Assessment in mathematics teaching in regards to the National Curriculum

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"The discipline of noticing what goes on in a classroom and reflecting on whether it is as good as it can be, improves the quality of a teacher's own teaching and their ability to share it with others." (Mason, in Lee, 2006, p10)

There are many methods of reflecting upon one's own work, but carried out critically and positively, all should lead to an ability to improve the quality of that work. The key word here is "should". Improvement will only take place if, following the reflection, some steps are identified and implemented to bring about improvement. It is the purpose of this paper to assist that reflection and contemplation process.

The title of the essay requires evaluation of my teaching. The word "evaluate" is taken to mean "to judge or determine the significance, worth, or quality of", as defined by Dictionary.com. The paper will evaluate, in a chronological order, the teaching I carried out over a two week period. In particular it will examine the assessment techniques that I employed at each stage, and will review the effectiveness of the assessment. Where appropriate the paper will suggest where things could have been done better.

As the title dictates, this paper is a reflection upon teaching that I carried out during my first placement, as part of my PGCE course. As such, it refers in the most part to personal experience and reflections upon that experience. The paper will therefore be written primarily in the first person, with references from other parties in the third person.

Having considered the assessment employed throughout the teaching, I will then consider a number of general findings, and how assessment may affect those findings.

What Is Assessment and Why Is It Important

Assessment is a part of all of our lives from the moment we are born, as the midwife places a baby on the scales. It follows us through early childhood (as mothers and health visitors assess what a baby can do at each stage in their development), into school and beyond into employment.

The aspect of assessment with which this paper is concerned is the assessment carried out within the mathematics classroom. The word "assessment" is used to denote any conscious activity intended to provide information about a pupil's achievement or attainment.

There are four main types of assessment, (as defined by Weeden, Winter and Broadfoot, 2002, p19)

Diagnostic - to identify pupils' current performance

Formative - to aid learning (including peer and self assessment)

Summative - for review, transfer and certification

Evaluative - to see how well teachers or institutions are performing.

Of these, formative and summative assessment will receive the most focus, with consideration being given to the effects these types of assessment have on pupils' learning. Evaluative assessment is not considered within the scope of this essay.

Whether assessment is beneficial to a pupil's learning or not depends on the use to which the information gained is put. For example, health visitors may identify a nutritional need of a young child which can be corrected with the appropriate input, and similarly teachers may identify an educational need of a pupil which they can take steps to address. William (in Weeden, Winter, Broadfoot, 2002, p29) suggests "all four functions of assessment require that evidence of performance or attainment is elicited, is then interpreted, and as a result of that interpretation, some action is taken". Weeden, Winter, Broadfoot (2002) also conclude that assessment becomes formative when the information gained is used by the teacher and pupil within the learning process. I would also add that it requires the results of the assessment to be acted upon within a short time frame, while feedback is still relevant.

Furthermore, "innovations that include strengthening the practice of formative assessment produce significant, and often substantial, learning gains". (Black et al. 2003, p9).

The aim, therefore, is to ensure that assessment, of all types, is used formatively wherever possible.

The Class That Was Taught

The class that I taught was a Year 9 class of 20 pupils. Although they were classified as a lower-attaining class (Set 3 out of 4), the range of abilities within the class and the special needs of a few individuals warrant some brief description.

One boy had arrived recently from Somalia and had a limited grasp of English, but no other special needs.

Several of the pupils had low reading ages, typically in the age 8 - 9 range, and were therefore challenged by some word problems. Additionally, one of these pupils had moved into mainstream classes at the start of year 9, after two years supported by the school's "core programme", and therefore sometimes required additional support as regards concentration.

Four of the pupils in the class were on the school's SEN register as BESD, i.e. they had behavioural, emotional or social difficulties which without careful handling could lead to disruption in lessons.

Half of the class entered year 7 with a National Curriculum assessment level of 3b or 3a, with the remainder at low level 4. By the end of year 8 all were accessed at level 4 or 5c. Several of the pupils moved up by only one or two level points during their first two years at Secondary school, indicating some cause for concern.

The class was routinely supported by a Teaching Assistant, whose role was primarily to support the pupil with EAL, but also to provide general support as required to other pupils.

The Topic Taught

The topic taught was Ratio and Proportion.

The Scheme of Work indicated five hours of teaching, the equivalent of six fifty minute lessons, and I set about producing a topic plan accordingly. The very first Standard which a trainee teacher must achieve is to "have high expectations of children and young people" (Training and Development Agency for Schools (TDA), 2010). With the benefit of hindsight, I can see that the word "high" in this standard is a word that requires considerable understanding and individual application to each pupil. At the time, however, I took this to mean that I should expect all my students to be able to grasp the basics of a given topic within a period of teaching time, determined by me, their teacher. "Low expectations by teachers are regarded as a much bigger problem than high expectations." (Weeden, Winter and Broadfoot, 2002, p64). Determined that my teaching would not be part of this "bigger problem", I set about my planning with high expectations for all my pupils. This view was proven to be rather too simplistic during the teaching, but this will be elaborated upon later. The topic of ratio and proportion, according to the scheme of work, is pre-dominantly a level 5 - 6 topic, so would by necessity be challenging for most of the pupils.

With a relatively unfamiliar class I felt it was important to start by assessing how well the class would cope with some of the fundamental mathematical concepts which would be necessary for the topic. The first activity in the teaching, therefore, was to re-cap fractions understanding. In particular, could the students reduce fractions to their simplest form, and could they find fractions of quantities? This was largely a diagnostic assessment, to establish a starting point for my teaching.

Having established that all pupils had a sound grasp of simplifying fractions, I introduced the concept of ratios and simplifying of ratios. This was followed by sharing a stated quantity by a given ratio, and finding a missing quantity given one quantity and a ratio.

The final teaching activity was to introduce the idea of proportion and to define the difference between proportion and ratio.

The topic was completed with a summative assessment, and by self- and peer-assessment in the form of production of a poster.

For the purposes of this paper, I will focus primarily on the start of the topic and the conclusion of the topic, and the assessment strategies employed at these points. Some mention will be made of the interim teaching, particularly with reference to the effectiveness or otherwise of assessment carried out at this stage.

The Teaching

Re-cap Fractions

My decision to start this topic with a re-cap of fractions was based on the similarities that I perceive between fractions and ratio. It therefore seemed logical, before starting on a new and potentially challenging topic, to establish how much the class already knew about fractions. As claimed by Ausubel, (in Clarke, 2005, p12), "The most important single factor influencing learning is what the learner already knows. Ascertain this and teach him accordingly."

Therefore, the purpose of this lesson was diagnostic assessment to be used to inform the teaching for the remainder of the topic. Although diagnostic assessment has been found to be ineffective if it is merely used for "setting" purposes, it has had benefits if used to identify the individual learning needs of a pupil. (Black et al., 2003).

A starter activity (of matching pairs of equivalent fractions) indicated that the majority of the class had an excellent grasp of this concept. I established this by allowing pupils to work in pairs to match the fractions, and then asking individuals, or their partners, for answers. I attempted to employ a "no hands up" approach, as recommended by Lee (2006), sometimes using lollipop sticks to select pupils to answer. "No hands up" was a new approach for the class, and many found it difficult to resist putting hands up or shouting out (or both). I would agree with Lee's assertion that putting hands up can lead to a competitive environment for some pupils. I would also suggest that it can lead to disinterest from other pupils who see no need to get involved.

Having established that the pupils possessed an ability to simplify fractions, the next relevant aspect of fractions was the ability to find a fraction of a quantity, e.g. ¼ of 36.

Using mini white boards, I was able to establish that all pupils could work out simple quantities. It was then important to know the methods they had employed and to understand their thought processes, and I used questioning and interactive discussion to establish pupils' methods. As Morgan, Watson and Tikly (2004, p133) put it, "knowing that they can produce the correct solution is a rather dead-end piece of knowledge for the teacher. It would be more helpful to find out the limitations of the way they currently see the task. Knowing how they see the process and what they say to themselves while doing it, provides you with a starting place to teach them to solve more complicated problems."

The popular method of finding ¼ was halving and halving again, which led me to revise the questioning by asking how to find 1/5th of 30. This in turn led to discussion of division and multiplication, and through the discourse it became apparent that many had difficulties with times tables. As it was not my objective in this lesson to revise times tables, multiplication grids were made available to pupils who asked for them. More pupils asked for the grids than I had expected, which I believe was indicative of a general lack of confidence with mental maths. As a form of differentiation this was effective for those who needed the grids, but may also have led to others seeking an "easy option". In the future I would seek to distribute such aids to only those I believed genuinely needed them.

However, with the use of the multiplication grids it was a straight-forward next step to move to non-unit fractions, and all pupils showed reasonable ability to find fractions of quantities.

I stated in the introduction that the purpose of this lesson was largely to perform a diagnostic assessment. As a result of the lesson I had gained a confidence in my pupils' ability to handle fractions sufficiently well to progress to the next stage. I had not, however, asked my pupils to self-assess whether they felt the same. As a result, when, towards the end of the whole topic, I asked the pupils to self-assess how they felt about the various parts of the topic, I was surprised to discover that nearly all felt that they were unable to "do fractions". It would seem apparent that although I believed my pupils had achieved the learning outcome, by the end of the topic they did not share that belief. If I had instigated self-assessment during this lesson, the result may have been different, but I will never know for certain. I agree, with the suggestion of Weeden, Winter and Broadfoot (2002, p73), that employing effective forms of self assessment will make pupils "more responsible for and involved in their own learning".

This is an area that I will need to look to improve as I move forward into my next phase of teaching, as "it is far more valuable for students themselves to participate in the assessment process than for teachers to be the sole monitors of progress." (Morgan, Watson and Tikly, 2004, p134).

Even though the stated purpose of the lesson was diagnostic assessment, I made some effective use of formative assessment during the lesson. For example, I used questioning to identify the pupils' methods, and then adjusted my questioning in response to their answers. While this was effective, there was scope for improved use of questioning, to probe deeper into their level of understanding.

Ratio and Proportion

The main teaching of ratio and proportion was spread over three lessons. Over that period we progressed from basic understanding of the algorithms to application of the algorithms to "real life" type problems.

As the work was spread over three lessons I was afforded the opportunity to review the work in pupils' books. Mindful of the research findings (Black et al.; Butler; Clarke; Thorndike; in Stobart, 2006), I restricted my marking to identifying mistakes and making comments.

The research of Thorndike, carried out nearly a hundred years ago, highlighted the comparative effect of assigning grades to work, i.e. the assignment of grades leading to pupils comparing themselves against one another. This is supported by more recent research summarised by Clarke, (2005), additionally making a link to the importance of assessing attainment against the learning objectives rather than against other pupils.

My intention in identifying pupils' mistakes was to be able to address errors and misconceptions in subsequent lessons. I was able to do this to some extent, but I do not have the evidence of a follow-up assessment to be able to confirm whether this was entirely successful.

The comment-only marking should have been followed up by allowing the pupils time to review the comments, and re-work their answers in line with the comments, as stressed by Black and William (in Clarke, 2005). I did not allow sufficient time for this to be done. It may also have been beneficial at this stage if I had asked the pupils to comment for themselves on their work, as recommended by Clarke (2005). She suggests this as a form of self assessment, whereby the pupil, having identified an area for improvement, would discuss the suggestion with the teacher. The agreed improvement would then be made either in lesson time or as a homework activity. In general, the planning of time to allow pupils to use the feedback they have received, and the subsequent assessment of the effects, is an area of weakness that I will need to improve if my feedback is to be truly formative.

Another area of assessment which I was attempting to use during this teaching was effective questioning. I found a number of difficulties with effective questioning in this class, which may well be common across most classes.

The first challenge was attempting to ensure all pupils were involved, without putting undue pressure on pupils who are not comfortable answering questions in front of the whole class. The use of paired activities helped in some cases, as pupils were able to give "team" answers and support each other if required. Team- or pair-working is a method which I should look to extend in the future, for use with higher-order questions, and to encourage the use of more mathematical language. The use of "Study Buddies" has been shown to encourage not only mathematical talk and expression, but also peer-assessment and self-reflection (Lee, 2006).

My attempts to ensure all pupils were involved in answering questions led to some very awkward silences as I allowed the recommended (Black et al., 2003) wait-time. With one particularly quiet pupil whom I never persuaded to answer a question, the issue turned out to be simply one of being "afraid to get it wrong" - this was a really unfortunate situation as it transpired from the end of topic test that she had an excellent grasp of the work. Not only was her reticence doing her a disservice, but the rest of the class would also have benefited greatly from her contribution. In addition, even if she had "got it wrong", this too would have been of benefit. As Lee (2006, p26 - 27) states "a wrong answer, perhaps more than the "right" one, helps the teacher assess what further learning pupils need."

End of Topic

At the end of the topic I decided to carry out two pieces of assessment - a summative test and a reflective self-assessment.

The summative test also included some items from the previous topic I had taught this group. Its purpose was to establish how well the pupils had remembered topics they had been taught throughout the previous six weeks.

I was fortunate that this test took place at the start of "Christmas week", when it was normal practice in the school for lessons to be more relaxed. This allowed me time to talk individually with each pupil, to discuss what they had done well and where they may have had misunderstandings. I was concerned that the pupils were more interested in the mark they had achieved than in the feedback I was able to give them about their learning. According to Weeden, Winter and Broadfoot, (2002, p115) this is not uncommon, as "the emphasis is always on mark or grade and seldom do pupils really care about what they actually achieved."

Although this was designed as a summative assessment, I was also able to use it formatively by identifying a common problem with algebra, and addressing this in the last lesson of term, thereby using "the aftermath of tests as an opportunity for formative work." (Black et al, 2003, p55) They also conclude that "summative tests should be… a positive part of the learning process." (2003, p56)

The self-assessment took the form of the production of posters summarising all they had learnt about Ratio and Proportion. The assessment showed a mixed level of understanding, and ability to communicate that understanding, within the class. One group showed an excellent understanding and had clearly referred back to their books to ensure that they included all the material correctly. The EAL student had difficulty with this activity, so I encouraged him to be very visual with his poster. Others had taken poor notes during the teaching and struggled to recall the information - this, in itself, was a useful learning point for them, as they consider the quality and usefulness of what they write in their books. (Two (contrasting) posters are included in Attachment A). Overall, this was an effective drawing together of the topic, and enabled the pupils to self-assess their own learning, as the production of posters "… requires students to reflect upon and organise their knowledge in order to communicate it" (Morgan, Watson and Tikly, 2004, p151)

Although this self-assessment enabled pupils to reflect upon their learning, as it took place at the end of the topic I was unable to provide an opportunity for pupils to put their reflections into practice. The assessment cannot therefore be described as truly formative. An extra lesson would have enabled the pupils to act upon their reflections, thereby making the assessment formative.

General Findings

Too Much Material

A recurring problem with many of my lessons was attempting to fit too much material into each lesson. This resulted in me hurrying to "get through" the lesson. I therefore allowed insufficient time for probing and higher-order questions. This was to the detriment of my teaching and my pupils' learning as "education is more than filling a child with facts. It starts with posing difficult questions" (Spendlove, 2009, p32). I was missing opportunities, not only to challenge and stimulate some of the pupils, but also to assess the depth of their understanding and identify misconceptions at an early stage.

Communication and Questioning

Much formative assessment can be achieved by effective communication and questioning. Black et al., found effective questioning "… led to richer discourse, in which the teachers evoked a wealth of information from which to judge the current understanding of their students." (2003, p41) However, many secondary school pupils appear to be unused to constructive discussion in a lesson environment, and there are considerable challenges involved in changing this situation, especially if expectations of communication are not consistent across a school. Add to this the extra language dimension of the "mathematics register" as Lee (2006) calls it, and it goes some way to explaining the difficulty I had in encouraging pupils to talk about their mathematics.

While it may be challenging, the benefits of effective communication to pupils' learning justify effort being spent to improve the quality of communication in lessons, as "learning cannot take place in a vacuum and it is at its best when there is a rich two-way dialogue between teacher and learner and learner and learner." (Spendlove, 2004, p44)

The areas of communication in which pupils are encouraged to become involved include answering questions, and explaining their ideas and methods. This can present further difficulties with the use language, particularly for pupils with EAL, who may find themselves completely excluded from the discourse. Another group of pupils who may have a similar problem are those for whom such communication is unfamiliar outside school. "Children ……. from households where English is not the first language may be disadvantaged by reliance on oral interaction." (Morgan, Watson and Tikly, 2004, p150)

The fact that all pupils are learning a new mathematical language together could be seen as a leveller for the EAL students, as all pupils " …need to learn how to use mathematical language to create, control and express their own mathematical meanings." (Pimm, in Lee, 2006, p18). Much of the language will need to be learnt by both EAL and non-EAL pupils.

As a teacher intent on encouraging the use of mathematical discourse as part of formative assessment in my teaching, I will need to meet these challenges creatively.

Expectations

As mentioned earlier, I set uniform "high expectations" for all my pupils, but I had given insufficient consideration to the relative nature of the word "high". Whilst I believe that I am right to expect all of my pupils to achieve an understanding of what they are being taught, how fast and to what level are things that need to be individually agreed. This then leads to the ability of pupils to assess their own progress against agreed targets and objectives.

I found that one pupil in the class never did any work in the lesson until his expectation for that lesson had been individually negotiated with him. He would then produce some excellent work, generally exceeding my expectations. Ollerton, (2003) relates the importance of pupils challenging themselves against their own targets, rather than competing against one another, and that in such a culture, expectation can be high but achievable, having been negotiated and agreed between pupil and teacher.

Assessment Against Levels

A difficulty which has permeated my first teaching practice has been the definition of "levels" at which pupils are working. There has been an emphasis placed upon pupils being aware of the level at which they are working, and being able to assess for themselves how they are progressing against those levels. There is a danger, however, that this can lead to "ticking boxes" for each item required within the level, without necessarily developing the understanding behind the subject material. This is similar to the issues raised by Skemp (1976), regarding the importance of Relational Understanding as opposed to Instrumental Understanding. It is surely possible for a pupil to be assessed, both by themselves and their teachers, to be working at level 5, for example, while lacking the relational understanding to apply their skills in other areas.

I tested this concern with a Year 6 pupil, with current maths level (teacher and test) assessed as level 4a/5c. Without any teaching about the subject of ratios (other than a definition of the word ratio), this pupil was able to apply existing mathematical knowledge and understanding to correctly work out ratio and proportion questions graded at level 6. My year 9 pupils on the other hand, while theoretically working at the same level, and after several lessons on the subject, were unable or unwilling to use their skills to tackle problems which differed from ones they had seen previously. (I say unwilling, as I suspect some of their reticence was due to a lack of confidence in their ability to tackle more challenging questions.)

So, while both the year 6 pupil and the year 9 pupils are assessed as working at the same level, my belief is that the year 6 pupil shows a far greater relational understanding, while the year 9 pupils show almost only instrumental understanding. This is supported by findings which indicate "teaching how to pass tests means that students may be able to pass even when they do not have the skills and understanding which the test is intended to measure" (Gordon and Reese in Harlen, 2006, p79).

The two groups come from different backgrounds and different learning experiences, so it is not possible to draw any conclusions, other than to say that the assignment of levels would appear to give little information regarding mathematical understanding and attainment. It is possible that the year 6 pupil has been encouraged to carry out more self discovery of mathematical concepts, an activity in which the year 9 group were reluctant to engage. It concerns me that pupils are encouraged to measure their achievement by being able to tick boxes rather than understand and apply their mathematics. This leads to the consideration of pupils' motivation - what they want to get from their education.

Motivation to Learn

A continuous challenge in all lessons has been encouraging pupils to want to learn. Pupils often ask the question, "Why do I need to know this?", and I confess to sometimes finding this a difficult question to answer. One of the primary areas I would seek to improve in my teaching is my ability to motivate my pupils to learn, but first I must understand motivation.

There are two main types of motivation which encourage pupils to be in lessons (as defined by Harlen, 2006) - extrinsic, where they are motivated by an external goal such as gaining a qualification, and intrinsic, where they are motivated by the goal of learning itself.

"Intrinsic motivation is seen as the ideal, since it is more likely to lead to a desire to continue learning, than learning motivated extrinsically." (Harlen, 2006, p62), and "intrinsic motivation is associated with levels of engagement in learning that lead to conceptual understanding and higher level thinking skills" (Kellaghan et al., in Harlen, 2006, p63).

So, perhaps I could have used assessment more effectively to increase the intrinsic motivation of my pupils.

The work of Harlen, (2006) identifies potential negative effects of assessment on pupil motivation, which largely relate to summative assessment and judgmental feedback. She also goes on to describe practices that maintain motivation levels, including involving pupils in self-assessment (Schunk), and the use of feedback from regular classwork.

None of this, however, leads to a conclusion that assessment can help increase intrinsic motivation. Further factors need to be considered in determining whether this is the case.

There is certainly evidence that formative assessment can raise achievement levels, as identified by Black et al (2003), but I do not believe that is necessarily the same as raising intrinsic motivation. Increased intrinsic motivation is most likely to arise from generating a genuine interest in what is being learnt, and therefore lead to a desire to continue that learning throughout life. As Clarke (2005) claims, involving students in their own learning process has had the proven effect of creating life-long, independent learners.

Harlen (2006) describes two types of interest - individual interest and situational interest. Pupils with individual interest in mathematics will persevere more and are likely to achieve well. Sadly, however, not all pupils will have individual interest. Either form of interest will encourage pupils to be involved in learning, so the teacher must attempt to create situational interest, for example, through games. Having encouraged the pupils' involvement through situational interest, not only will they start to learn, but they may also develop individual interest.

One of the keys to creating situational interest is ensuring the pupils know what they are learning and why, and then providing formative feedback to the pupil. Hence the importance of formative feedback in creating interest, leading to intrinsic motivation.

"no curricular overhaul, no instructional innovation, no change in school organisation, no toughening of standards, no rethinking of teacher training or compensation will succeed if students do not come to school interested in, and committed to, learning." (Steinberg, in Weeden, Winter, Broadfoot, 2002, p9)

It appears, therefore, to be of critical importance that teachers use formative techniques to increase the intrinsic motivation of their pupils to learn. At the same time, they should avoid actions which will reduce their intrinsic motivation.

Conclusions

There have been several studies over the years regarding the impact of assessment on pupils' learning, notably the work of Black and William, at the instigation of the Assessment Reform Group. These studies have concluded that for assessment to raise the standards of pupils' attainment it must be used formatively. (Black et al., 2003)

Reflecting upon my own teaching and my pupils' learning, I must acknowledge that there are many ways in which my practice can be improved. These include use of higher-order questions, allowing for self-discovery of mathematical concepts and allowing time for pupils to act upon feedback.

It is still a responsibility of the teacher to prepare pupils for external examinations, and to report progress to interested parties. As such it is necessary to strike a balance between the use of formative assessment to support learning, and summative assessment for reporting purposes, but also to prepare pupils for taking external examinations. However, "There is ample evidence that the changes involved (in improving formative assessment) will raise the scores of their students on normal, conventional tests." (Black et al., 2003, p2)

The saying goes, "the pig doesn't get fatter just by being weighed". According to Weeden, Winter, Broadfoot, 2002, p36, Formative Assessment could be an exception to this rule. In this particular case, repeated and on-going assessment could actually help to improve the quality of learning, if the information gleaned from the assessment is used to make it happen.

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