A Framework of Mathematics Inductive Reasoning

1. A Framework of Mathematics Inductive Reasoning Reporter: Lee Chun-Yi Advisor: Chen Ming-Puu Christou, C., & Papageorgiou, E. (2007). A framework of mathematics inductive reasoning. Learning and Instruction, 17, 55-66.

2. Introduction • Inductive reasoning is considered to consist of the educative ability, that is, the ability to generate the ‘‘new’’ ---the productive characteristic of human beings. • In recent proposals, the importance of having all students develop an awareness of inductive reasoning and applications has been recognized (NCTM, 2000). • This study addresses the development and validation of such a framework for inductive reasoning instruction.

3. Aims of the Research • This study attempted to • develop an initial framework for describing and predicting how children think in inductive mathematics situations, • refine and validate the framework using a test of inductive mathematics reasoning involving problems that require students to apply inductive processes. • The focus of our study was on the integration of various mathematics problem types into one common classification scheme prescribed by a framework.

4. Theoretical Considerations • Reasoning, in general, involves inferences that are drawn from principles and from evidence, whereby the individual either infers new conclusions or evaluates proposed conclusions from what is already known (Johnson-Laird & Byrne, 1993). • There are two main types of reasoning, namely deductive and inductive reasoning. • Deductive reasoning denotes the process of reasoning from a set of general premises to reach a logically valid conclusion • Inductive reasoning is the process of reasoning from specific premises or observations to reach a general conclusion or an overall rule

5. Theoretical Considerations • Neubert and Binko (1992) connected inductive reasoning in mathematics with finding patterns and relations among numbers and figures. • Inductive reasoning as a method involves four steps: • experiences with particular cases, • conjecture formulation, • conjecture proof and • verification with new particular cases (Po´lya, 1967).

6. Theoretical Considerations • Klauer defined inductive reasoning as the systematic and analytic comparison of objects aiming at discovering similarities and/or differences between attributes or relations (Klauer, 1988, 1992, 1999). • This definition results in the identification of six classes of inductive reasoning problems (generalization, discrimination, cross-classification, recognizing relationships, differentiating relationships and system construction), according to the kind of the inductive cognitive processes required for their solution. • The six classes of inductive problems are interrelated since all of them can be solved by a core strategy of inductive reasoning, namely the process of comparing (Hamers & Overtoom, 1997; Klauer, 1988, 1992, 1999).

7. Theoretical Considerations

8. The Proposed Mathematics Inductive Reasoning Framework

9. The Proposed Mathematics Inductive Reasoning Framework

10. The Proposed Mathematics Inductive Reasoning Framework

11. Method • Participants • Participants were 135 grade 5 students (69 females, 66 males), from seven existing classes of elementary schools in an urban district of Cyprus. • The mean age of students was 10 years and 6 months. • The school sample is representative of a broad spectrum of socioeconomic backgrounds. • In each intact class there were students of varying socioeconomic backgrounds as well as students of different levels of achievement. • Instrument • Each of six classifications consisted of 3 tasks. • Subgroups analysis • To determine whether the factor structure of the inductive reasoning of students is consistent across groups, the model was also subjected to a subgroups analysis. • Male-only, female-only model fits were tested. • Data analysis • Confirmatory factor analysis • The observed values for c2/df should be less than 2, the values for CFI should be higher than 0.9, and the RMSEA values should be close to or lower than 0.08.

12. Method

13. Results • The descriptive fit measures indicated support for the hypothesized first, second and third-order latent factors (c2/df =1.04, CFI =0.975, and RMSEA =0.02). • The fit of the model was very good and the values of the estimates were high in all cases, suggesting that the three-level architecture accurately captures the data. • Subgroup analyses were conducted to establish the validity of the proposed structure of inductive reasoning inmathematics. The male-only, and female-only models all fit the data well. The low RMSEA values (0.04 and0.005, for males and females, respectively) and the high CFI values (0.961 and 0.922, for males and females, respectively)suggest a good fit for the respective models. • All standardized coefficients were reasonable for both groups and consistent between the overall model, the male-only model, and the female-only model.

14. Results

15. Conclusion • The framework may enable children’s mathematics inductive thinking to be described and predicted in a coherent and systematic manner. • This model offers teachers a framework of students’ thinking while solving various formats of inductive mathematics problems. • From an assessment perspective, the framework appears to be valuable in providing teachers with useful background on students’ initial thinking and in enabling them to monitor general growth in inductive reasoning.

16. Conclusion • Further studies are needed to investigate whether the framework is appropriate for children from other cultural and linguistic backgrounds and to determine the extent to which it can actually be used to inform instructional and assessment programs in elementary school inductive reasoning. • Additional research is also needed to extend the framework to incorporate children’s thinking in inductive thinking.