Math 391 Exam 1
Teaching for problem solving Correct Answer: Frequently results in the instructor explaining a skill and
providing practice and application of the skill.
Teaching through problem solving Correct Answer: Requires a four-step approach to problem solving.
Selecting problem solving tasks that require higher levels of cognitive demand should include what?
Correct Answer: Use of complex and non-algorithmic thinking.
What are the characteristics of problem solving tasks that have multiple entry and exit points? Correct
Answer: Varying degrees of challenge and methods to approach a solution.
What are worthwhile features of tasks or problems for learning mathematics? Correct Answer:
Problematic, concepts and/or misconceptions, relevant.
What is something teachers should do when posing a worthwhile problem? Correct Answer: Teachers
should select problems that will help make relationships between mathematical concepts explicit for
students.
Instructional example of teaching through problem solving Correct Answer: 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.
What is one category of "worthwhile features"? Correct Answer: Relevant context.
Statement that is representative of practice more than drill Correct Answer: An increased opportunity
to develop conceptual ideas.
What do you need to know before committing to a solution of "just drill"? Correct Answer: The type of
drill that will build understanding.
What is one factor that would impact mathematical talk? Correct Answer: The level of English
proficiency of the students in the classroom.
A question that would require the student to reflect on their specific strategy Correct Answer: What did
you do to make sense of the problem?
Type of prompt to elicit student reasoning Correct Answer: "Who also used a similar strategy"
Type of information that teachers do need to tell Correct Answer: Help students clarify their ideas and
point out related ideas.
,Example of a student's conscious monitoring of how and why they are doing something Correct Answer:
Looking back at problems previously worked incorrectly to examine the mistakes.
Teaching through problem solving benefits all students in what way? Correct Answer: Focusing students
on ideas and sense making.
What common pattern of questioning fosters a greater chance of classroom discussion? Correct Answer:
Focusing - uses probing questions to negotiate a classroom discussion and help students understand the
mathematics.
What is meant by a process as referred to in the Principles and Standards process standards? Correct
Answer: The mathematical processes through which students should acquire and use mathematical
knowledge.
The five process standards Correct Answer: 1) Problem Solving, 2) Reasoning and Proof, 3)
Communication, 4) Connections, 5) Representation.
Problem Solving Correct Answer: the vehicle through which students develop mathematical ideas.
Reasoning and Proof Correct Answer: emphasize the logical thinking that helps us decide if and why our
answers make sense.
Communication Correct Answer: being able to talk about, write about, describe, and explain
mathematical ideas.
Connections Correct Answer: connect within and among mathematical ideas, and connect to the real
world and other disciplines.
Representation Correct Answer: the use of symbols, charts, graphs, manipulatives, and diagrams as
powerful methods of expressing mathematical ideas and relationships.
Standards for Mathematical Practice Correct Answer: 1) Make sense of problems and persevere in
solving them; 2) Reason abstractly and quantitatively; 3) Construct viable arguments and critique the
reasoning of others; 4) Model with mathematics; 5) Use appropriate tools strategically; 6) Attend to
precision; 7) Look for and make use of structure; 8) Look for and express regularity in repeated
reasoning.
How do the Standards for Mathematical Practice relate to the Common Core State Standards content
expectations? Correct Answer: They relate to the Common Core State Standards content expectations
in that the Standards for Mathematical Practice need to be met alongside the content expectations.
How would you describe what it means to "do mathematics"? Correct Answer: I would describe what it
means to "do mathematics" as looking at the problem, understanding what it's asking you to do,
developing a strategy to solve that problem, and then checking to see if your answer makes sense.
, What is important to know about relational understanding? Correct Answer: That is means to know
what to do and why, and ways to nurture relational understanding, which are: use and connect different
representations, and explore with tools.
What are the distinctions between the three ways to approach problem solving (teaching for problem
solving, teaching about problem solving, teaching through problem solving)? Correct Answer: Teaching
for problem solving is the teacher presents the mathematics, the students practice the skill, and
students solve story problems that require using that skill (they apply the skill).
Teaching about problem solving is the teacher giving guidance on how to problem solve, which includes
the process of problem solving and learning strategies that can help in solving problems.
Teaching through problem solving is students learning mathematics through inquiry. They explore real
contexts, problems, situations, and models. The problem or task is presented at the beginning of the
lesson and related knowledge or skills emerge from exploring the problem.
What is meant by multiple entry and exit points? Why are they important? Correct Answer: Multiple
entry points means a problem can be approached in a variety of ways and has varying degrees of
challenge within it.
Multiple exit points means various ways to express solutions.
Multiple entry and exit points are important because they accommodate the diversity of learners by
encouraging students to use a variety of strategies that are supported by their prior experiences. They
also reveal a range of mathematical sophistication and have the potential to generate new questions.
What are some important considerations in effectively implementing classroom discourse? Correct
Answer: The "level" of questions asked, the type of understanding that is targeted by questions asked,
the pattern of questioning, who is doing the thinking - questions must hold everyone accountable to
think about the question posed, and how you respond to an answer.
What two factors influence the teaching of Mathematics effectively? Correct Answer: Knowledge of
standards and practices.
The mathematical needs in society have changed and are influencing what should be taught in Pre-K - 8
mathematics classrooms. What is a key factor in the change? Correct Answer: Data on the performance
of U.S. students in national and international studies.
What did the National Assessment of Education Progress (NAEP) report in 2013? Correct Answer: That
less than half of all 4th and 8th grade students performed on a standardized mathematics test at the
desirable levels of "proficient" or "advanced".
Equity principle Correct Answer: High expectations for all. Intertwined with every other principle.
Learning principle Correct Answer: Learning is strongly enhanced when students are encouraged to
make and test their own mathematical conjectures.
Teaching for problem solving Correct Answer: Frequently results in the instructor explaining a skill and
providing practice and application of the skill.
Teaching through problem solving Correct Answer: Requires a four-step approach to problem solving.
Selecting problem solving tasks that require higher levels of cognitive demand should include what?
Correct Answer: Use of complex and non-algorithmic thinking.
What are the characteristics of problem solving tasks that have multiple entry and exit points? Correct
Answer: Varying degrees of challenge and methods to approach a solution.
What are worthwhile features of tasks or problems for learning mathematics? Correct Answer:
Problematic, concepts and/or misconceptions, relevant.
What is something teachers should do when posing a worthwhile problem? Correct Answer: Teachers
should select problems that will help make relationships between mathematical concepts explicit for
students.
Instructional example of teaching through problem solving Correct Answer: 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.
What is one category of "worthwhile features"? Correct Answer: Relevant context.
Statement that is representative of practice more than drill Correct Answer: An increased opportunity
to develop conceptual ideas.
What do you need to know before committing to a solution of "just drill"? Correct Answer: The type of
drill that will build understanding.
What is one factor that would impact mathematical talk? Correct Answer: The level of English
proficiency of the students in the classroom.
A question that would require the student to reflect on their specific strategy Correct Answer: What did
you do to make sense of the problem?
Type of prompt to elicit student reasoning Correct Answer: "Who also used a similar strategy"
Type of information that teachers do need to tell Correct Answer: Help students clarify their ideas and
point out related ideas.
,Example of a student's conscious monitoring of how and why they are doing something Correct Answer:
Looking back at problems previously worked incorrectly to examine the mistakes.
Teaching through problem solving benefits all students in what way? Correct Answer: Focusing students
on ideas and sense making.
What common pattern of questioning fosters a greater chance of classroom discussion? Correct Answer:
Focusing - uses probing questions to negotiate a classroom discussion and help students understand the
mathematics.
What is meant by a process as referred to in the Principles and Standards process standards? Correct
Answer: The mathematical processes through which students should acquire and use mathematical
knowledge.
The five process standards Correct Answer: 1) Problem Solving, 2) Reasoning and Proof, 3)
Communication, 4) Connections, 5) Representation.
Problem Solving Correct Answer: the vehicle through which students develop mathematical ideas.
Reasoning and Proof Correct Answer: emphasize the logical thinking that helps us decide if and why our
answers make sense.
Communication Correct Answer: being able to talk about, write about, describe, and explain
mathematical ideas.
Connections Correct Answer: connect within and among mathematical ideas, and connect to the real
world and other disciplines.
Representation Correct Answer: the use of symbols, charts, graphs, manipulatives, and diagrams as
powerful methods of expressing mathematical ideas and relationships.
Standards for Mathematical Practice Correct Answer: 1) Make sense of problems and persevere in
solving them; 2) Reason abstractly and quantitatively; 3) Construct viable arguments and critique the
reasoning of others; 4) Model with mathematics; 5) Use appropriate tools strategically; 6) Attend to
precision; 7) Look for and make use of structure; 8) Look for and express regularity in repeated
reasoning.
How do the Standards for Mathematical Practice relate to the Common Core State Standards content
expectations? Correct Answer: They relate to the Common Core State Standards content expectations
in that the Standards for Mathematical Practice need to be met alongside the content expectations.
How would you describe what it means to "do mathematics"? Correct Answer: I would describe what it
means to "do mathematics" as looking at the problem, understanding what it's asking you to do,
developing a strategy to solve that problem, and then checking to see if your answer makes sense.
, What is important to know about relational understanding? Correct Answer: That is means to know
what to do and why, and ways to nurture relational understanding, which are: use and connect different
representations, and explore with tools.
What are the distinctions between the three ways to approach problem solving (teaching for problem
solving, teaching about problem solving, teaching through problem solving)? Correct Answer: Teaching
for problem solving is the teacher presents the mathematics, the students practice the skill, and
students solve story problems that require using that skill (they apply the skill).
Teaching about problem solving is the teacher giving guidance on how to problem solve, which includes
the process of problem solving and learning strategies that can help in solving problems.
Teaching through problem solving is students learning mathematics through inquiry. They explore real
contexts, problems, situations, and models. The problem or task is presented at the beginning of the
lesson and related knowledge or skills emerge from exploring the problem.
What is meant by multiple entry and exit points? Why are they important? Correct Answer: Multiple
entry points means a problem can be approached in a variety of ways and has varying degrees of
challenge within it.
Multiple exit points means various ways to express solutions.
Multiple entry and exit points are important because they accommodate the diversity of learners by
encouraging students to use a variety of strategies that are supported by their prior experiences. They
also reveal a range of mathematical sophistication and have the potential to generate new questions.
What are some important considerations in effectively implementing classroom discourse? Correct
Answer: The "level" of questions asked, the type of understanding that is targeted by questions asked,
the pattern of questioning, who is doing the thinking - questions must hold everyone accountable to
think about the question posed, and how you respond to an answer.
What two factors influence the teaching of Mathematics effectively? Correct Answer: Knowledge of
standards and practices.
The mathematical needs in society have changed and are influencing what should be taught in Pre-K - 8
mathematics classrooms. What is a key factor in the change? Correct Answer: Data on the performance
of U.S. students in national and international studies.
What did the National Assessment of Education Progress (NAEP) report in 2013? Correct Answer: That
less than half of all 4th and 8th grade students performed on a standardized mathematics test at the
desirable levels of "proficient" or "advanced".
Equity principle Correct Answer: High expectations for all. Intertwined with every other principle.
Learning principle Correct Answer: Learning is strongly enhanced when students are encouraged to
make and test their own mathematical conjectures.
