# Designing Innovative Learning Activities to Face Difficulties in Algebra of Dyscalculic Students: Exploiting the Functionalities of *AlNuSet*

## Abstract

In this chapter I discuss students’ difficulties in algebra, considering in particular those students affected by *developmental dyscalculia* (DD) (Butterworth in *Handbook of Mathematical Cognition.* Hove, UK: Psychology Press, 2005; Dehaene in *The number sense: How the mind creates mathematics.* New York, Oxford University Press, 1997). Focusing on algebraic notions such as *unknown*, *variable*, *algebraic* *expression*, *equation* and *solution of an equation*, I will describe possible processes of meaning making in students with low achievement in mathematics, or even diagnosed with DD including adult learners. This involves considering algebra not only in its syntactic aspects but also in its semantic ones. The assumption on which the work is based, is that some difficulties in learning algebra could be due to the lack of meaning attributed by the students to the algebraic notions. Basing the analyses on studies both in the domain of cognitive psychology and in the domain of mathematics education, I will show how students with DD can make sense of the algebraic notions considered above, thanks to tasks designed within AlNuSet exploiting its semiotic multi-representations based on visual, non-verbal and kinaesthetic-tactile systems. AlNuSet (Algebra of Numerical Sets) is a digital artifact for dynamic algebra, designed for students of lower and upper secondary school.

## Keywords

Algebra Developmental dyscalculia AlNuSet Task Variable Solution of an equation Algebraic expression## References

- Artigue, M., & Perrin-Glorian, M. J. (1991). Didactic engineering, research and development tool: some theoretical problems linked to this duality.
*For the learning of Mathematics*,*11*(1), 13–17.Google Scholar - Arzarello, F. (2006). Semiosis as a multimodal process.
*Relime, Revista latinoamericana de investigación en matemática educativa*,*9*(1), 267–299.Google Scholar - Arzarello, F., & Edwards, L. (2005). Gesture and the construction of mathematical meaning (research forum 2). In
*Proceedings of 29th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 122–145). Melbourne, AU: PME.Google Scholar - Arzarello, F., & Robutti, O. (2001). From body motion to algebra through graphing. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.),
*Proceedings of the 12th ICMI Study Conference*(Vol. 1, pp. 33–40). Australia: The University of Melbourne.Google Scholar - Arzarello, F., Bazzini, L., & Chiappini, G. P. (1994). Intensional semantics as a tool to analyse algebraic thinking.
*Rendiconti del Seminario Matematico dell’Università di Torino,**52*(1), 105–125.Google Scholar - Baccaglini-Frank, A., & Robotti, E. (2013). Gestire gli studenti con DSA in classe: alcuni elementi di un quadro commune. In
*Convegno Grimed 18 “Per piacere voglio contare-difficoltà, disturbi di apprendimento e didattica della matematica*”,*Padova*, (pp. 75–86).Google Scholar - Butterworth, B. (2003).
*Dyscalculia screening.*London, Uk: nferNelson Publishing Company Limited. (ISBN: 0 7087 0366 6).Google Scholar - Butterworth, B. (2005). Developmental Dyscalculia. In
*Handbook of mathematical cognition*(pp. 455–467). Hove, UK: Psychology Press.Google Scholar - Chiappini, G., Robotti, E., & Trgalova, J. (2010). Role of an artifact of dynamic algebra in the conceptualization of the algebraic equality. In
*Proceedings of CERME 6*,*Lyon, France*(pp. 619–628). www.inrp.fr/editions/cerme6. - Chaachoua, H., Chiappini, G., Croset, M. C., Pedemonte, B., & Robotti, E., (2012). Introduction de nouvelles rerpésentations dans deux environnements pour l’apprentissage de l’algèbre.
*Recherche en Didactique des mathématiques*, 253–281.Google Scholar - Dehaene, S. (1997).
*The number sense: How the mind creates mathematics*. New York, Oxford Unicersity Press.Google Scholar - DeThorne, L. S., & Schaefer, B. A. (2004). A guide to child nonverbal IQ measures.
*American Journal of Speech-Language Pathology*,*13*(4), 275–290.Google Scholar - Fandiño Pinilla, M. I. (2005).
*Le frazioni, aspetti concettuali e didattici*. Bologna: Pitagora Editrice.Google Scholar - Gallese, V., & Lakoff, G. (2005). The brain’s concepts: The role of the sensory-motor system in conceptual knowledge.
*Cognitive Neuropsychology,**22*(3/4), 455–479.CrossRefGoogle Scholar - Hittmair-Delazer, M., Sailer, U., & Benke, T. (1995). Impaired Arithmetic Facts But Intact Conceptual Knowledge a Single—Case Study of Dyscalculia.
*Cortex*,*31*(1), 139–147.Google Scholar - Kieran, C. (2006). Research on the learning and teaching of algebra. In G. Gutierrez, & P. Boero (Eds.),
*Handbook of Research on the Psychology of Mathematics Education. Past, Present and Future*(pp. 11–49). Rotterdam, Taipei: Sense Publishers.Google Scholar - Landy, D., & Goldstone, R. L. (2010). Proximity and precedence in arithmetic.
*The Quarterly Journal of Experimental Psychology (Colchester),**63*, 1953–1968.CrossRefGoogle Scholar - Leung, A., & Bolite-Frant, J. (2015). Designing mathematics tasks: The role of tools. In A. Watson & M. Ohtani (Eds.),
*Task design in mathematics education: The 22nd ICMI study (new ICMI study series)*(pp. 191–225). New York: Springer.CrossRefGoogle Scholar - Mammarella, I. C., Giofrè, D., Ferrara, R., & Cornoldi, C. (2013). Intuitive geometry and visuospatial working memory in children showing symptoms of nonverbal learning disabilities.
*Child Neuropsychology*,*19*(3), 235–249.Google Scholar - Mammarella, I. C., Lucangeli, D., & Carnoldi, C. (2010). Spatial working memory and arithmetic deficits in children with non verbal learning difficulties.
*Journal of Learning Disability*,*43*, 455–468.Google Scholar - Mariani, L. (1996). Investigating learning styles, perspectives.
*Journal of TESOL-Italy,**XXI, 2*/*XXII, 1*, (pp. 35–49). Spring.Google Scholar - Mason, J., & Johnston-Wilder, S. (2006).
*Designing and using mathematical tasks*. Tarquin.Google Scholar - Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.),
*Proceedings 27th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 103–135). Honolulu, Hawaii: PME. C. K. Ogden, & I. A. Richards (1923).Google Scholar - Núñez, R. (2000). Mathematical idea analysis: What embodied cognitive science can say about the human nature of mathematics. In XXX (Eds.),
*Proceedings of the 24 PME Conference*(Vol. 1, pp. 3–22). Japan: Hiroshima University.Google Scholar - Radford, L. (2005). Body, tool, and symbol: semiotic reflections on cognition. In E. Simmt, & B. Davis (Eds.),
*Proceedings of the 2004 Annual Meeting of the Canadian Mathematics Education Study Group*(pp. 111–117). Toronto, Canada.Google Scholar - Raghubar, K. P., Barnes, M. A., & Hecht, S. A. (2010). Working memory and mathematics: A review of developmental, individual difference, and cognitive approaches.
*Learning and individual differences*,*20*(2), 110–122.Google Scholar - Robotti, E. (2013). Dynamic representations for algebraic objects available in AlNuSet: How develop meanings of the notions involved in the equation solution. In C. Margolinas (Ed.),
*Task design in mathematics education: Proceedings of ICMI study 22*(pp. 99–108). UK: Oxford.Google Scholar - Robotti, E., & Baccaglini-Frank, A. (2016). Using digital environments to address students’ mathematical learning difficulties. In E. Faggiano, F. Ferrara, & A. Montone (Eds.),
*Innovation and technology enhancing mathematics education. Perspectives in the digital era.*Springer Publisher (accepted).Google Scholar - Robotti, E., & Ferrando, E. (2013). Difficulties in algebra: new educational approach by AlNuSet. In E. Faggiano, & A. Montone (Eds.),
*Proceedings of ICTMT11*, 250–25. Italy: ICTMT.Google Scholar - Robutti, O. (2005). Hearing gestures in modelling activities with the use of technology. In F. Olivero, & R. Sutherland (Eds.),
*Proceedings of the 7th international conference on technology in mathematics teaching*(pp. 252–261). University of Bristol.Google Scholar - Rourke, B. P., & Conway, J. A. (1997). Disabilities of arithmetic and mathematical reasoning perspectives from neurology and neuropsychology.
*Journal of Learning disabilities*,*30*(1), 34–46.Google Scholar - Sfard, A., & Linchevsky, L. (1992). Equations and inequalities: Processes without objects? In W. Goeslin, K. Graham (Ed.),
*Proceedings PME XVI, Durham, NH*(Vol. 3, p. 136).Google Scholar - Stella, G., & Grandi, L. (2011).
*Conoscere la dislessia e i DSA*. Giunti Editore.Google Scholar - Szucs, D., Devine, A., Soltesz, F., Nobes, A., & Gabriel, F. (2013). Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment.
*Cortex*,*49*(10), 2674–2688.Google Scholar - Temple, C. M. (1991). Procedural dyscalculia and number fact dyscalculia: Double dissociation in developmental dyscalculia.
*Cognitive Neuropsychology,**8*, 155–176.CrossRefGoogle Scholar - Thomas, M. O. J., & Tall, D. O. (2001). The long-term cognitive development of symbolic algebra. In H. Chick, K. Stacey, J. Vincent & J. T. Zilliox (Ed.),
*International Congress of Mathematical Instruction (ICMI) Working Group Proceedings—the future of the teaching and learning of algebra,**Melbourne**(*Vol. 2, pp. 590–597).Google Scholar - Vergnaud, G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems.
*Addition and subtraction: A cognitive perspective*, 39–59.Google Scholar