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2017. In P. Preciado Babb, L. Yeworiew, & S. Sabbaghan (Eds.). Selected Proceedings of the

IDEAS Conference: Leading Educational Change, pp. 169-178. Calgary, Canada: Werklund

School of Education, University of Calgary.

USING VARIATION TO CRITIQUE AND ADAPT MATHEMATICAL

TASKS

Martina Metz, Paulino Preciado-Babb, Soroush Sabbaghan, Brent Davis, and Alemu Ashebir

Werklund School of Education

We report on four key ideas we have found important in our work with teachers

based on almost five years of research with the Math Minds Initiative. These ideas

combine the Variation Theory of Learning with a strong focus on continuous

assessment to inform the way teachers adapt task sequences offered in the resource

used by project teachers. In doing so, we expect that teachers aim to better serve

both struggling students and those who need extension as they develop coherent

mathematical knowing. We elaborate on each one of these ideas, with examples

from the Initiative in this paper.

Keywords: Variation Theory of Learning, continuous assessment, mastery learning

For almost five years, the Math Minds Initiative has centered on improving mathematics instruction

at the elementary level in Calgary. Part of this work has focused on teacher professional learning,

which is the focus of this paper. The initiative integrates research on formative assessment (Wiliam,

2011), intrinsic motivation (Pink 2011), mastery learning (Guskey, 2010) cognitive load (Clark,

Kirschner, & Sweller, 2012), and variation theory (Marton, 2015), which we have integrated in a

four-part framework to support the feedback we offer teachers (see Figure 1).

Here, what we call ribboning refers to an alternating pattern of tasks that draw attention to key

ideas and assessment of each key idea. Ribboning draws on both variation theory and cognitive

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IDEAS 2017 170

load theory to effectively direct attention to key ideas without overwhelming working memory.

Clear patterns of variation among ideas that may be held together in working memory makes it

more likely that connections between those ideas may be discerned. Formative assessment with

careful monitoring of student feedback and teaching that is responsive to that feedback are essential

to ensure that intended distinctions are in fact made before we ask students to extend or connect

them. In this paper, we focus primarily on variation, but it is important to emphasize how variation

supports and is supported by the other elements in the protocol.

Ribboning

Monitoring

Adapting

Connecting

Use structured

variation to draw

attention to

potentially novel

discernments

necessary to a

concept.

Ensure every student

is able and obligated

to provide feedback

to the teacher in

response to each

ribboned query.

Revise/devise tasks,

explanations, and

other engagements to

fit with demonstrated

understandings.

Move between “part”

and “whole” when

ribboning to ensure

that learners do not

lose sight of the

concept(s) under

study.

Figure 1: Math Minds Principles (adapted from Davis, 2016)

In previous work, we have described some of the challenges teachers face in (a) supporting

struggling students and (b) offering meaningful extensions (cf. Preciado-Babb, Aljarrah,

Sabbaghan, Metz, Pinchbeck, & Davis, 2016). We have also highlighted some of the difficulties

they experience when creating their own patterns of variation (Metz, Preciado-Babb, Sabbaghan,

Pinchbeck, Aljarrah, & Davis, 2016). We have since asked teachers to consider the following

questions as they examine the patterns of variation offered in the resource used by all project

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IDEAS 2017 171

teachers: (a) What features of the concept are separated for attention? (b) Are the features we wish

to draw attention to systematically varied? (c) Is variation set against a constant background? and

(d) Are key ideas juxtaposed in a manner that highlights the desired pattern of variation?

Here, we highlight four ideas that have emerged as significant in our work with teachers and

students around these questions. The first reiterates the importance of contrast: When too many

things change or the wrong things change, students may not make intended distinctions. The second

stresses that even tightly controlled variation may not be noticed if relevant items are not clearly

juxtaposed and if students are not invited to notice what changes and what stays the same from

example to example. In the third, we further distinguish between parsing work into manageable

chunks and using clear patterns of variation to draw attention to key concepts within those chunks.

Finally, in the fourth idea, we address concerns that tightly structured variation may allow students

to merely extend patterns without understanding why they occur. In the following sections, we

offer selected examples from our data set to exemplify each of these four ideas. Throughout, we

maintain an emphasis on continuous assessment and adaptive response: None of the identified

patterns should be taken as ideal or sufficient unto themselves. When tasks are organized such that

each piece builds on understanding of the previous, assessing understanding at the end of a lesson

or even at several intervals during a lesson is not enough.

ATTENDING TO DIFFERENCE: THE POWER OF CONTRAST

A subtle but powerful insight may be found in the distinction between systematic variation that

varies the feature we want students to notice while holding other features constant (i.e., contrast)

from systematic variation that varies everything else while holding the key idea constant (i.e.,

generalization). Marton (2015) noted that we tend intuitively toward the latter; i.e., when we want

to help someone understand something, we offer many examples. But this only works if the idea

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IDEAS 2017 172

has already been discerned and we need only to broaden perception of what it may encompass. It

does not work well for introducing new ideas. Appreciating this distinction can shift how we adapt

explanations and tasks to support struggling learners.

In presenting rounding at Grade 5, the resource used by the teacher first uses a number line to

prompt attention to whether various numbers are closer to 0 or 10, then 10 or 100. It then offers a

common procedure that involves checking the digit after the one being rounded to; if 5 or up, round

up; if less than 5, round down. Practice includes a collection of numbers to round to 10, a different

collection to round to 100, and another to round to 1000. This pattern was unproblematic for some.

One student, however, didn’t make sense of this until asked to contrast the same number rounded

to different place values, as in Figure 2. This student was particularly intrigued by the staircase

pattern of zeroes that forms in the first set and by the fact that ten million rounded to zero. He was

also interested in the third set, where all but the last case rounds to 2,000,000. What began as

support for a struggling student went significantly beyond the original task of rounding to 1000.

Figure 2: Using contrast to deepen understanding of rounding.

JUXTAPOSITION: THE IMPORTANCE OF DIRECT CONTRAST

Even when patterns of variation seem clear and teaching is responsive, however, there are times

when patterns go unnoticed or unmarked. Marton (2015) did not directly address cognitive load

(Clark, Kirschner, & Sweller, 2012). However, he stressed the importance of holding ideas

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IDEAS 2017 173

simultaneously in consciousness, which requires attention to working memory as well as careful

consideration of what it is that needs to be held together: “[T]he experience of sameness and

difference is not only a function of what there is to be experienced but of what things are

experienced simultaneously” (Marton, 2015, p. 66).

Effective juxtaposition supports simultaneous experience, and there are various tools and strategies

that can be used to support this. Mathematical representations often juxtapose ideas in ways that

allow particular relationships to be discerned. Consider the number line, the intersection of number

lines that form the axes of a Cartesian plane, or the plotting of a unit circle on a mere 2 x 2 portion

of that plane: Each allows new ways to experience mathematical relationships. We have also

observed pedagogical strategies that support the clear and uncluttered juxtaposition of ideas in time

and space.

In the rounding example offered earlier (and in other examples that follow), notice that the

sequences were presented on grid paper. While this helped keep work neat and organized, it also

did something more powerful: It made tracking variation easier by making it easier to see what

changed and what stayed the same from example to example. It was still important to ask “What

changes? What stays the same?” and to explore “Why?” in response to the student excited to see

emerging patterns such as the staircase pattern of zeroes in the first set of rounding tasks.

The resource used by Math Minds teachers includes a series of slides for each lesson. The simple

act of laying those slides side by side can help prompt attention to significant contrasts between

subsequent examples. In the original lesson highlighted in Figure 3, each of the three images

representing 28 and the T-Table were presented on different slides.

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IDEAS 2017 174

Figure 3: Representing 28 (Adapted from Mighton, Sabourin, & Klebanov, 2012, Slides 2-4).

After juxtaposing the images, the teacher asked “What is the same about the three charts on the

smartboard?” Initial responses focused on the open 29 and 30 in each chart. Her next question

prompted attention to difference: “Do they all look exactly the same way?” The students then noted

the changing number of tens and ones blocks, which she recorded in the T-Table beside the images.

ATTENDING TO LIMITS ON WORKING MEMORY: PARSING WITHOUT

FRAGMENTING

At times, demands on working memory (Clark, Kirschner, & Sweller, 2012) make it useful to

separate parts of a more complex procedure such that they may be learned in manageable pieces

and then put back together. While important, this alone does not draw attention to the mathematical

ideas embedded in those steps. In the resource used by teachers in our study, long division at Grade

4 is parsed into steps designed to reduce load on working memory. Students first work with two-

digit dividends and a one-digit divisor; the procedure is parsed into steps that are explained

conceptually, then practiced for fluency before adding the next step. For some, this was sufficient.

Others struggled to remember the steps; among those who did, some could not explain them. We

then offered a practice sequence that systematically varied the divisor, as in Figure 4.

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IDEAS 2017 175

Figure 4: Systematic variation of practice for long division.

Because only one thing changed, students could now see (when prompted to look) how various

parts of the algorithm changed (or not) each time the dividend went up by one (again, note the use

of grid paper to allow easier tracking of changes from example to example); they could also see

that the remainder went up by one each time. When asked why, they could more easily trace the

cause of the changes. Further, because some of the requisite number facts were constant from

question to question, less attention was diverted from understanding the algorithm. At 80/5, the

remainder cycles back to zero, which prompted further insights. It became clearer to students that

the quotient is counting groups (alternatively, it could have been seen as representing an amount

per group), while the remainder is counting singles; i.e., there were important opportunities for

noticing that wouldn’t have been there with more random practice. Significantly, it didn’t take long

before the students using these patterns started looking for connections. Again, what began as a

support for a struggling student soon exposed a pattern that prompted deeper insights into division

than the original task sequence.

HIGHLIGHTING CONNECTIONS: BRIDGING BIG IDEAS

Tightly structured sequences in which only one thing changes sometimes create predictable

sequences that allow students to predict answers based on observed patterns rather than

understanding of those patterns. However, it is easy to trivialize what to students may be important

insights foundational for deeper understanding.

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IDEAS 2017 176

A Grade 5 class was trying to find all the ways to make 45 cents with dimes and nickels. They had

been instructed to start with zero dimes and work their way up to 4 dimes, each time figuring out

how many nickels would be needed to make up the balance and recording answers in a T-Table.

Two students struggled to find the balance of nickels needed. To support them, they were asked to

make marks for nickels, counting by fives to know when they had 45 cents. When asked, “How

many nickels?” they at first wanted to count by fives again and had to be reminded that each mark

was a nickel; they then counted nine sticks. Next, they were instructed to circle two nickels and to

think of this as trading two nickels for a dime. They counted the total again, this time starting with

10 for the dime and adding five for each nickel: 10, 15, 20, 25, 30, 35, 40, 45. When asked how

many nickels remained, both counted the remaining sticks and wrote the total in their T-Tables.

They were then asked to make another dime, count the money to ensure the total was still 45 cents,

and to count the remaining nickels. While this may seem like rote repetition of a procedure, after

three repetitions, one of the students excitedly noted, “The nickels are going down by two every

time!” and related this to the circled nickels. For him, this was a big insight. The other was excited

to discover that the leftover stick at the end couldn’t be used to make a dime. The students needed

help setting this up again for 75 cents, but they started using observed patterns in ways that they

hadn’t done before; i.e., they could see more clearly that every time they added a dime, they had to

take away two nickels. They then worked with quarters and nickels, nickels and pennies, and dimes

and pennies.

IMPLICATIONS AND NEXT STEPS

By using clear patterns of variation coupled with prompts to attend to that variation, bridges as

appropriate when perceptual jumps are too big, and breaks in patterns that push understanding

beyond predictable patterns, we have asked teachers to move beyond what the resource often

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IDEAS 2017 177

describes (appropriately but insufficiently) in terms of scaffolding and attending to limits on

working memory. Teachers in the initiative have relied heavily on the resource to define and

sequence critical features for learning particular topics. However, they are assuming greater

responsibility for what is set side by side in the moment-by-moment sequence of a lesson.

To date, much of our work with teachers has been on connections within lessons. Moving forward,

we will further emphasize how careful attention to patterns of variation might support stronger

connections over time—i.e., between lessons, units, and grades. Like scaffolding, review of

material addressed in previous lessons is emphasized and supported by the resource. As with

scaffolding, conceptualizing review in terms of using variation to carefully juxtapose the old and

the new can support strong connections between key ideas. Contrast is most powerful when directly

experienced and not reliant on memory of a lesson done last year, last week, or even earlier in a

lesson. As doing so requires a broader awareness of what students are expected to understand over

time, we will consider ways to support teachers who are often most familiar with a single grade or

a small range of grades to look beyond the boundaries of their teaching assignments.

References

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Educator 36(1), 6-11.

Davis, B. (2016, November 28). Keynote speech presented at The Art and Science of Math

Education, Carleton University, Ottawa, Ontario, Canada.

Guskey, T. (2010). Lessons of mastery learning. Educational Leadership 68(2), 52-57.

Marton, F. (2015). Necessary conditions of learning. New York, NY: Routledge.

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