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Generalizations are what I would consider as a "big idea" for students to capture when being taught a new lesson. Generalizations are considered multiple different concepts, in which they are truthful statements supported by facts but can have key words like "often, can, may" in them to make them up to debate for not all different instances in the statement presented.

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Principles have similar notations to generalizations in which they form different statements related to mathematics however the statements created are usually theorems. These theorems have factual evidence to being true and do not contain words such as "often, can, may".

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Students are able to use the new math knowledge given to them through generalization, however they do not know why it works.

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Students have an understanding of why it works and or how it works. Students can use this knowledge to develop more critical thinking skills for future problems but do not know if it applies to anything else yet.

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Students can now understand why it works and its affect (the math content learned) beyond the foundations learned before or in other words, what type of problems were done before to begin learning the concept.

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During Teaching, Teachers must have different appropriate ways to develop understanding for students. Below, is an example of one of those ways through an inductive-inquiry led activity.

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This website provides the different generalizations once again but also is able to give examples of what it looks like in the classroom.

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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...