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Multiplicative reasoning
Several theorists have proposed models of multiplicative reasoning (see for example, Brown, 1981; Schwartz, 1988; Tourniaire and Pulos, 1985). The argument put forward in this paper primarily draws on the idea of ‘conceptual field’ as developed by Vergnaud, chosen because it establishes the link with proportional reasoning and explicitly contrasts multiplicative reasoning with additive reasoning.
In Vergnaud’s (1997) analysis a key feature of additive reasoning is the transformation of one variable occurring in one of three ways:
1. An initial quantity is changed either through combining or separating off a like quantity: for example, a bowl of apples has more apples added to it.
2. Part-part-whole considerations of*…show more content…*

Simple proportions: for example the same number of stamps on each page of an album. 2. Cartesian product of two measures: for example for a given number of t-shirts and jeans how many different outfits are possible? 3. Multiple proportions: for example, the amount of milk drunk in a cats’ shelter for a given number of cats over a given number of days. A glance at any page of multiplication problems for primary school students suggests that the majority of multiplicative problems that they meet are of the first type, that is simple, proportional problems involving two variables in a fixed-ratio to each other. Evidence from classroom observations in England suggests, not just that children’s working with these kinds of problems occurs frequently through the use of repeated addition, but further, that the objective of teaching multiplication as a form of repeated addition was interpreted by teachers not as ‘multiplication is more efficient than repeated addition’ but as ‘if you cannot do the multiplication, then add repeatedly’ Millet, Askew and Brown*…show more content…*

Reading and working down the columns of the ratio-table, the vertical unary operation is simpler for learners to understand in that the x3 is a scalar operator arising from the multiplicative comparison that just as three pencils is three times one pencil so the total price must also be three times the price of one pencil. This ‘vertical’ reasoning can be thought of in terms of repeated addition; the cost of three pencils equals three times the price of one pencil, so we can add 4 + 4 + 4 to get 12

Simple proportions: for example the same number of stamps on each page of an album. 2. Cartesian product of two measures: for example for a given number of t-shirts and jeans how many different outfits are possible? 3. Multiple proportions: for example, the amount of milk drunk in a cats’ shelter for a given number of cats over a given number of days. A glance at any page of multiplication problems for primary school students suggests that the majority of multiplicative problems that they meet are of the first type, that is simple, proportional problems involving two variables in a fixed-ratio to each other. Evidence from classroom observations in England suggests, not just that children’s working with these kinds of problems occurs frequently through the use of repeated addition, but further, that the objective of teaching multiplication as a form of repeated addition was interpreted by teachers not as ‘multiplication is more efficient than repeated addition’ but as ‘if you cannot do the multiplication, then add repeatedly’ Millet, Askew and Brown

Reading and working down the columns of the ratio-table, the vertical unary operation is simpler for learners to understand in that the x3 is a scalar operator arising from the multiplicative comparison that just as three pencils is three times one pencil so the total price must also be three times the price of one pencil. This ‘vertical’ reasoning can be thought of in terms of repeated addition; the cost of three pencils equals three times the price of one pencil, so we can add 4 + 4 + 4 to get 12

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