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.

# Tag: Structured variation

## 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

**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.”

**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

**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

**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.

https://werklund.ucalgary.ca/graduate-programs/topics/teaching-learning-mathematics

## Welcome to the Math Minds Blog

The purpose of this blog is to further illustrate and elaborate on the RaPID model. This is the first of many to come.

## 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.

## Math Games: Sleeping Queens

Used well, games can support both conceptual development and procedural fluency. Used poorly, they can reinforce weak strategies and bad habits and perpetuate achievement gaps.