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Takes scaffolding further and asks, “So what?”

Provides the significance or effect of learning something

Expresses the “real-world value of mathematics”

This level has the deepest extent of understanding and application

“Mathematically extends” the understanding of Level 2 with “greater conceptual specificity”

An example includes, “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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Dear Parents,

Math is very useful in everyday life. Math can help us do many things that are important in our everyday lives. Here are some daily tasks for which math is important:

Managing money $$$

Balancing the checkbook

Shopping for the best price

Preparing food

Figuring out distance, time and cost for travel

Understanding loans for cars, trucks, homes, schooling or other purposes

Understanding sports (being a player and team statistics)

Playing music

Baking

Home decorating

Sewing

Gardening and landscaping

Parents can help teens connect math they learn in school and their everyday lives. As a parent, you could talk to your teen about how you use math in your daily life. You could also ask family members and friends how they use math in their daily lives. Please talk to your teens about these math connections to real world. Share with your child the examples of everyday math applications, which are listed below. When your teens hear how math can be used every day, they will be more likely to view math as important and valuable. They may also become more interested in mathematics. Remember that you as a parent can greatly influence how your child thinks about mathematics.

The testimonials included on this website give brief examples of how people use math in their daily lives. Please watch these. You can share information from these videos with your teen.

Examples of Math Connections to Daily life

Managing Money

Your teen will learn skills in algebra class that will help them with money. One important skill they will learn is how to calculate interest and compound interest. Your teen can use this skill to manage their money now and when they grow up. This skill also will help them pick the best bank account. It will also help them decide which credit card is best to have. People who take out loans need to understand interest. It will also help them figure out the best ways to save and invest money.

Recreational Sports

Geometry and trigonometry can help your teens who want to improve their skill in sports. It can help them find the best way to hit a ball, make a basket or run around the track. Basic knowledge of math also helps keep track of sports scores.

Home Decorating and Remodeling

Calculating areas is an important skill. It will be useful for your teen in remodeling future homes and apartments. It will help your teen find how much paint they need to buy when repainting a room. It is also an important skill for anyone who wants to install new tiles in a bathroom or a kitchen. Knowing how to calculate perimeters can help your child when deciding how much lumber to buy for floor or ceiling trim.

Cooking

People use math knowledge when cooking. For example, it is very common to use a half or double of a recipe. In this case, people use proportions and ratios to make correct calculations for each ingredient. If a recipe calls for 2/3 of a cup of flour, the cook has to calculate how much is half or double of 2/3 of a cup. Then the cook has to represent the amount using standard measures used in baking, such as ¼ cup, 1/3 cup, ½ cup or 1 cup.

Shopping

Your teen will use math when buying different items. When buying a new computer, your child will need to figure out which store offers the best price or best financing. Math is useful in finding the best deal for food items. For example, your teen will need to decide which pack of soda to buy when given a choice of 20 oz., 2-liter, 12 pack, or 24-pack. Stores often have sales that give a percentage off an original price. It is helpful for people to know how to figure out the savings. This math skill is very useful because it helps us calculate discounts so we can buy an item for the best price offered.

Note. Partial content of this web page is adapted from Making Connections: Helping Your Teen With the Choices Ahead brochure (Harackiewicz, Hyde & Hulleman, n.d.) and Making Connections: Helping Your Teen Find Value in School brochure (Hulleman, Harackiewicz & Hyde, 2007). Please refer to the links to the brochures under the useful links for parents tab if you would like access to the full brochures.

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Math Matters in Everyday Life | Northern Illinois University

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Dear Parents, Math is very useful in everyday life. Math can help us do many things that are important in our everyday lives.

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Guided by the questions “Why?” and “How?”

Scaffolded further than a Level 1

Encompasses critical understanding and should be the main target for teachers

An example of a Level 2 is “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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Are “weak” and do not aid students in understanding “what and why” they are learning

Often include “no no” verbs such as is, affect, and influence

An example of a Level 1 is “the coefficients of a quadratic function in a connected equation affect the quadratic formula.”

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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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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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To make connections and recognize patterns

Deepen their understanding, and take their mathematical fluency beyond procedural understanding

To “release” the natural “powers” of a student

Give a higher sense of purpose to their learning

Encourage application of mathematics in other fields

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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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Laws or foundational truths

Hold “all the attributes of generalizations”

Never contain qualifiers (key difference!)

Are often more specific and less broad than a generalization

At times, can be tied to a specific person or group

One example of a principle is the Pythagorean Theorem

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Two or more statements “stated as a sentence of relationship”

Transfer through “time, cultures, and across situations”

May contain qualifiers (e.g. may or can)

Contain no proper nouns, past tense verbs, or pronouns that "associate them" with a group or person

For example, the angle sums in triangles “may” add to 180 degrees

One example of a generalization is the quadratic formula determines the zeros of functions, the roots of equations, and the x intercepts graphically

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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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What are generalizations in mathematics? What is the difference between a principle and a generalization in mathematics? 

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What are generalizations in mathematics? What is the difference between a principle and a generalization in mathematics? 