1. Middle school children's cognintive perceptions of constructive and deconstructive generalizations involving linear figural patterns Rivera, F. & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM: International Journal in Mathematics Education, 40, 65-82. Presenters: Wei-Chih Hsu Professor: Ming-Puu Chen Date: 06/12/2008

2. 2 Introduction zThis paper discusses the content and structure of generalization involving figural patterns of middle school students. (i.e., Grades 6–8, ages 11–14) zThree forms of generalization involving such figural linear patterns y constructive standard (Fig. 4) y constructive nonstandard y deconstructive (Fig. 6)

3. 3 Literature review Recognizing regularities in patterns zThe initial stage in generalization involves y ‘‘focusing on’’ or ‘‘drawing attention to’’ a possible invariant property or relationship (Lobato, Ellis, & Muñoz, 2003) y ‘‘grasping’’ a commonality or regularity (Radford, 2006) y ‘‘noticing’’ or ‘‘becoming aware’’ of one’s own actions in relation to the phenomenon undergoing generalization (Mason, Graham, & Johnston-Wilder, 2005). zDuval (1998) identifies at least two ways in which learners manifest their recognition of the figure. y Perceptual apprehension x seeing the figure as a single gestalt. y Discursive apprehension x seeing the figure as a configuration of several constituent gestalts or as sub-configurations.

4. 4 Literature review zPatterning tasks involve figural cues, we note the most important perception types is visual perception. y Visual perception involves the act of coming to see. y Two types, sensory perception and cognitive perception. x Sensory (or object) perception is when individuals see an object as being a mere object-in-itself. x Cognitive perception goes beyond the sensory when individuals see or recognize a fact or a property in relation to the object. continued

5. 5 Literature review zRealistic Mathematics Education (RME): Freudenthal, 1991 y Learners use models of their informal mathematical processes to assist them to develop models for more formal processes. y Being able to successfully transition is an indication that they are constructing a new mathematical reality. zAccording to Treffers (1987) y Horizontal mathematization x transforming real and experientially real problems into mathematical ones by using strategies such as schematizing, discovering relations and patterns, and symbolizing. y Vertical mathematization x reorganizing mathematical ideas using different analytic tools such as generalizing or refining of an existing model. continued

6. 6 Methodology zUsing pre-and post-interviews and videos of intervening teaching experiments. y In the Fall 2005 experiment xUsed two algebra units in the Mathematics-in-Context (MiC) curriculum that adhere to RME. y In the Fall 2006 three-phase experiment. zThe students were asked to deal with problem situations that consistently contained the following tasks relevant to generalizing: y Extending a near generalization problem physically (for example: drawing or demonstrating with the use of available manipulatives) and/or mentally (reasoning about it logically); y Calculating a far generalization1 task using either a figural or a numerical strategy; y Developing a general formula recursively and/or in closed, functional form, and; y Solving problems that involve inverse or reverse operations.

7. 7 Constructive versus deconstructive generalizations zA constructive generalization of a linear figural pattern yRefers to those direct or closed degree-1 polynomial forms which learners easily abduce or induce from the available cues. zA deconstructive generalization of a linear figural pattern y Seeing overlapping sub-configurations in the structure of the cues.

8. 8 Constructive versus deconstructive generalizations zFactors affecting students’ ability to develop constructive generalizations y Language x Language is an important factor (MacGregor & Stacey, 1992; Radford, 2000, 2001; Stacey & MacGregor, 2001) y Capacity to use analogies continued

9. 9 Conclusion zMiddle school students’ capability in justifying constructive generalizations (Becker & Rivera, 2007). y All of them employed extension generation, that is, they used more examples to verify the correctness of their rules; y Some used a generic case to show the perceived structural similarity; y Some employed formula projection, a type of figural based reasoning in which they demonstrated the validity of their formulas as they see them on the given figures, and; y Some used formula appearance match, a type of numerical-based reasoning in which they merely fit the formula onto the generated table of values that they had drawn from the figural cues

10. 10 Conclusion zWhat is the nature of the content and structure of generalization involving figural patterns of middle school learners? y Students’ generalizations shift from the recursive to the closed, constructive form. y The general forms are further classified according to strategy complexity x constructive standard as being the easiest for most middle school children to establish. x constructive nonstandard as being slightly difficult, x deconstructive as the most difficult to achieve. zTo what extent are such learners capable of establishing and/or justifying more complicated generalizations? y We note changes in their representational skills and fluency x from being verbal (situated) to symbolic (formal). y Four types of inductive justifications continued

11. Expansion “Rising to the challenge”: Using generalization in pattern problem to unearth the algebraic skills of talented pre-algebra students. Amit, M., & Neria, D. (2008). “Rising to the challenge”: Using generalization in pattern problem to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40, 111-129.

12. 12 Introduction zThis study focuses on the generalization methods used by talented pre-algebra students in solving linear and non- linear pattern problems. zPaper will illustrate y How capable students (aged 11–13) generalize pattern problems, y Describe what kind of generalization strategies they use, y How they communicate and justify their generalizations, y What algebraic thinking notations they demonstrate. zPattern Generalization Strategy y Numerical y Visual y Combination

13. 13 Discussion zCharacterizations of generalization methods y Mental flexibility x The processes of generalization revealed a high level of ‘‘problem solving management’’ and decision making (Radford, 2006). x Students shifted from graphical representation to numerical representation, and later to verbal representation and symbolic ones. y Manifold reflections x Reflection is a cornerstone of the generalization process (Doerfler 1991; Ellis 2007; Harel and Tall 1991; Sriraman 2003). x Three forms of reflection were evident in this study. Commonalities in the pattern sequence’s structure, The generalization method, The ‘‘tentative generalization’’ through verification of the pattern sequence. y Stages in the generalization process, following two strategies: x The recursive–operational–local strategy. x The functional–conceptual–global strategy.