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Using Variation to Critique and Adapt Mathematical Tasks

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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.
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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
Metz, Preciado-Babb, Sabbaghan, Davis, & Ashebir
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
Metz, Preciado-Babb, Sabbaghan, Davis, & Ashebir
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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... We hypothesize that this approach respects limitations on students' working memories ( Engle et al. 1992) and makes key ideas explicit. However, we have found that modifications to the JUMP Math materials based on systematic variation have often resulted in more effective ways of drawing students' attention to critical features or connecting key mathematical ideas ( Metz et al. 2017). ...
... While the JUMP Math resource has carefully identified critical discernments to be noticed by learners and has sequenced topics coherently, we have observed that adapting the resource using more clearly structured variation (Marton 2015;Pang et al. 2016;Watson 2017) and remaining mindful of broader learning targets can provide better opportunities for students' learning. Doing so has opened pathways that both supported the weakest students and challenged even the most capable students ( Metz et al. 2017;Preciado-Babb et al. 2017). ...
Chapter
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The use of systematic variance and invariance has been identified as a critical aspect for the design of mathematics lessons in many countries where different forms of lesson study and learning study are common. However, a focus on specific teaching strategies is less frequent in the literature. In particular, the use of systematic variation to inform teachers’ continuous decision-making during class is uncommon. In this chapter, we report on the use of variation theory in the Math Minds Initiative, a project focused on improving mathematics learning at the elementary level. We describe how variation theory is embedded in a teaching approach consisting of four components developed empirically through the longitudinal analysis of more than 5 years of observations of mathematics lessons and students’ performance in mathematics. We also discuss the pivotal role of the particular teaching resource used in the initiative. To illustrate, we offer an analysis of our work with a Grade 1 lesson on understanding tens and ones and a Grade 5 lesson on distinguishing partitive and quotitive division.
... tion are supported and partially constrained by what is offered in the teachers' guide, prepared slides, and student materials. Teachers, however, must be aware of the distinctions made in the resource so that they can effectively highlight them with students and so that they can effectively adapt the materials to support diverse student needs (cf.Metz, Preciado-Babb, Sabbaghan, & Davis, 2017); we discuss this further in the section on responsive teaching.We have recently begun to use the term " ribboning " to describe the way key lesson elements are separated for attention; this phrase stems from work with teachers to colour-code video-taped recordings of their own lessons according to whether a selected moment involves ins ...
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For four years we have invested in improving mathematics teaching at the elementary level. By drawing from diverse research emphases in mathematics education and by considering the impact of lessons in terms of student engagement and performance, we have identified four key elements impacting learning in mathematics. Here, we describe the protocol currently used to structure feedback for teachers in the Math Minds Initiative. The key elements that comprise the protocol are: (1) effective variation, (2) continuous assessment, (3) responsive teaching, and (4) engagement.
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Keynote speech presented at The Art and Science of Math Education
  • B Davis
Davis, B. (2016, November 28). Keynote speech presented at The Art and Science of Math Education, Carleton University, Ottawa, Ontario, Canada.
Teachers' Guide 2:1 Number Sense 2-25
  • J Mighton
  • S Sabourin
  • A Klebanov
Mighton, J., Sabourin, S., & Klebanov, A. (2012). Teachers' Guide 2:1 Number Sense 2-25 [Cdn. SmartBoard slides].
Drive: The surprising truth about what motivates us
  • D Pink
Pink, D. (2011). Drive: The surprising truth about what motivates us. New York: Riverhead.
Teachers' perceived difficulties for creating mathematical extensions at the border of students' discernments
  • M B Turner
  • E E Civil
  • M Eli
Teachers' perceived difficulties for creating mathematical extensions at the border of students' discernments. In Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (Eds.). Proceedings of the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 514-517). Tucson, AZ: The University of Arizona.