Following are few selected presentations given by the members of mathematics education group during various workshops, conferences and teacher education programs.

#### Policy Initiatives

Constructivism, NCF and Mathematics teaching

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#### Teaching and Teacher development

Knowing content through reading our textbook-Fractions

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We read our textbook at different occasions, often with multiple objectives and with different approaches of analyzing textbooks. various styles like, going through the textbook, solving the examples or for sequencing the teaching instruction. In this session teachers read the book to learn more about nature of mathematics in the textbook. We chose one particular topic for our discussion that was Fractions. Attached is the worksheet solved and discussed by the teachers. So if you are keen on knowing our textbooks better, have a look at the following worksheet, try to solve and answer each question. The textbook stanzas taken in the worksheet are from NCERT grade 4th to 7th Textbook. This session was conducted by Shweta Naik.

Mathematical Knowledge required for teaching

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Shikha Takker

Knowing content through reading our textbooks – case of Algebra

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Use of context in teaching of Mathematics: How and Why?

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Context have been used in teaching mathematics for various purposes ranging from introduction to a concept to giving application problems after students are well versed with the procedure to solve. This talk tried to present an alternative way of using a context to develop mathematics using students own reasons. These reasons are developed while correlating the actions students performed within a context to mathematical language.

Teachers listed various types of common misconceptions related to fractions and tried to discuss their sources. The common misconceptions included representing fractions in terms of shaded/unshaded and various other ways, adding numerators and denominators, 1/7 as bigger than 1/5, etc. Many misconceptions were due to the fact that students tried to deal with just numbers without ascribing any meaning to it. Thus need for using context was established to make mathematical concepts meaningful to students.

During this talk teachers discussed how an activity like playing a game of “Fraction wheel activity” ( Class V NCERT textbook, page No. 60) can become useful context for learning various concepts related to fraction using students own observations. The concepts that can be dealt using fraction wheel activity are: -unit fraction and its comparison -composite fraction and its equivalence with unit fraction -comparison of various fraction with 1/2 -addition and subtraction of fractions using different fractions – which fractions are nearer to whole and thus greater.

Similarly Fraction chart can also be used in the similar manner as fraction wheel. It is advisable to use both fraction chart and fraction wheel as the sense of fractions that students get while working in both context is different and useful too.In fraction wheel students get clearer idea about completing a whole while in fraction chart making subdivisions of smaller unit is much easier. Using fraction chart students can also generalize that as the number of pieces (subunit) increase the size of the piece gets smaller. Teachers can compare these kind of reasons to the rules taught in class like “As the denominator increase the fraction becomes smaller”. Such rules are counter-intuitive to students because they don’t know why fraction is smaller when the number is bigger. This justification can however be developed using context of fraction wheel and fraction chart.Some examples of student reasoning using contexts are given in the slides (pdf).

Another context prevalently used in classroom is that of “Marks”. Students mostly say 4/5 is 4 marks out of 5. However one needs to look critically at use of ” out of” in classrooms for fractions. When marks of different subjects are added- the numerators are added with other numerators and denominator with denominator ( taking it as ratio) where as in fractions this procedure is not valid mathematically. Thus there is need to critically look at context as to how it aids learning before using it in the class.

This talk used videos of teaching fractions as an example using context of measuring and sharing to develop mathematics. It can also be considered as an exemplar of ” mathematization of child’s thought” a phrase often referred in NCF-2005. For other examples of use of context see teachers presentation in the same workshop for “Integers”.

Teaching through open ended approach

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#### Presentations used in Workshops

#### 1. Enhancing Pedagogical Skills o Impact Classroom Transactions for Kendriya Vidyalaya Teachers (Nov, 2016)

- Understanding_students’_Error_HR
- Learning_decimal_numbers_KS
- Numbers_&_place_value_KS
- Research Reading Presentations by Teachers

#### 2. Design Training Programme for Lecturers of DIET, Gujarat (Feb, 2017)

- Nature_of_Mathematics_KS
- PCK_for_Teaching_Mathematics_SN
- Learning_through_problems_DP

#### Assessment

#### Assessing our questions

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In this session we discussed the various attributes of a ‘good’ question. After a brief presentation and discussion, the participants did analysis of various questions based on a framework discussed in the presentation.

Assessment questions for analysis

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In this session we discussed the various attributes of a ‘good’ question. After a brief presentation and discussion, the participants did analysis of various questions based on a framework discussed in the presentation.

#### Mathematical topics

Thinking algebraically by Aaloka Kanhere

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In this session the role of Algebra in school mathematics is discussed. Speaker provides the background of the field through research perspective highlighting students errors and their sources. Elaborate discussion was done on concept of “equality” and “algebraic structures”.

#### Content knowledge for educational leaders

On Constructivism and Learning by Prof K Subramaniam

Subject matter knowledge for mathematics teachers by Shweta Naik

Role of Education Officers: Assessment Issues by Dr Ritesh Khunyakari,/a>

Tools for Analyzing Classroom by Shikha Takker

Science Education by Dr Abhijeet Bardapurkar

Supporting Teacher Professional Development by Ruchi Kumar

#### Talks given by visiting faculty

Critical Mathematics Education by Prof. Ole Skovsmose

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Abstract given by Prof. Ole Skovsmose

Many issues are relevant in order to formulate preoccupations of critical mathematics education. I limit myself to mention the following three. (1) A certain ‘prototypical mathematics classroom’ seems to have dominated mathematics education research. I find it important that critical mathematics education challenges the dominance of the discourse created around this prototype and addresses the variety of sites for teaching and learning mathematics. (2) Mathematics operates as part of very many different work practices and technical settings. Often mathematics is integrated in work practices in a form that is not transparent to people involved in the professional practice. It is, however, important to address such variety of practices to provide a critical investigation of how mathematics might provide ‘horrors’ as well as ‘wonders’; and it is important to investigate possible relationships between out of school mathematical practices and how mathematics might be contextualised in a school setting. (3) Educational possibilities can be explored through notions like ‘empowerment’, ‘social justice’, and ‘mathemacy’. Such notions are, however, explosive, as they are not confined within strict definitions. They are important, anyhow, in order to formulate pedagogical imaginations as part of a critical approach. Such a approach, then, comes to reflect a deep uncertainty.)

Conducting In-service training of teachers: some experiences by S. C. Agarkar

Mathematics Laboratory by Amol Parab

Approaches to learning fractions by Shweta Naik

In this session different meanings of fractions depending on their context were discussed. It was followed by the discussion on a curricular sequence for teaching fractions which takes students’ outside school knowledge and contexts into account.

Learning from Artefacts of Teaching

Fractions, Ratios and Proportional Reasoning