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Photo taken of book

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By which teachers "transmit" knowledge for students to regurgitate in tests

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Sets up a synergy between the factual and the conceptual processing centers of the brain

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Macro: Algebra and Algebraic Expressions

Meso: functions

Micro: quadratics, arithmetic sequences, geometric sequences, complex numbers, variables and symbols, linear function, xy-plane, domain, range, graphs, inverse functions, composite funcations

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It is also a further breakdown of macro concepts, but it gives us the supportive depth of understanding

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It is a further breakdown of the macro concepts such as trigonometry

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It is our broader constructs that encompass a huge body of knowledge (e.g., geometry, algebra, and calculus)

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Being able to explore unfamiliar mathematical situations.

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They are ways to express different mathematical ideas such as graphs, tables, geometric figures and more.

Number talks are a combination of communicating and creating representations processes.

The purpose of representation in threefold:

To help learners organize, record, and communicate mathematical ideas

To select, apply, and translate different forms of representation to solve problems

To use representations to model and interpret physical, social, and mathematical phenomena

"Successful mathematical problem solving was positively correlated with use of schematic representations; conversely, it was negatively correlated with use of pictorial representations" (p. 246).

Pictorial Representations - include drawings that contain irrelevant aspects of the problem

Schematic Representations - purposeful for solving the problem and are codes or symbols to show relationships and connections

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Refers to the learner's ability:

To see connections between facts and how they relate to one another

To make connections between symbols and procedures

To make connections between what they are learning and the real world

To connect new problems to old

To see different concepts and how they relate to one another

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By writing mathematics in the form of words and explanations, it was able to help students to learn mathematic concepts more effectively, encouraging higher order thinking skills.

Explain what you have proved.

What are the applications and significance of these identities?

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This states people construct their own understanding and knowledge of the world through experiences, reflecting on those experiences, and collaboration among learners. With this, learners cognitively develop in a context of socialization.

Social Constructivism - emphasizes assisted discovery through teacher-student and student-student interactions

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It is a tool to help students develop computational fluency by comparing number relationships and looking at the different ways to add, subtract, multiply and divide.

Can be characterized as:

Class Discussion

Classroom Environment and Community

The Teacher's Role

Mental Mathematics

Purposeful Computation Problems

Research shows that learners who have a variety of different methods are more proficient in math.

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Refers to talking and writing about mathematics.

Learners need a chance to both listen and discuss.

This includes:

Discussion

Sharing Ideas

Collaboration

Number Talks

Social Constructivist Approach

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The ability to make generalizations and provide explanations and justification for arguments is key to this skill.

Leads to:

Generalizations (mathematical knowledge)

Building mathematical memory to understand concepts

Gives learners opportunities for mistakes

It really forces critical thinking and the "why" of math!

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1. Problem Solving

2. Reasoning and Proof

3. Communicating

4. Making Connections

5. Creating Representations

6. Investigating

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Strategies - a number of skills that learners use in a methodical and systematic way to support learning

Skills - small operations or actions that are embedded in strategies, and when appropriately applied, they "allow" the strategies to work

Mathematical Processes - complex, sophisticated performances, which are composed of strategies, algorithms, and skills (broad techniques)

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Fundamental building blocks for mathematics

It is a defined series of skills and strategies that form mathematical techniques that a learner employs when faced with an UNfamiliar situation

Involves thinking logically and making decisions in systematic ways. Principles are as follows:

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Review and extend

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Formula - an equation that uses mathematical symbols or variables to show a relationship and is represented by facts

Theorems - statements that have been proven and connect explanations of conceptual understanding

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Lowest level of the structure of knowledge

Includes rote memorization/basic knowledge of definition or formulas

Does not guarantee conceptual depth of understanding

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We know Pythagorean theorem is a^2+b^2=c^2, but what is the conceptual understanding?

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The fundamental theorem falls under Principle Generalization where limits, summation and bounded area would be Concepts, and integration is a Topic.