procedural fluency Model 

The Collaborative for Teaching and Learning has developed our Procedural Fluency Model to take advantage of what we know about effective learning of basic facts and number sense.

Why our model? Computational fluency relies heavily on automaticity with basic math facts. Teaching addition facts has traditionally neglected two of the four components of fluency (strategy use and flexibility) and instead focused only on accuracy and speed despite strong evidence that students in strategy-focused interventions outperform their peers on using strategies, automaticity and accuracy. Strategy use is an important outcome itself, not just a means to learning and retaining basic facts because strategies provide a foundation for general fluency. Visual images and strategy-focused games support student reasoning and learning.


Key Components

Developmental Progressions: Working with teachers, we have identified the building blocks of learning progressions for math fact development. The addition flow chart provides a learning progression for addition, acknowledging the prerequisite facts sets needed in order to implement derived fact strategies (Bay-Williams & Kling, 2019).

Explicit Strategy Instruction: Making connections explicit is a well-established way to support conceptual understanding and procedural fluency. Unfortunately, typical elementary mathematics classrooms in the United States focus on low-level skills and rarely attend explicitly to the important mathematical relationships to build student understanding and capacity (Hiebert & Grouws, 2007b, p. 2).

Our model uses an intentional progression of representations to help students develop an understanding of number relations through linear modeling with the number line and concrete set models using counters to models like tens-frames that help students connect to the abstract digit representations. Developing these skills in kindergarten and early first grade allow students to build their knowledge of more complex strategies when they are ready.

Game-based practice Having students practice their facts is an important step in helping them become fluent, but American mathematics classrooms have been using approaches that create more anxiety and stifle the learning process rather than promote stronger conceptual understanding (Boaler, 2019). By reducing the focus on correctness and allowing students to practice in low-stress games students are able to use strategies, develop a deeper understanding of the strategies, and generalize them.

Formative Assessment: Teacher use of formative assessments positively influences student learning (Frye, Et al., 2013; NMAP, 2008; Wiliam, 2007) and was one of the key recommendations of the Teaching Math to Young Children Panel (Frye, et al., 2013). The panel recommended (a) use of introductory activities, observations, and formal assessments, (b) instruction tailored to each student, and (c) formatively assessing each child.

Our model provides teachers the tools to assess student learning quickly and routinely while providing teachers a plan for addressing student needs. Formative assessment is key only if it affects how teachers respond to where students are in the learning progression.

Our curriculum kits reflect all four of these components. For example, Quick Looks and other tools (e.g., number lines) are incorporated into mini-lessons to introduce addition strategies. Also, games designed for each fact strategy are implemented in small groups based on readiness. As students play the games or engage in the writing activities, teachers use observation tools and interview protocols to formatively assess students’ attainment of fact sets and strategies to determine if they are ready to progress to the next fact strategy.

Professional Learning

CTL is responsive to research into effective professional learning, and while we will always focus on meeting the teacher where they are and helping them increase their understanding of the content as well as instructional approach. This model incorporates the most recent findings that show a combination of curriculum and professional learning produce the greatest impact on student learning (Lynch, Gonzalez, & Pollard, 2018). Each kit provides scripted lessons for teachers to follow as they learn the process and gain better insights into how students will react to the approach. The formative assessment scripts provide models for the kinds of questions that provide the greatest insights into student understanding of the fact strategy. Professional learning experiences focus on teacher understanding of the conceptual progression and analysis of evidence from students as well as analysis of instruction from teachers.

 The model includes:

  • Foundational training provided via a 2-day summer institute
  • Capacity building throughout the year via job-embedded coaching (modeling, co-teaching, pre/mid/post-observation, co-planning, etc.)
  • Where applicable, CTL offers job-embedded coaching for school and district mathematics coaches as they develop their capacity to lead and support teachers in the program’s design

 All of CTL’s professional learning models meet the Learning Forward Professional Learning Standards for effective student learning.

Personalized Professional Coaching

While the  brings educators together to regularly engage in collaborative professional learning, the program honors the individual needs of participating practitioners. CTL coaching provides a space for assessment, reflection, and refinement that is educator-driven, and proven to enhance engagement, motivation and instructional practices. Whether virtually or in-person, CTL staff engage in side-by-side instructional planning, pre/mid/post observations, reflection cycles, analysis of student work protocols, and other timely and relevant coaching experiences with teachers. CTL’s personalized professional coaching helps to create a school-wide culture of continuous improvement, which pays dividends both in the classroom and in the professional learning communities.