spiral with a series of investigative questions; Ujjwal Rane proves Fagnano’s Theorem in several innovative ways. CoMaC describes an unusual way to bisect an angle and also manages to pull yet another 3−4−5 triangle which has long connected math with art pops up in Kepler’s triangle — read more about it in Marcus Bizony’s article. And in How To Prove It, Shailesh Shirali uses Ptolemy’s theorem to reveal all kinds of fascinating relationships in cyclic quadrilaterals.

Tech Space features the first part of a two-part series on constructive definitions; Michael de Villiers shows you how to do so with a GeoGebra activity centered on the golden rhombus. Truely cutting edge math pedagogy!

Our review this time is by Kamala Mukunda who shares her views on Liping Ma’s classic Knowing and Teaching Elementary Mathematics. This is a must-have for every school library and a must-read for mathematics teachers of all classes.

Finally, it’s Time — this Pull Out by Padmapriya Shirali will give you several new ideas to introduce this all important concept and help students quantify something which impinges on their consciousness long before they come to school.

The teacher is central to the teaching — learning process, whatever a school’s policies, whether private or public, whatever board it has chosen to adopt. The teacher is an essential component of the success of the process. This being the case it is essential to consider what ​‘creates’ a good, effective teacher and what sustains her and , through her, the framework of the school. This Issue is aimed at not only teachers and teacher-educators, but also policy makers and observers, in short, anyone with an interest in education. The focus articles deal with the policy and practice of teacher development and suggests alternatives while two more examine gender sensitisation and pre-service teacher preparation discussing in considerable detail the curriculum and its realities. This is followed by articles from practitioners across the field, from both private and public institutions. Teachers have recounted their experiences on the ground. There are critiques of the present system holding up practices against the light of theory, while some teachers have described their positive encounters, testifying to the dedication of colleagues working with enthusiasm and vigour in remote areas, without the help of sophisticated teaching aids.

Engage with 2 themes — ​‘Emerging Trends in Physics’ and ​‘Indian Science Facilities’ — by exploring methods & processes defining future research in areas as fascinating as the discovery of black holes, understanding dark matter, using radio-astronomy to probe the universe, enabling space travel, the search for exoplanets, & establishing human settlements on Mars. In ​‘Origins’ & ​‘Annals of History’, piece together current understanding of our shared history through the beginnings of space-time & planetary worlds, the notion of elements, or that of life. Discover simple classroom activities to reveal & challenge mental models of force in ​‘The Science Lab’, encourage the use of art to study ecology in ​‘The Science Teacher at Work’, or integrate social justice in the science curriculum in ​‘Research to Practice’. Also discover little-known aspects of ​‘The Origins of Composting’ & ​‘Ocean Microbes’ in our pull-out posters!