Tag: Structured variation

Math Games: Froggy Boogie

In this post, we highlight how a game might be adapted so that it requires learners to make the particular distinction we want them to make. We also highlight the importance of first presenting this distinction to them. The original game is a memory game; the adapted game has to do with distinguishing left from right.

Math Minds Publications & Related Courses

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.

Transcending Dichotomies

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

Graduate Studies

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.


Math Games: Lost Cities

Lost Cities is a game that offers opportunities for meaningful practice at varying levels of expertise—provided strategies are taught in meaningful ways. In particular, it offers a motivating context for practicing skip counting, making tens, and flexible re-grouping for adding two and three-digit numbers.

Math Games: Sleeping Queens

Sleeping Queens can offer meaningful practice with regrouping to add, recognizing equality, and dividing by two and three. It can be played in a way that makes it accessible for children in K-3 or challenging for children in Grades 4-6 (and beyond). It can also be adapted for play with a traditional deck of cards.