Elementary school students engaging in making generalisation

1. Elementary school students engaging in making generalisation Presenters: Wei-Chih Hsu Professor : Ming-Puu Chen Date : 08/19/2008 Yeap, B.H. & Kaur, B. (2008). Elementary school students engaging in making generalisation. ZDM: International Journal in Mathematics Education, 40, 55-64.

2. Outline • Introduction • Literature review • Methodology • Discussion & Conclusion

3. Introduction • This article reports on • the generalisation strategies used by students in a grade five class in an elementary school in Singapore. • the factors influence generalisation strategy use.

4. Literature review (1/4) • Generalisation and school mathematics • Algebraic concepts be introduced to students in elementary and middle school years. (Kaput, 1995) • Expressing generality as one of the roots of algebra. (Mason, Graham and Johnston-Wilder, 2005) • Algebraic thinking ‘‘involves acts of deliberate generalisation and expression of generality. (Lins and Kaput, 2004) • Generalisation is the heartbeat of mathematics. (Mason, 1996) • The Singapore mathematics curriculum and the teaching of algebra • The Singapore mathematics curriculum does not include much formal algebra in the elementary school. • Students are introduced to formal algebra only in grade six (aged 12)(Ministry of Education of Singapore, 2006).

5. Literature review (2/4) • The Singapore mathematics curriculum and the teaching of algebra • Figure 1 shows two tasks found in one of the textbooks(Collars, Koay,Lee, Ong & Tan, 2006). • Tasks that require students to generalise and use algebraic expressions to describe general terms are not common.

6. Literature review (3/4) • The Singapore mathematics curriculum and the teaching of algebra • In an analysis of 190 recent released test items, only two items were found to include some kind of generalisation (Yeap, 2007). • One of them (Fig. 3) required students to find a given term in a repeated pattern. • The other item (Fig. 4) embedded a pattern into the context of a word problem.

7. Literature review (4/4) • The generalisation strategies • Children used to come up with a four-category framework to describe generalisation strategies.(Lannin, Barker & Townsend, 2006) • (1) Recursive • describe a relationship that occurs in the situation between consecutive values of the independent variable. • (2) Chunking • build on a recursive pattern by building a unit onto known values of the dependent variable. • (3) Unitising • use a portion as a unit to construct a larger unit using multiples of the unit. • (4) Explicit • construct a rule that allows for immediate computation of the value of the dependent variable for a given independent variable value.

8. Methodology (1/2) • Participants • A grade five(aged 11 years) class from a typical school in Singapore was selected for the study. • 38 students(although the students had differing abilities in problem solving, they had acquired the basic skills in arithmetic.) • Instruments • Students were given a novel task that comprises of several subtasks that required generalising. • The selected task called Odd Numbers is shown in Fig. 6. • It was not a typical textbook task.

9. Methodology (2/2) • Instruments: The Odd Numbers task. • Required students to find a method for finding the sum of consecutive odd numbers for a small number of terms. • Three types of subtasks were presented • Recognise the given pattern of using relevant square numbers. • Developing a generalisation similar to the ones given in the example. • near-transfer task. • Developing a generalisation that was less similar to the ones in the example. • far-transfer task. • Procedure • Students were asked to describe what they did after they had completed each subtask. • They were also asked a few additional questions in the post-task interview. • The data for each student was analysed at two levels • (1) to identify the strategy used, • (2) to identify factors that facilitated the ability to generalize.

10. Conclusions & Discussion (1/2) • The following factors were evidently important in the use of the generalisation strategies: • (1) the ability to see structures and relationships, • (2) prior knowledge, • (3) meta-cognitive strategies, • (4) critical thinking strategies, • (5) the use of organizing heuristics such as a table, • (6) use of simplifying heuristics such as trying out simpler cases, • (7) task familiarity, • (8) technology.

11. Conclusions & Discussion (2/2) • Lannin, Barker and Townsend (2006) have proposed three categories of factors that influence strategy selection in generalising: • (1) cognitive factors; • (2) task factors; • (3) social factors. • In the present study, we focused on the cognitive factors and task factors as the students were observed individually. • Future research should focus on • generalising strategies of mathematically able students and average, or even lessable, • students to determine how the latter can reach the level of thinking of mathematically able students. • A more comprehensive body of knowledge on • how to make the ability to generalise accessible to all has important curricular and instructional implications and applications.