The focus of the Formative Assessment project was on on-going, classroom-based formative assessment: assessment for learning, rather than end-of-phase summative assessment or assessment of learning. Formative assessment takes place all the time as we teach and interact with our learners, and it may be quite informal and ephemeral. Our object in this project was to capture this, and then to reflect on and develop the use that we make of assessment as we worked with our learners.
Assessment is at the heart of teaching and learning in all phases and at all levels. The project aimed to work on assessment with teachers from very different backgrounds working with learners all the way through the school system, from Early Years/Foundation Stage through to Key Stage 5. The intention was to identify and focus on the general points that are found in common across the phases, rather than on the particular details of any one phase or year group.
The structure of the project
Fourteen teachers from two sets of schools took part in the project. The first set comprised four linked schools: a nursery, a key stage 1, a key stage 2 and a secondary school. These were all located on one site in a predominantly white middle class area. The second set was an all-age primary school and the secondary school to which many of its leavers went, in an area with a high proportion of Asian and Black residents.
The project manager held a series of start-up meetings in the different schools in the summer of 2010 to explain the objectives of the project and to ensure that all the teachers involved had access to the NCETM Portal and the Project Community. Copies of some literature, including Jo Boaler's The Elephant in the Classroom and John Mason and Anne Watson's Questions and Prompts for Mathematical Thinking, had been bought for the project and were distributed.
The start-up meetings were followed in the Autumn term by three whole-team twilight sessions at monthly intervals, led by Professor John Mason of the University of Oxford and the Open University. John provided a series of tasks and activities, engaging all members of the group in mathematical reasoning and problem solving at their own level in order to stimulate discussion and to help them reflect on the nature of mathematical thinking and understanding.
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The first session focused on learners’ powers
. The development of our learners' powers is what really underlies their ability to think mathematically, so the ways in which learners invoke these powers for themselves, initiating actions, is really what is worth assessing. The approach was to engage participants in their own mathematical thinking in an effort to get them to recognise their use of their own powers, so that they would be sensitised to recognising their learners' use of those same powers.
concerned multiplication, which is often based solely on the notion of repeated addition, whereas it is a much broader notion with scaling as one of the most common manifestations. The task involved the use of a piece of elastic which was stretched, and the amount of stretching gauged by where ‘clothes pegs’ attached to the elastic were positioned before and after stretching. An applet
that displays the same possibilities as the elastic is available on John Mason's Projects website.
Most important was our reflection in response to the questions What did we do?
and What powers did we use?
involved attribute sequences, stressing the importance of distinguishing ‘what is the same and what different’ about various mathematical objects. Different groups then considered different construction tasks in which participants are asked to construct objects that are the same (eg hexagons) but that differ in various attributes. See the files Same Difference
, Triangular attributes
, and Trigonometric identities
During discussion a distinction came to the fore between listening to what learners might be trying to express as distinct from listening for a pre-determined answer.
Session 2 focused on generalisation
as a core ingredient of any mathematical lesson.
involved a sequence of arithmetical facts: 1 + 2 = 3; 4 + 5 + 6 = 7 + 8; 9 + 10 + 11 + 12 = 13 + 14 + 15; with invitations to predict the next two such ‘facts’ and to make a conjecture about what others might be, and even to justify that conjecture.
involved ‘painted wheel’ tasks, for which draft notes
involved Sundaram’s Sieve (aimed at secondary). There is an applet
for this together with a similar one designed for primary school on John Mason's Projects website, along with other related grid applets.
The emphasis was on expressing generality. The conjecture was made that “a lesson without the opportunity for learners to generalise is not a mathematics lesson”. The slogan ‘Watch What You Do’ was offered as a strategy for contacting a generality through watching what you do with a particular example, and then trying to bring those actions to articulation.
We also spent time describing to each other things we had tried out as a result of the previous session. We aimed for ‘brief-but-vivid’ reports so that others could join in. Some of these were posted on the project group's portal community.
Session 3 focused on ‘Listening for’ versus 'Listening to'
. This session was largely based around the Stacks
activity from the Nuffield Assessing Mathematical Processes
website. It was supported by an introductory sheet
designed to give access to teachers working with much younger children as well as those in secondary school. Everyone could get started by manipulating something, and developing a notation for keeping track of the moves. Participants experienced themselves engaging in logical reasoning; reaching for and trying to express generality; making conjectures and constructing counter-examples. All of these call upon powers that learners have displayed before they start school, and whose development will provide the basis for learning mathematical concepts effectively.
The paper by Cuoco et al (1996) was made available to participants as it offered a different articulation of the same idea concerning the use of natural powers in exploring and learning mathematics.
Cuoco, A. Goldenburg, P. & Mark, J. (1996). Habits of Mind: an organizing principle for mathematics curricula. Journal of Mathematical Behavior
, 15, 375-402.
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Some outcomes and observations
Opening up assessment
A key point to emerge and develop throughout the project was the richness of the opportunities created by opening up assessment, using open-ended, not closed, tasks and activities. These could be very simple – Give me two numbers that have a difference of 2 was used by teachers effectively at key stages 1, 2 and 3 to prompt learners to push the edges of their understanding and explore wider possibilities. This is clearly very different to a mark-based assessment, in which any correct answer would have to be accepted and there could be no distinction drawn between pupils who, for example, chose to cross a place value boundary (99 and 101; 9 999 and 10 001) and those who did not.
Classroom snapshot – Two numbers with a difference of 2 task
One member of the group described her use of this task:
Comments from teachers working in different phases indicated that developing open questions could be a useful strategy for assessment in very different contexts.
I tried the 'Give me two numbers with a difference of two' task with my group of G & T pupils ranging from Year 4 to Year 6. The results were very interesting. Most pupils rose to the challenge. Some, having worked on problem solving tasks before, started methodically, moving through the number system. I then asked them to find the 'hardest' two numbers they could.
Some pupils chose to work over a place value threshold explaining that this was a difficult area for some pupils, e.g. 99 and 101 or 9 999 and 10 001. Others went for the biggest numbers they could find revealing some inaccuracies in naming larger numbers (my own included!) The older children in the group chose to use decimals while others revealed they didn't understand decimals yet.
As a first mathematical activity for a new group it was very revealing to show where each child was working mathematically. An excellent assessment activity.
Opening up assessments – Teachers' comments
Focus on classroom practice has been to develop practitioners' open questioning skills to extend children’s learning and establish their understanding and thought processes during small group activities.
We included much more open ended questioning in our mental maths starter. For example instead of saying 4 add 6 we ask the children to give calculations with the answer 10.
(Key Stage 1)
In the classroom I included (and continue to include) more open-ended activities… Observation of pupils in e.g. the ‘difference between’ activity was incredibly useful for assessment purposes.
(Key Stage 2)
The Golden Ratio, as mentioned in the book [Jo Boaler: The Elephant in the Classroom]. I set as a homework for all year groups to go away and find out what it was. Some pupils just did the bare minimum and I got a couple of lines but some got really into it.
(Key Stage 3-4).
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Our objective in working on Assessment for Learning rather than Assessment of Learning was to focus on the development of our learner' understanding of key mathematical concepts. An approach that was adopted by several members of the project group was to invite learners to write their own assessment questions, and then to share these with other members of the class. Learners were encouraged to set 'really hard' questions (but ones which they themselves understood and could respond to).
Classroom snapshots – Learners developing their own questions
A teacher working with pupils in key stage 1 described a lesson using this approach:
My Y2 class had struggled with "missing number" problems during their half termly assessment, eg:
3 + = 7 or + 6 =11
I readdressed this in class today, using a + sign, an = sign and an empty box typed onto the IWB. I asked the children to give me two numbers between 1 and 10 and wrote these up on the IWB so that I could move all the elements around:
2 + 3 = They replied "5".
2+ = 3. They understood this and replied "1".
+ 2 = 3. Also understood.
3 + = 2. They replied "1".
I then used an interactive hundred square to prove or disprove their answers, showing the difference between the numbers as "how many jumps would we need from 2 to 3?" etc. We repeated the process with progressively larger numbers, using the hundred square to illustrate the missing number as the "space" or "distance" between the first number and landing on the total.
When I was certain that the majority of the children had understood the concept, I asked them to help me by writing their own missing number problems so that I could use their ideas to make a special worksheet for another class. I only gave them blank paper, asking them to write their problem on one side and the answer "secretly" on the back.
They loved it and after a very noisy but productive session, I collected the results and promised to type them up into a worksheet.
The results (from a class which had found this concept almost impossible to understand) were varied but promising:
5 + = 10
10 + = 20
100 + = 106
45 + = 72
200 + = 300
16 + = 33
are a few examples of their ideas for "really hard sums" and I have now been able to assess not only their understanding of the concept but also the complexity of the numbers they feel comfortable working with. The incorrect answers serve the same assessment purpose and will be a good way to introduce tomorrow's session. We had a really enjoyable lesson and I feel that they gained a good deal of understanding about what the + and = signs mean.
A similar approach to trigonometry identities in Year 13 worked well:
My year 13’s were struggling with the trig identities and the exam questions for these. John [Mason] suggested getting the pupils to start off with something and then use the functions on that until they had created their own exam question. This made them realise that the questions weren’t as hard as they first appeared.
(Key Stage 5)
As the first teacher says, by asking her pupils to write their own questions she was able to assess not only their understanding of the concept but also the complexity of the numbers they feel comfortable working with and thus to gain a measure of the 'level' at which each pupil was working. So, learners developing their own questions offers an excellenttool for both teaching and assessment, whatever the phase, level or topic. First, it allows the learners to rehearse new concepts and skills. Second, it allows them to explore the new mathematical relationships in a safe environment – they do not feel that they were being 'tested', but they are meeting the challenge of finding a 'really hard problem'so they are encouraged to push at the boundaries of their own understanding. This approach supported our efforts to find ways to assess learners' understanding, rather than their routine knowledge of techniques and methods.
Assessing Understanding – Teachers' comments
The project has given us a much clearer view as to what to look for, questions to ask and ideas to draw out understanding and misconceptions.
(Key Stage 1-2)
The gains have come from highlighting the need to assess children’s thinking and what lies behind it.
Impact on teaching and learning
Although project focused on Assessment for Learning, much of our time was spent working on issues relating to teaching and learning rather than on assessment as such. One member of the project team expressed her disappointment with this:
Another commented that:
I thought the project would be more in line with the work of Shirley Clarke and formative assessment, which I found interesting and useful in practice. Her work seems based on literacy and I have found turning her ideas into numeracy quite difficult, so was interested in taking part in the project as I thought it would help me do this. It did not.
It was a lot more activity driven than I thought it was going to be. If I was going to change anything I would make it a quicker pace, at times the activities although interesting took too long.
(Key stage 3-5)
However, while it is certainly true that a lot of time was spent working on and discussing teaching and learning rather than assessment as such, many of the effects on classroom practice that were noted by members of the project group were those that might support effective AfL. For example, teachers in different phases described an increased level of discussion between learners in the classroom. In parallel with this some teachers noticed that they were holding back and keeping silent in order to allow learners to think problems through for themselves. This focus on learning as something that learners do, rather than on teaching as something that is done to learners, is at the heart of AfL and so is very much in keeping with the project objectives.
Impact on teaching and learning – Teachers' comments
[We are] using silence and pauses during group work so children can demonstrate their understanding rather than joining in with the adult and to really focus them on the task. For example if counting objects into a box with the group the teacher will pause before counting in the next one (pause is usually for 2 – 5 seconds). Another tactic has been for the teacher to remain quiet after 2 – 3 objects have been placed in the basket but to continue placing the objects so the children ‘group’ count independently. Prior to this the teacher would have continued counting with the children or carried on at the same rate of counting objects.
I have included time for conversations to be done with the children over the week on topics previously covered. The children have been more responsive towards the tasks and gaining confidence. They are beginning to see relationships within topics.
(Key Stage 1)
Teachers are definitely listening to the children a lot more as a result of providing greater opportunities for them to talk.
(Key Stage 1-2)
Learners are encouraged to talk to each other and I only facilitate were absolutely necessary.
(Key Stage 2)
I use the idea of helpers and trouble shooters, sharing of ideas is now given more priority in my lessons….I have started making them talk a lot in the lessons so I will normally not give them or explain to them the answer but will make them think for themselves what would be the method of solving.
(Key Stage 3-4)
[There is] a more independent work ethic.
(Key Stage 3-5)
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Thinking about ourselves as maths learners
All three of the main project meetings in the autumn of 2010 were based around a series of mathematical activities, in which we participated at our own level in order to develop and reflect upon our own mathematical thinking and learning. Some teachers were surprised by this, but welcomed it.
Did not expect myself to be challenged in my own mathematical thinking – unexpected but in a good way!!
However, the cross-phase nature of the project led some teachers to feel out of their depth, particularly when asked to develop abstract, algebraic representations of mathematical relationships.
John [Mason's] emphasis on encouraging us to think about ourselves as maths learners was valuable.
(Key Stage 2)
I found a lot of what we did was aimed at too high a level for what we do in Reception, also for my own level of maths understanding.
Even when they felt confident with the mathematics, some teachers suggested that a number of the activities were not relevant to their own context.
The project was pretty much what I expected, but if I were to change something it would probably have been to split the group into primary and secondary sub-groups. That way we could have really concentrated on skills and activities directly applicable to the key stage in which we were working.
(Key Stage 1-2)
On the other hand, as some of the earlier comments have perhaps indicated, many of the key points – the value of open questions and activities, the need to assess learners' mathematical understanding, and the way in which Assessment for Learning may impact on classroom practice – were identified by several teachers who were working in different phases. To have split the group up, or to have restricted the project to just one phase, would undoubtedly have made it easier to manage, but it is possible that this would have detracted from the outcomes. Working as a cross-phase group certainly presented a particular challenge, and this might in some cases have been better met with activities that had a lower entry point to ensure that all our colleagues, including those from an Early Years/Foundation background, could become engaged with confidence in the mathematics. But overall, perhaps, more was gained by sticking with the cross-phase structure which helped us to identify and work on some of the common issues that may arise from the use of formative assessment in any context.
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