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Chapter 3: What are Generalizations in Mathematics? 

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Chapter 3: What are Generalizations in Mathematics? 

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1. What are Generalizations in Mathematics? What is the Difference Between a Principle and a Generalization in Mathematics?

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1. What are Generalizations in Mathematics? What is the Difference Between a Principle and a Generalization in Mathematics?

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Generalizations in math are broad statements about patterns.

A Principle is a specific rule, and a generalization is a bigger idea that includes many rules.

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1. Overarching enduring understandings are big ideas that last beyond a single unit.

2. Topical understandings are specific to a particular subject or topic.

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They are larger, transferable insights reflecting year-long program understandings. Also known as overarching generalizations or statements of inquiry.

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They focus on a specific insight within a unit of study. Also known as topical generalizations.

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1. Algebra solves complex problems.

2. Scalar products find vector angles and magnitudes.

3. Students struggle to understand logarithms.

4. A quadratic formula finds function zeros and roots.

5. The quadratic formulas discriminate shows the roots nature.

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Making Generalizations in Math.

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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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2. How do we Craft Quality Mathematics Generalizations?

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2. How do we Craft Quality Mathematics Generalizations?

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Letting students generalize is key to learning math. Math concepts can be expressed differently, depending on the student's level and the teacher's goal.

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Teachers should work together to write generalizations as part of planning a concept-based unit. When we write generalizations, we want to focus on understanding, not just skills. Starting with "Students will understand that..." is a great first step.

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Level 1: "Students will understand that . . ."

Level 2: "How or why?"

Level 3: "So what is the significance or effect?"

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The Levels of Writing Generalizations

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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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Level 1 generalizations are weak and lack clarity.

Example from the book: "The quadratic formula is one method to solve a quadratic equation".

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Level 2 generalizations help learners understand what they are learning.

Example from the book: "The quadratic formula describes the roots or zeros of the function and helps solve a quadratic equation".

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Level 3 generalizations explain the purpose and extend learning.

Example from the book: "The expression underneath the square root in the quadratic formula, the discriminant, conveys the nature of the roots, which highlight geometrical features".

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3. How do we Draw Out Conceptual Understandings From our Students? 

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3. How do we Draw Out Conceptual Understandings From our Students? 

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We draw out conceptual understandings from students by asking open-ended, thought-provoking questions that encourage them to think deeply and make connections.

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Inductive teaching involves giving students specific examples to help them form general ideas.

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Conceptual Understanding

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5 Effective Strategies For Conceptual Understanding In Students | ClassPoint

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Learn how to shift from rote memorization to conceptual understanding with 5 strategies that foster critical thinking and real-world application in students.

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1. Students should form conceptual understanding through writing in math.

2. Learning experiences should involve explaining specific examples, identifying patterns, and writing generalizations.