Assessing Students' Numerical Ways of Operating Using Written Instruments
The assessment project began in 2007, as a partnership between Anderson Norton and Jesse Wilkins. The goal of the project was to use students' written responses to numerical tasks--whole number multiplication and fractions tasks--to draw inferences about their ways of operating. Prior to this project, such inferences were drawn from clinical interviews and teaching experiments, which allow for interaction with the student and richer behavioral observations but are very time intensive. Validation studies have affirmed high correlations between assessments using clinical interviews and our method, using written instruments.
More recently, other researchers have joined the project to extend its work to include new assessments and new populations. These researchers include Catherine Ulrich at Virginia Tech, and a research team at James Madison University. Our instruments have been used by other projects across the United States, including Indiana University and the University of Colorado at Denver. They have also been used with children in China and South Korea.
Collective publications are included below, as are our tasks, which researchers are welcome to use in their own projects.
These tasks have been used primarily with 6th grade students to assess the following operations and fractions schemes:
· Iterating Operation: 1-4
· Partitioning Operation: 5-8
· Partitive Fraction Scheme: 9-12
· Partitive Unit Fraction Scheme: 13-16
· Splitting Operation: 17-20
See Wilkins, J. L. M., & Norton, A. (2011). The splitting loope. Journal for Research in Mathematics Education, 42(4), 386-406.
These tasks have been used primarily with 7th grade students, to assess the following fractions schemes:
· Partitive Fraction Scheme: 4-7
· Partitive Unit Fraction Scheme: 10-13
· Reversible Partitive Fraction Scheme: 16-19
· Splitting Operation: 20-23
· "Robustness tasks": 1-3, 8, 9, 14, 15 (used to examine other relationships)
See Norton, A. H., & Wilkins, J. L. M. (2010). Students’ partitive reasoning. Journal of Mathematical Behavior, 29(4), 181-194.
These tasks have been used primarily with 8th grade students, to assess the following fractions schemes:
· Iterative Fraction Scheme: 6-9
· Reversible Partitive Fraction Scheme: 10-13
· Splitting Operation: 14-17
· Units coordination with three levels of fractional units: 19-22
· "RobustnessTasks": 1-5, 18 (used to examine other relationships)
See Norton, A., & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43(5), 557-583.
These tasks include many of the tasks listed above, translated into Chinese. They also include four part-whole tasks: 7, 10, 16, and 21.
See Norton, A., Wilkins, J. L. M., & Xu C. Z. (2018). A progression of fraction schemes common to Chinese and U.S. students. Journal for Research in Mathematics Education, 49(2), 210-226.
Johnson, A., Siegfried, Z., Lovin, L. A., Busi, R. P., Wilkins, J. L. M., & Norton, A. (2018). Promoting sophisticated fractions constructs through instructional changes in a mathematics course for PreK-8 prospective teachers. Journal of Mathematics Teacher Education, 1-29. DOI: 10.1007/s10857-018-9415-5
Lovin, L. A., *Stevens, A., Siegfried, Z., Wilkins, J. L. M., & Norton, A. (2018). Pre-K-8 prospective teachers’ understanding of fractions: An extension of fractions schemes and operations research. Journal of Mathematics Teacher Education, 21(3), 207-235. DOI: 10.1007/s10857-016-9357-8
Norton, A., *Boyce, S., *Phillips, N., *Anwyll, T., Ulrich, C., & Wilkins, J. (2015). A written instrument for assessing students’ units coordination structures. Journal of Mathematics Education, 10(2), 111-136. DOI: 10.12973/mathedu.2015.108a
Norton, A., & Wilkins, J. (2009). A quantitative analysis of children’s splitting operations and fractional schemes. Journal of Mathematical Behavior, 28(2/3), 150-161.
Norton, A., & Wilkins, J. L. M. (2010). Students’ partitive reasoning. Journal of Mathematical Behavior, 29(4), 181-194.
Norton, A., & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43(5), 557-583.
Norton, A., & Wilkins, J. L. M. (2013). Supporting students’ constructions of the splitting operation. Cognition & Instruction, 31(1), 2-28.
Norton, A., Wilkins, J. L. M., & *Xu, C. Z. (2018). A progression of fractions schemes common to Chinese and U.S. classrooms. Journal for Research in Mathematics Education, 49(2), 210-226. DOI: 10.5951/jresematheduc.49.2.0210
Wilkins, J. L. M., & Norton, A. (2011). The splitting loope. Journal for Research in Mathematics Education, 42(4), 386-406.
Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5(27).
Wilkins, J. L. M., Norton, A., & *Boyce, S. (2013). Validating a written instrument for assessing students’ fractions schemes and operations. The Mathematics Educator, 22(2), 31-44.