Concept-Based Mathematics by Jenna Mendez

Ch.2: What Are the Levels of the Structures of Knowledge and Process for Mathematics?

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1. The Levels of the Structure of Knowledge

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The levels of the Structure of Knowledge refer to how mathematical knowledge is organized into different levels.

2. The Levels of the Structure of Process

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The levels of the structure of the Process refer to the sequential stages of mathematical thinking.

a. The Factual Level

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The lowest level in the structure of knowledge and it includes rote memorization and does not guarantee conceptual depth of understanding.

b. The Topics and Concepts Levels

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Topics and concepts refer to the specific areas or subjects and their underlying ideas, principles, and relationships within mathematics.

c. The Generalizations and Principles Levels

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Generalizations and principles represent a deep conceptual understanding that enables connections and broad applications.

a. Mathematical Processes, Algorithms, Strategies, and Skills.

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Mathematical Processes: Mental Actions used in problem-solving.

Algorithms: Step-by-Step procedures for solving problems.

Strategies: Deliberate approaches to solving problems.

Skills: Proficiency in executing mathematical procedures.

b. The Iterative Stages When Learning Mathematics

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Using math processes, strategies, and skills helps students better understand math concepts. As they learn mote concepts, they also learn new strategies and skills.

i. Facts are Not Transferable and Includes Definitions

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Knowing a definition or a formula does not imply understanding.

ii. Example

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The Pythagorean Theorem is a^2 + b^2 = c^2

This is a memorized fact and it does not reflect conceptual understanding.

i. Example

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A student who understands the principle of symmetry can apply it to mathematical concepts like similar right triangles. They share common ratios of corresponding sides.

d. Visual

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Video Explanation (General)

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i. 6 Mathematical Process Broken Down Into Skills and Strategies.

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Problem-Solving
Reasoning and Proof
Communication
Making Connections
Creating Representations
Investigating

ii. Example

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Solving 4 x 9

Mathematical Process: Reasoning

Algorithm: Standard Multiplication Steps

Strategy: Breaking down the problem (4 x 10 = 40 then subtract 4)

Skill: Recalling Multiplication Facts

c. Visual

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Symbiotic Relationship between The Structure of Knowledge and The Structure of Process

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Knowledge and Process work together:

Knowledge guides problem-solving.
Problem-solving deepens knowledge.
They reinforce each other.

This connection helps students:

Understand math concepts better.
Improve problem-solving skills.
Become more confident in math.

e. Useful Source

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3. Macro, Meso, and Micro Concepts in Mathematics

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Macro, Meso, and Micro concepts in mathematics form a hierarchical structure, ranging from board principles to connected topics to specific details.

a. Conceptual Relationships

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Macro: Overall mathematical principles and concepts.

Meso: Connections and relationships between math topics.

Micro: Specific math facts, formulas, and procedures.

i. Example

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Macro: Calculus

Meso: Differentiation and Integration

Micro: Slope, Rate of Change, Velocity, Limits, Derivative, and Summation

d. Useful Source

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The chapter of this book explains the Structure of Process.

b. Visual

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c. Useful Source

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