## Abstract

Written tasks that request an explanation can offer detailed information concerning students' knowledge to the classroom teacher. The examination of students' written responses for the mathematical systems that are employed (referents), the connections that are displayed (relationships) and the communication methods that are utilized (modes) provides evidence that suggests how the student is making sense of a problem. The purpose of this article is to describe the constructs of referents, relationships and modes and to illustrate how these constructs may be reflected in students' written responses to a decimal task that requests an explanation. This article will also examine sets of responses from two classrooms using the proposed framework to illustrate the type of information that teachers may acquire through the application of this framework.

### Similar content being viewed by others

## REFERENCES

Brookhart, S.M.: 1999,

*The Art and Science of Classroom Assessment: The Missing Part of Pedagogy,*ASHE-ERIC Higher Education Report (Vol. 27, No.1).Washington, DC: The George Washington University, Graduate School of Education and Human Development.Brown-Herbst, K.: 1999, 'So math isn't just answers',

*Mathematics Teaching in the Middle School*4(7), 448-455.Bruner, J.S. and Kenney, H.: 1966, 'Multiple ordering', in J.S. Bruner, R.R. Oliver and P.M. Greenfield (eds.),

*Studies in Cognitive Growth*, Wiley, New York.Burton, L.: 1984, 'Mathematical thinking: The struggle for meaning',

*Journal for Research on Mathematics Education*15(1), 35-49.Cobb, P., Wood, T., Yackel, E. and McNeal, B.: 1992, 'Characteristics of classroom mathematics traditions: An interactional analysis',

*American Educational Research Journal*29, 573-604.Firestone, W., Fitz, J. and Broadfoot, P.: 1999, 'Power, learning and legitimization: Assessment implementation across levels in the United States and the United Kingdom',

*American Educational Research Journal*36(4), 759-793.Fraivillig, J.L., Murphy, L.A. and Fuson, K.C.: 1999, 'Advancing children's mathematical thinking in Everyday Mathematics classrooms',

*Journal for Research in Mathematics Education*30(2), 148-17.Goldin, G.A.: 1992, 'Towards an assessment framework for school mathematics', in R. Lesh and S.J. Lamon (eds.),

*Assessment of Authentic Performance in School Mathematics*, AAAS Press, Washington, DC, pp. 64-89.Goldin, G.A.: 1987, 'Cognitive representational systems for mathematical problem solving', in C. Janvier (ed.),

*Problems of Representation in the Teaching and Learning of Mathematical Problem Solving*, Erlbaum, Hillsdale, NJ, pp. 64-89.Hancock, L.C.: 1995, 'Implementing the assessment standards for school mathematics: Enhancing mathematics learning with open-ended questions',

*Mathematics Teacher*88(6), 496-499.Hart, K., Johnson, D.C., Brown, M., Dickson, L. and Clarkson, R.: 1989, in D.C. Johnson (ed.),

*Children's Mathematical Frameworks 8-13: A Study of Classroom Teaching*, NFER-Nelson, Berkshire, England.Hiebert, J.: 1984, 'Children's mathematics learning: The struggle to link form and understanding',

*The Elementary School Journal*84(5), 497-513.Hiebert, J. and Lefeve, P.: 1986, 'Conceptual and procedural knowledge in mathematics: An introductory analysis', in Hiebert (ed.),

*Conceptual and Procedural Knowledge: The Case of Mathematics*, Erlbaum, Hillsdale, NJ, pp. 1-27.Hiebert, J. and Wearne, D.: 1986, 'Procedures over concepts: The acquisition of decimal number knowledge', in J. Hiebert (ed.),

*Conceptual and Procedural Knowledge: The Case of Mathematics*, Erlbaum, Hillsdale, NJ, pp. 199-223.Hiebert, J. and Wearne, D.: 1983, 'Students' conceptions of decimal numbers', Paper presented at the annual meeting of the American Educational Research Association, Montreal [ERIC document: Ed230415].

Hiebert, J., Wearne, D. and Taber, S.: 1991, 'Fourth graders' gradual construction of decimal fractions during instruction using different physical representations',

*The Elementary School Journal*91(4), 321-341.Laborde, C.: 1993, 'Language and mathematics', in P. Nesher and J. Kilpatrick (eds.),

*Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education*, Cambridge University Press, New York, pp. 53-69.Lane, S., Silver, E.A., Ankenmann, R.D., Cai, J., Finseth, C., Liu, M., Magone, M.E., Meel, D., Moskal, B., Parke, C.S., Stone, C.A., Wang, N. and Zhu, Y.: 1995,

*QUASAR Cognitive Assessment Instrument (QCAI)*, University of Pittsburgh, Learning Research and Development Center, Pittsburgh, PA.Magone, M.E., Cai, J., Silver, E. and Wang, N.: 1994, 'Validating the cognitive complexity and content quality of a mathematical performance assessment', R.J. Shavelson's (Guest Editor), Special Issue on Performance Assessment,

*International Journal of Educational Research*21(3), 317-34.Magone, M.E., Wang, N., Cai, J. and Lane, S.: 1993, April, 'An analysis of the cognitive complexity of QUASAR's performance assessment tasks and their sensitivity to measuring changes in students' thinking', Paper presented in the symposium

*Assessing Performance Assessments: Do They Withstand Empirical Scrutiny?*at the Annual meeting of the American Educational Research Association, Atlanta, GA.Markovits, Z. and Sowder, J.T.: 1991, 'Students' understanding of the relationship between fractions and decimals',

*Focus on Learning Problems in Mathematics*13(1), 3-11.Middleton, J.A., Van Den Heuvel-Panhuizen, M. and Shew, J.A.: 1998, 'Using bar representations as a model for connecting concepts of rational number',

*Mathematics Teaching in the Middle School*3(4), 302-312.Moskal, B.: 2000a, 'Understanding student responses to open-ended tasks',

*Mathematics Teaching in the Middle School [Focus issue on communication]*7(5), 500-505.Moskal, B.: 2000b, 'Scoring rubrics: What, when and how?',

*Practical Assessment, Research and Evaluation*7(3) [Available Online: http://ericae.net/pare/getvn.asp?v=7andn=3].Moskal, B.: 1997, 'Written open-ended decimal tasks: What information do teachers acquire and use through the examination of the modes, referents and relationships that students employ?', Unpublished Dissertation, University of Pittsburgh, Pittsburgh, Pennsylvania.

Moskal, B. and Leydens, J.: 2000, 'Scoring rubric development: Validity and reliability',

*Practical Assessment, Research and Evaluation*, 7(10) [Available Online: http://ericae.net/pare/getvn.asp?v=7andn=10].Moss, J. and Case, R.: 1999, 'Developing children's understanding of rational numbers: A new model and an experimental curriculum',

*Journal for Research in Mathematics Education*30(2), 122-147.National Council of Teachers of Mathematics: 2000,

*Principles and Standards for School Mathematics*, NCTM, Reston, Virginia.National Council of Teachers of Mathematics: 1995,

*Assessment Standards for School Mathematics*, NCTM, Reston, Virginia.National Council of Teachers of Mathematics: 1991,

*Professional Standards for Teaching Mathematics*, NCTM, Reston, Virginia.National Council of Teachers of Mathematics: 1989.

*Curriculum and Evaluation Standards for School Mathematics*, NCTM, Reston, Virginia.Parke, C.S. and Lane, S.: 1993, April, 'Designing performance assessments: An examination of changes in task structure on student performance', Paper presented at the meeting of the National Council on Measurement in Education, Atlanta, GA.

Peressini, D. and Webb, N.: 1999, 'Analyzing mathematical reasoning in students' responses across multiple performance tasks', in Lee V. Stiff (ed.),

*Developing Mathematical Reasoning in Grades K-12, NCTM Yearbook*, NCTM, Reston, Virginia, pp. 156-174.Pike, M. and Murray, L.: 1991, 'Assessing Open Ended Tasks',

*Mathematics in School*20(2), 32-33.Resnick, L.B., Nesher, P., Leonard, F., Magone, M., Omanson, S. and Peled, I.: 1989, 'Conceptual bases of arithmetic errors: The case of decimal fractions',

*Journal for Research in Mathematics Education*20(1), 8-27.Romberg, T.A. and Wilson, L.D.: 1995, 'Issues related to the development of an authentic assessment system for school mathematics',

*Reform in School Mathematics and Authentic Assessment*, State University of New York, Albany, NY, pp. 1-18.Snow, R.E. and Lohman, D.F.: 1989, 'Implications of cognitive psychology for educational measurement', in R.L. Linn (ed.),

*Educational Measurement*, (3rd ed), American Council on Education, New York, pp. 263-332.Stiggins, R.J.: 1993, 'Two disciplines of educational assessment',

*Measurement and Evaluation in Counseling and Development*26(1), 93-104.Streefland, L.: 1992, 'Thinking strategies in mathematics instruction: How is testing possible?' in R. Lesh and S.J. Lamon (eds.),

*Assessment of Authentic Performance in School Mathematics*, American Association for the Advancement of Science, Washington, DC, pp. 215-245.Suzuki, K. and Harnisch, D.: 1995, 'Measuring cognitive complexity: An analysis of performance based assessment in mathematics', Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, California [ERIC document: Ed390924].

Usiskin, Z.: 1993, 'Lessons from the Chicago Mathematics Project',

*Educational Leadership*50(8), 14-18.Vance, J.H.: 1986, 'Ordering decimals and fractions: A diagnostic study',

*Focus on Learning Problems in Mathematics*8(2), 51-59.Van den Heuvel-Panhuizen, M.: 1995, April, 'The tension between openness and certainty: An example of developmental research on assessment', Paper presented as part of the symposium,

*Theory and Practice of Authentic Assessment in Realistic Mathematics Education*, at the annual meeting of the American Educational Research Association, San Francisco, California.Van den Heuvel-Panhuizen, M.: 1994, 'Improvement of (didactical) assessment by improvement of problems: An attempt with respect to percentage',

*Educational Studies in Mathematics*27(4), 341-72.Wearne, D. and Hiebert, J.: 1988, 'A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers',

*Journal for Research in Mathematics Education*19, 371-384.Webb, N., Gold, K. and Qi: 1989, March, 'Mathematical problem-solving processes and performance', Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.

Wertheimer, M.: 1959,

*Productive Thinking*, Harper and Row, New York.Wiggins, G.: 1993, 'Assessment: Authenticity, context and validity',

*Phi Delta Kappan*75(Nov), 200-214.

## Author information

### Authors and Affiliations

## Rights and permissions

## About this article

### Cite this article

Moskal, B.M., Magone, M.E. Making Sense of What Students Know: Examining the Referents, Relationships and Modes Students Displayed in Response to a Decimal Task.
*Educational Studies in Mathematics* **43**, 313–335 (2000). https://doi.org/10.1023/A:1011983602860

Issue Date:

DOI: https://doi.org/10.1023/A:1011983602860