Download MATH 012 (Fall 2005)

SJSU Mathematics Department Course Outline
Fall 2005
Course Title: Number Systems
SJSU Number: Math 12
General Education Category: B4
Prerequisites: two years of high school algebra, one year of high school geometry, satisfaction
of ELM requirement
Course Description
Mathematics 12 is a course designed for prospective elementary and middle school teachers.
Students explore and develop understanding of mathematical concepts and processes taught at
those levels. In particular, students study problem solving techniques, numeration systems, the
structure of the real number system, and elementary number theory. Throughout the course,
students experience mathematics learning in the way that we want their future students to
experience mathematics learning. In addition, students analyze their learning experiences from
the perspective of a future teacher. Technology is integrated as appropriate.
Note: This is the first course in a three-course sequence of mathematics courses for future
elementary and middle school teachers. For an overview of the three-course sequence, see p. 12.
Bibliography – Knowledge Base
Textbook
Musser, Burger, & Peterson’s Mathematics for Elementary School Teachers, 7th edition. (Note
that Math 105 Concepts in Mathematics, Probability, and Statistics, and Math 106 Intuitive
Geometry, uses the same textbook.)
Required Topics and Suggested Schedule
Chapter 1
(1 week)
Chapter 2
(1.5 weeks)
Chapter 3
(1.5 weeks)
Chapter 4
(1.5 weeks)
Introduction to Problem Solving
 Section 1.1: The Problem Solving Process and Strategies (Emphasize the strategies guess
and test and draw a picture. The strategy use a variable will be emphasized in Math
105.)
 Section 1.2: Three Additional Strategies
 (Emphasize the strategies make a list and solve a simpler problem. The strategy look for
a pattern will be emphasized in Math 105.)
Sets, Whole Numbers, and Numeration (up through section 2.3)
 Section 2.1: Sets as a Basis for Whole Numbers
 Section 2.2: Whole Numbers and Numeration
 Section 2.3: The Hindu-Arabic System (may be integrated into sections 2.1 and 2.2)
Whole Numbers: Operations and Properties
 Section 3.1 Addition and Subtraction
 Section 3.2 Multiplication and Division
 Section 3.3 Ordering and Exponents
Whole-Number Computation – Mental, Electronic, and Written (may be integrated in ch. 3)
 Section 4.1 Mental Math, Estimation, and Calculators
 Section 4.2 Written Algorithms for Whole-Number Operations
 Section 4.3 Algorithms in Other Bases (The principle of regrouping in other bases is
more important than mastery of algorithms in other bases.)
1
Chapter 5
(1.5 weeks)
Chapter 6
(2 weeks)
Chapter 7
(0.5 week)
Chapter 8
(1 week)
Chapter 9
(2.5 weeks)
Miscellaneous
Total Time
Allocation
Number Theory
 Section 5.1 Primes, Composites, and Tests for Divisibility
 Section 5.2 Counting Factors, Greatest Common Factor, Least Common Multiple
Fractions
 Section 6.1 The Set of Fractions
 Section 6.2 Fractions: Addition and Subtraction
 Section 6.3 Fractions: Multiplication and Division
Decimals, Ratio, Proportion, and Percent
 Section 7.1 Decimals
 Section 7.2 Operations with Decimals
Integers
 Section 8.1 Addition and Subtraction
 Section 8.2 Multiplication, Division, and Order
Rational Numbers and Real Numbers with an Introduction to Algebra
 Section 9.1 The Rational Numbers
 Section 9.2 The Real Numbers (Omit the last subsection of 9.2 Introduction to Algebra.
This will be covered in Math 105.)
Review, quizzes, exams (2 weeks)
15 weeks
Journal Articles
1. Robert, Margo F., “Problem-Solving and At-Risk Students: Making Mathematics for All a
Classroom Reality” Teaching Children Mathematics 8 (January 2002): 290-295.
2. Zazlavsky, Claudia, “Developing Number Sense: What Can Other Cultures Tell Us?”
Teaching Children Mathematics 7 (February 2001): 312-319.
3. Whitenack, Joy W.; Knipping, Nancy; Novinger, Sue; Underwood, Gail; “Second Graders
Circumvent Addition and Subtraction Difficulties” Teaching Children Mathematics 8
(December 2001): 228-233.
4. Weidemann, Wanda; Mikovich, Alice K.; Hunt, Jane Braddock; “Using a Lifeline to Give
Rational Numbers a Personal Touch” Mathematics Teaching in the Middle School 7
(December 2001): 210-215.
5. Reys, Barbara J.; Arbaugh, Fran; “Clearing Up the Confusion over Calculator Use in Grades
K-5” Teaching Children Mathematics 8 (October 2001): 90-94.
6. Jackson, C.D., and Leffingwell, R.J., “The Role of Instructors in Creating Math Anxiety in
Students from Kindergarten through College.” The Mathematics Teacher 7, (1999): 583586.
7. Watanabe, T., “Let’s Eliminate Fractions from Primary Curricula!” Teaching Children
Mathematics 7 (October 2001): 70-72.
Instructors’ References
Principles and Standards for School Mathematics, National Council of Teachers of
Mathematics, 2000.
Mathematics Framework for California Public Schools, California Department of Education,
1999.
Supplementary Activity Packet, SJSU Math Education Committee
2
Goals and Objectives – Mathematical Content*
1. Problem Solving – Analyze and solve real-life problems involving whole numbers, integers,
rational numbers, and real numbers.
 Apply Polya’s Four-Step Problem Solving Process to real-life problems.
 Utilize multiple problem-solving strategies, including making a model, acting out the
situation, drawing a diagram, guess-check-revise, simplify the problem, make a list, look
for a pattern, make a table, use a specific case, work backward, and use reasoning.
2. Numeration – Explain patterns in notation, in naming of numbers, and in number
relationships within the Hindu-Arabic number system.
 Give geometric, numerical, and verbal patterns, in the Hindu-Arabic Numeration system.
In particular, demonstrate understanding of base-ten place value through base-ten block
models and comparison with numeration systems with different bases (e.g. base-five).
 Demonstrate understanding of the concepts of base, place value, and the use of 0 by
comparing several numeration systems, including Hindu-Arabic, Egyptian, and Roman.
3. Whole Numbers – Understand structure of the whole number system. Explain patterns,
number relationships, physical models, and operation relationships of whole numbers. Use
and explain algorithms to find estimates and exact answers to problems involving whole
numbers.
 Give real-world and mathematical situations that illustrate the need for 0. Give examples
of ways whole numbers are used in the real world (cardinal, ordinal, nominal). Give realworld or physical referents for the absolute size of large numbers and for comparisons
between large numbers.
 Place whole numbers accurately on a number line.
 Given a one-step word problem, tell the key action reflected by the real-world situation.
Given the key action reflected by a real-world situation, write a story problem that
embodies this action.
 Demonstrate fluency in standard algorithms for computation and evaluate the
correctness of nonstandard algorithms, using symbols, pictures, and physical models.
Describe relationships between the operations of and algorithms for addition,
subtraction, multiplication, and division. Understand properties of the whole number
system and their relationship to the algorithms, both standard and alternative. Identify
the informal thinking strategies children use to find basic facts and show how properties
of whole numbers justify this thinking.
 Use and explain mental calculation techniques (counting, compatible numbers,
compensation, break apart, equal additions) to obtain exact answers. Use and explain
estimation techniques (rounding, front-end, substitute compatible numbers, clustering) to
estimate the results of calculations.
 Use and explain various calculation techniques for complex calculations, using
technology. Demonstrate understanding of the order of operations. Perform operations
with positive, negative, and fractional exponents, as they apply to whole numbers.
*
Goals and objections in italic are directly quoted from the mathematics content specifications described in The
California Commission on Teacher Credentialing document Standards of Program Quality and Effectiveness for the
Subject Matter Requirement for the Multiple Subject Teaching Credential. Goals and objectives in italic must be
covered in the course in order for the course to remain compliant with this document. See pp. 9-11 for the full text
of these mathematics content specifications.
3
4. Number Theory – Understand number theory concepts and analyze quantitative situations
involving elementary number theory.
 Give real-world and mathematical situations that illustrate the need for common factors
and common multiples.
 Know divisibility tests for 2, 3, 5, 9, and 10 and give a mathematical justification for at
least one of these tests. Apply these and other divisibility tests to factor a number.
 Determine whether a number is prime or composite. Find the prime factorization of a
number, and represent the number using exponential notation. Find the LCM and GCF
using numbers in prime-factor form. Find the number of factors of a number and express
all factors in prime-factor form.
5. Integers – Understand structure of the integer number system. Explain patterns, number
relationships, physical models, and operation relationships of integers. Use and explain
algorithms to find estimates and exact answers to problems involving integers.
 Give real-world and mathematical situations that illustrate the need for negative integers.
 Order integers and place integers accurately on a number line.
 Use physical models to explain integer operations. Create numerical patterns that show
why the rules for adding, subtracting, multiplying, and dividing integers make sense.
Make connections between opposites and multiplication. Develop and explain how to
add, subtract, multiply, and divide integers using absolute value. Explain how integer
addition and subtraction are connected and how integer division and multiplication are
connected.
 Understand, identify, and apply properties of the integer number system and understand
their relationship to the algorithms.
6. Rational Numbers – Understand structure of the rational number system. Explain patterns,
number relationships, physical models, and operation relationships of rational numbers. Use
and explain algorithms to find estimates and exact answers to problems involving rational
numbers.
 Give real-world and mathematical situations that illustrate the need for non-integer
rational numbers.
 Develop fraction sense. Write the fraction represented by region, set, and number line
models. Given a fraction, create a region, set, and number line representation. Give
fractions equivalent to a particular fraction and explain various ways to show why they
are equivalent. Give different ways 1 can be represented and illustrate the importance of
being able to represent 1 in different ways. Use benchmark fractions and number sense
to compare and order fractions and to find fractions between other fractions.
 Interchange among real-world situations, symbolic representations, pictures, and physical
models for addition, subtraction, multiplication, and division of fractions. Use
benchmarks to estimate fraction sums, products, differences, and quotients. Add and
subtract special-case fractions mentally. Perform operations with positive, negative, and
fractional exponents, as they apply to fractions.
 Explain and illustrate different equivalent representations for a particular rational
number. Convert between fractions and decimals. Given a fraction in lowest terms,
examine the prime factors of the denominator to explain why the fraction can be
represented as a terminating or repeating decimal. Order integers, mixed numbers, and
rational numbers (including fractions, decimals, and percents), and place accurately on a
number line. Represent numbers in scientific notation.
4

Use and explain traditional and alternative algorithms for operations with rational
numbers. Understand properties of the rational number system and their relationship to
the algorithms.
7. Real Numbers – Understand structure of the real number system. Explain patterns, number
relationships, physical models, and operation relationships of real numbers. Use and explain
algorithms to find estimates and exact answers to problems involving real numbers.
 Give real-world and mathematical situations that illustrate the need for irrational
numbers.
 Explain the set-subset relationships among natural numbers, whole numbers, integers,
rational numbers, irrational numbers, and real numbers.
 Order integers, mixed numbers, rational numbers, and real numbers, and place numbers
accurately on a number line.
 Understand properties of the real number system and their relationship to the algorithms.
5
Goals and Objectives – Mathematical Processes
One of the ways in which we weave together the three mathematics courses for prospective
teachers is to have well-defined, long-term goals that guide the implementation of the syllabi
throughout the sequence. These are goals that go above and beyond the learning of specific
content topics, and which take students longer than one semester to achieve. These goals
correspond roughly to the National Council of Teachers of Mathematics’ process standards –
they “highlight ways of acquiring and using content knowledge” (NCTM, 2000, p. 29). Each
course builds upon these goals in a progressive fashion.
1. Understand understanding
 Recognize the validity of different approaches
 Recognize the equivalence of different answers
 Analyze errors to identify misunderstandings
 Analyze levels of understanding
 Explain multiple ways of understanding the same idea
 Recognize when language use is ambiguous, well-defined, or meaningless
 Recognize examples and non-examples
2. Utilize representations and connections
 Identify situations that can be modeled using mathematics
 Represent situations appropriately using mathematics
 Translate from one representation to one another
 Explain how representations are connected to one another
3. Develop new reasoning and problem-solving skills
 Experiment, conjecture, verify
 Recognize patterns, recognize recurring ideas in different settings
 Reason by analogy, infer in situations of uncertainty
 Synthesize, deduce
4. Communicate mathematical ideas
 Use mathematical terminology, notation, and language effectively and accurately
 Express ideas logically and clearly
 Model English with mathematics, interpret mathematics into English
 Illustrate and support ideas graphically, numerically, symbolically, or verbally as
needed
5. Develop positive attitudes and beliefs about mathematics
 Learn and articulate how mathematics is useful outside of school
 Identify potential sources of negative attitudes and beliefs
 Model helpful attitudes and beliefs while working with fellow students
6. Use technology appropriately
 Appreciate the role of technology as a tool for learning and problem solving
 Recognize the capabilities and limitations of computational aids
 See how elementary mathematics and technology interact
6
California Commission on Teacher Credentialing Standards
The California Commission on Teacher Credentialing requires all approved subject matter
programs to meet certain standards of program quality and effectiveness, as described in the
September 2001 document Standards of Program Quality and Effectiveness for the Subject
Matter Requirement for the Multiple Subject Teaching Credential. Here, we describe how
SJSU’s Math 12 contributes to meeting the content specifications in mathematics. (See pp. 9-11
for the mathematics content specifications.)
Part I: Content Domains for Subject Matter Understanding and Skill in Mathematics
Course Topics
Math 12
Problem Solving
Numeration
Whole Numbers
Number Theory
Integers
Rational Numbers
Real Numbers
Domain 1
Number
Sense
Domain 2
Algebra &
Functions
1.1
2.1
1.2
2.2
Domains
Domain 3
Measurement &
Geometry
3.1
3.2
3.3
Domain 4
Statistics, Data
Analysis, &
Probability
4.1
4.2
4.3
x
x
x
x
x
x
x
x
x
x
x
x
Domain 1.1: Numbers, Relationships Among Numbers, and Number Systems
The elements of this domain form the primary content of Math 12, the first course of our threesemester sequence of courses for future elementary and middle school teachers. Through
activities and projects involving manipulatives and group work, our future teachers develop
multiple understandings of the concepts in this domain.
Domain 1.2 Computational Tools, Procedures, and Strategies
The elements of this domain are introduced in Math 12 with particular emphasis placed on
nonstandard algorithms, understanding multiple strategies, and making connections between
various strategies. These elements are incorporated throughout the second and third courses of
our three-semester sequence of courses for future elementary and middle school teachers, with
particular emphasis placed on the use of technology.
Part II: Subject Matter Skills and Abilities Applicable to the Content Domain in
Mathematics
Subject matter skills and abilities in mathematics are developed in a progressive fashion over the
three-semester sequence of mathematics courses for future elementary school teachers. Broadly,
our goals are for our students to (1) understand understanding, (2) utilize representations and
connections, (3) develop new reasoning and problem-solving skills, (4) communicate
7
mathematical ideas, (5) develop positive attitudes and beliefs about mathematics, and (6) use
technology appropriately. We start down the path towards these goals in Math 12 by including a
unit on problem solving.
The problem-solving unit in Math 12 requires students to examine explicitly the nature and
importance of problem solving, strategies for solving problems, and Polya’s 4-step ProblemSolving Model. Through several collaborative problem-solving activities, students learn to
represent problems in multiple ways (including the use of concrete models, words, numbers,
charts, graphs, tables, diagrams, and symbols), understand connections between different
representations of a problem, seek out alternative solutions, and analyze the reasonableness of
their answers. In other activities, they create problems that meet certain conditions and they
modify problems to make them either more accessible or more challenging for their future
students. They analyze and evaluate actual solutions written by elementary school students.
Subject Matter Competency in Mathematics for Multiple-Subjects Credential Candidates
All future multiple-subjects credential candidates will have to pass the CSET exam in multiple
subjects prior to entering a credential program in California. Subtest II covers mathematics and
science. Sample questions can be viewed at the web site http://www.cset.nesinc.com/.
Instructors might wish to go over sample mathematics questions from the CSET on occasion.
8
Content Specifications in Mathematics*
Part I: Content Domains for
Subject Matter Understanding and Skill in
Mathematics
Domain 1: Number Sense
1.1
Numbers, Relationships Among Numbers, and Number Systems. Candidates for
Multiple Subject Teaching Credentials understand base ten place value, number theory
concepts (e.g., greatest common factor), and the structure of the whole, integer, rational,
and real number systems. They order integers, mixed numbers, rational numbers
(including fractions, decimals, and percents) and real numbers. They represent numbers in
exponential and scientific notation. They describe the relationships between the algorithms
for addition, subtraction, multiplication, and division. They understand properties of
number systems and their relationship to the algorithms, [e.g., 1 is the multiplicative
identity; 27 + 34 = 2 X 10 + 7 + 3 X 10 + 4 = (2 + 3) X 10 + (7 + 4)]. Candidates perform
operations with positive, negative, and fractional exponents, as they apply to whole
numbers and fractions.
1.2
Computational Tools, Procedures, and Strategies. Candidates demonstrate fluency in
standard algorithms for computation and evaluate the correctness of nonstandard
algorithms. They demonstrate an understanding of the order of operations. They round
numbers, estimate the results of calculations, and place numbers accurately on a number
line. They demonstrate the ability to use technology, such as calculators or software, for
complex calculations.
Domain 2: Algebra and Functions
2.1
Patterns and Functional Relationships. Candidates represent patterns, including
relations and functions, through tables, graphs, verbal rules, or symbolic rules. They use
proportional reasoning such as ratios, equivalent fractions, and similar triangles, to solve
numerical, algebraic, and geometric problems.
2.2
Linear and Quadratic Equations and Inequalities. Candidates are able to find
equivalent expressions for equalities and inequalities, explain the meaning of symbolic
expressions (e.g., relating an expression to a situation and vice versa), find the solutions,
and represent them on graphs. They recognize and create equivalent algebraic expressions
(e.g., 2(a+3) = 2a + 6), and represent geometric problems algebraically (e.g., the area of a
triangle). Candidates have a basic understanding of linear equations and their properties
(e.g., slope, perpendicularity); the multiplication, division, and factoring of polynomials;
and graphing and solving quadratic equations through factoring and completing the square.
*
Reprinted from California State Standards of Program Quality and Effectiveness for the Subject Matter Requirement for the Multiple Subject
Teaching Credential, September, 2001.
9
They interpret graphs of linear and quadratic equations and inequalities, including solutions
to systems of equations.
Domain 3: Measurement and Geometry
3.1
Two- and Three-dimensional Geometric Objects. Candidates for Multiple Subject
Teaching Credentials understand characteristics of common two- and three-dimensional
figures, such as triangles (e.g., isosceles and right triangles), quadrilaterals, and spheres.
They are able to draw conclusions based on the congruence, similarity, or lack thereof, of
two figures. They identify different forms of symmetry, translations, rotations, and
reflections. They understand the Pythagorean theorem and its converse. They are able to
work with properties of parallel lines.
3.2
Representational Systems, Including Concrete Models, Drawings, and Coordinate
Geometry. Candidates use concrete representations, such as manipulatives, drawings, and
coordinate geometry to represent geometric objects. They construct basic geometric
figures using a compass and straightedge, and represent three-dimensional objects through
two-dimensional drawings. They combine and dissect two- and three-dimensional figures
into familiar shapes, such as dissecting a parallelogram and rearranging the pieces to form a
rectangle of equal area.
3.3
Techniques, Tools, and Formulas for Determining Measurements. Candidates estimate
and measure time, length, angles, perimeter, area, surface area, volume, weight/mass, and
temperature through appropriate units and scales. They identify relationships between
different measures within the metric or customary systems of measurements and estimate
an equivalent measurement across the two systems. They calculate perimeters and areas of
two-dimensional objects and surface areas and volumes of three-dimensional objects. They
relate proportional reasoning to the construction of scale drawings or models. They use
measures such as miles per hour to analyze and solve problems.
Domain 4: Statistics, Data Analysis, and Probability
4.1
Collection, Organization, and Representation of Data. Candidates represent a collection
of data through graphs, tables, or charts. They understand the mean, median, mode, and
range of a collection of data. They have a basic understanding of the design of surveys,
such as the role of a random sample.
4.2
Inferences, Predictions, and Arguments Based on Data. Candidates interpret a graph,
table, or chart representing a data set. They draw conclusions about a population from a
random sample, and identify potential sources and effects of bias.
4.3
Basic Notions of Chance and Probability. Candidates can define the concept of
probability in terms of a sample space of equally likely outcomes. They use their
understanding of complementary, mutually exclusive, dependent, and independent events
to calculate probabilities of simple events. They can express probabilities in a variety of
ways, including ratios, proportions, decimals, and percents.
10
Part II: Subject Matter Skills and Abilities
Applicable to the Content Domains in Mathematics
Candidates for Multiple Subject Teaching Credentials identify and prioritize relevant and
missing information in mathematical problems. They analyze complex problems to identify
similar simple problems that might suggest solution strategies. They represent a problem in
alternate ways, such as words, symbols, concrete models, and diagrams, to gain greater insight.
They consider examples and patterns as means to formulating a conjecture.
Candidates apply logical reasoning and techniques from arithmetic, algebra, geometry, and
probability/statistics to solve mathematical problems. They analyze problems to identify
alternative solution strategies. They evaluate the truth of mathematical statements (i.e., whether
a given statement is always, sometimes, or never true). They apply different solution strategies
(e.g., estimation) to check the reasonableness of a solution. They demonstrate that a solution is
correct.
Candidates explain their mathematical reasoning through a variety of methods, such as words,
numbers, symbols, charts, graphs, tables, diagrams, and concrete models. They use appropriate
mathematical notation with clear and accurate language. They explain how to derive a result
based on previously developed ideas, and explain how a result is related to other ideas.
11
Mathematics Course Sequence at SJSU for Prospective Elementary and Middle School Teachers
Math 12, Math 105, Math 106
The San Jose State Mathematics Department offers a three-semester sequence of courses designed for prospective
elementary and middle school teachers. In these courses, students explore and develop understanding of
mathematical concepts and processes taught at those levels. Throughout the three-course sequence, students
experience mathematics learning in the way that we want their future students to experience mathematics learning,
using technology, as appropriate. In addition, students analyze their own learning experiences from the perspective
of a future teacher. Students are expected to grow in mathematical sophistication, scholarly responsibility, and
pedagogical perspectives over the three-course sequence.
Math 12 Number Systems
In Math 12 is the first course in the three-course sequence. Several local community colleges offer equivalent
courses. In this course, students study problem solving techniques, numeration systems, the structure of the real
number system, and elementary number theory.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 303)
Two years of high school algebra, one year of high school geometry, satisfaction of ELM requirement
Math 105 Concepts in Mathematics, Probability, & Statistics
Math 105 is the second course in the three-course sequence. This is an upper division class that cannot be taken at a
local community college. Students study problem-solving techniques, functions and algebraic reasoning, ratio and
proportions, probability, data, graphs, and statistics.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 304)
Two years of high school algebra, one year of high school geometry, SJSU’s Math 12 with a C- or better
Math 106 Intuitive Geometry
Mathematics 106 is the third course in the three-course sequence. This is an upper division class that cannot be
taken at a local community college. Students analyze characteristics and properties of two- and three-dimensional
geometric shapes; develop mathematical arguments about geometric relationships; apply transformations and use
symmetry to analyze mathematical situations; represent geometric objects using representational systems such as
concrete models, drawings, and coordinate geometry; and use techniques, tools, and formulas for determining
measurements. In general, students are encouraged to think about geometry as the study of objects in a plane or in
space. They are asked to investigate situations involving geometric objects, state conjectures, and provide
explanations that support their conjectures. Technology is integrated extensively. In particular, students will use a
dynamic geometry system, such as Geometer’s Sketchpad or Cabri, as a tool for visual investigations.
Course Prerequisites (reprinted from the SJSU 2004-2006 catalog, p. 304)
Two years of high school algebra, one year of high school geometry, SJSU’s Math 12 and Math 105 with grades of
C- or better
General Notes
 There are no exceptions to the prerequisites. This means that students must take the three courses
sequentially. (The level of mathematical sophistication, extent of scholarly expectations, and breadth of
pedagogical perspectives increase substantially from one course to the next in this sequence. The
prerequisites are designed to provide students with the greatest opportunity of success in the three-course
sequence as well as the best possible preparation for teaching mathematics at the elementary and middle
school levels.)
 Grades of C or better MAY be required by some students’ majors in order for them to graduate. Students
are expected to determine this in consultation with their major advisors.
 The topics in the three-course sequence include all of the topics covered in the mathematics portion of
Subtest II of the California Subject Examination for Teachers: Multiple Subjects. Students preparing for
teaching careers in California are strongly encouraged to take this three-course mathematics sequence in
preparation for this exam, even if their majors do not require all or any of the courses.
12