If you’d like to know more about Math Minds and its evolution, you may wish to explore some of our published work, here sorted by category.
When initially engaging with the Math Minds approach, it is easy to associate some of its features with familiar approaches, either traditional or reform. The articles linked below further elaborate the significance of the distinctions made by Math Minds. You may find it helpful to read them now and perhaps revisit them again at the end of Unit 1.
Note that our language has shifted away from that of “small steps” (in our earlier writings) to that of “critical discernments” (in later writings) as we recognized the potential for confusion with “procedural steps.” Similarly, we have moved away from talk of “minimizing cognitive load,” which can be interpreted simply as reducing how much we offer, toward “recognizing the limits of working memory,” which makes it easier to emphasize the importance of not reducing what we offer to the point that we lose the very contrasts needed to effectively prompt attention.
Transcending Traditional/Reform Dichotomies in Mathematics Education
Transcending Contemporary Obsessions: The Development of a Model for Teacher Professional Development
From Blocks to Ribbons
The article linked below highlights an early shift in one teacher’s practice as she began to engage with Math Minds ideas. This shift was initially based on a move from a blocked lesson structure to a ribboned lesson structure, which allowed (a) more opportunities for learners to engage and (b) more opportunities for the teacher to interpret learner understanding.
Again, note that our own language has evolved since the writing of this paper, in which we still refer to breaking content into small steps. We have since found it helpful to distinguish “small steps” from “critical discernments.”
Epiphanies in Mathematics Teaching: The Personal Learning of an Elementary Teacher in the Math Minds Initiative
The Development of the RaPID Model
In the paper linked below, we use the language of ribboning, monitoring, adapting, and connecting to describe key elements of the Math Minds approach. Although the selected examples remain relevant, our language has shifted to raveling, prompting, interpreting, and deciding as we have more clearly recognized nuances significant to effective practice. Ribboning now speaks to the broader lesson structure and is relevant to each of the RaPID elements.
Attending and Responding to What Matters: The Development of an Observation Protocol
The following paper is a more recent description of the model:
The RaPID Approach For Teaching Mathematics: An Effective Evidence-Based Model
Forming a Teacher-Resource Partnership
The articles linked here highlight some of the ways that teachers shifted from using a resource to working in partnership with a resource. This idea is further developed in the readings associated with Session 4.
Insights on the Relationships Between Mathematics Knowledge For Teachers and Curricular Material
Teachers’ Awareness of Variation
Since the time of writing, the Math Minds partnership has further considered ways that resources might further support teachers’ awareness of the intended ravel. This is an ongoing endeavor.
Using Structured Variation to Prompt Attention
The articles linked here offer a variety of classroom examples highlighting the potentially powerful impact of subtle—but carefully structured—variation.
Using Variation to Critique and Adapt Mathematical Tasks
How Variance and Invariance Can Inform Teachers’ Enactment of Mathematics Lessons
Attending to a Long-Term Ravel
While critical discernments can often be distinguished from procedural steps by their emphasis on understanding, they’re much more than procedural steps with conceptual explanations attached. An effective long-term ravel uses variation to prompt to differences at higher and higher levels.
Procedural Steps, Conceptual Steps, and Critical Discernments: A Necessary Evolution of School Mathematics in the Information Age
Interpreting Learner Understanding
The articles linked below highlight some ways that teachers have used to gather information from all students.
Fine-Grained, Continuous Assessment: A Key Factor to Increase Performance in Mathematics
The Role of Continuous Assessment and Effective Teacher Response in Engaging All Students
Meeting Diverse Needs
Finding ways to adapt lessons in response to observed needs can be challenging. The articles linked analyze some of the challenges and successes we’ve observed in Math Minds classrooms.
Dynamic Responsive Pedagogy: Implications of Micro-Level Scaffolding
Addressing the Challenge of Differentiation in Elementary Mathematics Classrooms
One Step Back, Three Forward: Success Through Mediated Challenge
Juxtaposing Mathematical Extensions With Cognitively Loaded Questions in the Mathematics Classroom
Teachers’ Perceived Difficulties For Creating Mathematical Extensions at the Border of Students’ Discernments
If you’re interested in graduate-level studies that take Math Minds findings into account, you may be interested in the Werklund School of Education’s Teaching and Learning Mathematics program.