Volume 10 · Number 3 · Pages 321–330
Building Bridges to Algebra through a Constructionist Learning Environment

Eirini Geraniou & Manolis Mavrikis

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Abstract

Context: In the digital era, it is important to investigate the potential impact of digital technologies in education and how such tools can be successfully integrated into the mathematics classroom. Similarly to many others in the constructionism community, we have been inspired by the idea set out originally by Papert of providing students with appropriate “vehicles” for developing “Mathematical Ways of Thinking.” Problem: A crucial issue regarding the design of digital tools as vehicles is that of “transfer” or “bridging” i.e., what mathematical knowledge is transferred from students’ interactions with such tools to other activities such as when they are doing “paper-and-pencil” mathematics, undertaking traditional exam papers or in other formal and informal settings. Method: Through the lens of a framework for algebraic ways of thinking, this article analyses data gathered as part of the MiGen project from studies aiming at investigating ways to build bridges to formal algebra. Results: The analysis supports the need for and benefit of bridging activities that make the connections to algebra explicit and for frequent reflection and consolidation tasks. Implications: Task and digital environment designers should consider designing bridging activities that consolidate, support and sustain students’ mathematical ways of thinking beyond their digital experience. Constructivist content: Our more general aim is to support the implementation of digital technologies, especially constructionist learning environments, in the mathematics classroom.

Key words: Algebraic generalisation and language, transition, exploratory learning, microworlds, bridging tasks.

Citation

Geraniou E. & Mavrikis M. (2015) Building bridges to algebra through a constructionist learning environment. Constructivist Foundations 10(3): 321–330. http://constructivist.info/10/3/321

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References

Abboud-Blanchard M. (2014) Teacher and technologies: Shared constraints, commons. In: Clark-Wilson A., Robutti O. & Sinclair N. (eds.) The mathematics teacher in the digital era. Springer, Dordrecht: 297–317. ▸︎ Google︎ Scholar
Artigue M. (2002) Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. The International Journal of Computers for Mathematical Learning 7: 245–274. ▸︎ Google︎ Scholar
Beach K. D. (1999) Consequential transitions: A socio-cultural expedition beyond transfer in education. Review of Research in Education 28: 46–69. ▸︎ Google︎ Scholar
Beach K. D. (2003) Consequential transitions: A developmental view of knowledge propagation through social organisations. In: Tuomi-Gröhn T. & Engeström Y. (eds.) Between school and work: New perspectives on transfer and boundary-crossing. Elsevier Science, Amsterdam: 39–62. ▸︎ Google︎ Scholar
Bransford J. D. & Schwartz D. L. (1999) Rethinking transfer: A simple proposal with multiple implications. Review of Research in Education 24: 61–100. ▸︎ Google︎ Scholar
Broudy H. S. (1977) Types of knowledge and purposes of education. In: Anderson R. C., Spiro R. J. & Montague W. E. (eds.) Schooling and the acquisition of knowledge. Erlbaum, Hillsdale NJ: 1–17. ▸︎ Google︎ Scholar
Clark-Wilson A., Robutti O. & Sinclair N. (eds.) (2014) The mathematics teacher in the digital era. Springer, Dordrecht. ▸︎ Google︎ Scholar
Cobb P. & Steffe L. P. (1983) The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education 14(2): 83–94. ▸︎ Google︎ Scholar
Cobb P. (1987) An analysis of three models of early number development. Journal of Research in Mathematics Education 18: 163–179. ▸︎ Google︎ Scholar
Cobb P., Confrey J., diSessa A. A., Lehrer R. & Schauble L. (2003) Design experiments in educational research. Educational Researcher 32(1): 9–13. ▸︎ Google︎ Scholar
Confrey J. & Lachance A. (2000) Transformative teaching experiments through conjecture-driven research design. In: Kelly A. & Lesh E. R. (eds.) Handbook of research design in mathematics and science education. Lawrence Erlbaum, Mahwah NJ: 231–265. ▸︎ Google︎ Scholar
Cuoco A., Goldenberg E. & Mark J. (1996) Habits of mind: An organizing principle for mathematics curriculum. Journal of Mathematical Behavior 15(4): 375–402. ▸︎ Google︎ Scholar
Davis R. B. (1985) Algebraic thinking in the early grades. Journal of Mathematical Behavior 4: 195–208. ▸︎ Google︎ Scholar
Dörfler, W. (2006) Inscriptions as objects of mathematical activities. In: Maasz J. & Schloeglmann W. (eds.), New Mathematics Education Research and Practice. Sense Publishers, Rotterdam: 97–111. ▸︎ Google︎ Scholar
De Bock D., Deprez J., Van Dooren W., Roelens M. & Verschaffel L. (2011) Abstract or concrete examples in learning mathematics? A replication and elaboration of Kaminski, Sloutsky, and Heckler’s study. Journal for Research in Mathematics Education 42: 109–126. ▸︎ Google︎ Scholar
Design-Based Research Collective (2003) Design-based research: An emerging paradigm for educational inquiry. Educational Researcher 32(1): 5–8. ▸︎ Google︎ Scholar
diSessa A. A. & Wagner J. F. (2005) What coordination has to say about transfer. In: Mestre J. (ed.) Transfer of learning from a modern multi-disciplinary perspective. Information Age Publishing, Greenwich CT: 121–154. ▸︎ Google︎ Scholar
Doerr H. M. & Pratt D. (2008) The learning of mathematics and mathematical modeling. In: Heid M. K. & Blume G. W. (eds.) Research on technology and the teaching and learning of mathematics. Volume 1: Research syntheses. Information Age Publishing, Charlotte NC: 259–285. ▸︎ Google︎ Scholar
Duke R. & Graham A. (2007) Inside the letter. Mathematics Teaching Incorporating Micromath 200: 42–45. ▸︎ Google︎ Scholar
Education, Audiovisual and Culture Executive Eurydice Report (2011) Key data on learning and innovation through ICT at school in europe. http://eacea.ec.europa.eu/education/eurydice
Fuson K. (1990) Conceptual structures and multi-digit numbers. Cognition and Instruction 7: 343–404. ▸︎ Google︎ Scholar
Geraniou E., Mavrikis M., Hoyles C. & Noss R. (2011) Students’ justification strategies on the equivalence of quasi-algebraic expressions. In: Ubuz B. (ed.) Proceedings of the 35th International Conference for the Psychology of Mathematics Education, 10–15 July 2011, Ankara, Volume 2: 393–400. ▸︎ Google︎ Scholar
Godwin S. & Beswetherick R. (2003) An investigation into the balance of prescription, experiment and play when learning about the properties of quadratic functions with ICT. Research in Mathematics Education 5(1): 79–95. ▸︎ Google︎ Scholar
Gurtner J.-L. (1992) Between Logo and mathematics: A road of tunnels and bridges. In: Hoyles C. & Noss R. (eds.) Learning mathematics and Logo. MIT Press, Cambridge MA: 247–268. ▸︎ Google︎ Scholar
Hart K. (1981) Children’s understanding of mathematics: 11–16. John Murray, London. ▸︎ Google︎ Scholar
Haskell R. E. (2001) Transfer of learning: Cognition, instruction and reasoning. Academic Press, San Diego. ▸︎ Google︎ Scholar
Hershkowitz R., Schwarz B. B. & Dreyfus T. (2001) Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education 32(2): 195–222. ▸︎ Google︎ Scholar
Hewitt D. (2014) A symbolic dance: The interplay between movement, notation, and mathematics on a journey toward solving equations. Mathematical Thinking and Learning 16: 1–31. ▸︎ Google︎ Scholar
Hiebert J. & Lefeivre P. (1986) Conceptual and procedural knowledge in mathematics: An introductory analysis. In: Hiebert J. (ed.) Conceptual and procedural knowledge: The case for mathematics. Lawrence Erlbaum Associates, Hillsdale NJ: 1–27. ▸︎ Google︎ Scholar
Hiller J., Gurtner J. & Kieran C. (1988) Structuring and destructuring a solution: An example of problem-solving work with the computer. In: Proceedings of the Twelfth International Conference for the Psychology of Mathematics Education, 20–25 July 1988, Budapest: 402–409. ▸︎ Google︎ Scholar
Hoyles C. & Noss R. (1992) A pedagogy for mathematical microworlds. Educational Studies in Mathematics 23(1): 31–57. ▸︎ Google︎ Scholar
James W. (1950) Principles of psychology. Volume 1. Dover, New York. Originally published in 1890. ▸︎ Google︎ Scholar
Jones I. & Pratt D. (2012) A substituting meaning for the equals sign in arithmetic notating tasks. Journal for Research in Mathematics Education 43: 2–33. ▸︎ Google︎ Scholar
Kaminski J. A., Sloutsky V. M. & Heckler A. F. (2008) The advantage of abstract examples in learning math. Science 320: 454–455. ▸︎ Google︎ Scholar
Kaput J. J. (1992) Technology and mathematics education. In: Grouws D. A. (ed.) Handbook of research on mathematics teaching and learning. Macmillan, New York: 515–556. ▸︎ Google︎ Scholar
Küchemann D. (2010) Using patterns generically to see structure. Pedagogies: An International Journal 5(3): 233–250. ▸︎ Google︎ Scholar
Kirshner D. (2001) The structural algebra option revisited. In: Sutherland R., Rojano T., Bell A. & Lins R. (eds.) Perspectives on School Algebra. Kluwer Academic Publishers, London: 83–99. ▸︎ Google︎ Scholar
Mason J. & Davis J. (1989) The inner teacher: The didactic tension, and shifts of attention. In: Vergnaud G., Rogalski J. & Artigue M. (eds.) Proceedings of the 13th Annual Conference of the International Group for the Psychology of Mathematics Education, Paris, France, 9–13 July 1989, Volume 2: 274–281. ▸︎ Google︎ Scholar
Mason J. (2002) Researching your own practice: The discipline of noticing. Routledge, London. ▸︎ Google︎ Scholar
Mason J. (2005) Developing thinking in algebra. Sage, London. ▸︎ Google︎ Scholar
Mason J. (2005) Frameworks for learning, teaching and research: Theory and practice. In: Lloyd G. M., Wilson M., Wilkins J. L. M. & Behm S. L. (eds.) Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Virginia Tech, Roanoke VA. http://www.pmena.org/proceedings/PMENA%2027%202005%20Proceedings.pdf
Mason J., Graham A., Pimm D. and Gowar N. (1985) Routes to, roots of algebra. The Open University Press, Milton Keynes. ▸︎ Google︎ Scholar
Mavrikis M., Noss R., Hoyles C. & Geraniou E. (2013) Sowing the seeds of algebraic generalisation: Designing epistemic affordances for an intelligent microworld. In: Noss R. & diSessa A. (eds.) Special issue on knowledge transformation, design and technology. Journal of Computer Assisted Learning 29(1): 68–84. ▸︎ Google︎ Scholar
McLuhan M. (1964) Understanding media: The extensions of man. McGraw Hill, New York. ▸︎ Google︎ Scholar
McNeil N. M. & Alibali M. W. (2005) Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development 76: 883–899. ▸︎ Google︎ Scholar
McNeil N. M. (2008) Limitations to teaching children 2 + 2 = 4: Typical arithmetic problems can hinder learning of mathematical equivalence. Child Development 79: 1524–1537. ▸︎ Google︎ Scholar
Nemirovsky R. (2002) On guessing the essential thing. In: Gravemeijer K., Lehrer R., van B. Oers & Verschaffel L. (edss) Symbolizing, modeling and tool use in mathematics. Kluwer Academic, Dordrecht: 233–256. ▸︎ Google︎ Scholar
Noss R. & Hoyles C. (1996) Windows on mathematical meanings: Learning cultures and computers. Kluwer, Dordrecht. ▸︎ Google︎ Scholar
Noss R., Poulovassilis A., Geraniou E., Gutierrez-Santos S., Hoyles C., Kahn K., Magoulas G. D. & Mavrikis M. (2012) The design of a system to support exploratory learning of algebraic generalisation. Computers and Education 59(1): 63–81. ▸︎ Google︎ Scholar
Papert S. (1972) Teaching children to be mathematicians vs. teaching about mathematics. International Journal of Mathematics Education and Science Technology 3: 249– 262. ▸︎ Google︎ Scholar
Papert S. (1980) Mindstorms: Computers, children and powerful ideas. Basic Books, New York. ▸︎ Google︎ Scholar
Pólya G. (1945) How to solve it: A new aspect of mathematical method. Princeton University Press, Princeton NJ. ▸︎ Google︎ Scholar
Perkins D. N. & Salomon G. (1988) Teaching for transfer. Educational leadership 46(1): 22–32. ▸︎ Google︎ Scholar
Pratt D. & Noss R. (2002) The microevolution of mathematical knowledge: The case of randomness. The Journal of the Learning Sciences 11(4): 453–488. ▸︎ Google︎ Scholar
Pratt D. & Noss R. (2010) Designing for mathematical abstraction. International Journal of Computers for Mathematical Learning 15(2): 81–97. ▸︎ Google︎ Scholar
Radford L. (2014) The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal 26: 257–277. ▸︎ Google︎ Scholar
Ruthven K. (2007) Teachers, technologies and the structures of schooling. In: Pitta-Pantazi D. & Philippou G. (eds.) Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (CERME 5) University of Cyprus, Larnaca: 52–67. ▸︎ Google︎ Scholar
Salomon G. & Perkins D. N. (1987) Transfer of cognitive skills from programming: When and how? Journal of Educational Computing Research 3: 149–169. ▸︎ Google︎ Scholar
Simon M. A. (2013) Issues in theorizing mathematics learning and teaching: A contrast between learning through activity and DNR programs. Journal of Mathematical Behavior 32: 281–294. ▸︎ Google︎ Scholar
Simon M. A. (2014) Models of students’ mathematics and their relationship to mathematics pedagogy. Constructivist Foundations 9(3): 348–350. http://www.univie.ac.at/constructivism/journal/9/3/348.simon
Stacey K. & Macgregor M. (2002) Curriculum reform and approaches to algebra. In: Sutherland R., Rojano T., Bell A. & Lins R. (eds.) Perspectives on school algebra. Springer, New York: 141–153. ▸︎ Google︎ Scholar
Steffe L. P. (2003) Fractional commensurate, composition, and adding schemes: Learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior 22(3): 237–295. ▸︎ Google︎ Scholar
Steffe L. P. (2004) On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning 6(2): 129–162. ▸︎ Google︎ Scholar
Thompson P. W. (1982) Were lions to speak, we wouldn’t understand. Journal of Mathematical Behavior 3(2): 147–165. ▸︎ Google︎ Scholar

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