INDICATORS OF MULTIPLICATIVE REASONING/THINKING

1. INDICATORS OF MULTIPLICATIVE REASONING/THINKING Zuhal Yilmaz North Carolina State University

2. Lets’ Try to Understand

3. How do children figure out how many dots in the picture?

4. 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)

5. 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)

6. 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) • 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

7. 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.

8. 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).

9. 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 (Fischbeinet al., 1985) • In multiplicative thinking students can think in terms of both units of one and units of more than one (Clark & Kamii, 1996)

10. 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)

11. 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)

12. Level of Progressions of Multiplicative Thinking in Existing Literature

13. 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)

14. 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)

15. 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)