Skip to main content
Log in

The problem of assessing problem solving: can comparative judgement help?

Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

School mathematics examination papers are typically dominated by short, structured items that fail to assess sustained reasoning or problem solving. A contributory factor to this situation is the need for student work to be marked reliably by a large number of markers of varied experience and competence. We report a study that tested an alternative approach to assessment, called comparative judgement, which may represent a superior method for assessing open-ended questions that encourage a range of unpredictable responses. An innovative problem solving examination paper was specially designed by examiners, evaluated by mathematics teachers, and administered to 750 secondary school students of varied mathematical achievement. The students’ work was then assessed by mathematics education experts using comparative judgement as well as a specially designed, resource-intensive marking procedure. We report two main findings from the research. First, the examination paper writers, when freed from the traditional constraint of producing a mark scheme, designed questions that were less structured and more problem-based than is typical in current school mathematics examination papers. Second, the comparative judgement approach to assessing the student work proved successful by our measures of inter-rater reliability and validity. These findings open new avenues for how school mathematics, and indeed other areas of the curriculum, might be assessed in the future.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  • ACME. (2005). Assessment in 14–19 Mathematics. London: Advisory Committee on Mathematics Education.

    Google Scholar 

  • ACME. (2011). Mathematical needs: Mathematics in the workplace and in higher education. London: Advisory Committee on Mathematics Education.

    Google Scholar 

  • AQA. (2010). GCSE Foundation Tier Mathematics Paper 1 (Specification A). Monday 7 June 2010. Manchester: Assessment and Qualifications Alliance.

    Google Scholar 

  • Berube, C.T. (2004). Are standards preventing good teaching? Clearing House, 77, 264–267.

    Article  Google Scholar 

  • Black, P. (2008). Strategic decisions: Ambitions, feasibility and context. Educational Designer, 1(1). Retrieved from http://www.educationaldesigner.org/ed/volume1/issue1/article1/

  • Black, P. at al. (2012). High-stakes examinations to support policy. Educational Designer, 2(5). Retrieved from http://www.educationaldesigner.org/ed/volume2/issue5/article16/

  • Borsboom, D., Mellenbergh, G.J., & van Heerden, J. (2004). The concept of validity. Psychological Review, 111, 1061–1071.

    Article  Google Scholar 

  • Bramley, T. (2007). Paired comparison methods. In P. Newton, J.-A. Baird, H. Goldstein, H. Patrick, & P. Tymms (Eds.), Techniques for monitoring the comparability of examination standards (pp. 264–294). London: Qualifications and Curriculum Authority.

    Google Scholar 

  • Bramley, T., Bell, J., & Pollitt, A. (1998). Assessing changes in standards over time using Thurstone paired comparisons. Education Research and Perspectives, 25, 1–24.

    Google Scholar 

  • Burkhardt, H. (2009). On strategic design. Educational Designer, 1(3). Retrieved from http://www.educationaldesigner.org/ed/volume1/issue3/article9/

  • Crisp, V. (2008). Exploring the nature of examiner thinking during the process of examination marking. Cambridge Journal of Education, 38, 247–264.

    Article  Google Scholar 

  • Cronbach, L.J. (1988). Five perspectives on the validity argument. In H. Wainer & H.I. Braun (Eds.), Test validity (pp. 3–17). Hillsdale: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Derrick, K. (2012). Developing the e-scape software system. International Journal of Technology and Design Education, 22, 171–185.

    Article  Google Scholar 

  • Duncan, A. (2010). Beyond the bubble tests: The next generation of assessments. Alexandria, VA, Secretary Arne Duncan’s Remarks to State Leaders at Achieve’s American Diploma Project Leadership Team Meeting. Retrieved from http://www.ed.gov/news/speeches/beyond-bubble-tests-next-generation-assessments-secretary-arne-duncans-remarks-state-l

  • Gewertz, C. (2012). Consortia provide preview of common assessments. Education Week, 32, 18–19.

    Google Scholar 

  • Heldsinger, S., & Humphry, S. (2010). Using the method of pairwise comparison to obtain reliable teacher assessments. The Australian Educational Researcher, 37, 1–19.

    Article  Google Scholar 

  • James, C. (1974). The consistency of marking a physics examination. Physics Education, 9, 271–274.

    Article  Google Scholar 

  • Jones, I., & Alcock, L. (2014). Peer assessment without assessment criteria. Studies in Higher Education, 39, 1774–1787.

    Article  Google Scholar 

  • Jones, I., Inglis, M., Gilmore, C., & Hodgen, J. (2013). Measuring conceptual understanding: The case of fractions. In A.M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 113–120). Kiel: PME.

    Google Scholar 

  • Jones, I., Swan, M., & Pollitt, A. (2014). Assessing mathematical problem solving using comparative judgement. International Journal of Science and Mathematics Education, 13, 151–177.

    Article  Google Scholar 

  • Kimbell, R. (2012). Evolving project e-scape for national assessment. International Journal of Technology and Design Education, 22, 135–155.

    Article  Google Scholar 

  • Koretz, D. (2008). Measuring up: What educational testing really tells us. Cambridge: Harvard University Press.

    Google Scholar 

  • Laming, D. (1984). The relativity of “absolute” judgements. British Journal of Mathematical and Statistical Psychology, 37, 152–183.

    Article  Google Scholar 

  • McLester, S., & McIntire, T. (2006). The workforce readiness crisis: We’re not turning out employable graduates nor maintaining our position as a global competitor—why? Technology and Learning, 27, 22–28.

    Google Scholar 

  • McMahon, S., & Jones, I. (2014). A comparative judgement approach to teacher assessment. Assessment in Education: Principles Policy and Practice. doi:10.1080/0969594X.2014.978839

    Google Scholar 

  • McVey, P.J. (1976). The “paper error” of two examinations in electronic engineering. Physics Education, 11, 58–60.

    Article  Google Scholar 

  • MEI. (2012). Integrating mathematical problem solving: Applying Mathematics and Statistics across the curriculum at level 3. End of project report. London: Mathematics in Education and Industry.

    Google Scholar 

  • Messick, S. (1980). Test validity and the ethics of assessment. American Psychologist, 35, 1012–1027.

    Article  Google Scholar 

  • Messick, S. (1989). Meaning and values in test validation: The science and ethics of assessment. Educational Researcher, 18, 5–11.

    Article  Google Scholar 

  • Murphy, R. (1979). Removing the marks from examination scripts before re-marking them: Does it make any difference? British Journal of Educational Psychology, 49, 73–78.

    Article  Google Scholar 

  • Murphy, R. (1982). A further report of investigations into the reliability of marking of GCE examinations. British Journal of Educational Psychology, 52, 58–63.

    Article  Google Scholar 

  • NCETM. (2009). Mathematics matters: Final report. London: National Centre for Excellence in the Teaching of Mathematics.

    Google Scholar 

  • Newton, P. (1996). The reliability of marking of general certificate of secondary education scripts: Mathematics and English. British Educational Research Journal, 22, 405–420.

    Article  Google Scholar 

  • Newton, P., & Shaw, S. (2014). Validity in educational and psychological assessment. London: Sage.

    Book  Google Scholar 

  • Noyes, A., Wake, G., Drake, P., & Murphy, R. (2011). Evaluating Mathematics pathways final report. DfE Research Report 143. London: Department for Education.

    Google Scholar 

  • Ofsted. (2008). Mathematics: Understanding the score. London: Office for Standards in Education.

    Google Scholar 

  • Pollitt, A. (2012a). The method of adaptive comparative judgement. Assessment in Education: Principles Policy and Practice, 19, 281–300.

    Article  Google Scholar 

  • Pollitt, A. (2012b). Comparative judgement for assessment. International Journal of Technology and Design Education, 22, 157–170.

    Article  Google Scholar 

  • Pollitt, A., & Murray, N. (1996). What raters really pay attention to. In M. Milanovic & N. Saville (Eds.), Performance testing, cognition and assessment: Selected papers from the 15th language testing research colloquium (pp. 74–91). Cambridge: Cambridge University Press.

    Google Scholar 

  • Popham, W.J. (2001). Teaching to the test? Educational Leadership, 58, 16–20.

    Google Scholar 

  • QCA (2007). National curriculum 2007. Coventry: Qualifications and curriculum authority.

  • Research Committee, N.C.T.M. (2013). New assessments for new standards: The potential transformation of mathematics education and its research implications. Journal for Research in Mathematics Education, 44, 340–352.

    Article  Google Scholar 

  • Seery, N., Canty, D., & Phelan, P. (2012). The validity and value of peer assessment using adaptive comparative judgement in design driven practical education. International Journal of Technology and Design Education, 22, 205–226.

    Article  Google Scholar 

  • Shepard, L.A. (1997). The centrality of test use and consequences for test validity. Educational Measurement: Issues and Practice, 16, 5–24.

    Article  Google Scholar 

  • Silver, E.A., Ghousseini, H., Gosen, D., Charalambous, C., & Font Strawhun, B.T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.

    Article  Google Scholar 

  • Suto, I. (2013). 21st Century skills: Ancient, ubiquitous, enigmatic? Research Matters: A Cambridge Assessment Publication, 15, 2–8.

    Google Scholar 

  • Suto, I., & Greatorex, J. (2008). What goes through an examiner’s mind? Using verbal protocols to gain insights into the GCSE marking process. British Educational Research Journal, 34, 213–233.

    Article  Google Scholar 

  • Suto, I., & Nadas, R. (2009). Why are some GCSE examination questions harder to mark accurately than others? Using Kelly’s repertory grid technique to identify relevant question features. Research Papers in Education, 24, 335–377.

    Article  Google Scholar 

  • Swan, M. (2014). Improving the alignment between values, principles and classroom realities. In Y. Li & G. Lappan (Eds.), Mathematics curriculum in school education (pp. 621–636). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Swan, M., & Burkhardt, H. (2012). Designing assessment of performance in mathematics. Educational Designer, 2(5). Retrieved from http://www.educationaldesigner.org/ed/volume2/issue5/article19/

  • Taggart, G.L., Phifer, S.J., Nixon, J.A., & Wood, M. (1998). Rubrics: A handbook for construction and use. Lancaster: Technomic Publishing.

    Google Scholar 

  • Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 273–286.

    Article  Google Scholar 

  • Truss, E. (2012). Elizabeth Truss calls for a renaissance in maths. Norfolk: Speech to the National Education Trust. Retrieved from https://www.gov.uk/government/speeches/elizabeth-truss-calls-for-a-renaissance-in-maths

  • Turner, H., & Firth, D. (2005). Bradley-Terry models in R: The BradleyTerry2 package. Journal of Statistical Software, 12(1). Retrieved from http://www.jstatsoft.org/v12/i01

  • van Aalst, J., & Chan, C.K.K. (2007). Student-directed assessment of knowledge building using electronic portfolios. Journal of the Learning Sciences, 16, 175–220.

    Article  Google Scholar 

  • Vordermann, C., Porkess, R., Budd, C., Dunne, R., & Rahman-Hart, P. (2011). A world-class Mathematics education for all our young people. London: The Conservative Party.

    Google Scholar 

  • Walport, M., Goodfellow, J., McLoughlin, F., Post, M., Sjøvoll, J., Taylor, M., et al. (2010). Science and Mathematics secondary education for the 21st century: Report of the science and learning expert group. London: Department for Business, Industry and Skills.

    Google Scholar 

  • Wiliam, D. (2001). Reliability, validity, and all that jazz. Education 3–13: International Journal of Primary Elementary and Early Years Education, 29, 17–21.

    Google Scholar 

  • Willmott, A.S., & Nuttall, D.L. (1975). The reliability of examinations at 16+. London: Macmillan Education.

    Google Scholar 

Download references

Acknowledgments

This work was supported by a Royal Society Shuttleworth Research Fellowship to IJ, a Royal Society Worshipful Company of Actuaries Research Fellowship to MI, and the Nuffield Foundation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ian Jones.

Electronic supplementary material

Below is the link to the electronic supplementary material.

ESM 1

(PDF 6.01 mb)

ESM 2

(PDF 216 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jones, I., Inglis, M. The problem of assessing problem solving: can comparative judgement help?. Educ Stud Math 89, 337–355 (2015). https://doi.org/10.1007/s10649-015-9607-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-015-9607-1

Keywords

Navigation