Variables & Expressions: Generalization

MODULE Variables & Expressions: Generalization

BACKGROUND The transition from arithmetic to algebra is primarily one of moving from concrete to abstract thinking. Abstract thinking in algebra is often the generalization of specific instances in arithmetic to a rule or formula that captures the pattern algebraically.

1) SET: Engage with a problem or problems that help teachers consider students' algebraic thinking (teachers’ prior knowledge)

A problem/scenario in which teachers need to struggle with the concept of generalizing a series of computations, such as: after solving three or more “straw in the juice box problems” (until they are frustrated and want the “need for algebra” to make it “easier”) the expectation is that the Preservice teachers then generalize how to find the length of the straw.

2) STUDENTS: Watch video clips of students describing their thinking as they engage with problems What do you learn from what you are hearing or seeing regarding students' thinking?

Video of formative assessments below. 1) ANSWER THE FOLLOWING: 65 + 49 = 58 + 79 = 126 + 199 = Tell me how you solved this mentally.

[We would hope for something like: 65 + 49 = (64 + 1) + (50 - 1) = 64 + 50 = 114.]

( From arithmetic processes to algebraic thinking )

2) Jason uses a simple method to work out problems like 27 + 15 and 34 + 19 in his head.

a) Show how to use Jason’s method to work out 298 + 57 b) Show how to use Jason’s method to work out 35.7 + 9.8 c) Use Jason’s method to work out what goes in the space: 58 + n = 60 + ______d) Use Jason’s method to work out what goes in the space: 9.9 + k = 10 + ______e) Use Jason’s method to work out what goes in the space: a + b = (a + c) + ______

(From Generalization of Patterns) Find the next term:

a) 1, 2, 4, 8, … b) 1, 3, 6, 10, … c) 1, 4, 9, 16, … d) 1, 1, 2, 3, 5, …

Ask the students to write this process algebraically.

We would hope for something like: x + y = (x + a) + (y – a); •••

Making meaning from the video excerpts: Preservice students: Discussion about sequencing problems to give to students; [problems where one of the units is 9, so the compensation was to take one from the other addend to the other and make a “ten”] – do you now move to units of 8? Or units of 1?

3) RESEARCH: Examine/discuss research (encyclopedia entries) Variables: Generalization -- From arithmetic processes to algebraic thinking; Variables: Expressions -- Teaching algebra through arithmetic Variables: Teaching Strategies -- Using Spreadsheets to gain algebraic thinking Variables: Generalizing from Patterns

Students must be exposed to the structure of algebraic expressions. However, it must be done in a way that enables them to develop structure sense. This means they will be able to use equivalent structures of an expression flexibly and creatively.

The research confirms the assumption that difficulties revealed in children’s understanding of structural properties of the algebraic system originate in their understanding of the number system.

From an educational point of view, what matters is not to learn to find a numerical solution to algebraic problems but rather to understand the nature and the power of the theoretical problem solving method of algebra.

There can be three approaches to increasing the ability to think algebraically; using arithmetic strategies, using variables, expressions and equations, and using spreadsheets. There is a developmental progression of understanding generalization. See “Generalization of Patterns” in Variables: Generalization.

Learning to generalize means changing students’ level of reasoning and communication to get them to focus on the patterns, procedures, structures, and relations across and between cases, rather than on a single case.[vi]

Researchers suggest that generalizing occurs in three stages: 1) seeing (recognizing the pattern), 2) saying (verbalizing a description), and 3) recording (creating a written representation).[viii]

Given a pattern such as “1, 4, 7, 10, …”, students are typically asked to figure out the next few terms, as in Problem #1, then predict the tenth or fiftieth term. Finally, they are asked to predict the nth term, in which n could be any number. These represent stages of development:[x] Stage 0: no progress. Stage 1: next term provided. Stage 2: next and tenth terms provided. Stage 3: next, tenth, and fiftieth terms provided. Stage 4: next, tenth, fiftieth, and nth terms provided. Stage 4a: a correct verbal statement Stage 4b: a creditable attempt at an algebraic expression Stage 4c: a correct algebraic representation, but not necessarily the simplest

4) ASSESSMENT: Consider assessments (Formative Assessment Database)

a) Show how to use Jason’s method to work out 298 + 57 b) Show how to use Jason’s method to work out 35.7 + 9.8 c) Use Jason’s method to work out what goes in the space: 58 + n = 60 + ______d) Use Jason’s method to work out what goes in the space: 9.9 + k = 10 + ______e) Use Jason’s method to work out what goes in the space: a + b = (a + c) + ______

5) SUGGESTIONS FOR TEACHING: Consider strategies based on research (including apps) With students in teams of 3 or 4, ask them to work a few problems independently then have them compare their results, explaining the order of their operations. When teammates have different results, expect them to discuss their differences and come to a common result.

6) Did the preservice teachers understand? How do you know? Evidence

REFERENCES Britt, M., & Irwin, K. (2008). Algebraic thinking with and without algebraic representation: a three-year longitudinal study. ZDM, 40(1), 39-53. Linchevski, L., & Livneh, D. (1999). Structure Sense: The Relationship between Algebraic and Numerical Contexts. Educational Studies in Mathematics, 40(2), 173-196.

Dettori, G., Garuti, R., & Lemut, E. (2001). From arithmetic to algebraic thinking by using a spreadsheet. In R. Sutherland (Ed.), Perspectives on School Algebra (pp. 191-208). Netherlands: Kluwer Academic Publishing.

[iv] Mason (1996), p. 74. [v] Mason (1996), p. 65. [vi] Kaput (1999), p. 137. [vii] Blanton & Kaput (2002), p. 25.