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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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Opportunities for students to demonstrate and communicate their conceptual understandings. 

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Opportunities for students to demonstrate and communicate their conceptual understandings. 

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-Students can use graphic organizers to visualize how concepts are related, then be able to draw generalizations.
-At the end of the unit you can give ask students to complete the sentence, "I understand that..."
-Students can create a concept math.
-You can ask students to give a headline for the lesson.

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-They are specific topics or subjects.
- Particular understanding from within the unit.

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-Beyond the specifics of the unit.
-They are the larger understandings that usually take longer periods.

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What is the difference between Level 1, 2, and 3 generalizations?

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What is the difference between Level 1, 2, and 3 generalizations?

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Level 1: Students will understand that....
Level 2: How or why..
Level 3: What is the significance or effect?

The teaching target is at Level 2, and Level 3 is for extra clarification or extensions when needed. Level 1 is the helping step to use the right verbs and grammar.

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"Adapted from original Structure of Process figure from Transitioning to Concept-Based Curriculum and Instruction, Corwin Press Publishers, Thousand Oaks, CA."
Concept-Based Mathematics
Teaching for Deep Understanding in Secondary Classrooms
by Jennifer T.H. Wathall

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Why do we wish our students to understand principles and generalizations in mathematics?

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Why do we wish our students to understand principles and generalizations in mathematics?

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-Guiding students to particular principles or generalizations is through inquiry is effective for learning in mathematics.
-Promoting mathematical generalizations is at the core of learning the subject.
-It helps students understand how or why they are learning what they are learning.

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-Foundational truths or laws
-Theorems in math are known as 'principles'.
-Generalizations are more universal than Principles

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-Statements of conceptual understandings.
-They give us explanations of why and what we want our students to comprehend in terms of the relationship between two or more concepts.
-They are the 'big ideas' of the unit.
-They provide goals when unit planning and help learners make connections between concepts.
-They're two types of generalizations
-AKA: enduring understandings, statements of inquiry