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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

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an equation using mathematical symbols or variables to show a relationship

(represented by facts)

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proven statements that connect explanations of conceptual understanding

(represented by principles)

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topics are broader concepts broken down into specific micro concepts

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The Generalizations and Principles Level

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The Generalizations and Principles Level

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both are statements of conceptual understanding that allow students to make connections between two or more concepts

in math, principles are theorems-- the truths

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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

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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

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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

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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

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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

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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

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Explore the effect of applying a composition of translation, rotation, and reflection transformations to objects.

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being able to explore unfamiliar mathematical situations

ask yourself questions

research and extract relevant information

explore different methodologies

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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