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Teaching and Classroom Practice

1252

Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (Eds.). (2016). Proceedings of the 38th annual meeting of the

North American Chapter of the International Group for the Pyschology of Mathematics Education. Tucson, AZ:

The University of Arizona.

TRANSCENDING TRADITIONAL/REFORM DICHOTOMIES IN MATHEMATICS

EDUCATION

Martina Metz A. Paulino Preciado-Babb Soroush Sabbaghan

University of Calgary University of Calgary University of Calgary

martina.metz@ucalgary.ca appreciado@ucalgary.ca ssabbagh@ucalgary.ca

Brent Davis Geoffrey Pinchbeck Ayman Aljarrah

University of Calgary University of Calgary University of Calgary

abdavi@ucalgary.ca ggpinchb@ucalgary.ca ajarrah74@yahoo.com

Here, we report on the development of a theoretical framework for mathematics teaching that looks

aside from common traditional / reform dichotomies. Attention is focused on managing the amount of

new information students must attend to, while structuring that information in a manner that allows

discernment of key features and continuous extension of meaning. Drawing on and combining ideas

based on mastery learning and on the variation theory of learning, we propose an alternative where

fluency and emergent knowing are inseparable and mutually reinforcing, and motivation is based on

success and the challenge of pressing the boundaries of knowing. Gains in student achievement

associated with this framework have been significantly faster than national norms.

Keywords: Mathematical Knowledge for Teaching, Design Experiments, Instructional Activities and

Practices, Elementary School Education

Introduction and Purpose

Recent calls to find a balance between allegedly opposite instructional approaches—commonly

dichotomized as “traditional” and “reform” approaches—have at times posited these extreme

approaches as complementary, suggesting that the two may productively interact. For instance,

Ansari (2015) claimed that “[I]t is time to heed the empirical evidence coming from multiple

scientific disciplines that clearly shows that math instruction is effective when different approaches

are combined in developmentally appropriate ways” (para. 14). In this paper, we argue that such

seemingly contradictory approaches are in fact very similar in significant ways and propose a ‘third

way’ that addresses important features not stressed by either approach.

The work reported here developed from our work with the Math Minds initiative, which is a five-

year partnership between a large school district, a mathematical charity, a children’s support group, a

university education faculty, and a funder. It is aimed at improving mathematics teaching and

learning at the elementary level. More specifically, it aims to deepen understanding of relationships

between teachers’ knowledge, curricular resources, professional development, and students’

performance, a combination not commonly addressed in the literature. In particular, we are working

to identify features of resources that can support the development of teachers’ mathematical

knowledge for teaching. Here, we outline key principles that we have identified as significant for

teaching mathematics and then contrast these principles with other approaches. While our primary

aim here is theoretical, we also provide a brief statement regarding impact on student achievement.

Theoretical Perspectives

While there are widely varying approaches to teaching mathematics, most approaches may be

placed somewhere along a continuum with respect to the degree of emphasis they place on (A)

mastery of fixed algorithms as a means of achieving procedural fluency and (B) conceptual

understanding and mathematical process (cf. Star, 2005). They may similarly be placed along a

second continuum according to the degree of (A) teacher guidance or (B) student responsibility for

developing their own strategies and procedures (cf. Chazan & Ball, 1999). While both A’s are often

Teaching and Classroom Practice

1253

Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (Eds.). (2016). Proceedings of the 38th annual meeting of the

North American Chapter of the International Group for the Pyschology of Mathematics Education. Tucson, AZ:

The University of Arizona.

associated with traditional approaches to teaching mathematics and both B’s are often associated

with reform approaches (cf. National Council of Teachers of Mathematics [NCTM], 1989), even

attempts to balance (A) and (B) must by definition buy into one or both dichotomies (cf. Ansari,

2015; NCTM, 2006; Common Core State Standards Initiative [CCSSI], 2015). What we propose is

not a compromise, but something that shifts attention to features seldom addressed by either: In this

light, A’s and B’s emphases turn out to have some surprising similarities. Perhaps most notably, both

A and B (and points between) typically require students to attend simultaneously to multiple pieces

of new information.

Although the notion of cognitive load has been primarily used to critique Type (B) (cf. Kirschner,

Sweller, & Clark, 2007), it can also lead to difficulty in Type (A) approaches that present complex

procedures that require simultaneous attention to multiple pieces of new information. Here, Marton’s

(2015) Variation Theory of Learning provides a helpful framework for analysis. Marton argued that

“we can only find a new meaning through the difference between meanings” and that “the secret of

learning is to be found in the pattern of variance and invariance experienced by learners” (p. xi).

More specifically, these patterns of variation are divided into three main categories: contrast,

generalization (or induction), and fusion. Contrast allows separation or discernment of a critical

feature by varying only the thing one wants to draw attention to; generalization separates the ways a

previously discerned feature or object can vary, and fusion recombines features that have been

separated. When variation is not carefully structured, learners may overlook significant discernments

critical to the intended object of learning. In addition, individual items may be perceived as either

unique and difficult or as boring and repetitive, with no further meaning to be potentially gleaned

from the juxtaposition of different items or from their combination in increasingly complex

arrangements.

Type (B) approaches typically aim to engage students in mathematical contexts that help them

make sense of mathematical relationships through processes such as problem solving, reasoning,

proof, communication, representation, and making connections (cf. CCSSI, 2015; Kilpatrick,

Swafford, & Findell, 2001; NCTM, 2000; Western and Northern Canadian Protocol, 2006). While it

is possible to attend to these features in conjunction with mastery and careful variation, the

importance of doing so is typically not made explicit. For example, in our own work outside of this

project, we have noted that with an emphasis on multiple strategies, students have often remained

unaware of the connections between them. Problems too complex for students to unpack on their

own have prompted teachers to do so for them. Further, some students have engaged in ways that

allowed them to bypass key learning objectives.

In documenting the difficulties teachers sometimes face when transitioning away from

transmission-based models of math instruction (Type A), Swan, Peadman, Doorman, and Mooldjik

(2013) noted that teachers may “at first withdraw support from students and then recognise the need

to redefine their own role in the classroom” (p. 951). In our work, we have not asked teachers to

begin with complex problems nor to withdraw key supports but have instead focused directly on

what different supports might look like, and how these might contribute to both sense-making and

continuous extension of mathematical knowing.

A key motivating principle of this study is the conviction that virtually all students are capable of

learning challenging material (in our case, mathematics) if given appropriate supports (Bruner,

1960). From the outset, we discussed with teachers the importance of nurturing a growth mindset

(Blackwell, Trzesniewski, & Dweck, 2007), which includes the belief that mathematical ability is

learnable rather than innate. We explicitly advocated a mastery approach to learning (Guskey, 2010),

with an emphasis on parsing material into manageable pieces, assessing continuously (also see

Wiliam, 2011), moving forward as students demonstrate independent mastery, and extending as

needed to ensure challenge for all. We further distinguish our approach through use of what we call

micro-variation, in which we treat variations that might otherwise be seen as trivial as legitimate

Teaching and Classroom Practice

1254

Wood, M. B., Turner, E. E., Civil, M., & Eli, J. A. (Eds.). (2016). Proceedings of the 38th annual meeting of the

North American Chapter of the International Group for the Pyschology of Mathematics Education. Tucson, AZ:

The University of Arizona.

obstacles (Metz et al., 2015). By structuring variation in a responsive manner, we have worked to

nurture classroom environments where all can succeed and be challenged. Motivation is thus based

on success and continuous growth (Malone & Lepper, 1987; Pink, 2007).

Building on the Marton’s Variation Theory of Learning (cf. Gu, Huang, & Marton, 2004;

Kullberg, Runesson, & Mårtensson, 2014; Marton, 2015; Park, 2006; Runesson, 2005; 2006; Watson

& Mason, 2005; 2006), we have worked to avoid the fragmentation that can happen when curricula

are parsed into tiny pieces. By using systematic variation (Park, 2006) to draw attention to key ideas

that are often overlooked and by considering how these may change within and eventually between

topics, we have aimed to support the development of rich webs of interconnected understanding (for

both teachers and students). These webs then allow the continued emergence of new understanding

(Davis & Renert, 2014).

Mode of Inquiry

We began the study with the intent of exploring the impact of a mastery approach to learning

with an explicit focus on structured variation. The study draws on multiple sources of data

(classroom observation, video-taped classes, teacher and student interviews, informal conversations

with teachers, and standardized tests) to inform next steps. The principles that currently guide our

work have been both informed by and used to inform a supporting resource and an associated

professional development program for participant teachers. The primary study site is a small K-6

elementary school with a history of low achievement. Our work is consistent with Cobb, Confrey,

diSessa, Lehrer, and Schauble’s (2003) description of design-based research in that it involves “both

‘engineering’ particular forms of learning and systematically studying those forms of learning within

the context defined by the means of supporting them” (p. 9). Also consistent with a design focus, our

work has been subject to ongoing test and revision, continuously informed by classroom observations

and interactions with teachers and students. It has also developed in response to insights gained

through regular meetings among members of the research team and meetings involving the research

team, school leaders, and a representative from the teaching resource used to support the initiative.

Relationships among initiative partners are deeply reciprocal: School leaders contribute their

expertise and inform the other partners of school-based needs as well as learn from the expertise of

the others; the person who represents the resource offers support to both teachers and to the research

team in accessing key features of the resource and maximizing their potential, while also gathering

feedback that will be used to improve the resource. The research team relies on the insights and

expertise of the school-based leaders and the resource representative, while offering feedback based

on research observations.

As part of the initiative, teachers were provided with guides, resource materials, and student

materials that support our emphases on growth mindset, mastery learning, continuous assessment,

and careful attention to variation. With these in place, ongoing professional development has

emphasized the importance of appropriate parsing and pacing of instruction, continuous assessment

and feedback, and strategies for extending work beyond that provided in the resources. In working

with teachers as they attempt to use these ideas and materials, we have found it useful to clarify

distinctions between this approach and the more familiar (A) and (B) approaches described earlier as

dichotomies; while perhaps overgeneralized if taken all together and taken in their extreme forms,

these have provided useful points of contrast for what we here describe as approach (C).

Distinguishing a Third Way

We now offer a summary of our current thinking on these matters. Here, we refer to (A) and (B)

to elaborate the dichotomies we presented earlier, while (C) represents our own current thinking.

Importantly, we see (C) not as a balance but as something significantly different from either (A) or

(B). Perhaps most notably, (C) asks students to attend to one new idea at a time, where both (A) and

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The University of Arizona.

(B) tend to require simultaneous attention to multiple topics, whether those be (A) elements of

complex procedures or (B) multiple topics / strategies. By systematically varying one thing at a time,

(C) invites continuous extension and elaboration of meaning rather than the articulation of meaning

found in given applications (as is common in A) or broad contexts (as in B). Therefore, in (C),

fluency and understanding are mutually supporting rather than competing. Considerations of the role

of practice follow closely from these views, with (A) using repetitive practice to build fluency, (B)

using embedded practice presumably made meaningful through context, and (C) offering practice

through continuous extension. Positions (A) and (B) tend to result in a wide range of achievement: In

(A), some students successfully master content, while others do not, where (B) intentionally allows

multiple entry points and open-ended paths. Position (C) supports the mastery of a common base of

understanding that may be personalized through extension.

In working with teachers, we have found it important to draw attention to the manner in which

selected resources offer variation, both so that teachers can draw student attention to such variation

and so that they can adapt given examples in ways that support struggling learners and those who

require extension. In Tables 2 and 3, we offer examples to clarify the distinctions we are attempting

to make; here, Task Sets A, B, and C correspond to the same distinctions we have described in terms

of a traditional (A) / reform (B) dichotomy and a proposed alternative (C).

In Table 1, Task Set A offers focused practice with grouping coins to make particular values, but

both the values and the numbers of coins vary from item to item. As a result, there is limited value in

looking for connections between individual items: The sequence does little to scaffold such relational

thinking. For instance, a student fluent in calculating differences might note that moving from $0.30

to $0.45 with two additional coins would require an additional dime and nickel. However, a similar

transition makes less sense in moving from $0.45 to $0.55 with three additional coins. In addition to

regrouping coins, working effectively with Set B requires a systematic approach to finding all

combinations. If both re-grouping money values and exploring combinations are new to students, this

question likely varies too much at once; importantly, only students who work systematically are

exposed to meaningful variation among items, while others may choose a more random approach to

finding possible coin values. In Set C, only the number of coins varies, and students are not (yet)

responsible for creating the variation that prompts attention to relationships between subsequent

items (as in Set B).

Table 1: Three ways to vary coin-grouping tasks

A

B

C

Using quarters, dimes, and /

or nickels:

Make $0.30 with 3 coins.

Make $0.45 with 5 coins.

Make $0.55 with 8 coins.

Make $0.60 with 6 coins.

Find all the ways to make

$0.45 with quarters, nickels,

and dimes.

Using quarters, dimes, and

nickels:

Make $0.45 with 3 coins.

Can you do it with 4 coins?

Can you do it with 5 coins?

Can you do it with 7 coins?

In Table 2, the distinctions between the sets are perhaps even more pronounced. In Set A, the

numbers were chosen to allow variation in the size of number and number of factors (though of

course bigger numbers do not always have more factors); while this may offer practice with creating

factor trees, it does little to direct attention to ways that prime factors can make particular number

structures visible, as too many features change from item to item. To successfully complete Set B,

students must discern many relationships between prime factors and factors, not the least of which is

that a particular pattern of unique and non-unique prime factors will always yield the same number of

factors (e.g. 2 x 2 x 3 has the same number of factors as 3 x 3 x 5, as both follow an a x a x b

pattern); here, the onus is on students to organize their work in a manner that allows exploration of

Teaching and Classroom Practice

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The University of Arizona.

particular patterns of variation. If they fail to do so, the variation they experience will be largely

random. By carefully controlling how much is changing, Set C1 can be used to draw attention to the

fact that regardless of how a number is factored, the prime factors are always the same (which is not

obvious to many students). Similarly, C2 draws attention to patterns in the structure of numbers;

teachers and students alike are sometimes surprised to discover that doubling a number does not

double the number of prime factors.

Table 2: Three ways to explore prime factors

A

B

C1

C2

Make factor trees for

each of the following:

15

18

40

60

What is the smallest

number with 14

divisors?

Use patterns in prime

factors to help you

explore this problem.

Make a factor tree for

36.

Can you do it another

way?

Another?

Another?

What is the

same/different about

your solutions?

Make a factor tree for

each of the following.

In each set, what

changes from one to

the next?

8, 16, 32, 64

5, 25, 125, 625

10, 100 1000, 10,000

25, 50, 75, 100

In sessions for professional development, we have observed that teachers who worked through

these examples noted that the tasks in the (C) sets made it easier for them to recognize important

ideas as well as to create their own extensions (something that most have struggled with): They, too,

were prompted to think differently by this arrangement of content. In other words, when a resource

offers clearly structured variation, both teachers and students engage differently with the content.

This is not to say that the tasks in A and B have no place if properly contextualized within an

appropriate sequence; rather, our key point is that C serves a critical purpose that is often

overlooked.

While our aim here is primarily theoretical, we note that evidence based on weekly classroom

observations shows students who previously struggled have become more willing to take part, and

many students have become excited to keep pushing their understanding to new levels. We have also

noticed a consistent improvement in mathematics basic skills. The school’s total math scores on the

Canadian Test of Basic Skills (Nelson, 2014) improved over a two-year period at a rate that was

significantly faster than the national normed population gains [F (2,70)=6.977, p=.002]. Overall, the

model indicates that the mean total math score increased from an estimated average score of 43.5

(national average=50) to that of 47.0 [t(60.42)=3.732, p<.001], with roughly equal gains over the

national normed population in each year (year one = +1.7%, year 2 = +1.8%).

Conclusions & Significance

The approach briefly described in this paper resembles current research emerging in the UK that

has emphasized combining mastery learning with structured variation (cf. National Centre for

Excellence in Teaching Mathematics, 2014; Schripp, 2015). Two features that set our work apart

from these efforts are (a) the emphasis on what we have been referring to as micro-variation and (b)

attention to intrinsic motivation by using micro-variation to continuously elaborate understanding

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The University of Arizona.

and increase levels of challenge in ways that are accessible to all. We argue that it is not enough to

combine instructional approaches; rather, it is necessary to transcend the borders of restrictive

dichotomies. As we suggested in the beginning, we see (C) as an alternative to approaches that, when

considered in terms of how content is structured, turn out to be more similar than typical traditional

vs. reform dichotomies suggest. Based on results from the first three years of the program, we

propose that combining mastery learning and structured variation in a context that attends closely to

continuous assessment, intrinsic motivation, and emergent knowing holds great promise for

improving student achievement in mathematics and for supporting teachers in achieving these goals.

Considering this third way opens up new avenues for research. It will be important to investigate

strategies to support teachers who are using this approach. We are interested in the interaction

between educational curricular material and teachers’ knowledge.

Acknowledgement

We acknowledge the generous support of Canadian Oilsands, Ltd. to Math Minds.

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