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Chapter 3: Teaching Through Problem Solving
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
1) Which statement about the teaching for problem solving approach is not accurate?
a) It is the method most frequently used in traditional textbooks.
b) It usually involves students learning an algorithm and then being asked to apply the algorithm to a
story problem.
c) The approach has traditionally been a very effective way to help students gain conceptual
understanding.
d) It frequently results in the likelihood that a student won’t attempt a new problem without explicit
instructions on how to solve it.
2) Which statement about the teaching through problem solving approach is most accurate?
a) It is a less demanding process for the teacher.
b) The productive struggle that it frequently provides to students leads to enhanced learning.
c) Any problem-solving task will help students to develop their own strategies and learn the content.
d) It is pretty much the same process as teaching for problem solving.
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
3) Explicitly teaching students the four-step problem solving process does not usually improve their ability
to reason mathematically. It is better to let them discover the process on their own.
4) When teaching students about problem-solving strategies, it is best not to explicitly tell students which
strategy to pick and how to use it.
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
5) One commonly used problem-solving strategy used with story problems is writing an
_______________________________.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
6) Which is the most accurate statement regarding posing a problem?
a) Teachers should select the problems that will help make relationships between mathematical
concepts explicit for students.
b) Any problem used should have the potential to be solved by students using a memorized
procedure.
c) There should be agreement between students that there is one correct answer.
d) Problems should involve words.
7) All of the following are features a problem should have EXCEPT
a) Its design must take into consideration the students’ current level of understanding.
b) It should have an engaging context.
c) It must require some kind of justification for methods and answers.
d) It must be at a level that would make a student unable to solve it alone.
8) Which of the following is an example of teaching through problem solving?
a) Providing students with a list of area formulas and asking them to find the area of a given
rectangle
b) 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
c) Teaching students the algorithm for fraction division and then asking them to find out how many
servings of 1/3 pizza could be made from 3-2/3 pizzas
d) Having students develop their own word problems that use a recently learned algorithm
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
9) Characteristics of a worthwhile problem include the level of cognitive demand, the potential of the task
to have ______________________________ entry and exit points, and whether the task is relevant to
students.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
10) Which of the following is the weakest example of providing students with relevant contexts for
problems?
a) Using a piece of children’s literature to introduce a situation that requires some kind of solution
b) Asking students to reduce the amount of each ingredient that is needed for a recipe that is divided
in half to accommodate a small family
c) Asking a student to explain how he chose to divide the numerator and denominator of a fraction
by 2 in order to simplify it
d) Asking students to determine, given the area and population of two countries, which would be less
crowded
11) Traditional textbooks
a) Have been designed using the teaching for problem solving model.
b) Should be avoided, unless they are required by your district.
c) Are often not being designed around the teaching through problem solving model.
d) Rarely contain problems that can be adapted to the teaching through problem solving model.
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
12) Facilitating classroom discourse is easy for the teacher, because the students are doing all the talking.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
13) Examples of effective use of classroom discussion include all of the following EXCEPT
a) Giving students with disabilities the opportunity to leave the room for certain interventions
without missing valuable new mathematics content.
b) Requiring students to use mathematical vocabulary to express their ideas and take ownership of
making sense of mathematics.
c) Providing students with limited language skills the opportunity to articulate their thinking.
d) Allowing students the opportunity to discover multiple approaches to solving problems.
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
14) Provide two specific examples of research-based recommendations that contribute to effective classroom
discourse and an example of what you might say or do to implement each recommendation.
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
15) The impact of questioning on student learning is determined by the level of the question, the type of
knowledge targeted, the pattern of questioning, and how you ________________________.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
16) Which of the following is the best example of metacognition?
a) Taking a timed multiplication test
b) Taking two numbers in a word problem and adding them because your class worked on addition
problems the previous day
c) When a student looks back at problems he previously worked incorrectly to examine his mistakes
d) When two students are playing a game of Integer War, during which they each flip a card and the
student who calls out the higher of the two numbers first wins the round
17) When determining how much to tell, researchers suggest that teachers tell students all of the following
EXCEPT
a) Certain mathematical conventions, such as the meaning behind some symbols
b) Alternative strategies that don’t emerge naturally from students
c) The answer to any problem that has required more than 10 minutes of students’ time to solve
d) Any needed clarification of students’ methods
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
18) Describe at least 4 ways to use writing in the mathematics classroom.
19) Describe at least 4 ways that teaching through problem solving helps ensure equity in the classroom.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
20) In the before phase of a lesson, the teacher has as one of her goals
a) Showing students a worked example similar to what they will be doing.
b) Telling them the problem solving strategies that would be best for the task.
c) Answering every question the students might have about the task.
d) Establishing clear expectations.
21) In the during phase of the lesson, teachers must do all but which of the following?
a) Avoid the urge to provide too much assistance to students
b) Leave students completely alone to explore their thinking
c) Notice and record students’ processes
d) Provide any necessary extensions
