Conference PaperPDF Available
The RaPID approach for teaching mathematics:
An effective, evidence-based model
Preciado-Babb, Armando Paulino; Metz, Martina; Davis, Brent
University of Calgary
Paper presented at the Canadian Society for the Study of Education Annual Conference
June 1-5, 2019, University of British Columbia, Vancouver, BC.
Abstract: The use of systematic variance and invariance has been identified as a critical aspect
of mathematics lessons in many countries with top results in international assessments;
however, the literature on teaching strategies is less frequent. In particular, the use of
systematic variation to inform teachers’ continuous decision-making during class is uncommon.
We elaborate on the five-year longitudinal results from an initiative targeted at elementary level
and involving collaboration among two school districts and a university in Alberta along with a
resource developer. Data for this study include students’ performance in mathematics,
classroom observation, interviews with student and teachers, and analysis of video-recorded
lessons. Based on this data, we proposed the Raveling, Prompting, Interpreting, and Deciding
(RaPID) model for teaching mathematics, which is informing our efforts to scale up teacher
professional learning for teachers across the province. In this presentation, we describe the
model and the data supporting its development.
Keywords: Mathematical knowledge for teachers; Mastery Learning; Formative Assessment;
Teacher Education
Paper presented at the Canadian Society for the Study of Education, June 1-5, 2019
University of British Columbia, Vancouver, BC.
2
1. Purpose
Amid national and international efforts to improve mathematics education at K to 12 levels, the
province of Alberta has prioritized access and ability to “high-quality professional development
and training opportunities specific to the teaching of Mathematics” (O’ Connor, de Vries, Goldie,
Beltaos, Bica & Lagu, 2016, p. 8). Since 2012, the Math Minds initiative has focused on
improving mathematics instruction at the elementary level. The initiative aims to: (1) identify
effective features of mathematics lessons, including resources and teaching strategies, and (2)
develop a corresponding teacher professional learning model. Here, we describe the Raveling,
Prompting, Interpreting, and Deciding (RAPID) model resulting from the study and elaborate on
the empirical evidence informing its development.
2. Theoretical framework
The Math Minds initiative draws on a variety of insights from the literature. We build on previous
insights from Mastery Learning, such as the work of Guskey (2010) who suggested classroom
assessments to identify students who have already mastered the learning goals and those who
need further support. The Math Minds initiative’s approach to Formative Assessment can be
described as all-student response systems (William, 2011), with a particular focus on
continuous assessment to all students during class using specific questions designed to surface
potential misunderstanding (Wylie & Wiliam, 2007). Intrinsic Motivation (Pink 2011) informs our
attention to students’ engagement in mathematical tasks that represent a new challenge at the
appropriate students’ level. Marton’s (2015) Variation Theory of Learning refers to the critical
aspects that a learner must discern in order to learn something. The use strategic variation and
juxtaposition of examples and tasks can orient students’ attention to such critical features. We
have found in the project that effective variation can help to address the limits of Working
Memory identified by Clark, Kirschner, and Sweller (2012). The RAPID framework is informed
by these approaches and based on the empirical evidence of the research in the initiative.
Paper presented at the Canadian Society for the Study of Education, June 1-5, 2019
University of British Columbia, Vancouver, BC.
3
3. Methods
The Math Minds is a design-based study (Cobb et al. 2003) that involved the collaboration
among two school districts, a university in Alberta, and a teaching material developer: JUMP
Math. All participating teachers and students used materials from this developer. Participant
teachers received professional development on a regular basis, changing from one year to the
next in response to both emerging findings and school policies. In order to track students’
learning, we analyzed longitudinal results in performance in mathematics, as measured by the
Canadian Test for Basic Skills (CTBS; Nelson 2018). We used a Linear Mixed Model (LMM) to
accommodate this unbalanced study design. In this way, “not all individuals need to have the
same number of observations and not all individuals need to be measured at the exact same
time points” (West 2019, p. 212). CTBS scores were converted to t-scores and normalized
before conducting the analysis. Additionally, we conducted classroom observations, interviews
with student and teachers, and analyses of video-recorded lessons. The observation of the
lessons and the video recordings served to identify common features in the most successful
classrooms, which has informed the iterative development of the RAPID model (Preciado-Babb,
Metz, Davis, Sabbaghan, In Press).
4. Data sources
Phase 1 (2012-2017) of the initiative included three different schools. Data from this phase
include longitudinal results in the mathematics component of CTBS for 363 students, weekly
mathematics lesson observations in 31 classrooms, 300 video-recorded lessons, 44 teacher
interviews and 228 student interviews.
5. Results and Conclusions
The LMM analysis showed a significant improvement in student performance in mathematics,
with national percentile rankings rising from 27 to 55, evidencing an important improvement in
students’ mathematical understanding. By contrasting results and observations from different
classrooms, we were able to identify four elements that have become the pillars of the model.
Paper presented at the Canadian Society for the Study of Education, June 1-5, 2019
University of British Columbia, Vancouver, BC.
4
Raveling refers to the complexity of mathematics regarding how concepts are connected one to
each other. A well-raveled lesson focuses on clearly identified and connected mathematical
discernments (e.g. critical aspects that learners have to discern). Raveling informs cycles of
prompting students’ attention to a key idea, requiring all learners to engage with that idea.
During this process, it is important that the teacher interpret all responses and make decisions
about appropriate next steps. Cycles of prompting, interpreting, and deciding take place multiple
times in a single lesson and must be rapid enough to maintain the momentum needed for
learners to experience ideas as connected.
6. Educational Importance of the Study
The RaPID model, based on empirical evidence from the classroom, and consistent with key
insights from the literature, has proven to be effective for teaching mathematics. The use of
JUMP Math material has played an important role in raveling content for project teachers and in
providing initial prompts for students. Teachers assumed the primary responsibility for
interpreting students’ understanding and making responsive decisions during class. In this
model, teaching has been a responsibility shared between curricular materials and teachers.
We have identified challenges in the implementation of the model grounded on common
associations related to both direct instruction and reform- oriented teaching styles. The RaPID
model is not situated in the middle, or as a balance, of these two seemly opposite approaches:
Fine-grained attention to critical features of mathematical content combined with prompting
strategies that recognize a necessary tension between variation and working memory have
resulted in a model that requires both a strong directive role for the teacher and a clear
emphasis on making sense of increasing connected mathematical ideas.
Paper presented at the Canadian Society for the Study of Education, June 1-5, 2019
University of British Columbia, Vancouver, BC.
5
References
Clark, R., Kirschner, P., & Sweller, J. (2012). The case for fully-guided instruction. American
Educator 36(1), 6-11.
Cobb, P., diSessa, A., Lehrer, R., Schauble, L. (2003). Design experiments in educational
research. Educational Researcher, 32(1): 9-13.
Guskey, T. (2010). Lessons of mastery learning. Educational Leadership 68(2), 52-57.
Marton, F. (2015). Necessary conditions of learning. New York, NY: Routledge.
Nelson (2018). Canadian tests of basic skills (CTBS). Nelson. www.assess.nelson.com/
Pink, D. (2011). Drive: The surprising truth about what motivates us. New York: Riverhead.
Preciado-Babb, A. P., Metz, M., & Davis, B. (In press). How variance and invariance can inform
teachers’ enactment of mathematics lessons? In R. Huang, A. Takahashi & J. P. da
Ponte (Eds.), Theory and practice of lesson study in mathematics: An international
perspective. Springer. West, B. T. (2009). Analyzing longitudinal data with the linear
mixed models procedure in SPSS. Evaluation & the Health Professions, 32(3), 207-228.
Wiliam, D. (2011). Embedded formative assessment. Bloomington, IN: Solution Tree.
Wylie, C., & Wiliam, D. (2007). Analyzing diagnostic items: What makes a student response
interpretable? Paper presented at Annual Meeting of the National Council on
Measurement in Education (NCME), Chicago, IL. Retrieved from
http://www.dylanwiliam.org/Dylan_Wiliams_website/Papers_files/
ResearchGate has not been able to resolve any citations for this publication.
Chapter
Full-text available
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.
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Canadian tests of basic skills (CTBS). Nelson. www.assess.nelson.com/ Pink
  • Nelson
Nelson (2018). Canadian tests of basic skills (CTBS). Nelson. www.assess.nelson.com/ Pink, D. (2011). Drive: The surprising truth about what motivates us. New York: Riverhead.
Theory and practice of lesson study in mathematics: An international perspective
Ponte (Eds.), Theory and practice of lesson study in mathematics: An international perspective. Springer. West, B. T. (2009). Analyzing longitudinal data with the linear mixed models procedure in SPSS. Evaluation & the Health Professions, 32(3), 207-228.