Course Type | Course Code | No. Of Credits |
---|---|---|

Foundation Elective | SES203107 | 4 |

**Semester and Year Offered: **Semester III (Monsoon Semester 2020)

**Course Coordinator and Team:** Dr. Gunjan Sharma (C) and Mr. Sanjay Raghav (adjunct faculty) (TBC)

**Email of course coordinator:** gunjan@aud.ac.in

**Pre-requisites: **Mathematics as a regular subject at senior secondary school level

**Aim:**

- To critically engage the participant teachers with pre-algebraic competencies to develop algebraic thinking for their students.
- To critically engage the participant teachers with patterns in arithmetic and geometry and help the teachers to analyze and generalize them in more formal algebraic expressions
- To help the participant teachers to critically engage with students’ conceptions and errors in algebra in middle school and develop a structured understanding of student’s error and misconceptions.
- To give useful guide to the participant teachers to plan strategies to help to develop algebraic thinking for their students.

**Course Outcomes:**

- Identify the essential components in algebraic thinking in middle and secondary school.
- Develop effective lessons to develop or assess the pre-algebraic competencies for their students.
- Critically engage with and make sense of children's misconceptions and errors in algebra in middle and secondary school and develop strategies to help the children in the same.
- Identify the essential components in mathematical proofs and proving in middle school geometric problems and theorems.
- Design effective lessons on proving and proofs in middle school geometry for their students.

**Brief description of modules/ Main modules:**

The course aims to equip participants to understand the concerns in the teaching and learning of algebra and the theory of proving in middle and secondary school geometry and design lessons to develop the necessary mathematical skills and dispositions for their students. This is achieved through two modules that are further organised in sub-modules.

**Module 1:** Pre-algebra and Algebraic Thinking (24 hours): This module is organised in three parts: Overview and analysis of pre-algebraic skills; Patterns, generalizations and stages in development of algebraic thinking focus on fundamentals of algebraic thinking; and Common conceptual errors in algebra in middle school.

**Module 2: **Levels in geometric thinking and its implications to proving and proofs in geometry (24 hours): This module has two components: Stages in geometric thinking; Steps in proving geometric statements: The syntax and the theory; and

**Assessment Details with weights:**

- Attendance and participation (20%)
- Activities Design (40%)
- Making a handbook /Teacher/student manual/TLM for the teachers or students to develop algebraic understanding and geometric understanding in middle and secondary school students/teachers (40%)

**Reading List:**

**Essential (may be revised):**

- Kaput, J. J., Carraher, D. W. and Blanton, M. L. (Eds.). (2008). Algebra in the early grades. LEA/ NCTM. New York, NY: Lawrence Erlbaum. (Chapter 1 and 2).
- Walle, V. D., John, A. (2010). Developing concepts of exponents, integers and real numbers, elementary and middle school mathematics: teaching developmentally. Boston: Pearson. (Chapters 16, 17, 23).
- Post (ed.), T. R., & Post, T. R. (1988). Teaching mathematics in grades K-8: Research based methods. Boston, MA: Allyn & Bacon. (Chapter 13).
- Subramaniam, K. and Banerjee, R. (2011). The arithmetic-algebra connection: A historical pedagogical perspective. In J. Cai and E. Knuth (Ed.), Early algebraization: A global dialogue from multiple perspectives (pp. 87- 105). Springer.
- Kaput, J. J., Carraher, D. W. and Blanton, M. L. (Eds.). (2008). Algebra in the early grades. LEA/ NCTM. New York, NY: Lawrence Erlbaum. (Chapter 3 and 8).
- Osborne, A. & Wilson, S. (1992). Moving to algebraic thought. In T. R. Post (Ed.), Teaching mathematics in grades K-8: Research based methods (pp. 421- 442). Boston, MA: Allyn & Bacon.
- Walle, V. D. and John, A. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson (Chapter 14).
- Patterns and algebra: student workbook G series. Retrieved from:
- https://drive.google.com/open?id=1rZcAL3LIpPAx9eXEm75dqalRozr25w9z
- Bush, S. B. (2011). Analyzing common algebra-related misconceptions and errors of middle school students. Electronic Theses and Dissertations. Paper 187, (pp 33- 61, 327-338).
- NCEERA. (2015). Teaching strategies for improving algebra knowledge in middle and high school students. U.S. Department of Education. Retrieved from: https://ies.ed.gov/ncee/wwc/docs/practiceguide/wwc_algebra_040715.pdf
- Crowley, Mary L. (1987). The van Hiele Model of the Development of Geometric Thought. In Mary Montgomery Lindquist (Ed.), Learning and teaching geometry, K-12: 1987 yearbook of the national council of teachers of mathematics, pp.1-16. Reston, Va.: National Council of Teachers of Mathematics. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.456.5025&rep=rep1&type=pdf
- Post, T. R. (Ed.). (1992). Geometry and visual thinking: Teaching mathematics in grade K-8. Boston: Allyn & Bacon.
- Walle, V. D., John, A. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson. (Chapter 14).
- Stylianou, D. A., Blanton, M. L. and Knuth, E. J. (Eds.). (2009). Teaching and learning proof across grades: a K-16 perspective. New York: Routledge. (Chapter 1, 3, 5).
- Chigonga, B. (2016). Learners' misconceptions in deductive geometry proofs and remedial strategies. Retrieved from: https://www.researchgate.net/publication/314285221_Learners'_misconceptions_i n_deductive_geometry_proofs_and_remedial_strategies
- Connolly, S. (2010). The impact of van Hiele-based geometry instruction on student understanding. Mathematical and computing sciences masters, Paper 97 (pp 8-24). https://fisherpub.sjfc.edu/cgi/viewcontent.cgi?article=1096&context=mathcs_etd_ masters
- Wilson, J. (2007). Proof and mathematical reasoning: A primer on mathematical proof. Retrieved from: http://www.math.lsa.umich.edu/~jchw/PrimerOnProof.pdf NCERT. (2008). Mathematics textbook class 9 appendix 1
- NCERT. (2008). Mathematics textbook class 10 appendix 1.
- Videos Four techniques of proving in mathematics retrieved from https://www.youtube.com/watch?v=V5tUc-J124s&list=WL&index=36&t=0s

**Workbook**

- Arcidiacono, M. J., & Maier, E. (1993). Picturing algebra. Unit 9, Math in the mind's eye. Math learning center.
- Maier, E. & Shaughnessy, M. (1999). Graphing algebraic relationships part 1 and 2.
- Unit 9, Math and the mind's eye activities. Math learning Centre
- Shirali, S. (2002). Adventures in problem solving. Delhi Universities Press. (Chapter 3, pp. 44- 48)
- Posamentier, A. S. & Salkind, C. T. (1996). Challenging problems in geometry. Dover Books. (pp. 1-28)

**Supplementary**

- Ameron, B. V. (2002). Learning and teaching of school algebra. In Reinvention of early algebra: Developmental research on the transition from arithmetic to algebra (pp 17-28). Dordrech, Netherlands. Retrieved from: https://dspace.library.uu.nl/bitstream/handle/1874/874/full.pdf
- Larson, C. L. (1983). Problem-solving through problems. In P. R. Halmos (Ed.), Problem books in mathematics (pp 120 - 144). Springer: New York.
- Bush, S. B. (2011). Analyzing common algebra-related misconceptions and errors of Middle school students. Electronic Theses and Dissertations, (pp 42-100). https://drive.google.com/drive/folders/1oRyHwmnyR93QyzU19rmw2qtxSsffK
- Banerjee, R. (2011). Is arithmetic useful for the teaching and learning of algebra? Contemporary Education Dialogue, 8 (2), 137-159.
- Banerjee, R. and Subramaniam, K. (2012). Evolution of a teaching approach for beginning algebra. Educational Studies in Mathematics, 81(2), 351-367.
- Kaput, J. J., Carraher, D. W. and Blanton, M. L. (Eds.). (2008). Algebra in the early grades. LEA/ NCTM. New York, NY: Lawrence Erlbaum.
- Gilmore, C., Gobel, S. M. and Inglis, M. (2018). An introduction to mathematical cognition. Routledge: Abingdon. (Chapters 8 and 9).
- Ameron, B. V. (2002). Re-inventing initial algebra: development research around the transition from arithmetic to algebra. (pp 17-28) (Chapter 2). Retrieved from: https://dspace.library.uu.nl/bitstream/handle/1874/874/full.pdf
- Gilmore, C., Gobel, S. M. and Inglis, M. (2018). An introduction to mathematical cognition. Routledge (Chapters 8 and 9).
- Houston, K. (2009). Generalization and specialization. (pp 248- 251). (Chapter 6).
- Booth, J., Mcginn, K., Barbieri, C. & Young, L. (2017). Misconceptions and learning algebra. 10.1007/978-3-319-45053-7_4.
- Judah, P. M. & Nzima, S. (2016). Eliciting learner errors and misconceptions in simplifying rational algebraic expressions to improve teaching and learning.
- International Journal of Educational Sciences, 12(1), 16- 28, DOI: 10.1080/09751122.2016.11890408
- Solow, D. (2014) How to read and do proofs: An introduction to mathematical thought processes. Wiley: London (Chapters 3, 9, 12, 16, 17).
- David, F. D. (2010). Basic proof techniques. Retrieved from, https://www.cse.wustl.edu/~cytron/547Pages/f14/IntroToProofs_Final.pdf
- Douek, N. (2009). Approaching proof in school: From guided conjecturing and proving to a story of proof construction. Lin, F-L, Hsieh, F-J, Hanna, G. and de Villers, M. (Eds.). Proofs and proving in mathematics education: The 19 ICMI study. Vol. 1, pp. 142-147.
- Guerrier, D. V & Arsac, G. (2009). Analysis of mathematical proofs: some questions and first answers. Lin, F-L, Hsieh, F-J, Hanna, G. and de Villers, M. (Eds.). Proofs and proving in mathematics education: The 19 ICMI study. Vol. 1, pp. 148-153.
- Gilmore, C., Gobel, S. M. and Inglis, M. (2018). An introduction to mathematical cognition. Routledge: Abingdon. (Chapter 10).
- Kunimune, S., Fujita, T., & Jones, K. (2009). “Why do we have to prove this?” Fostering students’ understanding of ‘proof’ in geometry in lower secondary school. Lin, F-L, Hsieh, F-J, Hanna, G. and de Villers, M. (Eds.). Proofs and proving in mathematics education: The 19 ICMI study. Vol. 1, pp. 256-261.