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This paper considers the impact of Assessment for Learning on children’s progress in a particular strand of the Primary Maths Curriculum. It does so firstly through a review of the relevant literature, and then employs some empirical examples to illustrate how the cycle had helped to secure learning points in a particular context. The specific strand under consideration is the solving of multi-step problems, ‘and problems involving fractions, decimals and percentages; choose and use appropriate calculation strategies at each stage, including calculator use.’ (DCFS 2009).
Changes in the professional framework for the teaching and assessment of Primary maths have been reflected in a constantly expanding literature. This is now so expansive, that it can only really be reviewed here through some representative examples. There are two principal sub-genres which feature here: specifically, these are official publications, and range of commercially produced texts which may be characterised as critical, professional, or vocational self-help literature. It is also the case that some generic texts on the subject of Primary Assessment for Learning may be pertinent here, although they do not relate specifically to mathematics.
The official literature emphasises the holistic nature of assessment by asserting that ‘assessment of children’s achievements and progress should be based on the expected learning outcomes identified through the learning objectives. In mathematics, assessing children’s progress in a core strand of learning should be informed by the objectives in the strand.’ (DCFS 2009). The fruition of this process may be visualized in the motivation and empowerment of the learners themselves, supported by ‘Constructive feedback that identifies how children’s work and responses have led to success’ this, it advises, should provide a ‘shared understanding of the achievements on which to build to make further progress. It helps children to see how the next steps take account of this success and are attainable.’ (DCFS 2009). There is a sense in which this acknowledges that Assessment for Learning has an importance, over and above what is revealed in outcome-based results, i.e. those from standardised tests. In other words, the latter no longer implies that it can stand as ‘proxy for other kinds of learning.’ (Campbell et al. 2004: p.119)
The commercially published literature is constantly being updated by texts which engage with official policy and curriculum changes, interpreting them for practitioners and parents. However, the majority of these, although they make some reference to assessment, do not do so in the terms now prescribed by the DCFS, i.e., day-to-day and periodic assessment. This is possibly because these models have only been operating in the official discourse for a relatively short period. Overall, this genre may itself be split into sub-groups, the most significant of which are the reflective or critical genre, and the vocational or self-help group. One of the most prolific authorities within this group is Sharon Clarke, whose Targeting Assessment in the Primary Classroom: Strategies for Planning, Assessment, Pupil Feedback and Target Setting (1998), Unlocking Formative Assessment: Practical Strategies for Enhancing Pupils’ Learning in the Primary Classroom, (2001), and Active Learning Through Formative Assessment (2008) straddle successive developments in the teaching and assessment of Primary mathematics. Also helpful in these areas is Hansen’s Primary Mathematics: Extending Knowledge in Practice (Achieving QTS Extending Knowledge in Practice) (2008), and David Clarke’s Constructive Assessment in Mathematics: Practical Steps for Classroom Teachers (Key Resources in Professional Development), (1999).
As Shirley Clarke indicates, the ‘sharing of a learning intention is more complex than simply repeating what is in the teacher’s plan. In order for the learning intention to be shared effectively, it needs to be clear and unambiguous, so that the teacher can explain it in a way which makes sense.’ (2001: p.20) This may be taken as supportive of the official position: it endorses the idea that planning should draw not only on the learning outcome, but also on the prior knowledge of the students in question. If they are expected to objectively assess their own progress, they must understand the frame of reference, and be able to envisage the learning outcome, even if they haven’t yet attained it. This idea is also implicit in the ideas of David Clarke: as he points out, earlier approaches to assessment focussed on ‘â€¦measuring the extent to which students possess a set of tools andâ€¦the extent to which they can apply them.’ However, he further indicates that ‘â€¦to be mathematically equipped, a student must also understand the nature of mathematical tools and be able to select the correct tool for a given problem-solving situation.’ (1999: p.11) This perspective is also endorsed in the reflections of Hansen, who argues that, ‘â€¦it is possible to help children to learn mathematical content through effectively integrating problem-solving, reasoning and communication into mathematics lessons.’ (Hansen 2008: p.5)
Texts such as Gardner’s edited collection, , Assessment and Learning, (2006), Gipps and Murphy’s A Fair Test? Assessment, Achievement and Equity, (1994), and Taber’s Classroom-based research and evidence-based practice, (2007), go some way to bridging the gap between the official and the educational literature, specifically by looking at how policy and curriculum matters are linked by research and ideology. These are, however, not specifically devoted to Primary mathematics, and neither are they wholly accepting of the orthodoxies which pervade the official literature. Gipps and Murphy make the point that evaluating assessment is ‘â€¦not just a question of looking at the equity in the context of assessment but also within the curriculum, as the two are intimately related.’ (1994: p.3) As Taber points out, practitioners are at the end of a very long and often remote supply chain when it comes to weighing the evidence on what is ‘best practice’. As they put it, ‘â€¦teachers are told what research has found out during their initial “training”, and are updatedâ€¦through courses and staff development days, but largely through centralised official “guidance”.’ (2007: p.4) This is reinforced by commentators such as Rist, who argues that, ‘We are well past the time when it is possible to argue that good research will, because it is good, influence the policy process.’ (2002: p.1002).
These are academic but not unimportant points in terms of the overall discussion, even if they are not particularly prominent in the day to day responsibilities of the class teacher. The point is that, as reflective practitioners, we might all benefit from some awareness of what shapes the frameworks which inform our approach to teaching and learning. With regard to the current Assessment for Learning conventions, the ideas in Assessment for Learning, Beyond the black box
(Assessment Reform Group, 1999), are acknowledged by the QCA to have been constructive of the whole approach. (QCA 2003: p.1). As the latter state, ‘The study posed three questions: is there evidence that improving formative assessment raises standards?; is there evidence that there is room for improvement in the practice of assessment?; and is there evidence about how to improve formative assessment? This research evidence pointed to an unqualified ‘yes’ as the answer to each of these questions.’ (QCA 2003: p.1). These are important points, as the teaching, learning and assessment frameworks which define contemporary practice are profoundly adaptive of them.
Discussion/Example from Experience.
A strand of the Primary curriculum where day to day and periodic assessment was found to be particularly important in the overall Assessment for Learning approach, was securing number facts, relationships and calculating. The examples used here are from Year 6 block E, especially Ma2, Written and calculator methods, and Ma2, solving numerical problems from Unit One, and focused on dealing with errors and misconceptions. One context where assessment was found to be particularly relevant was in dealing with upper school (i.e. Years 4, 5 and 6) learning of multiplication and division. The assessment process had to be multi-faceted, taking in all of the associated knowledge and skills, the errors and misconceptions which arose, and the modelling of questions to identify the origin of such problems. This may be illustrated by focusing on one example, taken from Year 6 Key Objective 2, Multiplying and dividing by powers of ten and the associative law, where commonly, the unprepared or confused learner ‘â€¦Misuses half understood rules about multiplying and dividing by powers of ten and the associative lawâ€¦’ (2009). The important thing about multiplication and division through successive addition or subtraction respectively, is that, once mastered, they can demonstrate to learners that the application of basic skills will enable them to break down seemingly complex problems into a manageable format. Multiplying or dividing a three digit number by a two digit number depends on the use of a number of skills: knowledge of number facts, i.e. times tables, place value, to quickly assess the viability of an answer, and organisational skills, i.e. being able to apply the correct steps in the appropriate order. It may also be useful to augment these with calculator use, in order to verify answers.
The important point here is that day to day and periodic assessment – and reflective feedback from the learners themselves – was indispensable in the planning, pitching and delivery of this input. The interdependence of each step in these calculations meant that the failure to execute one step, often resulted in the failure to complete the overall objective. For example, if times tables and multiplication by 10 and 100 were not securely in place, the learner would get bogged down in the arithmetic. Conversely, the securing of one of the incremental skills involved in these calculations was a positive factor in the learners’ overall approach: i.e., if they knew their times tables facts, place value, or multiplication by 10 and 100 were in place, it gave them a starting point from which to analyse errors or problems. For some learners, this had the generic effect of making them realise that their long-term work in achieving these positions of strength had a positive outcome, rather than being an abstract, stand-alone process. This in turn made them more interested in acquiring other general mathematics skills. Looking beyond specific mathematics skills, this may also have the propensity to develop the students’ own capacities for self-realisation and self-motivation. As the QCA points out, ‘â€¦In many classrooms, pupils do not perceive the structure of the learning aims that give meaning to their work. Therefore they are unable to assess their own progress.’ (QCA 2003: p.3) Achievement in a multi-step process such as long multiplication or division might therefore enable them to map out where they are within the overall standards.
However, it was only through a combination of day to day and periodic assessment that the practitioner could be confident of planning effectively with regard to these tasks. There was no point in assembling sessions which relied on a range of skills when they were not secure, either in individual learners, or sufficiently across the cohort as a whole. In mixed ability groups, this approach was obviously the key the necessary differentiation. The logical corollary to this is that discursive feedback from the learners themselves was also important in defining the next stage of planning, i.e. what worked, what didn’t, who tried which method, were there any preferences etc. The appeal of this activity also lays in its fine balance of mental and pencil and paper methods, and the way in which estimation is the necessary accompaniment to concrete calculation. Overall, these experiences may be deemed supportive of the proposals of commentators such as Clarke and Hansen, (see above) in that they emphasize the need for the continuous reinforcement of planning with assessment.
Summary, Analysis and Reflection: Implications for Future Teaching.
In summary, the conclusion of this paper is that both the literature and practical experience discussed here are mutually supportive of the need for complimentary assessment and planning. Outcome orientated results can illustrate individual and whole school performance in certain contexts, but practitioners need to be aware of assessment in a holistic way, as a daily part of their approach to teaching and learning. As the QCA expresses it, ‘â€¦Teachers are experiencing an increased sense that pupils are working with them rather than for them. For example, pupils are asking for more questions or examples to practice applying their understanding of a topic or to repeat homework or tests if they have not met the standard and the objectives that they and the teacher have set.’ (QCA 2009: p.48). Whilst this dynamic sounds very positive, practitioners have new and different responsibilities within it. In terms of assessment, these can be itemised in the following waysâ€¦
Day to day: within this level of assessment, specific learning objectives should explicitly communicated, and augmented with both peer and self assessment as appropriate.
Periodic: ideally, this should assemble a broader overview of progress across the subject for both learner and teacher. It is also an opportunity to interweave the national standards in a sensitive way with classroom practice. The practitioner can use the insights gained from this process to inform both long and medium term planning.
Overall, it should be recognised that the ideal situation, i.e. of self-motivated, self-actuating learners, involved in their own self-assessment, is unlikely just to ‘happen’. Considered superficially, it might seem that the practitioner’s role in assessment has lessened, whilst the remainder has been taken up by the learners themselves. The reality is rather different: pupils will only become adequate and effective assessors of their own progress if they are provided with the appropriate support and guidance. In a sense, this facilitating role is a much more challenging and subtle one than that implied in a more top-down, didactic model. Also, there are obvious problems in considering the ‘learner’ as a passive or generalised aspect of this approach: it is much more likely that there is a staggered and variegated uptake of the model, as different learners are engaged at their own pace and level. This in turn indicates that, as with all aspects of the curriculum, the social and emotional aspects of learning should be taken into consideration.
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