# Fostering Middle School Students’ Number Sense Through Contextualized Tasks

## Main Article Content

## Abstract

Number sense has been considered as one of the most important mathematical notions to be addressed in school mathematics in the 21st century. In this paper, we identify how students of a public middle school, located in a rural area in Mexico, develop several aspects of number sense by performing tasks involving arithmetic operations in a shopping context. Students invented mental and written calculation strategies that provide evidence of understanding regarding numbers and operations. Students used fantasy money as a representation system to decompose quantities in non-standard ways, or referred to strategies employed by unschooled people. Strategies that involve modifying the quantities before applying the written algorithm for subtraction, in order to perform a subtraction without regrouping, also appeared. That is, it was identified a transfer from mental calculation strategies to a paper and pencil scenario for facilitating calculations during problem solving, which is an indicator of student’ creativity and number sense. Additionally, the proposed tasks promoted the understanding of decimal numbers.

### Downloads

## Article Details

**International Electronic Journal of Elementary Education**, [S.l.], v. 12, n. 1, p. 75-86, oct. 2019. ISSN 1307-9298. Available at: <https://iejee.com/index.php/IEJEE/article/view/800>. Date accessed: 19 sep. 2020.

## References

Anghileri, J. (2006). Teaching number sense (2nd ed.). London: Continuum.

Anthony, G., & Walshaw, M. (2009). Effective pedagogy in mathematics. Educational series 19. Brussels: International Academy of Education; Geneva: International Bureau of Education.

Ausubel, D. P. (2000). The acquisition and retention of knowledge: A cognitive view. Springer.

Beishuizen, M., & Anghileri, J. (1998). Which mental strategies in the early number curriculum? A comparison of British ideas and Dutch views. British Education Research Journal, 24(5), 519-538. https://doi.org/10.1080/0141192980240503

Berger, W. (2014). A more beautiful question: The power of inquiry to spark breakthrough ideas. New York, NY: Bloomsbury.

Blöte, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10, 221-247. https://doi.org/10.1016/S0959-4752(99)00028-6

Boaler, J. (1993). The role of contexts in mathematics classrooms: Do they make mathematics more “real”? For the learning of mathematics, 13(2), 12-17. https://www.jstor.org/stable/40248079

Burger, W. F. & Shaughnessy, J. M. (1986). Characterizing the van Hiele Levels of Development in Geometry. Journal for Research in Mathematics Education, 17(1), 31-48. DOI: 10.2307/749317

Burns, M. (1994). Arithmetic: The last holdout. Phi Delta Kappan, 75(6), 471-476. https://www.jstor.org/stable/20405143

Carpenter, T. P., Franke, M. L., Jacobs, V. J., Fennema, E., & Empson, S. B. (1997). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3-20. DOI: 10.2307/749715

Chimhande, T., Naidoo, A., & Stols, G. (2017): An analysis of Grade 11 learners’ levels of understanding of functions in terms of APOS theory. Africa Education Review, 14(3-4), 1-19. https://doi.org/10.1080/18146627.2016.1224562

Clarke, D., & Roche, A. (2018). Using contextualized tasks to engage students in meaningful and worthwhile mathematics learning. The Journal of Mathematical Behavior, 51, 95-108. https://doi.org/10.1016/j.jmathb.2017.11.006

Clements, D. H., & Sarama, J. (2013). Rethinking early mathematics: What is research-based curriculum for young children? In L. D. English y J. T. Mulligan (Eds.), Reconceptualizing Early Mathematics Learning (pp. 121-147). Dordrecht: Springer.

Cobb, P., Wood, T., Yackel, E., Nichols, J., Wheatley, G., Trigatti, B., & Perlwitz, M. (1991). Assessment of a problem-centered second grade mathematics project. Journal for Research in Mathematics Education, 22, 3-29. DOI: 10.2307/749551

Devlin, K. (2017, January 1). All the mathematical methods I learned in my university math degree became obsolete in my lifetime. The Huffington Post. https://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131. https://doi.org/10.1007/s10649-006-0400-z

Fuson, K. C. (1992). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhard, R. Putman, and R. A. Hattrup (Eds.), Analysis of Arithmetic for Mathematics Teaching (pp. 53-187). Hillsdale, N.J.: Lawrence Erlbaum Associates.

Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22(3), 170-218. https://www.jstor.org/stable/749074

Harel, G. & Sowder, H. (1998). Students’ proof schemes: Results from exploratory studies. In E. Dubinsky, A. Schoenfeld and J. Kaput (Eds.), Issues in mathematics education: Vol. 7. Research in collegiate mathematics education III (pp. 234-282). Providence, RI: American Mathematical Society.

Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students' learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393-425. https://doi.org/10.3102/00028312030002393

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Jones, G. A., Thornton, C. A., Putt, I. J., Hill, K. M., Mogill, A. T., Rich, B. S., & Van Zoest, L. R. (1996). Multidigit Number Sense: A Framework for Instruction and Assessment. Journal for Research in Mathematics Education, 27(3), 310-336. DOI: 10.2307/749367

Kolb, D. A. (1984). Experiential learning: experience as the source of learning and development. Englewood Cliffs, NJ: Prentice Hall.

Lago, R. M. (2007). Examining the psychometrics of number sense among kindergarten students. Unpubliseh Doctoral Dissertation. The Pennsylvania State University.

Lithner, J. (2003). Students’ mathematical reasoning in university textbooks exercises, Educational Studies in Mathematics, 52 (1), 29-55. https://doi.org/10.1023/A:102368371

Lubienski, S. (2000). Problem solving as a means towards mathematics for all: An exploratory look through a class lens. Journal for Research in Mathematics Education, 31 (4), 454–482. DOI: 10.2307/749653

Malloy, C. E. (1999). Perimeter and area through the van Hiele model. Mathematics Teaching in the Middle School, 5(2), 87-90.

Marcovits, Z., & Sowder, J. (1994). Developing number sense: an intervention study in grade 7. Journal for Research in Mathematics Education, 25(1), 4-29. DOI: 10.2307/749290

McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2-8. https://flm-journal.org/Articles/94F594EF72C03412F1760031075F2.pdf

Mulligan, J., & Mitchelmore, M. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28, 309-330. DOI: 10.2307/749783

Munisamy, S. & Doraisamy, L. (1998) Levels of understanding of probability concepts among secondary school pupils. International Journal of Mathematical Education in Science and Technology, 29 (1), 39-45. https://doi.org/10.1080/0020739980290104

National Council of Teachers of Mathematics [NCTM] (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

National Governors Association Center for Best Practices, Council of Chief State School Officers [NGACBP & CCSSO] (2010). Common Core State Standards for Mathematics. Washington D.C.: NGACBP & CCSSO. http://www.corestandards.org/Math/

Nickerson, S. D., & Whitacre, I. (2010). A local instruction theory for the development of number sense. Mathematical Thinking and Learning, 12(3), 227-252. http://dx.doi.org/10.1080/10986061003689618

Niss, M. (1987). Applications and modelling in the mathematics curriculum — state and trends. International Journal of Mathematical Education in Science and Technology, 18(4), 487-505. http://dx.doi.org/10.1080/0020739870180401

Paulos, J. A. (2000). El hombre anumérico: el analfabetismo matemático y sus consecuencias. Madrid: Túsquets.

Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26 (2-3), 165-190.

Pitta-Pantazi, D., Christou, C., & Zachariades, T. (2007). Secondary school students’ levels of understanding in computing exponents. Journal of Mathematical Behavior 26, 301–311. https://doi.org/10.1016/j.jmathb.2007.11.003

Plunkett, S. (1979). Decomposition and all that rot. Mathematics in School, 8 (3), 2-7.

Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.

Qualifications and Curriculum Authority [QCA] (1999). The National Numeracy Strategy. Teaching mental calculation strategies. London: QCA.

Reys, B. J. (1994). Promoting number sense in middle grades. Teaching Mathematics in the Middle School, 1, 114–120.

Reys, R. E., Reys, B. J., Nohda, N., & Emori, H. (1995). Mental computation performance and strategy use of Japanese students in grades 2, 4, 6, and 8. Journal for Research in Mathematics Education, 26(4), 304-326. DOI: 10.2307/749477

Reys, R., Reys, B., McIntosh, A., Emanuelsson, G., Johansson, B., & Yang, D. C. (1999). Assessing number sense of students in Australia, Sweden, Taiwan, and the United States. School, Science and Mathematics, 99(2), 61-70.

Santos-Trigo, M. (2018). Problematizar los contenidos y la tecnología digital. C2 Ciencia y cultura. Recuperado de http://www.revistac2.com/problematizar-los-contenidos/ el 26 de julio de 2018.

Sarama, J., & Clements, D.H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.

Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando, FL: Academic Press.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.

Segura, J. (2015). La utilización de los algoritmos de sustracción en educación primaria. Edma 0-6: Educación Matemática en la Infancia, 4(2), 73-88. http://funes.uniandes.edu.co/8395/1/Edma0-6_v4n2_73-88.pdf

Simon, M. A. (1994). Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational studies in mathematics, 26(1), 71-94. https://doi.org/10.1007/BF01273301

Sowder, J. T. (1992). Making Sense of Numbers in School Mathematics. In G. Leinhard, R. Putman, and R. A. Hattrup (Eds.), Analysis of Arithmetic for Mathematics Teaching (pp. 1–51). Hillsdale, N.J.: Lawrence Erlbaum Associates.

Steen, L. A. (1988). The science of patterns. Science, 240(29), 616. DOI: 10.1126/science.240.4852.611

Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation: An International Journal on Theory and Practice, 2(1), 50-80. http://dx.doi.org/10.1080/1380361960020103

Steinle, V. (2004). Detection and remediation of decimal misconceptions. In B. Tadich, S. Tobias, C. Brew, B. Beatty, & P. Sullivan (Eds.), Towards excellence in mathematics (pp. 460–478). Brunswick: The Mathematical Association of Victoria.

Sullivan, P., Clarke, D., & Clarke, B. (2013). Teaching with tasks for effective mathematics learning. New York: Springer.

Swan, P. & Sparrow, L. (2001). Strategies for going mental. In Proceedings of the Eighteenth Biennial Conference of The Australian Association of Mathematics Teachers (pp. 236-243). Canberra: Australian National University. http://www.aamt.edu.au/content/download/21441/285094/file/tdt_MC_swan1.pdf

Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50, 29-47. https://doi.org/10.1023/A:102057280

United Nations Educational, Scientific and Cultural Organization [UNESCO] (1989). Multigrade Teaching in Single Teacher Primary Schools. Bangkok: UNESCO.

Valencia, E. (2013). Desarrollo del cálculo mental a partir de entrenamiento en combinaciones numéricas y estrategias de cálculo. Números. Revista de Didáctica de las Matemáticas, 84, 5-23. http://www.sinewton.org/numeros/numeros/84/Volumen_84.pdf

van den Heuvel-Panhuizen, M. (2005). The role of contexts in assessment problems in mathematics. For the Learning of Mathematics, 25 (2), 2–9. https://www.jstor.org/stable/40248489

Vygotski, L. S. (2006). El desarrollo de los procesos psicológicos superiores. Barcelona: Crítica.

Wearne, D. (1990). Acquiring meaning for decimal fraction symbols: A one year follow-up. Educational Studies in Mathematics, 21, 545-564. https://doi.org/10.1007/BF00315944

Wenger, E., McDermott, R., & Snyder, W. M. (2002). Cultivating communities of practice. Boston, MA: Harvard Business School Press.

Wertsch, J. V. (1993). Voices of the Mind: A Sociocultural Approach to Mediated Action. Cambridge, MA: Harvard University Press.

Yang, D.-C., & Wu, W.-R. (2010). The study of number sense: Realistic activities integrated into third-grade math classes in Taiwan. The Journal of Educational Research, 103(6), 379-392. http://dx.doi.org/10.1080/00220670903383010

Zull, J. E. (2002). The art of changing the brain: Enriching the practice of teaching by exploring the biology of learning. Sterling, VA: Stylus.