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Chapter 2: Exploring What It Means to Know and Do Mathematics
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
1) Which of the following is an example of a statement spoken in the language of doing
mathematics?
A) “Memorize these steps.”
B) “Compute this answer.”
C) “Explain how you solved the problem.”
D) “Copy down these steps into your notebooks.”
2) Which statement below best describes the idea of mathematics as engaging in the science of
pattern and order?
A) 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.
B) Students develop conceptual understanding and become less confused when they are given a
specific set of logical, orderly steps to solve each type of math problem.
C) Students best acquire efficient methods for computing with timed drills.
D) To avoid confusion and be successful in all mathematics, students must develop conceptual
understanding of topics in a very specific order.
3) Doing mathematics begins with posing worthwhile tasks. Which verbs align with activities
that lead to higher-level thinking?
A) Memorize, drill and practice.
B) Investigate, construct and formulate.
C) Listen, copy and compute.
D) Recall, application and calculate.
4) To set up an environment for “doing” mathematics, teachers need to
A) Develop and demonstrate rules.
B) Efficiently manage time and materials.
C) Quickly provide corrected answers, so students are not embarrassed by mistakes.
D) Allow students to make engage in “productive struggle.
5) 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?
A) Generalizing relationships.
B) Experimenting and explaining.
C) Search for a pattern.
D) Analyzing a situation.
6) Which of the following represents an example of an effective way a teacher might help
facilitate students’ construction of mathematical relationships?
A) Asking students, “How is today’s topic related to the fraction multiplication we investigated
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last week?”
B) Prompting a student with a hint as soon as he pauses while offering an explanation of a
process.
C) Conveying the message to families and students that, as long as the concepts are taught
correctly, math is usually easy.
D) Never intervening in students’ construction of meaning. The process is much more effective if
they are just left alone.
7) What does it mean to be mathematically proficient? Identify the statement below that is true of
students becoming mathematically proficient.
A) The student will need to be mathematically proficient to graduate.
B) The student will begin the process of mathematical proficiency in high school.
C) The student will become mathematical proficient by following daily expectations for doing
mathematics.
D) The student becomes mathematically proficient by following the directives of just the
mathematics teacher.
8) 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?
A) I consider the traditional algorithm my go to solution strategy.
B) When I encounter a wrong path I generally skip that problem.
C) I always look for shortcuts and never consider alternative methods.
D) When I complete a problem I wonder if there are other answers that could be right.
9) 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.”
A) Class activities and lessons should be designed with students’ prior experiences in mind.
B) Students’ mistakes should be minimized in order to build their confidence and enjoyment in
the mathematics classroom.
C) New concepts should only be presented to students through teacher-centered presentations, in
order to help them build the necessary background knowledge.
D) Most students learn best in a quiet class that consists primarily of direct-instruction, so that
they can focus and won’t be distracted by others.
10) Which statement best represents a method to expose students to multiple approaches to
problem solving?
A) Students should see a variety of inferior methods so that they can better appreciate the one
best method to solve a problem.
B) Class discussions lead to confusion for students and inhibit alternative problem-solving
strategies.
C) Multiple strategies are not very useful for simple mathematical ideas, such as basic
computation facts.
D) Exposure to multiple approaches and the subsequent connections help students to recall the
steps to complete mathematical processes.
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11) Your teaching of mathematics is controlled by what factor below?
A) Behaviorism theory.
B) Personal theory and beliefs.
C) Cognitivism and constructivism.
D) Sociocultural theory.
12) Identify the statement below that would represent a constructivist approach to a problem
solving activity.
A) Students are given resources that they can watch, touch and listen in order to build new
understandings.
B) Students are given the strategy and procedure to follow to connect to their prior knowledge.
C) Students are directed to follow a clear path to problem solve and develop new understanding.
D) Students are given a specific time frame to complete the problem and a correct solution is
given at the end of the time period.
13) Vygotsky believed that learning was better achieved through social interaction. What
statement best identifies a learning environment that represents this belief?
A) Students are seated in rows and given problems to solve in a quiet environment.
B) Students are encouraged to work in groups and share problem solving strategies and
solutions.
C) Students are assigned problems as homework and directed not to talk with other classmates.
D) Students are not expected to solve all of the problems, only the ones that they have the
knowledge to complete.
14) What theory(s) allow a classroom culture to access prior knowledge, use tools to build
knowledge?
A) Zone of proximal development (ZPD).
B) Sociobehaviorist.
C) Constructivist.
D) Sociocognivist.
15) Making connections among mathematical relationships improves student conceptual
understanding. What statement is a tenet of this belief?
A) Teacher requires a single strategy solution.
B) Teacher does not want students to struggle in mathematics.
C) Teacher provides solutions when students make errors.
D) Teacher scaffolds new content through the use of tools and peer assistance.
16) Classroom culture influences the individual learning of students. What statement is an
example of how a teacher can honor diversity?
A) Value student ideas and approaches.
B) Value only your method or approach.
C) Value traditional strategies over inventive strategies.
D) Value the teacher “telling” approach.
17) Complete this statement, “Classrooms where students are making sense of mathematics do
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not happen by accident they happen because…”
A) The administration requires this of their teachers.
B) The parents set high expectations for their child’s learning.
C) The teacher has practices and expectations that foster risk taking, reasoning and sharing.
D) The students expect a classroom where they have control over their own learning.
18) The focus of connecting the dots between theory and practice require teachers to focus in
opportunities. All of the statements below are true EXCEPT:
A) Plan and design instruction based on prior learning.
B) Designate time for student reflection.
C) Plan tasks that reflect the social and cultural makeup of the classroom.
D) Design tasks that will require students to use only the standard algorithm.
19) 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?
A) Distributing materials with guidance on how to use them to construct models.
B) Demonstrating at least one connection between the model and the mathematical concept.
C) Encouraging students to converse about the model without knowledge of what the
mathematical goal they are working on.
D) Maintaining a balance between the appropriate amounts of guidance and student exploration.
20) All of the following statements regarding teaching for mathematical proficiency are true
EXCEPT:
A) It requires students to memorize less.
B) It allows students to more easily make connections to new concepts.
C) It increases student enjoyment and attitudes towards mathematics.
D) It takes much less effort and time than teaching traditionally.
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
21) 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.
22) 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.
23) 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.”
