math camppp 2011

Math CAMPPP 2011

Draft Ideas for Plenary SessionsAlgebraic Reasoning

Ruth Beatty & Cathy Bruce

Overview

• 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

• 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

Multiplicative Thinking• Patterning, even at a primitive 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).

Multiplicative Thinking

• Transitioning from additive reasoning to multiplicative reasoning is difficult for students

• In terms of algebraic reasoning, multiplicative thinking is fundamental to understanding the idea of function – or the co-variation between two sets of data – also known as one-to-many correspondence

• Many traditional approaches to modeling functional relationships (linear growing patterns, ordered tables of values) seem to support additive thinking

Input Output

1 4

2 7

3 10

4 13

5 16

What is Additive Thinking?

1

2

3

+3+3

+3+3

When students use additive thinking, they consider the change in only one set of data. 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.

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)

Multiplicative Thinking• Understanding the co-variation of two sets of data• For instance, in this pattern, the mathematical structure can be

articulated initially by a pattern rule, number 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

1 2 3

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

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

• Graphical representations are a powerful way to support this understanding (we’ll explore this when we think about multiple representations)

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 being able to find and articulate a “pattern rule” or equation that represents the mathematical structure of a pattern

• Another form of generalizing is making “far predictions”

Generalizing• Productive problems offer the opportunity to discover multiple

solutions, or generalizations, based on how the pattern is visually perceived

Students can then discover how all of the resulting algebraic expressions relate to one another

Generalizing• Another level of generalizing is recognizing the

structural similarity between two seemingly different problems

This is a 4 dot triangle where each side has 4 dots. It is made using a total of 9 dots.

How many would you need for a 16 dot triangle?How many for a 100 dot triangle?Can you figure out the number of dots you’d need for any triangle?

This is 3x3 grid of squares with only the border squares shaded

If you had a 17x17 grid with only the border squares shaded, how many would be shaded?Can you figure out the number of shaded squares for any grid?

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+3

Generalizing

• Working with patterns allows students to develop rigour 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

Multiple Representations

• Observations of patterns is a powerful way of acquiring a deep understanding of algebraic relationships

• 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)


• 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)


• 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


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


• 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…

Tiles = position number x1+1Tiles = position number x3+1Tiles = 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?

Tiles = position number x3+2Tiles = position number x3+6Tiles = 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?


• 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


This line [3x+5] and this line [2x+6] are going to intersect for sure because 3x+5 for [x-value] one is going to be 8 and for 2x+6 it’s also going to be 8.

So, the zeroth position is going to be 5 for this one [drawing a point at 5 on the y-axis] and 6 for this one [drawing a point at 6 on the y-axis] and then it’s going to go something like this [sketches the two trend lines].

But I know it’s going to intersect here [pointing at (1, 8)] for sure because for position 1, both answers are 8!


• Grounding an understanding of algebraic relationships in concrete, abstract and graphical representations allows students to construct symbolic expressions

• For instance, an understanding of both linear growing patterns and graphs can support students in solving equations of the form ax+b=cx+d

0 1 2 3

number of tiles = position number x5+3

0 1 2 3



This is an initial understanding of the point of intersection. The pattern rules are represented by both the linear growing patterns AND by the trend lines. The point of intersection indicates where the two patterns would have the same number of blocks. This is a preliminary understanding that the point of intersection indicates the point where the rules, calculated for the same position number (x-value) result in the same number of tiles (y-value).

Anne: The x5+3 is going to start higher because the plus 3 is a higher number so it would start higher but not grow as fast. The x6+2 would be steeper because x6 is a higher value, and so would mean a steeper line on the graph because it’s growing faster.

Jack: So they will intersect like pow, smack, boom! They’re going to intersect at 1 and then keep going!


{3 spaces apart

{2 spaces apart

{1 space apart

intersection

Pete: For each one (trend line) you have to think about how far apart they’re starting on the graph and how long it will take them come together. So if the rules are x3+5 and x4+2, they start three spaces apart and get together by one space each time, so it would take them to the third position to intersect.

number of tiles = position number x3+5 number of tiles = position number x4+2

Students develop a sense of the difference between where the trend lines “start” on the y-axis, as indicated by the values of the constant, and the rate at which they “come together” as indicated by the multipliers. For the two rules above, the trend lines start 3 spaces apart, but come together by 1 space each time, therefore they intersect at (3,14). Students figure this out both numerically (by comparing the values of the parameters) and by considering the trajectories of the trend lines.

{{

4 spaces apart

2 spaces apart

intersection



Alan: So then say you have multipliers that are two apart. The trend lines would come together by 2 spaces each time. So if they start 4 spaces apart, and then come together by 2 each time, they’ll intersect at position 2.

John: So if you know how far apart they start, and you know how fast they come together by comparing the multipliers, you can predict where the trend lines will intersect!


For the two rules above, the trend lines start 4 spaces apart, but come together by 2 spaces each time, therefore they intersect at (2,12).

• This helps us construct an understanding of how to solve equations of the form ax+b=cx+d

difference of 3

difference of 15

15 ÷ 3 = 52x+16 = 5x+1


Josh: Well I would see that the difference between these two [the coefficients] is 3 and that the difference between these two [constants] is 15. I know that 15 divided by 3 is 5, so I think it’s going to intersect on the 5 th position.

Teacher: How would you check?

Josh: Try it out. So 5x2 is 10 plus 16 is 26, and 5 times 5 is 25 plus 1 is 26.

Teacher: What does it mean when you get 26 for both rules?

Josh: Um, that’s the amount of each pattern and that’s where they would intersect.

Teacher: But I thought you said they intersect at 5?

Josh: The 5th position! At the 5th position they would both equal 26 tiles – that’s

like the number they would intersect on. They’d both end up there.

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.


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)

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

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)

Some Suggested Tasks

• Graphical Tasks– Given a graph of 5x+3, find all the “rules” or

expressions that will result in trend lines that intersect at (3, 18). What do you notice about the rules? [There is a lot of room for exploration with this task. Some student might start to incorporate negative values for the coefficient or the constant – can they represent these graphically? How can students know with certainty that they have discovered all possible rules that include only whole positive values? What do students notice about the rules or expressions? How can this help them to start to think about predicting the point of intersection for a given set of rules/expressions? What does the point of intersection represent?]


• Graphical Tasks– Given a graph of 5x+3 (or any other expression) can

you think of a rule or expression that will result in• A parallel trend line• A trend line with the same y-intercept• A trend line that will intersect with the given trend line

[Again this task offers a lot of scope for exploration. Do students offer specific rules or a more generalized understanding that, for instance, rules with the same coefficient will be parallel. Do they offer any examples with negative values? Do they offer examples with rational numbers? Is the point of intersection always in the first quadrant, or other quadrants – for instance, what if the rule/expression is 6x+4?]


• Other suggestions most welcome!!

math camppp 2011

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