The Mathematics Pre-Service Teachers Need to Know docx

The Mathematics Pre-Service
Teachers Need to Know
R. James Milgram
Department of Mathematics, Stanford University, Stanford,
California, 94305
E-mail address:
The author was supported in part by a Grant from the U.S. Department of
Education. Any opinions, findings, conclusions, or recommendations
expressed in this publication are those of the author and do not necessarily
reflect the views of the United States Department of Education.
Copyright 2005, R. James Milgram. Permission granted to reproduce for educational
purp oses and distribute for cost of reproduction and distribution.
For my students at the University of New Mexico and Skip Matthes. To my wife Judy,
my son Jules, my daughter Jean, and dedicated to the memory of Martha, my sister.

Contents
Preface vii
Chapter 1. Introduction 1
1. Why is math important? 3
2. The critical role of mathematics in modern society 6
3. Common misconceptions by pre-service teachers about
mathematics 8
4. The mathematics K - 8 teachers need to know 10
5. Mathematical problem solving 11
6. Chapter 2 - the mathematics students need to know 13
7. Chapter 3 - the core material central to all school mathematics 14
8. The detailed course discussions, chapters 4 - 8 16
9. Chapter 9, experiences teaching pre- and in-service teachers 21
10. The book The Mathematical Education of Teachers 21
Chapter 2. The Basic Topics in K - 8 Mathematics 23
1. Introduction 23

2. Place Value and Basic Number Skills 24
3. Fractions 27
4. Ratios, Rates and Percents 33
5. The Core Processes of Mathematics 37
6. Functions and Equations 46
7. Real Measurement and Measurement in Geometry 51
8. Course Outline for the First Course 57
9. Course Outline for the Fractions Course 70
Chapter 3. Topics Needing Special Attention in all Four Courses 73
1. Introduction 73
2. Precision 74
3. Making Sense of Mathematics for Students 79
4. Abstraction 82
5. Definitions 85
6. Problem Solving: Overview 92
7. Well-Posed and Ill-Posed Problems in K-8 Mathematics 96
8. Problems with Hidden Assumptions 100
9. Problems where Psychology Affects the Outcome 101
10. Patterns in School Mathematics 104
iii
iv CONTENTS
11. Parsing Word Problems 107
12. Real World Problems 110
13. Polya’s Four Step Problem Solving Model 111
14. Working With Problems for Elementary Teachers 114
Chapter 4. Issues in the Basics Course 121
1. Introduction: Foundational Mathematics in the Early Grades 121
2. Whole Numbers: First Steps 122
3. Addition and Subtraction 128
4. Multiplication and Division 146

5. Magnitude and Comparison of numbers 153
6. Place value 157
7. Decimals 168
8. Bringing in the Number Line 171
9. Other systems for writing numbers 175
10. Algorithms and their Realizations 177
11. Algorithms: Addition and Subtraction 187
12. Algorithms: Multiplication 197
13. Rounding, Approximation, and Estimation 207
14. Algorithms: Division 209
15. Factoring, Multiplication and Division 215
16. Fractions: First Steps 218
17. Average, Rates, Ratios, Proportions, and Percents 221
Chapter 5. Fractions, Ratios, Percents, and Proportion 227
1. Definition of fractions and immediate consequences 230
2. Negative fractions 240
3. Arithmetic operations 240
4. Complex fractions 248
5. Percent 250
6. Ratios and Rates 252
7. Alternative Development of Ratios, Rates, and Percents 255
8. Finite decimals 263
9. Infinite decimals 265
10. False Periods for Repeating Fractions 270
11. The two-sided number line and the rational numbers 271
12. The arithmetic operations on rational numbers 272
13. Ordering rational numb ers 279
14. The Fundamental Assumption of School Mathematics 280
15. Sample Problems from Other Nations 282
Chapter 6. The Role of Technology in Mathematics Instruction 289

1. Introduction 289
2. Introducing the Graphing Calculator and Its Functions 290
3. Introducing Calculators Into The Classroom 292
4. Mathematical Activities 315
CONTENTS v
5. References 334
Chapter 7. Discussion of Issues in the Geometry Course 337
1. Introductory Comments 337
2. Lines, Planes and Figures in Space 338
3. Length and Perimeter 342
4. Angles and Arc Length on the Circle 349
5. Polygons in the plane 353
6. Measurement, Perimeter, Area, and Volume 354
7. Congruence and Similarity 366
8. Grade 8: Scale Factors 372
9. Coordinate Geometry 374
10. The Euclidian Group: I 377
11. Euclidean Group II: Reflections and Applications 381
12. Optional discussion of relation to optics 386
13. Similarity and Dilations 388
14. Geometric Patterns - Symmetry 389
15. Geometry in Space 392
16. Length and Euclidian Group in Space 393
17. A Problem Solving Example in Plane Geometry 394
Chapter 8. Discussion of Issues in the Algebra Course 401
1. Introduction 401
2. Objectives of the Algebra Course and Key Definitions 402
3. Variables and Constants 405
4. Decomposing and Setting Up Word Problems 408
5. Symbolic Manipulation 416

6. Functions 417
7. Graphs of Equations Contrasted with Graphs of Functions 422
8. Symbolic Manipulation and Graphs 424
9. Linear Functions 425
10. Polynomials 430
11. Rational Functions 439
12. Inductive Reasoning and Mathematical Induction 450
13. Combinations, Permutations and Pascal’s Triangle 454
14. Problem Solving Applications of Binomial Coefficients 466
15. Compound interest 473
Chapter 9. Experiences in Teaching Math to Pre-Service and
In-Service Teachers 475
1. Comments on the need for mathematician involvement in
pre-service teacher training 475
2. Some points to consider in teaching pre-service elementary
teachers 478
3. A Mathematician’s Thoughts on Teacher In-service Learning 482
4. Comments on the Issues of Pre-Service Teachers 484
vi CONTENTS
5. Mathematics for elementary teachers: “Explaining why” in ways
that travel into the school classroom 488
6. Teaching Math for elementary Ed majors 490
7. The Geometry of Surprise 495
8. Vermont Mathematics Initiative (VMI) 497
Appendix A. Singapore Grade Level Standards Arranged by Topic 499
1. Multiplication and division 499
2. Decimals 500
3. Standard multiplication algorithm 501
4. Rounding, approximation, and estimation 501
5. Standard division algorithm 502

6. Fractions 502
7. Rates, ratios, proportion, and percent 504
8. Lines, planes, space 505
9. Length and perimeter 505
10. Angles and arc-length on the circle 506
11. Length, perimeter, area, volume 509
12. Congruence and similarity 510
Appendix B. Algorithms from the Education Perspective 511
Appendix C. The Foundations of Geometry 515
1. A Model for Geometry on the Line 516
2. A Model for the Plane and Lines in the Plane 520
3. Distance in the Plane and Some Consequences 523
4. Further Properties of Lines in the Plane 526
5. Rays and Angles in the Plane 528
6. Euclid’s Axioms in the model for Plane Geometry 531
Appendix D. The Sixth Grade Treatment of Geometry in the Russian
Program 533
Appendix E. The Sixth Grade Treatment of Algebra in the Russian
Program 543
Preface
It has long been felt that the mathematical preparation of pre-service
teachers throughout the country has been far too variable, and often too
skimpy to support the kind of outcomes that the United States currently
needs. Too few of our K - 12 graduates are able to work in technical areas or
obtain college degrees in technical subjects. This impacts society in many
and increasingly harmful ways, and it is our failure in K - 8 mathematics
instruction that is at the heart of the problem.
This is esp ecially true when we compare outcomes in the United States
with outcomes in countries that do a better job of teaching mathematics,
countries such as Poland, Hungary, Bulgaria, Romania, Singapore, China,

and Japan, to name a few.
It has also been increasingly recognized that if we are to improve our
performance in K - 8 mathematics instruction, pre-service teachers should
take focused, carefully designed courses directly from the mathematics de-
partments, and not, as is often the case, just a single math methods course
taught in the Education School. A focused two year sequence in the basic
mathematics teachers have to know is the minimal mathematics sequence
that pre-service teachers need in order to to successfully teach students in
K - 8.
The United States Department of Education under the guidance of Sec-
retary Paige awarded an FIE (Fund for the Improvement of Education) grant
to Doug Carnine, Tom Loveless and R. James Milgram in 2002 to analyze
the reasons for the success of these foreign programs and produce a book,
designed for the use of mathematics departments in constructing a two year
sequence sequence of courses that will achieve this goal.
A critical part of the project was an advisory committee comprised of
many of the top people in this country concerned with the issues of K - 12
mathematics education and outcomes. Their advice has been critical in the
development of this book.
The members of the advisory committee:
Prof. Richard Askey, Department of Mathematics, University of Wisconsin
(emeritus)
Prof. Deborah Ball, School of Education, University of Michigan
Prof. Hyman Bass, Department of Mathematics and School of Education,
University of Michigan
Prof. Sybilla Beckmann, Department of Mathematics, University of Georgia
vii
viii PREFACE
Dr. Tom Fortmann, Mass Insight Education, Boston, Massachusetts
Prof. Sol Friedberg, Department of Mathematics, Boston College

Prof. Karen Fuson, School of Education, Northwestern University (emerita)
Prof. Ken Gross, Department of Mathematics, University of Vermont
Prof. Roger Howe, Department of Mathematics, Yale University
Kathi King, Messalonskee High School, Oakland, Maine
Prof. Jim Lewis, Department of Mathematics, University of Nebraska
Prof. David Klein, Department of Mathematics, California State University,
Northridge
Prof. Stan Metzenberg, Department of Biology, California State University,
Northridge
Prof. Ira Papick, Department of Mathematics, University of Missouri
Prof. Tom Parker, Department of Mathematics, Michigan State University
Prof. Paul Sally, Department of Mathematics, University of Chicago
Prof. Uri Treisman, Department of Mathematics, University of Texas at
Austin
Prof. Kristin Umland, Department of Mathematics, University of New Mex-
ico
Prof. H H. Wu, Department of Mathematics, University of California,
Berkeley
We have also benefitted from the advice of Barry Garelick and Karen
Jones-Budd.
Prof. Klein played a critical role in the writing of most of the chapters
3 - 8. Prof. H H. Wu also deserves special thanks for help beyond the call,
as do Prof. Beckmann, Prof. Fuson Prof. Parker, and Prof. Umland.
A second component of the FIE grant was to study the issues needed to
construct successful in-service mathematics training. Both Prof. Sally and
Prof. Gross have been running long-term in-service training and the grant
has helped them collect data on their outcomes, though, at this time, the
data is still being analyzed.
We would like to thank Susan Sclafani, Assistant Secretary of Education,
and above all Pat Ross of the U.S. Department of Education for their help

and support.
We would also like to thank Tom Kelly at Cappelli Miles [Spring] for
assistance with design and layout, as well as the people at Direction Service
who managed the grant, particularly Aimee Taylor and Marshall Peter.
CHAPTER 1
Introduction
It is well known that for many years mathematics outcomes for K - 12
students in this country have lagged far behind what they should be. This is
clearly illustrated by the results of the TIMSS tests, which show our students
about average internationally in grade 4, significantly below average in grade
8, and near the bottom by grade 12. It is also illustrated by the very low
numbers of United States students who graduate from college with degrees
in technical areas.
The level and quality of the highest mathematics courses that students
successfully take in K - 12 is the greatest single predictor of degree com-
pletion in college, and the data clearly show that Algebra II is the college
“gatekeeper.”
1
Mathematics is the key component of success in any technical area. If we
are to prepare our students to maximize their opportunities to succeed in
today’s society, then improving their backgrounds in mathematics is the key.
Moreover, there is only so much that can be done to improve outcomes by
improving the quality of the texts they use and focusing instruction on the
most critical topics. In California, in 1997 - 1998, for the first time in many
years, mathematicians were asked to write the state mathematics standards
1
Clifford Adelman, Answers in the Tool Box, Academic Intensity, Attendance Patterns, and
Bachelor’s Degree Attainment, U.S. Dept. of Education, 1999, p. 17
1
2 1. INTRODUCTION

and the California Mathematics Framework that guides instruction and cur-
riculum selection. The initial results were very promising bringing California
math outcomes from second from the bottom among all states to something
much more respectable:
but over the next two years the results flattened out. Nobody knows for cer-
tain what the cause was, and it is clear that student outcomes in California
were still nowhere near where they should be. But it is highly plausible that
the reason lies in limitations in teacher math content knowledge, especially
in the lower grades.
2
It is perhaps surprising to a number of people that teacher content knowl-
edge matters for student outcomes even in the first grade, but recent research
of Deborah Ball, et. al.
3
shows exactly this. They point out that
“Many kindergarten and first grade teachers explain their choice
of grade level by referencing both their love of young children and
lack of mathematics knowledge. However, our analysis suggests
that mathematical knowledge for teaching is important, even at
this grade level, in our sample schools.”
The effect of teacher content knowledge by grade three is very dramatic
according to this study -
2
Our teachers, as a group, are remarkable people, doing an extremely difficult job with
dedication, intelligence, and care, but the preparation they are given for the task is the issue in
this book. Comments on their limitations, such as the one this footnote refers to, are not directed
at the teachers, but at the job we do in preparing them. In this regard, Chapter 9 consists of
essays by a number of research mathematicians who have worked with pre-service teachers. Their
experiences have generally been very positive.
3

Heather C. Hill, Brian Rowan, Deborah Loewenberg Ball, Effects of teachers’ mathematical
knowledge for teaching on student achievement.(2004)
1. WHY IS MATH IMPORTANT? 3
“In third grade, its effect size rivals that of SES and students’ ethnic
and gender affiliation and in the first grade models, the size is not
far off. This suggest that knowledgeable teachers can positively and
substantially affect student learning of mathematics, and the size
of this effect is, at least in this sample, in league with the effects of
student background characteristics.”
 Course instructors need to instill a sense of “mission” to help
pre-service and in-service teachers believe they must prepare
students at all levels for jobs that will require more mathe-
matical education. Pre-service and in-service teachers need
to become as concerned about students who have difficulty
with math as they are for students who have difficulty with
reading. This is currently not the mindset of K-12 teachers.
4
Among the things that have to be done to help change this mind-set
5
are
• to clarify for pre-service and in-service teachers the reasons why
mathematics is essential for todays students,
• to give pre-service and in-service K - 8 teachers a much better
grounding in the subject.
1. Why is math important?
There is a common perception that “math is for nerds” and that being
good at mathematics is not important. This misperception is also shared
by some faculty members in our education schools and some of our K -
12 teachers. In fact even misconceptions about the benefits of studying
mathematics are common and tend to focus on advantages that may have

been important many years back but are not nearly as critical today. Liping
Ma, one of the best known mathematics educators in the United States,
gave a presentation at the International Congress of Mathematics Education
which had as one of its main themes, the reasons for studying mathematics.
4
The dangerous bend symbol is used throughout this book to indicate a point where special
care is needed.
5
It is important to keep in mind that the attitudes that teachers bring to the job are moulded
by a combination of their own K-12 educations, their family backgrounds, and their college prepa-
ration for the task. When we talk ab out “changing this mind-set” we are strictly discussing their
college preparation, more specifically, their preparation to teach the mathematics component of
the material in K - 8. In particular, it is our perspectives and those of their education school
professors that are at issue.
4 1. INTRODUCTION
This is what she came up with:
and “practical” is expanded as “counting sheep, measuring land, compiling
the calendar, collecting tax, paying salaries, merchant.”
This situation comes from our earliest history and continues to the
present day. From a short history of mathematics education in this country
by Alan Tucker we have the following remarks: “The country’s first colleges,
created to train ministers, taught no mathematics or science. There was no
training for teachers and theirs was one of the lowest ranked professions in
early America.
“Some founding fathers argued that a voting citizenry needed a deeper
education. For example, George Washington wrote, ‘The science of figures.
. . is not only indisp ensably requisite in every walk of civilized life, but the
investigation of mathematical truths accustoms the mind to correctness in
reasoning.’ However, Washington’s type of education was associated with
landed aristocracy, whose learning and power in Europe most immigrants

to the U.S. despised. Further, it was deemed of little value in business. Our
country quickly developed a tradition of anti-intellectualism in parallel to
its support of basic education for all citizens.
“While other countries were developing academically demanding goals in
the early 1900’s for high school education, focused on either vocational train-
ing or college preparation, U.S. high schools had vague educational goals.
To many reformers, they were foremost semi-custodial institutions to keep
young people out of dangerous factories. The NCTM
6
was formed around
1920 to fight efforts to eliminate any mathematics course as a requirement
for high school graduation.”
6
NCTM is the National Council of Teachers of Mathematics, the national umbrella orga-
nization that plays an analogous role to that of the MAA and AMS for K - 12 mathematics
teachers.
1. WHY IS MATH IMPORTANT? 5
Diane Ravitch, the noted education historian points out “At every level
of formal education, from nursery school to graduate school, equal opportu-
nity became the overriding goal of postwar
7
educational reformers. Some-
times those who led the battles seemed to forget why it was important to
keep students in school longer; to forget that the fight for higher enroll-
ments was part of a crusade against ignorance, and that institutions would
be judged by what their students had learned as well as by how many were
enrolled.”
8
Beyond the historical antecedents, there are teacher perceptions and
beliefs that have to be dealt with. A recent poll of high school teachers

asked them how many of their students they thought went on to college
after high school graduation. The average response was 23%. There is also
a very common belief that college preparatory math such as calculus in
high school is only for the top 10% or 12% of students. In actuality, over
75% of current high school graduates attempt college within two years of
graduation. Moreover, among the total population of this country over the
age of 18, over 61% of high school graduates have enrolled in either a two
or a four year college. The following data comes from the U.S. Department
of Education and breaks this 61% down more exactly.
A 2003 update of these data showed that the rate of enrollment in college
had increased to 63.7% for people between 20 and 24 years of age in 2002
and to 61.7% overall, (Digest of Education Statistics 2003, Table 9, page
22, NCES, Dec. 2004). Consequently, it is imperative that, in teaching
these courses, the pre-service teachers be made aware of the critical role of
mathematics in society, and that, far from being of limited use, mathematics
is one of the most critical factors in our lives.
Additionally, there is every reason to believe that far more students than
currently major in high tech areas in our colleges have the ability to do this
level of work. For example, when we compare our outcomes with those of
higher achieving countries we have
7
Post 1945
8
Diane Ravitch, The Troubled Crusade: American Education 1945-1980, Basic Books, 1983,
pp. xi - xii.
6 1. INTRODUCTION
Thus, the most advanced mathematics students in the United States,
about 5 percent of the total age cohort, performed similarly to 10
to 20 percent of the age cohort in most of the other countries.
9

2. The critical role of mathematics in modern society
 Current estimates are that over the next 15 years at least
3.3 million jobs and $136,000,000,000 in wages will move to
East Asia Forrester Research, Cambridge, MA. The major
component of this situation traces to the poor mathematics
backgrounds of our citizens.
Often we mathematicians are inarticulate when challenged to explain
why mathematics is such a critical aspect of K - 12 education today. We
know it is, but are so involved in the subject that we have not looked at
the broader perspective of the way in which mathematics actually interacts
with today’s society. So what follows are some observations that may well
be useful in this regard. It is strongly suggested that at a number of points
in these courses for pre-service teachers, the instructor should take a few
minutes to explain some of the (real) core applications of mathematics, so
as to break the mind-set that mathematics is only good for amusement,
aesthetics, mental discipline, counting sheep, and collecting tax.
The applications of mathematics. The usual reasons given in school
mathematics for studying mathematics are because it is beautiful, for “men-
tal discipline,” or a subject needed by an educated person. These reasons
are naive. It doesn’t matter if students find the subject beautiful or even
like it. Doing mathematics isn’t like reading Shakespeare, something that
every educated person should do, but that seldom has direct relevance to
an adult’s everyday life in our society. The main reason for studying math-
ematics is that our society could not even function without the applications
of a very high level of mathematical knowledge. Consequently, without a
real understanding of mathematics one can only participate in our society
in a somewhat peripheral way. Every student should have choices when
he or she enters the adult world. Not learning real mathematics closes an
inordinate numb er of doors.
The applications of mathematics are all around us. In fact, they are the

underpinnings of our entire civilization, and this has been the case for quite
a long time. Let us look at just a few of these applications. First there are
buildings, aqueducts, roads. The mathematics used here is generally avail-
able to most people, but includes Euclidean geometry and the full arithmetic
9
S. Takahira, P. Gonzales, M. Frase, L.H. Salganik, Pursuing Excellence: A Study of U.S.
Twelfth-Grade Mathematics and Science Achievement in International Context, U.S. Dept. of
Education, 1998, p. 44
2. THE CRITICAL ROLE OF MATHEMATICS IN MODERN SOCIETY 7
of the rationals or the reals. Then there are machines from the most primi-
tive steam engines of 150 years back to the extremely sophisticated engines
and mechanisms we routinely use today.
Sophisticated engines could not even happen until Maxwell’s use of dif-
ferential equations in order to stop the engines of that time from flying apart,
stopping, or oscillating wildly, so the mathematics here starts with advanced
calculus. Today’s engines are far more sophisticated. Their designs require
the solutions of complex non-linear partial differential equations and very
advanced work with linear algebra.
Today a major focus is on autonomous machines, machines that can do
routine and even non-routine tasks without human control. They will do
the most repetitive jobs, for example automating the assembly line and the
most dangerous jobs.
Such jobs would then be gone, to be replaced by jobs requiring much more
sophisticated mathematical training. The mathematics needed for these
machines, as was case with engines, has been the main impediment to actual
wide-scale implementation of such robotic mechanisms. Recently, it has
become clear that the key mathematics is available, (the mathematics of
algebraic and geometric topology, developed over the last 80 - 90 years),
and we have begun to make dramatic progress in creating the programs
needed to make such machines work. Because of this, we have to anticipate

that later generations of students will not have the options of such jobs,
and we will have to prepare them for jobs that require proportionately more
mathematical education.
But this only touches the surface. Computers are a physical implementation
of the rules of (mathematical) computation as described by Alan Turing
and others from the mid 1930’s through the early 1940’s. Working with a
computer at any level but the most superficial requires that you understand
algorithms, how they work, how to show they are correct, and that you
are able to construct new algorithms. The only way to get to this point
is to study basic algorithms, understand why they work, and even why
these algorithms are better (or worse) than others. The highly sophisticated
standard algorithms of arithmetic are among the best examples to start.
But one needs to know other algorithms, such as Newton’s method, as well.
What is essential is real knowledge of and proficiency with algorithms in
general, not just a few specific algorithms.
And we’ve still only touched the surface. Students have to be prepared to
live effective lives in this world, not the world of 500 years back. That world
is gone, and it is only those who long for what never was who regret its
passing. Without a serious background in mathematics one’s options in our
present society are limited and become more so each year. Rob ert Reich
described the situation very clearly in a recent article.
10
10
This article is from The American Prospect Online, December, 2003, and the URL is
/>8 1. INTRODUCTION
“The problem isn’t the number of jobs in America; it’s the quality of
jobs. Look closely at the economy today and you find two growing
categories of work – but only the first is commanding better pay and
benefits. This category involves identifying and solving new prob-
lems. Here, workers do R&D, design and engineering. Or they’re

responsible for high-level sales, marketing and advertising. They’re
composers, writers and producers. They’re lawyers, bankers, fi-
nanciers, journalists, doctors and management consultants. I call
this ‘symbolic analytic’ work because most of it has to do with ana-
lyzing, manipulating and communicating through numbers, shap es,
words, ideas. This kind of work usually requires a college degree.

“The second growing category of work in America involves personal
services. Computers and robots can’t do these jobs because they
require care or attentiveness. Workers in other nations can’t do
them because they must be done in person. Some personal-service
workers need education beyond high school – nurses, physical ther-
apists and medical technicians, for example. But most don’t, such
as restaurant workers, cabbies, retail workers, security guards and
hospital attendants. In contrast to that of symbolic analysts, the
pay of most personal-service workers in the U.S. is stagnant or de-
clining. That’s because the supply of personal-service workers is
growing quickly, as more and more people who’d otherwise have
factory or routine service jobs join their ranks.”
Here is how things change. Originally, the public school curriculum was
designed under the assumption that students would, in the main, work on
assembly lines, or do physical labor. But assembly lines today are highly
mechanized, and much of the current assembly line work demands high
level programming and maintenance of robotic mechanisms. Similar changes
have occurred with respect to direct physical labor. For example, even
40 years back, dock-work was brutal - lifting and carrying. Today, the
vast majority of this work is done by huge robotic mechanisms, and the
dock-worker of today spends most of his or her time controlling a very
expensive and complex machine or smaller fork-lifts. The usual requirement
is two years of college to handle the big machines, because running these big

machines entails extensive non-routine problem solving. Thus, pre-service
teachers have to be carefully educated in what we know and what we don’t
know about solving problems.
3. Common misconceptions by pre-service teachers about
mathematics
There are three main areas where mathematicians teaching mathematics
courses for pre-service teachers have to contend with serious misconceptions.
3. COMMON MISCONCEPTIONS BY PRE-SERVICE TEACHERS ABOUT MATHEMATICS 9
The first is the widely held view that mathematics is, somehow, innate
11
.
Pre-service teachers will often indicate that they do not see the need to learn
the material being covered because, when the time comes that they actually
need it, they will be able to dredge it up. As a result, it is quite impor-
tant that, throughout these courses, the instructor should keep reminding
students of the nature of mathematics and the fact that mathematics is,
entirely, a human construction that has developed over thousands of years
and reflects the contributions of many of the most intelligent people from
past civilizations all the way to recent times.
Speaking of the nature of mathematics, this is another point where stu-
dent misconceptions get in the way. Often they will have been taught that
mathematics is “the study of patterns,” or as was some people’s view in the
mid-nineteenth century, “the science of numbers, and the art of computing
by means of them.” If either of these were ever true, they are certainly
not true today, as mathematics has grown in breadth over the centuries. It
is impossible to “define” mathematics today. About the best we can do is
roughly describe it but when we do two things stand out:
(1) Precision (precise definitions of all terms, operations, and the
properties of these operations)
When we look at typical

U.S. elementary school
texts and workbooks,
one of the most striking
things is the lack of
precision, especially
when contrasted with
texts from high
achieving countries.
Thus, for most of the
pre-service students in
these courses, precision
is one of the most
critical things we can
teach them.
(2) Stating well-posed problems and solving them. (Well-p osed
problems are problems where all the terms are precisely defined and
refer to a single universe where mathematics can be done.)
The students should constantly have this perspective on mathematics in
front of them throughout these courses.
The third misconception is, in many ways the hardest to deal with.
Many subjects in K - 12 appear to students to be little more than learning
lists of facts and repeating them as necessary. Instruction in mathematics
usually tends to have the same character in the United States. “Methods” of
solving certain classes of problems are carefully categorized and then taught
separately. For example there are one step equations, two step equations,
three step equations and four step equations when looking at linear equations
in one variable. Likewise, multiplying two binomials is usually not taught
as a consequence of the distributive rule, but rather as a rigid process called
“foiling” (first, outer, inner, last). Consequently, pre-service teachers expect
to learn the content of these courses in exactly the same way. Thus, when

the focus of discussion deals with underlying principles as is usually the case
in the recommended content for these courses, the audience will, at least
initially, be likely to be totally confused. This situation has to be handled
with patience.
11
An interesting article on this recently appeared in Science, P. Pica, C. Lemar, V.Izard,
S. Dehaene, Exact and approximate arithmetic in an Amazonian idigene group, Science, 306
(2004), 499-503. They study addition and estimation in a tribe with a language that does not
have expressions for numbers larger than 5. What was found is that these people could estimate
as well as native French speakers using less than, greater than, and equal to, but they had extreme
difficulty with exact addition and subtraction with numbers larger than 5.
10 1. INTRODUCTION
4. The mathematics K - 8 teachers need to know
In the mid 1980’s Lee Shulman and others began the study of the math-
ematical knowledge that teachers need to know. This was broken down
This book covers the
issues in designing a
four course sequence for
pre-service teachers who
do not intend to teach
high school
mathematics. In many
states there is a single
certificate for K-8
teachers. Consequently,
the material here
represents the minimal
amount of mathematics
that teachers in these
grades must know. In

states where there are
separate K-6 and
middle school
certificates, it is likely
that mathematics
beyond this core could
be required for the
middle school
certification. However,
we do not discuss such
material here.
into
(1) Content knowledge, which includes both facts and concepts as well
as the reasons why the facts are true
(2) Pedagogical content knowledge, which goes beyond content knowl-
edge to the subject of the content that is needed for teaching which
includes such things as what makes learning specific content easy
or difficult
(3) Curriculum knowledge, which involves knowing how topics are ar-
ranged over the course of a academic year or over the years and
ways of using such things as textbooks to organize their teaching.
Since then this area has been developed much further by a number of re-
searchers, among them being Deborah Ball and Liping Ma.
In this work we take the view that the best foreign programs in math-
ematics are so overwhelmingly successful with such a high percentage of
students that we can learn a great deal about the issues above by studying
how these programs handle the core topics and their development. Fore-
most among these programs is the Russian mathematics program that was
developed during the 1930’s and 1940’s. It was used in Israel from the late
1940’s to the early 1970’s during which time their outcomes were the best

in the world. It was adopted by China during the 1950’s, and from China
was modified for use in Singapore. It is also the core program underlying
the current programs in former iron countries such as Bulgaria, Hungary,
Poland, and Romania. Thus, we pay a great deal of attention to the way
in which the Russian program develops the core concepts in mathematics
during the early years, and we also reference the Singapore program exten-
sively to learn about how the three topics above are treated in countries
where instruction in mathematics is successful.
Two things are notable when looking at these programs. The first is
that aside from whole numbers, everything is precisely, though grade appro-
priately, defined for students. Thus, when it comes time to define even and
odd numbers in the second grade Russian program, we find
From the series of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, first give those which are divisible by 2, and then
those which are not divisible by 2.
Numbers that are divisible by 2 are called even.
Numbers that are not divisible by 2 are called odd.
One could quibble that instead of talking about “numbers” they should
have said “whole numbers,” but fractions have not even been defined at this
5. MATHEMATICAL PROBLEM SOLVING 11
point in the second grade course. Before continuing we should clearly note
the following
 Who gets hurt when definitions are not present? The em-
phasis on precision of language and definitions matters most for
exactly the most vulnerable of our students. It is these students
who must b e given the most careful and precise foundations. The
strongest students often seem able to fill in definitions for them-
selves with minimal guidance. On the other hand, foreign outcomes
clearly show that with proper support along these lines, all students
can get remarkably far in the subject.

The second thing that is notable in the programs of the high achieving
countries is the level of abstraction that is present in the problems students
are expected to do in these early grades. Variables have been introduced
and are routinely used, so students set up and solve simple equations in
second grade and quite sophisticated equations in third grade. There is a
strong belief in the education community in the United States that “young
students learn strictly in context,” which means that they do not believe
young students can handle abstraction. This belief is not supported by
research, nor is it supported by the outcomes in the high achieving countries.
We will discuss the mathematics programs used in the high achieving
countries in detail in chapters 3 - 8.
Summary. In summary, we need to first describe the core mathematics
that should be covered in, say, grades K - 7. This turns out to be far less
than the content that is typically required by state standards in this country.
It is consistent in all the programs in successful foreign countries that the
number of topics covered in the early grades is far less than is covered here.
But these are the key topics, and since they are covered to far greater depth
than is the case here, those foreign students have a much more solid and
dependable base, which makes it far more likely that they will be able to
succeed in the more advanced topics in mathematics, science and related
areas that are so essential today.
5. Mathematical problem solving
To start, we must deal directly with the widespread belief that we know
how to teach students how to solve problems. Everyone needs to be aware
of this basic truth:
Problem solving is currently an arcane art.
We do not know how to reliably teach problem solving. The most effective
method for communicating this process seems to be to have a mathematician
stand in front of a class and solve problems. Many students seem to be able
to learn something of this multi-faceted area in this way, but, as we will see,

the ground has to be carefully prepared before students can take advantage
of this kind of experience.
12 1. INTRODUCTION
What will be discussed now is what virtually all serious research mathe-
maticians believe, and most likely most research scientists as well. This is
not what will be found in a typical math methods textbook. Other theories
about mathematical problem solving are current there. It could be that
the focus of the views on problem solving in these texts is concerned with
routine problems where the biggest effort might be in understanding what
the problem is asking. This can be a difficult step, but here we are talking
about solving a problem where the answer is not immediate and requires a
novel idea from the student. It is exactly this level of problem solving that
should be the objective for every student, because, at a minimum, this is
what virtually all non-routine jobs require today.
The hidden part of problem solving. There is a hidden aspect to
problem solving: something that happens behind the scenes, something that
we currently do not know how to measure or explain. It is remarkable, when
you read the biographies of great mathematicians and scientists that they
keep saying of their greatest achievements, I was doing something else and
the answer to my problem just came to me.
12
This is not only true for the
greatest, it seems to be true for every serious research mathematician or
scientist.
Answers and ideas just seem to come out of the blue. But they don’t!
There are verbal and non-verbal aspects to problem solving. Successful
researchers seem to have learned how to involve non-verbal mechanisms in
their brains in analyzing and resolving their problems, and it is very clear
that these non-verbal regions are much more effective at problem solving
than the verbal regions.

In order to engage the non-verbal areas of the brain in problem solving,
extensive training seems to b e needed. This is probably not unlike the
processes that one uses to learn to play a musical instrument.
13
Students
must practice! One of the effects, and a clear demonstration that the process
is working, is when students become fluent with the basic operations and
don’t have to think about each separate step.
For school mathematics, students must practice with numbers. They
must add them until basic addition is automatic. The same for subtraction
and multiplication. They must practice until these operations are automatic.
This is not so that they can amaze parents and friends with mathematical
parlor tricks, but to facilitate the non-verbal processes of problem solving.
At this time we know of no other way to do this, and it is a grim thing to
watch otherwise very bright students struggle with more advanced courses
because they have to figure everything out at a basic verbal level. What
happens with such students, since they do not have total fluency with basic
12
H H. Wu points out that the first example of this that he is aware of in print is due to H.
Poincar´e.
13
It is probably not a coincidence that an inordinate number of professional mathematicians
are also skilled musicians.
6. CHAPTER 2 - THE MATHEMATICS STUDENTS NEED TO KNOW 13
concepts, is that - though they can often do the work - they simply take far
too long working through the most basic material, and soon find themselves
too far behind to catch up.
Skill and automaticity with numbers is only part of the story. Students
must also bring abstraction into play. This is also very commonly an uncon-
scious process. There are huge numbers of choices for what to emphasize

and what to exclude in real problems so as to focus on the core of what
matters. Indeed, it is often far from clear what the core actually is. As was
the case before, one has to practice to facilitate abstraction. How?
One explores the situation, focusing on one area, then another, and accu-
mulates sufficient data so that non-verbal tools in the brain can sort things
out and focus on what matters. But in order to do this, the groundwork has
to be laid. That is what algebra does (or is supposed to do). That is why
students should practice with abstract problems and symbolic manipulation.
Moreover, as we have seen in section 1, Algebra I and more particularly Al-
gebra II are the gate keepers for college. When we think of problem solving
in this way, that is not so surprising.
The need for further study. Our knowledge here is fragmentary and
anecdotal. What has been stated above is highly plausible, and most re-
search mathematicians tend to agree that it fits their experiences. However,
it is not yet possible to assert this knowledge as fact. Basic research needs
to be done, much as was done for reading. The medical and psychological
sciences almost certainly have the to ols to begin such research now. Indeed,
the NIH has recently begun to support work in this direction.
6. Chapter 2 - the mathematics students need to know
In the first seven sections of chapter 2 we discuss the core mathematics
that students should learn in grades K - 7. By the end of grade 8 they should
have also been exposed to a considerable amount of algebra. However, the
discussion here is restricted to the core pre-algebra material that students
must know. This core material breaks up into six main topics:
(1) Basic number skills and place value
(2) Fractions
(3) Ratios, rates and percents
(4) Symbols, equations, symbolic manipulation, solving linear equa-
tions in one variable
(5) Functions and equations

(6) Measurement: the exact measurements of geometry and the mea-
surements with errors that occur in real situations
The final two sections of chapter 2 give outlines for the first two courses or
the first year in the four course sequence that we recommend.
This discussion reflects the mathematics that students need to know,
and consequently the core mathematics that pre-service K - 8 teachers must
know, though it goes without saying that they must know more than just
14 1. INTRODUCTION
this. They must know enough more, at minimum, to accommodate students
who are able to learn more. Also, they must have sufficient further knowl-
edge to be able to explain the mathematics they are teaching as well as why
it is important for students to know, and how it will help them.
However, as pointed out above, teachers must also know more about
this mathematics than just the key mathematical points and issues. They
must be able to diagnose student errors and misunderstandings, and they
must be able to explain these topics to a wide range of students. Thus
it is necessary to explore in much greater detail the material discussed in
chapter 2, expanding on it mathematically as well as discussing the most
important pedagogical issues involved in presenting it. This is what the next
five chapters do.
7. Chapter 3 - the core material central to all school
mathematics
Chapter 3 is key. It develops the central concepts of mathematics, and
covers the following topics:
It appears to be the norm in K - 8 mathematics instruction in the United
States that the material is treated as lists of facts and techniques for students
to learn. Students do not see definitions, and never are exposed to the
7. CHAPTER 3 - THE CORE MATERIAL CENTRAL TO ALL SCHOOL MATHEMATICS 15
general principles underlying school mathematics. For example, the teaching
of the standard algorithms is purely mechanical. Students are taught how to

apply each algorithm and then are drilled on various special cases - perhaps
where special carrying procedures have to be used - so that they can arrive at
the correct answer in all cases. But they are not exposed to the underlying
reasons why the algorithm works.
Such instruction was entirely appropriate many years ago, when it was
essential that people be able to fluently and accurately do such calculations
by hand. However, today, though skill with the basic algorithms is still
important, the need for hand calculation is not nearly what it was. It is
also more imp ortant today that students learn the underlying reasons why
algorithms work. Consequently, the material in chapter 3 must become a
basic part of K - 8 mathematics instruction, and this means that pre-service
teachers not only have to be exposed to it, but actually have to internalize
it.
The problem solving sections in chapter 3. Another critical aspect
of chapter 3 is the detailed discussion of problem solving which takes up
sections 5 - 13. There is very wide agreement that students have to learn
as much as can be taught to them about problem solving. However, the
existing discussions of this extremely recondite topic are not very good,
and the outcomes for today’s students have been discouraging. Thus it is
critical that actual problem solving be a core component of each of these
four courses. Pre-service teachers should clearly understand that, though
there are aspects of problem solving that can be isolated and taught, there
are other aspect which are key, but which nobody really understands today.
The best we can say is that these deeper lying aspects of problem solving
appear to be developed by practice and by study. They depend on fluent
technical skill and a wide background in the area of the problems being
considered.
16 1. INTRODUCTION
8. The detailed course discussions, chapters 4 - 8
The next four chapters concentrate on the four main mathematical topics

to be covered in the two year sequence. Chapter 4 covers foundational
material
Chapter 4 breaks naturally into four parts.
The first part, comprising §2 - 5 covers the introduction of the whole
numbers the basic operations, and magnitude.
The second part, §6 - 9 is a careful discussion of the base-10 place-
value system. This is a core area, and perhaps the key area where student
misunderstandings cripple their further development in mathematics. Place
value is critical in the early years, and too many pre-service teachers only
have a very sketchy understanding of it. Consequently, this should be a
point of particular emphasis in this course.
The third part, §10 - 14 covers algorithms It is very important that
pre-service teachers understand that algorithms play a special role in math-
ematics and particularly in the applications of mathematics. They should