a. Math tells us something about the way it should be taught.
b. Brains are plastic.
c. Working memory is limited.
d. Minds are attuned to change.
e. Minds link overlapping events.
a. A RaPID lesson is ribboned as opposed to blocked.
b. The RaPID model is not a balance of discovery and rote, traditional and reform, teacher and student-centered.
c. A RaPID lesson is raveled and contains multiple PID cycles of prompting attention, interpreting understanding, and deciding how the lesson should proceed.
d. Responsibility for raveling, prompting, interpreting, and deciding may be productively shared between a skilled teacher and a well-raveled resource.
a. A well-raveled lesson is both internally coherent and part of a coherent sequence of lessons that weave together mathematical discernments…over minutes, days, months, and years.
b. Mathematical discernments are distinct from procedural steps.
c. Generally, mathematical discernments should be clearly understood before attempting to weave them together.
a. Prompting uses contrast and juxtaposition to highlight key principles (distinctions), associations (patterns), and relationships.
b. Prompting requires learners to make key distinctions, associations, and relationships.
a. Interpreting involves (a) asking questions that distinguish understanding from lack of understanding and (b) attending to the responses of every learner.
b. Deciding involves identifying, clarifying, extending, and/or linking key ideas in a manner that supports the continuous extension of understanding for all learners.
a. Math Minds does not attempt to balance traditional and reform approaches to teaching mathematics.
b. Math Minds does re-interpret many ideas currently popular in what might be considered both traditional and reform approaches to teaching mathematics
a. Is the lesson ribboned or blocked?
b. Does the lesson have clear PID cycles?
c. Is the lesson well-raveled, “stepped,” or meandering?
a. Effective raveling bridges and integrates discernments over time.
b. Although a raveled progression is experienced in linear time, discernments are woven together into multiple higher-order discernments that are themselves gradually woven into increasingly dense ideas.
c. New principles and relationships emerge from and anticipate multiple ideas.
d. Understanding the long-term ravel can help teachers prompt attention to key relationships.
a. Allowing everyone to work only with their own personal ways of knowing can be disabling, anxiety-inducing, and isolating.
b. Personal ways of knowing can serve as anchors to shared ways of knowing.
a. A well-raveled resource offers a coherent weave of ideas over time. A strong teacher is aware of how ideas within and between lessons support the emergence of a broader whole.
b. A strong resource offers helpful contrasts and sequences. A strong teacher must draw attention to these and ensure that learners engage with them in intended ways.
c. A strong resource offers questions that help distinguish understanding from lack of understanding. A strong teacher attends to and knows how to interpret those responses.
d. A strong resource offers potential extensions. A strong teacher adjusts as needed.
a. Elaborates on the variation theory of learning, emphasizing the importance of (a) using change to prompt attention and (b) requiring learners to make relevant distinctions.
b. Focus on change and choice emphasizes subtle changes that can have a profound impact on student learning.
a. Introduces the notion of raveled difference, which is a way of looking at mathematical structure that focuses on continuous deepening and elaborating of important ideas—and the task structures that support this. Participants are invited to engage with an extended mathematical engagement that highlights five levels of raveled difference and associated task structures.
a. Emphasizes connections between Level 1-3 and traditional approaches to mathematics—as well as ways that traditional structures may be adapted to support deeper understanding.
a. Emphasizes connections between Levels 4-5 and reform approaches to mathematics—as well as ways that reform approaches may be adapted to support deeper understanding. Importantly, all five levels are essential.
a. Further explores the notion of a teacher-resource partnership. Participants are once again invited to engage in a mathematical sequence that prompts to a particular series of critical discernments.
b. Participants are then invited to contrast how four different lessons approach a similar topic—and what a teacher's role would be if partnering with each resource.