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The lived experience of linear algebra: a counter-story about women of color in mathematics

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Abstract

This paper focuses on the mathematical sensemaking by women of color in the USA as part of the global effort of dismantling deficit narratives about historically marginalized groups of students. Following Adiredja’s anti-deficit framework for sensemaking, this cognitive study invited a group of women of color to share their understanding of basis from linear algebra to construct a sensemaking counter-story. Extending the framework, this study examines a task that explores the boundaries and nuances of a concept to support the effort of going beyond students’ deficits. Eight women extended the concept of basis (and vector spaces) to 22 distinct everyday contexts, drawing from their everyday lives as well as topics from their academic experiences. Their explanations revealed analytical codes describing roles and characteristics of a basis. These codes suggest ways that students can mobilize the concept of basis beyond its logical underpinnings. Contrasting interpretations using a deficit and an anti-deficit perspective construct a counter-story that showcases these women’s creativity and flexibility in understanding the concept, and potential resources for the teaching and learning of linear algebra.

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Notes

  1. In the USA, women of color include Black and African American women, Latinx, Asian American, Pacific Islander women, and Indigenous women. We consider them together as a group to recognize their shared underrepresentation in STEM higher education and careers.

  2. All names are pseudonyms.

References

  • Adiredja, A. P. (2019). Anti-deficit narratives: Engaging the politics of mathematical sense making. Journal for Research in Mathematics Education, 50(4), 401–435.

    Google Scholar 

  • Adiredja, A. P., Bélanger-Rioux, R., & Zandieh, M. (2020). Everyday examples from students: An anti-deficit approach in the classroom. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 30(5), 520–538.

    Google Scholar 

  • Adiredja, A. P., & Louie, N. (2020). Understanding the web of deficit discourse in mathematics education. For the Learning of Mathematics, 40(1), 42–46.

    Google Scholar 

  • Adiredja, A. P., & Zandieh, M. (2017). Using intuitive examples from women of color to reveal nuances about basis. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 20thAnnual Conference on Research in Undergraduate Mathematics Education (pp. 346–359), San Diego, CA.

  • Apple, M. W. (1992). Do the standards go far enough? Power, policy, and practice in mathematics education. Journal for Research in Mathematics Education, 23(5), 412–431.

    Google Scholar 

  • Aydin, S. (2014). Using example generation to explore students’ understanding of the concepts of linear dependence/independence in linear algebra. International Journal of Mathematical Education in Science and Technology, 45(6), 813–826.

    Google Scholar 

  • Bagley, S., & Rabin, J. M. (2016). Students’ use of computational thinking in linear algebra. International Journal of Research in Undergraduate Mathematics Education, 2(1), 83–104.

    Google Scholar 

  • Cárcamo, A., Fortuny, J., & Fuentealba, C. (2017). The emergent models in linear algebra: An example with spanning set and span. Teaching Mathematics and its Applications: An International Journal of the IMA, 37(4), 202–217.

    Google Scholar 

  • Carlone, H. B., & Johnson, A. (2007). Understanding the science experiences of successful women of color: Science identity as an analytic lens. Journal of Research in Science Teaching, 44(8), 1187–1218.

    Google Scholar 

  • Çelik, D. (2015). Investigating students’ modes of thinking in linear algebra: The case of linear independence. International Journal for Mathematics Teaching and Learning. Retrieved from http://www.cimt.org.uk/journal/index.htm. Accessed 7 Nov 2017.

  • D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44–48.

  • Darragh, L., & Valoyes-Chávez, L. (2019). Blurred lines: Producing the mathematics student through discourses of special educational needs in the context of reform mathematics in Chile. Educational Studies in Mathematics, 101, 425–439 1-15.

    Google Scholar 

  • de Abreu, G. (1995). Understanding how children experience the relationship between home and school mathematics. Mind, Culture, and Activity, 2, 119–142.

    Google Scholar 

  • De Freitas, E., & Sinclair, N. (2017). Concept as generative devices. In E. De Freitas, N. Sinclair, & A. Coles (Eds.), What is a mathematical concept? New York, NY: Cambridge University Press.

  • Delgado, R., & Stefancic, J. (2001). Critical race theory: An introduction. New York, NY: New York University Press.

  • Delgado-Gaitan, C. (1994). Socializing young children in Mexican-American families: An intergenerational perspective. In P. M. Greenfield & R. R. Cocking (Eds.), Cross-cultural roots of minority child development (pp. 55–86). Hillsdale, NJ: Lawrence Erlbaum.

  • diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10(2–3), 105–225.

    Google Scholar 

  • diSessa, A. A. (2014). The construction of causal schemes: Learning mechanisms at the knowledge level. Cognitive Science, 38(5), 795–850.

    Google Scholar 

  • diSessa, A. A., Sherin, B., & Levin, M. (2016). Knowledge analysis: An introduction. In A. A. diSessa, M. Levin, & N. J. S. Brown (Eds.), Knowledge and interaction: A synthetic agenda for the learning sciences (pp. 30–71). New York, NY: Routledge.

  • Dogan, H. (2017). Differing instructional modalities and cognitive structures: Linear algebra. Linear Algebra and its Applications, 542, 1–648.

    Google Scholar 

  • Ertekin, E. (2010). The effects of formalism on teacher trainees’ algebraic and geometric interpretation of the notions of linear dependency/independency. International Journal of Mathematical Education in Science and Technology, 41(8), 1015–1035.

    Google Scholar 

  • Frade, C., Acioly-Régnier, N., & Jun, L. (2013). Beyond deficit models of learning mathematics: Socio-cultural directions for change and research. In M. A. K. Clements, A. Bishop, C. Keitel-Kreidt, J. Kilpatrick, & F. K.-S. Leung (Eds.), Third international handbook of mathematics education (pp. 101–144). New York, NY: Springer.

  • Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177.

    Google Scholar 

  • Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68.

    Google Scholar 

  • Harel, G. (2017). The learning and teaching of linear algebra: Observations and generalizations. The Journal of Mathematical Behavior, 46, 69–95.

    Google Scholar 

  • Hunter, R., & Hunter, J. (2017). Maintaining a cultural identity while constructing a mathematical disposition as a Pāsifika learner. In E. A. McKinley & L. T. Smith (Eds.), Handbook of indigenous education (pp. 1–19). Singapore: Springer.

    Google Scholar 

  • Jilk, L. M. (2016). Supporting teacher noticing of students' mathematical strengths. Mathematics Teacher Educator, 4(2), 188–199.

    Google Scholar 

  • Kleiner, I. (2007). A history of linear algebra. In I. Kleiner (Ed.), A history in abstract algebra (pp. 79–89). Boston, NJ: Burkhäuser.

  • Ladson-Billings, G., & Tate, W. F. (1995). Toward a critical race theory of education. Teachers College Record, 97(1), 47–68.

    Google Scholar 

  • Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.

  • Leikin, R., & Pitta-Pantazi, D. (2013). Creativity and mathematics education: The state of the art. ZDM, 45(2), 159–166.

    Google Scholar 

  • Leonard, J., & Martin, D. B. (Eds.). (2013). The brilliance of Black children in mathematics: Beyond the numbers and toward new discourse. Charlotte, NC: Information Age Publishing.

  • Lerman, S. (2000). The social turn in mathematics education perspectives in mathematics education research. In J. Boaler (Ed.), Multiple perspectives in mathematics teaching and learning (pp. 19–44). Westport, CT: Ablex.

  • Lewis, K. E. (2014). Difference not deficit: Reconceptualizing mathematical learning disabilities. Journal for Research in Mathematics Education, 45(3), 351–396.

    Google Scholar 

  • Leyva, L. A. (2016). An intersectional analysis of Latin@ college women’s counter-stories in mathematics. Journal of Urban Mathematics Education, 9(2), 81–121.

    Google Scholar 

  • Louie, N. L. (2017). The culture of exclusion and its persistence in equity-oriented teaching. Journal for Research in Mathematics Education, 28(5), 488–519.

    Google Scholar 

  • Louie, N. L. (2018). Culture and ideology in mathematics teacher noticing. Educational Studies in Mathematics, 97(1), 55–69.

    Google Scholar 

  • Malloy, C. E., & Jones, M. G. (1998). An investigation of African American students' mathematical problem solving. Journal for Research in Mathematics Education, 29, 143–163.

    Google Scholar 

  • Martin, D. B., Gholson, M. L., & Leonard, J. (2010). Mathematics as gatekeeper: Power and privilege in the production of knowledge. Journal of Urban Mathematics Education, 3(2), 12–24.

    Google Scholar 

  • McGee, E., & Martin, D. (2011). “You would not believe what I have to go through to prove my intellectual value!” Stereotype management among academically successful black mathematics and engineering students. American Educational Research Journal, 48(6), 1347–1389.

    Google Scholar 

  • Nasir, N. I. S., Snyder, C. R., Shah, N., & Ross, K. M. (2013). Racial storylines and implications for learning. Human Development, 55(5–6), 285–301.

    Google Scholar 

  • Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 40(4), 396–426.

  • Ong, M., Wright, C., Espinosa, L. L., & Orfield, G. (2011). Inside the double bind: A synthesis of empirical research on undergraduate and graduate women of color in science, technology, engineering, and mathematics. Harvard Educational Review, 81(2), 172–209.

    Google Scholar 

  • Plaxco, D., & Wawro, M. (2015). Analyzing student understanding in linear algebra through mathematical activity. The Journal of Mathematical Behavior, 38, 87–100.

    Google Scholar 

  • Rasmussen, C., Wawro, M., & Zandieh, M. (2015). Examining individual and collective level mathematical progress. Educational Studies in Mathematics, 88(2), 259–281.

    Google Scholar 

  • NCTM Research Committee. (2018). Asset-based approaches to equitable mathematics education research and practice. Journal for Research in Mathematics Education, 49(4), 373–389.

    Google Scholar 

  • Solórzano, D. G., & Yosso, T. J. (2002). Critical race methodology: Counter storytelling as an analytic framework for education research. Qualitative Inquiry, 8(23), 23–44.

    Google Scholar 

  • Stewart, S., Andrews-Larson, C., & Zandieh, M. (2019). Linear algebra teaching and learning: Themes from recent research and evolving research priorities. ZDM, 51(7), 1017–1030.

    Google Scholar 

  • Stewart, S., & Thomas, M. O. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology, 41(2), 173–188.

    Google Scholar 

  • Straehler-Pohl, H., Gellert, U., Fernandez, S., & Figueiras, L. (2014). School mathematics registers in a context of low academic expectations. Educational Studies in Mathematics, 85(2), 175–199.

    Google Scholar 

  • Teo, T. (2008). From speculation to epistemological violence in psychology: A critical-hermeneutic reconstruction. Theory & Psychology, 18(1), 47–67.

    Google Scholar 

  • Trigueros, M., & Possani, E. (2013). Using an economics model for teaching linear algebra. 16th ILAS Conference Proceedings, Pisa 2010, 438(4), 1779–1792.

  • Valero, P. (2004). Socio-political perspectives on mathematics education. In P. Valero & R. Zevenbergen (Eds.), Researching the socio-political dimensions of mathematics education (pp. 5–24). Norwell, MA: Kluwer.

  • Varma, R., Prasad, A., & Kapur, D. (2006). Confronting the “socialization” barrier: Crossethnic differences in undergraduate women’s preference for IT education. In J. M. Cohoon & W. Aspray (Eds.), Women and information technology: Research on underrepresentation (pp. 301–322). Cambridge, MA: MIT Press.

  • Wagner, J. (2006). Transfer in pieces. Cognition and Instruction, 24(1), 1–71.

    Google Scholar 

  • Zandieh, M., Adiredja, A. P., & Knapp, J. (2019). Exploring everyday examples to explain basis from eight German male graduate STEM students. ZDM, 51(7), 1153–1167.

    Google Scholar 

  • Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning. The Journal of Mathematical Behavior, 29(2), 57–75.

    Google Scholar 

  • Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. The Journal of Mathematical Behavior, 25(1), 1–17.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ, 85721-0089, USA

    Aditya P. Adiredja

  2. Faculty of Science and Mathematics, College of Integrative Sciences and Arts, Arizona State University, 6073 S. Backus Mall, MC 2780, Mesa, AZ, 85212-6420, USA

    Michelle Zandieh

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Appendix

Appendix

1.1 Basis interview protocol

Q1. (a) What does a basis mean to you?

Follow-up:

(i) [If students felt like their course did not cover basis formally]: Some students have said that a basis of a vector space is a linear independent set that spans the vector space. What do you think that student means by that? Then go to (iii).

(ii) [If they only mention one but not the other]: Some students have said that a basis of a vector space is a linear independent set that spans the vector space. What do you think that student means by that? Then go to (iii).

(iii) [If they mention both span or linear independence]: What does each of those things mean to you?

Q1. (b) How would you explain it to a student who is about to take a Linear Algebra course?

Q2. (a) Can you think of an example from your everyday life that describes the idea of a basis?

(b) How does your example reflect your meaning of basis? What does it capture and what does it not?

Q3. Could basis be relevant for any of the tasks you did? If so, how?

Q4. Can you see a basis as a way to describe something? If so, what is the something? How?

Q5. Can you see basis as a way to generate something? If so, what is the something? How?

Q6. Go through each task, and ask if they CAN possibly see basis in them.

Follow up: Can you express #3 in parametric form?

[If time permits] Q7. (a) Some students say that a basis is a minimal spanning set, what do you think the student means by that?

(b) Some students say that a basis is a maximal linear independent set, what do you think the student means by that?

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Adiredja, A.P., Zandieh, M. The lived experience of linear algebra: a counter-story about women of color in mathematics. Educ Stud Math 104, 239–260 (2020). https://doi.org/10.1007/s10649-020-09954-3

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