# Opportunities for learning given to prospective mathematics teachers: between ritual and explorative instruction

- 528 Downloads
- 3 Citations

## Abstract

The mathematics education field, including prospective teacher education program, has seen a continuous effort to change teaching practices to be more cognitively demanding, conceptually oriented and student centred. Our goal in this study was to examine how certain underlying assumptions about mathematical learning, as reflected in a skilled instructor’s discourse, align with opportunities to learn. The data included a set of fully transcribed 11 lessons from an introductory algebra course. The method of analysis was built upon the communicational (commognitive) framework and included discerning between the instructor’s mathematizing and identifying talk. This framework was extended to quantify the instructor’s identifying talk over the whole set of lessons. Our findings showed that at the surface level, the instruction in the class seemed to align with “explorative” goals. On a deeper level, however, it was more aligned with “ritual” goals that are concerned with producing narratives about people, not about mathematics.

## Keywords

Communicational (commognitive) framework Opportunities to learn Prospective teachers Explorative learning Ritual learning Identity## Notes

### Acknowledgments

This study is supported by the Israel Science Foundation, No. 446/10. The first author wishes to thank Prof. Lauren Resnick from the University of Pittsburgh for providing a postdotctoral fellowship that supported part of this work.

## References

- Adler, J., Ball, D., Krainer, K., Lin, F. L., & Novotna, J. (2005). Reflections on an emerging field: Researching mathematics teacher education.
*Educational Studies in Mathematics,**60*(3), 359–381. doi: 10.1007/s10649-005-5072-6.CrossRefGoogle Scholar - Ashcraft, M., Krause, J., & Hopko, D. (2007). Is math anxiety a mathematical learning disability. In
*Why is math so hard for some children*(pp. 329–348). Baltimore: Paul H Brooks. Retrieved from http://scholar.google.com/scholar?hl=en&btnG=Search&q=intitle:Is+math+anxiety+a+mathematical+learning+disability?#0 - Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education.
*The Elementary School Journal,**90*(4), 449. doi: 10.1086/461626.CrossRefGoogle Scholar - Bandura, A. (1977). Self-efficacy: Toward a unifying theory of behavioral change.
*Psychological Review,**84*(2), 191–215. doi: 10.1037/0033-295X.84.2.191.CrossRefGoogle Scholar - Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement.
*Proceedings of the National Academy of Sciences of the United States of America,**107*(5), 1860–1863. doi: 10.1073/pnas.0910967107.CrossRefGoogle Scholar - Bill, V. L., Leer, M. N., LE Reams, & Resnick, L. B. (1992). From cupcakes to equations: The structure of discourse in a primary mathematics classroom.
*Verbum,**1*(2), 63–85.Google Scholar - Blackwell, L. S., Trzesniewski, K. H., & Dweck, C. S. (2007). Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention.
*Child Development,**78*(1), 246–263. doi: 10.1111/j.1467-8624.2007.00995.x.CrossRefGoogle Scholar - Blanton, M. L. (2002). Using an undergraduate geometry course to challenge prospective teachers’ notions of discourse.
*Journal of Mathematics Teacher Education,**5*(2), 117–152. doi: 10.1023/A:1015813514009.CrossRefGoogle Scholar - Blömeke, S., Suhl, U., Kaiser, G., & Döhrmann, M. (2012). Family background, entry selectivity and opportunities to learn: What matters in primary teacher education? An international comparison of fifteen countries.
*Teaching and Teacher Education,**28*(1), 44–55. doi: 10.1016/j.tate.2011.08.006.CrossRefGoogle Scholar - Boaler, J., & Greeno, J. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.),
*Multiple perspectives on mathematics education*(pp. 171–200). Westport, CT: Ablex.Google Scholar - Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside School.
*The Teachers College Record*,*110*(3), 608–645. Retrieved from http://www.tcrecord.org/Content.asp?contentid=14590 - Boston, M. D. (2012). Assessing instructional quality in mathematics.
*The Elementary School Journal,**113*(1), 76–104.CrossRefGoogle Scholar - Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers’ classrooms.
*Journal for Research in Mathematics Education*,*40*(2), 119–156. Retrieved from http://www.jstor.org/stable/40539329 - Boston, M. D., & Smith, M. S. (2011). A “task-centric approach” to professional development: Enhancing and sustaining mathematics teachers’ ability to implement cognitively challenging mathematical tasks.
*ZDM,**43*(6–7), 965–977. doi: 10.1007/s11858-011-0353-2.CrossRefGoogle Scholar - Brown, P., & Levinson, S. (1987).
*Politeness: Some universals in language*. Cambridge: Cambridge University Press. Retrieved from http://scholar.google.com/scholar?q=brown+levinson+politeness&btnG=&hl=iw&as_sdt=0,5#0 - Brown, T., & McNamara, O. (2011). Becoming a mathematics teacher, (2008), 31–48. doi: 10.1007/978-94-007-0554-8.
- Brown, T., McNamara, O., Hanley, U. N. A., & Jones, L. I. Z. (1999). Primary student teachers’ understanding of mathematics and its teaching.
*British Educational Research Journal,**25*(3), 299–322.CrossRefGoogle Scholar - Brownlee, J., Purdie, N., & Boulton-Lewis, G. (2001). Changing epistemological beliefs in prospective teacher education students.
*Teaching in Higher Education,**6*(2), 247–268.CrossRefGoogle Scholar - Bursal, M., & Paznokas, L. (2006). Mathematics anxiety and prospective elementary teachers’ confidence to teach mathematics and science.
*School Science and Mathematics,**104*(6), 173–180.CrossRefGoogle Scholar - Chapin, S. H., O’Connor, C., & Anderson, N. (2009).
*Classroom discussions: Using math talk to help students learn, grades K*-*6*.*2009*. Math Solutions. Retrieved from http://books.google.com/books?hl=iw&lr=&id=2NX4I6mekq8C&oi=fnd&pg=PR15&dq=sarah+michaels+catherine+o’connor+nancy&ots=4W3FYxrjB4&sig=ubfPNY0skac15MuPWyc2YwHQqkU - Chazan, D., & Ball, D. (1999). Beyond being told not to tell.
*For the Learning of Mathematics,**19*(2), 2–10. doi: 10.2307/40248293.Google Scholar - Chi, M. T. H. (1997). Quantifying qualitative analyses of verbal data: A practical guide.
*The Journal of the Learning Sciences*,*6*(3), 271–315. Retrieved from http://www.tandfonline.com/doi/abs/10.1207/s15327809jls0603_1 - Clarke, S. N. (2015). The right to speak. In L. B. Resnick, C. A. Asterhan, & S. N. Clarke (Eds.),
*Socializing intelligence through academic talk and dialogue*. Washinton, D.C.: American Educational Research Association.Google Scholar - Cobb, P., Yackel, E., & Wood, T. (2011). Young children’s emotional acts while engaged in mathematical problem solving. In E. Yackel, K. Gravermeijer, & A. Sfard (Eds.),
*A journey in mathematics education research: Insights from the work of Paul Cobb*(pp. 117–148). Dordrecht: Springer.Google Scholar - Dweck, C. (2000).
*Self-theories: Their role in motivation, personality, and development 2000*. Lillington, NC: Taylor & Francis.Google Scholar - Eggins, S., & Slade, D. (1997).
*Analysing casual conversation*. Cassel: London & New York.Google Scholar - Esmonde, I., Brodie, K., Dookie, L., & Takeuchi, M. (2009). Social identities and opportunities to learn: Student perspectives on group work in an urban mathematics classroom.
*Journal of Urban Mathematics Education*,*2*(2), 18–45. Retrieved from https://tspace.library.utoronto.ca/handle/1807/32194 - Even, R., & Ball, D. L. (Eds.). (2009).
*The Professional education and development of teachers of mathematics*—*The 15th ICMI study*(Vol. 11). Springer. doi: 10.1007/978-0-387-09601-8 - Frykholm, J. A. (1996). Prospective teachers in mathematics: Struggling with the standards.
*Teaching and Teacher Education,**12*(6), 665–681. doi: 10.1016/S0742-051X(96)00010-8.CrossRefGoogle Scholar - Frykholm, J. A. (1999). The impact of reform: Challenges for mathematics teacher preparation.
*Journal of Mathematics Teacher Education,**2*, 79–105. doi: 10.1023/A:1009904604728.CrossRefGoogle Scholar - Gresalfi, M. (2009). Taking up opportunities to learn: Constructing dispositions in mathematics classrooms.
*The Journal of the Learning Sciences,**18*(3), 327–369.CrossRefGoogle Scholar - Gresham, G. (2007). A study of mathematics anxiety in prospective teachers.
*Early Childhood Education Journal,**35*(2), 181–188. doi: 10.1007/s10643-007-0174-7.CrossRefGoogle Scholar - Hannula, M. S. (2012). Exploring new dimensions of mathematics-related affect: Embodied and social theories.
*Research in Mathematics Education, 14*(2), 137–161.Google Scholar - Harper, N. W., & Daane, C. J. (1998). Causes and reduction of math anxiety in prospective elementary teachers.
*Action in Teacher Education,**19*(4), 29–38.CrossRefGoogle Scholar - Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety.
*Journal for Research in Mathematics Education*,*21*(1), 33–46. Retrieved from http://www.jstor.org/stable/10.2307/749455 - Henningsen, M., & Stein, M. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning.
*Journal for Research in Mathematics Education*,*28*(5), 524–549. Retrieved from http://www.jstor.org/stable/749690 - Herbel-Eisenmann, B., & Cirillo, M. (Eds.). (2009).
*Promoting purposeful discourse: Teacher research in mathematics classrooms*. Reston: NCTM.Google Scholar - Heyd-Metzuyanim, E. (2013). The co-construction of learning difficulties in mathematics—Teacher–student interactions and their role in the development of a disabled mathematical identity.
*Educational Studies in Mathematics,**83*(3), 341–368.CrossRefGoogle Scholar - Heyd-Metzuyanim, E. (2015). Vicious cycles of identifying and mathematizing—A case study of the development of mathematical failure.
*Journal of the Learning Sciences*. doi: 10.1080/10508406.2014.999270. - Heyd-Metzuyanim, E., & Sfard, A. (2012). Identity struggles in the mathematics classroom: On learning mathematics as an interplay of mathematizing and identifying.
*International Journal of Educational Research,**51–52*, 128–145.CrossRefGoogle Scholar - Horn, I. S. (2007). Fast kids, slow kids, lazy kids: Framing the mismatch problem in mathematics teachers’ conversations.
*Journal of the Learning Sciences,**16*(1), 37–79. doi: 10.1080/10508400709336942.Google Scholar - Huang, Y. (2007).
*Pragamatics*. Oxford: Oxford University Press.Google Scholar - Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community.
*Journal for Research in Mathematics Education,**35*(2), 81. doi: 10.2307/30034933.CrossRefGoogle Scholar - Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure.
*Journal of the Learning Sciences,**21*(1), 45–83.CrossRefGoogle Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics.
*Book Reviews,**34*, 6.Google Scholar - Lampert, M. (1990). When the problem is no the question and the solution is not the answer: Mathematical knowing and teaching.
*American Eductaional Research Journal,**27*, 29–63.CrossRefGoogle Scholar - Lave, J., & Wenger, E. (1991).
*Situated learning: Legitimate peripheral participation*. New York: Cambridge University Press.CrossRefGoogle Scholar - Leder, G., & Forgasz, H. J. (2006). Affect and mathematics education. In A. Gutierrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*(pp. 403–427). Rotterdam: Sense Publishers.Google Scholar - Lerman, S. (2000). The social turn in mathematics education. In J. Boaler (Ed.),
*Multiple perspective on mathematics education*(pp. 19–44). Westport, CT: Ablex.Google Scholar - Malinsky, M., Ross, A., Pannells, T., & McJunkin, M. (2006). Math anxiety in prospective elementary school teachers.
*Education*,*127*(2), 274–279. Retrieved from http://eric.ed.gov/ERICWebPortal/recordDetail?accno=EJ765826 - McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 575–596). New York: Macmillan Publishing Company.Google Scholar - Meece, J. L., Wigfield, A., & Eccles, J. S. (1990). Predictors of math anxiety and its influence on young adolescents’ course enrollment intentions and performance in mathematics.
*Journal of Educational Psychology*,*82*(1), 60–70. Retrieved from http://doi.apa.org/getdoi.cfm?doi=10.1037/0022-0663.82.1.60 - Nasir, N. S., & Hand, V. (2008). From the court to the classroom: Opportunities for engagement, learning, and identity in basketball and classroom mathematics.
*The Journal of the Learning Sciences,**17*, 143–179.CrossRefGoogle Scholar - Nathan, M. J., Eilam, B., & Kim, S. (2007). To disagree, we must also agree: How intersubjectivity structures and perpetuates discourse in a mathematics classroom.
*Journal of the Learning Sciences,**16*(4), 523–563. doi: 10.1080/10508400701525238.CrossRefGoogle Scholar - Oliveira, H., & Hannula, M. S. (2008). Individual prospective mathematics teachers—Studies on their professional growth. In
*the international handbook of mathematics teacher education*(Vol. 3, pp. 13–33). Retrieved from http://www.ie.ul.pt/pls/portal/docs/1/298339.PDF - Pekrun, R., Goetz, T., Titz, W., & Perry, R. (2002). Academic emotions in students’ self-regulated learning and achievement: A program of qualitative and quantitative research.
*Educational Psychologist,**37*(2), 91–105. doi: 10.1207/S15326985EP3702.CrossRefGoogle Scholar - Rowland, T., Turner, F., & Thwaites, A. (2014). Research into teacher knowledge: A stimulus for development in mathematics teacher education practice.
*ZDM,**46*(2), 317–328. doi: 10.1007/s11858-013-0564-9.CrossRefGoogle Scholar - Sfard, A. (2008).
*Thinking as communicating*. New York: Cambridge University Press.CrossRefGoogle Scholar - Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grown-ups cannot see as different?—Early numerical thinking revisited.
*Cognition and Instruction,**23*(2), 237–309.CrossRefGoogle Scholar - Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity.
*Educational Researcher,**34*(4), 14–22.CrossRefGoogle Scholar - Smith, M. S. (2001).
*Practice-based professional development for teachers of mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell.
*Mathematical Thinking and Learning*,*10*, 313–340. Retrieved from http://www.tandfonline.com/doi/abs/10.1080/10986060802229675 - Stein, M. K., Grover, B. R., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms.
*American Educational Research Journal*,*33*(2), 455–488. Retrieved from http://aer.sagepub.com/content/33/2/455.short - Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E. (2000).
*Implementing standards*-*based mathematics instruction: A casebook for professional development.*New York, NY: Teachers College Press. Retrieved from http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=ED442659 - Tatto, M. T., & Senk, S. (2011). The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics.
*Journal of Teacher Education,**62*(2), 121–137. doi: 10.1177/0022487110391807.CrossRefGoogle Scholar - Trask, R. (1999).
*Key concepts in language and linguistics*. London & New York: Routledge.Google Scholar - Trujillo, K., & Hadfield, O. (1999). Tracing the roots of mathematics anxiety through in-depth interviews with prospective elementary teachers.
*College Student Journal*,*33*(2). Retrieved from http://www.freepatentsonline.com/article/College-Student-Journal/62839422.html - Uusimaki, L., & Nason, R. (2004). Causes underlying prospective teachers’ negative beliefs and anxieties about mathematics. In
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 369–376). Retrieved from http://www.emis.ams.org/proceedings/PME28/RR/RR141_Uusimaki.pdf - Vacc, N. N., & Bright, G. W. (1999). Elementary prospective teachers’ changing beliefs and instructional use of children’s mathematical thinking.
*Journal for Research in Mathematics Education,**30*(1), 89–110. doi: 10.2307/749631.CrossRefGoogle Scholar - Vygotsky, L. S. (1978).
*Mind in society: The development of higher psychological processes*. Cambridge, MA: Harvard University Press.Google Scholar - Wagner, D. (2007). Students’ critical awareness of voice and agency in mathematics classroom discourse.
*Mathematical Thinking and Learning*,*9*(1), 31–50. Retrieved from http://www.tandfonline.com/doi/abs/10.1080/10986060709336604 - Walshaw, M. (2004). Prospective mathematics teaching in the context of schools: An exploration into the constitution of identity.
*Journal of Mathematics Teacher Education,**7*(1), 63–86. doi: 10.1023/B:JMTE.0000009972.30248.9c.CrossRefGoogle Scholar - Watts, R. J. (2003).
*Politeness*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Weinberger, A., & Fischer, F. (2006). A framework to analyze argumentative knowledge construction in computer-supported collaborative learning.
*Computers and Education*,*46*(1), 71–95. Retrieved from http://linkinghub.elsevier.com/retrieve/pii/S0360131505000564 - Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics.
*Journal for Research in Mathematics Education*,*22*(5), 390–408. Retrieved from http://www.jstor.org/stable/749187