

Manipulative
and representational models of mathematical concepts create a
reference framework through which abstract mathematical knowledge
and procedures can be introduced, exemplified and understood in
the classroom. Analysis of a NEMP task involving a contextually
based ‘sharing’ problem that used circular representations
of pizzas enabled an exploration of the influence of children’s
informal knowledge on their developing understanding of fractions.




The
videotaped responses of 60 Year 4 and 60 Year 8 students to five questions
were studied. Two model pizzas (one ham/one pepperoni) cut into sections
of four and set on plates were provided.
The questions were:
1. How much of the pepperoni pizza is left?
2. How much of the ham and pineapple is left?
3. All together, how much pizza is left?
4. Let’s now think about two different ways of
using up the pizza that’s left over.
If four children had a quarter piece
of pizza each, how much would be left?
Prompt, if answer not given
as fraction: What fraction or part is left?
Year 8 students were asked two extra questions:
1. Now imagine that the two of us are going to have
an equal share of the pizza that is left.
What fraction or part of a whole
pizza do we each get?
2. Can you explain to me how you worked that out?
Prompt: You can move the
pieces of pizza around to help you work it out.



•
Fractional understanding builds incrementally, over considerable time,
rather than being an allornothing occurrence and is very much a
function associated with educational experiences.
• Context played a significant part for Year 4 students, with
about 12% of the answers to Q1 and Q2 revolving around the orientation
or specific toppings of the pizza.
• About one quarter of Year 4 students responded with a natural
number response to Q1 and Q2.


•
There was marked development between the age groups in understanding
part/whole distributions (81% versus 35%).
• Few Year 4 students exhibited knowledge of fraction addition
of parts greater than one (Q3). Ten percent of Year 4 students adequately
dealt with the whole pizza but could not name the remaining fraction.
• Year 8 students solved the sharing questions by physically
sharing the pieces, division by 2, halving, or estimation. Most students’
preferred approach involved referencing whole number partitioning
strategies rather than more formal fraction operations. 



The
findings emphasise the important role of students’ contextual
and informal knowledge. For Year 4 students who have yet to receive
extensive ‘formal’ instruction about fractions, informal
knowledge dominated their solution strategies. Their solutions were
more often dependent on the situation with its concrete and visual
supports, rather than on symbolic manipulation.
Year 8 students were more able to mathematise the problem situation,
generally using mathematical language to provide an answer. However,
some Year 8 students demonstrated a lack of ‘operation sense’
with regards to fractions. Their formal knowledge took precedence
over their informal, and for those students who relied on partially
constructed and remembered algorithms, mathematically nonsensible
answers were proffered.


Developing
a productive mathematical disposition requires frequent opportunities
for students to experience the rewards of sensemaking in mathematics.
Unless children are given sufficient opportunity to ‘make
sense’ of realistic problem examples in appropriate contexts,
they are unlikely to connect their informal knowledge of rational
number concepts to their knowledge of formal symbols and procedures,
or develop flexible understandings of fractions.



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