Concept-Based Mathematics #3 by Jenna Mendez

Chapter 3: What are Generalizations in Mathematics?

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1. What are Generalizations in Mathematics? What is the Difference Between a Principle and a Generalization in Mathematics?

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Generalizations in math are broad statements about patterns.

A Principle is a specific rule, and a generalization is a bigger idea that includes many rules.

a. Two Types of Enduring Understanding

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Overarching enduring understandings are big ideas that last beyond a single unit.
Topical understandings are specific to a particular subject or topic.

i. Overarching Enduring Understandings

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They are larger, transferable insights reflecting year-long program understandings. Also known as overarching generalizations or statements of inquiry.

ii. Topical Understandings

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They focus on a specific insight within a unit of study. Also known as topical generalizations.

b. Generalization Examples

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Algebra solves complex problems.
Scalar products find vector angles and magnitudes.
Students struggle to understand logarithms.
A quadratic formula finds function zeros and roots.
The quadratic formulas discriminate shows the roots nature.

c. Visual

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d. Useful Source

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Making Generalizations in Math.

2. How do we Craft Quality Mathematics Generalizations?

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Letting students generalize is key to learning math. Math concepts can be expressed differently, depending on the student's level and the teacher's goal.

a. Writing Generalizations

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Teachers should work together to write generalizations as part of planning a concept-based unit. When we write generalizations, we want to focus on understanding, not just skills. Starting with "Students will understand that..." is a great first step.

b. Three Step Guide.

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Level 1: "Students will understand that . . ."

Level 2: "How or why?"

Level 3: "So what is the significance or effect?"

c. Useful Source

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The Levels of Writing Generalizations

d. Visual

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i. Level 1

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Level 1 generalizations are weak and lack clarity.

Example from the book: "The quadratic formula is one method to solve a quadratic equation".

ii. Level 2

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Level 2 generalizations help learners understand what they are learning.

Example from the book: "The quadratic formula describes the roots or zeros of the function and helps solve a quadratic equation".

iii. Level 3

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Level 3 generalizations explain the purpose and extend learning.

Example from the book: "The expression underneath the square root in the quadratic formula, the discriminant, conveys the nature of the roots, which highlight geometrical features".

3. How do we Draw Out Conceptual Understandings From our Students?

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We draw out conceptual understandings from students by asking open-ended, thought-provoking questions that encourage them to think deeply and make connections.

a. Inductive Teaching

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Inductive teaching involves giving students specific examples to help them form general ideas.

b. Useful Source

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Conceptual Understanding

i. Key Principles

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Students should form conceptual understanding through writing in math.
Learning experiences should involve explaining specific examples, identifying patterns, and writing generalizations.

c. Visual

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