URI/School of Education in partnership with

Rhode Island Teacher Education Renewal (Project RITER)


Course Title:  MTH 420 Re-examining Mathematical Foundations for Teachers

(A shortened version prints on the transcriptRe-exam Mth Foundatns)


Credits:  3 (Graduate Level)


Course Description:  Connects ideas covered in upper level math courses to topics taught in secondary school.  Designed for teachers.


Instructor:    Dr. Nancy Eaton, Professor   

E-mail: eaton@math.uri.edu    

Phone: (401) 874-4439


Location:   Shepard Building, URI Providence Campus


Meeting Times:  Tuesday, Thursday, 4:00 – 7:45, May 22 – June 21


Course Web Page:  http://www.math.uri.edu/~eaton/Mth420Su07Syllabus.htm


Course Goals:

§         Provide our pre-service teachers with a capstone course that makes connections between and among the ideas covered in upper level math courses to topics taught in secondary school.  Core topics of a typical high school math program are presented from an advanced perspective, revealing their relationships to college level math.  This perspective and analysis of topics provides teachers with a deeper understanding of high school math and fosters an appreciation of its structures and beauty.

§         Provide a course for in-service teachers that better prepares them to handle the challenges of developing teaching strategies and assessment tools for the secondary school standards. 

§         Better prepare pre-service teachers for the newly proposed licensure exams and provide a mechanism for making improvements once pass rates on state exams become available.


Course Objectives and Assessment:

The NCTM has set standards for the mathematics preparation of all mathematics teacher candidates.  This course addresses several of these standards. 

1.      Knowledge of Problem Solving.  In this course, candidates demonstrate that they know, understand and apply the process of mathematical problem solving by successfully completing the problem set for the course.

2.      Knowledge of Reasoning and Proof.  Candidates get further experience with reasoning, constructing, and evaluating mathematical arguments in this course.

3.      Knowledge of Mathematical Communication.  This course is structured to allow many opportunities for the candidates to communicate their ideas, orally and in writing.  Assignments include classroom discussion, presentations, and papers.

4.      Knowledge of Mathematical Connections.  The main emphasis of this course is to give candidates a structure in which to make connections between different mathematical ideas and build a more complex mathematical understanding.  A class project will be completed to demonstrate an understanding of such a connection.

5.      Knowledge of Mathematical Representation.  The main representations are verbal, graphical, numerical, and analytic.  Each type of representation is included in the problem list for the course.  Emphasis will be given to proper use of math symbols and terms. 

6.      Knowledge of Technology.  The Mathematics Content Knowledge exam that will be required for teacher candidates entering the work force requires the use of graphing calculators.  In this course we will use a graphing calculator as often as possible to further enhance the solution and illustration of problems.

7.      Dispositions.  An appreciation of mathematics often comes after much knowledge is acquired and has become intuitive.  When connections are made, moments of greater understanding are reached, “Aha” moments.  There is a great deal of satisfaction in this understanding that is often hard sought.   We make the connections in this course and hope that the chances that the teachers themselves become lifelong learners are increased.

8.      Knowledge of Mathematics Pedagogy.  This course will serve to deepen the understanding of how students learn mathematics, by providing examples of questions given to students in the secondary level courses and seeing what concepts are being addressed.  In-service and pre-service teachers will be given assignments to design questions that address the items given in the Grade Span Expectations (GSEs) and Grade Level Expectations (GLEs).

9.      through 15.  The content areas of knowledge for secondary level mathematics teachers are provided in their other courses, with the exception of Discrete Mathematics.  In this course, we review many of those ideas as they come up in examples.  Teacher candidates need to have knowledge of the math that students will encounter when they leave high school, either in college, vocational school, or on the job.


Required Text:  Anthony Peressini, Zalman Usiskin, Elena Anne Marchisotto, and Dick Stanley Mathematics for High School Teachers: An Advanced Perspective, Prentice Hall 2003.


Optional Readings


1.      Mathematics Magazine. MAA Publication.

2.      The College Mathematics Journal.  MAA Publication.

3.      The Mathematical Monthly.  MAA Publication.

4.   Math Horizons.  MAA Publication.

5.   The Newsletter of the MAA, Focus.  MAA Publication.



1.      Posamentier, Smith, and StepelmanTeaching Secondary Mathematics: Techniques and Enrichment Units.  Pearson Prentice Hall, 2006.

2.      Ameis and Ebenezer.  Mathematics on the Internet:  A Resource for K-12 Teachers.  Pearson Merrill Prentice Hall, 2006.  

3.      Dunham.  Euler: The Master of Us All.  Mathematical Association of America, 1999.

4.      Nelsen.  Proofs Without Words.  Mathematical Association of America, 1993.

5.      Nelsen.  Proofs Without Words II.  More Exercises in Visual Thinking.  The Mathematical Association of America, 2000.

6.      PolyaHow to Solve It:  A New Aspect of Mathematical MethodPrinceton University Press, 2004.

7.      Calinger and Reynolds.  Vita Mathematica: Historical Research and Integration with Teaching.  The Mathematical Association of America, 1997.

8.      Cuoco.  Mathematical Connections:  A companion for Teachers and Others.  The Mathematical Association of America, 2005.

9.      Katz and MichalowiczHistorical Modules for the Teaching and Learning of Mathematics.  The Mathematical Association of America, 2005.

10.  Swetz, et. Al.  Learn From the Masters (Classroom Resource Material).  The Mathematical Association of America, 1997.

11. Courant, Robbins.  Revised by Stewart.  What is Mathematics?  Oxford University Press, 1996.


Web Sites:

1.      Ask Dr Math - http://mathforum.org/dr.math/

2.      Tower of Hanoi - http://www.lhs.berkeley.edu/Java/Tower/towerhistory.html

3.      The Prime Number Pages - http://primes.utm.edu/

4.      Author’s Web Site for Mathematics for High School Teachers – An advanced Perspective. - http://mtl.math.uiuc.edu/math-hst/

5.      Quantum Magazine, The Magazine of Math and Science - http://www.nsta.org/quantum/

6.      GLEs & GSEs: http://www.tivschools.com/curriculum/webaccount/Math/Math GLEs.htm



Course Requirements and Grading:

Exercises:  You will be given a problem list from the textbook to work on at home and during class.  These problems provide a mechanism for reviewing advanced math and making connections among mathematical ideas.

Reports:  Two written assignments will be given, using sources other than our text to address issues related to the teaching of mathematics.   Discussion of the reports will be held in class.

World Wide Web Assignment:  Two WWW assignments will be given that will help teachers to explore the usefulness of the web for teaching mathematics.  They will be sent by the instructor to the class members by e-mail at the beginning of the course and the completed work is to be sent back to the instructor by e-mail by the last day of class.

Exams:  Short midterm and final exams will be given.  The questions are similar in scope to those that might be given on a licensure exam for beginning secondary mathematics teachers.

Projects:  From a list of topics, Investigate and analyze the topic and explore connections between and among mathematical concepts and applications.  These are to be presented to the class.  A handout will be given to each class member and a final report will be given to the instructor.  Students may work in groups, but each class member is responsible for an entire project including the report and presentation.  Design two GSE or GLE questions related to your topic.  You are required to complete a 15-20-minute midterm and a 25-30-minute final project (No longer!)


The final grade for the course will be based on the following:

10% - Presentations of four solutions to exercises on the board.

20% - Written solutions to ten exercises

15% - 2 reports 2-3 pages each

05% - World Wide Web assignment

05% - Midterm exam – take home Download  (Just do 25 out of 50)

15% - Final exam – in class

10% - Midterm project 

20% - Final Project

A(92-100) A-(90,91) B+(87,88,89) B(82-86) B-(80,81) C+(77,78,79) C(72-76) C-(70,71) D+(65-69) D(60-64)


Special Considerations:

If you have a documented disability, which may require individual accommodations, please make an appointment with me prior to the class meeting. We will discuss how to meet your needs to ensure your full participation and fair assessment procedures.



  Topics Covered


Exercises from the book

Session 1

Tues 05/22


Chapter 1:  What is meant by an advanced perspective? 


Presentation:  The number e


Chapter 2:  Real Numbers and Complex Numbers

Entry Survey

Page 15: 9, 10, 12


2.1.1:  2, 12

2.1.2:  3, 4, 5

2.2.1:  1, 4

2.2.2:  2, 4

Session 2

Thurs 05/24

Presentation:  Combinatorics.  Download.


Chapter 3:  Functions



Problems on Board

3.1.1:  1, 3

3.1.2:  5, 6, 7

3.2.1:  1, 3, 6

3.2.2:  5, 11, 12

3.2.4:  1, 3

3.3.3:  1, 2, 5

Session 3

Tues 05/29

Chapter 4: Equations


Presentation:  Graphing Calculators.  Help.

Problems on Board



4.1.1:  4, 5, 6

4.3.1:  2, 5, 6a

4.3.2:  1, 5

4.3.3:  2, 4, 13

4.3.4:  7, 8, 14

Session 4

Thurs 05/31

Discussion – Report 1


Chapter 5:  Integers and Polynomials


Problems on Board


Report 1 due

5.1.1:  5

5.1.4:  2, 5

5.2.1:  5

5.2.2:  7

5.2.5:  1, 4, 5

5.3.1:  1, 3, 4, 5

5.3.2:  1, 3, 4, 5

Session 5

Tues 06/05

Chapter 6:  Number System Structures


Chapter 7:  Congruence and Geometry

Midterm Projects

6.1.1:  5, 6


7.1.1:  2, 3, 5

Session 6

Thurs 06/07

Chapter 8:  Distance and Similarity


Midterm Exam-take home


Problems on Board


WWW due

8.2.1:  1, 3, 4, 5

8.2.3:  1, 2

Session 7

Tues 06/12

Chapter 9:  Trigonometry


Discussion – Report 2


Problems on Board


Report 2 due

9.1.2:  1, 7

9.1.3:  1

9.3.1:  2, 3

9.3.2:  1, 4, 5

Session 8

Thurs 06/14

Chapter 10:  Area and Volume

Chapter 11:  Euclidean Geometry

Problems on Board


Portfolio – 10 problems

10.1.3:  1, 2, 8

10.3.4:  1, 2


11.1.1:  3, 4, 5

Session 9

Tues 06/19

Presentation: Limits

Final Exam


Session 10

Thurs 06/21


Final Projects


Exit Survey



Class Period Structure


Go over exercises from the previous night in groups


Put selected problems on the board




New material


Begin exercises on new material


Reports:  Pick a question from this list, consult at least one source other than our book (see material in class), and write a 2-3 page report.  All quotes are to be properly cited and your source material needs to be well documented. 


Topics for Reports:


1.     There are many issues related to the topic of cheating. How can we prevent it in the classroom?  How do we deal with it when it happens?  How do you feel about cheating?

2.     Besides knowledge of topics in mathematics, what outcomes do we want our students to achieve as a result of their high school education?  Ideas:  properly using and understanding symbolism, having an open mind to new ideas, be able to understand multiple forms of a correct answer and alternative equivalent definitions, be motivated to continue in mathematics, basic mathematical knowledge becomes part of the students intuition, understanding that mathematics is a system of statements built on logic, be able to apply deductive reasoning, understanding that mathematical statements are either true or false.

3.     Do students want to know WHY?   Take, for example, the quadratic formula.  We want high school students to learn to apply it but also understand why it is true.  Take the formula for the derivative of a power function in calculus.  Can you explain why it is true?  Is it important to you to know why?

4.     College level students are often weak in using fraction, exponents, logarithms, and trig functions.  Drill in these areas can start by into using specific numbers and build to simplifying examples using formulas with variables and constants.

5.     Can anyone learn math?  What issues are common among students who say they are not good at math, or not math people?  It takes effort to learn the subject.  Some students have no motivation to do so.  Can we change their minds?  Do we need to convince more students to continue to study math in college?  Find some articles.

6.     What questions could a high school student ask that may require college level understanding of mathematics to answer.  Give examples of questions and answers that are broken down to the high school level.

7.     What is abstract thinking?  Give examples.  Compare to metaphors.

8.     What is the difference between forms of functions, types of functions and families of functions?  Give examples of each.

9.     As teachers, you have to be good at math.  What does that mean?

10. How can you find time for each individual student to find his or her weaknesses? 

11. What are some of the major stumbling blocks from elementary education?  Fractions!

12. Use the GSEs and GLEs for this question.  Choose one goal.  What is the best way to learn this concept or topic?  What learning materials can you use to teach this concept or topic?

13. What are some ideas for courses that are not generally taught in high school?  How would you fit this course into the high school curriculum? 

14. The development of a mathematical idea may have been initiated by an application.  Name some mathematical concepts and their applications.  For instance, parallel.  Using the definition: two curves in 2-dimensional space are parallel if they are equidistant apart, train tracks are parallel.  What about lines of longitude and latitude?

15. The more skilled and practiced you are at mathematics, the more of it is in your subconscious.  You have more intuition.  This happens over time.  Your subconscious is always at work.  How can you use this to problem solve?  Can you do this when you are experiencing math anxiety?

16. Can we teach abstract thinking?  Give examples from math and real life that may illustrate abstract thinking.  Example from math the number system of the real numbers under addition and the positive real numbers under multiplication are both groups. 

17. How is deductive reasoning applied in other subjects outside of math?

18. Name some words that mean something in English and are co-opted to mean something else in math.  Name some words that have no meaning outside of math.

19. What is a number?  Is a telephone number a number? What is a number system?

20.   I have heard students say, “I hate proofs”, even teacher candidates.  What is behind this statement?  How does learning to prove theorems help teacher candidates to be better teacher?

21.  Other…


Ideas for Projects


1.      Given a function f(x) and a number a in the domain of f(x), find the formula for a polynomial P(x) of degree 3 that has the following properties.  The first, second, and third derivatives of f(x) and P(x) at x=a are the same.  Compare your answer to the Taylor Polynomial of degree 3 for f(x) at x=a.  (This question leads to many others and can be turned into a project.)

2.      There is a division theorem about the integers that says, for any 2 integers, a and b, when dividing a by b, there is a remainder r, which is an integer such that 0 ≤ r < b.  Similarly, for polynomials, there is a theorem that says, for any 2 polynomials, p(x) and g(x), when dividing p(x) by g(x), there is a remainder r(x), which is a polynomial that such that the degree of r(x) is less than that of g(x).   Explain what Integers have in common with polynomials that would explain this similarity.

3.      Applications often drive the development of new mathematics.  What problems were being worked on that led to the development of the derivative and integral?

4.      Other questions involving applications.

5.      Give examples of symbols that are used to mean different things.

6.      Consider the usual definition and characterization theorem of combinatorial trees that gives 4 alternative definitions.  Under what circumstances would one of the alternative definitions be more intuitive than the usual definition?

7.      You learned to derive the volume of a cone in Calculus.  Can you use high school math to explain the formula for the volume of a cone?

8.      We often try to sidestep the rigorous definition of the limit of a function of x as x approaches infinity by using a more intuitive explanation.  Give a description of the meaning of the limit using pictures that is useful and accurate. 

9.      Use the epsilon-delta definition of the limit to prove that limit x -> 2 of x^2  =  4.

10.  Project - Factoring polynomials.  What did you learn from calculus and abstract algebra about factoring of polynomials?  What type of polynomials can we ask high school students to factor (quadratics, cubic, higher powers)?  What type would they have no chance to factor and why?  Make a lesson that explores using the graphing calculator to graph and find roots of polynomials.  What does the graph say about the roots being real or imaginary numbers? Break down into elemental pieces and drill.

11.  Write a lesson that teaches using the quadratic formula to solve quadratics.  Apply to examples like:  a x^2 + a x + b.  Break down into elemental pieces and drill.

12.  Write a lesson that teaches the difference between variables and constants.  Break down into elemental pieces and drill.

13.  Project - In your college courses, you learned the definition of a linear function.  Is a general line a linear function?  What functions are linear?  Give examples of some that are and some that aren’t.  Prove your answers.  Relate this to shifting, compressing, and stretching of graphs of functions.  Write a series of lessons for high school students on this topic.  Break down into elemental pieces and drill.

14.  Define even and odd functions.  For each type, give examples of functions that have the property and some that don’t.  Can a function be both even and odd?  For even and odd functions, what properties do their respective Taylor Series have?

15.  Make up some examples of problems for high school students that can be solved using arithmetic or algebra.  Devise extensions of these problems that require algebra.  For example:  Jane has an average of 87 after 4 tests.  What score does she need on the fifth test to average 90 for all 5?  If the test grade is at least 0 and at most 100, what are her possible overall averages?  Draw a graph.

16.  Some questions that they have probably never seen in any college course.  How to solve?  Brainstorm, free flow of ideas.

17.  Solve all questions in the sample Content Knowledge test in Mathematics for Secondary School Teachers.  Relate each question to the college course in which you learned the topic.

18.  Explain the relationship between real, complex, and imaginary numbers in the context of college abstract algebra.  Give examples of each.  Find a way to motivate learning about complex numbers to high school students.

19.  Illustrate an example given in one of the books:  Proofs Without Words, and Proofs Without Words II.

20.  Problems from Teaching secondary Mathematics, M. Jordan and S. O’Neal, pages 6-10, number 9 on page 15.

21.  Describe the timeline of the development of number systems.

22.  Real numbers can be defined in two ways.  Describe the top down approach of using complete ordered fields and the bottom up approach using rational numbers.

23.  Show that the square root of [ 3 + the square root of 7 – the square root of (8 – 2 times the square root of 7) ]  is rational.   I haven’t solved this yet!

24.  Model the following situation with an equation….. There are many examples of this type.

25.  Make this into many exercises:  Solve some challenging problems in Geometry, involving proofs.  There are many to select from in the following book list.  All of these are being ordered for the library. Methods of Geometry, by Smith; A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, by Rosenfeld, Grant, and Shenitzer; Challenging Problems in Geometry,  by Posamentier, and Salkind; Symmetry, Shape and Space, by Kinsey and Moore; Active Geometry, by Thomas; Transformation Geometry: An Introduction to Symmetry, by Martin;  Elementary Mathematics from an Advanced Standpoint: Geometry, by Klein, Hedrick, and Noble;  Euclidean and Non-Euclidean Geometry:  Development and History, by Greenberg.