Problem solving in mathematics: 
                 realising the vision through better 
                 assessment
                 June 2016
               Problem solving in mathematics: realising the vision through better assessment                                                           1
       Contents 
       1. Introduction .......................................................................... 3 
       2. Problem solving in mathematics assessment ................................................ 5
       3. The assessment of problem solving ....................................................... 7
        3.1 Problem solving assessment design ....................................................... 7
        3.2 What do the characteristics of questions mean in terms of assessment and test design? ............... 8
        3.3 What do the characteristics of questions mean in terms of mark schemes?  ........................ 8
       4. The next steps for the assessment of problem solving  ....................................... 9
        4.1 Recommended actions .................................................................9
       5. Conclusion ...........................................................................10
       Acknowledgements ......................................................................11
       2                          Problem solving in mathematics: realising the vision through better assessment
          1      Introduction 
                 Problem solving is an important component of mathematics across all  
                 phases of education. 
                 In the modern world, young people need to be able to                                    Box 1: Summary of ACME’s recent work on 
                 engage with and interpret data and information.                                         problem solving 
                 They need to become flexible thinkers capable of                                         
                 dealing with novel problems and situations and                                          Since 2010 ACME’s work on problem solving  
                 analysing their own and others’ solutions to these.                                     has included:
                 ACME has long recognised the centrality of problem 
                 solving in the effective teaching, learning and                                         •  a call for exemplars of effective assessment of 
                 assessment of mathematics and this has been reflected                                       problem solving in mathematics in spring 2015;
                 in its policy work, as set out in Box 1.                                                •  a series of conference workshops and 
                                                                                                             discussion sessions on problem solving across 
                 The current education system prioritises assessment                                         all phases from 2013 – 2015;
                 and accountability. This means that assessment will                                     •  policy advice on recent reforms across all  
                 impact on how mathematics is taught in the classroom.                                       key stages;
                 Therefore, high-quality assessment of problem solving 
                                                            1
                 in public tests and assessments  is essential in order to                               •  sections in ACME reports, for example 
                 ensure the effective learning and teaching of problem                                       Mathematical Needs: Mathematics in the 
                 solving throughout primary and secondary education.                                         workplace and in Higher Education and 
                 Although the focus here is on the assessment of problem                                     Mathematical Needs of learners. 
                 solving in mathematics, many of the ideas will be directly                              Further information can be found at  
                 transferable to the assessment of problem solving in                                    www.acme-uk.org/policy-advice/
                 other subjects, e.g. in the sciences and beyond.2
                                                                                                         qualifications-and-assessment.
                 Problem solving is a broad encompassing term and 
                 providing a single prescriptive definition is of limited                             •  describes the desirable characteristics of questions 
                          3
                 merit.  For the purposes of this report it is more useful                                 used to assess problem solving (section 3);
                 to consider the desirable characteristics of questions                               •  sets out actions for policymakers, awarding 
                 used to assess problem solving. In this report ACME:                                      organisations and the mathematics community 
                 •  considers the assessment of problem solving in                                         to ensure that improvements in the quantity and 
                      public tests and examinations across all key stages                                  quality of problem solving in mathematics tests and 
                      of mathematical development in the light of recent                                   assessments are realised over time (section 4). 
                      reforms (section 2); 
                 1    The focus in this report is on summative assessment, that is assessment          3   Schoenfeld, A. H. 1992, ‘Learning to think mathematically: Problem 
                      that sums up what has been achieved at the end of a period of time,                  solving, metacognition, and sense-making in mathematics’ in D. Grouws 
                      relative to the learning aims and national standards.                                (Ed.) Handbook for Research on Mathematics Teaching and Learning New 
                                                                                                           York: MacMillan (pp 334 – 370).
                 2    Many new A level subjects also have specific mathematical requirements. 
                      See ACME and the Royal Statistical Society (RSS) 2015 Embedding Statistics 
                      at A level: a report on statistical requirements and assessment across A 
                      level courses in Biology, Business, Chemistry, Geography, Psychology 
                      and Sociology (see http://www.acme-uk.org/media/32719/embedding-
                      statistics-at-a-level.pdf accessed 25 April 2016). 
                 Problem solving in mathematics: realising the vision through better assessment                                                                                    3
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         4                                   Problem solving in mathematics: realising the vision through better assessment