# Designing algebraic tasks for 7-year-old students – a pilot project inspired by Davydov’s learning activity concept

### Abstract

The issue of this article is to identify and discuss what conditions may be necessary to build into tasks to make it likely for students to be involved in an algebraic Learning Activity inspired by Davydov. Data from a pilot study was used in which a group of students (N=28) in grade 1 (7-year-olds) were invited to participate in discussions and laborations of how to decide whether two or more variables are equal or not, and making unequal “variables” equal by the help of measurement, abstract symbols and relational material. Three tasks were designed and from the analysis we will highlight five requirements for tasks that have the potential to enable students to engage in an algebraic learning activity.### References

Adolfsson Boman, M., Eriksson, I., Hverven, M., Jansson, A. & Tambour, T. (2013). Att introducera likhetstecknet i ett algebraiskt sammanhang (Introducing the equal sign within an algebraic context). Forskning om undervisning och lärande, 10, 29-49.

Anderson, T., & Shattuck, J. (2012). Design-based research: A decade of progress in education re¬search? Educational Researcher, 41(1), 16–25.

Adams, P. (2008). Considering 'best practice': The social construction of teacher activity and pupil learning as performance. Cambridge Journal of Education, 38(3), 375-392.

Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990. In N. Balacheff, M. Cooper, R. Sutherland & V. Warfield. Dordrecht: Kluwer Academic Publishers.

Brown, A. L. (1992). Design experiment: Theoretical and methodological challenges in creating complex interventions in classroom settings. The Journal of the Learning Sciences, 2(2), 141-178.

Chaiklin, S. (2002). A developmental teaching approach to schooling. In G. Wells & G. Claxton (Eds). Learning for life in the 21st century. Oxford, UK: Blackwell Publishers.

Cobb, P. (1987). An investigation of young children’s academic arithmetic contexts. Educational Studies in Mathematics, 18, 109-124.

Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32, 9-13.

Davydov, V. V. (2008). Problems of developmental instruction: A theoretical and experimental psychological study. New York: Nova Science Publishers, Inc.

Dougherty, B. (2004). Early algebra: Perspectives and assumptions. For the Learning of Mathematics, 24(3), 28-30.

Elliot, J. (1991). Action research for educational change. Milton Keynes: Open University Press.

Eriksson, I. (2015). Constitution of objects in DWR activity. In T. Hansson (Ed.) Contemporary approaches to activity theory. Interdisciplinary perspectives on human behavior. USA: IGI Global. (pp. 304-321).

Eriksson, I. & Lindberg, V. (2016). Enriching ‘learning activity’ with ‘epistemic practices’ - enhancing students’ epistemic agency and authority. NordSTEP 2016, 2: 32432 - http://dx.doi.org/10.3402/nstep.v2.32432.

Eriksson, I. & Lindberg, V. (2007). Matematikundervisningens innehåll (The content of mathematical education). Avrapportering av ett kollaborativt forskningsprojekt om att utveckla redskap och innehåll i arbetet med att realisera strävansmålen i matematik. Lärarhögskolan i Stockholm.

Engeström, Y. (2011). From design experiments to formative interventions. Theory & Psychology, 21(5), 598 - 628.

Falkner, L. & Falkner, K. P. (1999). Children’s understanding of equality: A foundation for Algebra. Teaching Children Mathematics, 6(4), 232-237.

Harn, B., Parisi, D. & Stoolmiller, M. (2013). Balancing fidelity with flexibility and fit: What do we really know about fidelity of implementation in schools? Exceptional Children, 79(2), 181-193.

Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. London: Routledge.

Illyenkov, E. V. (1977). The concept of the ideal. Philosophy in the USSR: Problems of dialectical materialism (pp. 71-99). Moscow: Progress.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317-326.

Kinard, J. T., & Kozulin, A. (2008). Rigorous mathematical thinking: Conceptual formation in the mathematics classroom. Cambridge: Cambridge University Press.

Leontʹev, A. N. (1978/1975). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice-Hall. (Original work published 1975)

Leung, A., & Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In A. Watson & M. Ohtani, (Eds), Task design in mathematics education: An ICMI study 22. Berlin: Springer.

Neuman, D. (1989). Räknefärdighetens rotter (The roots of counting) Stockholm: Utbildningsförlaget

Radford, L. (2012). On the development of early algebraic thinking. PNA, 6(4), 117-133.

Radford, L. (2013). Sensuous cognition. In D. Martinovic, V. Freiman, & Z. Karadag (Eds.), Visual mathematics and cyberlearning (pp. 141-162). New York: Springer.

Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26(2), 257-277.

Repkin, V.V. (2003). Developmental teaching and.learning activity Journal of Russian and East European Psychology, 41(5) 10-33.

Rubtsov, V. (1991). Learning in children: Organization and development of cooperative actions. NY, Nova Science Publishers, Ins.

Rubtsov, V. (2013). The concept of joint activity as a unit of activity theory. Presented ISCAR summer school.

Schmittau, J. (2003). Cultural-historical theory of mathematics education. In A. Kozulin, B. Gindis, V. S. Ageyev & S. Miller (Eds) Vygotsky’s educational theory in cultural context (pp. 225-245). Cambridge University Press.

Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, XIX(I), 19-43.

Schmittau, J. (2005). The development of algebraic thinking. A Vygotskian perspective. ZDM, 37(1), 16-22.

Schmittau, J. & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V.V. Davydov. The Mathematics Educator, 8(1), p. 60-87.

Sophian, C. (2007). The origins of mathematical knowledge in childhood. Lawrence Erlbaum Associates.

Stetsenko, A. (1999). Social interaction, cultural tools, and the zone of proximal development: In search of a synthesis. In M. Hedegaard, S. Chaiklin, S. Boedker, & U. J. Jensen (Eds.), Activity theory and social practice (pp. 235-253). Aarhus: Aarhus University Press.

Stetsenko, A. & Arievitch, I. (2002). Teaching, Learning, and Development: A Post-Vygotskian Perspective. In G. Wells & G. Claxton (Eds). Learning for Life in the 21st Century: Sociocultural Perspectives on the Future of Education (pp. 84-96). Oxford, UK: Blackwell Publishers Ltd.

Stigler, J. & Hiebert, J. (1999). The teaching gap. Best ideas from the worlds teachers for improving education in the classroom. New York: The Free Press.

van den Akker, J. (1998). The science curriculum: Between ideals and outcomes. In B. Fraser & K. Tobin (Eds.), International Handbook for Science Education. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Veneciano, L. & Dougherty, B. (2014). Addressing priorities for elementary school mathematics. For the Learning of Mathematics, 34, 1.

Wiliam, D. (2011). Embedded formative assessment. Bloomington, IN: Solution Tree Press.

Vygotskij, L. (1963/1934). Learning and mental development at school age. In B & J Simon (Eds) Educational psychology in the U.S.S.R. London: Rutledge & Kegan Paul. (Original work published 1934)

Watson, A. & Ohtani, M. (red.) (2015). Task design in mathematics education: An ICMI study 22. Berlin: Springer.

Waremö, M. (2016). Enculturation into inclusion, protecting what ‘is’, and changed acting: Exploring children’s break-time table tennis playing. Learning Culture and Social Interaction.

Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.

Zittoun, T. (2009). Dynamics of life-course transitions – A methodological reflection. In J. Valsiner, P. C. M. Molenaar, M. C. D. Lyra, & N. Chaudhary (Eds.), Dynamic process methodology in the social and developmental sciences (pp. 405–430). New York: Springer.

Zuckerman, G. (2004). Development of reflection through learning activity. European Journal of Psychology of Education, XIX(1), 9-18.

Zuckerman, G. (2011). Developmental education. A genetic modelling experiment. Journal of Russian and East European Psychology.

Anderson, T., & Shattuck, J. (2012). Design-based research: A decade of progress in education re¬search? Educational Researcher, 41(1), 16–25.

Adams, P. (2008). Considering 'best practice': The social construction of teacher activity and pupil learning as performance. Cambridge Journal of Education, 38(3), 375-392.

Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990. In N. Balacheff, M. Cooper, R. Sutherland & V. Warfield. Dordrecht: Kluwer Academic Publishers.

Brown, A. L. (1992). Design experiment: Theoretical and methodological challenges in creating complex interventions in classroom settings. The Journal of the Learning Sciences, 2(2), 141-178.

Chaiklin, S. (2002). A developmental teaching approach to schooling. In G. Wells & G. Claxton (Eds). Learning for life in the 21st century. Oxford, UK: Blackwell Publishers.

Cobb, P. (1987). An investigation of young children’s academic arithmetic contexts. Educational Studies in Mathematics, 18, 109-124.

Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32, 9-13.

Davydov, V. V. (2008). Problems of developmental instruction: A theoretical and experimental psychological study. New York: Nova Science Publishers, Inc.

Dougherty, B. (2004). Early algebra: Perspectives and assumptions. For the Learning of Mathematics, 24(3), 28-30.

Elliot, J. (1991). Action research for educational change. Milton Keynes: Open University Press.

Eriksson, I. (2015). Constitution of objects in DWR activity. In T. Hansson (Ed.) Contemporary approaches to activity theory. Interdisciplinary perspectives on human behavior. USA: IGI Global. (pp. 304-321).

Eriksson, I. & Lindberg, V. (2016). Enriching ‘learning activity’ with ‘epistemic practices’ - enhancing students’ epistemic agency and authority. NordSTEP 2016, 2: 32432 - http://dx.doi.org/10.3402/nstep.v2.32432.

Eriksson, I. & Lindberg, V. (2007). Matematikundervisningens innehåll (The content of mathematical education). Avrapportering av ett kollaborativt forskningsprojekt om att utveckla redskap och innehåll i arbetet med att realisera strävansmålen i matematik. Lärarhögskolan i Stockholm.

Engeström, Y. (2011). From design experiments to formative interventions. Theory & Psychology, 21(5), 598 - 628.

Falkner, L. & Falkner, K. P. (1999). Children’s understanding of equality: A foundation for Algebra. Teaching Children Mathematics, 6(4), 232-237.

Harn, B., Parisi, D. & Stoolmiller, M. (2013). Balancing fidelity with flexibility and fit: What do we really know about fidelity of implementation in schools? Exceptional Children, 79(2), 181-193.

Hattie, J. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. London: Routledge.

Illyenkov, E. V. (1977). The concept of the ideal. Philosophy in the USSR: Problems of dialectical materialism (pp. 71-99). Moscow: Progress.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317-326.

Kinard, J. T., & Kozulin, A. (2008). Rigorous mathematical thinking: Conceptual formation in the mathematics classroom. Cambridge: Cambridge University Press.

Leontʹev, A. N. (1978/1975). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice-Hall. (Original work published 1975)

Leung, A., & Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In A. Watson & M. Ohtani, (Eds), Task design in mathematics education: An ICMI study 22. Berlin: Springer.

Neuman, D. (1989). Räknefärdighetens rotter (The roots of counting) Stockholm: Utbildningsförlaget

Radford, L. (2012). On the development of early algebraic thinking. PNA, 6(4), 117-133.

Radford, L. (2013). Sensuous cognition. In D. Martinovic, V. Freiman, & Z. Karadag (Eds.), Visual mathematics and cyberlearning (pp. 141-162). New York: Springer.

Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26(2), 257-277.

Repkin, V.V. (2003). Developmental teaching and.learning activity Journal of Russian and East European Psychology, 41(5) 10-33.

Rubtsov, V. (1991). Learning in children: Organization and development of cooperative actions. NY, Nova Science Publishers, Ins.

Rubtsov, V. (2013). The concept of joint activity as a unit of activity theory. Presented ISCAR summer school.

Schmittau, J. (2003). Cultural-historical theory of mathematics education. In A. Kozulin, B. Gindis, V. S. Ageyev & S. Miller (Eds) Vygotsky’s educational theory in cultural context (pp. 225-245). Cambridge University Press.

Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, XIX(I), 19-43.

Schmittau, J. (2005). The development of algebraic thinking. A Vygotskian perspective. ZDM, 37(1), 16-22.

Schmittau, J. & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V.V. Davydov. The Mathematics Educator, 8(1), p. 60-87.

Sophian, C. (2007). The origins of mathematical knowledge in childhood. Lawrence Erlbaum Associates.

Stetsenko, A. (1999). Social interaction, cultural tools, and the zone of proximal development: In search of a synthesis. In M. Hedegaard, S. Chaiklin, S. Boedker, & U. J. Jensen (Eds.), Activity theory and social practice (pp. 235-253). Aarhus: Aarhus University Press.

Stetsenko, A. & Arievitch, I. (2002). Teaching, Learning, and Development: A Post-Vygotskian Perspective. In G. Wells & G. Claxton (Eds). Learning for Life in the 21st Century: Sociocultural Perspectives on the Future of Education (pp. 84-96). Oxford, UK: Blackwell Publishers Ltd.

Stigler, J. & Hiebert, J. (1999). The teaching gap. Best ideas from the worlds teachers for improving education in the classroom. New York: The Free Press.

van den Akker, J. (1998). The science curriculum: Between ideals and outcomes. In B. Fraser & K. Tobin (Eds.), International Handbook for Science Education. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Veneciano, L. & Dougherty, B. (2014). Addressing priorities for elementary school mathematics. For the Learning of Mathematics, 34, 1.

Wiliam, D. (2011). Embedded formative assessment. Bloomington, IN: Solution Tree Press.

Vygotskij, L. (1963/1934). Learning and mental development at school age. In B & J Simon (Eds) Educational psychology in the U.S.S.R. London: Rutledge & Kegan Paul. (Original work published 1934)

Watson, A. & Ohtani, M. (red.) (2015). Task design in mathematics education: An ICMI study 22. Berlin: Springer.

Waremö, M. (2016). Enculturation into inclusion, protecting what ‘is’, and changed acting: Exploring children’s break-time table tennis playing. Learning Culture and Social Interaction.

Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.

Zittoun, T. (2009). Dynamics of life-course transitions – A methodological reflection. In J. Valsiner, P. C. M. Molenaar, M. C. D. Lyra, & N. Chaudhary (Eds.), Dynamic process methodology in the social and developmental sciences (pp. 405–430). New York: Springer.

Zuckerman, G. (2004). Development of reflection through learning activity. European Journal of Psychology of Education, XIX(1), 9-18.

Zuckerman, G. (2011). Developmental education. A genetic modelling experiment. Journal of Russian and East European Psychology.