PBMCF is a framework for the learning and teaching of mathematics. Built on prior frameworks such as the KOM Project, PISA Project, Adding It Up, and the NCTM Standards, it relies on 11 competencies and 6 phases of learning for each topic of focus. PBMCF’s phase-based approach was inspired by the phase-based approach of training for distance running developed by Arthur Lydiard and Bill Bowerman. More generally, the underlying structure of PBMCF can be applied to the learning and teaching of any subject.

Competencies:

Symbols and formalism (KOM, PISA) Procedural fluency (AIU, TIMSS) Conceptual understanding (AIU, TIMSS) Representations (NCTM, KOM, PISA) a. Models (KOM, PISA)

b. Connections (NCTM)

c. Thinking mathematically (KOM, PISA) Content-dependent problem solving (NCTM, KOM, PISA, MMPSP) Problem posing (KOM, PISA) Content-independent problem solving (MMPSP) Logic and reasoning (NCTM, KOM, PISA, AIU, TIMSS, MMPSP) Communication (NCTM, KOM, PISA, MMPSP) Productive disposition (AIU) Aids and tools (KOM, PISA)

Phases:

Exposure

Maximize exposure to both the topic as well as the associated symbolisms and formalisms required for subsequent phases in an informal way. Produces familiarity with the topic.

Procedure

This phase focuses on procedural methods and rote memorization relevant to the selected topic of focus.

Concrete

In this phase the focus is on conceptual understanding grounded in concrete examples.

Abstract

The focus shifts from concrete examples to abstracting the concrete examples to a more general conceptual understanding.

Dependent

In the sense of Woods (1997), problem solving is focused on in this phase but primarily as it presents in problems that are dependent on the content.

Independent

This phase develops more general problem solving skill in a content-independent context in the sense of Woods (1997).

Structure of PBMCF: