INDICATORS OF
MULTIPLICATIVE
REASONING/THINKING
Zuhal Yilmaz
North Carolina State University
Lets’ Try to Understand
How do children figure out
how many dots in the picture?
Importance of Multiplicative
Thinking
Multiplicative thinking forms a base for advanced
mathematical thinking (Confrey, 1994; Dreyfus, 1991;
Harel & Sowder, 2005; Lamon, 1994; Tall, 1991)
Strong ground for multilicative thinking is important
for success in advance mathematics (Harel & Sowder,
2005; Lamon, 1994; Tall, 1991)
What is Multiplicative Thinking
Reconceptualization of the notion of unit (Hiebert &
Behr, 1988)
“Multiplication is established when the whole is
defined in relation to the objects after the split, and
division is defined when the whole is not reinitialized
after the split” (Confrey, 1994)
Continuous relation between two quantities (Nunes,
2001)
Cognitive Roots for Multiplicative
Thinking
Students’ difficulties as they first introduced
the multiplication concept (Piaget; 1987; Kamii
& Livingston, 1994; Clark & Kamii, 1996;
Mulligan & Mitchelmore, 1997; Anghileri,
1999)
o
o
o
In the context of a problem that requires multiplicative
students add instead of multiplying
cannot figure it out product of facts from a known fact
attaching the meaning of multiplication in problem
setting
Additive Roots
Fischbein et al. 1985 have conjectured that “Each
fundamental operation of arithmetic generally remains
linked to an implicit, unconscious, and primitive
intuitive model” (p.4) and repeated addition model is
intuitively attached multiplication.
Identification of Problems with
Additive Roots
Nunes and Bryant (1996) pointed an important problem:
We have argued that children’s first step in multiplicative
reasoning follow directly from their experiences in additive
reasoning but we have also tried to show that children who
rely entirely on the continuity between two types of
reasoning will begin to make serious blanders in
multiplicative tasks. So they must at some stage come to
grips with differences between two kinds of reasoning
(p.195).
Why Repeated Addition is
Insufficient?
Repeated addition is insufficient for constructing
multiplicative understanding (Clark and Kamii, 1996;
Tall, 1991; Thompson & Saldanha, 2003)
Dealing with single unit in terms of so many more
This model fails when students work with decimals or
fractions (Fischbein et al., 1985)
In multiplicative thinking students can think in terms
of both units of one and units of more than one (Clark
& Kamii, 1996)
Why Repeated Addition is
Insufficient Cont…
Repeated addition is also in sufficient for advance
mathematical thinking such as exponential functions,
covariation, ratio (Confrey, 1994; 1995)
“times” concept is not clear for the students who just
think additively (Confrey, personnal communication;
Clark & Kamii, 1996; Carrier, 2010)
Multiplicative Roots
According to Piaget multiplication and addition differs as (1983,
1987) “being in the number of levels of abstraction and the
number of inclusion relationships the child has to make
simultaneously” (as cited in Clark & Kamii, 1996; p.42).
Repeated addition can be used to evaluate the result of
multiplying, but cannot support conceptualizations of
multiplication (Confrey, 1994; Steffe, 1988; Thompson &
Saldanha, 2003 as cited in Carrier, 2010).
Iterating composite units (Steffe, 1988)
Identification and construction of multiplicand and the
multiplier, coordination of those factors (Davydov, 1992; Clark &
Kamii, 1996; Lamon, 1996)
Level of Progressions of
Multiplicative Thinking in
Existing Literature
Broad Phases of Multiplicative
Thinking
Level 1: 1-1 counting
Level 2 Additive composition (Skip counting or repeated
addition)
Level 3 Many to one counting
Level 4 Multiplicative Relations ( recognize multiplicative
situations 1) groups of equal size (multiplicand), number of
groups (multiplier) and total amount (the product)
Level 5 Operating on the operator (Jacop & Willis, 2001)
More …
Level 1 Non- quantifier, Level 1 Spontaneous Guesser
Level 2 Keyword Finder, Level 2 Counter, Level 2
Adder, Level 2 Quantifier, Level 2 Measurer
Level 3 Repeated Adder, Level 3 Coordinator
Level 4 Multiplier, Level 4 Splitter
Level 5 Predictor (adapted from Clark & Kamii, 1996;
Carrier, 2010)
Final Thoughts and
Suggestions
use appropriate assessment tasks that fosters multiplicative
thinking such as Piaget’s Fish Task to recognize and remediate
students’ misconception or difficulties
attach students informal knowledge of mathematics (Fuson,
2004; Gingsburg, 1977; Piaget & Inhelder, 1967) to the
formal mathematics in early grades through fair sharing,
qualitative quantification (Piaget, 1970; Clark and Kamii,
1996; Thompson, 1994; Park and Nunes, 2001)
use fair sharing tasks in order to emphasize multiplicative
thinking instead of addition which underlies basis in counting
LT based instruction will help teachers to understand students’
progression and help them to remediate their weakness
(Confrey et al., 2008; 2009; Yilmaz; 2011)