DataPDF Available
Teaching and Classroom Practice
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.
Martina Metz A. Paulino Preciado-Babb Soroush Sabbaghan
University of Calgary University of Calgary University of Calgary
Brent Davis Geoffrey Pinchbeck Ayman Aljarrah
University of Calgary University of Calgary University of Calgary
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 approachescommonly
dichotomized as “traditional” and “reform” approacheshave 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
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
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
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 involvesboth
‘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
Teaching and Classroom Practice
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.
(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
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
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
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.
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
Make factor trees for
each of the following:
What is the smallest
number with 14
Use patterns in prime
factors to help you
explore this problem.
Make a factor tree for
Can you do it 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
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
Teaching and Classroom Practice
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.
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.
We acknowledge the generous support of Canadian Oilsands, Ltd. to Math Minds.
Ansari, D. (2015). No more Math Wars: An evidence-based, developmental perspective on math education. Canada
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Technical Report
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