Math 391 Exam 1 - Questions With Solutions
Teaching for problem solving ✔️Ans - Frequently results in the
instructor explaining a skill and providing practice and application of the
skill.
Teaching through problem solving ✔️Ans - Requires a four-step
approach to problem solving.
Selecting problem solving tasks that require higher levels of cognitive
demand should include what? ✔️Ans - Use of complex and non-
algorithmic thinking.
What are the characteristics of problem solving tasks that have multiple
entry and exit points? ✔️Ans - Varying degrees of challenge and
methods to approach a solution.
What are worthwhile features of tasks or problems for learning
mathematics? ✔️Ans - Problematic, concepts and/or misconceptions,
relevant.
What is something teachers should do when posing a worthwhile problem?
✔️Ans - Teachers should select problems that will help make relationships
between mathematical concepts explicit for students.
Instructional example of teaching through problem solving ✔️Ans -
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"? ✔️Ans - Relevant
context.
Statement that is representative of practice more than drill ✔️Ans - An
increased opportunity to develop conceptual ideas.
,What do you need to know before committing to a solution of "just drill"?
✔️Ans - The type of drill that will build understanding.
What is one factor that would impact mathematical talk? ✔️Ans - The
level of English proficiency of the students in the classroom.
A question that would require the student to reflect on their specific
strategy ✔️Ans - What did you do to make sense of the problem?
Type of prompt to elicit student reasoning ✔️Ans - "Who also used a
similar strategy"
Type of information that teachers do need to tell ✔️Ans - 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 ✔️Ans - Looking back at problems previously worked
incorrectly to examine the mistakes.
Teaching through problem solving benefits all students in what way?
✔️Ans - Focusing students on ideas and sense making.
What common pattern of questioning fosters a greater chance of classroom
discussion? ✔️Ans - 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? ✔️Ans - The mathematical processes through which
students should acquire and use mathematical knowledge.
The five process standards ✔️Ans - 1) Problem Solving, 2) Reasoning
and Proof, 3) Communication, 4) Connections, 5) Representation.
Problem Solving ✔️Ans - the vehicle through which students develop
mathematical ideas.
Reasoning and Proof ✔️Ans - emphasize the logical thinking that helps
us decide if and why our answers make sense.
, Communication ✔️Ans - being able to talk about, write about, describe,
and explain mathematical ideas.
Connections ✔️Ans - connect within and among mathematical ideas,
and connect to the real world and other disciplines.
Representation ✔️Ans - the use of symbols, charts, graphs,
manipulatives, and diagrams as powerful methods of expressing
mathematical ideas and relationships.
Standards for Mathematical Practice ✔️Ans - 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? ✔️Ans - 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"? ✔️Ans - 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? ✔️Ans -
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)? ✔️Ans - 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 for problem solving ✔️Ans - Frequently results in the
instructor explaining a skill and providing practice and application of the
skill.
Teaching through problem solving ✔️Ans - Requires a four-step
approach to problem solving.
Selecting problem solving tasks that require higher levels of cognitive
demand should include what? ✔️Ans - Use of complex and non-
algorithmic thinking.
What are the characteristics of problem solving tasks that have multiple
entry and exit points? ✔️Ans - Varying degrees of challenge and
methods to approach a solution.
What are worthwhile features of tasks or problems for learning
mathematics? ✔️Ans - Problematic, concepts and/or misconceptions,
relevant.
What is something teachers should do when posing a worthwhile problem?
✔️Ans - Teachers should select problems that will help make relationships
between mathematical concepts explicit for students.
Instructional example of teaching through problem solving ✔️Ans -
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"? ✔️Ans - Relevant
context.
Statement that is representative of practice more than drill ✔️Ans - An
increased opportunity to develop conceptual ideas.
,What do you need to know before committing to a solution of "just drill"?
✔️Ans - The type of drill that will build understanding.
What is one factor that would impact mathematical talk? ✔️Ans - The
level of English proficiency of the students in the classroom.
A question that would require the student to reflect on their specific
strategy ✔️Ans - What did you do to make sense of the problem?
Type of prompt to elicit student reasoning ✔️Ans - "Who also used a
similar strategy"
Type of information that teachers do need to tell ✔️Ans - 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 ✔️Ans - Looking back at problems previously worked
incorrectly to examine the mistakes.
Teaching through problem solving benefits all students in what way?
✔️Ans - Focusing students on ideas and sense making.
What common pattern of questioning fosters a greater chance of classroom
discussion? ✔️Ans - 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? ✔️Ans - The mathematical processes through which
students should acquire and use mathematical knowledge.
The five process standards ✔️Ans - 1) Problem Solving, 2) Reasoning
and Proof, 3) Communication, 4) Connections, 5) Representation.
Problem Solving ✔️Ans - the vehicle through which students develop
mathematical ideas.
Reasoning and Proof ✔️Ans - emphasize the logical thinking that helps
us decide if and why our answers make sense.
, Communication ✔️Ans - being able to talk about, write about, describe,
and explain mathematical ideas.
Connections ✔️Ans - connect within and among mathematical ideas,
and connect to the real world and other disciplines.
Representation ✔️Ans - the use of symbols, charts, graphs,
manipulatives, and diagrams as powerful methods of expressing
mathematical ideas and relationships.
Standards for Mathematical Practice ✔️Ans - 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? ✔️Ans - 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"? ✔️Ans - 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? ✔️Ans -
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)? ✔️Ans - 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).
