### Elementary Children’s Multiplicative Reasoning: Initial Validation of a Written Assessment

#### Abstract

Multiplicative reasoning is essential for students’ engagement with various mathematical concepts. Although the field’s understanding of children’s multiplicative concepts has grown over the past 30 years, relatively few studies have examined the development of multiplicative concepts with whole numbers, and even fewer have studied this phenomenon at scale. The present study reports on the development of an assessment of elementary students’ multiplicative concepts with whole numbers that can be used at a large scale. Findings suggest the initial version of the assessment has sufficient reliability and validity. Further, less than 20% of second grade students and approximately 50% of third grade students participating in the study engage in tasks with at least the first multiplicative concept.

#### Full Text:

PDF#### References

Brickwedde, J. (2011). Transition from additive to multiplicative thinking: A design and teaching experiment with third through fifth graders (Doctoral dissertation, University of Minnesota). Retrieved from https://conservancy.umn.edu/

Confrey, J., & Harel, G. (1994). Introduction. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. vii-xxvii). Albany, NY: State University of New York.

Crocker, L, & Algina, J. (2006). Introduction to classical and modern test theory. Mason, OH: Thomson Wadsworth.

Eliustaoglu, E. (2016). Investigation of fraction schemes and models as a means to understand how sixth grade students make sense of fractions (Masters thesis, Kent State University). Retrieved from http://www.ohiolink.edu/etd/

English, L. D. (1991). Young children’s combinatoric strategies. Educational Studies in Mathematics, 22(5), 451-474.

Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383-432.

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), 1-18.

I, J. Y., & Dougherty, B. J. (January, 2014). Linking multiplication models to conceptual understanding in measurement approach. Paper presented at the 11th Hawaii International Conference of Education, Honolulu, HI. Retrieved at https://www.researchgate.net/profile/Ji_Yeong_I/

publication/270279409_Linking_Multiplication_Models_to_Conceptual_Understanding_in_Measurement_Approach/links/54a5e2870cf267bdb9082af5.pdf

Izsàk, A., Jacobson, E., & de Araujo, Z. (2012). Measuring mathematical knowledge for teaching fractions with drawn quantities. Journal for Research in Mathematics Education, 43(4), 391-427.

Izsàk, A., Orrill, C. H., Cohen, A. S., & Brown, R. E. (2010). Using the mixture Rasch model to assess middle grades teachers’ reasoning about rational numbers. Elementary School Journal, 110, 279-300.

Kilpatrick, J. (2001). Where’s the evidence? Journal for Research in Mathematics Education, 32(4), 421-427.

Kosko, K. W., & Singh, R. (2016). Children’s linguistic and numeric unit coordination. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th annual meeting of the North American Chapter for the Psychology of Mathematics Education (pp. 605-612). Tucson, AZ: The University of Arizona.

Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics, 33, 159-174.

Mulligan, J., & Mitchelmore, M. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28(3), 309-330.

National Governors Association Center for Best Practices, & Council of Chief State School Officers (2010). Common core state standards for mathematics. Washington, DC: Author.

Norton, A., Boyce, S., Phillips, N., Anwyll, T., Ulrich, C., & Wilkins, J. (2015). A written instrument for assessing students’ units coordination structures. IEJME - Mathematics Education, 10(2), 111-136.

Norton, A., Boyce, S., Ulrich, C., & Phillips, N. (2015). Students’ units coordination activity: A cross-sectional analysis. Journal of Mathematical Behavior, 39, 51-66.

Norton, A., & Hackenberg, A. J. (2010). Continuing research on students’ fraction schemes. In L. P. Steffe & J. Olive (Eds.), Children’s fractional knowledge (pp. 341-352). New York: Springer.

Olive, J. (2001). Children’s number sequences: An explanation of Steffe’s constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator, 11(1),4-9.

Singh, R., & Kosko, K. W. (2016). Elementary students’ understanding of equivalence and multiplicative reasoning. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th annual meeting of the North American Chapter for the Psychology of Mathematics Education (p. 212). Tucson, AZ: The University of Arizona.

Smith, S. Z., & Smith, M. E. (2006). Assessing elementary understanding of multiplication concepts. School Science and Mathematics, 106(3), 140-149.

Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259-309.

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.

Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20, 267-307.

Steffe, L. P. (2017). Psychology in mathematics education: Past, present, and future. Plenary Talk given at the 39th annual meeting of the North American Chapter for the Psychology of Mathematics Education, Indianapolis, IN.

Tillema, E. S. (2013). Relating one and two-dimensional quantities: Students’ multiplicative reasoning in combinatorial and spatial contexts. Journal of Mathematical Behavior, 32(3), 331-352.

Tzur, R., Johnson, H., McClintock, E., Xin, Y. P., Si, L., Kenney, R., Woodward, J., Hord, C., & Jin, X. (2012). Children’s development of multiplicative reasoning: A scheme and tasks framework. In T. Y. Tso (Ed.), Proceeding of the 36th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 155-162). Taipei: PME.

Wilkins, J. L. M., Norton, A., & Boyce, S. J. (2013). Validating a written instrument for assessing students’ fractions schemes and operations. The Mathematics Educator, 22(2), 31-54.

von Glasersfeld, E. (1995). Sensory experience, abstraction, and teaching. In L. P. Steffe and J. Gale (Eds.), Constructivism in Education (pp. 369-383). Hillsdale, NJ: Lawrenc

### Refbacks

- There are currently no refbacks.