Assessing knowledge and understanding in mathematics with a focus on

This brief introduction to assessment in number and arithmetic provides some reflections on the issues which abound when children’s knowledge and

understanding of number and arithmetic are assessed.

2.5.1 What is mathematical understanding?

It seems important to try to think about this question before reflecting on how knowledge and understanding in number and arithmetic might be assessed.

Hiebert and Wearne (1996, p.253) noted that:

Efforts to improve our understandings of how conceptual

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several conceptual and methodological problems. A first problem is that clear definitions of understanding and skill are difficult to formulate and even more difficult to operationalize. The notion of conceptual understanding has been especially difficult to define and measure. Without clear definitions that can be operationalized through assessment tasks, it is difficult to interpret empirical findings.

Following on from this, Hiebert and Wearne (1996, p.253) considered “understanding from both a cognitive and a mathematical point of view. We borrowed from the

common cognitive view of understanding as the internal construction of connections or relations between representations of mathematical ideas.”

This view of understanding has been developed further by Barmby, Bilsborough, Harries and Higgins (2009) who proposed a model for mathematical understanding where “understanding is built up from connections between mental representations, the connections being made by the reasoning processes that we carry out” (Barmby, Bilsborough, Harries and Higgins, 2009, p.3). This definition is useful as it includes reasoning processes, but it is also a challenge because we do not know how this reasoning should be recognised or expressed. Hiebert and Wearne (1996) found that the children that they termed “understanders” were not only better able to be flexible in their strategy use, but were also able to explain their thinking verbally.

While this seems quite clear, it needs to put within the context of the relationship between conceptual understanding and procedural knowledge. If we use the definition of “conceptual understanding” as the one described above, we now need to define “procedural knowledge”. Hiebert and Wearne (1996) describe procedural knowledge as knowledge of procedures which can be applied in order to solve problems in mathematics. They found that children who learned procedures only may at first appear to make more progress. However, these children may also forget the procedures quickly and be less able to apply them in unexpected circumstances which differ from those which have been practised. Rittle-Johnson and Siegler (1998) investigated four possible relationships between what they called “procedural knowledge” and “conceptual knowledge”:

1. Procedural knowledge develops before conceptual knowledge. 2. Procedural knowledge develops after conceptual knowledge.

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3. Procedural and conceptual knowledge develop concurrently. 4. Procedural knowledge and conceptual knowledge develop

iteratively, with small increases in one leading to small increase in the other, which trigger new increase in the first.

(Rittle-Johnson and Siegler, 1998, p.77)

In evaluating a range of studies, Rittle-Johnson and Siegler (1998) concluded that most of the research they explored showed that either procedural knowledge develops before conceptual knowledge or procedural knowledge develops after conceptual knowledge. It was hard to find studies that explored the development of the two processes together or iteratively. It could be argued, though, that it is very hard to know where the boundaries are between understanding and the ability to apply a learned procedure (Hiebert and Wearne, 1996).

It is important to point out that in studies exploring children’s understanding, “understanding” has usually been assessed by the children’s performance in

particular tasks. For example, in the study carried out by Hiebert and Wearne (1996, p.253) defined understanding “for purposes of investigating multidigit addition and subtraction, as the construction of connections between the key ideas of the base-10 number system”. Skill was defined as “performing addition and subtraction

procedures” (Hiebert and Wearne, 1996, p.254). In recognition of the fact that this is not very satisfactory, Hiebert and Wearne (1996) interviewed the children as they were working on the problems, in order to gain more insight into what they were thinking. The fact that Hiebert and Wearne chose to interview the children in addition to assessing their performance as right or wrong, challenges the view that a

conventional experimental approach alone provides rich enough data from which conclusions can be drawn.

2.5.2 What does assessment in mathematics tell us?

Williams and Ryan (2000) found that while standardised tests for 7 year old children were useful in providing some information about areas of difficulty in mathematics, they did little to provide information on the nature of the difficulty. They also found

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that the children left out many questions and often failed to provide any written evidence of their calculations. Consequently, even if children got the answers correct, it was hard to identify whether or not they actually understood the

mathematics. Williams and Ryan also noted that it was hard to distinguish between poor test design and poor mathematical understanding.

Denvir and Brown (1987) carried out a study with children in Years 3 to 6 (aged 7 to 11 years) which explored the impact of environment on outcomes when children were assessed. In the first assessment, a whole class was required to take a formal assessment. Later a group of six children, who had performed at a similar level in the test were asked to answer similar questions, but this time the assessment was more like an interview and included discussion with the assessor. When Denvir and Brown compared their results, they noticed some discrepancies. In order to explain these, they suggested that perhaps when there was discussion, learning might actually take place. They also proposed that testing in a formal environment might yield different behaviours than testing in a less formal set-up. For example,

Seb wrote in the class assessment: 52 – 36 = 24, but in the interview he said:

S: Sixteen. I took 30 from 50 that gives 20, then took two from the six and took away four from the 20.

(Denvir and Brown, 1987, p.105)

Denvir and Brown explored the factors that they thought could account for the

differences in outcomes between the two assessment conditions. They came up with a number of different causes which included: the expectation of feedback in an interview situation; poor short-term memory (important in whole class assessments); greater anxiety in the whole class assessment situation; and inability to see or hear or attend properly in the whole class assessment.

Their conclusions, as can be seen, are very similar to those of Williams and Ryan (2000). They too believed that the more formal assessment provided useful information which could form the basis of a more detailed diagnostic assessment.

The idea of attempting to develop a schema which outlined specific developmental stages was explored by Denvir and Brown (1986a). Denvir and Brown worked with

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children aged 7 to 9 years. They identified 47 skills and then attempted to explore the links between these (for example, which skills appeared to be pre-requisites for other skills). The skills were assessed through an interview and seven of the children were then assessed regularly over a 2 year period. It was observed that all the

children did make progress, but that a less sensitive test may not have identified this, as the progress was very slow. Denvir and Brown (1986b) also argued that this detailed assessment made it possible for interventions to be appropriately targeted.

Standardised tests enable us to observe how children perform in relation to their typically developing peers in a particular test. It is also possible to compare children according to identified trajectories, such as those proposed by Sarama and

Clements (2009, p.17):

Learning trajectories have three parts: a goal (that is, an aspect of a mathematical domain children should learn), a developmental progression, or learning path through which children move through levels of thinking, and instruction that helps them move along that path.

Knowledge of these developmental paths, according to Sarama and Clements (2009) should support teachers to better understand children’s thinking and assess their understanding. This approach is also underpinned by a view of mathematical learning as being hierarchical and predictable. However, even if these typical

trajectories exist, given that children with genetic conditions such as Down syndrome and Williams syndrome, appear to have atypical trajectories when compared with a typically-developing population (Ansari and Kamiloff-Smith, 2002), it is possible that children with Apert syndrome will also have their own trajectory. More significantly though, the suggestion that children do follow particular trajectories conflicts with Denvir and Brown’s (1987) study of mathematical skills in 7 – 9 year old children. Denvir and Brown attempted to provide a framework which could represent children’s acquisition of understanding in number and early arithmetic. While they were able to identify a potential hierarchy of skills, the relationships between these skills were quite complex and very interrelated. With the framework created by Denvir and Brown, no clear trajectories can be identified. Finally, in an 18 month longitudinal study of 29 7 – 9 year old children, Jordan, Mulhern and Wylie (2009) found that the children’s trajectories were very different in terms of “initial status, final

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status, growth trajectories, and growth rates” (Jordan, Mulhern and Wylie, 2009, p.466). Taken all together, these findings suggest that learning in number and arithmetic is complex and varied, thus making assessment challenging and

dependent on individual children’s particular strengths and weaknesses as identified by Denvir and Brown (1986b) and Dowker (2009).