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. A well-structured mathematics program must attend to both short and long-term raveling.
b. Once critical discernments have been identified, we can use change to prompt attention.
c. To engage learners in relevant distinctions, we need to ask questions that require them to make choices regarding those distinctions.
d. Effective practice involves practice discerning, not just practice doing.
a. Differences at one level become mathematical objects to be contrasted at the next.
b. A well-structured understanding of even and odd can support understanding of division and divisibility.
a. “Basics” may be understood in terms of critical discernments that can be woven into more complex understandings.
b. “Grounding metaphors of number” are basic to the rest of mathematics.
c. We can understand subtraction in terms of the grounding metaphors.
d. Mathematical patterns don’t necessarily prompt to the relationships that underlie them; both are important.
e. It’s important to distinguish what is arbitrary (i.e., could have been otherwise) from what is necessary in mathematics.
a. Within careful constraints, learners can and should be responsible for generating their own variations. This is a form of problem solving.
b. Problem solving may also involve applying a known structure to a new situation or integrating familiar discernments.
a. Designing a well-structured mathematics and well-prompted mathematics program is a monumental undertaking and is best supported by a strong resource.
b. Teachers must take on different responsibilities with different resources.