Innovations in Mathematics Education: Current Trends and Issues
A national conference for mathematics teachers and teacher educators
Organized by
Department of Elementary Education, Lady Shri Ram College for Women
Supported by
National Board of Higher Mathematics, Department of Atomic Energy, Government of India
3rd& 4th March, 2017
A national conference for mathematics teachers and teacher educators
Organized by
Department of Elementary Education, Lady Shri Ram College for Women
Supported by
National Board of Higher Mathematics, Department of Atomic Energy, Government of India
3rd& 4th March, 2017
A REPORT
World over, Mathematics education has witnessed many paradigm shifts in the last few decades. Recent trends show that the emphasis in the mathematics classroom has shifted from developing manipulative skills and memorization of procedures to learning through visualization and exploration. Rather than being viewed as static knowledge bounded by skills which are detached from everyday events, learning mathematics is viewed as problem solving, problem posing, reasoning, communicating and decision making – skills that empower the student to think mathematically and relate mathematics to life experiences.
While the world is witnessing major changes in mathematics education, a large majority of schools in India continues to focus on rigid methods and on drill and practice of mathematics. The emphasis on mastery of procedural knowledge leaves very little opportunity for visualization, exploration or discovery. This emphasizes the need for a multidimensional approach for making mathematics accessible to all students – an approach which integrates multiple resources in the classroom in terms of curricular materials such as books, hands on manipulatives, technology tools and multimedia. The traditional teaching methods have to be replaced by more innovative methods and pedagogical practices which engage the learner in the processes of mathematics learning.
In order to meet the changing demands of mathematics education, we need to address the multiple issues and challenges pertaining to changes in the curriculum, integration of appropriate technology and teacher preparation on a large scale. This conference aimed to bring together school and college mathematics teachers, mathematics educators, practicing mathematicians and individuals interested in mathematics education to deliberate on various aspects of mathematics education in our country. It attempted to provide opportunities to participants to learn about the contributions and innovative work done by individuals as well as groups in various areas of mathematics education. It is hoped that the deliberations will enable participants to enhance their own work and develop new ideas for solving the problems of mathematics education.
The following themes were identified to steer the discussions of the conference
Theme 1: Innovative practices in mathematics classrooms
Theme 2: The role of technology in mathematics teaching and learning
Theme 3: Teacher knowledge and professional development
Theme 4: Innovative mathematical practices in non-formal spaces
The Department of Elementary Education organized the conference. Dr. Jonaki B Ghosh was the convener and Ms. Deepika Papneja was the co-convener. The conference was held in the Ramkrishna Dalmia Auditorium. In his inaugural address, Prof. Ramanujan, working at the Institute of Mathematical Sciences, Chennai, presented interesting anecdotes from his work with adult learners. He talked about how adult neo-literate learners with little formal schooling could engage with mathematical ideas which emanated from their rich life experiences. He cited interesting examples of vegetable sellers who dealt with fairly complex problems in their day to day life. With appropriate scaffolding they could create efficient strategies to optimize their earnings. He also highlighted the example of kolams to show that it is indeed possible to introduce mathematically sophisticated notions to learners with little mastery of school mathematics. According to him, this has implications for school children learning mathematics as well as on our expectations from them. Is it possible to reorient school mathematics at all whereby we de-emphasize procedures but enable children to learn mathematics through life experiences? His talk was indeed insightful and helped to set the tone of the conference.
Prof. K Subramaniam, Director, Homi Bhabha Centre for Science Education, in his plenary talk, highlighted that there are significant or systematic gaps in the way mathematics teacher preparation programmes have been designed. He cited examples from classroom mathematics teaching, which were drawn from research studies done at the Homi Bhabha Centre. He elaborated on the PCK framework of which SCK and KCS are important components. He analyzed the possible reasons for these gaps and on the systemic changes that are needed to fill these gaps. In his talk he cited several examples of instances from the classroom where learners’ alternative conceptions came to the fore. It is through the process of reflection and engagement of students’ work that leads to the development of PCK
Prof. Amitabha Mukherjee, Head, Department of Physics, Delhi University, in his talk highlighted how mathematics can be made accessible to all levels of learners. He began with the question of ‘Why we should learn mathematics?’ and went on to analyze the nature of school mathematics vis-a-vis the nature of mathematics required in daily life. He ended his talk by suggesting that games and puzzles can play a critical role in generating and sustaining the learner’s interest in mathematics.
In the afternoon, three panelists presented their views on the theme ‘Innovative Practices in Mathematics Classrooms’. Dr. Anup Rajput, NCERT, felt that teachers should be made aware of the usefulness of activity-based teaching in mathematics. Instead of stating facts to children, they should be provided with opportunities that help them in creating their own rules and formulae. According to him, strong pre-service and in-service teacher orientation programs can help in achieving the goals of mathematics education as envisaged in NCF 2005.
Ms. Anita Sharma, Principal, S D Public school shared some of the innovative practices used by her to address some of the challenges related to children’s mathematical learning. In particular she cited the example of Frayer’s model, role of open-ended questions, computer games, and craft work in dealing with some of these challenges. She highlighted some examples of student’s written work highlighting various aspects of their thinking.
Arousing and maintaining students’ interest in Mathematics is a major challenge for the teacher. According to Mr. Amit Bajaj of CRPF School, Rohini, a good Mathematics Teacher can involve his students in a number of co-curricular activities such as Mathematics Club, Mathematics Fair, and Mathematics Laboratory etc. According to him a Mathematics Club can play an important role in sustaining interest in learning Mathematics among students. He cited interesting problems which stimulate their mathematical thinking and also enables learners in engaging with mathematical processes of generalization, estimation, approximation and visualization.
In the last session of the first day of the conference, there were three invited presentations by teacher educators. Dr. Asha Mathur’s presentation focused on visualization. Visualization has the potential of concretizing abstraction, to be able to relate to something through thinking. It could be done through recalling an image, or relationships to other things, or simply through properties possessed by that thing. Visualization helps in pattern recognition, which is another important activity in mathematics. According to Dr. Mathur, visualization can provide a bridge between content learnt and knowledge acquired by facilitating the thinking process that connects the two.
Dr. Haneet Gandhi, in her presentation, shared some interesting results of an ongoing project on probability in which children’s conceptions related to randomness were understood through eight partial or perfectly symmetric objects. The results were drawn from the arguments of two groups of students who worked in small group situations.
Dr. Jyoti Sharma highlighted the challenges faced by high ability learners in regular classrooms. She discussed the dilemma of math teachers in dealing with such students within the purview of standard curriculum. Her presentation suggested the pathways for teachers on how to interpret learning behavior of students and how to build up true math learning communities using differentiation.
The first plenary talk on the second day of the conference was by Prof. Inder Kumar Rana which focused on Gorge Polya’s suggestions, that a mathematics teacher should spend her time in engaging the learner in meaningful problem solving, and in stimulating their thinking by asking appropriate questions. Using a Panchantra tale and several mathematical problems, he highlighted the difference between lower order and higher order thinking skills. He attempted to provide a recipe for creating and enriching the teaching and learning of mathematics. He demonstrated the use of technological tools in creating such examples.
Prof. Parvin Sinclair talked about the role of a mathematics teacher, and her own understanding of mathematics as a discipline. She raised some important issues in her talk. How much of this fear is due to the teacher’s understanding of the nature of mathematics? How much is due to her understanding, and experience, of the nature of learning? Does the situation change when we move from primary schooling to secondary schooling, and further to undergraduate teaching? If so, in what ways does it change? Does the Open and Distance Learning system offer any solutions?
Two parallel hands-on workshop sessions on GeoGebra were conducted by Sangeeta Gulati, Sanskriti School and Dr. Jonaki B Ghosh respectively. Participants were divided into two groups and each participant attended one of the sessions. In both sessions participants were made to experience the power of GeoGebra by creating and exploring applets on Geometry, Algebra and Calculus. Dr. Ghosh showed how geometrical inquiry may be encouraged by designing appropriate tasks in GeoGebra, and demonstrated how Variation Theory supports the learning of concepts in a Dynamic Geometry Environment. A hands-on booklet on GeoGebra activities, prepared by the resource persons, was given to each participant.
The conference ended with a panel discussion on Bridging School and College Mathematics. Professor Geetha Venkataraman, Ambedkar University, pointed out that mathematics teachers at the college level often lament at the lack of foundational knowledge of students which are required to pursue mathematics, and on their inability to understand rigor, or express themselves mathematically. So, what can be done to ensure that students in college are better equipped? How can we align the school mathematics curriculum with the requirements of mathematics courses at the undergraduate level? Professor Amber Habib, Shiv Nadar University, emphasized the role of project based learning in enabling students both at school and college level to engage with mathematical ideas. These projects may be related to exploring application of mathematics or modeling real world problems. Professor Amitabha Tripathi, IIT Delhi, shared his experience of his interaction with students in the Mathematical Olympiad program. He felt that gaps exist at all levels of school and college education.
The conference ended with the Valedictory session, where Dr. Suman Sharma, Principal, LSR, gave away the certificates of participation. Dr. Jonaki B Ghosh proposed the vote of thanks and acknowledged the contribution of the faculty, students and staff members towards the success of the conference. She particularly lauded the contribution of student volunteers of the B.El.Ed department, and also the first-year students, who had put up an exhibition of various mathematics projects. These were greatly appreciated by the speakers as well as the participants.
While the world is witnessing major changes in mathematics education, a large majority of schools in India continues to focus on rigid methods and on drill and practice of mathematics. The emphasis on mastery of procedural knowledge leaves very little opportunity for visualization, exploration or discovery. This emphasizes the need for a multidimensional approach for making mathematics accessible to all students – an approach which integrates multiple resources in the classroom in terms of curricular materials such as books, hands on manipulatives, technology tools and multimedia. The traditional teaching methods have to be replaced by more innovative methods and pedagogical practices which engage the learner in the processes of mathematics learning.
In order to meet the changing demands of mathematics education, we need to address the multiple issues and challenges pertaining to changes in the curriculum, integration of appropriate technology and teacher preparation on a large scale. This conference aimed to bring together school and college mathematics teachers, mathematics educators, practicing mathematicians and individuals interested in mathematics education to deliberate on various aspects of mathematics education in our country. It attempted to provide opportunities to participants to learn about the contributions and innovative work done by individuals as well as groups in various areas of mathematics education. It is hoped that the deliberations will enable participants to enhance their own work and develop new ideas for solving the problems of mathematics education.
The following themes were identified to steer the discussions of the conference
Theme 1: Innovative practices in mathematics classrooms
Theme 2: The role of technology in mathematics teaching and learning
Theme 3: Teacher knowledge and professional development
Theme 4: Innovative mathematical practices in non-formal spaces
The Department of Elementary Education organized the conference. Dr. Jonaki B Ghosh was the convener and Ms. Deepika Papneja was the co-convener. The conference was held in the Ramkrishna Dalmia Auditorium. In his inaugural address, Prof. Ramanujan, working at the Institute of Mathematical Sciences, Chennai, presented interesting anecdotes from his work with adult learners. He talked about how adult neo-literate learners with little formal schooling could engage with mathematical ideas which emanated from their rich life experiences. He cited interesting examples of vegetable sellers who dealt with fairly complex problems in their day to day life. With appropriate scaffolding they could create efficient strategies to optimize their earnings. He also highlighted the example of kolams to show that it is indeed possible to introduce mathematically sophisticated notions to learners with little mastery of school mathematics. According to him, this has implications for school children learning mathematics as well as on our expectations from them. Is it possible to reorient school mathematics at all whereby we de-emphasize procedures but enable children to learn mathematics through life experiences? His talk was indeed insightful and helped to set the tone of the conference.
Prof. K Subramaniam, Director, Homi Bhabha Centre for Science Education, in his plenary talk, highlighted that there are significant or systematic gaps in the way mathematics teacher preparation programmes have been designed. He cited examples from classroom mathematics teaching, which were drawn from research studies done at the Homi Bhabha Centre. He elaborated on the PCK framework of which SCK and KCS are important components. He analyzed the possible reasons for these gaps and on the systemic changes that are needed to fill these gaps. In his talk he cited several examples of instances from the classroom where learners’ alternative conceptions came to the fore. It is through the process of reflection and engagement of students’ work that leads to the development of PCK
Prof. Amitabha Mukherjee, Head, Department of Physics, Delhi University, in his talk highlighted how mathematics can be made accessible to all levels of learners. He began with the question of ‘Why we should learn mathematics?’ and went on to analyze the nature of school mathematics vis-a-vis the nature of mathematics required in daily life. He ended his talk by suggesting that games and puzzles can play a critical role in generating and sustaining the learner’s interest in mathematics.
In the afternoon, three panelists presented their views on the theme ‘Innovative Practices in Mathematics Classrooms’. Dr. Anup Rajput, NCERT, felt that teachers should be made aware of the usefulness of activity-based teaching in mathematics. Instead of stating facts to children, they should be provided with opportunities that help them in creating their own rules and formulae. According to him, strong pre-service and in-service teacher orientation programs can help in achieving the goals of mathematics education as envisaged in NCF 2005.
Ms. Anita Sharma, Principal, S D Public school shared some of the innovative practices used by her to address some of the challenges related to children’s mathematical learning. In particular she cited the example of Frayer’s model, role of open-ended questions, computer games, and craft work in dealing with some of these challenges. She highlighted some examples of student’s written work highlighting various aspects of their thinking.
Arousing and maintaining students’ interest in Mathematics is a major challenge for the teacher. According to Mr. Amit Bajaj of CRPF School, Rohini, a good Mathematics Teacher can involve his students in a number of co-curricular activities such as Mathematics Club, Mathematics Fair, and Mathematics Laboratory etc. According to him a Mathematics Club can play an important role in sustaining interest in learning Mathematics among students. He cited interesting problems which stimulate their mathematical thinking and also enables learners in engaging with mathematical processes of generalization, estimation, approximation and visualization.
In the last session of the first day of the conference, there were three invited presentations by teacher educators. Dr. Asha Mathur’s presentation focused on visualization. Visualization has the potential of concretizing abstraction, to be able to relate to something through thinking. It could be done through recalling an image, or relationships to other things, or simply through properties possessed by that thing. Visualization helps in pattern recognition, which is another important activity in mathematics. According to Dr. Mathur, visualization can provide a bridge between content learnt and knowledge acquired by facilitating the thinking process that connects the two.
Dr. Haneet Gandhi, in her presentation, shared some interesting results of an ongoing project on probability in which children’s conceptions related to randomness were understood through eight partial or perfectly symmetric objects. The results were drawn from the arguments of two groups of students who worked in small group situations.
Dr. Jyoti Sharma highlighted the challenges faced by high ability learners in regular classrooms. She discussed the dilemma of math teachers in dealing with such students within the purview of standard curriculum. Her presentation suggested the pathways for teachers on how to interpret learning behavior of students and how to build up true math learning communities using differentiation.
The first plenary talk on the second day of the conference was by Prof. Inder Kumar Rana which focused on Gorge Polya’s suggestions, that a mathematics teacher should spend her time in engaging the learner in meaningful problem solving, and in stimulating their thinking by asking appropriate questions. Using a Panchantra tale and several mathematical problems, he highlighted the difference between lower order and higher order thinking skills. He attempted to provide a recipe for creating and enriching the teaching and learning of mathematics. He demonstrated the use of technological tools in creating such examples.
Prof. Parvin Sinclair talked about the role of a mathematics teacher, and her own understanding of mathematics as a discipline. She raised some important issues in her talk. How much of this fear is due to the teacher’s understanding of the nature of mathematics? How much is due to her understanding, and experience, of the nature of learning? Does the situation change when we move from primary schooling to secondary schooling, and further to undergraduate teaching? If so, in what ways does it change? Does the Open and Distance Learning system offer any solutions?
Two parallel hands-on workshop sessions on GeoGebra were conducted by Sangeeta Gulati, Sanskriti School and Dr. Jonaki B Ghosh respectively. Participants were divided into two groups and each participant attended one of the sessions. In both sessions participants were made to experience the power of GeoGebra by creating and exploring applets on Geometry, Algebra and Calculus. Dr. Ghosh showed how geometrical inquiry may be encouraged by designing appropriate tasks in GeoGebra, and demonstrated how Variation Theory supports the learning of concepts in a Dynamic Geometry Environment. A hands-on booklet on GeoGebra activities, prepared by the resource persons, was given to each participant.
The conference ended with a panel discussion on Bridging School and College Mathematics. Professor Geetha Venkataraman, Ambedkar University, pointed out that mathematics teachers at the college level often lament at the lack of foundational knowledge of students which are required to pursue mathematics, and on their inability to understand rigor, or express themselves mathematically. So, what can be done to ensure that students in college are better equipped? How can we align the school mathematics curriculum with the requirements of mathematics courses at the undergraduate level? Professor Amber Habib, Shiv Nadar University, emphasized the role of project based learning in enabling students both at school and college level to engage with mathematical ideas. These projects may be related to exploring application of mathematics or modeling real world problems. Professor Amitabha Tripathi, IIT Delhi, shared his experience of his interaction with students in the Mathematical Olympiad program. He felt that gaps exist at all levels of school and college education.
The conference ended with the Valedictory session, where Dr. Suman Sharma, Principal, LSR, gave away the certificates of participation. Dr. Jonaki B Ghosh proposed the vote of thanks and acknowledged the contribution of the faculty, students and staff members towards the success of the conference. She particularly lauded the contribution of student volunteers of the B.El.Ed department, and also the first-year students, who had put up an exhibition of various mathematics projects. These were greatly appreciated by the speakers as well as the participants.