Educational Studies in Mathematics

, Volume 89, Issue 3, pp 337–355 | Cite as

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

  • Ian Jones
  • Matthew Inglis


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.


Assessment Problem solving Validity Comparative judgement 



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.

Supplementary material

10649_2015_9607_MOESM1_ESM.pdf (6 mb)
ESM 1 (PDF 6.01 mb)
10649_2015_9607_MOESM2_ESM.pdf (217 kb)
ESM 2 (PDF 216 kb)


  1. ACME. (2005). Assessment in 14–19 Mathematics. London: Advisory Committee on Mathematics Education.Google Scholar
  2. ACME. (2011). Mathematical needs: Mathematics in the workplace and in higher education. London: Advisory Committee on Mathematics Education.Google Scholar
  3. AQA. (2010). GCSE Foundation Tier Mathematics Paper 1 (Specification A). Monday 7 June 2010. Manchester: Assessment and Qualifications Alliance.Google Scholar
  4. Berube, C.T. (2004). Are standards preventing good teaching? Clearing House, 77, 264–267.CrossRefGoogle Scholar
  5. Black, P. (2008). Strategic decisions: Ambitions, feasibility and context. Educational Designer, 1(1). Retrieved from
  6. Black, P. at al. (2012). High-stakes examinations to support policy. Educational Designer, 2(5). Retrieved from
  7. Borsboom, D., Mellenbergh, G.J., & van Heerden, J. (2004). The concept of validity. Psychological Review, 111, 1061–1071.CrossRefGoogle Scholar
  8. 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
  9. 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
  10. Burkhardt, H. (2009). On strategic design. Educational Designer, 1(3). Retrieved from
  11. Crisp, V. (2008). Exploring the nature of examiner thinking during the process of examination marking. Cambridge Journal of Education, 38, 247–264.CrossRefGoogle Scholar
  12. 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
  13. Derrick, K. (2012). Developing the e-scape software system. International Journal of Technology and Design Education, 22, 171–185.CrossRefGoogle Scholar
  14. 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
  15. Gewertz, C. (2012). Consortia provide preview of common assessments. Education Week, 32, 18–19.Google Scholar
  16. Heldsinger, S., & Humphry, S. (2010). Using the method of pairwise comparison to obtain reliable teacher assessments. The Australian Educational Researcher, 37, 1–19.CrossRefGoogle Scholar
  17. James, C. (1974). The consistency of marking a physics examination. Physics Education, 9, 271–274.CrossRefGoogle Scholar
  18. Jones, I., & Alcock, L. (2014). Peer assessment without assessment criteria. Studies in Higher Education, 39, 1774–1787.CrossRefGoogle Scholar
  19. 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
  20. Jones, I., Swan, M., & Pollitt, A. (2014). Assessing mathematical problem solving using comparative judgement. International Journal of Science and Mathematics Education, 13, 151–177.CrossRefGoogle Scholar
  21. Kimbell, R. (2012). Evolving project e-scape for national assessment. International Journal of Technology and Design Education, 22, 135–155.CrossRefGoogle Scholar
  22. Koretz, D. (2008). Measuring up: What educational testing really tells us. Cambridge: Harvard University Press.Google Scholar
  23. Laming, D. (1984). The relativity of “absolute” judgements. British Journal of Mathematical and Statistical Psychology, 37, 152–183.CrossRefGoogle Scholar
  24. 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
  25. 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
  26. McVey, P.J. (1976). The “paper error” of two examinations in electronic engineering. Physics Education, 11, 58–60.CrossRefGoogle Scholar
  27. 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
  28. Messick, S. (1980). Test validity and the ethics of assessment. American Psychologist, 35, 1012–1027.CrossRefGoogle Scholar
  29. Messick, S. (1989). Meaning and values in test validation: The science and ethics of assessment. Educational Researcher, 18, 5–11.CrossRefGoogle Scholar
  30. 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.CrossRefGoogle Scholar
  31. Murphy, R. (1982). A further report of investigations into the reliability of marking of GCE examinations. British Journal of Educational Psychology, 52, 58–63.CrossRefGoogle Scholar
  32. NCETM. (2009). Mathematics matters: Final report. London: National Centre for Excellence in the Teaching of Mathematics.Google Scholar
  33. Newton, P. (1996). The reliability of marking of general certificate of secondary education scripts: Mathematics and English. British Educational Research Journal, 22, 405–420.CrossRefGoogle Scholar
  34. Newton, P., & Shaw, S. (2014). Validity in educational and psychological assessment. London: Sage.CrossRefGoogle Scholar
  35. Noyes, A., Wake, G., Drake, P., & Murphy, R. (2011). Evaluating Mathematics pathways final report. DfE Research Report 143. London: Department for Education.Google Scholar
  36. Ofsted. (2008). Mathematics: Understanding the score. London: Office for Standards in Education.Google Scholar
  37. Pollitt, A. (2012a). The method of adaptive comparative judgement. Assessment in Education: Principles Policy and Practice, 19, 281–300.CrossRefGoogle Scholar
  38. Pollitt, A. (2012b). Comparative judgement for assessment. International Journal of Technology and Design Education, 22, 157–170.CrossRefGoogle Scholar
  39. 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
  40. Popham, W.J. (2001). Teaching to the test? Educational Leadership, 58, 16–20.Google Scholar
  41. QCA (2007). National curriculum 2007. Coventry: Qualifications and curriculum authority.Google Scholar
  42. 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.CrossRefGoogle Scholar
  43. 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.CrossRefGoogle Scholar
  44. Shepard, L.A. (1997). The centrality of test use and consequences for test validity. Educational Measurement: Issues and Practice, 16, 5–24.CrossRefGoogle Scholar
  45. 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.CrossRefGoogle Scholar
  46. Suto, I. (2013). 21st Century skills: Ancient, ubiquitous, enigmatic? Research Matters: A Cambridge Assessment Publication, 15, 2–8.Google Scholar
  47. 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.CrossRefGoogle Scholar
  48. 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.CrossRefGoogle Scholar
  49. 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.CrossRefGoogle Scholar
  50. Swan, M., & Burkhardt, H. (2012). Designing assessment of performance in mathematics. Educational Designer, 2(5). Retrieved from
  51. Taggart, G.L., Phifer, S.J., Nixon, J.A., & Wood, M. (1998). Rubrics: A handbook for construction and use. Lancaster: Technomic Publishing.Google Scholar
  52. Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
  53. Truss, E. (2012). Elizabeth Truss calls for a renaissance in maths. Norfolk: Speech to the National Education Trust. Retrieved from
  54. Turner, H., & Firth, D. (2005). Bradley-Terry models in R: The BradleyTerry2 package. Journal of Statistical Software, 12(1). Retrieved from
  55. 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.CrossRefGoogle Scholar
  56. 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
  57. 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
  58. 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
  59. Willmott, A.S., & Nuttall, D.L. (1975). The reliability of examinations at 16+. London: Macmillan Education.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Mathematics Education CentreLoughborough UniversityLoughboroughUK

Personalised recommendations