Scenario: Heidi revising percentages

Context – national, curricular, professional, other

The National Curriculum for mathematics in England includes work on fractions, decimals and percentages in Key Stage 3 (years 7-9, pupil age 11-14). Heidi was revising the four operations with fractions, before moving on to discuss some word problems related to percentages. Heidi was a graduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her one-year teacher preparation.

Scenario

Heidi, a graduate student-teacher, was teaching in an open entry school (pupil age 11-18) in a village in the UK. The school divides the year group into maths sets (by ability) and Heidi was teaching one of the two parallel top sets in Year 8 (pupil age 12-13). This was a revision lesson in which she was considering fractions and percentages with the class. Calculators were available within the classroom.

After an initial activity with a fraction magic square, Heidi moved on to examples of word problems involving percentages. In working through the problems, Heidi used an interactive white board (IWB) to record the suggestions that the pupils made – she would often have a range of suggestions or methods on the board at any time.

Her second example concerned her mean uncle: “My mean uncle wants to decrease my allowance by 7% each month. If I used to get £450, how much will I get after 3 months?” One pupil offered the suggestion of finding 7% by calculating 450 x 0.07 and then multiplying by 3. Heidi asked what would need to be done first (i.e. before multiplying by 3) and the same pupil said “take it away from 450”. Heidi recorded the two steps on the board () saying that this gave the amount after the first month of decrease. Then she said that she would need to do the same process again and again to get the amount after three months.

A second pupil asked “Could you do 450 times 0.93?” Heidi agreed with him and said that he was “absolutely right” and clarified - that “by taking 7% away you end up with 93%” of the previous month’s allowance. She highlighted that this gave the result in one step. Having asked the class to check on their calculator that they get £418.50, Heidi continued to say (and record on the IWB) that they could use the ANS function on their calculators to repeat the calculation twice more (the ANSwer button inserts, into a current calculation, the result from the last press of the ‘=’ button).

Then Oliver suggested that “instead of timesing by 0.93 loads of times, you could times by 2.79.” With a hint of surprise, Heidi asked him where the 2.79 had come from and he said that “you get it after three months, with the decrease as well.” When she sought further clarification about the 2.79 he explained that it was “93 plus 0.93 add 0.93.” Accepting this suggestion, Heidi asked half the class to calculate  and the others to calculate , , . She pointed to the , saying that there is a decimal number (0.93) implying that the result is decreasing. Oliver began to realise that there was a problem with his approach and pointed out that his way you are adding not ‘timesing’. Another pupil then offers that 0.93 to the power 3 would achieve the same result. Heidi recorded this saying “0.93 times 0.93 times 0.93”. She also highlighted that multiplying by a number greater than 1 will result in a larger number. Eventually Oliver explained that he had been reading the question as being how much you get after three months - in total, rather than how much in the third month!

In this cycle of our research, we were interviewing the student-teacher after the lesson. During this meeting we viewed episodes from the video of the lesson and one member of the team led a discussion with the student-teacher in the spirit of stimulated-recall. This discussion was audio recorded and included the following:

AT                   … so, you had Oliver ... who suddenly suggests 2.79. … you accepted what he was saying, you … responded to him, you gave him space, and you kind of incorporated what he was doing ... why did you run with that?

Heidi               ... I guess it’s awful, but Oliver in particular, but the whole class are very bright, [laughter] most things he says are usually true.  And … I think I had a complete ... dip in confidence right there and I was thinking, ‘Gosh, I’m not sure.  I want to sit down and work this out,’ but you can't when you’re at the front of the class. ... I think maybe if it had been a class I hadn’t known I might have been quicker to actually dismiss it.  … I realise now the complete ambiguity of the question, which hadn’t even occurred to me.  … I think it was mainly because I knew Oliver.

Researcher      ... were you hoping that somebody was going to come up with a power of three?

Heidi               No. In fact, I think I’d overlooked it completely myself. … I think because of the level of planning I was doing at the time … I don't tend to actually do the problems anymore myself … because I was quite confident with this topic, because I’d taught it before to the class I don't feel any need to practise the numbers.  … but I hadn’t even really thought about it at all, so I was very pleased when they said that because … I hadn’t really considered it.

Researcher      But then Oliver comes back again over his 2.79 and he’s sort of... he’s quite persistent.

Heidi               Yes, he is, yeah.

Researcher      ... you incorporated what he was talking about, which was I think at the point where you suddenly sort of said, ‘Oh, are you meaning the total amount...

Heidi               Yeah, it clicked. [laughs]

Researcher      ... after three months?’

Knowledge Quartet Coding Commentary

Scenario: Heidi revising percentages

Here we have a classic case of the unexpected response from a pupil to a question from the teacher. Heidi has been gathering different methods to solve problems and having started on the problem of her ‘mean uncle’, she had two successful methods recorded on the board. But Oliver then came up with the suggestion of multiplying by 2.79. From the tone of her voice and her subsequent questioning, it is evident that Heidi could not see where this 2.79 has come from.

In looking at scenarios where pupils offer unexpected responses, we have found that there are three types of responses – ignore; acknowledge but sideline; and respond and incorporate (Rowland et al, 2009). Here Heidi has both responded to Oliver’s suggestion and then incorporated it into the next part of the lesson. This seems quite surprising as she is all too aware that the monthly allowance is going down and multiplication by 2.79 will result in a substantial increase.

The post lesson interview was very helpful in coming to an understanding of Heidi’s reasoning in the midst of the lesson. Here is a situation where Heidi receives a suggestion from a pupil (Oliver), she has a feeling that it isn’t correct but can’t immediately put her finger on why it is incorrect (“…I think I had a complete ... dip in confidence right there.”). She also knows Oliver well and recognises that his contributions are usually meaningful (“… Oliver in particular, but the whole class are very bright, [laughter] most things he says are usually true.”). So she makes the ‘in-action’ decision to accept his suggestion and see if it gives rise to anything. In some respects this is giving her some thinking time, and in fact it is also giving Oliver further time to realise that there are two different ways of reading the question. Although Heidi quite clearly felt uneasy, she makes the decision to trust her own mathematical knowledge, and her knowledge of the pupils, to allow Oliver’s suggestion to be explored.

In contrast, the suggestion of raising 0.93 to the power 3 is one that Heidi immediately sees as being directly related to the previous work that she had been doing with the pupils even though she had not anticipated it (“ … I was very pleased when they said that because … I hadn’t really considered it.). So although she had not expected that approach, she is very prepared to incorporate it into the efficient ways of solving the problem, another instance of responding and incorporating.

References

Rowland, T., Turner, F., Thwaites, A. and Huckstep, P. (2009) Developing Primary Mathematics Teaching: reflecting on practice with the Knowledge Quartet. London: Sage.