Math CAMPPP 2011 PLENARY 1

1. Math CAMPPP 2011PLENARY 1 Draft Ideas for Plenary Sessions Algebraic Reasoning Ruth Beatty & Cathy Bruce

2. Overview of Two Days • We would like to ground our discussions within the contexts of different tasks in order to consider how to use tasks effectively to support algebraic thinking (and what makes an effective task) • We will practice responding to student work with respect to our three focus concepts • We will explore how multiplicative thinking, generalizing, and understanding multiple representations develops at different grade levels • We will consider how to incorporate concrete, abstract, graphical and symbolic representations at all grade levels • We will also be looking to research in order to better understand how students develop algebraic thinking

3. Three Key Patterning Concepts • The plenary sessions will focus on three fundamental interconnected concepts of algebraic reasoning: • Multiplicative thinking • Generalizing • Multiple representations • We’ve selected these concepts because they continue to be difficult for students and are foundational ideas that underpin algebraic reasoning • During plenary sessions we will also look at responding to student work in the moment (using video of student thinking) and responding to written work

4. Additive and Multiplicative Thinking • Patterning, even at a basic level of skip counting, can play an important role in the development of multiplicative reasoning (Mulligan & Mitchelmore, 1997; Nunes, Bryant, Burman, Bell, Evans, & Hallett, 2009). • The early patterning experiences of young children often involve simple repetition using one variable (e.g., blue, red, blue, red) (Papic, 2007). • Focusing only on repeating patterns in the younger grades may account for difficulties that older students face in recognizing and generalizing patterns and relationships. • Often, the teaching of patterning skills focuses on additive thinking, rather than on multiplicative thinking, which is necessary for developing composite units in complex repetitions, and on constructing growing patterns and functional relationships (Mulligan & Mitchelmore, 1997).

5. What is Additive Thinking? When students use additive thinking, they consider the change in only one set of data, and rely on the previous iteration for information. For instance, in the examples below, students can recognize that the pattern increases by 3 blue tiles each time, or that the value in the right column increases by 3 each time. Students who utilize only additive thinking do not recognize the co-variation between the term number and tiles, or between the two columns in the table. 1 2 3 +3 +3 +3 +3

6. Limitations of Additive Thinking • While using additive thinking allows student to describe the pattern (e.g., add three blue tiles each time) and extend the pattern sequentially, it does not allow for the prediction of terms far down the sequence, for instance the 100th or 375th term • Additive thinking does not allow for finding the mathematical structure of a pattern, and articulating this as a rule (or equation)

7. What is Multiplicative Thinking? In numeration, multiplicative situations involve three ideas: number of items in a group, number of groups, and a total amount Total amount 1 2 3 Number of groups Number of items in each group (equal groups) number of tiles = position number x3

8. What is Multiplicative Thinking? • In Algebra, multiplicative thinking also involves understanding the co-variation of two sets of data [the three ideas here are set 1 (term number or x value), set 2 (total amount of tiles or y value), and the rule between set 1 and set 2] Set 2 Set 1 1 2 3 The mathematical structure of a pattern rule: # of tiles = term number x3+1 In older grades more formal symbolic notation can be used, y=3x+1 This allows students to confidently predict the number of tiles for any term of the pattern.

9. Pattern Building 1 2 3 number of tiles = position number x3

10. Pattern Building 1 2 3 number of tiles = position number x2+3

11. Your Turn!

12. Multiplicative Thinking • Working with patterns (either building patterns, or discovering the mathematical structure of a pattern and expressing it as a “pattern rule”) offers students a concrete context for developing multiplicative thinking • Students develop a concept of “unitizing” – for instance, in the pattern below four green tiles arranged in a checkerboard configuration become the “unit” or “core” of the pattern – the four tiles are simultaneously “four” tiles and “one” pattern core • The number of iterations of the unit depends on the term number • This can also support a developing understanding of the relationship between an independent variable (term number) and a dependent variable (tiles) 1 2 3

13. Multiplicative Thinking • In older grades, students construct an understanding of the role of the multiplier in an equation of the form y=mx or y=mx+b • In addition to patterns, graphical representations are also a powerful way to support this understanding (we’ll explore this when we think about multiple representations)

14. Generalizing • Patterns offer an initial way for students to grapple with the notion of generalizing • Algebra can be thought of as generalizations of laws about relationships between and among numbers and patterns • One form of generalizing is making “far predictions” • Another form of generalizing is being able to find and articulate a “pattern rule” or equation that represents the mathematical structure of a pattern

15. Generalizing • A difficulty for many students is understanding that the “pattern rule” or equation must hold for any iteration of the pattern • Students will often consider only one term or iteration of the pattern when coming up with a rule, and once they select a rule they tend to persist that their rule is correct even when finding a counter example (they tend to refute the example, rather than their rule) • For example, for this pattern most students consider the rule to be 4x+2

16. Generalizing • Working with patterns allows students to develop rigour in their mathematics thinking and a commitment to providing justifications • Students learn to make connections among the actions they take during problem solving (for instance, physically or mentally configuring the matchsticks), the resulting output (number of matchsticks needed for the 10th term, the 20th term etc) and the expression of a generalized rule in the form of a symbolic equation

17. Multiple Representations • Past research indicates that working with visual representations and deconstructing these in order to identify the relationship between variables is a more successful method of developing generalized algebraic formulae than either working with number sequences (using ordered tables of values) or memorizing rules for transforming equations (Beatty & Bruce, 2011)

18. Multiple Representations • Representations fall along a continuum from concrete to symbolic • Concrete (e.g., building linear growing patterns with tiles; this is also geometric) • Abstract (visual representations of patterns, diagrams) • Graphical (graphs) • Symbolic (pattern rules, equations)

19. Multiple Representations • Math education researchers recommend that students be introduced to various representations (concrete, abstract, graphical, symbolic) in order to develop a deep conception of algebraic relationships • Students require opportunities to explore the interaction between relationships in order to make connections, and predict how changes in one representation result in transformations of other representations

20. Multiple Representations • For instance, students can start to make connections between linear growing patterns and graphs

21. Multiple Representations • Students can then start to explore the connections among the representations, for example, how changing the value of the multiplier or the constant in the pattern rule affects • The linear growing pattern • The graphical representation • Students are encouraged to make predictions about what the patterns and graphs will look like • Students are also encouraged to find similarities and differences within and among the different sets of rules…

22. Tiles = position number x1+1 Tiles = position number x3+1 Tiles = position number x5+1 What is similar in the 3 rules? What is different? What is similar in the 3 patterns? What is different? What is similar about the trend lines on the graph? What is different?

23. Tiles = position number x3+2 Tiles = position number x3+6 Tiles = position number x3+9 What is similar in the 3 rules? What is different? What is similar in the 3 patterns? What is different? What is similar about the trend lines on the graph? What is different?

24. Multiple Representations • This then allows students to start to form conjectures, such as: • The multiplier is responsible for the slope of the trend line • The constant tells you “where the line starts” [the y-intercept] • A higher multiplier results in a steeper trend line • Rules that have the same multiplier result in parallel trend lines – there is no x-value (position number) at which these lines would intersect (sets with no solution) • Students can theorize about the graphical representation of rules that have different multipliers and different constants, for instance, 3x+5 and 2x+6

25. Multiple Representations Context or story problems offer another way for students to think about the relationships between quantities. Students need to recognize the dependent and independent variables, represent the quantities in various ways (diagrams, graphs) find a rule, make predictions, and offer solutions to problems.

27. Struggling Learners • We will examine how using a continuum of representations (concrete -> symbolic), sequential tasks, and the integration of online learning objects (CLIPS) can support struggling learners to develop algebraic reasoning (Beatty & Bruce, 2011)

28. Some Suggested Tasks • Paired tasks (tasks that offer the opportunity to recognize structural similarity) • Cube Sticker and Trapezoid Tables • Border Problem and Triangle Dot Problem • Handshake Problem and Pattern Kingdom (non-linear) • Other Tasks • Butterfly House • Swimming Pool Problem (similar to Border Problem but offers the opportunity to compare linear and non-linear growth using patterns, tables, and graphs) • Crossing the River Problem (non-visual – an interesting task because first the pattern must be identified, and then a rule constructed)