# Pedagogy

## Aligning Teaching to Learning

The Carnegie Learning pedagogical approach focuses on how students think, learn, and apply new knowledge. The Standards for Mathematical Practice have always been the foundation of Carnegie Learning worktexts.

Carnegie Learning Pedagogy:

**Accommodates multiple learning styles**by inspiring students to learn with and from each other.**Promotes a student-centered classroom**where teachers facilitate learning and coach students to master concepts and procedures with class time centered on active engagement and problem-solving.**Encourages mathematical discourse**where students are prompted to explain their thoughts and processes for solving math problems.

Perhaps the most important feature of the lessons is the thoughtful, authentic questions embedded throughout the text. This has been extremely helpful for teachers who are from a very diverse background such as alternative preparation programs, new to teaching, or not experienced with designing mathematical questions.

### Promote Higher Order Thinking

Carnegie Learning materials are built on an over-arching questioning strategy that promotes analysis and higher order thinking beyond simple "yes" or "no" responses. Students internalize the processes and reasoning behind the mathematics. Questions, instruction, and worked examples interweaved in student lessons help students develop their own mathematical reasoning. Lessons are also structured to provide students with various opportunities to reason, model, and expand on explanations about mathematical ideas.

Students are asked to:

- Look for patterns
- Estimate
- Predict
- Describe
- Represent
- Compare and contrast
- Calculate
- Solve
- Write a rule
- Generalize
- Explain reasoning

### Real-World Problem Types

Carnegie Learning materials teach students that math is relevant because it provides a common and useful language for discussing and solving complex problems in everyday life. For this reason, our worktext lessons include a variety of problem types which support students in developing a rich understanding of mathematical ideas.

#### Peer Analysis: Thumbs Up, Thumbs Down

These problems help students make inferences about correct responses. Research shows that providing only positive examples will not eliminate some of the things students may think, and therefore negative examples are included as well.

#### Who’s Correct?

Students will determine what is correct and what is incorrect, and then explain their reasoning. These types of problems will help students analyze their own work for errors and correctness.

#### Worked Examples

Many students need a model to know how to engage effectively with worked examples. Students need to be able to question their understanding, make connections with the steps, and ultimately self-explain (the progression of the steps and the final outcome).

#### Models

Real-world contexts confirm concrete examples of mathematics. The scenarios in the lessons help students recognize and understand that the quantitative relationships seen in the real world are no different than the quantitative relationships in mathematics.

#### Manipulatives

Manipulatives are used throughout the curricula to foster a conceptual understanding of mathematical concepts. These activities provide students with opportunities to develop strategies and reasoning that will serve as the foundation for learning more abstract mathematics.

#### Using Technology

Step-by-step instructions provide students with opportunities to understand how to use graphing calculators.

#### Organizing Concepts: Talk the Talk

Open-ended questions require students to summarize and generalize their mathematical understandings and key concepts.

#### Matching, Sorting, and Exploring

Students will experience various hands-on activities that match or sort verbal descriptions, tables, and graphs. These activities help develop skills recognizing and categorizing patterns in mathematics.

#### Graphic Organizers

Students will use graphic organizers to create their own representations of key mathematical concepts.

#### View Complete Lessons

View a complete Student Worktext chapter to see how these problem types are used together.