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 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.
2) 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.”
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
3) Teacher directions spoken in the language of doing mathematics center around verbs that require lower–
level thinking, such as listen and memorize.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
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.”
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
5) In order to avoid distractions, teachers should minimize talking between students during math
class.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
6) Which of the following represents an example of an effective way a teacher may help facilitate students’
construction of mathematical relationships?
a) Asking students, “How is today’s topic related to the fraction multiplication we investigated 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
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
7) The big idea behind ___________________________ is that learners are creators of their own learning.
8) One of the big ideas behind ___________________________ is that learning occurs through interactions that are
influenced by classroom culture.
9) The _________________________________ refers to a range of knowledge that may be out of
reach for a learner to learn on her own but that can be accessible with support.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
10) One of the educational implications of the learning theory discussed in Chapter 2 of the text
is a) Students’ mistakes should be minimized in order to build their confidence and enjoyment in the
mathematics classroom.
b) 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.
c) Class activities and lessons should be designed with students’ prior experiences in mind.
d) New concepts should only be presented to students through teacher-centered presentations, in
order to help them build the necessary background knowledge.
11) Which of the following is a true statement regarding the practice of exposing students to multiple approaches to
solving problems?
a) The most useful aspect of the strategy is that 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 are not a valuable way for students to investigate alternative problem-solving
strategies and make connections between mathematical ideas.
c) The strategy is not very useful for very simple mathematical ideas, such as basic computation facts.
d) Exposure to multiple approaches and the subsequent connections that develop can help students to
recall the steps to complete mathematical processes.
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
12) The process of scaffolding involves providing students with a learning task that requires them to
seek assistance outside of the classroom.
13) Because student learning should build upon previous knowledge and life experiences, teachers should become
familiar with and respect the various forms of diversity in their classrooms.
14) If a student makes the statement “I can simplify 6/8 to 3/4,” the teacher can conclude the student probably
conceptually understands the idea of equivalent fractions.
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
15) In contrast to a student who has understanding closer to the instrumental end of the continuum, students with
understanding closer to the ______________________ end are more likely to be able to provide their own real-life
examples of a concept.
16) _____________________________ are physical objects that students and teachers can use to
illustrate and discover math concepts. Sometimes they are created specifically for mathematical purposes (Unifix
cubes), and sometimes they are originally created for other purposes (buttons).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
17) Which of the following is an example of ineffective use of manipulative materials?
a) Not showing students exactly every step of how to use them
b) Making a connection between the model and the mathematical concept
c) Encouraging students to converse about the model without knowledge of what the mathematical
goal is
d) Maintaining a balance between the appropriate amounts of guidance and student exploration
18) Which of the following is NOT a true statement regarding the five strands of mathematical proficiency?
a) They are the foundation for the Standards for Mathematical Practice in the Common Core State
Standards.
b) Conceptual understanding is not critical to developing procedural proficiency.
c) Students should be skilled at choosing, testing, and evaluating problem-solving strategies within
contexts.
d) Students should be skilled at reflecting, evaluating, and adapting their work, as needed.
19) Which of the following statements regarding teaching towards mathematical proficiency is NOT true?
a) It requires students to memorize less.
b) It allows students to more easily make connections to new concepts.
c) It takes much less effort and time than teaching traditionally.
d) It increases student enjoyment and attitudes towards mathematics.
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
20) Name at least two examples of a tool that could help students to do mathematics and gain relational
understanding of a concept. Describe a specific example of a way each tool could help develop this
understanding.
Chapter 2