Published 6 months ago by Ann Baker

I am trying to get rid of accumulated, unfiled bits of paper and recently discovered a stack of photocopied work samples from a Year 6 demonstration lesson.

As I went through them, I remembered how surprised the teacher and I were at the students' responses to the task that I gave them. The class had just finished a lengthy unit on fractions, decimals and percentages and the teacher wanted a post-assessment task for her class.

I brainstormed with the class where, when, why and how factions are used out of school. It was a slow start, but this is the list generated:

- Food — pie, pizza
- Money — bills, %, sharing, time-sequences
- Measurement — capacity, mass, L, ml, cups, paint, decimals
- Krypto keys (a current fad).

Students were asked to 'show off' their fraction knowledge so that we could see how much they had learnt. They were asked to:

- create a fraction problem
- solve it
- present a problem-solving strategy to show a friend how to solve it

Students could create as many problems as they had time for and many students managed 4 or 5, although some only managed one. The outcome was very revealing and, for the teacher, disappointing.

We know that there are three forms of connections that we are not helping students make as well as we need to. These are connections between:

- In-school maths and maths in the real world
- Different representations of the same mathematical idea
- Maths strands (e.g. Fractions and measurement, data, chance) and other curriculum areas

We saw shortcomings in each of those areas in the work samples students presented.

- Students had difficulty identifying contexts for fractions and in creating worthwhile, realistic problems for fractions.
- Procedural inflexible strategies based on limited deep understanding were the most prevalent.
- There were no links to STEM or any other curriculum areas at all. The two connections made across strands linked division with fractions or measurement, capacity and mass (not linear measurements so no use of empty number lines)

I have inked over some of the photocopied work samples so that they can be clearly seen.

I have colour coded this sample - match the colours to the text for my first impression of what is being demonstrated.

The fractions do not all have related denominators. 60 would be the lowest compatible denominator.

Should we assume that the fractions relate to cups? In which case, the units should be named.

The fractions do not reflect normal proportions, in particular, three quarters would be a lot of sugar.

A visual strategy was selected. The area model for a measurement problem perhaps helping the student to make connections between the familiar and the unfamiliar. Look carefully at the diagrams. One sixth is actually the size of one eighth. Two thirds is actually the size of a quarter. The whole circle has not been fully shaded but it says =1.

It's not quite clear but it looks as though a connection has been made between fifteenths and thirds and that five fifteenths are equivalent to about one third in the diagram. The label one fifteenth is misaligned (accident or error?).

Finally, it is not clear how the answer one and eleven fifteenths was arrived at and of course an improper fraction with no named units makes no sense in this context.

As a formative assessment task we can identify some links to powerful maths. For instance, using the area model to connect what is familiar to what is less familiar, and the connections between fifteenths and thirds.

There is evidence of a connection between fractions and cooking and perhaps measuring cups although I have never seen one marked in fifteenths. I wonder if this student could form a visual in her head or show on paper her proportional thinking about this.

Over all, there is a jagged profile that needs probing and supporting. There seems to be a misconception about the size of the denominator and the size of the fraction, one sixth is larger than one eighth, a third is larger than a quarter. The 'fill up' strategy has been used but not correctly. We can begin to see what needs to be targeted and plan some learning intentions. While we can create personalised intentions for this student, we can in this instance apply those same intentions to about a third of the class.

**The following three questions from one child demonstrate informed assessment**.

Question 1 is about quarters and demonstrates use of the area model (correctly drawn) and counting in quarters to (in essence) carry out a subtraction of fractions with like denominators.

Question 2 is a simple halving money question. The answer $17. 50 is given but we cannot see how it was arrived at. The answer was checked with a traditional method. Neither of these questions is particularly challenging and like the majority of other questions posed by the students only involved unit fractions, a numerator of 1.

Question 3 applies fractions to capacity and demonstrates knowledge of litres and millilitres.

Again, a traditional method has been used to divide by 5 to find a fifth of 2000 ml. Does this mean that this student does not have a mental strategy for this? The diagram shows the fifths marked and half correctly marked on the scale. The answer to what is one fifth was found and checked but the answer how much is left was not found.

As with the previous Work Sample, we can see that some targeted intentions need to be identified, for example creating and solving similar problems to those above but using non-unit fractions, fractions to add or subtract with related denominators but not the same denominators.

**There were several examples along the lines of the one below**.

Even students who the teacher identified as her good mathematicians, could not initially write a word problem, rather they began with procedural sums and had to be pushed to create problems.

Notice that Question 3 is another unit denominator question. Look closely at the diagrams which show that a visual strategy could be used to find a third and then use that model to find a sixth. But you will see that the second diagram is not the same size or shape as the first. We need to probe a little here to ensure that this is not masking a misconception.

The final question is not really a fraction problem at all. It is a time division question with missing information. We don't know how long the stop for water lasted. Of course, it could be treated as an open-ended question in which case there are many possible answers.

**This simple impromptu task revealed that**:

- Students chose quite easy fractions even though asked to show off.
- Fraction misconceptions and common error patterns were unearthed.
- Students found it difficult to identify contexts (real world connections) for their problems.
- Actually creating a problem with a realistic outcome was hard for many.
- Models for working with fractions and operations were procedural.
- The area model was the go-to model for creating visual representations

'**Know thy impact**' says John Hattie.

If this post-test had been given as a pre-test, it is possible and probable that the students would have responded in a similar style prior to the unit. Using an assessment such as this at the beginning of a unit allows clear, targeted intentions to be designed to match the students’ entry points and to inform planning.

It can also be repeated at any stage during a unit to give the teacher a clear picture showing that they really do 'know thy impact'.

By the way, I have more work samples around this fraction assessment. Just let me know if you want me to post them on our Facebook page. Why not try a similar problem-posing assessment (chose any topic) and see what it reveals. I look forward to hearing from you!

Check out our website for more fractions help: Fraction Mats, a guide to introducing Fractions, and courses on the recommended opening activity for a maths lesson: Mental Routines and the central learning process: Problematised Situations.

**Cheers, Ann**

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