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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/