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Rote memorization

If a student does not know what they fact they memorized means, it has no value

Has no depth of understanding

Facts are not transferable

Definitions

Formulas (y=mx+b)

Names of Polygons (hexagon, octagon)

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Topics in math are broader concepts that break down to specific micro-concepts at the next level

ex: Properties of parallelograms (topic) can be supported by congruency, supplementary angles, complementary angles, perpendicular, etc (micro-concepts).

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3. The Generalizations and Principles Level

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3. The Generalizations and Principles Level

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Statement of conceptual understanding (Theorems)

Allow students to make connections between two or more concepts

ex: Similar right-angled triangles share a common acute angle

ex: Fundamental Theorem of Calculus allows us to solve a definite integral

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Why do two negatives make a positive?

Why can we not divide by zero?

Having only the factual level has no depth of understanding and cannot transfer the knowledge to other concepts

Additionally, students are much more likely to retain knowledge they fully understand and can apply!

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Several skills that students in a methodical and systematic way to support learning

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Operations and actions that support strategies.

For example, you need the skills of utilizing a four quadrant graph to graph functions.

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Complex, sophisticated performances whoch are composed of strategies, algorithms, and skills.

broad techniques students utilize in understanding mathematics

problem solving, mathematical inquiry, representation, and communication

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1. Understand the problem

2. Come up with a plan

3. Execute the plan

4. Review and extend

Skills and strategies needed include the ability to look for a pattern, identifying all possibilities, guessing and checking, and interpreting what the question is truely asking.

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Make generalization and provide explanations and justifications for the argument

Builds mathematical memory

gives opportunities to make AND LEARN from mistakes

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Number talks: communication tool to help students develop computational fluency

Writing Mathematics: helps students learn concepts more effectively, encouraging higher order thinking skills

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Math Generation: Number Talks - Toby Johnson MS

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See connections between facts and there relation to one another

Make connections between symbols and procedures

Make connections between what they are learning and the real world

Connect new problems to old

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Ways to express different mathematical ideas such as graphs, tables, geometric figures, etc.

1. Pictoral Representations: drawings containing irrelevant aspects of the problem

2. Schematic Representations: purposeful to solving the problem and are codes or symbols to show relationships.