INSIGHTS ON THE RELATIONSHIPS BETWEEN MATHEMATICS KNOWLEDGE FOR
TEACHERS AND CURRICULAR MATERIAL
University of Calgary
University of Calgary
University of Calgary
University of Calgary
This study reports teachers’ insights and challenges after one year of adopting a curricular material
designed to move students through carefully engineered, small steps and encourage learners through
success and accessible challenges. The analysis of interviews showed that teachers ‘followed’ the
material in different ways, not necessary in-line with its underlying principles. Two of these
principles—bonusing and breaking down concepts into smaller elements—were particularly difficult
for many teachers, suggesting the need of a specific teachers’ mathematical knowledge.
Keywords: Mathematical Knowledge for Teaching; Elementary School Education; Teacher
Education-Inservice (Professional Development); Teacher Knowledge
While there is extensive research on both mathematics teachers’ knowledge and the quality of
curricular materials, the number of studies combining these two factors is limited. In an effort to
address this gap, Charalambous and Hill (2012) reported a multiple case study suggesting that
curricular materials can increase quality of instruction if they are supported and followed properly.
Understanding the relationships between mathematics teachers’ knowledge, curricular materials, and
student performance would inform policy decisions regarding adoption and implementation of new
resources, as well as the design of corresponding professional learning opportunities for teachers.
This paper analyzes one case of an elementary school adopting new curricular material and
engaging teachers in corresponding professional learning over the course of one year. The study was
conducted as part of a broad, longitudinal project, the Math Minds Initiative, involving a school
district in western Canada, researchers from the University of Calgary, and the JUMP Math (2015)
organization. The initiative focused on a particular school with a history of low performance in
mathematics. The purpose of the initiative was to improve mathematics teaching and learning at the
elementary level and to understand the relationship between curricular resources, teachers’
knowledge and students’ performance. We are interested in what teachers need to know in order to
teach mathematics well, and how this knowledge can be supported through access to particular
resources and related teacher professional development. As design-based research, this study draws
on multiple sources of data informing next steps in the initiative. However, the focus of this paper is
on teachers’ experience of adopting the JUMP Math program. Specifically, we address the question:
what were the insights and challenges perceived by teachers during the first year that all teachers at
the school adopted the JUMP material?
Understanding teachers’ insights sheds light on teachers’ learning through the year, as well as
knowledge required to adopt the JUMP Math materials. Teachers’ challenges during this project
provide information about the knowledge required not only for the adoption of the material, but also
for quality mathematics instruction in general.
Curricular Material and Mathematics Knowing for Teachers
Teachers’ disciplinary knowledge of mathematics has been a focus of research since the 1970s.
With an initial emphasis on formal mathematics content, over the last few decades, the main interest
has shifted to more varied aspects of mathematics knowing such as access to a diversity of meanings
for concepts, beliefs on the nature of the subject matter, and how knowledge is enacted in the
classroom (Davis & Renert, 2014; Thompson, 2015). While there are efforts to measure this knowing
through tests, such as the instrument proposed by Thompson, we concur with Davis and Renert’s
argument that such knowing includes an open disposition and cannot therefore be readily measured
with tests and other instruments. Two features of this disposition are relevant for this report. First,
teachers have to be responsive to students’ mathematical conceptions and misconceptions. They
should be continuously aware of students’ potential interpretations of a concept. Second, school
mathematics is not limited to standard definitions, notations and algorithms such as those reflected in
a program of studies. Teachers should be open to enact mathematics as a creative, emergent activity,
which involves mathematical explorations and inquiry beyond textbooks that may result in insights
not only for students, but also for teachers.
Teachers draw from a variety of resources including textbooks, teachers’ guides, online material,
electronic devices, and the community (Clark-Wilson et al. 2014; Gueudet, Pepin, & Trouche, 2013;
Gueudet & Trouche, 2009). Following Gueudet, Pepin and Trouche, we conceive the adoption of
curricular material as a creative act: “teachers’ work with resources includes selecting, modifying,
and creating new resources, in-class and out-of-class” (p. 1003). Gueudet and Trouche proposed the
term documental genesis for the evolving process of the manner in which teachers use a resource. A
document, for a particular teacher in given moment, consists of a resource and a scheme of
utilization. As the scheme of utilization changes over time, a document is dynamic, whereas the
resource may remain unchanged. The process of document genesis is twofold: “The
instrumentalization dimension conceptualizes the appropriation and reshaping processes … .The
instrumentation dimension conceptualizes the influence onthe teacher’s activity of the resources she
draws on [emphasis added]” (Guedet & Trouche 2009, p. 205). Most recently, Gueudet et al. (2013)
considered a collective dimension of document genesis including joint work on selecting and
adapting educational resources. We extend the idea of document genesis to a more ecological
perspective in which the community includes not only other teachers, but also the research team and
professional learning facilitators. The teacher participants are coupled with the researchers and
facilitators in a process of mutual influence (Preciado Babb, Metz, Marcotte, 2015). In this sense, our
perception as researchers of curricular material is also influenced by our interactions with teachers
and informed by the data collectively gathered and analyzed for research purposes.
The Math Minds Initiative
The Math Minds Initiative is a five-year project started in 2012. While the school district
provided a research school as a main focus for the study, the team from the University of Calgary
provided professional support to teachers from this school as well as from other schools in the
district. The JUMP Math organization contributed the mathematics program as well as further
support for professional learning. During the first year of the initiative, two teachers started mid-year
to use the program with no further support. In 2013 all the teachers were required to adopt JUMP
Math as official curricular material and to attend the corresponding professional development
sessions through the year.
The curricular material provided by JUMP Math consisted of teachers’ guides, an assessment and
practice book for each student, and access to pre-designed SmartBoard slides. Additionally, students
were provided with individual mini-whiteboards—a suggestion from the research team to assist with
the continuous assessment recommended by the resource package.
JUMP Math Principles
The Canadian version of JUMP Math is based on both the Western-Northern Canadian Protocol
for Collaboration in Education (WNCP, 2006), which provides guidelines for the curriculum in
several provinces in Canada, and the Ontario program of studies. The teachers’ guide (Mighton,
Sabourin, & Klebanov,2010) provides lesson plans with references to each particular outcome in the
corresponding program of studies—WNCP or Ontario. The lesson plans correspond to the
assessment and practice book and include individual and group activities and explanations. The guide
shows teachers how to introduce one concept at a time, explore concepts and make connections in a
variety of ways, assess students quickly, extend learning with extra bonus questions and activities,
and develop problem-solving skills. It also provides support material for each strand.
The JUMP Math program is based on a number of principles, including confidence building,
guided practice, guided discovery, continuous assessment, rigorously scaffolded instruction, mental
math, and deep conceptual understanding. While the assessment and practice book consists of
sequences of exercises, the teachers’ guide has numerous suggestions for engaging students in
discovery and problem solving. The guide also encourages students’ independent thought and work:
“When you feel your students have sufficient confidence and the necessary basic skills, let them
explore more challenging or open problems” (Mighton et al. 2010, p. A-5). The JUMP material
shows teachers how to break the material into steps and assess component skills and concepts. It
teaches “fundamental rules, algorithms, and procedures of mathematics for mastery, but students are
enabled to discover those procedures themselves (as well as being encouraged to develop their own
approaches) and are guided to understand the concepts underlying the procedures fully” (p. A-6).
Despite the seemingly direct approach to instruction, every lesson in the teachers’ guide refers to
at least one problem solving strategy, including: looking for patterns; changing into a known
problem; reflecting on other ways to solve a problem; doing a simpler problem first; making and
investigating conjectures; using mental math and estimation; representing; guessing, checking and
revising; selecting tools and strategies; using logical reasoning; justifying the solution; and revisiting
conjectures that were true in one context. An important component of the program is bonusing,
which involves extensions of concepts and skills in each lesson. The teachers’ guide advises teachers
to “be ready to write bonus questions on the board from time to time during the lesson for students
who finish their quizzes or tasks earlier” (Mighton et al. 2010, p. A-8). Lessons in the teachers’ guide
include examples of such questions, and teachers are encouraged to create their own. Strategies to
create bonus question include: change to larger numbers or introduce new terms or elements; ask
students to correct mistakes; ask students to complete missing terms in a sequence; vary the task or
the problem slightly; look for applications of the concept; look for patterns and ask students to
The Math Minds Initiative is design-based research (Cobb, Confrey, diSessa, Lehrer, &
Schauble, 2003) that includes the implementation of the curricular material, as well as professional
development aimed at improving mathematics literacy in a school with a long history of low
achievement—as well as other schools in the district. The initiative also aims at further research and
theory on mathematics teacher knowledge. The research project includes multiple sources of data
such as video-recorded lessons, class observations, longitudinal results of the Canadian Test of Basic
Skills (CTBS, Nelson 2014), and interviews with teachers and students. In this paper we present the
analysis of six semi-structured interviews with teachers who taught during the school year 2013-2014
at the research school. Examples of the interview questions are: What specific advice would you give
to new teachers joining Math Minds? Have you found [JUMP Math] materials to be helpful?
Restrictive or difficult? To what extent did you follow the teachers’ guide? SmartBoard lessons?
Workbook? In what ways did you improvise / extend / elaborate? Have you found [JUMP Math]
principles helpful? Restrictive or difficult? What are your goals or priorities for improving your
teaching of math?
Transcripts of the interviews were coded using NVivo with a particular focus on the manner
teachers used the resources to capture the documentation process. The initial codes were compared
each other, forming broader categories. Four major categories resulted from the analysis, which
included the codes with higher prevalence. These categories are consistent with class observations
conducted by different members of the research team.
We present the results in four sections, corresponding to each major category. While the first
category refers to how teachers used the material in general, the other three are more specific to
JUMP Math principles. Excerpts from the interviews are included as evidence to support our
Teachers claimed that they followed the teachers’ guide and used JUMP Math materials
consistently at the beginning. Some tried to fully complete all the pages in the assessment and
practice books that applied to the official program of studies and to use all of the associated lessons
in the teachers’ guide and, often, all of the associated SmartBoard slides. This is evident in the
Teacher: Whereas I think when you first begin, you feel like, okay, I’ve got to go through each
one, and it just wasn’t working. So again, it’s just the experience and sort of knowing,
okay—and obviously previewing the slides and saying, okay, we don’t—we can skip this
one, or this—unless they’re really struggling or—and just being able to know where can I
stop and how much do I really need to go through all of this.
While it was clear for teachers that some slides or parts of a lesson would have to be selected, the
motivation for such decisions varied. The previous excerpt suggests that the teacher made the
decisions based on assessment of students’ struggles. However, other motivations included both time
pressures and a-priori judgments that some steps were not needed, as is evident in the following
Teacher: Towards the end, when I was trying to catch up a bit, I was taking the teacher guide and
I was looking at the outcomes and what our curriculum outcomes were, and if it was like …
number sense … in four lessons, then I would look at those four lessons, see what the big
picture was, because then I could condense them maybe to two lessons instead of four.
A third type of motivation identified in the interviews was familiarity with another, previously used
resource. One teacher commented that it was easier to use a resource she was already familiar with,
as long as it was similar to what was suggested in the JUMP Math materials:
Teacher: Well I have one that’s very similar that will still teach the same outcome, but it’s a
different game in a little bit of a different way…. Taking what I’ve had from my past as a
teacher, because it worked, it was good. Is it the activity in the JUMP lesson? No, but it
worked. And so it would save me some time that way, because it does take a lot of time to
prep for these, so I would have something like that, maybe use that game instead.
Finally, another manner in which teachers used the material was to select pages from the practice
book for bonusing:
Teacher: I try to follow [the teachers’ guide] exclusively. The SmartBoard lessons, like I say,
some of them—if they’re very hands-on, I will use a lot of them. … I just make sure that
I’ve looked through [the material] and then I just pull up those two or three that I need. And
the workbook, I look at it: is this going to be for everybody or is it going to be a bonus page?
This last excerpt shows a decision based on two JUMP Math principles: continuous assessment and
All teachers mentioned continuous assessment in the interviews. They also consistently referred
to the use of the small whiteboards to assess students in-the-moment. Overall, the material seemed to
impact teachers’ knowledge regarding this fine-grained presentation of concepts and procedures, as
well as the corresponding assessment practice. Continuous assessment not only served to break the
content into small pieces, so everybody understands the concept, skill or instruction in class, but also
to inform decisions about whether to skip parts that might already be mastered. These decisions,
however, seemed to be more difficult to make, as suggested in the following teacher’s comment:
Teacher :I feel like I need to speed up. I don’t know. I need to become better at just moving on
and not getting hung up on things and being able to recognize when we can move on and at
the same time—and so it’s not—at the same time, not compromising that in-depth study of
things. Like knowing where, hey, they got it, we can go. We don’t need to keep doing this.
Breaking down into smaller steps and constantly assessing students was particularly relevant to a
teacher who had been a successful mathematics student:
Teacher: I was very successful in math as a student, and I just get it, and I find it difficult to do
those microsteps back as to how to make it simpler for the kids and simplify it. And when I
taught it that way, I’m like, oh my God, I don’t know how to teach it a different way, because
I just get it. And so I don’t see a different way to get there, and I think that’s my biggest
challenge because I’ve never struggled with math. As a student, I was very, very successful,
but that makes teaching math harder, because I don’t know how to attack a problem from a
This excerpt is consistent with Davis and Renert’s (2014) notion of the teacher being an expert who
is able to appreciate the struggles of a novice.
All teachers made reference to bonusing. However, all but one claimed that finding and creating
bonus questions and tasks was challenging. Although the teachers’ guide shows how to create bonus
questions and the assessment and practice book has bonus questions, one teacher perceived the need
to find bonus questions beyond the material:
Teacher: You need to find bonus questions often from a variety of other sources beyond the
JUMP resource in order to find the proper challenge for each individual child.
This comment also highlights a perceived need to personalize bonus questions. The following
statement reflects a similar assumption:
Teacher: Coming up with really good ones has taken a lot of time, a lot of effort. But I feel now
I’ve got a better idea of kind of what works for the kids as well and also just realizing not
every kid’s going to have the same bonus question, right? Like you’re going to change the
bonus question based on the kid and kind of the extra challenges that they need.
In contrast, the teacher who found it easy to create bonus questions claimed:
Teacher: Everybody is so engaged in the workbooks and so it gives me an opportunity to
continually assess their learning and because there—there is generally enough in the
workbook that everybody has enough to do, and it’s easy. Having said that, it’s very easy to
create challenges from—because of the way that the questions are structured, because of the
way the work is structured. It’s very, very easy to just create challenges on the spot for those
who need it. And in a lot of cases, the students will create challenges on their own—their
For this teacher, bonus questions and tasks were easy to create on the spot by following the
structure of questions in the material. The excerpt also suggests a culture of self-bonusing in her
Inquiry and Problem Solving
Teachers consistently indicated a lack of opportunity for problem solving or inquiry in the JUMP
Math approach. However, most of them indicated that going through the mini-steps was necessary,
and that the program did this very well. An example of a teacher’s perception on inquiry in the
Teacher: And when I—and as far as inquiry goes, that is our direction in education in the next
ten years, and as soon as I heard that, I thought, well, JUMP doesn’t lend itself to inquiry.
But in thinking about it, it certainly can, it just has to—it’s maybe how we’re going to start
praising things but once again, I still think we—we need the foundation before we can even
[missing word?] an inquiry.
And so my struggle this year is sometimes should it—should I just do like an inquiry lesson
or should I stick with my microsteps, but I want to do the microsteps because I’m learning so
much about what I missed teaching them. So to me, right now, that’s more important and
maybe we throw in an inquiry day on Fridays or something. Throw in everything and just
give them an open-ended question and maybe change that next year. It’s just this year I’m
just sticking to my recipe.
In the previous excerpt, the teacher gave second thought to the possibility of including inquiry in the
JUMP Math approach. However, the last comment regarding sticking to the recipe suggests that she
did not see inquiry addressed in the material.
There was a particular comment regarding students not being used to more complex, or multistep
Teacher: So all of a sudden, when [students] had to do this sort of a—in a way, it was a multistep
problem, whereas the vast majority of this program is very one step questions and these
microsteps. So as soon as you throw a multistep problem at them, I was very surprised at how
many kids were just like, whoa, what am I going—how do I solve this? And there was just
no—not even an attempt to work through the problem.
Overall, teachers’ perceptions of inquiry and problem solving seem to be contrary to the problem
solving strategies included in the teachers’ guide.
The analysis of teachers’ interviews presented in this paper yields several conclusions regarding
the interactions of the classroom resources and mathematics knowledge for teaching. First, the
analysis of document geneses showed that teachers’ interpretation of what it means to follow JUMP
Math were very different. The initial approach of having all students cover all the material contrasts
with the approach based on assessing students and selecting pages from the assessment and practice
book for bonus. The latter approach seems to be more aligned with the philosophy of the program.
Second, all the teachers made reference to the incremental steps and to continuous assessment. In
particular, the use of the mini-whiteboards supported continuous assessment during class. It is
particularly interesting that the teachers who claimed having no problem with mathematics when she
was a student found it difficult to break concepts into smaller steps. The use of the resources enabled
this realization; however, the resource did not seem to enable her to deconstruct concepts
appropriately. This suggests that even if teachers know that the resources were designed around
microsteps, they may experience difficulty in breaking concepts and skills into smaller elements
themselves. This research finding is consistent with a most recent analysis of teachers’ perception of
scaffolding in the year proceeding the interviews reported in this paper. Sabbaghan, Metz, Preciado
Babb, and Davis (in press) found that teachers with less experience in the initiative tended to use
traditional strategies for scaffolding—such as modeling and coaching—in contrast to teachers with
more than one year in the initiative who considered micro-level scaffolding strategies.
Third, even though the teachers’ guide provides advice on bonusing, most teachers found this
very challenging. The idea of bonusing has been evolving during the Math Minds initiative. The
research team has compiled examples from teachers implementing the program. The team has also
identified connections to the literature on intrinsic motivation, shaping the collective understanding
of the bonus principle of the JUMP Math program. Moreover, in contrast to the teachers’ guide’s
emphasis on creating bonus for early finishers, Mighton (2007) also advised to consider bonus
questions for everyone: “I always make up special bonus questions for the most challenged students,
too, so they can feel that they are doing harder work as well” (p. 106). We have come to perceive
bonusing as a strategy for both fostering a positive attitude towards mathematics and deepening
The longitudinal results for student performance on the CTBS tests—omitted in this paper due to
limited space—showed a significant improvement after one year of adopting the JUMP Math
program (Metz, Sabbaghan, Preciado Babb, & Davis, in press). This was particularly reflected in
students who initially had low performance. However, scores leveled or decreased for some students
with initially high performances. Our principal hypothesis for this situation is that it might be
attributed to teachers’ lack of confidence in creating bonus questions for students. This hypothesis is
supported by the fact that the students of the one teacher who reported confidence with bonusing
showed significant improvement across the board (Preciado Babb, McInnis, Metz, Sabbaghan, Davis,
Finally, the general agreement that a resource like JUMP Math does not include inquiry contrasts
with the problem solving strategies included in each lesson in the teachers’ guide. This is probably
due to a strong focus on the assessment and practice book instead of the suggested activities in the
guide and the SmartBoard slides. The research team considers that both bonusing and the selected
sequence of tasks (Metz, et al., in press) in the assessment and practice book afford opportunities for
mathematical inquiry. There is, therefore, a need to better understand the mathematical knowledge
required for bonusing and for breaking down concepts into smaller elements.
We acknowledge the support provided by Canadian Oil Sands Limited. Without it, this
partnership wouldn’t be possible.
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