Math 391 Test 1

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

"Explain how you solved the problem."

Problem solving provides opportunities for students to demonstrate mathematical proficiency. Identify two ways that teaching through problem solving works for all students.

-Approaches mathematical learning through inquiry -Explore real contexts, problems, situations, and models -Learning through doing shifts the focus on the students -Problems have multiple entry and exit points -Links to other disciplines

When determining the mathematics content and learning goals for a lesson, the teacher should do all of the following:

-Consult his or her state's curriculum standards -Ask, "What is it my students should be able to do at the end of this lesson?" -Be sure to focus on the mathematics, rather than just the activity.

This chapter pointed out the difference between drill and practice. What do these differences look like in a classroom?

-Drill is used to increase facility of an already learned procedure -Drill is used to help the retention of facts and procedures -Practice is used to develop alternative and flexible strategies -Practice can deepen conceptual knowledge through more useful connections.

Identify three strategies that teachers can use to build strong rapport with all families to support the learning of mathematics.

-Honor different strategies for doing mathematics -Communicate with positive notes and phone calls -Host informal gatherings to discuss mathematics teaching and learning -Incorporate homework that involves the family -Translate letters that are sent home -Put homework on your webpage

What are two advantages of flexible grouping related to differentiation of instruction?

-Increase the chance to communicate about mathematics -build conceptual understanding -Improves confidence -pairs students with varied needs effectively -builds collaboration skills -builds accountability

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

-It requires students to memorize less -It allows students to more easily make connections to new concepts -It increases student enjoyment and attitudes towards mathematics.

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".

-Models and compares possible outcomes -Models the relationship between frequency and all possible outcomes -Creates relative frequencies by observing the sections of the spinner -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.

-Persistence, effort, and concentration are valued -Sharing of ideas among students -Students listen to each other -Students look for and discuss connections

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

-Plan and design instruction based on prior learning -Designate time for student reflection -Plan tasks that reflect the social and cultural makeup of the classroom.

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.

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

How we do mathematics is culturally determined. For example, in the US recording each step is valued. Three of the questions below address culturally relevant instruction:

-Will you require students to show steps disregarding the way they learned? -Will you ask students to elaborate on how they did it? -Will you have students show other students their way?

What are two ways that learning progressions can inform instructional planning?

-informing teachers of what came before -what to expect next as students reach key points -identity interim goals along the path to learning mathematical concepts -inform the order of instructional experiences

Which of the following is an instructional example of teaching through problem solving?

After students have conceptual understanding of the area of a rectangle, asking them to find the area of a triangle that was constructed by cutting a given rectangle in half and then to generalize their process to how they might find the area of any given triangle.

The NCTM position statement on Access and Equity in Mathematics states that we should hold true which statement below?

All students should be held to high levels of performance regardless of race, ethnicity, gender, and socioeconomic status and linguistic background.

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

Allow students to make engage in "productive struggle".

Drill and practice is present to some degree in every classroom. The definitions of drill and practice demonstrates how they are different and how they link. One of the statements below is representative of practice more than drill.

An increased opportunity to develop conceptual ideas.

Three of the reflective questions focus on content relevant culturally response mathematics instruction. Which one is about valuing students' identities?

Are individual student approaches presented and showcased?

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?"

Selecting, designing, or adapting a worthwhile task requires that teacher to ask what important question?

Does this task have potential for students to demonstrate Mathematical Practices?

Complete this statement, "To make mathematics instruction more culturally relevant, teachers should..."

incorporate students' identities

Complete this statement, "Parental involvement in students' mathematical development..."

Can be positive when parents provide their children with a quiet workplace and rules about homework completion.

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

There are research proven strategies that can support students in tier 2. Identify the targeted instruction below that models the strategy, uses self-instructive prompts for student to use through the model.

Explicit Instruction

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.

What are examples of effective assessment techniques that will guide your understanding of student knowledge?

Feedback, on-going observation and continuous data gathering.

Some gender-specific issues that continue to be present in the classroom include which of the following?

Female teachers with math anxiety positively influencing female math achievement.

Asking effective questions requires planning and teachers become more adept at asking quality questions throughout their careers. What common pattern of questioning fosters a greater chance of classroom discussion?

Focusing

Teaching through problem solving provides opportunities for all students to become mathematically proficient. Teaching through problem solving benefits all students in what way?

Focusing on the single strategy needed to solve most problems. *Focusing students on ideas and sense making.

What is the purpose of establishing clear expectations for a lesson before the lesson transpires?

Guide students on the products they need to prepare.

Which statement best reflects the approach of teaching for problem solving?

It frequently results in the instructor explaining a skill and providing practice and application of the skill

Teachers must provide a successful mathematics instruction for all students and address the needs of students' learning disabilities. All but one of the statements below is an appropriate strategy for students with identified learning disabilities.

Make content relevant

What is the most critical aspect of the after phase of the lesson?

Make the main ideas of the lesson visible to others through drawings, notations, and writing.

The focus of an inquiry lesson should start with what method?

Mathematical content and Mathematical practices.

Which statement below best describes the idea of mathematics as engaging in the science 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.

To provide an educational environment that ensures equity for boys and girls a teacher must do what of the following?

Opportunities to act out or model mathematical situations.

Your teaching of mathematics is controlled by what factor below?

Personal theory and beliefs

Which statement below best represents worthwhile features of tasks or problems for learning mathematics?

Problematic, concepts and/or misconceptions, relevant

Common features across RTI tiers include all of the following. Identify the statement that is related to instructions.

Prompts, cues, and environmental arrangements.

Researchers have found that teachers benefit from using an intentional selection process to determine whether a task they chose is worthwhile. One of the criteria of the selection guide is "worthwhile features". Which statement below represents one of the categories in "worthwhile features"?

Relevant context

Tiered lessons and Parallel Tasks are examples of strategies that teachers can use to differentiate mathematical instruction. Which problem below would provide students with choice?

Represent one of numbers in the list at least four different ways.

Which statement about the teaching through problem solving approach is most accurate?

Requires a four-step approach to problem solving.

Learning center and short tasks can be adapted to use the three-phase lesson design. Identify the statement below that would be part of the after portion of a short task.

Small groups share and compare strategies.

There are four basic categories for adapting mathematics content for gifted students. Identify the category that a greater depth is pursued through connecting mathematics to other subject areas.

Sophistication

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 statement reflects the Connections standard of the Five Process Standards from Principles and Standards for School Mathematics?

Students should be given opportunities to see how various mathematical ideas relate to one another and to a variety of real-life experiences.

Parents may be concerned when their child struggles with a single mathematics problem. Often parents see "fast" as successful. What is a reason that fast does not mean smarter?

Students should have productive struggle to help them with making connections.

The goal of equity is to offer all students access to important mathematics during the regular mathematics instruction in the classroom. Which group listed below would need a modification versus an accommodation?

Students who are English Language Learners.

The act of writing is a reflective process and involves students in metacognition. What statement is an example of a students' conscious monitoring of how and why they are doing something?

Taking a timed multiplication test. *Looking back at problems previously worked incorrectly to examine the mistakes.

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.

What are the main tenets of the Curriculum principle in the NCTM Principles to Action?

The tenets of the Curriculum Principle include a coherent curriculum build around big ideas in daily classroom instruction. The big ideas are links from one idea to another and help students see the integration and growth over grades versus an isolated series of bits and pieces.

Drill of skills has a place in helping students improve procedural knowledge but not conceptual knowledge. Before committing to a solution of "just drill" what do you need to know?

The type of drill that will build understanding

Children's literature can be a method of creating tasks with context students can connect to other subjects. What statement below demonstrates how a teacher made a connection?

Today we are going to find out how numbers double with a Chinese magical pot.

Selecting problem solving tasks that require higher levels of cognitive demand should include which one of the following?

Use of complex and non-algorithmic thinking.

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.

Problem solving tasks that have multiple entry and exit points exhibit what characteristics below?

Varying degrees of challenge and methods to approach a solution.

Identify the statement that completes this statement, "Learning progressions reflect current research and practice and they guide teachers in understanding..."

What came before as well as what to expect next.

Effective use of classroom discussion has the teacher monitoring the students' strategies and asking questions. What question below would require the student to reflect on their specific strategy?

What mathematical terms, symbols, or tools did you use to find a solution?

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 another answers that could be right.

Flexible grouping means that the size and makeup of the small group vary in a purposeful and strategic manner. Below are examples of considerations teachers must make in putting together mixed-ability groups. What statement reflects the shared responsibility?

Whether groups members participate in team building activities to set the standard for collaboration.

Asking quality questions are critically important to the potential learning. What question below would solicit higher-level responses from students?

Which rectangle would an architect prefer for a dance stage rather than a play stage?

Teachers can prompt productive classroom discussion through the use of five teacher talk actions. What statement below reflects the type of prompt to elicit student reasoning?

Who also used similar strategy as Ricardo?

What statement below is an example of an accommodation?

Write directions in larger font.

There are stumbling blocks for some students as they are developing conceptual and procedural mathematical knowledge and practices. Identify the statement below that might be a student that is struggling with organizational skills and the ability to self-regulate.

Writes computations in a random way and hard to follow.

The National Assessment of Educational Progress...

reported in 2013 that less than half of all 4th and 8th graders performed on a standardized mathematics test as the desirable levels of "proficient" or "advanced".

What statement represents how a teacher listens without evaluation?

Do others agree with Caroline's strategy?

The Standards for Mathematical Practice are...

A component of the Common Core State Standards that outlines a series of important mathematical processes all students should be able to do regularly.

The following statements represent the "thinking" behind the creation of the Common Core Standards for School Mathematics. Which statement is also true of the NCTM Principle and Standards for School Mathematics?

A focused set of Mathematics content standards and practices.

What statement below is the best description of what is meant by the term RTI?

A multi-tiered student support model that focuses on prevention, providing student with support before they begin to fail.

Identify the statement below that best represents students who are mathematically gifted.

A student with a passion for mathematics even though they have to work hard to learn.

The during phase of the lesson the teacher has to ask this question when determining what students need.

Does my action lead to deeper-thinking or is taking away the thinking?

Students with identified learning disabilities have specific challenges with perceptual and cognitive processing. Three of the statements below relate to the support needed related to the environment. Identify the one that is about clarity, not environment.

Adapt delivery mode-materials, images, and examples

A three-phase lesson format provides a structure for students to have inquiry on a topic, engage in the content through action and discussion and time to reflect and make connections. What statement below demonstrates the before related agendas?

Be sure the task is understood

When facilitating engagement for ELL's in mathematics activities and discussion teachers should include what method?

Build background on prior knowledge

One of the statements below represents a method for developing the Reasoning and proof process standard.

Create opportunities for students to evaluate conjectures.

One of the statements below is a strategy that should be employed with gifted students to accelerate mathematics content.

Curriculum compacting

The mathematical needs in society have changed and are influencing what should be taught in pre K-8 mathematics classrooms. What statement below reflects a key factor in the change?

Data on performance of U.S. students in national and international studies.

What statement below would be a method of activating prior knowledge?

Design a warm up to use the vocabulary needed for the task.

How much to tell or not tell is a constant dilemma for teachers. What statement would be the type of information that teachers do need to tell?

Help students clarify their ideas and point out related ideas

What is an example of the type of assessment question that is used in the during phase of the lesson?

How did you figure out the area of that rectangle?

What statement below can inadvertently shut down student thinking and possibly affect their self-esteem?

How did you solve the problem?

Differentiation of instruction in mathematics lessons is needed to support and challenge the learning for all students. One statement below is an example of a differentiating process:

Identify what thinking will engage students in the mathematical content.

Teachers may be guilty of setting lower expectations for students when they suggest that the group is too unruly to work together or they don't have the reading skills to do problem solving. What "gap" does this represent?

Instructional gap

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

Investigate, construct, and formulate

21st Century teachers need to support students and parents in understanding the way mathematics is taught and learned. Identify the question below that a teacher should address about their pedagogy before it is asked by a parent.

Is my child doing their own work when they are in groups?

Planning a lesson for a classroom where inquiry and problem solving are emphasizing requires a shift in the type of lessons being used. What statement below represents this shift in the agenda of a lesson?

It should be a "thought experiment" to consider what might happen.

The teacher plays an important role in establishing a positive classroom environment. A major role is establishing and distributing power. Three of the statements below demonstrate teacher power. Identify the one that is about student power.

Justifying the correctness of solutions.

Which choice reflects two factors that influence the teaching of Mathematics effectively?

Knowledge of standards and practices.

What is true about the learning principle?

Learning is strongly enhanced when students are encouraged to make and test their own mathematical conjectures.

Which statement below captures the big idea from one of the Eight Guiding Principles outlines in the NCTM Principles to Action?

Look for both horizontal and vertical alignment to build coherence.

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.

Which is the most accurate statement regarding posing a worthwhile problem?

Teachers should select the problems that will help make relationships between mathematical concepts explicit for students.

As a practicing or prospective teacher each face the challenge of teaching mathematics from a problem solving perspective. This often requires some to confront personal beliefs about:

Teaching conceptual understanding and procedural fluency.

Facilitating classroom discourse requires that the teacher be aware of what factors that will affect the level of mathematical talk in the classroom. What statement below would impact the mathematical talk?

The level of English proficiency of the students in the classroom.

The six Principles and Standards for School Mathematics articulate high-quality mathematics education. What statement below represents the equity principle?

The message of high expectations for all is intertwined with every other principle.