A (Pre)Calculus and Discrete Math Course for Pre-Service Middle School Teachers Tim Flood Pittsburg State University Pittsburg, KS Preliminary report

January 5, 2007MAA/AMS Joint Mathematics Meetings2 Teacher Education Math Standards For Kansas Educators (Grades 5-8) #1The teacher of mathematics has conceptual and procedural understanding of mathematics. #2The 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings3 Teacher Education Math Standards For Kansas Educators (Grades 5-8) #3The 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings4 Teacher Education Math Standards For Kansas Educators (Grades 5-8) #4The 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

January 5, 2007MAA/AMS Joint Mathematics Meetings5 Teacher Education Math Standards For Kansas Educators (Grades 5-8) #5The 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings6 Teacher Education Math Standards For Kansas Educators (Grades 5-8) #6The 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. #7The 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings7 Teacher Education Math Standards For Kansas Educators (Grades 5-8) #8The 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings8 MINOR IN TEACHING MATHEMATICS (GRADES 5-8) CourseHours 126 Pre-Calculus4 143 Elementary Statistics3 304 Mathematics for Education II3 307 Geometry for Education3 Includes historical and cultural perspectives 471 Manipulatives for Teaching Mathematics1 472 Calculators in Teaching Mathematics1 473 Mathematical Software1

January 5, 2007MAA/AMS Joint Mathematics Meetings9 MINOR IN TEACHING MATHEMATICS (GRADES 5-8) CourseHours 479 Techniques for Teaching Mathematics3 503 Introduction to Advanced Mathematical 3 Concepts for Education A computer programming course which will 3 satisfy the General Education computing requirement Total25

January 5, 2007MAA/AMS Joint Mathematics Meetings10 Teacher Education Math Standards For Kansas Educators (Grades 5-8) #6The 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. #7The 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings11 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

January 5, 2007MAA/AMS Joint Mathematics Meetings12 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings13 MATH 503 Introduction to Advanced Mathematical Concepts for Education Text  Functions and Change: A Modeling Approach to College Algebra, 2 nd Ed, by Crauder, Evans, and Noell, Houghton Mifflin Co., Note: 3 rd 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

January 5, 2007MAA/AMS Joint Mathematics Meetings14 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings15 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings16 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. a.Explain in practical terms the meaning of. b.Where might you be when is a large positive number? c.You reach a point where is briefly zero. Where might you be? d.Where might you be when is a large negative number?

January 5, 2007MAA/AMS Joint Mathematics Meetings17 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.

January 5, 2007MAA/AMS Joint Mathematics Meetings18 Sample Test Questions When an average-size man with a parachute jumps from an airplane, he will fall feet in t seconds. a.Plot a graph of S versus t over at least the first 10 seconds of the fall. b.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.

January 5, 2007MAA/AMS Joint Mathematics Meetings19 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

January 5, 2007MAA/AMS Joint Mathematics Meetings20 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]

January 5, 2007MAA/AMS Joint Mathematics Meetings21 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]

January 5, 2007MAA/AMS Joint Mathematics Meetings22 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]

January 5, 2007MAA/AMS Joint Mathematics Meetings23 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]

January 5, 2007MAA/AMS Joint Mathematics Meetings24 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 ]

January 5, 2007MAA/AMS Joint Mathematics Meetings25 Additional Information If you have any questions or would like a copy of these slides, please contact me at