What is Algebraic About Children’s Numerical Operating?

1. What is Algebraic About Children’s Numerical Operating? Leslie P. Steffe University of Georgia lsteffe@coe.uga.edu Presented by John Olive University of Georgia jolive@coe.uga.edu

2. Early Algebra • We start from the premise that "early algebra" is about exploiting the potentially algebraic nature of arithmetic rather than about moving students from arithmetic to algebra more quickly or efficiently.

3. The Algebraic Nature of Arithmetic • Rather than consider arithmetic as the arithmetic of the whole or of the rational numbers, we interpret it in terms of children’s quantitative schemes. • The potentially algebraic nature of arithmetic concerns children's awareness of how they operate in numerical situations and the symbolic nature of that operating.

4. The Algebraic Nature of Arithmetic • von Glasersfeld (1991) explained reflection as that mysterious capability "that allows us to step out of the stream of direct experience, to re-present a chunk of it, and to look at it as though it were direct experience, while remaining aware of the fact that it is not" (p. 47).

5. The Algebraic Nature of Arithmetic • Our view of algebraic reasoning is not one of “generalized arithmetic processes” but of a reflective awareness of those processes and relations among quantities that enable a child to reorganize their quantitative schemes so that they will work with new quantitative relations.

6. Example: Construction of a Fractional Composition Scheme • Context: A 3-year constructivist teaching experiment with 12 children working in pairs with a teacher-researcher, using our Tools for Interactive Mathematical Activity (TIMA) computer environments. • TIMA: Sticks provided children with actions on line segments that they could create on the computer screen by dragging the computer mouse. For example they could partition a stick into a specified number of equal parts. Furthermore, they could repartition any one of those parts.

7. Teaching Episode with Jason & Laura Exploring the Fraction 3/4 • While they were in the context of showing the teacher all they knew about three fourths, an entirely unplanned event occurred.

8. Teaching Episode with Jason & Laura Exploring the Fraction 3/4

9. Protocol: Making a Fraction of a Fraction. • J: (Makes a copy of a 4/4-stick and colors three parts of it. He then pulls these three colored parts out of the 4/4-stick, releases the mouse, and sits back in his chair.) • T: Ok, so now you have three fourths. So now I want to find another way to use PullParts to do three fourths . • L: You can make it smaller! • T: Go ahead. I don’t see what you mean so let’s see. • L: (Takes the mouse and pulls one part out of the 4/4-stick.) • T: Now, you can use that one to make three fourths!

10. L: (Dials Parts to “4” and clicks on the pulled part. Following this, she colors three of the four parts and uses Pull Parts to pull them from the stick as shown at the bottom of Figure 2. The three green parts below the 4/4-stick were those that Jason pulled out.)

11. T: Wait, wait, wait, wait. Now I want to ask you a question because what you did was so nice! Can you give a name, a fractional name, can you tell me how much this is (the three fourths of the one fourth) out of the whole? • L: Three tenths. • J: (Puts his hand under his chin and thinks for a rather longish period of time.) Three sixteenths. • T: Because you have different answers, explain to each other until you get to one solution. • L: Oh oh! (meaning, “we are in trouble”) well, we had … I don’t know…. • J: (Points at the 3/16-stick Laura made) See, if we would have had it in that (points to each part of the 3/4-stick he made by pulling parts) four, four, four, and four—sixteen. But you colored three, so it is three sixteenth!

12. Recursive Partitioning • Jason’s goal to find how much three fourths of one fourth would be of the whole stick, when coupled with the visible results of Laura’s act of partitioning, evoked his productive act of distributing the operation of partitioning across the results of the first partition. We call this operation Recursive Partitioning.

13. Recursive Partitioning • Recursive partitioning is quite different than simply repartitioning a partition. For a composition of two partitionings to be judged as recursive, there must be good reason to believe that the child, given a partial result of repartitioning, can produce the numerosity of the full result. • But this is not all, because the child must also use the second of the two partitions (the one that is not fully implemented) in the service of a goal that is not a partitioning goal.

14. A Unit Fractional Composition Scheme

15. Jason’s Unit Fractional Composition Scheme • The goal: find how much a fraction of a unit fraction is of a fractional whole. • The Situation: the result of taking a fractional part out of a unit fractional part of the fractional whole, hence the name “composition”. • The Activity: the reverse of the operations that produced the fraction of a fraction, with the important addition of the sub-scheme, recursive partitioning. • The Result: the fractional part of the whole constituted by the fraction of a unit fraction

16. The Algebraic Nature of Jason’s Fractional Composition Scheme • The concept of function is implicit in Jason’s partitive fractional scheme. • The concept of inverse function as well as the composition of two functions is evident (to us) in Jason’s unit fractional composition scheme. • From Jason’s point of view, he was neither composing two functions nor was he multiplying two fractions. Rather, he was solving a problem presented to him by Laura’s actions and the question of the teacher.

17. The Symbolic Nature of Jason’s Activity • Jason was explicitly aware of how he operated to solve his problem—“See, if we would have had it in that (points to each part of the 3/4-stick he made by pulling parts) four, four, four, and four—sixteen. But you colored three, so it is three sixteenths!” • Jason mentally partitioned each of the three fourths into four parts each. So, to make the explanation that he did entails him stepping out of the stream of his thought, regenerating it in visualized imagination, and “looking” at it as if it were a direct experience (von Glasersfeld’s notion of reflective awareness). • We claim that Jason’s operations symbolized actually partitioning each of the three one-fourths that he made into four parts each.

18. Interiorization • To be constituted as symbolic, the sensory material that is recorded in the mental operations would need to be recorded at a level of interiorization. • Jason’s numerical concept of four served him in his partitioning activity, and it is what enabled him to proceed mentally without actually partitioning the three one-fourths in his visual field into four parts each. Jason could use his concept of four to disembed a composite unit of four from another composite unit containing four elements. • Jason could produce the containing unit as a unit of units of units. That is, his concept of four was constructed at the third level of interiorization. As a consequence, he could distribute a composite unit of four across the elements of another composite unit of four.

19. Symbolic Thought in Action • Jason used his concept of four in considering the stick partitioned into four equal parts. That these parts were not simply four separated perceptual unit items is indicated when he willfully united the one fourth of the stick Laura pulled out with the three fourths that he pulled out and regarded them as together comprising a whole stick. Jason was operating above the sensory plane and essentially didn’t need the material that was in his visual field in order to operate.

20. Symbolized Operations • The figurative material that Jason used to make a 4/4-stick on which he operated further was symbolic in nature. It symbolized the operations he used to make the 4/4-stick, and the operations could be said to be embedded in the figurative material in that the figurative material was the result of operating. • Moreover, we claim that partitioning one of these four parts into four parts symbolized partitioning each of the remaining three parts into four parts. This symbolic nature of operating apparently induced his reflective awareness of the products of operating without actually operating on sensory material.

21. The Algebraic Nature of Children’s Numerical Knowledge • We construe children’s numerical knowledge as the schemes of acting and operating that are functioning reliably and effectively, and consider them, along with their symbolic aspects, the subject of our study of the algebraic nature of children’s numerical knowledge.