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,

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 mathematicsGeneralisation 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 algebraThe 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 algebraThe 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 algebraThe 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.