1. INTRODUCTION
Although as teachers we use many student activities and teaching methods in our classroom teaching, all students in the class do not fully achieve the learning objective of the lesson. Therefore, assessments may be employed for the purpose of evaluating student attainment, planning future work, assigning student grades and comparing performance. There are many types of assessment that we use for this purpose. Summative, formative and ipsative are some of the assessments that are commonly used among the teachers. Summative assessment is an assessment which is frequently used at the end of a lesson or a particular period for ‘summarizing the achievement status of a student'(Sadler, 1989, p. 120).
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Although it gives us feedback about the students' subject knowledge, sometimes it is too late thus action cannot be taken to improve the learning of the student. More seriously teachers continue without really noticing the real setback of the child, thus the need of a more effective evaluation tool arises. Black & Wiliam(1998a) state that “Assessment for learning” which is also known as “Formative Assessment” is one of the key factors that helps the students to improve their standard of achievement.
The main focus of this essay is how assessment for learning can be used in an effective way in mathematics lessons. First I will briefly define assessment for learning. Then I will provide an overview of the main steps involved in the implementation and the approaches to improve the quality of assessment for learning and finally this paper will focus on the change of the role of the teacher and the student. The ideas in this essay are aimed at improving the quality of mathematics teaching by using assessment for learning in the most effective way.
2. Definitions of Assessment for Learning/ Formative assessment
Assessment for learning is defined by a number of researchers over the past years. Sadler(1989, p. 120) defines assessment for learning as an assessment that ‘ is concerned with how judgment about the quality of student responses (performances, pieces, or works) can be used to shape and improve the student's competence by short-circuiting the randomness and inefficiency of trial-and -error learning'.
Tunstall & Gipps(1996, p. 389) also takes a similar view by suggesting that formative assessment is a ‘process of appraising, judging or evaluating students' work or performance and using this to shape and improve their competence. In everyday classroom terms this means teachers using their judgments of children's knowledge or understanding to feed back into the teaching process and to determine for individual children whether to re-explain the task/concept, to give further practice on it, or move on the next stage'
P. J. Black(2003, p. 2) explains the basic qualities in this assessment as : ‘an assessment activity can help learning if it provides information to be used as feedback by teachers, and by their students in assessing themselves and each other, to modify the teaching and learning activities in which they are engaged. Such assessment becomes formative assessment when the evidence is used to adapt the teaching work to meet learning needs'.
He elaborates further and says that assessment for learning occurs several times in every lesson therefore it is an “ integral and intimate part of a teachers daily work” (P. J. Black, 2003, p. 2)
In the light of the definitions given above, assessment for learning can be described as a process of judging and evaluating the student performance and providing feedback for the improvement of the learning process. The feedback is used to assess and modify the teaching techniques to meet the learning needs, so that the teacher can judge the student's knowledge and decide on the next step to improve the student competencies.
3. Implementation of Assessment for learning
Consciously or unconsciously teachers continuously assess their students while teaching. But for proper judgement to be made and for the judgement to be helpful in the learning process of the student, the assessment needs to be done in a particular manner. It is important to identify the key factors in implementation of assessment for learning for it to be productive. Therefore the implementation process is divided into four sections: setting and sharing goals, collecting evidence, interpreting evidence and deciding the next step. Each section is not mutually exclusive but for ease of clarification I have separated them.
3.1 Setting and sharing goals
When planning the lesson it is the teacher's responsibility to set clear learning goals (objectives). According to Lee (2006) learning objective may be a short phrase or a sentence. It is indeed a challenge for the teacher to set out objectives because she needs to take into account the curriculum, examination syllabus and the students' prior knowledge. Most importantly the objectives need to be shared with the student as well. In doing so both the student and the teacher get a clear picture of what should be attained by the end of the lesson. It is important to follow up the objective and it makes it easier for the students to evaluate their progress of attainment. Lee(2006, p. 45) notes that ‘it is important that the learning objectives are shared with pupils and that they are discussed in the start, during and at the end of the lesson, and that the pupils can refer to them during the lesson'. This gives the student the opportunity to monitor themselves as they progress.
Setting clear objectives is important for the teacher as well. P. J. Black(2003, p. 91) points out that the teachers are aware of what they want the students to learn, they are able to find ‘what the ‘gap' was between the state of the students' current learning and the learning goal and to be able to monitor that gap as it closed'.
It is important to set mastery (learning) goals rather than performance goals. P. Black & Wiliam(1998a, p. 22) reveals the results in a study done with the 3rd and 6th grades mathematical problem solving. Newman and Schwager(1995) did this study where one group was given the goal of learning which had emphasised on the understanding of the method of solving the problem (‘this will help you to learn new things') whereas the other was given performance goals where the main importance was completing many problems correctly (‘how would you help us to know how smart you are and what kind of grade you will get?'). When the two groups were compared the performance goal students showed maladaptive questioning patterns. Therefore when learning goals are given, the children concentrate more on mastering the learning aspect rather than obtaining the final answers. Yorke(2003, p. 488) states that students who work to learning goals, would recognise failure as task information ‘to be assimilated or accommodated (using Piagetian terminology), whereas for children working to performance goals it was a crushing blow'.
According to Lee(2006) as objectives are broader and often generic, it can be broken down in to simpler steps and made more specific to the lesson. These steps are called success criteria. Success criteria should be planned in advance by the teacher by closely studying the criteria the student should master. In another instance the teacher could get the students to explore and discuss the learning objective and get the children to partake in setting up the success criteria so that the students have ownership of the success criteria. When the student progresses along the list, mastering each success criterion they gradually achieve the learning objective and most importantly they are aware of their learning progress. It helps the students to build up their confidence in learning, especially in mathematics as mastering one mathematical topic would involve mastering in a series of mathematical operations and procedures beforehand. ‘Pupils in knowing what they are learning, how well they are learning and that they are learning and therefore are important in helping the pupils know that they are successful learners'(Lee, 2006, p. 49).
Lee(2006, p. 47) gives an example of using learning objectives and success criteria in teaching Pythagoras theorem .
Learning objective
State Pythagoras' theorem and know which sides of the triangle the letter refer to Use it to calculate the hypotenuse of 5 different right-angled triangles with different orientations use it to find a short side of 5 different triangles use it to find the height of 2 isosceles triangles.
As it is shown in the above example the objective of learning Pythagoras' theorem is quite general and it is learning oriented. By breaking down the objectives into success criteria it is made more specific to the lesson. The children first learn what a right angled triangle is then draw squares on each side. Then they would find the area of each square, put them in a table and obtain data from other students as well. Then they would find the relationship between the areas and finally come up with the algebraic expression.
At the end of the lesson the students can answer the question whether they have mastered the Pythagoras' theorem by referring to the success criteria. As an example: Do I know what a right angled triangle is? What have I learnt about Pythagoras' theorem?
According to Lee, the learning task is achieved by mastering a sequence of success criteria. Students can interpret the criteria in different ways. Wiliam (2005) supports this view by saying that ‘the words do not have the meaning for the student that they have for the teacher'(p.29). He suggests that the student should be given time to think and discuss as to what the ‘criteria' really mean. Therefore, it is very important for the teacher to know whether the children are gaining the required knowledge at each stage or whether they are able use the knowledge in a different environment. One way we can overcome this is by getting them to do some examples at the end of the success criteria.
For example: which one of the triangles make a right angled triangle? A- 6cm, 8cm, 10cm
B- 3cm, 2cm, 4 cm C- 5m, 12m, 13m
When this kind of a question is given, the students can use the knowledge of Pythagoras theorem to answer so that the teacher gets a feedback as to whether they have mastered the criteria.
However, listing out the criteria may limit the knowledge gained by the student to a specific direction and level. It may be more appropriate to let the student experiment with the information and find their path of learning, giving them the liberty of deciding the limits so that some students may be able to learn more than the listed criteria.
For an example: the teacher could ask them to check whether Pythagoras theorem is valid for semi-circles drawn on the sides of the triangle. Then extend it to different shapes such as isosceles triangles, pentagons etc. Then guide the children to explore so that they could come up with a broader idea of the theorem.
3.2 Collecting evidence
The teacher needs to have a detailed knowledge of the abilities and the ‘developmental needs' of the learner to assist the student to ‘move to the next stage of their development'(Harris, 2007, p. 253). Teachers must plan the lesson well because in formative assessment, the emphasis is on ‘how I am going to teach this and what are pupils going to learn?' rather than on ‘what am I going to teach and what are the pupils going to do?'(P. Black, Harrison, Lee, Marshall, & Wiliam, 2002, p. 19).
It is also important to explore the student's thinking before concluding that the student ‘understood' because very often what children ‘understand' is not the same as what the teacher intended to teach. Thus ‘assessment is the bridge between teaching and learning' and quality questions give teachers this ‘window into thinking' (Wiliam, 2005, p. 22).
In two questions used in the Third International Mathematics and science study (TIMSS), though the questions were similar the success rates were different. In Israel, 88% of the students answered the first question correctly, while 46% answered the second correctly, with 39% choosing response (b). The reason for this is that many students develop a concept that ‘the largest fraction is the one with the smallest denominator and vice-versa'. Although this leads to the correct answer to the first question, it does not lead to the correct answer to the second question. Furthermore if we add 46% and 39% it is very close to 88% which provides evidence that many students who got the first question right may have used the incorrect strategy. Therefore, a student answering a question correctly does not always mean that the student perception matches the teacher's. Thus the questions should be well planned and rich in substance. However a ‘rich question' would not only provide what the student can do, but also what needs to be done next, to broaden or deepen understanding' (p23).
(Wiliam, 2005, p. 21)
Wiliam (2005) describes the solving of the following pair of simultaneous equations:
2a = 24
a + b =16
Many students found this difficult and the teachers might conclude that the children need help in solving equations in this sort, but it was found that the difficulty was not with the skills but their beliefs that each algebraic letter stands for a different number. Thus it is important to use questions that ‘reveal unintended conceptions' (Wiliam, 2005, p. 22) if we intend to improve students' mathematical thinking. Lee(2006) notes that asking questions that enable students to think and explore so that misunderstandings of the concepts are revealed is an important part of assessment for learning.
Children should be encouraged to listen to the ideas of others and ‘support one another to develop a common understanding' (Lee, 2006, p. 51). Challenging questions should be given so that students would have to think a great deal and take risks when answering. Lee states that even when questions could be answered quickly it can be explored lengthily by the changing questioning style.
For example: instead of asking, “Are all prime numbers odd numbers?” which requires a yes/no answer, the teacher can frame the question as a statement - “All prime numbers are not odd numbers” - and ask the students to discuss in small groups before presenting the reasons and their conclusion to the class.
It is important that the students know that their contribution is ‘valued as an important step on the road to understanding'(Lee, 2006, p. 51) so that everyone contributes to the discussion. I feel discussions are very useful especially in higher classes as it gives them the liberty to explore broadly in that topic while thinking critically; moreover the knowledge gained by this type of learning is long lasting.
Torrance & Pryor (1998, p. 18)defines appropriateness of the responses in relation to classroom management as ‘ keeping the lesson moving along rather than a narrowly constructed notion of a ‘correct' answer'. This indicates that the students no longer need to be afraid to give wrong answers but need to express their thoughts. By looking at the explanation of the answer whether right or wrong both the teacher and the student will be able to identify the perception of the students and whether there are any misconceptions. Furthermore this type of learning would ‘focus on pupils' learning and not getting through the content at any cost'(Lee, 2006, p. 52). In practice this could be very difficult from the teacher's point of view as the curriculum is fixed and syllabus needs to be covered within the given time. Especially in examination classes, teachers have the burden of finishing the syllabus and revising as schools are more interested in producing good results.
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P. J. Black (2003, p. 33) point outs that Rowe(1974) had looked at the effects of increase in ‘Wait time'. She had found that ‘answers were longer, failure to respond decreased, responses were more confident, students challenged and/or improved the answers of the other students and more alternative explanations were offered'. Lee (2006) suggests that its more appropriate to call this ‘think time' and states that ‘no hands up' unless you want to ask a question creates an atmosphere where everyone contributes and listen to the others. Another way to increase the participation of students is to ask them to ‘brainstorm ideas' (P. J. Black, 2003), group discussions and encourage students to formulate their own questions and ask each other (Harris, 2007). As most students are not used to this practice in classrooms they need to be gradually trained to feel comfortable to voice their opinion and carry out discussions. Especially in lower classes teachers would need to work hard to get the ‘climate right'(Lee, 2006, p. 51).
All this needs a great deal of preparation. Wiliam (2005) states that in most Anglophone countries, teachers spend most of their preparation time marking books alone and in some other countries the majority of the lesson preparation time is spent in planning how new topics could be introduced, which context and examples can be used. However, in Japan most of their preparation time is spent on working together to devise questions to evaluate the success of their teaching, through ‘the process known as ‘lesson study' (Fernandez & Makoto, 2004)' (Wiliam, 2005, p. 22).
It is important that teachers have a set of rich questions and would be more useful if they share them among the other teachers of the school.
As an example: simplify (if possible): 2a + 5b (where you do not expect to get the same as the answer) , which fraction is the largest 3/7 or 3/11 (whether the students are able to select larger fraction of two ‘ordinary' fractions) (Wiliam, 2005, p. 23).
However, a rich question is a type of a question that would address common misconception and enable the student's thinking and exploring ability. Although having a set of rich questions is important, the teachers need to rethink and renew them each time they plan the lesson and most importantly change them appropriately as the lessons proceed. I think no teacher can fully plan a lesson because different children could respond differently to the same question. Therefore, depending on the students' responses the teacher needs to decide the next step to close the gap.
3.3 Interpreting evidence
P. Black & Wiliam (1998a, p. 16) states that ‘feedback between those taught and the teacher, and this is entailed in the quality of their interactions which is at the heart of pedagogy. The nature of these interactions between teachers and students, and of students with one another, will be key determinants for the outcome of any changes'. This shows the importance of quality feedback irrespective of the source (teacher or peer). Sadler (1998)states that the ultimate intention of feedback should be to make the student an independent learner thus peer assessment and self assessment gives great opportunity. In this section I will discuss how feedback can be given in an effective way by teachers, peers and the learner himself.
3.3.1 Teacher feedback
‘Feedback to students should focus on the task, should be given regularly and while still relevant, and should be specific to the task' (P. Black & Wiliam, 1998a, p. 8). Feedback is not formative if it is not understood, cannot be used by the learner to improve their learning or given at the end of a module as it is too late be used. Therefore, it is more useful if the students are given the opportunity to read the written comments in class, so that they can make any further clarifications. Furthermore, the teacher should follow up as to find out whether the student has taken remedial measures. If not sometimes students may not take any action.
Butler(1988) conducted an experimental study about the link between intrinsic motivation and the type of evaluation given to the students. This involved 48, 11 year old Israeli students who were divided into three groups. Butler gave three different feedbacks to the learners- grades, comments and a combination of both grades and comments. The study showed that the group who were given comments gained significant improvement. Furthermore, the other two groups showed a decline in the learning (scores). This not only revealed that giving comments is the best way of providing feedback, but when comments are combined with ‘normative feedback' (grades) it eliminates the beneficial effects of the comment as well. (P. Black & Wiliam, 1998a)
According to Wiliam (2005) when students get a higher score they do not see a need to read the comment whereas when they get low scores they do not want to read the comment as they are de-motivated. When students are given marks it affects their self-esteem furthermore there is no indication as to what they have/have not achieved or what remedial action needs to be taken.
Wiliam (2005) illustrates another study done by Butler(1987) where students were given comments, grades, praise and no feedback. It revealed that the students who received comments had improved, whereas those who received grades or praise made no more progress than those who did not receive any feedback. The questionnaire given to the students revealed that the students who got comments had higher level of task-involvement, whereas those who got grades or praise had substantially higher ego-involvement. Therefore, if grades or praise are given we would only be able to ‘increase the sense of ego-involvement without increasing achievement'(Wiliam, 2005, p. 25).
When investigating about praise, researchers found that feedback could have adverse effects if it is focused on ‘self-esteem' or ‘self-image'( praise or grades) (Wiliam, 2005). Although praise can increase motivation, it is necessary to maintain praise throughout to maintain the motivation which is rather difficult thereby praising can cause adverse effects on a student. However, Wiliam (2005, p. 26) states that Brophy(1981) has pointed out that the quality is more important than the quantity of praise; ‘teachers praise is far more effective if it is infrequent, credible, contingent, specific and genuine'. I feel that a student who is gifted being praised for his born talent can have a negative effect on others, especially on the less able student's self-esteem. Thus, feedback has to focus on what can be done to improve rather than how well they have done. For feedback to be formative it should ‘contain a recipe for further action' (p28).
As an example: instead of commenting ‘good' or ‘well done', it would be more appropriate for a teacher to write a comment as ‘ Susan, you have got the right idea here about trying to explain your rule. Think: does it apply to all triangles?' (P. J. Black, 2003, p. 45). This comment helps the child to move a step forward.
This does not mean the more feedback you get the better it is. Wiliam (2005) states that Day and Cordon(1993) looked at the learning of a group of students where half of the students were given a ‘scaffolding' when they got stuck and the other half was given a complete answer. The students who received the ‘scaffolding' learned more and retained the knowledge longer than those who received the full answer.
3.3.2 Peer-assessment
Peer-assessment is assessment of students by their peers, providing information that will help them to move on to the next step. One way that peer-assessment can be done successfully is by getting the students to assess and mark the work of others. As they know they are marking the work of another, they take interest and responsibility in doing so. Moreover, when there are disagreements about the answer they discuss and come to agreements. Lee (2006) states that students are honest and challenging and accept criticism by one another than with the teacher. I have experienced that students sometimes while going through another person's explanation or method may broaden their understanding of the concept or even challenge and argue to prove or disapprove with their own colleagues.
As an example: Measure the angles of the triangle given below using a protractor.
a angle a = 32⁰
angle b = 43⁰
angle c = 65⁰
b
c
If a student is marking the work of a colleague he might discuss with his peer that when adding all three angles they do not add up to 180⁰ and that angle c is an obtuse angle therefore it cannot be 65⁰. They would find through discussion that the student has measured the exterior angle instead of the interior angle when measuring the angle c of the triangle. They would together measure the angle c so that the student learns his mistake.
Getting students to work in groups is another strategy that promotes peer-assessment which helps them to develop communication skills as they talk and share their ideas. Especially when they discuss about the objectives and what has been done and what needs to be done, it helps them to develop skills needed for self-assessment(P. J. Black, 2003).
Training students how to discuss strategies with others, language to use when they critique the work of the others (P. J. Black, 2003)and how to interpret feedback so that they can connect it to their future work, is equally important as providing information(Sadler, 1998). They also need to be trained to refer to success criteria and identify what has been done and what needs to be done. I feel that peer assessment needs close monitoring and training as it can affect another persons' self-esteem. Especially when dealing with low achievers unless they are comfortable they might feel their work is been compared and evaluated against another which in return may cause adverse an effect.
3.3.3 Self-assessment
According to Lee(2006) Self-assessment is an important form of assessment which engage pupils to talk about their own learning which makes them self-critical and independent. As teachers it is important that we train the students to understand what they are meant to learn. However, according to P. Black & Wiliam(1998b), most students are unaware of the learning objectives, but if they are aware of their targets (objectives), ‘their own assessment become an object of discussions with their teachers and with one another'(p10).
Getting the students to review their own work and record their progress is a method of self-assessment. Thus, the student becomes ‘independent and confident learner' (Brookhart, Andolina, Zuza, & Furman, 2004, p. 214). Lee suggests that asking the children to decide the level of confidence in their work as good way of developing self-assessment skills, especially in lower classes. ‘Traffic lights' and ‘thumbs up' method can be used to get the students to assess their confidence level. If the students are confident about the concept they use green or thumbs up, if they are still unsure amber or thumb horizontal or if they are very unsure red or thumb down. When they are more comfortable they might say “I'm going a bit red on this” so that the teacher can take remedial measures easily. Through experience I feel that this is a very good method as the students can report difficulties without being noticed by other peers. This also minimizes the distraction of the other learners.
Getting the students to assess exemplar pieces of work which contain common misconceptions or errors would help the student to understand how to assess their peers and their own work. For example, draw a graph with an inconsistent scale on the y-axis and plot points badly (Lee, 2006). When the students start discussing about the graph they not only understand the concept but ‘develop communication skills and math vocabulary' (Brookhart, et al., 2004, p. 214).
3.4 Deciding on the next step
In a traditional classroom, assessment is used as a ‘tool for the control or modification of behavior, for rewards and punishment' (Sadler, 1989, p. 141), but in a classroom where formative assessment is practiced, assessment is used to evaluate the gap between the ‘present position' and the ‘desired goal' and to understand ‘the way to close the gap' (P. Black & Wiliam, 1998b). The teacher guides the student in taking the next step, but eventually the students ‘become independent of the teacher and intelligently engage in and monitor their own development'(Sadler, 1989, p. 141).
If the student has not understood the concept, the teacher spends more time or even try different approaches to make sure that the student fully understood the concept. In an instance where everyone has mastered the concept the teacher can ask the student to decide what needs to be done next to reach the next learning criteria. However, with time students become more independent in deciding what the next step should be to finally attain the objective.
However, in a real classroom this is not an easy step for the teacher. The level of attainment and individual needs would be different from one child to another. Especially in a classroom with a large number of students it would be a great challenge for the teacher. Monitoring the level of attainment, decide the next step that would drive each child in class towards the target and finally covering the syllabus within the given time frame would indeed be a great challenge for the teacher. Therefore, having group and whole class discussions, peer and self assessment would reduce the burden on the teacher.
4. Changing the teacher's role
Formative assessment makes the teacher move away from being dominated by the completing of the curriculum to being a facilitator of student learning. Teachers think of the content to be taught as a series of learning goals rather than a series of activities to be completed at any cost (P. J. Black, 2003). The teacher's responsibility is to help the students to close the gap between the current state and the learning goal while building their understanding. P. J. Black (2003) states that in the King's-Medway-Oxfordshire Formative Assessment Project (KMOFAP) research, teachers were prepared to teach two thirds of the curriculum as this is a time consuming process, but in a real class room teachers do not have the liberty to decide, but instead they are compelled to cover the whole syllabus. Therefore, teachers may face practical issues due to the nature of the curriculum. Especially teachers who have been in the profession for many years may find these innovations hard to accept and understand moreover they consider this as additional work to their daily teaching. To overcome this situation teachers should not practice this method of teaching in addition to the normal teaching, but should be practice as a part of the normal teaching. They could also practice this method gradually taking section by section of the syllabus so that they finally become comfortable to teach the whole syllabus using the method. Thus, this would take time for the teachers to adjust and practice this method of teaching.
The students are encouraged to discuss and express mathematical Ideas. Lee (2006) states that teachers look at a classroom as a ‘discourse community' which means they share common goals, that there is information and feedback and that everyone understands the mathematical language. Students are stimulated to talk and think and the teacher guides them in these rich discussions. However, these discussions could lead to the disclosure of concepts. Therefore, a teacher should have a good knowledge of the subject for him to explore the ideas confidently. A teacher should not take a lesson or topic in isolation, but instead he should consider it as a whole subject so that the discussion would relate the areas that had already been done or would be completing in future.
P. J. Black(2003) states that the teacher's expectations cha
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