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2016. In M. A. Takeuchi, A. P. Preciado Babb, & J. Lock (Eds.). Proceedings of the IDEAS:

Designing for Innovation, pp. 182-191. Calgary, Canada: University of Calgary.

TEACHERS’ AWARENESS OF VARIATION

Martina Metz Paulino Preciado Babb Soroush Sabbaghan Geoffrey Pinchbeck

Ayman Aljarrah Brent Davis

University of Calgary

Here, we report on a study of teachers’ evolving awareness of how they work with

patterns of variation to structure and teach mathematics lessons. We identify a

number of critical features regarding teachers’ awareness of variation.

Keywords: Mathematical Knowledge for Teaching; Enactivism; Variation Theory

of Learning; STEM; Design Experiments

The Math Minds Initiative is a five-year initiative exploring how critical use of high quality

curriculum resources might help mathematics teachers support student achievement and

confidence. As part of this work, we have identified three key principles as significant: (a)

mastery learning, which emphasizes success-for-all, formative assessment, and enrichment

(Guskey, 2010); (b) structured variation (Marton, 2015; Watson & Mason, 2006); and (c)

intrinsic motivation (Blackwell, Trzesniewski, & Dweck, 2007; Pink, 2011). We treat these

features as design principles that allow flexibility and responsiveness to evolving relationships

between teachers, students, and mathematics. While the resources used in the initiative embody

important features of variation, teachers must select from, highlight, and extend what is offered.

Here, we propose five aspects critical to teachers’ awareness of variation.

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THEORETICAL FRAMEWORK: THE VARIATION THEORY OF LEARNING

Our approach is based on Marton’s (2015) Variation Theory of Learning (VTL). The core

principle of the VTL is that new understanding requires experiencing difference against a

background of sameness, rather than experiencing sameness against a background of difference.

Note that experience of variation is of key significance here; just because certain elements are

available to be experienced does not mean that they will be. This is consistent an enactivist

perspective of knowing (cf. Simmt & Kieren, 2015), in that both enactivism and the VTL

emphasize the significance of an observer and of perceptually guided action in bringing forth a

world of significance. Marton has used the VTL to develop practical implications for directing

attention in ways that make it more likely for learners to attend to particular aspects of a chosen

object of learning. This offers a way to build sufficient commonality for learners’ worlds of

significance to interact and continue to co-evolve with each other and with the object of learning.

The VTL has similarities with the theory of concept attainment elaborated by Bruner,

Goodnow, and Austin, (1967), but it further elaborates particular patterns of variation for

drawing learners’ attention to aspects needed to perceive a particular situation in a particular way.

These patterns include separation of key aspects (through contrast), generalization (through

induction), and fusion (through combination). For example, to discern “green,” we must separate

it by contrast with other colours (which is easier when colour is the only thing that varies),

generalize “green” to other objects that share that colour, then fuse it with other aspects that must

be discerned simultaneously to perceive something in a certain way (e.g. “green” and “plant”).

Clear focus on an object of learning is central to the VTL. Each such object has critical aspects

that learners must discern and that teachers must present in a way that makes them

distinguishable from the background and eventually simultaneously discernable. Marton (2015)

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and Runesson (2005) emphasized that variation offers an important lens for analyzing what is

possible for students to learn, which may be different from what is actually learned.

METHODOLOGY

In this study we wanted to better understand how teachers’ use of variation evolved over time.

Our data were based on weekly classroom observations (typically by a single researcher) of six

teachers’ mathematics classes, ranging from Grade 1 to 6. As we observed, we attended

particularly to (a) task sequences, (b) the ways teachers drew attention to variation within those

tasks, and (c) student engagement with and responses to those tasks. Observations were

structured such that each time the teacher presented a task for which student feedback was

invited, it was recorded, along with visible student responses. Sometimes individual students

would respond to a prompt, while at other times all students would respond in a way that the

teacher could receive feedback via their responses (for example, answers written on

mini-whiteboards, fingers in the air to indicate a particular number). In some cases, the teacher

and a researcher met afterwards to discuss the lessons, with a particular focus on task sequence

and student engagement. Task sequences were analysed in terms of the patterns of variation

available for discernment in the sequences, for student responses to those patterns of variation,

and for how student feedback was used to inform further evolution of the sequences. In the

following section, we describe five features of teachers’ awareness of variation that were

identified through this analysis.

FINDINGS: CRITICAL FEATURES FOR AWARENESS OF VARIATION

To present our results, we elaborate on contrasting task sequences that draw attention to

particular features of teachers’ use of variation. In some cases we contrast a particular teacher’s

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IDEAS 2016 185

sequences at different points in time (with different objects of learning); in others we contrast the

lesson-as-taught with a potential alternative. Through these contrasts, we wish to draw attention

to five features that we deem critical to teachers’ awareness of variation: (a) a focus on a clear

object of learning; (b) attention to mathematical structure; (c) prompts that highlight change

against a background of sameness; (d) efforts to draw attention to variation; and (e) a distinction

between micro-steps and micro-discernments.

Clear Object of Learning

Figure 1 shows a task sequence used by Teacher 1 and a modified version of that sequence.

A

B

5+7=12

5+8=13

5+9=14

4+9=13

3+9=12

2+9=11

3+10=13

4+11=15

5+12=17

50+120

500+1200

5000+12000

5+7=12

5+8=13

5+9=14

4+9=13

3+9=12

2+9=11

3+10=13

4+11=15

5+12=17

105+12 = 117

205+12 = 217

305+12 = 317

405+112 = 517

505+112 = 617

Figure 1: What is the object of learning? (Teacher 1: Apr. 13, 2015)

In Sequence A in Figure 1, the intent was to separate the impact of each addend on the sum, to

then generalize this effect, and finally to fuse the impact of varying both addends. In the last three

items, however, the object of learning shifted to consider the effect of shifting place values.

Sequence B offers an alternative that retains a focus on the original object of learning.

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Attention to Mathematical Structure

Figure 2 shows two sequences used by the same teacher on different days. In Sequence A,

spotting a pattern (Hewitt, 1992) took precedence over insight into mathematical structure,

whereas in Sequence B, attention was drawn to the impact on the sum of altering both addends.

A

B

3 + 7 = 10

4 + 7 = 11

5 + 7 = 12

5 + 8 = 13

5 + 10 = 15

T: Predict the next answer without

knowing the question.

0 + 7 = 7

1 + 6 = 7

2 + 5 = 7

S1: One side is adding 1 number and the other

side is losing one number [many students started

to extend the sequence at this point; S2 extended

to 7+0]

T: What happens to the sum?

S3: The beginning is going up, the middle is

going down, and the total staying the same!

S2: There’s nowhere left to count. [after 7 + 0]

S3: You could switch to 8 – 1, then 9 – 2.

T: Is there a limit?

S1: There’s no limit.

S3: You could go up to 100!

T: Here’s a bonus: 53 + 15 = 68. What is 54 +

14? [several students extended to 68 + 0]

Figure 2: Pattern spotting vs. mathematical structure (Teacher 1: May 1, May 6, 2015)

In Sequence A, the teacher’s focus on the patterns in the sums may be seen in her prompt to

“predict the next answer without knowing the question.” The discussion following Sequence B,

on the other hand, explored the implications of one child’s observation that when one addend

goes up by 1 and other goes down by 1, the sum stays the same. After noting the structure of the

initial sequence, several students spontaneously extended it beyond what the teacher had asked

and one noted a way it might be continued past zero. Later, the teacher made a choice to offer a

similar problem with larger numbers. Had the emphasis in Sequence B remained on “predict the

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next answer,” it would have been a trivial exercise for the students to merely note the repeating

“sum of 7” pattern.

Change against a background of sameness (one thing changing)

Figure 3 shows two sequences used by the same teacher at different times.

A

B

6 x 6 = 36

6 x 7 = 42 (another 6)

6 x 8 = 48 (another 6; most students

successful)

8 x 8 = 64 (2 more 8s; several students

stumped)

7 x 2 = 14

7 x 3 = 21 (another 7)

7 x 4 = 28 (another 7)

7 x 6 = 42 (two more 7s)

7 x 12 = 84 (twice as many 7s)

7 x 24 = 168 (twice as many 7s)

7 x 48 = 336 (twice as many 7s)

7 x 47 = 329 (one less 7)

7 x 23 = 161 (one less 7 than 7 x 24)

7 x 22 = 154 (one less 7)

(high success and engagement)

Figure 3: Changing 2 things vs. changing 1 thing (Teacher 2: Mar. 30, May 1, 2015)

In Sequence A of Figure 3, the fourth item introduced two elements of variation that were not

first varied on their own (i.e. not separated): There was a switch from increasing by 6s to

increasing by 8s, and there was a switch from jumping by 1 multiple to jumping by 2 multiples.

Had the sequence moved from 6 x 8 to 7 x 8 (highlighting a shift from another 6 to another 8) and

then to 8 x 8 (highlighting a shift from one more 8 to two more 8s), this may have been more

effective. In Sequence B, the teacher offered a pattern of variation that separated the effects of

adding or subtracting 7s and doubling the number of 7s. Once students can confidently work with

such changes, they might work with both simultaneously (an example of fusion), perhaps by

doubling and adding 7; e.g. 7 x 22 ! 7 x 43. Or they might try varying either factor: 7 x 22 ! 7

x 23 (one more 7) ! 7 x 25 (two more 7s) ! 8 x 25 (one more 25) ! 12 x 25 (four more 25s) !

24 x 25 (twice as many 25s), etc.

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Sequence A in Figure 4 shows a sequence used by Teacher 3.

A

B

C

D

E

F

2 + 3 = __

6 = 4 + __

7 = __ + 2

4 + 4 = __

6 + 2 = __

3 + 1 = __

2 + 1 = __

2 + 2 = __

2 + 3 = __

3 + 3 = __

4 + 3 = __

5 + 4 = __

6 + 5 = __

5 + 5 = __

4 + 5 = __

4 + 2 = __

__ = 4 + 2

__ = 3 + 5

5 + 3 = ___

4 + 2 = __

__ + 2 = 6

4 + __ = 6

3 + 5 = __

__ + 5 = 8

3 + __ = 8

3 + 4 = __

7 = __ + 4

__ + 3 = 7

__ = 2 + 3

2 + __ = 5

5 = __ + 3

4 + __ = 7

4 + __ = 6

4 + __ = 5

3 + __ = 5

2 + __ = 5

__ + 1 = 3

__ + 1 = 4

__ + 1 = 5

__ + 2 = 5

__ + 3 = 5

Figure 4: Separating critical features (Teacher 3: Nov. 18, 2015)

Here, we see evidence that the teacher was aware of critical features pertaining to students’

understanding of equivalency in addition. This was evident both in her choice of examples that

highlighted various patterns and in her use of verbal cues to draw attention to the features she

wanted students to notice. Nonetheless, these features were not separated through contrast in the

chosen task sequence. In the lesson leading up to Sequence A, the teacher reviewed the meaning

of the equal sign and the plus sign, then worked with the entire class on a sequence intended to

draw attention to the significance of the placement of the missing addend. When presented with

Sequence A, students were asked to model each statement with counters, draw a picture of their

model, and write a number sentence to represent the picture. While explaining the instructions,

the teacher juxtaposed the various representations to highlight the connection between them and

worked through one example of each kind.

For this task, representations changed while a particular addition statement was held constant.

While most students quickly discerned the distinction between representations, the shifting

position of the missing addend and equal sign remained difficult for a significant number of

students. Some students struggled with even the first question in Sequence A (2 + 3 = ____); for

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IDEAS 2016 189

them, Sequence B may have helped draw attention to the structure of the relationship between

sum and addends prior to engaging with varying placement of the addends (Sequence C), the

equal sign (Sequence D), and both (Sequence E). Sequence F offers possibilities for fusing ideas

in B and D.

Attention to Variation

Teachers in our study further drew attention to variation by commenting on or asking children to

consider what was changing and what was staying the same, and by visually juxtaposing

particular features in a manner that allowed this sort of comparison—approaches similar to those

described by Kullberg, Runesson, and Mårtensson (2014). Initially, Teacher 1 (Figure 1) would

erase each item in her sequences prior to putting up another item. As she reflected on one lesson

after class, she recognized that students would likely be better able to detect variation if they

could see relevant examples one above the other. Similarly, positioning variations in such a

manner that the aspects that were similar / different were visually aligned proved helpful in some

cases. Teacher 3 was particularly attentive to highlighting significant changes with visual (e.g.

colour) and / or verbal (e.g. tone) cues. Direct requests to attend to what changed and what stayed

the same were also effective in some cases.

Distinguishing Micro-steps from Micro-discernments

The resource being used in the project places a high emphasis on breaking ideas into small steps,

assessing at every step, having the class move together to greater challenges (with further

extensions for those who need them), and stepping back if it becomes clear that too big a leap has

been made. Invoking the language of variation has helped us to draw attention to the significance

of micro-discernments rather than merely micro-steps. In other words, separating material into

manageable pieces is not just about using smaller pieces or easier numbers; it is about creating

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the conditions necessary for discernment of critical features. While the resource often models

this, it is not made explicit: Micro-discernments are themselves a critical feature that initially was

not adequately separated from micro-steps. Although it is important to start where all students

can engage, to assess continuously, and to adapt task sequences in response to student responses,

it is also important that such adaptations offer meaningful mathematical variation, which is not

defined by difficulty. Often, it is failure to distinguish critical mathematical features that impede

learning, and simply using (for example) smaller numbers does not solve this problem.

SUMMARY

We have observed that combining the practices of mastery learning with careful attention to

mathematical variation allows students to make important distinctions in mathematical

understanding. Teachers’ awareness of this variation is therefore of central importance to

effective teaching, and we believe that the five critical features presented in this paper can inform

teachers’ daily practices. While the examples given here involved very elementary number

sequences, our continuing work with teachers will also support the extension and combination of

concepts to allow for consideration of increasingly complex problems.

References

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Bruner, J., Goodnow, J. J., & Austin, G. A. (1967). A study of thinking. New York: Science

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

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Hewitt, D. (1992). Train spotters' paradise. Mathematics Teaching, 140, 6–8.

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

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