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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 best reflects the approach of teaching for problem solving?
A) It is the method rarely used in traditional textbooks.
B) It usually involves students exploring alternative methods of solving a story problem.
C) It frequently results in the instructor explaining a skill and providing practice and application
of the skill.
D) The approach has traditionally been a very effective way to help students gain conceptual
understanding.
2) Which statement about the teaching through problem solving approach is most accurate?
A) Requires a four-step approach to problem solving.
B) Rarely requires students to use alternative methods for problem solving.
C) Can only be used with certain problem solving tasks.
D) It is closely associated with teaching for problem solving.
3) Selecting problem solving tasks that require higher levels of cognitive demand should include
which one of the following?
A) Use of a specific procedure.
B) Are routine and students can use previously learned procedure.
C) Use of complex and non-algorithmic thinking.
D) Are straightforward with little ambiguity of what needs to be done to solve.
4) Problem solving tasks that have multiple entry and exit points exhibit what characteristics
below?
A) Are focused on producing correct answers.
B) Varying degrees of challenge and methods to approach a solution.
C) Are best done independently to address diverse learning styles.
D) Varying levels of written explanation are required.
5) 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?
A) Today we are going to learn about doubling numbers.
B) Today we are going to compute using doubles.
C) Today we are going to solve problems with numbers that double.
D) Today we are going to find out how numbers double with a Chinese magical pot.
6) Which is the most accurate statement regarding posing a worthwhile 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.
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D) Problems should involve words.
7) Which statement below best represents worthwhile features of tasks or problems for learning
mathematics?
A) Problematic, memorization, no connection.
B) Problematic, concepts and or misconceptions, relevant.
C) Problematic, require correct answer, rules and formulas.
D) Problematic, algorithmic, routine.
8) Which of the following is an instructional 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) Teaching students the algorithm for fraction division and then asking them to find out how
many servings of pizza could be made from 3 pizzas.
C) Having students develop their own word problems that use a recently learned algorithm.
D) 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.
9) 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”?
A) Relevant context.
B) Problematic task.
C) Requires tools and models to represent the mathematics.
D) Mathematically interesting.
10) Drill and practice is present to some degree in every classroom. The definitions of drill and
practice demonstrate how they are different and how they link. One of the statements below is
representative of practice more than drill.
A) An increased facility with a procedure.
B) A focus on a singular method.
C) An increased opportunity to develop conceptual ideas.
D) A focus on rule oriented view of mathematics.
11) 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?
A) The amount of time the student has put in on learning basic facts.
B) The type of drill that will build understanding.
C) The student’s attitude toward mathematics learning.
D) The level of procedural knowledge the student has demonstrated.
12) Facilitating classroom discourse requires that the teacher be aware of what factors that will
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affect the level of mathematical talk in the classroom. What statement below would impact the
mathematical talk?
A) The level of English proficiency of the students in the classroom.
B) The depth of mathematical concepts required to meet the curriculum guidelines.
C) The level of teacher knowledge of the mathematical concepts.
D) The depth of the student questioning skills.
13) 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?
A) What did you do to make sense of the problem?
B) What numbers or information in the problem did you not need to use?
C) What mathematical terms, symbols or tools did you use to find a solution?
D) What other strategies did your classmates use to solve the problem?
14) 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?
A) You used a hundreds chart.
B) Ricardo said that he used a draw a picture strategy.
C) Who also used similar strategy as Ricardo?
D) Let’s take some time to think about the strategies.
15) 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?
A) The level of cognitive demand that the students need to demonstrate.
B) Specific problem solving strategies needed to solve the problem.
C) Information needed to keep students from struggling to solve the problem.
D) Help students clarify their ideas and point out related ideas.
16) 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?
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) Looking back at problems previously worked incorrectly to examine the mistakes.
D) 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) Teaching through problem solving provides opportunities for all students to become
mathematically proficient. Teaching through problem solving benefits all students in what way?
A) Focusing students on ideas and sense making.
B) Focusing on the single strategy need to solve most problems.
C) Focusing on procedures that will effectively find answers.
D) Focusing on the technology that will guide them to solutions.
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18) 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?
A) Funneling.
B) Focusing.
C) Initiating, responding, feedback.
D) Correcting.
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
19) Problem solving provides opportunities for students to demonstrate mathematical
proficiency. Identify two ways that teaching through problem solving works for all students.
20) This chapter pointed out the difference between drill and practice. What do these differences
look like in a classroom?
