Wathall Chapter 3 Concept Map by Kimball Ray

Types of Generalizations

kray26_1 — 2025-09-28 22:59:24 UTC

Level 1 Generalizations - "Students will understand that..."

kray26_1 — 2025-09-28 23:00:13 UTC

Typically weak and don't help learners fully grasp what and why they are learning

Ex: The coefficients of a quadratic function in a connected equation affect the quadratic formula.

Level 2 Generalizations - "How or why?"

kray26_1 — 2025-09-28 23:00:43 UTC

Should be used to develop teaching targets
Ex: Whether arithmetic and geometric sequences and series share a common difference or a common ratio distinguish one from another

Level 3 Generalizations - "So what? What is the significance or effect?"

kray26_1 — 2025-09-28 23:01:27 UTC

Can be used to express the real world value of mathematics
Arithmetic and geometric sequences and series describe patterns in numbers and supply algebraic tools that help to solve real-life situations

What is a generalization?

kray26_1 — 2025-09-28 23:08:00 UTC

two or more concepts stated as a sentence of relationship
transfer through time, across cultures, and situations
Supported by factual examples
Overarching - Math reveals patterns that might have remain unseen
Enduring - Statistical analysis and graphical displays often reveal patterns in seemingly random data or populations, enabling predictions

Strategies to Draw Generalizations from Students

kray26_1 — 2025-09-28 23:09:00 UTC

Give concepts and ask students to connect concepts

kray26_1 — 2025-09-28 23:10:05 UTC

Example: Write the following concepts in a related sentence. This should demonstrate your understanding of how the concepts are connected:

Parallel lines, gradient, transversal, angles

At the end of the lesson, ask students to complete the sentence, "I understand that..."

kray26_1 — 2025-09-28 23:10:54 UTC

Example:

I understand that solving a quadratic equation by using the quadratic formula graphically displays the x-intercepts of a quadratic function

Concept Maps (woah, what a concept!)

kray26_1 — 2025-09-28 23:12:07 UTC

After a lesson/unit, ask students to create a concept map which includes the concepts they've learned and how they are connected, with statements explaining them

The END GOAL

kray26_1 — 2025-09-28 23:14:00 UTC

We need to understand how to guide students to particular principles/generalizations through inductive inquiry. This encourages students to form conceptual understandings from their own learning experiences