Concept Based Mathematics by Izabel Gutierrez

Structure of Knowledge

igutie12 — 2025-02-10 23:04:20 UTC

The Factual Level

igutie12 — 2025-02-10 23:09:55 UTC

includes habitual memorization and doesn't guarantee conceptual depth of understanding
in math, students needs to know the facts to reinforce understanding of related concepts

Formula...

igutie12 — 2025-02-10 23:15:34 UTC

an equation using mathematical symbols or variables to show a relationship

(represented by facts)

Theorems...

igutie12 — 2025-02-10 23:16:26 UTC

proven statements that connect explanations of conceptual understanding

(represented by principles)

The Topic and Concepts Levels

igutie12 — 2025-02-10 23:19:04 UTC

topics are broader concepts broken down into specific micro concepts

The Generalizations and Principles Level

igutie12 — 2025-02-10 23:23:04 UTC

both are statements of conceptual understanding that allow students to make connections between two or more concepts
in math, principles are theorems-- the truths

Structure of Process

igutie12 — 2025-02-10 23:29:50 UTC

Mathematical Processes

igutie12 — 2025-02-10 23:37:25 UTC

complex, sophisticated performances composed of algorithms, strategies, and skills
- example: problem solving
- being able to recognize connections and utilize different tools to create representations
Strategies: multiple skills that students use in a methodical and systematical way to learn
Skills: small operations/actions that are embedded in strategies that, when applied properly, "make" the strategy work
Algorithms: consist of skills allowing students to follow a set of rules that lead to a specific outcome

1. Problem Solving

igutie12 — 2025-02-10 23:45:50 UTC

a defined series of skills and strategies that form mathematical techniques that a students uses when facing an unfamiliar situation

understand/interpret the question
look for a pattern
guess and check
work backwards
solve a simpler or similar problem

2. Reasoning and Proof

igutie12 — 2025-02-10 23:48:08 UTC

ability to make generalizations and provide explanations and justification for arguments, can lead to building mathematical memory to understand concepts

ask "why"
recognize mistakes
thinking critically

3. Communicating

igutie12 — 2025-02-11 01:08:17 UTC

refers to talking and writing about mathematics-- to explain and describe their ideas while also given opportunities to learn descriptions and explanations

write reflectively
use correct mathematical terminology as a language
question, criticize, clarify

4. Making Connections

igutie12 — 2025-02-11 01:14:49 UTC

refers to the student's ability to

see connections between facts + how they relate
make connections between symbols and procedures
make connections between what they're learning and the real world
connect new problems to old + how they relate

5. Creating Representations

igutie12 — 2025-02-11 01:18:19 UTC

ways to express different mathematical ideas that allow student's to understand from a different perspective/representation to solve problems; to organize and communicate mathematical ideas; and interpret physical, social, mathematical phenomena

graphs/diagrams
tables/charts
geometric figures
technology

6. Investigating

igutie12 — 2025-02-11 01:23:42 UTC

being able to explore unfamiliar mathematical situations

ask yourself questions
research and extract relevant information

explore different methodologies

Macro, Meso, Micro Modeling

igutie12 — 2025-02-11 02:08:14 UTC

macro concepts: give breadth of understanding and possess the greatest transferability across most disciplines
- broader constructs that encompass a huge body of knowledge, such as algebra, geometry, calculus, etc.
meso concepts are broken down within macro, such as trigonometry or differentiation and integration
micro conepts: give supportive depth of understanding, such as ration, angles, and similarity