# A critical discourse analysis of practical problems in a foundation mathematics course at a South African university

- 523 Downloads
- 1 Citations

## Abstract

Mathematical problems that make links to the everyday and to disciplines other than mathematics—variously referred to as practical, realistic, real-world or applied problems in the literature—feature in school and undergraduate mathematics reforms aimed at increasing mathematics participation in contexts of inequity and diversity. In this article, we present a micro- and macro-analysis of a prototypical practical problem in an undergraduate mathematics course at a South African university. This course offers an alternative route to a mathematics major for students considered disadvantaged by enduring educational inequalities in South Africa. Using a socio-political practice perspective on mathematics and critical discourse analysis—drawn from Norman Fairclough’s critical linguists—we describe what mathematics and mathematical identities practical problems make available to students and compare this to what is valued in school mathematics and other university mathematics courses. Our analysis shows that these practical problems draw in complex ways on sometimes contradictory practices in the wider context, requiring the student to work flexibly with the movement of meaning within and across texts. We raise for further consideration the possible consequences of this complexity and offer suggestions for practice that take into account issues of power.

## Keywords

Access Advanced mathematics Calculus reform Critical discourse analysis Equity Practical problems Socio-political practice perspective## Notes

### Acknowledgments

We thank Lucia Thesen for her insightful comments, and reviewers whose feedback has enhanced the clarity. This study has been supported by the National Research Foundation in South Africa under Grant number TTK2006040500009. Any opinion, findings and conclusions or recommendations expressed in this article are those of the authors, and the National Research Foundation does not accept any liability in regard thereto.

## References

- Bansilal, S. (2009). The use of real life contexts in the CTA: Some unintended consequences.
*Pythagoras, 69*, 17–27.Google Scholar - Bergsten, C., Jablonka, E., & Klisinska, A. (2010). Reproduction and distribution of mathematical knowledge in higher education: Constructing insiders and outsiders. In U. Gellert, E. Jablonka, & C. Morgan (Eds.),
*Proceedings of the Sixth International Mathematics Education and Society Conference*(pp. 150–160). Berlin: Freie Universität Berlin.Google Scholar - Boaler, J. (1993). Encouraging transfer of “school” mathematics to the “real world” through interpretation of process and content, context and culture.
*Educational Studies in Mathematics, 25*, 341–373.CrossRefGoogle Scholar - Chouliaraki, L., & Fairclough, N. (1999).
*Discourse in late modernity: Rethinking critical discourse analysis*. Edinburgh: Edinburgh University Press.Google Scholar - Cooper, B., & Dunne, M. (2000).
*Assessing children’s mathematical knowledge: Social class, sex and problem-solving*. Buckingham, England: Open University Press.Google Scholar - Council on Higher Education (2013).
*A proposal for undergraduate curriculum reform in South Africa: The case for a flexible curriculum structure (Discussion document).*Pretoria: Council on Higher Education. Retrieved from www.che.ac.za/sites/default/files/publications/Full_Report.pdf - De Freitas, E., & Zolkower, B. (2009). Using social semiotics to prepare mathematics teachers to teach for social justice.
*Journal of Mathematics Teacher Education, 12*, 187–302.Google Scholar - Department of Basic Education. (2011).
*National Curriculum Statement Grades 10–12 Curriculum and Assessment Policy Statement: Mathematics, Grade 10–12*. Pretoria: Department of Basic Education.Google Scholar - Department of Education. (2003).
*National Curriculum Statement Grades 10–12 (General): Mathematics*. Pretoria: Department of Education.Google Scholar - Douglas, R. G. (1986). Introduction: Steps toward a lean and lively calculus. In R. G. Douglas (Ed.),
*Toward a lean and lively calculus*(pp. iv–vi). Washington, DC: Mathematical Association of America.Google Scholar - Dowling, P. (1998).
*The sociology of mathematics education: Mathematical myths/pedagogic texts*. London: Falmer Press.Google Scholar - Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 25–41). Dordrecht: Kluwer.Google Scholar - Engelbrecht, J., Harding, A., & Phiri, P. (2010). Are OBE‐trained learners ready for university mathematics?
*Pythagoras, 72*, 3–13.Google Scholar - Fairclough, N. (1995).
*Critical discourse analysis: The critical study of language*. London: Longman.Google Scholar - Fairclough, N. (2001).
*Language and power*(2nd ed.). Harlow, England: Longman.Google Scholar - Fairclough, N. (2003).
*Analysing discourse: Textual analysis in social research*. London: Routledge.Google Scholar - Fairclough, N. (2005). Peripheral vision: Discourse analysis in organization studies—the case for critical realism.
*Organization Studies, 26*, 915–939.CrossRefGoogle Scholar - Fairclough, N. (2010).
*Critical discourse analysis: The critical study of language*(2nd ed.). Harlow, UK: Pearson.Google Scholar - Garner, B. E., & Garner, L. E. (2001). Retention of concepts and skills in traditional and reformed applied calculus.
*Mathematics Education Research Journal, 13*(3), 165–184.CrossRefGoogle Scholar - Gellert, U., & Jablonka, E. (2009). “I am not talking about reality”: Word problems and the intricacies of producing legitimate text. In L. Verschaffel, B. Greer, W. Van Dooren, & S. Mukhopadhyay (Eds.),
*Words and worlds: Modelling verbal descriptions of situations*(pp. 39–53). Rotterdam: Sense.Google Scholar - Gerofsky, S. (2004).
*A man left Albuquerque heading east: Word problems as genre in mathematics education*. New York: Peter Lang.Google Scholar - Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A “perceptual” view of simple arithmetic.
*Journal for Research in Mathematics Education, 25*, 116–140.CrossRefGoogle Scholar - Harel, G., & Kaput, J. (1991). The role of conceptual entities and their symbols in building advanced mathematical concepts. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 82–94). Dordrecht: Kluwer.Google Scholar - Hazzan, O. (2003). How students attempt to reduce abstraction in the learning of mathematics and the learning of computer science.
*Computer Science Education, 13*(2), 95–122.CrossRefGoogle Scholar - Herbel-Eisenmann, B. (2007). From intended curriculum to written curriculum: Examining the “voice” of a mathematics textbook.
*Journal for Research in Mathematics Education, 38*, 344–369.Google Scholar - Hoyles, C., Newman, K., & Noss, R. (2001). Changing patterns of transition from school to university mathematics.
*International Journal of Mathematical Education in Science and Technology, 32*, 829–845.CrossRefGoogle Scholar - Hughes-Hallet, D., Gleason, A. M., Flath, D. E., Gordon, S. P., Lomen, D. O., Lovelock, D., et al. (1994).
*Calculus*. New York: John Wiley & Sons.Google Scholar - Jablonka, E., Ashjari, H., & Bergsten, C. (2012). Recognising knowledge criteria in undergraduate mathematics education. In C. Bergsten, E. Jablonka, & M. Ramon (Eds.),
*Proceedings of the Eighth Swedish Mathematics Education Research Seminar*(pp. 101–110). Linköping: Swedish Society for Research in Mathematics Education.Google Scholar - Janks, H. (2010).
*Literacy and power*. New York: Routledge.Google Scholar - Kanes, C., Morgan, C., & Tsatsaroni, A. (2014). The PISA mathematics regime: Knowledge structures and practices of the self.
*Educational Studies in Mathematics, 87*, 145–165.CrossRefGoogle Scholar - Kessi, S. (2013, second quarter). Transforming historically white universities: Students and the politics of racial representation.
*New Agenda,*53–55.Google Scholar - Le Roux, C.J. (2011).
*A critical examination of the use of practical problems and a learner-centred pedagogy in a foundational undergraduate mathematics course*(Unpublished doctoral dissertation). University of the Witwatersrand, Johannesburg, South Africa.Google Scholar - Le Roux, K., & Adler, J. (2012). Talking and looking structurally and operationally as ways of acting in a socio-political mathematical practice. In T.Y. Tso (Ed.),
*Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 51–58). Taipei: PME.Google Scholar - Lubienski, S. T. (2000). Problem solving as a means toward mathematics for all: An exploratory look through a class lens.
*Journal for Research in Mathematics Education, 31*, 454–482.CrossRefGoogle Scholar - Maharaj, A. (2010). Students’ understanding of the concept of a limit of a function.
*Pythagoras, 71*, 41–52.Google Scholar - McBride, M. (1994). The theme of individualism in mathematics education: An examination of mathematics textbooks.
*For the Learning of Mathematics, 14*(3), 36–42.Google Scholar - Morgan, C. (1998).
*Writing mathematically: The discourse of investigation*. London: Falmer Press.Google Scholar - Morgan, C. (2014). Understanding practices in mathematics education: Structure and text.
*Educational Studies in Mathematics, 87*, 129–143.CrossRefGoogle Scholar - Moschkovich, J. (2002). An introduction to examining everyday and academic mathematics practices. In M. E. Brenner & J. Moschkovich (Eds.),
*Everyday and academic mathematics classrooms*(pp. 1–11). Reston, VA: NCTM.Google Scholar - Nyabanyaba, T. (2002).
*Examining examination: The ordinary level (O-level) mathematics examination in Lesotho and the impact of recent trends on Basotho students’ epistemological access*(Unpublished doctoral dissertation). University of the Witwatersrand, Johannesburg, South Africa.Google Scholar - O’Halloran, K. (2011). The semantic hyperspace: Accumulating mathematical knowledge across semiotic resources and modalities. In F. Christie & K. Maton (Eds.),
*Disciplinarity: Functional linguistic and sociological perspectives*(pp. 217–236). London: Continuum.Google Scholar - Raman, M. (2002). Coordinating informal and formal aspects of mathematics: Student behaviour and textbook messages.
*Journal of Mathematical Behavior, 21*, 135–150.CrossRefGoogle Scholar - Schoenfeld, A.H. (1995). A brief biography of calculus reform.
*Undergraduate Mathematics Education (UME) Trends*,*6*(6), 3–5. Retrieved from http://www.researchgate.net/profile/Alan_Schoenfeld2/publication/234705281_A_Brief_Biography_of_Calculus_Reform/links/00463528dae95538a1000000.pdf - Schwingendorf, K.E., & Dubinsky, E. (1990). Calculus, concepts, and computers: Innovations in learning. In W. C. Tucker (Ed.),
*Priming the calculus pump: Innovations and resources*, MAA Notes 17 (pp. 175–198). Washington, DC: Mathematical Association of America.Google Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*, 1–36.Google Scholar - Sfard, A. (2008).
*Thinking as communicating: Human development, the growth of discourses, and mathematizing*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Smith, J. P., III, & Star, J. R. (2007). Expanding the notion of impact of K-12 Standards-based mathematics and reform calculus programmes.
*Journal for Research in Mathematics Education, 38*, 3–34.Google Scholar - Soudien, C. (2008). The intersection of race and class in the South African university: Student experiences.
*South African Journal of Higher Education, 22*, 662–678.Google Scholar - Spaull, N. (2013).
*South Africa’s education crisis: The quality of education in South Africa 1994–2011 (Report commissioned by the Centre for Development Enterprise).*Retrieved from http://www.section27.org.za/wp-content/uploads/2013/10/Spaull-2013-CDE-report-South-Africas-Education-Crisis.pdf - Stewart, J. (2006).
*Calculus: Concepts and contexts*(3rd ed.). Belmont, CA: Thomson Brooks/Cole.Google Scholar - Straehler-Pohl, H., Fernández, S., Gellert, U., & Figueras, L. (2014). School mathematics registers in a context of low academic expectations.
*Educational Studies in Mathematics, 85*, 175–199.CrossRefGoogle Scholar - Swanson, D. M. (2005). School mathematics: Discourse and the politics of context. In A. Chronaki & I. M. Christiansen (Eds.),
*Challenging perspectives on mathematics classroom communication*(pp. 261–294). Greenwich, England: Information Age.Google Scholar - Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 495–511). New York: Macmillan.Google Scholar - Tall, D. (1996). Functions and calculus. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.),
*International handbook of mathematics education*(pp. 289–325). Dordrecht: Kluwer.Google Scholar - Tall, D. (1997). From school to university: The effects of learning styles in the transition from elementary to advanced mathematical thinking. In M. O. J. Thomas (Ed.),
*Proceedings of the Seventh Annual Australasian Bridging Network Mathematics Conference*(pp. 9–26). New Zealand: University of Auckland.Google Scholar - Tobias, B. (2009).
*From textual problems to mathematical relationships: Case studies of secondary school students and the discourses at play in interpreting word problems*(Unpublished doctoral dissertation). Johannesburg, South Africa: University of the Witwatersrand.Google Scholar - Valero, P. (2007). A socio-political look at equity in the school organization of mathematics education.
*Zentralblatt für Didaktik der Mathematik (ZDM), 39*, 225–233.CrossRefGoogle Scholar - Wagner, D. (2012). Opening mathematics texts: Resisting the seduction.
*Educational Studies in Mathematics, 80*, 153–169.CrossRefGoogle Scholar - Walkerdine, V. (1988).
*The mastery of reason: Cognitive development and the production of rationality*. London: Routledge.Google Scholar - Wood, L. N. (2001). The secondary-tertiary interface. In D. Holton (Ed.),
*The teaching and learning of mathematics at university level: An ICMI study*(pp. 87–98). Dordrecht: Kluwer.Google Scholar