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, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
ACME. (2005). Assessment in 14–19 Mathematics. London: Advisory Committee on Mathematics Education.
ACME. (2011). Mathematical needs: Mathematics in the workplace and in higher education. London: Advisory Committee on Mathematics Education.
AQA. (2010). GCSE Foundation Tier Mathematics Paper 1 (Specification A). Monday 7 June 2010. Manchester: Assessment and Qualifications Alliance.
Berube, C.T. (2004). Are standards preventing good teaching? Clearing House, 77, 264–267.
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.
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.
Bramley, T., Bell, J., & Pollitt, A. (1998). Assessing changes in standards over time using Thurstone paired comparisons. Education Research and Perspectives, 25, 1–24.
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.
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.
Derrick, K. (2012). Developing the e-scape software system. International Journal of Technology and Design Education, 22, 171–185.
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.
Heldsinger, S., & Humphry, S. (2010). Using the method of pairwise comparison to obtain reliable teacher assessments. The Australian Educational Researcher, 37, 1–19.
James, C. (1974). The consistency of marking a physics examination. Physics Education, 9, 271–274.
Jones, I., & Alcock, L. (2014). Peer assessment without assessment criteria. Studies in Higher Education, 39, 1774–1787.
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.
Jones, I., Swan, M., & Pollitt, A. (2014). Assessing mathematical problem solving using comparative judgement. International Journal of Science and Mathematics Education, 13, 151–177.
Kimbell, R. (2012). Evolving project e-scape for national assessment. International Journal of Technology and Design Education, 22, 135–155.
Koretz, D. (2008). Measuring up: What educational testing really tells us. Cambridge: Harvard University Press.
Laming, D. (1984). The relativity of “absolute” judgements. British Journal of Mathematical and Statistical Psychology, 37, 152–183.
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.
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
McVey, P.J. (1976). The “paper error” of two examinations in electronic engineering. Physics Education, 11, 58–60.
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.
Messick, S. (1980). Test validity and the ethics of assessment. American Psychologist, 35, 1012–1027.
Messick, S. (1989). Meaning and values in test validation: The science and ethics of assessment. Educational Researcher, 18, 5–11.
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.
Murphy, R. (1982). A further report of investigations into the reliability of marking of GCE examinations. British Journal of Educational Psychology, 52, 58–63.
NCETM. (2009). Mathematics matters: Final report. London: National Centre for Excellence in the Teaching of Mathematics.
Newton, P. (1996). The reliability of marking of general certificate of secondary education scripts: Mathematics and English. British Educational Research Journal, 22, 405–420.
Newton, P., & Shaw, S. (2014). Validity in educational and psychological assessment. London: Sage.
Noyes, A., Wake, G., Drake, P., & Murphy, R. (2011). Evaluating Mathematics pathways final report. DfE Research Report 143. London: Department for Education.
Ofsted. (2008). Mathematics: Understanding the score. London: Office for Standards in Education.
Pollitt, A. (2012a). The method of adaptive comparative judgement. Assessment in Education: Principles Policy and Practice, 19, 281–300.
Pollitt, A. (2012b). Comparative judgement for assessment. International Journal of Technology and Design Education, 22, 157–170.
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.
Popham, W.J. (2001). Teaching to the test? Educational Leadership, 58, 16–20.
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.
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.
Shepard, L.A. (1997). The centrality of test use and consequences for test validity. Educational Measurement: Issues and Practice, 16, 5–24.
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.
Suto, I. (2013). 21st Century skills: Ancient, ubiquitous, enigmatic? Research Matters: A Cambridge Assessment Publication, 15, 2–8.
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.
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.
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.
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.
Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 273–286.
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.
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.
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.
Wiliam, D. (2001). Reliability, validity, and all that jazz. Education 3–13: International Journal of Primary Elementary and Early Years Education, 29, 17–21.
Willmott, A.S., & Nuttall, D.L. (1975). The reliability of examinations at 16+. London: Macmillan Education.
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.
About this article
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