Scenario: Róisín teaching equivalence of fractions
Grade (student age): 3rd class (age 8-9)
Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland
Context – national, curricular, professional, other
The Irish primary curriculum proposes that “the child should be enabled to identify fractions and equivalent forms of fractions with denominators 2, 4, 8 and 10” first among a list of six objectives for children in third class. Róisín, who was a second year Bachelor of Education student, with a relatively strong background in mathematics devised an ambitious lesson to develop an understanding of equivalent fractions with this class of 21 girls in the third week of her spring teaching placement. The lesson lasted 40 minutes.
Róisín set the lesson in a pizzeria, with five friends sharing a “ten slice special” pizza. She introduced circular paper coasters, (diameter 10 cm) as resource materials, which she had cut up and labelled with appropriate fraction names. The materials were well thought out, even to having orange dots on the back of the tenths, which could later be used to distinguish between pepperoni and cheese-only slices. Róisín also provided worksheets - integral to the lesson - with matchstick figures drawn on top, which were designed to support children’s reasoning about the problem of sharing the ten slice special among three boys and two girls. She set the scene like this:
Róisín: Do you know when you order pizza and somebody goes “Oh I don’t like cheese, I don’t like mushrooms”? Well the guys love pepperoni. Shane, Brian and Ron all like pepperoni. The girls can’t stomach it at all. So Mario the pizza man says “OK I’ll put pepperoni on the boys’ slices”... So how many slices of the pizza do you think he’ll have to put it on? How much will have pepperoni? How many slices will Ron get? ... A ten slice special yeah. OK girls. How much does one person get? A ten-slice special. Five people walk in yeah.
Child: They each get two.
Róisín: Yeah, they each get two. So show me … show me how much will Ron get. Show me the two slices that Ron’s going to get. Yeah. So girls you know the three people that are going to get pepperoni. Can you turn over the amount of the pizza that you think will have pepperoni? Can you turn them over? Do you see the orange bit, that’s going to represent the pepperoni part of the pizza for us, OK? So three want pepperoni, and two want cheese only.
Despite the affinity between the number of children (five) and the number of slices of pizza (ten), and the students having already established that each person got two slices each, they appeared to have difficulty in deciding how many slices of pizza would be cheese flavoured and how many pepperoni. Róisín appeared in a hurry to record answers.
Róisín: OK girls, can I get you to fill out the sheet. The first question. Keep in mind the pieces of the circle. OK girls fill out the sheet quickly. How many pieces will each child get? One child?
The answer of “five”, possibly because there were five persons sharing the pizza, appeared to answer a different question. Róisín pursued this by asking “five, why five? How many pepperoni pieces are there on the pizza?” to which another child replied “half”. Róisín then established what the child actually meant and seemed to realise that the children were thinking along an equal sharing of the pizza with five pieces (5/10) being pepperoni and five pieces (5/10) being cheese. The three two divide between boys and girls appeared to have been forgotten by the class. Róisín however responded to the children’s idea and pursued it.
Child: Half of the pizza.
Róisín: But there’s more boys than girls. There’s three boys and two girls. Will they get half each? The girls will get more, won’t they? Or would they? There’s ten pieces and they get five each? They get five each?
Child: No … oh (laughter) No. Everybody gets more than one slice.
Róisín: Why would they? There’s three boys and two girls.
At this stage, Róisín chose five girls from the class to act out the scenario, and dividing the five into a group of three and a group of two, questioned the fairness of each group’s getting half the pizza. Turning to the first three, representing the boys she said:
Róisín: OK. So you get half, so I’ll get half a pizza for ye, for the three of ye and this person will get half you’ll get …
Child B: Three quarters.
Registering this error, Róisín used it as a further contingency opportunity and deviated from her initial agenda of the 3:2 split. However, instead of pursuing the source of the child’s error, she used it as a moment to teach the meaning of three quarters.
Róisín: Why would I get three quarters? What does the three mean?
(Writes 3/4 on the board)
What’s three over four? So if you write three over four, right? What does the three mean? Remember? Three out of the four of them so the three means …
Children: Three people want pepperoni.
Róisín: The amount you want and the four is …
Children: (Indistinct chorus)
Róisín: So there’s four people then, you’re saying? There’s four of them. But don’t forget Amber as well. We have five people here. So how will we say it? How many … OK girls you can [sit down]. How many people walk into the pizzeria?
Róisín: OK. So that’s going to be under the line anyway. So how many of us are the boys?
Róisín: OK. Three. So that’s the fraction of the circle we want?
Róisín: Pepperoni. So how many slices of the circle with pepperoni?
Róisín: Three, three slices? And how many slices would be cheese.
Child C: Seven.
Child: (Indistinct … would be cheese)
Róisín ignored this intervention (7 cheese slices) to continue with her argument about the 3:2 ratio of fifths.
Róisín: Well OK, three pepperoni, if you say there’s three pepperoni because there’s three boys … So there’s two girls, so how many cheese would there be?
Róisín: OK so there’s two … I’m not saying it’s right though ... That’s only half of the pizza. So what are we going to do with the other half?
Róisín appeared dedicated to pushing the mathematical agenda forward, but it may have been at the cost of impaired understanding. One child suggested “you could put two more slices of pepperoni on there”, which Róisín realised would not solve the problem.
Róisín: No let’s say we double it. The boys get three pieces.
Child: They all get two each.
Róisín: We double it. Then they all get two each, exactly.
Soon after the “let’s say we double it” suggestion Róisín called for a plenary session to reinforce for the whole class the work she had done with each group.
Róisín: OK girls, where are we? OK? Can I get you all looking up here? Girls …We’ve all figured out by now that four tenths of the pizza is cheese yeah. We’ve got as far as there yeah? We’ve got as far as there? (Writes 4/10 on the board) Four tenths of the pizza is cheese. Did anyone figure out another way of writing that?
Child: Two over five. (Adopting Róisín’s language pattern)
Róisín: Two over five? … Does anyone agree with her that two out of five …
(Writes 2/5 on the board) Two pieces of the five pieces were cheese? How many girls were there? How many people were there? So two out of five of them wanted cheese. So is that right?
At this point it seemed that the thinking had gone full circle and the class was back to the 3:2 divide of pizza again, but this time equivalence had been established between two fifths and four tenths.
Knowledge Quartet Coding Commentary
Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland.
Knowledge Quartet Dimension: Contingency
Knowledge Quartet Code: Responding to children’s ideas (RCI)
Scenario: Róisín teaching equivalence of fractions
Research findings indicate that children can talk the language of fractions without much understanding (Nunes and Bryant, 1996, p. 202). Róisín’s lesson included exploration of both the ‘extensive’ and ‘intensive’ aspects of fractions required for understanding and she repeatedly referred to the half as a referent which is recognised as a significant factor in children’s reasoning The extensive aspect of fractions relates to comparing parts of the same whole – the “how muchness?” aspect. The intensive aspect depends on the relationship between two variables and must be expressed numerically by two values. It has been found that pupils do not to reason about intensive quantities, in the absence of formal teaching (ibid). In this lesson however, Róisín placed greater emphasis on the intensive, part-part relations which may in this instance have been introduced before children had sufficient opportunity to reinforce more basic concepts of part-whole relationships based on equal sharing (Empson, 2002). By presenting children with circles already divided into named fractions and asking them to record their answers, instead of first posing problems asking children to partition the whole into equal parts, Róisín may have been expecting these children to move too quickly through important mathematical ideas. Nor did she spend time assessing children’s understanding of this fundamental aspect of fractions (Burns, 1996).
In response to someone’s suggestion that the pizza might be divided in half between the 3 boys and 2 girls, Róisín got five girls up to model the situation and appealed to the fairness of two girls getting the same amount as three. While thus demonstrating that a 2:3 divide of the pizza could not result in three children and two children getting equal amounts she mused aloud:
Róisín ... so Mario the pizza man decided to slice it in half ... so you get half, so I’ll get half a pizza for ye, for the three of ye and this person will get half you’ll get …
The “three quarters” response from Child B sounds like a wild guess and was not what Róisín expected. It possibly arose - like the “five” from Child A - because Róisín was relying solely on oral communication to demonstrate how the pizza would be shared. In the absence of anything more concrete than relatively imprecise teacher talk, Child B obviously misunderstood Róisín’s reasoning.
There are at least three pedagogical points at issue at the beginning of this Contingency exchange. First, the teacher demonstration strategy Róisín improvised to explain how the ten-slice pizza should be divided fairly between five people was not proving to be effective. She could have concretised it to aid understanding, by drawing five stick figures and ten pizza pieces on the board or by actually sharing a paper plate or a page so that each of the five children got an equal share. Second, if Róisín had posed the “how many slices will have cheese/ pepperoni?” as a ‘genuine’ question and allowed the class time to engage with solving it, a more productive discussion, in terms of mathematical meaning might have ensued. Instead, Róisín addressed a student’s suggestion of “three quarters” by demonstrating what ¾ would mean in terms of three out of four people choosing pepperoni, and then set that in the context of there actually being five people leading to 3/5 being the appropriate answer. Róisín continued to lead the plenary as she steered children towards reasoning about the ‘ten-slice special’ in terms of two fifths cheese and three fifths pepperoni.
Third, Róisín could have gone back to the previous episode to re-establish that each child got two tenths or one fifth of the pizza, but instead, she suggested that they could double the five portions, a leap for which she offered no explanation, beyond her reference to “that’s only half of the pizza.” Telling children the meaning of de-contextualised symbols is not as effective in teaching number as is purposeful recording of children’s own solutions (Mack, 1993). However, Róisín was sticking with her teaching agenda and her context. As she argued for a three fifth share of pepperoni pizza for the boys, the ten slices special appeared to be uppermost in the mind of the child who suggested 7cheese slices (3+7=10), without making the “two tenths equals one fifth” connection.
Bell, A. (1992) Problem Solving Mathematical Activity and Learning: The Place of Reflection and Cognitive Conflict, in J. Ponte (Ed) Mathematical Problem Solving and New Information Technologies: Research in Contexts of Practice (New York, Springer) 137-154.
Burns, M. (1996) 50 problem-solving lessons: Grades 1-6 (Sousalito, CA, Math Solutions).
Empson, S. (2002) Using Sharing Situations to Help Children Learn Fractions, in D. Chambers (Ed) Putting Research into Practice in the Elementary Grades: Reading from Journals of the NCTM (Reston, VA, NCTM) 122-127.
Mack, N. (1993) Learning rational numbers with understanding: The case of informal knowledge, in T. Carpenter, E. Fennema, and T. Romberg (Eds) Rational numbers: An integration of research (Mahwah, NJ: Lawrence Erlbaum) 85-105.
Nunes, T. and Bryant, P. (1996) Children Doing Mathematics (UK, Blackwell).