1. Indicators of Multiplicative Reasoning among Fourth Grade Students Carrier, J. (2010) Under the direction of Dr. Sarah B. Berenson and Dr. Kerri Richardson. University of North Carolina, Greensborough, NC

2. PROBLEMS Development of multiplicative thinking is a key construct for understanding advance mathematic topics (Confrey, 1994; Lamon, 1994; Harel & Sowder, 2005). Many US elementary school teachers have limited understanding about the mathematics they teach (Ball, 1990; Post, Harel, Behr & Lesh, 1993; Tall 1991). Students encounter difficulties as they move into more advance mathematical thinking.

3. COGNITIVE ROOTS OF MULTIPLICATIVE REASONING repeated addition( Steffe, 1994) splitting (Confrey &Smith, 1994) relational understanding(Clark & Kamii, 1991).

4. THEORETICAL FRAMEWORK an adapted framework (Thorton &Fuller, 1981; Karplus &Lawson, 1974; Clark & Kamii, 1996) that specified multiplicative reasoning levels. Level 1: Spontaneous Strategy Level 2: Additive Strategy Level 3: Multiplicative Strategy without Success Level 4: Multiplicative Strategy with Success Level 5: Proportional Strategy

5. RESEARCH QUESTIONS What are the indicators of multiplicative reasoning among fourth grade students? What strategies do fourth grade students utilize in solving multiplicative reasoning word problems? What multiplicative reasoning strategies do the items on the test instrument invoke in fourth grade students?

6. METHODOLOGY Piaget stated that students cannot present multiplicative reasoning until they reached the 4 th stage of the development; formal operations stage ages 11-12 (predominantly in 5 th grade). Author conducted a pilot study in 2008 with four promising 4 th grader students (ages 9 and 10) to test the five levels of multiplicative reasoning.

7. METHODOLOGY CONT.. a qualitative study : Conducted interviews with 15 4 th grade students (ages 9-10) from a charter I school in a large urban school district in NC 10 questions were developed using Microsoft Visio Computer was used as a medium for interviews. Cross case analysis of 10 items Response analysis of each student

8. FINDINGS research elaborated, diagnosed and classified indicators of multiplicative reasoning of 4 th grade students and categorized these indicators from least to most sophisticated ones

9. CRITIQUE clearly formulated research problems laid a body of evidences from the existing literature to address the problem Did not discuss in detail why multiplicative reasoning so important for advance mathematics understanding provided context for the research questions identified and critically analyzed the related works addressed the gaps

10. CRITIQUE CONT… author’s strict interpretation of formal operation development stage limits the extension of the study Did not clearly stated the audience of this research how other people use the findings of the study how this study help teachers, policy makers and researcher to make smart decisions on curriculum development and instruction

11. REFLECTION a search of literature is critical to identify what has been done, what are the gaps in existing literature needs to be filled and to define your research problems and questions. broader perspective to set up specific points of interest of the study theoretical background of the study is critical part of the literature review that serve a foundation for subsequent actions in the study; your methodology selection, your way of reporting findings

12. REFLECTION CONT… careful about the scope of the research a smart flow within the literature review toward accomplishing purpose of the study.

13. QUESTIONS?