# The nature of student predictions and learning opportunities in middle school algebra

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## Abstract

In this article, we describe how using prediction during instruction can create learning opportunities to enhance the understanding and doing of mathematics. In doing so, we characterize the nature of the predictions students made and the levels of sophistication in students’ reasoning within a middle school algebra context. In this study, when linear and exponential functions were taught, prediction questions were posed at the launch of the lessons to reflect the mathematical ideas of each lesson. Students responded in writing along with supportive reasoning individually and then discussed their predictions and rationale. A total of 395 prediction responses were coded using a dual system: sophistication of reasoning, and the mechanism students appeared to utilize to formulate their prediction response. The results indicate that using prediction provoked students to connect among mathematical ideas that they learned. It was apparent that students also visualized mathematical ideas in the problem or the possible results of the problem. These results suggest that using prediction in fact provides learning opportunities for students to engage in mathematical sense making and reasoning, which promotes students’ understanding of the mathematics that they learn.

## Keywords

Mathematical reasoning Prediction Learning opportunities Connections Visualization## References

- Ball, D., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.),
*A research companion to Principles and Standards for School Mathematics*(pp. 27–44). Reston: National Council for Teachers of Mathematics.Google Scholar - Battista, M. T. (1999). Fifth graders’ enumeration of cubes of 3D arrays: Conceptual progress in an inquiry-based classroom.
*Journal for Research in Mathematics Education, 30*(4), 417–448.CrossRefGoogle Scholar - Bishop, A. J. (1988). A review of research on visualization in mathematics education. In A. Borbas (Ed.),
*Proceedings of the 12th PME International Conference, 1*, 170–176.Google Scholar - Block, C., Rodgers, L., & Johnson, R. (2004).
*Comprehension process instruction: Creating reading success in grades K-3*. New York: The Guilford.Google Scholar - Boaler, J., & Humphreys, C. (2005).
*Connecting mathematical ideas/middle school video cases to support teaching and learning*. Portsmouth: Heinemann.Google Scholar - Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999).
*How people learn: Brain, mind, experience and school*. Washington, DC: National Academy.Google Scholar - Brown, D., & Wheatley, G. (1990). The role of imagery in mathematical reasoning. In G. Booker, P. Cobb & T. de Mendicuti (Eds.),
*Proceedings of the 14th PME International Conference, 1*, 217.Google Scholar - Brownell, W. A. (1935). Psychological considerations in the learning and teaching of arithmetic. In W. D. Reeve (Ed.),
*The teaching of arithmetic: Tenth yearbook of the National Council of Teachers of Mathematics*(pp. 1–31). New York: Teachers College, Columbia University.Google Scholar - Buendía, G., & Cordero, F. (2005). Prediction and the periodical aspect as generators of knowledge in a social practice framework.
*Educational Studies in Mathematics, 58*, 299–333.CrossRefGoogle Scholar - Clements, M. A. (1982). Visual imagery and school mathematics.
*For the Learning of Mathematics, 2*(2), 33–39.Google Scholar - Duffy, G. (2003).
*Explaining reading: A resource for teaching concepts, skills, and strategies*. New York: The Guilford.Google Scholar - Dunham, P. H., & Osborne, A. (1991). Learning how to see: Students’ graphing difficulties.
*Focus on Learning Problems in Mathematics, 13*(4), 35–49.Google Scholar - Gamoran, A. (2001). Beyond curriculum wars: Content and understanding in mathematics. In T. Loveless (Ed.),
*The great curriculum debate: How should we teach reading and math?*(pp. 134–142). Washington, DC: Brookings Institution.Google Scholar - Good, R. (1989).
*Toward a unified conception of thinking: Prediction within a cognitive science perspective*. Paper presented at the meeting of the National Association of Research in Science Teaching, San Francisco, CA.Google Scholar - Greeno, J. G. (1998). The situativity of knowing, learning, and research.
*American Psychologist, 53*(1), 5–26.CrossRefGoogle Scholar - Gunstone, R. F., & White, R. T. (1981). Understanding of gravity.
*Science Education, 65*(3), 291–299.CrossRefGoogle Scholar - Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
*The handbook of research on mathematics teaching and learning*(pp. 65–100). New York: Macmillan.Google Scholar - Hiebert, J., & Grouws, D. (2007). The effects of classroom mathematics teaching on students
*’*learning. In F. K. Lester Jr. (Ed.),*Second handbook of research on mathematics teaching and learning*(pp. 371–404). Charlotte: Information Age.Google Scholar - Kasmer, L., & Kim, O. K. (2011a). Using prediction to promote mathematical understanding and reasoning.
*School Science and Mathematics Journal, 109*(1), 20–33.CrossRefGoogle Scholar - Kasmer, L., & Kim, O. K. (2011b). Using prediction to motivate personal investment in problem solving. In D. J. Brahier & W. R. Speer (Eds.),
*The Seventy-third Yearbook of the National Council of Teachers of Mathematics, Motivation and disposition: Pathways to learning mathematics*(pp. 157–168). Reston: National Council of Teachers of Mathematics.Google Scholar - Kim, O. K., & Kasmer, L. (2009). Prediction as an instructional strategy.
*Journal of Mathematics Education Leadership, 11*(1), 33–38.Google Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: National Academy.Google Scholar - King, C., & Johnson, L. (1999). Constructing meaning via reciprocal teaching.
*Reading Research and Instruction, 38*(3), 169–186.CrossRefGoogle Scholar - Krutetskii, V. A. (1976).
*The psychology of mathematical abilities in school children*. Chicago: University of Chicago Press.Google Scholar - Lappan, G., Fey, J., Fitzgerald, W., Friel, S., & Phillips, E. D. (2004a).
*Connected mathematics. Moving straightforward*. Menlo Park: Dale Seymour.Google Scholar - Lappan, G., Fey, J., Fitzgerald, W., Friel, S., & Phillips, E. D. (2004b).
*Connected mathematics. Frogs, fleas, and painted cubes*. Menlo Park: Dale Seymour.Google Scholar - Lavoie, D. R. (1999). Effects of emphasizing hypothetico-predictive reasoning within the science learning cycle on high school student’s process skills and conceptual understandings in biology.
*Journal of Research in Science Teaching, 36*(10), 1127–1147.CrossRefGoogle Scholar - Lim, K., Buendia, G., Kim, O. K., Cordero, F., & Kasmer, L. (2010). The role of prediction in the teaching and learning of mathematics.
*International Journal of Mathematics Education in Science and Technology, 41*(5), 595–608.CrossRefGoogle Scholar - Mullis, I., Martin, M., Smith, T., Garden, R., Gregory, K., Gonzalez, E., et al. (2001).
*TIMSS assessment frameworks and specifications 2003*. Chestnut Hill: TIMSS International Study Center, Boston College.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: NCTM.Google Scholar - National Council of Teachers of Mathematics. (2009).
*Focus in high school mathematics: Reasoning and sense making*. Reston: NCTM.Google Scholar - Peirce Edition Project. (1998).
*The essential Peirce: Selected philosophical writings, 1893–1913*. Bloomington: Indiana University Press.Google Scholar - Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*(pp. 205–235). Rotterdam: Sense.Google Scholar - Reid, D. (2002). Conjectures and refutations in grade 5 mathematics.
*Journal for Research in Mathematics Education, 33*(1), 5–29.CrossRefGoogle Scholar - Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to Principles and Standards for School Mathematics*(pp. 289–303). Reston: National Council of Teachers of Mathematics.Google Scholar - Skemp, R. (1987).
*The psychology of learning mathematics*. Hillsdale: Erlbaum.Google Scholar