Mathematics content courses for prospective elementary school teachers at the
University of Georgia
There are 3 required mathematics content courses for prospective elementary teachers at
the University of Georgia (in addition to 2 mathematics methods courses and “core”
mathematics courses required of all students):
1) A course in arithmetic:
Course topics: Problem Solving (1-2 days on the handshake problem and possibly also
triangular numbers).
Numbers: (about 3 weeks) The natural numbers, the whole numbers, the rational numbers
(fractions), and the real numbers (decimals). The decimal system and place value.
Representing decimals with bundled objects. Representing decimals on a number line.
Comparing sizes of decimals. Finding decimals in between decimals. Rounding decimals.
The meaning of fractions. The importance of the whole associated with a fraction.
Improper fractions. Equivalent fractions. Simplest form of a fraction. Fractions as
numbers on number lines. Comparing sizes of fractions: by giving them common
denominators, by converting to decimals, and by cross-multiplying. Using other
reasoning to compare sizes of fractions. Solving fraction problems with the aid of
pictures. Percent. Benchmark percentages and their common fraction equivalents.
Solving percentage problems with the aid of pictures. Solving percentage problems
numerically.
Addition and subtraction: (about 4 weeks) Adding and subtracting fractions. Explaining
why we add and subtract fractions the way we do. The importance of the whole when
adding and subtracting fractions, especially in story problems. Recognizing and writing
story problems for fraction addition and subtraction. Recognizing story problems that are
not solved by fraction addition or subtraction. Mixed numbers. Understanding when
percentages should and should not be added. Calculating percent increase and decrease
with the aid of pictures. Calculating percent increase and decrease numerically. Percent
of versus percent increase or decrease.
Multiplication: (about 5 weeks) The meaning of multiplication. Ways of showing
multiplicative structure: with groups, with arrays, and with tree diagrams. Using the
meaning of multiplication to explain why various problems can be solved by multiplying.
Explaining why multiplication by 10 is easy in the decimal system. Why the commutative
and associative properties of multiplication and the distributive properties make sense
and how to illustrate them with arrays, areas of rectangles, and volumes of boxes. Using
properties of arithmetic in solving arithmetic problems mentally. Writing equations that
correspond to a mental method of calculation (to demonstrate the connection between
mental arithmetic and algebra). The distributive property and FOIL. Using multiplication
to estimate how many. The partial products multiplication algorithm. Using pictures and
the distributive property to explain why the standard and partial products procedures for
multiplying whole numbers are valid. Explaining why non-standard strategies for
multiplying can be correct or incorrect. The meaning of multiplication for fractions.
Recognizing and writing story problems for fraction multiplication. Recognizing story
problems that are not problems for fraction multiplication. Explaining why the procedure
for multiplying fractions works. Powers. Scientific notation. Multiplication of decimals:
explaining why the procedure for the placement of the decimal point is valid.
Multiplication of negative numbers. Understanding that multiplication does not always
``make bigger.''
Division: (about 2 weeks) The meaning of division (two interpretations, with or without
remainder). Understanding when the answer to a story problem solved by whole number
division is best expressed as a decimal, as a mixed number, or as a whole number with a
remainder. Why dividing by zero is undefined. The scaffold method of division.
Explaining why the scaffold and standard longhand procedure for dividing whole
numbers works. Explaining why some non-standard methods of division are valid. The
relationship between fractions and division. Calculating decimal representations of
fractions. Explaining the relationship among remainder, mixed number, and decimal
answers to division problems.
Course objectives: To strengthen and deepen knowledge and understanding of
arithmetic, how it is used to solve a wide variety of problems, and how it leads to algebra.
In particular, to strengthen the understanding of and the ability to explain why various
procedures from arithmetic work. To strengthen the ability to communicate clearly about
mathematics, both orally and in writing. To promote the exploration and explanation of
mathematical phenomena. To show that many problems can be solved in a variety of
ways.
2) A course in geometry:
Course topics:
Visualization: Visualizing cross-sections of solid objects. Explaining why there are time
zones. Explaining the phases of the moon. Using diagrams that show the Earth, Moon,
and Sun rays as seen from outer space in order to determine the phase of the Moon and
the time of day at a selected spot on the Earth.
Angles: angles as amount of rotation and as rays meeting at a point. How light reflects.
Angles that the Sun's rays make at the Earth's surface. Proving that the vertical angles
that are created when two lines intersect are equal.
Circles and spheres: definitions of circles and spheres. Ways that circles and spheres can
intersect. Story problems about circles.
Triangles: definition of triangles. Showing that the sum of the angles in any triangle is
180 degrees.
Quadrilaterals and other polygons: definitions of square, rectangle, rhombus,
parallelogram, and trapezoid. Explaining how the various kinds of quadrilaterals are
related. Understanding when relationships can be explained directly from the definitions
and when information that is derived from the definitions, but not stated directly in the
definitions, is needed. Showing relationships with Venn diagrams. Diagonals of
quadrilaterals, especially rhombuses. Angles in quadrilaterals, especially rhombuses. The
sum of the angles in an n-gon.
Constructions with straightedge and compass: bisecting a line segment and bisecting an
angle. Relating the constructions to properties of rhombuses.
Polyhedra and other solid shapes: definitions of prisms, cylinders, pyramids, and cones.
Making patterns for and analyzing prisms, cylinders, pyramids, and cones. The 5
Platonic solids. Explaining why there are no other Platonic solids.
Transformation geometry: reflections, translations, rotations. Defining reflection
symmetry, rotation symmetry, translation symmetry, and glide-reflection symmetry.
Creating designs with specified symmetry. Use of Geometer's Sketchpad.
Congruence: side-side-side triangle congruence and the structural stability of triangles.
Optional: angle-side-angle congruence and the size of one's reflected face in a mirror.
Similarity: explaining how to solve similarity problems by using a scale factor or by
considering “relative sizes.'” Angle-angle-angle criterion for triangle similarity.
Determining distances and heights by using similar triangles.
Measurement: the concept of measurement. The U.S. Customary and Metric systems of
measurement. Reporting and interpreting measurements (proper use of rounding).
Measurable attributes of objects and explaining why different objects in a collection can
be considered ``biggest'' depending on which attribute is used for comparison.
Informally: dimension (1, 2, 3). The distinction between length, area, and volume.
Explaining why we add to calculate perimeters and why we multiply to calculate areas of
rectangles and volumes of boxes.
Converting measurements: by reasoning about multiplication and division and by
“dimensional analysis.” Common errors in converting measurements.
Principles underlying calculations of areas. Calculating areas using only the principles
and the area formula for rectangles. Using the principles to prove the Pythagorean
Theorem. Using the Pythagorean Theorem.
Additional ways to determine areas: approximating areas. Cavalieri's principle about
shearing and area.
Area formulas: explaining why the formula for areas of triangles is valid. Explaining
why the area formula for parallelograms is valid. Understanding why there is no formula
for areas of parallelograms in terms of the lengths of the sides of the parallelogram.
Areas of circles and the number pi. Explaining why pi is between 3 and 4. Explaining
why the area of a circle of radius r is pi times r squared.
Area versus perimeter: understanding that perimeter does not determine area. The range
of possible areas for a given perimeter.
Principles underlying calculations of volume. Determining volume by submersing.
Optional: Archimedes's principle on floating objects. Understanding and using volume
formulas for prisms, cylinders, pyramids, and cones.
The behavior of area and volume under scaling.
Course objectives: To strengthen and deepen knowledge and understanding of
measurement and basic geometry and how they are used to solve a wide variety of
problems. In particular, to strengthen the understanding of and the ability to explain why
various procedures and formulas in mathematics work. To strengthen the ability to
communicate clearly about mathematics, both orally and in writing. To promote the
exploration and explanation of mathematical phenomena. To show that many problems
can be solved in a variety of ways.
3) A course in algebra, number theory, probability, and statistics:
Division of fractions and decimals: The meaning of division for fractions. Recognizing
and writing story problems for fraction division. Understanding the distinction between
dividing by 1/2 and dividing in half. Explaining why the “invert and multiply” procedure
for dividing fractions is valid. Explaining why the procedure for placement of the
decimal point in decimal division problems is valid. Understanding that division does not
always “make smaller.”
Ratio and proportion: the meanings of ratio and proportion. Solving ratio problems
using only multiplication, division, and logical thinking. Explaining the logic behind
setting up proportions by setting two fractions equal to each other and solving these
proportions by cross-multiplying. Understanding when problems can't be solved by a
proportion. Optional: the Consumer Price Index.
Number Theory: definitions of factors and multiples and concrete problems that use and
illustrate these concepts, definitions of greatest common factor and least common
multiple and concrete problems that use and illustrate these concepts. Prime numbers.
The Sieve of Eratosthenes for producing lists of prime numbers. The trial division
method for determining if a number is prime. Factoring counting numbers into products
of prime numbers. Optional: the proof that there are infinitely many prime numbers.
Lightly: consequences of the irrationality of the square root of 3 for making designs with
standard school pattern tile sets. Even and odd: different ways of defining even and their
equivalence. Divisibility tests: explaining the divisibility tests for 3, 5, and 9.
Algebra and functions: patterns, sequences, formulas, and equations. Creating numerical
sequences from picture sequences, creating picture sequences from numerical sequences.
Describing sequences in words and with formulas. Determining a specified entry (say, the
100th) in a repeating pattern. Lightly: sums of sequences (series).
Functions and their graphs. Relating qualitative descriptions of functions to their graphs.
Understanding that the graph of a function is a line exactly when there is a fixed increase
in the output for a given fixed increase in the input.
Basic descriptive statistics: Lightly: designing investigations and gathering data.
Common ways to display data. Three levels of questions about graphs: reading the data,
reading between the data, reading beyond the data. The average of a numerical set of
data. Understanding the average as “making even” or “leveling out.” The median of a
numerical set of data. Showing that different data sets can have the same median but a
different average. Showing that different data sets can have the same average but a
different median. Understanding that “more than half can be above average.”
Percentiles. Understanding the difference between percentile and percent correct.
Probability: Basic principles of probability. Simple probability calculations. Using the
meaning of fraction multiplication to understand simple probability calculations.
Critique of mathematics lessons: students should critique several mathematics lessons
that they taught to elementary school children and discuss ways that the lessons could be
improved or extended. The critique should focus on mathematics content.
Solving, posing and modifying problems: Because fractions, decimals, and percents are
traditionally difficult topics to teach, it is recommended that part of the course be devoted
to solving, posing, and modifying problems involving fractions, decimals, and percents,
especially problems that are relevant to and helpful for the teaching of these topics.
Course objectives: To strengthen and deepen knowledge and understanding of
probability and statistics, elementary number theory and algebra, and how they are used
to solve a wide variety of problems. In particular, to strengthen the understanding of and
the ability to explain why various procedures and formulas in mathematics work. To
strengthen the ability to communicate clearly about mathematics, both orally and in
writing. To promote the exploration and explanation of mathematical phenomena. To
show that many problems can be solved in a variety of ways. To learn to pose and modify
mathematical problems.
Textbook: All three courses use Mathematics for Elementary Teachers, 2nd edition, by
Sybilla Beckmann, published by Addison-Wesley.