Canvas by Kristen Kimpel

Concept-Based Mathematics

kjkimpel — 2025-09-06 06:02:26 UTC

A framework for teaching mathematics that emphasizes deep conceptual understanding, not just procedural skills.

Structure of Knowledge (SoK)

kjkimpel — 2025-09-06 06:06:08 UTC

The SoK is hierarchical. It moves from memorized facts to abstract theories.

Structure of Process (SoP)

kjkimpel — 2025-09-06 06:09:23 UTC

SoP focuses on how students think and solve, not just what they know.

Formulae vs. Generalizations

kjkimpel — 2025-09-06 06:13:53 UTC

Generalizations build conceptual connections, meanwhile formulas are tools.

SoK + SoP Symbiotic Relationship

kjkimpel — 2025-09-06 06:21:43 UTC

Wathall emphasizes that SoK + SoP must work together in teaching.

Processes, Algorithms, Skills

kjkimpel — 2025-09-06 06:23:22 UTC

Wathall stresses not to stop at skills/algorithms. Processes are where transfer happens.

Algorithms

kjkimpel — 2025-09-06 06:24:26 UTC

An algorithm is a set of ordered steps. For example, long division.

Processes

kjkimpel — 2025-09-06 06:25:32 UTC

Processes involve complex thinking. Some examples of processes are justifying, problem-solving, and modeling.

Skills

kjkimpel — 2025-09-06 06:26:22 UTC

Skills are simple and often memorized. A common example is basic multiplication facts.

Facts

kjkimpel — 2025-09-06 06:27:26 UTC

Some examples are symbols, vocabulary, and formulas.

Topics

kjkimpel — 2025-09-06 06:28:21 UTC

Some examples are word problems and multiplicative comparison

Concepts

kjkimpel — 2025-09-06 06:29:03 UTC

Some examples are multiplication as comparison and variables as unknowns.

Generalizations

kjkimpel — 2025-09-06 06:29:49 UTC

An example is mathematical statements of truth across contexts.

Principles

kjkimpel — 2025-09-06 06:30:26 UTC

Some examples are properties of operations and inverse relatonships.

Theory

kjkimpel — 2025-09-06 06:30:54 UTC

Some examples include the algebraic theory and the number theory.

Cognitive Processes

kjkimpel — 2025-09-06 06:32:03 UTC

These consist of generalizing, comparing, and classifying.

Mathematical Processes

kjkimpel — 2025-09-06 06:32:47 UTC

These consist of problem-solving, reasoning, representing, and communicating.

Knowledge without process

kjkimpel — 2025-09-06 06:33:43 UTC

This is equal to rote memorization.

Process Without Knowledge

kjkimpel — 2025-09-06 06:34:30 UTC

This is equivalent to inefficient trial and error.

Together

kjkimpel — 2025-09-06 06:35:03 UTC

SoK and SoP lead to deeper understanding when they work together.

Formulae

kjkimpel — 2025-09-06 06:36:22 UTC

Fixed procedures or equations. Some examples are shown in the link above.

Generalizations

kjkimpel — 2025-09-06 06:37:43 UTC

Transferable ideas across contexts. An example of this would be "area is the product of two dimensions."