Whathall Chapter 2 (Structures of Knowledge and Process for Mathematics) by Anthony Giliberto

Structure of Knowledge

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Factual Level

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Includes root memorization and does not guarantee conceptual depth of understanding. This includes facts which are not transferrable and include definitions, formulae in the forms of symbols, and different names of polygons. EX: Knowing the shape of a graph without necessarily understanding how it is generated

Formula

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Is an equation that uses mathematical symbols or variables to show a relationship and is represented by FACTS

Theorems

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Are statements that have been proven and connect explanations of conceptual understandings, represented by PRINCIPLES

Topic and Concepts Level

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Includes broader concepts that help break down specific micro concepts (EX): the topic of integration is supported by concepts such as limits, summation, etc

Generalization and Principles Level

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Statements that allow students to make connections between two or more concepts. (Principles are theorems in Mathematics) (EX): Fundamental Theorem of Calculus

Skills

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Small operations or actions that are embedded in strategies that when applied can allow the strategy to work (EX): Create a Table of Values, Know how to Plot Points

Mathematical Processes

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Broad techniques that students draw upon. They can be seen as the processes through which students acquire and apply mathematical skills, knowledge, and concepts.

1) Problem Solving

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Fundamental building block of math that is a defined series of skills and strategies that form math techniques when a learner face an unfamiliar situation. 1) Understand the Problem, 2) Devise a Plan, 3) Carry out Plan, 4) Review

2) Reasoning and Proof

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The ability to make generalizations and provides explanations for arguments. Helps build mathematical memory and allows to find out the "why" and critical thinking skills (EX): proving an identity such as double angle in Trig

3) Communicating

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Refers to talking and writing about mathematics

Number Talks - students compare relationships and look at different ways to add, subtract, multiply, divide

Social Constructivist - people construct their own understanding and knowledge through experiences and reflecting on them (emphasizes assisted discovery through teacher-student and student-student interactions)

4) Making Connections & 5) Creating Representations & 6) Investigating

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4) Learners can see the link between symbols and procedure, and between facts and how they relate to one another

5) expressions of math ideas such as graphs, tables, figures, etc (helps learners organize and communicate math ideas and helps interpret phenomena)

6) Explore mathematical situations (Ex) ask questions

Process of Mathematics Key Catagories

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Macro, Meso, and micro concepts in mathematics

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MACRO - give breadth of understanding and possess the greatest transferability across disciplines (EX) Geometry, Algebra, Calculus

MESO - breaks down macro concepts to more focused understanding (Ex): Trigonometry

MICRO - gives us the supportive depth of understanding (EX) Ratios, Angles, Similarity

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Algorithms

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Consist of skills, which allow learners to follow a set of rules to a specified outcome

Marriage of Structure of Knowledge and Structure of Process

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Creating synergy between facts, skills , and conceptual understanding is important, Lynn Erickson writes, "to stimulate more sophisticated, complex thinking we need to create a synergy between the simpler and more complex processing centers in the brain. This interactive synergy requires the mind to process information on two cognitive levels, the factual and conceptual. The conceptual mind uses facts as a tool to discern pattern, connections, and deeper understandings" (10).