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Generalizations

Give us explanations of why and what we want our students to know in terms of the relationship between two or more concepts

A.K.A enduring understandings, essential understandings, or big ideas

Summarize what students should be taking away from a unit

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"Generalizations are defined as two or more concepts stated as a sentence of relationship. Generalizations are truths supported by factual examples, but they may include a qualifier (often, can, may) when the idea is important but does not hold in all instances” (Erickson, 2014)

Generalizations

The overarching ideas we want our students to understand

Contain qualifiers

Provide goals when unit planning

Help learners construct and make connections when learning

Example: The sum of angles in a triangle *may* add up to 180 degrees. (We use the word "may" because this does not hold true in Euclidean geometry)

Principles

The theorems in mathematics

Never contain qualifiers (may, can, often)

Examples: Pythagorean theorem, Fermat's theorem, etc.

Principles are statements of relationship written in the Structure of Knowledge or the Structure of Process

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Level 1: Students will understand that...

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Level 1: Students will understand that...

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Level 1 generalizations are weak and do not help learners fully understand what and why they are learning. This level of generalization contains verbs that we should try to avoid such as: is, are, have, impact, influence, and affect.

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Level 2 generalizations ask the questions "How?" or "Why" and this scaffolds a level 1 generalization up to the next level. At this level, learners are given greater clarity and conceptual specificity, resulting in greater depth of knowledge.

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Level 3: So what? What is the significance?

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Level 3: So what? What is the significance?

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At this level, we are explaining the purpose of the generalization. Critical understanding is already expressed in level 2, but in level 3 we are expressing the real-world value of mathematics and expanding level 2 with greater conceptual specificity.

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Inductive Teaching

Providing students with examples from which to draw generalizations

Traditional curriculum focuses on deductive approaches: students are given a generalization/ equation and then are told to study specific examples

Concept-based curriculum prioritizes the inductive approach. It involves guiding students through inquiry to discover principles for themselves.

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Students need to have opportunities to form conceptual understanding from learning experiences

Learning should involve students looking at specific examples to find patterns and then create generalizations

Graphic Organizers

Gives students visual representations of how concepts are related, making it easier for them to visually form generalizations

Unit Webs

Unit webs are tools that provide strategies for including generalizations

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Generalizations and principles in unit plans provide clear focus for instruction

Giving students the ability to generalize is the essence of math

Leads to inductive methods of teaching that draw out greater conceptual understandings

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https://us.corwin.com/sites/default/files/upm-assets/78590_book_item_78590.pdf

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Understanding the Process of Generalization in Mathematics through Activity Theory Research Article

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Understanding the Process of Generalization in Mathematics through Activity Theory Research Article

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International Journal of Learning, Teaching and Educational Research Vol. 16, No. 12, pp. 46-69, December 2017 https://doi.org/10.26803/ijlter.16.12.4

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https://grantwiggins.wordpress.com/2014/04/23/conceptual-understanding-in-mathematics/

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https://blogs.ams.org/mathgradblog/2016/08/21/role-generalization-advanced-mathematical-thinking/

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The Role of Generalization in Advanced Mathematical Thinking

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Generalization and abstraction both play an important role in the minds of mathematics students as they study higher-level concepts. In the second chapter of the Springer book Advanced Mathematical Thinking , Tommy Dreyfus defines generalization as the derivation or induction from something particular to something general by looking at the common things and expanding their domains of validity.

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https://whatihavelearnedteaching.com/generalizations-in-math/

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Do you help your students make generalizations in math?

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https://math4teaching.com/teaching-making-generalizations-in-mathematics/

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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...