Fall 2007
Math 4417 Teaching of Secondary Mathematics
Schedule: 3:45 – 5:05 p.m. Tuesdays and Thursdays
Instructor: Michel Helfgott
Office: 308G Gilbreath Hall
Telephone/Email: (423) 439-6977
helfgott@etsu.edu
Office Hours: T-R 2-3 p.m. or by appointment
Course Description: Math 4417 is a course on Secondary School mathematics from a
higher point of view. We will deal both with methods and content areas, trying to reach a
balance between conceptual and computational aspects of mathematics.
Textbook: “Teaching Secondary Mathematics”, 7th edition, by A.S. Posamentier and J.
Stepelman. Merrill Prentice Hall, 2006.
References:
● Principles and Standards for School Mathematics, NCTM.
● Polya, G., Mathematical Discovery. John Wiley&Sons, New York 1962.
● S.F. Taback, Coordinate Geometry: A powerful tool for solving problems. The
Mathematics Teacher, April 1990, pp. 264-268.
● R. Newburgh, Real, Imaginary, and Complex Numbers: Where does the physics hide?
The Physics Teacher, Vol. 34, January 1996, pp. 23-25.
● M. Helfgott, Computer Technologies and the Phenomenon of Refraction, The Physics
Teacher, Vol. 36, April 1998, pp. 236-238.
Technology Recommended: TI-83 , or similar graphing calculators.
Secondary Mathematics Competency: At the beginning of the academic semester
students will have to take a test to demonstrate proficiency in High School Mathematics
(Algebra II, Geometry, and Trigonometry).
Lesson Presentation: The first part of the textbook on methods of teaching secondary
mathematics (pp. 1-230) will help students deliver one lesson, which should be
appropriate for a High School class. Everyone is advised to read the textbook carefully,
striving for understanding. Moreover, students are expected to write a lesson plan, related
to the above-mentioned lesson, which should be submitted for grading.
Grading Procedure:
Secondary Mathematics Competency Test
Lesson Presentation
Test 1 (parts I and II of content area)
Test 2 (parts III and IV of content area)
Final Test (comprehensive)
25%
15%
20%
20%
20%
Grading Scale:
A (93 and above), A- ( 90-92), B+ (87-89), B (83-86), B- (80-82), C+ (77-79), C(73-76),
C- (70-72), D+ (67-69), D (60-66), F (below 60).
Content Area:
Part I
1. Linear equations in one and two variables. Assorted real-life problems that lead to
linear equations. Basic linear programming.
2. Non-linear equations involving exponentials and logarithms. Annuities.
3. Quadratic equations. Quadratic formula. Justification by “completion of squares”.
Equations that can be reduced to quadratics by a suitable transformation.
4. Diverse applications to physics ( a boat going upstream and downstream, depth of
a well and speed of sound, and the like)
5. Complex numbers. Use of the complex field to solve problems in the real domain,
especially in the areas of number theory and geometry. Roots of cubics and
quartics (comparison between Ferrari’s and Descartes’ method). Examples from
geometry and physics that lead to polynomial equations of third or fourth degree.
Part II
1. Cartesian approach to parabolas and other conics.
2. Maxima and minima of quadratic and biquadratic functions. Solution of
optimization problems in plane and solid geometry.
3. Equation of the tangent to a parabola, ellipse, and hyperbola. Parabolic telescopes.
4. Basic kinematics and parabolas. Range of a projectile. Lambert W function and
closed solution of equations of the type (a  bx)e cx  d
5. Area of a segment of a parabola through a limit process.
6. Comparison between Euclidean and Cartesian methods in geometry.
Part III
1. Geometry of the circle.
2. Justification of the fact that for any circles with perimeters p, p  and radii r , r 
the equality p r  p r  is true. Definition of  . Radian measure of angles.
3. Approximation of  through Archimedes method and by simulation. How to
avoid the phenomenon of “cancellation”.
4. Area of a circle through a limit process.
5. Proof of an important equality: p 2r   360o . The conical hat problem.
6. Basic plane trigonometry. Ptolemy’s theorem. Sine and cosine of the sum and
difference of two angles (different proofs). Law of sines and cosines. Proof of
Heron’s formula using the law of cosines (comparison with a purely algebraic
approach). Application problems (building a tunnel, two-islands, antenna, et
cetera). Huygens and the phenomenon of refraction of light.
7. Volume of a pyramid, cone, and sphere. Alternative use of Cavalieri’s principle.
Part IV
1. Arithmetic-Geometric-Mean inequality (AGM), with particular emphasis on the
cases n  2 and  3 . Cauchy’s proof.
2. Use of AGM and Heron’s formula to solve the isoperimetric problem for any
triangle.
B
3. Finding the minimum of the function x  Ax  through AGM.
x
4. Diverse problems that lead to the need to maximize or minimize functions: tents,
the rectangular box with maximum volume (with and without a square base), the
cylinder with minimum surface area, hanging lamps, et cetera.
Technology
Besides learning how to make sound use of software incorporated in their graphing
calculators, students will learn about some programming capabilities of those electronic
devices. The following is a sample of technology-related topics to be discussed during the
academic semester:
● Pythagorean triples
● The birthday problem
● Recursive formulas
● Approximations of 
Attendance
Students are expected to attend all classes. More than two absences can adversely affect
grades. Please take notice that there will be no makeup tests.
Some Dates
Test 1: Thursday, October 11; Test 2: Thursday, November 29; Final Test: Thursday,
December 13 (10:30-12:30). The last week of classes is set aside for the lesson
presentations.