A (Pre)Calculus and Discrete Math
Course for Pre-Service Middle
School Teachers
Tim Flood (tflood@pittstate.edu)
Pittsburg State University
Pittsburg, KS
Preliminary report
Teacher Education Math Standards
For Kansas Educators (Grades 5-8)
#1 The teacher of mathematics has conceptual
and procedural understanding of
mathematics.
#2 The teacher of mathematics can demonstrate
conceptual and procedural understanding of
number and number systems and is able to
identify and apply these understandings within
a real world context.
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Teacher Education Math Standards
For Kansas Educators (Grades 5-8)
#3 The teacher of mathematics can demonstrate
the need for, uses of, and conceptual and
procedural understanding of patterns,
functions and algebra from both concrete
and abstract perspectives, and is able to
identify and apply these relationships in the
real world context, including the use of
appropriate technology.
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Teacher Education Math Standards
For Kansas Educators (Grades 5-8)
#4 The teacher of mathematics can demonstrate
the need for, uses of, and conceptual and
procedural understanding of geometry,
measurement, and spatial visualization from
both concrete and abstract perspectives, and
are able to identify and apply these
relationships in the real world context,
including the use of appropriate technology
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Teacher Education Math Standards
For Kansas Educators (Grades 5-8)
#5 The teacher of mathematics can demonstrate
conceptual and procedural understanding of
concepts of data, statistics and probability
and is able to identify and apply these
relationships within a real world context
including the use of appropriate technology.
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Teacher Education Math Standards
For Kansas Educators (Grades 5-8)
#6 The teacher of mathematics can demonstrate
conceptual and procedural understanding of
concepts of calculus and is able to identify
and apply these relationships within a real
world context including the use of appropriate
technology.
#7 The teacher of mathematics can demonstrate
conceptual and procedural understanding of
discrete processes and is able to identify and
apply these understandings within a real world
context including the use of appropriate
technology.
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Teacher Education Math Standards
For Kansas Educators (Grades 5-8)
#8 The teacher of mathematics can demonstrate
knowledge of the history of mathematics.
#9 The teacher of mathematics has a foundational
knowledge of students as learners and of
pedagogical strategies.
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MINOR IN TEACHING MATHEMATICS
(GRADES 5-8)
Course
Hours
126 Pre-Calculus
4
143 Elementary Statistics
3
304 Mathematics for Education II
3
307 Geometry for Education
3
Includes historical and cultural perspectives
471 Manipulatives for Teaching Mathematics
1
472 Calculators in Teaching Mathematics
1
473 Mathematical Software
1
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MINOR IN TEACHING MATHEMATICS
(GRADES 5-8)
Course
Hours
479 Techniques for Teaching Mathematics
3
503 Introduction to Advanced Mathematical
Concepts for Education
3
A computer programming course which will
satisfy the General Education computing
requirement
3
Total
25
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Teacher Education Math Standards
For Kansas Educators (Grades 5-8)
#6 The teacher of mathematics can demonstrate
conceptual and procedural understanding of
concepts of calculus and is able to identify
and apply these relationships within a real
world context including the use of appropriate
technology.
#7 The teacher of mathematics can demonstrate
conceptual and procedural understanding of
discrete processes and is able to identify and
apply these understandings within a real world
context including the use of appropriate
technology.
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MATH 503 Introduction to Advanced
Mathematical Concepts for Education
• Course Description
 An introduction into advanced topics in
mathematics including concepts of: matrices,
discrete and continuous functions, calculus, and
graph theory. The topics will be introduced using
appropriate technology.
• Prerequisites
 MATH 126 Pre-Calculus
 MATH 472 Calculators in Teaching Mathematics
 MATH 473 Mathematical Software
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MATH 503 Introduction to Advanced
Mathematical Concepts for Education
• Objectives: Students, upon successful
completion of this course, will be able to:
 Use graphing calculators to investigate the
concept of calculus including limit, continuity,
derivative, and integral.
 Use the concepts of calculus to investigate and
solve problems from a variety of disciplines.
 Investigate discrete processes using graphing
calculators and computer spreadsheets.
 Use technology to formulate and test conjectures
concerning discrete processes.
 Use discrete processes to model nature and
problems from other disciplines.
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MATH 503 Introduction to Advanced
Mathematical Concepts for Education
• Text
 Functions and Change: A Modeling Approach to
College Algebra, 2nd Ed, by Crauder, Evans, and
Noell, Houghton Mifflin Co., 2003.
Note: 3rd edition is available.
 Numerous handout and Web explorations on
discrete mathematics (graph theory)
• Course Breakdown
 2/3 of the semester on Calculus concepts
 1/3 of the semester on discrete math topics
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Exercises from Functions and Change: A
Modeling Approach to College Algebra
Growth: The length L, in inches, of a certain flatfish
is given by the formula
L  15  19  0.6t
and its weight W, in pounds, is given by the formula
W  (1  1.3  0.6t )3
Here t is the age of the fish, in years, and both formulas
are valid from the age of 1 year.
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Exercises from Functions and Change: A
Modeling Approach to College Algebra
a. Make a graph of the length of the fish against its age,
covering ages 1 to 8.
b. To what limiting length does the fish grow? At what age
does it reach 90% of this length?
c. Make a graph of the weight of the fish against its age,
covering ages 1 to 8.
d. To what limiting weight does the fish grow? At what age
does it reach 90% of this weight?
e. One of the graphs you made in parts a and c should have
an inflection point, whereas the other is always concave
down. Identify which is which, and explain in practical
terms what this means. Include in your explanation the
approximate location of the inflection point.
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Exercises from Functions and Change: A
Modeling Approach to College Algebra
Hiking: You are hiking in a hilly region and E  E (t )
is your elevation at time t.
dE
a. Explain in practical terms the meaning of
.
dt
dE
b. Where might you be when
is a large positive
dt
number?
dE
c. You reach a point where
is briefly zero. Where
dt
might you be?
dE
d. Where might you be when
is a large negative
dt
number?
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Velocity
Sample Test Questions
0
1
2
3
4
5
6
7
8
9 10 11
Time
The given graph shows your velocity on a car trip
along an east-west highway. You start from your
home driving west. Draw a graph of your distance
west of home.
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Sample Test Questions
When an average-size man with a parachute jumps
from an airplane, he will fall S (t )  12.5(0.2t  1)  20t
feet in t seconds.
a. Plot a graph of S versus t over at least the first 10
seconds of the fall.
dS
b. By direct calculation, estimate the value of
dT
at 2 seconds. (Use an increment of 0.01) Explain
what the number you calculated means in practical
terms.
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Additional Topics from Discrete Math
• Graph Theory
 Euler Paths and Cycles
 Hamiltonian Paths and Cycles
 Dijkstra's Shortest Path Algorithm
 Planar Graphs
• Trees
 Binary Search Trees
 Tree Traversal
 Polish Notation
 Reverse Polish Notation
• Modular Arithmetic
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Recommendations from The Mathe-
matical Education of Teachers
• Prospective middle grades teachers of mathematics should be required to take at least 21
semester-hours of mathematics, that includes
at least 12 semester-hours on fundamental
ideas of school mathematics appropriate for
middle grades teachers. [p. 8]
• One semester of calculus could be part of this
second group of courses if there is (or could be
designed) a calculus course that focuses on
concepts and applications, as opposed to the
traditional course offered to mathematics
majors and engineers. [p. 26]
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Recommendations from The Mathe-
matical Education of Teachers
• Number theory and discrete mathematics can
offer teachers an opportunity to explore in depth
many of the topics they will teach.
• A history of mathematics course can provide
middle grades teachers with an understanding
of the background and historical development
of many topics in the middle grades curriculum.
• A mathematical modeling course, depending on
the level and substance of the course, can
provide prospective teachers with understanding of the ways in which mathematics can
be applied. [All p. 26]
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Recommendations from The Mathe-
matical Education of Teachers
• Prospective middle grades teachers should:
 Understand and be able to work with algebra as a
symbolic language, as a problem solving tool, as
generalized arithmetic, as generalized quantitative
reasoning, as a study of functions, relations, and
variation, and as a way of modeling physical situations.
 Develop an understanding of variables and functions,
especially of different equivalent relationships between
variables.
 Understand linearity and how linear functions can
illustrate proportional relationships.
 Recognize change patterns associated with linear,
quadratic, and exponential functions.
 Demonstrate algebraic skills and be able to give a
rationale for common algebraic procedures. [p. 31]
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Recommendations from The Mathe-
matical Education of Teachers
• Functions, relations, and variation all play
important roles in school algebra. In middle
grades these roles are prominent as students
come to understand algebraic functions,
particularly linear functions and the manner in
which they can illustrate proportional
relationships. Using qualitative graphs, that is,
graphs that have no numbers but that “tell a
story,” can lead to a deeper understanding of
functional relationships. [p. 31]
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Recommendations from The Mathe-
matical Education of Teachers
• Middle grades teachers should know how to
deal with problems involving rate of change, for
example, of the growth of a plant, the decline of
an endangered species, or of the speed of a
car. The accumulation of miles traveled while
driving at a certain speed is also a problem of
change, and shows the relationship between
distance traveled and rate of travel (Noble,
Wright, Nemirovsky, & Tierney, 2001). [pp. 3132]
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Additional Information
• If you have any questions or would like a copy
of these slides, please contact me at
tflood@pittstate.edu.
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