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# Mathematical Generalizing

### Description

The Mathematical Generalizing Performance task introduces students to the steps for developing proof-related competencies with a variety of mathematical areas.

This task requires students to think and operate within the constraints of mathematical reasoning and proof, and to develop cogent arguments within the context of mathematical representations (algebraic, graphical, geometric, and tabular). These are non-trivial tasks and require students to think in sophisticated ways about what does and does not constitute an accurate proof, and the distinction between using empirical examples to provide evidence that a claim is true and the production of an argument that shows the claim is true in all cases.

• Math

### Opportunities for Student Choice

This task is not design to provide a choice as to the patterns that students will be exploring; these should be provided by the teacher as part of the task. However, throughout the process of generalizing patterns, students will need to choose between mathematical representations.  A pattern can be represented in multiple ways (algebraic, geometric, or pictorial); some representations may be easier to use as a basis for proving or refuting a claim than others.

These tasks introduce students to the kinds of reasoning used by mathematicians to determine the validity of claims. While the content of these tasks is strictly mathematical and does not involve “real-world” situations, the types of arguments that students must generate in order to complete these tasks are authentic in that they provide a window into the rules that real mathematicians use to advance the field. ​

### Opportunities for Exhibition to an Audience

Students are expected to work closely with one another as they complete these tasks, and may be required to present their work more formally to the rest of the class once they are finished.  However, as the content within these tasks is strictly mathematical and aligned with the CCLS standards, it is not suggested that students present their work in an assembly, letter to the editor, blog, or other form of exhibition to an audience outside of the classroom.

Models of high school-level mathematical proofs:

3-4 Levels below High School:

Student learning might focus on using empirical examples to make sense of what a claim is saying and to probe whether a claim could be true or not.​

2 Levels below High School:

Student learning might focus on how to represent numerical/empirical patterns in a more general way. If students are having a difficult time determining which representation to use, teachers should be prepared to help students choose a useful representation.

High School:

Student learning might focus on constructing a valid mathematical argument, articulating why this argument is valid, and extending the argument to a more general domain when applicable.

### Teacher Resources

Resource detailing the steps of constructing a valid mathematical argument:

Resources detailing different ways of presenting and critiquing mathematical arguments and reasoning:

Resource for helping teachers (and students) understand mathematical generalizing, with problem sets, comments, and reflection questions. (From the Open University):

Resource for helping teachers to identify formal aspects of mathematical proofs:

## Explore Mathematical Generalizing

What is the Mathematical Generalizing?

Develop a clear enough understanding of the characteristics of mathematical generalizing to be able to explain your understanding to others.

• CCSS.MP.4

1-3 Hours

## Explore The Validity Of A Mathematical Claim Through Empirical Experimentation

Does the information presented support the mathematical claim?

Use your knowledge of mathematics to discover whether the claim is supported by the information provided.

• CCSS.MP.7
• CCSS.MP.8

1-2 Hours

## Detect And Articulate Patterns

What can I infer about the relationship among the members of the set?

Find the similarities and extend them to other cases.

• CCSS.MP.6
• CCSS.MP.7
• CCSS.MP.8

2-3 Hours

## Conjecture About The Validity Of The Claim

Does it always work?

Take a stance about whether the claim is true or not?

• CCSS.MP.3

1 Hour

## Modify Or “Tinker” With The Claim To Make It Valid

What would make my claim valid?

Use a variety of tools and strategies to figure out if a mathematical claim can be made valid.​

• CCSS.MP.6
• CCSS.MP.1

1-2 Hours

## Model Using Mathematical Representations

What are various ways to represent numerical data?

Develop an appropriate mathematical representation that accurately reflects the patterns that you noticed.

• CCSS.MP.6
• CCSS.MP.4
• CCSS.MP.2

3-4 Hours

## Develop A Convincing Mathematical Argument

How do I know it works and how do I convince others?​

Provide evidence and reasoning to convince others that your claim is valid.

• CCSS.MP.6
• CCSS.MP.3

2-3 Hours

## Give and Receive Feedback (Mathematical Generalizing)

How can I improve my work?

Strengthen your work by giving and receiving feedback from a peer or teacher.

• CCSS.MP.6
• CCSS.MP.3

1 Hour

## Revise (Mathematical Generalizing)

How can I improve my draft?

Using feedback from a peer and/or teacher, strengthen your draft by ensuring your facts and details are relevant, your organization is clear, and your use of voice is strong and appropriate to the audience.

• CCSS.MP.6
• CCSS.MP.3

2-3 Hours

## Edit (Mathematical Generalizing)

How can I improve my draft?

Using feedback from a peer and/or teacher, strengthen your writing with a focus on conventions such as grammar and professional language.

• CCSS.MP.5
• CCSS.MP.6

1-2 Hours