



Math 401020, Mathematics for Elementary School Teachers, I, II
Class Work in Math 40104020
The main goal of the Math 40104020 sequence is to help prospective elementary school teachers develop
a conceptual understanding of the mathematics they will teach and through that understanding, an
awareness of young children's mathematical thinking and the various stages of development in their
mathematical growth. During class students listen to lectures, work in groups on specific mathematical
activities, watch and discuss videos of elementary school children working on mathematics. The class
work focuses first on helping the students develop their own mathematical understanding. Then, students
examine how this understanding relates to the elementary mathematics classroom as they consider
children's mathematical work through written synopses and videos of elementary mathematics classrooms.
Finally, as a part of a 6 hour practicum in a local elementary school, students work with a few pupils
on a particular mathematics concept. In this work the prospective teachers are asked to examine what
the pupil understood about the concept as well as what they did not understand and to explain the
reasons for their conclusions. In the first semester the practicum examines the concept of subtraction
in the lower grades and multiplication in the upper grades; in the second semester, concepts of
measurement, perimeter, area, volume are the focus. The main goal of the practicum is to help the
students develop their ability to listen to children's mathematical thinking and to appreciate the role of a
teacher's mathematical knowledge in promoting children's mathematical understanding across the grades.
Students in the course are graded on regular Assignments (46) done at home, closedbook Tests in class
(23), a portfolio of course work, a comprehensive Final Exam, and a typewritten report on their 6 hour
practicum at an elementary school during which they observe for three hours and work with a few pupils
for three hours. The assignments involve some exercises practicing the course material that has been
developed; some investigations to work through, analyze, and draw conclusions from their work; some
reflection on their own mathematical thinking/work or discussion of their classroom observations; some
research of articles or web resources in mathematics education. The practicum focuses on one or two of
the course concepts as observed in the elementary classroom and the practicum experience forms the
basis for two class room discussions and part of one of the Assignments.
Math 4010 (Prerequisite: Math 1050 (College Algebra) with a grade of C or higher)
Concepts Covered:
This course examines the Real Number System from a mature perspective. The emphasis is on developing
a conceptual understanding of number, arithmetic operations and their properties, and models for
explaining these concepts in the elementary school classroom. Students are expected to examine
nonroutine problems, carryout arithmetic operations in different bases, study basic number theory
related to prime numbers, divisibility tests, factoring.Students consider the different representations
of rational numbers as decimals, fractions, and percentages, examine the proof that the square root of
2 is irrational, and address beginning concepts of infinite sums and limits by considering that
1/3 = .333 . . . and discussing the irrational number e.
Throughout the course students examine how the concepts of number and operation grow across the
grades. In particular they examine the mathematical connections between concepts that teachers can make
to help children build on previous knowledge and to encourage mathematical thinking and reasoning. The
focus is primarily on the prospective teacher developing their own mathematical understanding within the
context of the elementary mathematics curriculum. For that reason, examples of questions from elementary
school children are presented throughout the course so that the propsective teachers can consider how
they would respond to the question and can discuss the mathematical aspects of the children's thinking
that is illustrated in the question.
Texts currently used: Mathematics for Elementary School Teachers, by Musser, Burger, Peterson
(Ch 19); The Graph Paper Fractions Book, by Herb Clemens.
Math 4020 (Prerequisite: Math 4010)
Concepts Covered:
This course focuses on two main topics: Data Analysis through Basic Probability & Statistics 
approximately 5 weeks; Elementary Geometry  approximately 10 weeks.
Data Analysis through Basic Probability & Statistics: Students examine ways to present and describe
data with charts and graphs, measures of the center, and measures of spread. The emphasis is on
interpreting data, comparing different measurements, examining conclusions based on the data. Through
simulations in simple experiments the students compare relative frequency and theoretical probabilty
and they use tree diagrams to study permutations and combinations. The basic Probability Laws including
conditional probability are developed.The normal distribution is introduced and the general idea of the
Central Limit Theorem with some applications is developed through experiments.
Elementary Geometry: Students examine basic properties of two and three dimensional shapes from
an informal perspective (regular polygons, platonic solids, symmetry). Mesurement ideas and the concepts
of length, area and volume are discussed and related to children's development of these concepts at
different ages. The irrational number Pi is examined, and the simple formulas for area and volume
developed. Several proofs of the Pythagorean Theorem are explored using manipulatives, geoboards, and
graph paper. The effect of scaling on area and volume is examined through experiments and practical
implications of scaling on the size and shape are considered. The notions of congruence, similarity and
basic Euclidean constructions are introduced. The simpler, basic geometric proofs are developed and
students are expected to justify the simpler constructions. The emphasis is not on formality but rather
on logical reasoning. Students consider the notion of proof from an elementary school persepctive as
well as from a formal prospective. Some connections between algebra and geometry are developed as the
students study the Cartesian coordinate system, the distance formula, slope of a line, and equation of
a circle.
Because Math 4010 is a prerequisite for Math 4020, the students in Math 4020 have a greater appreciation
of the goals of the sequence and of what is meant by conceptual understanding. However, the students
may be encountering many of the course concepts for the first time. Much of the course material is
introduced from a handson, experimental approach. Throughout the course the students discuss the idea
that many concepts in data analysis and geometry result from elementary school children's natural
explorations and that these explorations provide an important context for doing mathematics and
developing a quantitative sense.
Texts currently used: Mathematics for Elementary School Teachers, by Musser, Burger, Peterson
(Ch 1015); The Graph Paper Geometry Book, by Herb Clemens.
For more information on programs in mathematics education at the
University of Utah, please contact Marilyn Keir at
keir@math.utah.edu.
