# Exploring teachers’ attention to students’ responses in pattern generalization tasks

## Abstract

This paper explored the following: (1) teachers’ ways of attending to students’ written responses in pattern generalization tasks and (2) differences in the ways of attending to students’ responses associated with different factors. A questionnaire was developed to classify teachers’ ways of attending to students’ written responses. The questionnaire was given to a sample of 91 practicing mathematics teachers from different grade levels (grades 4–12). The structure of attention referring to “holding wholes, discerning details, recognizing relationships, perceiving properties, and reasoning on the basis of agreed properties” was used as a theoretical model. A mixed quantitative–qualitative approach was adopted to analyze the data. Analysis of data showed that teachers’ attention to students’ written responses seems to focus on distinguishing details of students’ responses in particular step(s) of the pattern without demonstrating awareness of relationships that could hold in different steps of the pattern. The analysis also showed that differences in the ways of attending to students’ written responses were associated with different factors. In particular, pattern generalization type (near and far generalization tasks), student strategy use (e.g., functional and recursive strategies), student reasoning approach (numerical and figural reasoning approaches), and teaching experience in particular school cycle were associated with significant differences in teachers’ forms of attention to students’ responses in pattern generalization. In contrast, the number of years of teaching was not associated with significant differences in teachers’ forms of attention to students’ responses.

## Keywords

Forms of attention Practicing teachers Student written work Pattern generalization## References

- Ainley, J., & Luntley, M. (2007). The role of attention in expert classroom practice.
*Journal of Mathematics Teacher Education,**10*(1), 3–22.CrossRefGoogle Scholar - Ball, D. L. (2001). Teaching, with respect to mathematics and students. In T. Wood, B. S. Nelson, & J. Warfield (Eds.),
*Beyond classical pedagogy: Teaching elementary school mathematics*(pp. 11–22). Mahwah, NJ: Lawrence Erlbaum Associates Inc.Google Scholar - BouJaoude, S., & El-Mouhayar, R. (2010). Teacher education in Lebanon: Trends and issues.
*International Handbook of Teacher Education World-wide, 2*, 309–332.Google Scholar - Callejo, M. L., & Zapatera, A. (2017). Prospective primary teachers’ noticing of students’ understanding of pattern generalization.
*Journal of Mathematics Teacher Education, 20*(4), 309–333.CrossRefGoogle Scholar - Center for Educational Research and Development (CERD). (1997).
*Manahij Ata’alim Ala’am wa Ahdafuha [Curricula and objectives of general education]*. Beirut: CERD.Google Scholar - Chamberlin, M. T. (2005). Teachers’ discussions of students’ thinking: Meeting the challenge of attending to students’ thinking.
*Journal of Mathematics Teacher Education,**8*(2), 141–170.CrossRefGoogle Scholar - Chua, B. L., & Hoyles, C. (2010). Generalisation and perceptual agility: How did teachers fare in a quadratic generalising problem?
*Research in Mathematics Education,**12*(1), 71–72.CrossRefGoogle Scholar - Cobb, P. (1988). The tension between theories of learning and instruction in mathematics education.
*Educational Psychologist,**23,*87–103.CrossRefGoogle Scholar - Crespo, S. (2000). Seeing more than right and wrong answers: Prospective teachers’ interpretations of students’ mathematical work.
*Journal of Mathematics Teacher Education,**3,*155–181.CrossRefGoogle Scholar - Dörfler, W. (2008). En route from patterns to algebra: Comments and reflections.
*ZDM Mathematics Education,**40*(1), 143–160.CrossRefGoogle Scholar - Dreher, A., & Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom.
*Educational Studies in Mathematics,**88*(1), 89–114.CrossRefGoogle Scholar - El-Mouhayar, R., & BouJaoude, S. (2012). Structural and conceptual foundations of teacher education programs in selected universities in Lebanon.
*Recherches Pédagogique : Revue éditée par la Faculté de Pédagogie de l’Université Libanaise, Beyrouth, 22*, 37–60.Google Scholar - El Mouhayar, R. R., & Jurdak, M. E. (2013). Teachers’ ability to identify and explain students’ actions in near and far figural pattern generalization tasks.
*Educational Studies in Mathematics, 82*(3), 379–396.CrossRefGoogle Scholar - El Mouhayar, R., & Jurdak, M. (2015a). Variation in strategy use across grade level by pattern generalization types.
*International Journal of Mathematical Education in Science and Technology, 46*(4), 553–569.CrossRefGoogle Scholar - El Mouhayar, R., & Jurdak, M. (2015b). Teachers’ perspectives used to explain students’ responses in pattern generalization. In K. Beswick, T. Muir, & J. Wells (Eds.),
*Proceedings of the 39th conference of the international group for the psychology of mathematics education*(Vol. 2, pp. 265–272). Hobart, Australia: PME.Google Scholar - El Mouhayar, R., & Jurdak, M. (2016). Variation of student numerical and figural reasoning approaches by pattern generalization type, strategy use and grade level.
*International Journal of Mathematical Education in Science and Technology, 47*(2), 197–215.CrossRefGoogle Scholar - Erickson, F. (2011). On noticing teacher noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 17–34). New York: Routledge.Google Scholar
- Fernández, C., Llinares, S., & Valls, J. (2012). Learning to notice students’ mathematical thinking through on-line discussions.
*ZDM Mathematics Education,**44*(6), 747–759.CrossRefGoogle Scholar - Glaser, B., & Strauss, A. (1967).
*The discovery of grounded theory: Strategies for qualitative research*. Chicago: Adeline. Google Scholar - Jacobs, V. R., Lamb, 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 - Jacobs, V. R., Lamb, L. L., Philipp, R. A., & Schappelle, B. P. (2011). Deciding how to respond on the basis of children’s understandings. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: Seeing through teachers’ eyes*(pp. 97–116). New York: Routledge.Google Scholar - Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.),
*Mathematics classrooms that promote understanding*(pp. 133–155). Mahwah, NJ: Erlbaum.Google Scholar - Kieran, C. (1996). The changing face of school algebra. In C. Alsina, J. Alvarez, B. Hodgson, C. Laborde, & A. Pérez (Eds.),
*8th international congress on mathematical education: Selected lectures*(pp. 271–290). Sevilla: S.A.E.M. Thales.Google Scholar - Kieran, C. (2004). Algebraic Thinking in the Early Grades: What Is It?
*The Mathematics Educator,**8*(1), 139–151.Google Scholar - Küchemann, D., & Hoyles, C. (2001). Investigating factors that influence students’ mathematical reasoning. In M. van den Heuvel-Panhuizen (Ed.),
*Proceedings of the 25th conference of the international group for the psychology of mathematics education*(Vol. 3, pp. 85–92). Utrecht: Utrecht University.Google Scholar - Küchemann, D., & Hoyles, C. (2005). Pupils’ awareness of structure on two number/algebra questions. In M. Bosch (Ed.),
*Proceedings of the 4th congress of the European society for research in mathematics education (CERME 4)*(pp. 438–447). Sant Feliu de Guixols: CERME.Google Scholar - Lannin, J., Barker, D., & Townsend, B. (2006). Algebraic generalization strategies: factors influencing student strategy selection.
*Mathematics Education Research Journal,**18*(3), 3–28.CrossRefGoogle Scholar - Maher, C. A., & Davis, R. A. (1990). Teacher’s learning: Building representations of children’s meanings.
*Journal for Research in Mathematics Education*(Monograph No. 4), 79–90.Google Scholar - Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention.
*For the Learning of Mathematics, 9*(2), 2–8.Google Scholar - Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.),
*Approaches to algebra: Perspectives for research and teaching*(pp. 65–86). Dordrecht: Kluwer Academic.CrossRefGoogle Scholar - Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention.
*Journal of Mathematics Teacher Education,**1,*243–267.CrossRefGoogle Scholar - Mason, J. (2002).
*Researching your own practice: The discipline of noticing*. London: Routledge.Google Scholar - Mason, J. (2008). Being mathematical with and in front of learners.
*The mathematics teacher educator as a developing professional*, 31-55.Google Scholar - Mason, J. (2011). Noticing: Roots and branches. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: Seeing through teachers’ eyes*(pp. 65–80). New York: Routledge.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: NCTM.Google Scholar - Radford, L. (2008). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts.
*ZDM Mathematics Education,**40*(1), 83–96.CrossRefGoogle Scholar - Rivera, F. D. (2010). Visual templates in pattern generalization activity.
*Educational Studies in Mathematics,**73*(3), 297–328.CrossRefGoogle Scholar - Rivera, F., & Becker, J. R. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra.
*The Journal of Mathematical Behavior,**26,*140–155.CrossRefGoogle Scholar - Rivera, F. D., & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns.
*ZDM Mathematics Education,**40*(1), 65–82.CrossRefGoogle Scholar - Schifter, D. (2001). Learning to see the invisible: What skills and knowledge are needed to engage with students’ mathematical ideas? In T. Wood, B. S. Nelson, & J. Warfield (Eds.),
*Beyond classical pedagogy: Teaching elementary school mathematics*(pp. 109–134). Mahwah, NJ: Lawrence Erlbaum Associates Inc.Google Scholar - Schoenfeld, A. H. (2011). Noticing matters. A lot. Now what? In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: Seeing through teachers’ eyes*(pp. 223–238). New York: Routledge.Google Scholar - Sherin, M. G., & Han, S. Y. (2004). Teacher learning in the context of a video club.
*Teaching and teacher Education,**20*(2), 163–183.CrossRefGoogle Scholar - Sherin, M. G., Jacobs, V., & Philipp, R. (2011).
*Mathematics teacher noticing: Seeing through teachers’ eyes*. New York and London: Routledge.Google Scholar - Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.
*Educational Researcher,**15*(2), 4–14.CrossRefGoogle Scholar - Simons, D., & Chabris, C. (1999).
*Selective attention test*. Visual Cognition Lab: University of Illinois.Google Scholar - Stacey, K. (1989). Finding and using patterns in linear generalizing problems.
*Educational Studies in Mathematics,**20*(2), 147–164.CrossRefGoogle Scholar - Star, J., & Strickland, S. (2008). Learning to observe: Using video to improve preservice mathematics teachers’ ability to notice.
*Journal of Mathematics Teacher Education,**11,*107–125.CrossRefGoogle Scholar - Stephens, A. (2008). What ‘‘counts’’ as algebra in the eyes of preservice elementary teachers?
*Journal of Mathematical Behavior,**27,*33–47.CrossRefGoogle Scholar - Strauss, A., & Corbin, J. (1998).
*Basics of qualitative research techniques*. Thousands Oaks: Sage publications.Google Scholar - Van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions.
*Journal of Technology and Teacher Education,**10*(4), 571–595.Google Scholar - Van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club.
*Teaching and Teacher Education,**24*(2), 244–276.CrossRefGoogle Scholar - Walkoe, J. (2010). Seeing algebraic thinking in the classroom: A study of teachers’ conceptualizations of algebra. In
*Proceedings of the 9th international conference of the learning sciences*(Vol. 1, pp. 1055–1062). International Society of the Learning Sciences.Google Scholar - Walkoe, J. (2015). Exploring teacher noticing of student algebraic thinking in a video club.
*Journal of Mathematics Teacher Education,**18*(6), 523–550.CrossRefGoogle Scholar - Wallach, T., & Even, R. (2005). Hearing students: The complexity of understanding what they are saying, showing, and doing.
*Journal of Mathematics Teacher Education,**8*(5), 393–417.CrossRefGoogle Scholar - Wickens, C. D., & Alexander, A. L. (2009). Attentional tunneling and task management in synthetic vision displays.
*The International Journal of Aviation Psychology,**19*(2), 182–199.CrossRefGoogle Scholar - Williams, A. M., & Davids, K. (1998). Visual search strategy, selective attention, and expertise in soccer.
*Research Quarterly for Exercise and Sport,**69*(2), 111–128.CrossRefGoogle Scholar - Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation.
*Educational studies in mathematics,**49*(3), 379–402.CrossRefGoogle Scholar