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A (Pre)Calculus and Discrete Math Course for Pre-Service Middle School Teachers

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### 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.

MAA/AMS Joint Mathematics Meetings

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.

MAA/AMS Joint Mathematics Meetings

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

MAA/AMS Joint Mathematics Meetings

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.

MAA/AMS Joint Mathematics Meetings

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.

MAA/AMS Joint Mathematics Meetings

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.

MAA/AMS Joint Mathematics Meetings

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

MAA/AMS Joint Mathematics Meetings

MINOR IN TEACHING MATHEMATICS (GRADES 5-8)

Course Hours

479 Techniques for Teaching Mathematics 3

503 Introduction to Advanced Mathematical 3 Concepts for Education

A computer programming course which will 3 satisfy the General Education computing requirement

Total 25

MAA/AMS Joint Mathematics Meetings

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.

MAA/AMS Joint Mathematics Meetings

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

MAA/AMS Joint Mathematics Meetings

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

MAA/AMS Joint Mathematics Meetings

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

and its weight W, in pounds, is given by the formula

Here t is the age of the fish, in years, and both formulas

are valid from the age of 1 year.

MAA/AMS Joint Mathematics Meetings

Exercises from Functions and Change: A Modeling Approach to College Algebra

- Make a graph of the length of the fish against its age, covering ages 1 to 8.
- To what limiting length does the fish grow? At what age does it reach 90% of this length?
- Make a graph of the weight of the fish against its age, covering ages 1 to 8.
- To what limiting weight does the fish grow? At what age does it reach 90% of this weight?
- 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.

MAA/AMS Joint Mathematics Meetings

Exercises from Functions and Change: A Modeling Approach to College Algebra

Hiking: You are hiking in a hilly region and

is your elevation at time t.

- Explain in practical terms the meaning of .
- Where might you be when is a large positive number?
- You reach a point where is briefly zero. Where might you be?
- Where might you be when is a large negative number?

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Sample Test Questions

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.

MAA/AMS Joint Mathematics Meetings

Sample Test Questions

When an average-size man with a parachute jumps from an airplane, he will fall feet in t seconds.

- Plot a graph of S versus t over at least the first 10 seconds of the fall.
- By direct calculation, estimate the value of at 2 seconds. (Use an increment of 0.01) Explain what the number you calculated means in practical terms.

MAA/AMS Joint Mathematics Meetings

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

MAA/AMS Joint Mathematics Meetings

Recommendations from The Mathe-matical Education of Teachers

- Prospective middle grades teachers of mathe-matics 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]

MAA/AMS Joint Mathematics Meetings

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 under-standing of the ways in which mathematics can be applied. [All p. 26]

MAA/AMS Joint Mathematics Meetings

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]

MAA/AMS Joint Mathematics Meetings

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]

MAA/AMS Joint Mathematics Meetings

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. 31-32]

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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.

MAA/AMS Joint Mathematics Meetings

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