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What Are the Levels of the Structures of Knowledge and Process for Mathematics?

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What Are the Levels of the Structures of Knowledge and Process for Mathematics?

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The Levels of the Structure of Knowledge

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The Levels of the Structure of Knowledge

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LOWEST LEVEL and it focuses on memorization and does not guarantee conceptual depth of understanding.

Example :

Facts are specific examples of people, places, situations, or things.

So in this case it could be formulas like

Y=mx + b or memorizing different names of polygons (e.g., pentagon, hexagon).

Keypoint: Knowing a definition or a formula does not imply understanding. Memorized formulae in mathematics are facts that support the broader concepts in mathematics.

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Extra note: The Difference Between Formulae and Theorems in the Structure of Knowledge

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Extra note: The Difference Between Formulae and Theorems in the Structure of Knowledge

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A formula is an equation that uses mathematical symbols or variables to show a relationship and is represented by the facts in the Structure of Knowledge.

Theorems are statements that have been proven and connect explanations of conceptual understandings. Theorems are represented by principles in the Structure of Knowledge.

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Mathematical topics encompass broader concepts that can be further divided into specific micro-concepts at a more detailed level.

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The Generalizations and Principles Level

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The Generalizations and Principles Level

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Generalizations and principles are ideas that help students see how different concepts are related. In math, these principles are known as theorems, which are fundamental truths.

Example:

Fundamental theorem of calculus

Theorems in calculus allow the evaluation of a definite integral without having to go back to the definition and take the limiting sum of a bounded area.

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

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

Mathematical processes are complex, sophisticated performances, which are composed of strategies, algorithms, and skills (Erickson & Lanning, 2014).

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1.Problem Solving

2.Reasoning and Proof

3.Communicating

4.Making Connections

5.Creating Representations

6.Inquiring

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Provided a link for further information regarding problem solving with examples

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Solving problems involves thinking logically and making decisions in a systematic way.

1.Understand the problem

2.Devise a plan

3.Carry out the plan

4.Review and extend

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Being able to make generalizations and explain your reasoning is essential for strong reasoning skills. This process creates a network of connections in math, helping to link different concepts together. Reasoning also helps strengthen mathematical memory and understanding. Additionally, learning through reasoning allows students to make mistakes and learn from them.

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Communicating involves talking and writing about math. It’s important for students to have chances to explain their thoughts and learning processes, as well as to listen to and discuss others’ ideas. Sharing different methods gives learners new perspectives and helps them develop a range of problem-solving strategies.

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Number talks are based on research by Kathy Richardson and Ruth Parker from the 1990s. They are important for building mental math skills that help students understand more advanced math later on. Number talks encourage students to discuss and compare number relationships, helping them improve their ability to add, subtract, multiply, and divide in different ways.

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Mathematics is a build-up of knowledge where everything is connected. Making connections means that learners can:

See how facts relate to each other,

Understand how symbols and procedures are linked,

Relate what they’re learning to the real world,

Connect new problems to what they already know,

Recognize how different concepts fit together.

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Representations are different ways to show mathematical ideas, like graphs, tables, and shapes. Using various representations can help learners understand concepts better and see how they connect.

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Investigating means exploring new and unfamiliar math situations. When you investigate, you ask yourself questions to deepen your understanding. It also involves being resourceful by looking up information and finding useful ideas. It’s important for teaching to encourage creativity when facing tough math problems. Students should be motivated to explore, ask questions, try different methods, and conduct their own research.

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Macro, Meso, and Micro Concepts in Mathematics

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Macro, Meso, and Micro Concepts in Mathematics

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Timeless

Universal

Transferable

Abstract to varying degrees

One or two words, or a short phrase

Have different examples that share common attributes (Erickson, 2007)