# Teachers’ reflection on PISA items and why they are so hard for students in Serbia

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## Abstract

The study explores how teachers perceive and go about students’ thinking in connection to particular mathematical content and how they frame the notion of applied mathematics in their own classrooms. Teachers’ narratives are built around two released PISA 2012 mathematics items, the ‘Drip rate’ and ‘Climbing Mount Fuji’ (will be referred to as the *Fuji* item). Teachers show concordance as to the reasons that could make either of the items difficult for students and are able to provide more examples justifying their reasoning for the ‘Fuji’ item. Suggestions linked to making the items more familiar to the students mostly relate to de-contextualization of the items’ content towards a more formal mathematical record. The teachers agree that students need only basic mathematical knowledge, at a level learned during elementary school, in order to solve these problems. Yet, at the same time, many teachers have difficulty clearly verbalising which procedures students are expected to follow to be able to solve the tasks. Disagreement among the teachers is noticeable when labelling the most difficult part(s) of each of the selected items. Mathematics teachers show openness for learning on how to create math problems we examined in this study, but question the purpose and meaning in incorporating more such problems in their own teaching.

### Keywords

Teachers’ narratives Mathematics Mathematics instruction PISA## Notes

### References

- Anić, I., & Pavlović Babić, D. (2015). How we can support success in solving mathematical problems?
*Teaching Innovations, 28*(3), 36–49.Google Scholar - Ball, D. L. (1997). What do students know? Facing challenges of distance, context, and desire in trying to hear children. In B. Biddle, T. Good, & I. Goodson (Eds.),
*International handbook on teachers and teaching*(Vol. II, pp. 679–718). Dordrecht: Kluwer Press.Google Scholar - Ball, D. L. (2001).
*Teaching, with respect to mathematics and students. Beyond classical pedagogy: Teaching elementary school mathematics*(pp. 11–22). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide?
*American Educator, 30*(3), 14–17 20–22, 43–46.Google Scholar - Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what makes it special?
*Journal of Teacher Education, 59*(5), 389–407.CrossRefGoogle Scholar - Baucal, A., & Pavlović Babić, D. (2010).
*PISA 2009 u Srbiji: prvi rezultati. Nauči me da mislim, nauči me da učim.*Institut za psihologiju Filozofskog fakulteta u Beogradu i Centar za primenjenu psihologiju.Google Scholar - Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., et al. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress.
*American Educational Research Journal, 47*, 133–180.CrossRefGoogle Scholar - Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2003).
*Assessment for learning: Putting it into practice*. Maidenhead: McGraw-Hill International.Google Scholar - Blömeke, S., Gustafsson, J. E., & Shavelson, R. (2015). Beyond dichotomies: Viewing competence as a continuum.
*Zeitschrift für Psychologie, 223*(1), 3–13.CrossRefGoogle Scholar - Borg, M. (2001). Key concepts: teachers’ beliefs.
*ELT Journal, 55*(2), 186–188.CrossRefGoogle Scholar - Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology.
*Qualitative Research in Psychology, 3*, 77–101.CrossRefGoogle Scholar - Cai, J., & Ding, M. (2017). On mathematical understanding: perspectives of experienced Chinese mathematics teachers.
*Journal of Mathematics Teacher Education, 20*, 5–29.CrossRefGoogle Scholar - Clarke, B. (2008). A framework of growth points as a powerful teacher development tool. In D. Tirosh & T. Wood (Eds.),
*International handbook of mathematics teacher education: tools and processes in mathematics teacher education*(Vol. 2, pp. 235–256). Rotterdam: Sense Publishers.Google Scholar - Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education.
*Mathematical Thinking and Learning, 6*(2), 81–89.CrossRefGoogle Scholar - Cobb, P., Wood, T., Yackel, E., & McNeal, E. (1993). Mathematics as procedural instructions and mathematics as meaningful activity: the reality of teaching for understanding. In R. Davis & C. Maher (Eds.),
*Schools, mathematics and the world of reality*(pp. 119–134). Needham Heights: Allyn & Bacon.Google Scholar - Crespo, S. (2000). Seeing more than right and wrong answers: prospective teachers’ interpretations of students’ mathematical work.
*Journal of Mathematics Teacher Education, 3*(2), 155–181.CrossRefGoogle Scholar - Didis, M. G., Erbas, A. K., Cetinkaya, B., Cakiroglu, E., & Alacaci, C. (2016). Exploring prospective secondary mathematics teachers’ interpretation of student thinking through analyzing students’ work in modelling.
*Mathematics Education Research Journal, 28*, 349–378.CrossRefGoogle Scholar - Doabler, C. T., Baker, S. K., Kosty, D. B., Smolkowski, K., Clarke, B., Miller, S. J., & Fien, H. (2015). Examining the association between explicit mathematics instruction and student mathematics achievement.
*Elementary School Journal, 115*(3), 303–333.CrossRefGoogle Scholar - Doyle, W. (1983). Academic work.
*Review of Educational Research, 53*, 159–199.CrossRefGoogle Scholar - Dyer, E. B., & Sherin, M. G. (2016). Instructional reasoning about interpretations of student thinking that supports responsive teaching in secondary mathematics.
*ZDM Mathematics Education, 48*, 69–82.CrossRefGoogle Scholar - Ellis, M. V., & Berry, R. Q. (2005). The paradigm shift in mathematics education: explanations and implications of reforming conceptions of teaching and learning.
*Mathematics Educator, 15*(1), 7–17.Google Scholar - Empson, S. B., & Jacobs, V. R. (2008). Learning to listen to children’s mathematics. In D. Tirosh & T. Wood (Eds.),
*The international handbook of mathematics teacher education: tools and processes in mathematics teacher education*(Vol. 2, pp. 257–281). Rotterdam: Sense Publishers.Google Scholar - Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 147–164.) New York.Google Scholar - Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction.
*Journal for Research in Mathematics Education, 27*(4), 403–434.CrossRefGoogle Scholar - Fives, H., & Buehl, M. M. (2012). Spring cleaning for the “messy” construct of teachers’ beliefs: what are they? Which have been examined? What can they tell us? In K. R. Harris, S. Graham, & T. Urdan (Eds.),
*Individual differences and cultural and contextual factors*(pp. 471–499). Washington, DC: APA.Google Scholar - Fives, H., & Gregoire-Gill, M. (2014).
*International handbook of research on teachers’ beliefs*. New York: Routledge.Google Scholar - Furtak, E. M., Kiemer, K., Kizil Circi, R., Swanson, R., de Leon, V., Morrison, D., & Heredia, S. H. (2016). Teachers’ formative assessment abilities and their relationship to student learning: findings from a four-year intervention study.
*Instructional Science, 44*, 267–291.CrossRefGoogle Scholar - Hattie, J. (2008).
*Visible learning: a synthesis of over 800 meta-analyses relating to achievement*. London: Routledge.Google Scholar - Hattikudur, S., Sidney, P. G., & Alibali, M. W. (2016). Does comparing informal and formal procedures promote mathematics learning? The benefits of bridging depend on attitudes toward mathematics.
*Journal of Problem Solving, 9*, 13–27.CrossRefGoogle Scholar - Hill, H. C., Blunck, M. L., Charalambos, Y. C., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: an exploratory study.
*Cognition and Instruction, 26*(4), 430–511.CrossRefGoogle Scholar - Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking.
*Journal for Research in Mathematics Education, 41*(2), 169–202.Google Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001).
*Adding it up: helping children learn mathematics*. Washington, D.C.: National Academy Press.Google Scholar - König, J., Blömeke, S., Klein, P., Suhl, U., Busse, A., & Kaiser, G. (2014). Is teachers’ general pedagogical knowledge a premise for noticing and interpreting classroom situations? A video-based assessment approach.
*Teaching and Teacher Education., 38*, 76–88.CrossRefGoogle Scholar - Krstić, K. (2015). Attachment in the student-teacher relationship as a factor of school achievement.
*Teaching Innovation, 28*(3), 167–188.Google Scholar - Leinhardt, G. (1986). Expertise in mathematics teaching.
*Educational Leadership, 43*, 28–33.Google Scholar - Li, Y., & Kaiser, G. (2011). Expertise in mathematics instruction: advancing research and practice from an international perspective. In Y. Li & G. Kaiser (Eds.),
*Expertise in mathematics instruction, an international perspective*(pp. 3–16). New York: Springer.CrossRefGoogle Scholar - Merriam, S. R. (1998).
*Qualitative research and case study applications in education*. San Francisco: Jossey Bass.Google Scholar - Mincu, M. E. (2009). Myth, rhetoric and ideology in eastern European education.
*European Education, 41*(1), 55–78.CrossRefGoogle Scholar - Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016).
*TIMSS 2015 international results in mathematics.*Retrieved from Boston College, TIMSS & PIRLS International Study Center.Google Scholar - OECD. (2013).
*PISA 2012 assessment and analytical framework: mathematics, reading, science, problem solving and financial literacy*. Paris: OECD Publishing.CrossRefGoogle Scholar - OECD. (2016).
*PISA 2015 results (volume I): excellence and equity in education*. Paris: OECD Publishing.Google Scholar - Palm, T. (2008). Impact of authenticity on sense making in word problem solving.
*Educational Studies in Mathematics, 67*(1), 37–58.CrossRefGoogle Scholar - Pavlović Babić, D., & Baucal, A. (2013). Inspiriši me, podrži me, PISA 2012 u Srbiji: prvi rezultati. Beograd: Institut za psihologiju.Google Scholar
- Pepin, B. E. U., Xu, B., Trouche, L., & Wang, C. (2016). Developing a deeper understanding of mathematics teaching expertise: an examination of three Chinese mathematics teachers’ resource systems as windows into their work and expertise.
*Educational Studies in Mathematics, 94*(3), 257–274.CrossRefGoogle Scholar - Pešić, J., & Stepanović, I. (2004). Škola kao sredina za učenje-učenička percepcija i njihove strategije [School as a learning space—Students’ perceptions and strategies]. In D. Plut & Z. Krnjaić (Eds.),
*Društvena kriza I obrazovanje – dokument o jednom vremenu [A social crises and education – documenting an era]*(pp. 24–69). Beograd: Institut za psihologiju.Google Scholar - Pierson, J. L. (2008).
*The relationship between patterns of classroom discourse and mathematics learning (unpublished dissertation)*. Austin: University of Texas at Austin retrieved at: https://repositories.lib.utexas.edu/bitstream/handle/2152/17898/piersonj.pdf?sequence=2&isAllowed=y Google Scholar - Radišić, J., & Baucal, A. (2015). Portret nastavnika matematike u srednjoj školi: kritička analiza dominantne prakse. Primenjena psihologija. (Portrait of high school math teachers: Critical analysis of dominant practice).
*Primenjena psihologija, 8*(1), 25-46.Google Scholar - Radišić, J., & Baucal, A. (2016). "What about when the majority is excluded?": A Critical Eye on Language and Math Classrooms in Serbia. In Surian, A. (Ed.),
*Open Spaces for Interactions and Learning Diversities*(pp. 167-178) Sense Publishers.Google Scholar - Radišić, J., Baucal, A., & Videnović, M. (2014). Unfolding the assessment process in a whole class mathematics setting.
*Psihološka istraživanja, 17*(2), 137-158.Google Scholar - Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics.
*Educational Psychologist, 47*, 189–203.CrossRefGoogle Scholar - Rittle-Johnson, B. (2017). Developing mathematics knowledge.
*Child Development Perspectives, 0*(0), 1–7.Google Scholar - Romberg, T. A. (1992). Perspectives on scholarship and research methods. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 49–64). New York: Macmillan.Google Scholar - Säljö, R. (2009). Learning, theories of learning, and units of analysis in research.
*Educational Psychologist, 44*(3), 202–208.CrossRefGoogle Scholar - Santagata, R., & Yeh, C. (2016). The role of perception, interpretation, and decision making in the development of beginning teachers’ competence.
*ZDM Mathematics Education, 48*, 153–165.CrossRefGoogle Scholar - Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge.
*Developmental Psychology, 47*, 1525–1538.CrossRefGoogle Scholar - Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 334–371). New York: Macmillan.Google Scholar - Schoenfeld, A. H. (2014).
*Mathematical problem solving*. London: Academic Press.Google Scholar - Schoenfeld, A. H., & Kilpatrick, J. (2008). Toward a theory of proficiency in teaching mathematics. In T. Wood & D. Tirosh (Eds.),
*International handbook of mathematics teacher education: Vol. 2. Tools and processes in mathematics teacher education*(pp. 321–354). Rotterdam: Sense Publishers.Google Scholar - Sherin, M. G., & van Es, E. A. (2009). Effects of video club participation on teachers’ professional vision.
*Journal of Teacher Education, 60*, 20–37.CrossRefGoogle Scholar - Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (Eds.). (2011).
*Mathematics teacher noticing: seeing through teachers’ eyes*. New York: Routledge.Google Scholar - Shulman, L. (1987). Knowledge and teaching: foundations of the new reform.
*Harvard Educational Review, 57*, 1–22.CrossRefGoogle Scholar - Smith, M. S. (2001).
*Practice-based professional development for teachers of mathematics*. Reston: National Council of Teachers of Mathematics.Google Scholar - Stein, M. K., Grover, B. W., & 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.CrossRefGoogle Scholar - Teuscher, D., Moore, K. C., & Carlson, M. P. (2016). Decentering: a construct to analyze and explain teacher actions as they relate to student thinking.
*Journal of Mathematics Teacher Education, 19*, 433–456.CrossRefGoogle Scholar - Verschaffel, V., Greer, B., & de Corte, E. (Eds.). (2000).
*Making sense of word problems*. Heereweg: Swets & Zeitlinger.Google Scholar - Verschaffel, L., Depaepe, F., & Van Dooren, W. (2015).
*Individual differences in word problem solving, The Oxford Handbook of Numerical Cognition*(pp. 1–17). Oxford: Oxford University Press.Google Scholar - Wijaya, A., van den Heuvel-Panhuizen, M., Doorman, M., & Robitzsch, A. (2014). Difficulties in solving context-based PISA mathematics tasks: an analysis of students’ errors.
*The Mathematics Enthusiast, 11*(3), 555–584.Google Scholar - Wyndhamn, J., & Säljö, R. (1997). Word problems and mathematical reasoning—a study of children’s mastery of reference and meaning in textual realities.
*Learning and Instruction, 7*(4), 361–382.CrossRefGoogle Scholar