pp. 111-136 | Article Number: mathedu.2015.008
Published Online: August 02, 2015
Article Views: 364 | Article Download: 391
Units coordination refers to students’ abilities to create units and maintain their relationships with other units that they contain or constitute. In recent research, units coordination has arisen as a key construct that mediates opportunities for student learning across several domains of mathematics, including fractions knowledge and algebraic reasoning. To date, assessments of students’ stages of units coordinating ability have relied upon clinical interviews or teaching experiments whose time-intensive nature precludes opportunities for conducting large-scale studies. We introduce a written instrument that teachers and researchers can use with large populations of students. We report on the reliability and validity of assessments based on the instrument.
Keywords: assessment; fractions; multiplicative reasoning; units coordination; written instrument
Boyce, S., & Norton, A. (in review). Co-construction of fractions schemes and units coordinating structures. Journal of Mathematical Behavior.
Clements, D. H., Battista, M. T., Sarama, J., & Swaminathan, S. (1997). Development of students' spatial thinking in a unit on geometric motions and area. The Elementary School Journal, 171-186.
Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20(1), 37-46.
Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155-159.
Ellis, A. B. (2007). The influence of reasoning with emergent quantities on students’ generalizations. Cognition & Instruction, 25, 439 – 478. doi:10.1080/07370000701632397
Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior, 26, 27-47. doi:10.1016/j.jmathb.2007.03.002
Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 1-50. doi:10.1080/07370008.2010.511565
Hackenberg, A. J. (2013a) Musings on three epistemic algebraic students. In: Moore K. C., Steffe L. P. & Hatfield L. L. (eds.) Epistemic algebraic students: Emerging models of students’ algebraic knowing. University of Wyoming: Laramie WY: 81–124.
Hackenberg, A. J. (2013b). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. The Journal of Mathematical Behavior, 32(3), 538-563. doi:10.1016/j.jmathb.2013.06.007
Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196-243.
Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. Journal of Mathematical Behavior, 28, 1-18. doi:10.1016/j.jmathb.2009.04.004
Hunting, R. P. (1983). Alan: A case study of knowledge of units and performance with fractions. Journal for Research in Mathematics Education, 14(3), 182-197. doi:10.2307/748381
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic Books.
Izsák, A., Jacobsen, E., de Araujo, Z., & Orrill, C. H. (2012) Measuring mathematical knowledge for teaching fractions with drawn quantities. Journal for Research in Mathematics Education, 43(4). doi:10.5951/jresematheduc.43.4.0391
Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics, 33, 159-174. doi:10.2307/2529310
Norton, A., & Boyce, S. (in review). Provoking the Construction of a Structure for Coordinating n+1 Levels of Units. Journal of Mathematical Behavior.
Norton, A., & Boyce, S. (2013). A cognitive core for common state standards. Journal of Mathematical Behavior, 32, 266-279. doi:10.1016/j.jmathb.2013.01.001
Norton, A., Boyce, S., Hatch, J. (in press). Coordinating units and constructing fractions at the Candy Depot. Teaching Mathematics in the Middle School.
Norton, A., Boyce, S., Ulrich, C., & Phillips. N. (2015). Students’ units coordination activity: A cross-sectional analysis.Journal of Mathematical Behavior, 39, 51-66. doi:10.1016/j.jmathb.2015.05.001
Olive, J. & Çağlayan, G. (2008). Learner’s difficulties with quantitative units in algebraic word problems and the teacher’s interpretations of those difficulties. International Journal of Science and Mathematics Education, 6, 269–292. doi:10.1007/s10763-007-9107-6
Olive, J., & Steffe, L. P. (2002). The construction of an iterative fractional scheme: the case of Joe. Journal of Mathematical Behavior, 20(4), 413-437. doi:10.1016/S0732-3123(02)00086-X
Piaget, J. (1970a). Genetic epistemology (E. Duckworth, Trans.). New York: Norton.
Piaget, J. (1970b). Structuralism. New York: Basic Books.
Piaget, J. (1972). The principles of genetic epistemology. London: Routledge & Kegan Paul.
Piaget, J., & Inhelder, B. (1967). The child’s conception of space (Trans. F. J. Langdon & J. L. Lunzer). New York: Norton (Original work published in 1948).
Reynolds, A., & Wheatley, G. H. (1996). Elementary students' construction and coordination of units in an area setting. Journal for Research in Mathematics Education, 564-581. doi:10.2307/749848
Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259-309. doi:10.1016/1041-6080(92)90005-Y
Steffe, L. P. (1994). Children's multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-39). Albany, NY: State University of New York Press.
Steffe, L. P. (2007, April). Problems in mathematics education. Paper presented for the Senior Scholar Award of the Special Interest Group for Research in Mathematics Education (SIG-RME) at the annual conference of the American Educational Research Association in Chicago, Illinois.
Steffe, L. P., & Olive, J. (2010). Children's fractional knowledge. New York: Springer.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Mahwah, NJ: Erlbaum.
Steffe, L. P. & Ulrich, C. (2014). The constructivist teaching experiment. In S. Lerman (Ed.), Encyclopedia of Mathematics Education: SpringerReference. Berlin, Germany: Springer. doi:10.1007/SpringerReference_313181
Tillema, E. S. (2013). A power meaning of multiplication: Three eighth graders’ solutions of Cartesian product problems.The Journal of Mathematical Behavior, 32(3), 331-352. doi:10.1016/j.jmathb.2013.03.006
Ulrich, C. (2012). Additive relationships and signed quantities. (Doctoral dissertation). Retrieved from http://www.libs.uga.edu/etd