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Generalizations (Big Ideas) in Mathematics

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Generalizations (Big Ideas) in Mathematics

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

-Provide clear goals for students

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In mathematics, generalizations are:

- Defined as two or more concepts that are stated as a sentence of relationship.

-Understandings that transfer through time, across cultures, and situations

-Truths supported by factual examples. They may include a qualifier when the idea is important but does not hold in all instances.

Example: In Euclidean Geometry, triangles always add up to 180 degrees, but in non-euclidean geometry, this generalization is not true.

-Provide goals when unit planning and help learners construct and make connections with what they are learning

In inductive, inquiry-led teaching, learners are guided to discover generalizations and answer the question: "Why am I learning this, and what is the point beyond this example?"

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In Mathematics, principles are laws or foundational truths that hold all attributes of generalizations and commonly describe real-life situations.

Example: Pythagorean Theorem

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Students should understand principles and generalizations in mathematics because they lay the foundation for mathematics. Students should be able to take generalizations and principles and connect them to new learnings.

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Include "no-no" verbs such as is, are, have, impact, affect, and influence. We don't want students to write level 1 generalizations.

-Example: The coefficients of a quadratic function in a connected equation affect the quadratic formula

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Level 2 generalizations showcase deeper understandings than level 1 generalizations. Teachers should ask questions like "How?" or "Why?" to help scaffold a generalization to the next level because it gives learners clarity and conceptual specificity, which results in deeper understanding.

-Ex: The quadratic formula utilizes the coefficients of a quadratic function in a connected equation and describes the roots of a quadratic function.

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Can be used to express the real-world value of mathematics or to mathematically extend level 2 understandings. They address the question, "So what is the significance or effect?''

-Ex: Different methods of solving quadratic equations distinguish the roots of the quadratic equation, which contribute to providing a graphical representation of the function, which, in turn, describes real-life problems.

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One of the Mathematics Teaching Practices from National Council of Teachers of Mathematics (NCTM, 2014) is:

Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses goals to guide instructional decisions.

Generalizations are summaries of thought and overriding ideas that we want our students to understand from their study. They are our mathematics goals to focus learning. Lynn Erickson (2007) first coined the term "generalizations" and in education circles generalizations are also known as enduring or essential understandings. It is a statement that connects two or more concepts in a sentence of relationship. Here are some examples of generalizations:
Students understand that:

Linear functions show relationships that exhibit a constant rate of change

A function represents a mapping (rule) that assigns each input (domain) with one output (range)

For right-angled triangles, the area of the square drawn from the hypotenuse represents the sum of the areas of the squares drawn from the other sides

Logarithm laws provide a means of changing multiplicative processes into additive processes and can be used to find inverses of exponential functions

If my students said any of the above statements I would be satisfied that they have fully understood the topic and actually have a deep understanding of the concepts involved.

To help craft quality generalizations Erickson (2007) suggests a scaffolding process.

Here is a specific example of how to scaffold a generalization in order to improve the quality of the generalization.

Level 1

Students will understand that..... Logarithm laws affect logarithmic expressions

Level 2

How or why?

Logarithm laws reduce logarithmic expressions utilizing the inverse process of exponential functions which represent continuous compounded growth.

Level 3

So what? What is the significance or effect? Logarithm laws provide a means of changing multiplicative processes into additive processes and this can provide the means to find inverses of exponential functions (continuous compounded growth).

In summary here are the three levels for crafting generalizations:

Level 1

Students will understand that...

Level 2

How or why? (Choose which question is most appropriate)

Level 3

So what? What is the significance or effect?

It is important to note here that critical understanding in mathematics is usually achieved through a level 2 generalization therefore the target for curriculum development and instruction is level 2 type of generalizations. Level 3 generalizations are used if we wish to extend an idea or apply to a real life situation.
How do we draw out conceptual understandings from our students?
John Mason (1996) encapsulates the essence of math teaching in the following quote: "a lesson without the opportunity for learners to generalise is not a mathematics lesson".

In a concept-based model of curriculum and instruction we do not tell students the generalizations at the beginning of a unit of work or at the beginning of a lesson. We guide students to communicate generalizations through a variety of strategies.

The following table illustrates some examples of how we can draw conceptual understandings from our students.

Strategy

Example

Give the concepts and ask students to connect these concepts in a statement that reflects the purpose of the unit of study.

Write the following concepts in a related sentence. This statement should demonstrate your understanding of how these concepts are connected:

zeros of a function, roots of an equation, x intercepts, quadratic formula
or
parallel lines, gradient, transversal, angles

At the end of a lesson or unit of work ask students to complete the sentence, "I understand that........"

I understand th at solving a quadratic equation by using the quadratic formula graphically displays the x intercepts of a quadratic function.

Give students a choice of a few statements and ask students to choose one that explains the big idea of the lesson or unit and to justify why.

The quadratic formula is

Solving a quadratic equation by using the quadratic formula graphically displays the x intercepts of a quadratic function.

The quadratic formula tells us what x is equal to and allows us to solve quadratic equations.

After a lesson or unit of study ask students to create a concept map, which includes the concepts they have learne d and how they are connected with statements explaining this.

With the word quadratics at the centre , create a concept map using the ideas you have learned and include statements to explain how they are connected.

Give students a close activity where they have to fill in the blanks, and provide them with a list of hints.

Hint Jar:

quadratic equation

quadratic formula

quadratic Function

x intercepts

Solving a ___________ ________

by using the _________ ________ graphically displays the __________ of a ______________.

Ask students to devise a headline which summarizes the key ideas (concepts) of a unit.

This can be used as a diagnostic or formative assessment.

Wathall, J. T. H. (2016) Concept-based mathematics: Teaching for deep understanding in secondary classrooms. Thousand Oaks, CA: Corwin.
Crafting generalizations is an important part of unit planning and if teachers know what their end goals are for their students then the rest of the unit planning process is enhanced.

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author: James B. Short and Stephanie Hirsh

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author: Paul Emerich France, Kurtis Hewson, Lorna Hewson, Heather Rubin

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Crafting Quality Math Generalizations: Establishing Goals to Focus Learning - Corwin Connect

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One of the Mathematics Teaching Practices from National Council of Teachers of Mathematics (NCTM, 2014) is: Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses goals to guide instructional decisions.

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Students look at specific examples and seek patterns to devise and write generalizations

Graphic organizers help draw out generalizations; they give students a visual representation of how the concepts are related

Students communicate conceptual understanding verbally and nonverbally

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Understanding beyond the specifics of a unit. They are larger, transferable insights we want students to acquire

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Subject or topic specific. They focus on a particular insight we want students to acquire within a unit of study.