Anticipation of complexity
AC: Scenario 3
6 years, 4 months ago Posted in: Anticipation of complexity 0

Scenario: Nóirín teaching fraction concepts

Country: Ireland

Grade (student age): 5th Class (age 10-11)

Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland

Context – national, curricular, professional, other

Nóirín is an undergraduate student in her final year in a teacher education programme engaged in a Learning to Teach Mathematics Using Lesson Study elective course. This lesson was a ‘reteaching’ of a lesson which the group had amended following discussion of its first iteration. The initial lesson in a middle-class school was supported by a worksheet with six equal sized circles representing pizzas to be divided among a group of friends. On this occasion, in a designated disadvantaged setting, it was decided to use the same ‘problems’, but half the page was left blank so children could represent the problem however they liked. There were ten evenly spaced lines at the bottom of the page for writing so children could record their problem solving strategies.


Nóirín’s lesson was taught in St Paul’s National School write out. Children were seated around four islands each created by pushing two tables together. The lesson was slightly less than the usual forty minutes duration. Nóirín’s lesson was based on that taught previously by her colleague Bríd with some few changes. Nóirín asked the class to imagine a boy named James who was hosting a party for seven of his friends and wanted to share six pizzas equally between them. Nóirín asked a girl to repeat the problem once, but the children appeared fazed by the task and sat immobile and in silence for a few minutes. Nóirín suggested drawing the pizzas as a start. The phrasing of the problem regarding James and his seven friends had been a deliberate move, to cause children to think about the number of people who would be getting equal shares. In the previous lesson, one group had thought the divisor might be seven but soon corrected themselves as they heard others talking about their solutions in terms of 8ths. In this class, no one thought the divisor might be eight and proceeded to attempt to cut up the six pizzas between seven people so that everyone would get an equal share.

Nóirín moved from group to group encouraging and prompting the pairs as they worked. After six or seven minutes two children were invited to the blackboard to demonstrate how they had solved the pizza problem. One child took the chalk, drew six circles, divided each one into quarters, explained how they had 24 pieces and by dividing 24 by 7 (the number of people at the party) each got 3 pieces and there were 3 pieces left over. Nóirín wrote this solution on the board without comment. Then she rubbed it off and invited another pair to come up and explain their solution. This time, on a child’s instructions, she drew six pizzas and quartered each, then handed the chalk to the child to finish sharing the pizzas. Nóirín tried to discuss their solution with reference to their work sheet, and then rubbed the board clean again. She asked for a third solution. Nóirín accepted this solution, wrote it in words on the board and repeated the solution “three pieces each and three left over” adding “so that’s going to go cold. Anyone that’s hungry can eat that, right?” Then she rubbed it off and asked if anyone else had got a similar answer. Nóirín then invited another pair who had “tried it a different way” to come up to the board and explain their solution. She engaged in respectful coaxing to get these children to articulate their thinking, which didn’t appear to be mathematically sound, but provided a Contingency opportunity. This time, on children’s instructions, Nóirín drew six pizzas and quartered each saying:

1 Nóirín:          Quarters. That means how many pieces so? Four pieces. That means they’re all equal                                 quarters, all the same size. Right. What did you do?

3 Girl: Then we have …

4 Nóirín:     Right. Ye can do it now. I’ve done enough … (as she handed the chalk to a child to finish sharing the pizzas)

6 Girl: Then we split them two into seven pieces.

7 Nóirín:     Let me see. So each one is it? Or just these two? So was it just these two split into seven? So, one two, three four, five six, seven, so OK, seven. Yeah. OK. So what did you do with the rest of them then? So how much did each person get eventually?

11 Boy:       Well you see there was seven people like and six pizzas like. Three of them got one each and the last two, they like they’re all split in six. See the three ... the other people that were left out, we split it all up like these two ... they were all split up.

15 Nóirín:        So three people got a whole pizza each, was it?

16 Boy:           No like, how many people was it? Seven?

17 Nóirín:        Seven.

18 Boy:       So three people got a pizza each like, and the last kids that were left like, yeah these two got all splittened (sic) up between them

20 Girl:            (pointing to two pizzas on bottom right hand side) Them two

21 Nóirín:   OK. Very interesting. And was there some leftover then? It says here (looking at their worksheet) everyone will get half but there is some left over. Is that the same or is that different?

24 Girl:            No because we started off with half.

25 Nóirín:        You started off with half and then you went on to that ... OK. Thank you.

The fact that everyone in the class mistook the number of partygoers, as seven instead of eight required an awareness of complexity for which Nóirín was surprisingly unprepared. Her avowed openness to children’s ideas prevented her from telling them they should have counted eight people, as she hoped they would realise.

Nóirín cleaned the board again, and asked for another way of dividing the pizzas. A girl suggested that they had forgotten the eighth member of the party: “the birthday boy himself,” as Nóirín agreed. Instead of revising the problem in the light of this insight, Nóirín suggested lightly that he might get to eat all these pieces that had been left over. She then introduced a new scenario of six pizzas and ten people.


Knowledge Quartet Coding Commentary

Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland

Knowledge Quartet Dimension: Connection

Knowledge Quartet Code: Anticipation of complexity (AC)

Scenario: Nóirín teaching fraction concepts

Nóirín recalled later that she had been “stuck” about whether to tell the children to count eight people at the party or seven. Such a feeling of ‘stuckness’ led to uncertainty about the best decision to make and resulted in momentary paralysis where she had turned her back to the class and to the camera. However, Nóirín was confident that she and the children could work just as easily with a denominator of 7 as 8.She continued the lesson by inviting children to the board to explain their solutions without questioning the assumption that it would be sufficient to share the pizza between seven people in order to cater to James and his seven friends. Nóirín did not anticipate the complexity of dividing six pizzas between seven persons. Unless one knows the relationship between fractions and division (expected to be taught in 3rd class) then children could only be expected to divide the pizzas into manageable sections, starting with the referent half (Nunes and Bryant, 1996). If everybody gets a half and a quarter then there are three quarters of a single pizza remaining to be divided between 7 people, requiring a denominator much bigger than the recommended maximum of 12 in the Irish curriculum

If Nóirín had realized how complicated it was going to be to share 6 pizzas between 7 people—she would have drawn children’s attention to the exact problem context rather than letting them go on with what she knew to be a misunderstanding - that dividing by 7 would lead to a reasonable solution, though not the correct one.


Cooper, B. and Dunne, M. (2000) Assessing Children’s Mathematical Knowledge: Social Class, Sex and Problem-solving (Buckingham, Open University).

Lubienski, S. T. (2007). What we can do about achievement disparities. Educational Leadership, 65(3), 54-59.

Nunes, T. and Bryant, P. (1996) Children Doing Mathematics (UK, Blackwell).

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