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The structure of knowledge is a structure composed of “levels” that are based on “intellectual thinking” and demonstration of deep understanding

It allows one to “classify and recognize similarities, differences, and relationships” in mathematics

The lowest level is the “factual level,” which consists of route memorization and factual knowledge (e.g. the different name of polygons)

The highest level is principle generalization, as this requires the application and understanding of mathematical concepts, and an engagement with math that goes beyond memorizing formulas and facts

This structure allows educators to identify whether their students are simply memorizing facts, or understanding math on a deep and conceptual level, which often provides key skills, insights, and applications.

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Macros:

Provide “breadth of understanding”

Large constructs that contain a vast entity of knowledge

Examples include algebra, statistics, and calculus

Meso:

Smaller than Macros, but bigger than micros

Can be thought of as a topic

Make macros digestible and allow the formation and organization of micros

One example of a meso is trigonometry

Micro:

Provide the “supportive depth of understanding”

The smallest of the scales, and often embody a small portion of the math language

Examples include ratio and angles

Ultimately, these concepts work together to establish the language of mathematics as a language of conceptual relationships by creating a logical and chain-like reaction that enhances the engagement with math, and allows people to understand and practice the language conceptually. In a sense, these concepts build and support one another to create a structure for a conceptual language, and by pushing a person to see the concept behind math, rather than the procedures.

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How is mathematics a language of conceptual relationships made of macro, meso, and micro concepts?

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How is mathematics a language of conceptual relationships made of macro, meso, and micro concepts?

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“Statements of conceptual relationships”

Often demonstrate a deeper level of understanding and are at the top of the Structure of Knowledge

Aligned with patterns and recognizing why or how a certain concept works

One examples of a generalization is trigonometry being based on similar right-angled triangles sharing acute angles and corresponding sides.

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Making generalizations is fundamental to mathematics. Developing the skill of making generalizations and making it part of the students' mental disposition or habits of mind in learning and dealing with mathematics is one of the important goals of mathematics education. Making generalizations is a skill, vital in the functioning of society. It is one of the reasons why mathematics is in the curriculum. Learning mathematics (if taught properly) is the best context for developing the skill of making generalizations.

What is generalization?

There are three meanings attached to generalization from the literature. The first is as a synonym for abstraction. That is, the process of generalization is the process of "finding and singling out [of properties] in a whole class of similar objects. In this sense it is a synonym for abstraction (click here to read my post about abstraction). The second meaning includes extension (empirical or mathematical) of existing concept or a mathematical invention. Perhaps the most famous example of the latter is the invention of non-euclidean geometry. The third meaning defines generalization in terms of its product. If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a theorem.

For further discussion on these meanings, read Michael Mitchelmore paper The role of abstraction and generalization in the development of mathematical knowledge. For discussion about the importance of generalization and some example of giving emphasis to it in teaching algebra, the book Approaches to Algebra - Perspectives for Research and Teaching is highly recommended. There is a chapter about making generalizations and with sample tasks that help promote this attitude.

Sample lessons

Mathematical investigations and open-ended problem solving tasks are ways of engaging students in making generalizations. The following posts describes lessons of this type:

1. Sorting number expressions

2. Lesson study: Teaching subtraction of integers

3. Math investigation lesson: polygons and algebraic expressions

4. Polygons and teaching operations on algebraic expressions

Of course it is not just the type of tasks or the design of the lesson but also the classroom environment that will help promote making generalization and make it part of classroom culture. Students will need a classroom environment that allows them time for exploration and reinvention. They will need an environment where a questioning attitude is promoted: "Does that always work?" ,"How do I know it works"? They will need an environment that accords respect for their ideas, simple or differing they may be.

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Making generalizations in mathematics - Mathematics for Teaching

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Making generalizations is fundamental to mathematics. Developing this skill and making it part of the students' mental disposition or habits of mind...

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Equations that use mathematical symbols or variables

These equations express a relationship that represents "facts" within the Structure of Knowledge

One example of a formula is the slope-intercept formula ( y = mx + b)

Formulas are also at the bottom of the Structure of Knowledge, as they mostly exemplify memorization and procedure, rather than deep understanding of a subject

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What is the difference between formulae and generalizations in mathematics? 

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What is the difference between formulae and generalizations in mathematics? 