Math Methods Chapter Two

Which of the following is an example of a statement spoken in the language of doing mathematics?

"Explain how you solved the problem."

Conceptual understanding is a flexible web of connections and relationships within and between ideas, interpretations and images of mathematical concepts. Give two ways that a spinner could be used to guide the understanding of the concept of "chance."

1. Models and compares possible outcomes 2. Models the relationship between frequency and all possible outcomes 3. Creates relative frequencies by observing the sections of the spinner 4. Configuration of the sections of spinner from easy to more complex

Name at least two examples of classroom culture that could be an environment for students to do mathematics and gain relational understanding of a concept.

1. Persistence, effort and concentration are valued 2. Sharing of ideas among students 3. Students listen to each other 4. Students look for and discuss connections

To set up an environment for "doing" mathematics, teachers need to

Allow students to make engage in "productive struggle."

Which of the following represents an example of an effective way a teacher might help facilitate students' construction of mathematical relationships?

Asking students, "How is today's topic related to the fraction multiplication we investigated last week?"

Identify the statement that reflects an educational implication of the learning theories discussed in Chapter 2 "Exploring What it Means to Know and Do Mathematics."

Class activities and lessons should be designed with students' prior experiences in mind.

What theory(s) allow a classroom culture to access prior knowledge, use tools to build knowledge?

Constructivist.

Manipulative materials have the potential to provide opportunities for connection and communication. What statement would be a non-example of how to utilize the materials?

Encouraging students to converse about the model without knowledge of what the mathematical goal they are working on.

There are many ways to model and solve problems and explore how others develop understanding. Which strategy would foster students examining multiple solutions to try other methods?

Experimenting and explaining.

Which statement best represents a method to expose students to multiple approaches to problem solving?

Exposure to multiple approaches and the subsequent connections help students to recall the steps to complete mathematical processes.

Which statement below best describes the idea of mathematics as engaging in the science of pattern and order?

Mathematical processes and concepts follow logical patterns and have a logical order. Students are capable of and should be allowed to explore this regularity and make their own sense of mathematics.

Your teaching of mathematics is controlled by what factor below?

Personal theory and beliefs.

Vygotsky believed that learning was better achieved through social interaction. What statement best identifies a learning environment that represents this belief?

Students are encouraged to work in groups and share problem solving strategies and solutions.

Identify the statement below that would represent a constructivist approach to a problem solving activity.

Students are given resources that they can watch, touch and listen in order to build new understandings.

What does it mean to be mathematically proficient? Identify the statement below that is true of students becoming mathematically proficient.

The student will become mathematical proficient by following daily expectations for doing mathematics.

Complete this statement, "Classrooms where students are making sense of mathematics do not happen by accident they happen because..."

The teacher has practices and expectations that foster risk taking, reasoning and sharing.

Classroom culture influences the individual learning of students. What statement is an example of how a teacher can honor diversity?

Value student ideas and approaches.

The standards for mathematical proficiency state that we should want students to not only know the concepts but also to how to use them to problem solve. What statement below reflects how a proficient mathematical student might think?

When I complete a problem I wonder if there are other answers that could be right.

A mathematically proficient student would approach a challenging problem solving task with a certain disposition. Describe at least two examples of what that disposition would look and sound like in a classroom.

1. Students asking questions not only of the teacher but each other 2. Students trying multiple approaches when problem solving 3. Students recognizing errors and continuing to try other methods 4. Students that can communicate the how and why their answer makes sense 5. Students that believe learning mathematics is worthwhile and useful

The focus of connecting the dots between theory and practice require teachers to focus in opportunities. All of the statements below are true EXCEPT:

Design tasks that will require students to use only the standard algorithm.

Doing mathematics begins with posing worthwhile tasks. Which verbs align with activities that lead to higher-level thinking?

Investigate, construct and formulate.

All of the following statements regarding teaching for mathematical proficiency are true EXCEPT:

It takes much less effort and time than teaching traditionally.

Making connections among mathematical relationships improves student conceptual understanding. What statement is a tenet of this belief?

Teacher scaffolds new content through the use of tools and peer assistance.