##### Lucy Rycroft-Smith, Cambridge University, UK

##### Bart Crisp, CUREE, UK

Last month, a student answer to a test question was placed on Twitter, asking maths teachers for their feedback as if they were marking it. Before you read ahead, try to imagine what the question may have been and what kind of answer was given. Here are some of the responses that the teachers gave:

‘Seems fine to me.’

‘It needs a conclusion.’

‘Great answer.’

‘I think it needs a conclusion or interpretation (written or verbal but I’d push for written).’

‘Shows confidence.’

‘Very efficient and simple.’‘Elegant and I love it.’

‘Depends on level of student, how abstract they can think.’

‘Anyone who writes this is probably good at maths.’

‘I don’t like it. A lot more could be shown here.’

‘I am very happy with this answer.’

‘Full marks but your answer would be improved by adding a sentence or two to explain your thinking.’‘Whilst these calculations are helpful and correct, you need to explain why they are relevant. Your answer should be able to convince someone else who has not attempted the problem – without words, they will still have to figure out why you did what you did.’

‘My mind is kind of blown that lots of people seem to think this is a good answer. Maths is about forming rigorous, logical and precise arguments – not just writing down some calculations.’

Confused yet?

Despite the view often expressed that mathematics – and, by extension, mathematics education – is neutral, is value-free, deals with the irrefutably ‘right and wrong’ and therefore is easy to mark, evidence would suggest otherwise. What makes a good mathematical test response is disputed, as is the inextricably linked notion of what makes a good mathematical test question.

Mathematics has been characterised as abstract, pure and value-free (Joseph, 2011); rigorous, logical and non-creative (Bibby, 2002); the pinnacle of human cognitive development (Walkerdine, 1998); permanent, eternal and universal (Aaboe, 1978); and, perhaps most importantly for assessment, something that one is either born with or not (Hottinger, 2016). When drawing on evidence to support these claims, people often reach for ideas such as ‘2+2 is always 4’ or ‘mathematical proof is absolute logical truth’ (and at least one of us has uttered these words in such a manner in the past). In current assessment practices in the UK and around the world, as well as in the responses of teachers to the student work here, these narrative threads are obviously present in various ways. Yet there are many important criticisms of these views that have a direct impact on how we might assess – and therefore teach – mathematics. Below, we briefly look at each, before moving on to possible alternatives.

### Mathematics is not an appropriate way to stratify society: Assessing it should not depend exclusively or primarily on a white, Western, middle-class ‘standard’

Maths education has been argued as being heavily racialised (Martin, 2009) having been mostly developed around a white, male, middle-class standard and narrative (Hottinger, 2016; Gutierrez, in Kastberg et al., 2018). Maths has organising power; it is the subject that has the highest prevalence of setting and streaming, which has few benefits and many disadvantages for learners, including social stratification and lowering of morale and expectations (Hallam and Parsons, 2013). Setting in mathematics is still prevalent in the UK, ‘despite the research evidence that such grouping or setting is of only limited benefit to children with higher levels of prior attainment and certainly damages the progress of children in lower groups or sets’ (Boyd and Ash, 2018, p. 220).

### Mathematics is not pure or value-free: Assessing it should acknowledge and take account of bias as well as ethical considerations

Maths is often viewed through an ideology of certainty and infallibility, which contributes to power differentials and social control (Borba and Skovsmose, 1997). This means that students and teachers may feel indoctrinated into an idea that only one correct answer or method – or both – is permitted within the conventions of mathematical practice. It also does not take account of the idea that humans perform, make discourse about and assess mathematics, and are inevitably biased. Challenging this view of mathematics as certain and infallible supports more ‘human’ pedagogies that are more uncertain, critical and creative (Boyd and Ash, 2018).

### Mathematics can be both logical and creative in its application to problems (in pedagogical and non-pedagogical contexts): Assessing it should mirror more closely the ways in which mathematicians solve problems outside of the classroom

Mathematical creativity exists as an area of research and practice – for example, Ervynck’s stages of mathematical creativity, as cited in Liljedahl and Sriraman (2006)**, **and Haylock’s definitions of creativity in mathematics as letting ‘the mind run freely’ over a range of available mathematical ideas (cited in Bibby, 2002). Mathematicians outside the school setting usually solve challenging problems in context, which should be reflected more closely in school mathematics assessment (Schoenfeld, 1989).

### Mathematical ‘ability’ is not fixed or innate: Assessing it should aim to understand the learner’s inclination towards mathematical thought instead of determining the presence or otherwise of mathematical ‘gifts’

Mathematics does not have a ‘gene’; it is a myth that only some people have ‘natural talent’ for it (Boyd and Ash, 2018). Mathematics likely evolved alongside language and is extremely social (Devlin, 2000). The factors that affect an individual’s comfort and progress with mathematical learning are many and varied, and there is conservable overlap between mathematical fluency, reading fluency and general cognitive skills (Hart et al., 2009). Therefore, in almost all cases, assessing it should happen in service of encouragement and development, not exclusion and elitism.

### Mathematics is not significantly different from other subjects: Assessing it should take account of a broader range of methods and approaches used elsewhere

Finally, combining the ideas above suggest that mathematics can be seen as contextual, dynamic, social, creative, plural and accessible. Assessing subjects like art, drama, music and design (but also creative approaches to assessment in less traditionally ‘creative’ subjects, such as history, geography and even law or economics) may allow for portfolios, narratives, creation of artefacts, oracy, in-depth investigation and a degree of personal choice that is often absent in maths assessment, perhaps without good reason. Some researchers have proposed ways in which to specifically assess ideas like divergent thinking in maths, such as finding as many ways as possible to solve a problem (Leikin, 2013).

### What might this mean for assessment?

Here we identify some alternative principles, drawn from thinking on assessment in the context of other disciplines (chiefly the arts and teacher professional development and learning) that do not start from the assumption that what is being assessed is a learner’s grasp of a topic that is fixed, innate and/or representative of a grand, deeper truth.

**Assessment depends on practitioners working to a consistent, coherent and purposeful approach**

Measuring learning progress over time depends more on creating a well-understood environment for assessment and for using its findings to revise practice and address gaps in learning, and this in turn relies on an understanding of whether the focus of assessment is, for example, knowledge or experience (Zimmerman, 1992).

**Assessment needs to expand its horizons beyond progress towards a pre-set goal orientation**

This can include the move away from exclusively pursuing goal-related outcomes (Beghetto, 2005), expanding to include those that focus on self-improvement, skill development, creativity and understanding, and emphasise making feedback accessible and developmental for learners.

**Assessment needs to establish a role for pupils within its procedures**

It is important to establish systems to gather and use learner feedback on their experiences of assessment – not all pupils experience the classroom environment in the same way (Beghetto, 2005), and classrooms with a goal structure that emphasises pupil progress relative to learners’ own prior achievements seem to foster creative expression better than those that represent a performance goal structure (Amabile, 1996; Collins and Amabile, 1999).

**Assessors should be shown the value in altering or adapting preconceptions around assessment of subjects**

It is important not to treat mathematics as a subject separate to the rest of the full curriculum (including areas such as drama, which are often seen as irrelevant to mathematical thinking), to focus on the process of the learning as well as or instead of the outcome(s) (Zimmerman, 1992) and to minimise the use of assessments in making social comparisons; when students focus on self-improvement, they are more likely to take risks, seek out challenges and persevere in the face of difficulty (Nickerson, 1999). While originally made in reference to the arts, we believe that the arguments set out here demonstrate the applicability of the same modes of thinking to mathematics; there is some evidence, for example, of journaling in mathematics, that supports this view (e.g. Boyd and Ash, 2018)

### What could applying these principles in mathematics assessment look like in practice?

The principles here suggest a wide range of alternatives to the use of relatively closed questioning in maths assessment, which is the default for so much of current practice: from changing the structure of the questions themselves to writing essays, making videos, creating 3D models, giving lectures, designing games, writing dialogue and composing music, for example. But even these are, unquestionably, limited in comparison to the wider range of possible approaches, and are drawn primarily from other educational philosophies, which are, themselves, also often white, Western and middle-class in origin, meaning that there is huge scope to expand the thinking on this topic by looking further afield. In short: the time is ripe for a radical rethink.

The piece of student work alluded to in the introduction to this piece is shown in **this Tweet**, with full permission. Given the evidence that we have considered here on the evolution of views on mathematics and how we might therefore assess it, how might you assess these kinds of ideas differently and what kinds of responses might you consider mathematically successful?

### References

Aaboe A (1978) *Episodes from the Early History Of Mathematics*. Washington D.C.: Mathematical Association of America.

Amabile, T. M (1996). *Creativity in context: Update to the social psychology of creativity. *Westview Press

Beghetto, R. (2005). Does Assessment Kill Student Creativity? *The Educational Forum, 69(3)*, 254-263

Bibby T (2002) Creativity and logic in primary-school mathematics: A view from the classroom. *For the Learning of Mathematics* 22(3): 10–13.

Borba MC and Skovsmose O (1997) The Ideology of certainty in mathematics education. *For the Learning of Mathematics* 17(3): 17–23.

Boyd P and Ash A (2018) Mastery mathematics: Changing teacher beliefs around in-class grouping and mindset. *Teaching and Teacher Education* 75: 214–223. DOI: 10.1016/j.tate.2018.06.016.

Collins, M. A., and Amabile, T. M. (1999). Motivation and creativity. In *Handbook of creativity*, ed. Sternberg, R. J., 297-312. Cambridge University Press

Devlin KJ (2000) *The Math Gene: How Mathematical Thinking Evolved and Why Numbers are Like Gossip*, 1st ed. New York: Basic Books.

Hallam S and Parsons S (2013) The incidence and make up of ability grouped sets in the UK primary school. *Research Papers in Education* 28(4): 393–420. DOI: 10.1080/02671522.2012.729079.

Hart SA, Petrill SA, Thompson LA et al. (2009) The ABCs of math: A genetic analysis of mathematics and its links with reading ability and general cognitive ability. *Journal of Educational Psychology* 101(2): 388–402. DOI: 10.1037/a0015115.

Hottinger SN (2016) *Inventing the Mathematician: Gender, Race, and our Cultural Understanding of Mathematics*. New York: Suny Press.

Joseph GG (2011) *The Crest of tThe Peacock: Non-European Roots of Mathematics*, 3rd ed. Princeton: Princeton University Press.

Kastberg SE, Tyminski AM, Lischka AE et al. (eds) (2018) *Building Support for Scholarly Practices in Mathematics Methods*. Charlotte, South Carolina: Information Age Publishing, Inc.

Leikin R (2013) Evaluating mathematical creativity: The interplay between multiplicity and insight. *Psychological Test and Assessment Modeling *55(4): 385–400, 55, 385–400.

Liljedahl P and Sriraman B (2006) Musings on mathematical creativity. *For the Learning of Mathematics* 26(1): 17–19.

Martin DB (2009) Researching race in mathematics education. *Teachers College Record* 111(2): 295–338.

Nickerson, R. S (1999). Enhancing Creativity. In *Handbook of creativity*, ed. Sternberg, R. J., 392-410. Cambridge University Press

Schoenfeld A (1989) Problem solving in context(s). In: Charles RI and Silver EA (eds) *The Teaching and Assessing of Mathematical Problem Solving. *London: Routledge, pp. 82–92.

Walkerdine V (1998) *Counting Girls Out: Girls and Mathematics*, new ed. London; Bristol, PA: Falmer Press.

Zimmerman E (1992) Assessing students’ progress and achievements in art. *Art Education *45(6): 14–24.