# Patterns of non-verbal social interactions within intensive mathematics intervention contexts

## Abstract

This study examined the non-verbal patterns of interaction within an intensive mathematics intervention context. Specifically, the authors draw on social constructivist worldview to examine a teacher’s use of gesture in this setting. The teacher conducted a series of longitudinal teaching experiments with a small number of young, school-age children in the context of early arithmetic development. From these experiments, the authors gathered extensive video records of teaching practice and, from an inductive analysis of these records, identified three distinct patterns of teacher gesture: behavior eliciting, behavior suggesting, and behavior replicating. Awareness of their potential to influence students via gesture may prompt teachers to more closely attend to their own interactions with mathematical tools and take these teacher interactions into consideration when forming interpretations of students’ cognition.

### Keywords

Gesture Numeracy Constructivism Teaching experiment Models### References

- Alibali, M. W., & Nathan, M. J. (2012). Embodiment in mathematics teaching and learning: evidence from learners’ and teachers’ gestures.
*The Journal of the Learning Sciences, 21*, 247–286.CrossRefGoogle Scholar - Alibali, M. W., Nathan, M. J., Church, R. B., Wolfgram, M. S., Kim, S., & Knuth, E. J. (2013). Teachers’ gestures and speech in mathematics lessons: forging common ground by resolving trouble spots.
*ZDM, 45*, 425–440.CrossRefGoogle Scholar - American Psychiatric Association (APA). (2013).
*Diagnostic and statistical manual of mental disorders*(5th ed.). Washington: American Psychiatric Association.Google Scholar - Aubrey, C. (1993). An investigation of the mathematical knowledge and competencies which young children bring into school.
*British Educational Research Journal, 19*, 27--41.CrossRefGoogle Scholar - Bauersfeld, H., Krummheuer,G., & Voigt, J. (1988). Interactional theory of learning and teaching mathematics and related microethnographical studies. In H. G. Steiner & A. Vermandel (Eds.), Foundations and methodology of the discipline of mathematics education (pp. 174--188). Antwerp: Proceedings of the Theory of Mathematics Education Conference.Google Scholar
- Bauersfeld, H. (1994). Theoretical Perspectives on Interaction in the Mathematics Classroom, in Biehler R. et al. (Eds.) The Didactics of mathematics as a Scientific Discipline, Kluwer, Dordrecht.Google Scholar
- Bryant, D. P., Bryant, B. R., Gersten, R., Scammacca, N., & Chavez, M. M. (2008). Mathematics intervention for first- and second-grade students with mathematics difficulties: the effects of tier 2 intervention delivered as booster lessons.
*Remedial and Special Education, 29*, 20–32.CrossRefGoogle Scholar - Carson, P. M., & Eckert, T. L. (2003). An experimental analysis of mathematics instructional components: examining the effects of student-selected versus empirically-selected interventions.
*Journal of Behavioral Education, 12*, 35–54.CrossRefGoogle Scholar - Chidsey, R. B., & Steege, M. W. (2010).
*Response to intervention: principles and strategies for effective practice*(2nd ed.). New York: Guilford.Google Scholar - Choy, B. H. (2013). Productive mathematical noticing: what it is, and why it matters. In V. Steinle, L. Ball, & C. Bardini (Eds.),
*Mathematics education: yesterday, today, and tomorrow*(pp. 186–193). Melbourne: MERGA. Proceedings of the 36th Annual Conference of the Mathematics Education Research Group of Australasia.Google Scholar - Clarke, B., Clarke, D., & Cheeseman, J. (2006). The mathematical knowledge and understanding that young children bring to school.
*Mathematics Education Research Journal, 18*, 78–102.Google Scholar - Clements, D. H., & Sarama, J. (2009).
*Learning and teaching early mathematics: the learning trajectories approach*. New York: Routledge.Google Scholar - Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research.
*Educational Psychologist, 31*, 175--190.Google Scholar - Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education. Dordrecht, The Netherlands: Kluwer.Google Scholar
- Cockroft, W. H. (1982).
*Mathematics counts: report of the committee of inquiry into the teaching of mathematics in schools*. London: HMSO.Google Scholar - Cook, S. W., Duffy, R. G., & Fenn, K. M. (2013). Consolidation and transfer of learning after observing hand gesture.
*Child Development, 84*, 1863–1871.CrossRefGoogle Scholar - Elia, I., Gagatsis, A., & van de Hueval-Panhuizen, M. (2014). The role of gestures in making connections between space and shape aspects and their verbal representations in the early years: findings from a case study.
*Mathematics Education Research Journal*. doi: 10.1007/s13394-013-0104-5. online first. - Ellemore-Collins, D. L., & Wright, R. J. (2008). Assessing student thinking about arithmetic: videotaped interviews.
*Teaching Children Mathematics, 15*, 106–111.Google Scholar - Erickson, F. (2006). Definition and analysis of data from videotape: Some research procedures and their rationales. In J. L. Green, G. Camilli, P. B. Elmore, A. Skukauskaite, & E. Grace (Eds.), Handbook of Complementary Methods in Education Research. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
- Flevares, L. M., & Perry, M. (2001). How many do you see? The use of nonspoken representations in first-grade mathematics lessons.
*Journal of Educational Psychology, 93*, 330–345.CrossRefGoogle Scholar - Fosnot, C., & Dolk, M. (2001).
*Young mathematicians at work: constructing number sense, addition and subtraction*. Portsmouth: Heinemann.Google Scholar - Fuchs, L. S., & Fuchs, D. (2007). A model for implementing responsiveness to intervention.
*Teaching Exceptional Children, 39*, 14–20.CrossRefGoogle Scholar - Fuchs, L. S., Fuchs, D., & Hollenbeck, K. N. (2007). Extending responsiveness to intervention to mathematics at first and third grades.
*Learning Disabilities Research and Practice, 22*, 13–24.CrossRefGoogle Scholar - Fuchs, L. S., Fuchs, D., Powell, S. R., Seethaler, P. M., Cirino, P. T., & Fletcher, J. M. (2008a). Intensive intervention for students with mathematics disabilities: seven principles of effective practice.
*Learning Disability Quarterly, 31*, 79–92.Google Scholar - Fuchs, L. S., Seethaler, P. M., Powell, S. R., Fuchs, D., Hamlett, C. L., & Fletcher, J. M. (2008b). Effects of preventative tutoring on the mathematical problem solving of third-grade students with math and reading difficulties.
*Exceptional Children, 74*, 155–173.CrossRefGoogle Scholar - Geary, D. C. (1990). A componential analysis of an early learning deficit in mathematics.
*Journal of Experimental Child Psychology, 33*, 386–404.Google Scholar - Geary, D. C. (1993). Mathematical disabilities: cognitive, neuropsychological, and genetic components.
*Psychological Bulletin, 114*, 345–362.CrossRefGoogle Scholar - Geary, D. C., Hoard, M. K., & Hamson, C. O. (1999). Numerical and arithmetical cognition: patterns of functions and deficits in children at risk for a mathematical disability.
*Journal of Experimental Child Psychology, 74*, 213–239.CrossRefGoogle Scholar - Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). Numerical and arithmetical cognition: a longitudinal study of process and concept deficits in children with learning disability.
*Journal of Experimental Child Psychology, 77*, 236–263.CrossRefGoogle Scholar - Gelman, R., & Gallistel, C. R. (1978).
*The child’s understanding of number*. Cambridge: Harvard University Press.Google Scholar - Gersten, R., & Chard, D. (1999). Number sense: rethinking arithmetic instruction for students with mathematical disabilities.
*Journal of Special Education, 33*, 18–28.CrossRefGoogle Scholar - Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009).
*Assisting students struggling with mathematics: response to intervention (Rtl) for elementary and middle schools*. Washington: U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance.Google Scholar - Glaser, B. & Strauss, A. (1967). The Discovery of the Grounded Theory: Strategies for Qualitative Research. New York, NY: Aldine de Gruyter.Google Scholar
- Goldin-Meadow, S. (1999). The role of gesture in communication and thinking.
*Trends in Cognitive Science, 3*, 419–429.CrossRefGoogle Scholar - Herbert, S., & Pierce, R. (2007). Video evidence: what gestures tell us about students’ understanding of rates of change. In J. Watson & K. Beswick (Eds.), Mathematics: essential research, essential practice (pp. 362–371). Adelaide: MERGA. Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia.Google Scholar
- Hickok, G., Bellugi, U., & Klima, E. S. (1996). The neurobiology of sign language and its implications for the neural basis of language.
*Nature, 381*, 699–702.CrossRefGoogle Scholar - Jacobs, V. A., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking.
*Journal for Research in Mathematics Education, 41*, 169–202.Google Scholar - Kentucky Department of Education. (2010).
*Commonwealth of Kentucky: school report card.*Retrieved December 3, 2015 from: https://applications.education.ky.gov/SchoolReportCardArchive/ - Kroesbergen, E. H., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special needs: a meta-analysis.
*Remedial and Special Education, 24*, 97–114.CrossRefGoogle Scholar - Lewis, K.E. (2014). Difference not deficit: Reconceptualizing mathematical learning disabilities.
*Journal for Research in Mathematics Education, 45,*351--396.Google Scholar - Malafouris, L., & Renfrew, C. (Eds.). (2010).
*The cognitive life of things: recasting the boundaries of the mind*. Cambridge: McDonald Institute Monographs.Google Scholar - Mason, J. (2002).
*Researching your own practice: the discipline of noticing*. London: Routledge Falmer.Google Scholar - McNeill, D. (1992).
*Hand and mind*. Chicago: University of Chicago Press.Google Scholar - Philipp, R. A. (2014). Research on teachers’ focusing on children’s thinking in learning to teach: teacher noticing and learning trajectories. In J. J. Lo, K. R. Leatham, & L. R. Van Zoest (Eds.),
*Research trends in mathematics teacher education*. New York: Springer.Google Scholar - Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: a semiotic analysis.
*Educational Studies in Mathematics, 42*, 237–268.CrossRefGoogle Scholar - Radford, L. (2014). The progressive development of early-embodied algebraic thinking.
*Mathematics Education Research Journal, 26*, 257–277.CrossRefGoogle Scholar - Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the general.
*Journal for Research in Mathematics Education, 38*, 507–530.Google Scholar - Schack, E., Fisher, M., Thomas, J., Eisenhardt, S., Tassell, J., & Yoder, M. (2013). Preservice teachers professional noticing of children’s early numeracy.
*Journal of Mathematics Teacher Education, 16*, 379–397.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 - Singer, M. A., & Goldin-Meadow, S. (2005). Children learn when their teacher's gestures and speech differ.
*Psychological Science, 16*, 85–89.CrossRefGoogle Scholar - Steffe, L. (1992). Learning stages in the construction of the number sequence. In J. Bideaud, C. Meljac, & J. Fischer (Eds.),
*Pathways to number: children’s developing numerical abilities*(pp. 83–88). Hillsdale: Lawrence Erlbaum.Google Scholar - Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 267–307). Mahwah: Erlbaum.Google Scholar - Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983).
*Children’s counting types: philosophy, theory, and application*. New York: Praeger Scientific.Google Scholar - Steffe, L. P., Cobb, P., & von Glasersfeld, E. (1988).
*Construction of arithmetical meanings and strategies*. New York: Springer.CrossRefGoogle Scholar - Thomas, J., & Harkness, S. S. (2013). Implications for intervention: categorizing the quantitative mental imagery of children.
*Mathematics Education Research Journal, 25*, 231–256.CrossRefGoogle Scholar - Thomas, J., & Tabor, P. D. (2012). Developing quantitative mental imagery.
*Teaching Children Mathematics, 19*, 174–183.CrossRefGoogle Scholar - Thomas, J., Tabor, P. D., & Wright, R. J. (2010). Three aspects of first-graders’ number knowledge: observations and instructional implications.
*Teaching Children Mathematics, 16*, 299–308.Google Scholar - Thomas, J., Fisher, M., Eisenhardt, S., Schack, E., Tassell, J., & Yoder, M. (2015). Professional noticing: a framework for responsive mathematics teaching.
*Teaching Children Mathematics, 21*, 295–303.Google Scholar - Thompson, P. W. (1979). The teaching experiment in mathematics education research. Paper presented at the NCTM Research Presession, Boston, MA.Google Scholar
- Thompson, P. W. (1982). Were lions to speak, we wouldn't understand.
*Journal of Mathematical Behavior, 3*, 147--165.Google Scholar - Valenzeno, L., Alibali, M. W., & Klatzky, R. (2003). Teachers’ gestures facilitate students’ learning: a lesson in symmetry.
*Contemporary Educational Psychology, 28*, 187–204.CrossRefGoogle Scholar - Wertsch, J. V. (1998).
*Mind as action*. New York: Oxford University Press.Google Scholar - Wright, R. J. (1994). A study of the numerical development of 5-year-olds and 6-year-olds.
*Educational Studies in Mathematics, 26*, 25–44.CrossRefGoogle Scholar - Wright, R. J., Martland, J., & Stafford, A. (2006a).
*Early numeracy: assessment for teaching and intervention*(2nd ed.). London: Paul Chapman/Sage.Google Scholar - Wright, R. J., Martland, J., Stafford, A., & Stanger, G. (2006b).
*Teaching number: advancing children’s skills and strategies*(2nd ed.). London: Paul Chapman/Sage.Google Scholar