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Micro concepts are the basis of understanding for meso and macro topics. Micro concepts supports the depths of understanding.
Example: Geometry
A student is able to find the magnitude of a vector which supports their understanding of vectors (meso) and geometry (macro).

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Meso Concepts are large areas of knowledge within a macro concept.
Example: Geometry
Vectors and Trigonometry are meso connects in geometry.

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Macro concepts are the broad construct or idea. It is usually the name of the class a student is taking.
For example: Geometry
Geometry is the study of the relationship between shapes, points, lines, surfaces, and arrangement.

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Mathematics has its own language. The language is made of macro, mesh and micro concepts. Students must understand the all levels of the language.

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Generalization require a student to synthesize what they have learned about a topic. This requires a conceptual understanding of the topic.
Example: Quadratic Formula
A student may use a generalization to explain when the quadratic formula is useful. They may say, "The quadratic formula is used to find the x-intercepts of a trinomial, especially when the a of the trinomial is more than 2 or it is unfactorable." This demonstrates that the child understands the purpose and has come to a valid conclusion.

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A student with only a procedural or factual understanding are able to recite a formula. They may not be able to explain the formula, the purpose of the formula, or how it is derived.
Example: Quadratic Formula
A student may be able to recite the quadratic formula; however, they must know when to use it.

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Formulae: a mathematical relationship or rule expressed in symbols
Generalizations: a statement that is true for a whole topic or mathematical idea

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Formulae and Generalizations represent different levels of understanding.

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Summary of Wathall Chapter 2

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Students learn in different ways. There are models to show students why the Pythagorean theorem works. Here is a song that explains the formula. This song can help your musically-inclined students.

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Colin Dodds - Pythagorean Theorem (Math Song)

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A conceptual understanding of a subjects means a student is able to explain why they can use a formula or why a fact is true. The student is able to apply their understanding of the information outside the context of the problem.
Example: Pythagorean Theorem
A student who understands Pythagorean Theorem can related the formula to the appropriate sides of a triangle. Their knowledge goes beyond memorization.

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A factual understanding of a subject means a student can recite rules or formulas. These does not demonstrate a deep understanding of the topic. The student's knowledge is trapped in that understanding.
Example: Pythagorean Theorem

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Students can have different levels of understanding. While a students may know the facts of a concept, it is important that they develop a deeper level of understanding.