Elementary School Mathematics Priorities

Introduction

The February, 2006, U.S. Department of Education study, The Toolbox Revisited, tells us
that 80% of the 1992 U.S. high school graduating class went on to college. Only about
half of those students graduated with a bachelor’s degree. The others dropped out.
Inadequate preparation for college mathematics was a major contributor to the drop out
rate. The foundation for K-12 mathematics is laid in the early years of elementary
school. To succeed in college, this foundation must be solid.

A guiding principle of No Child Left Behind is equal opportunity for all children. Every
child should learn the fundamental building blocks of mathematics. No child should be
denied the preparation for high school and college mathematics that opens up the
growing number of career opportunities that require mathematics.

Organization

We first describe some of the basic skills and knowledge that a solid elementary school
mathematics foundation requires. We do this first briefly and then, in indented
paragraphs, we elaborate. We follow our main points with a general discussion, some
final comments and some suggested research topics.

Overview

Provided below is a minimal list of core concepts that must be mastered. They are the
building blocks for all higher mathematics. This is not a curriculum or set of standards
and it is certainly not all that students should learn in elementary school. It is also not
just a list of skills to acquire. Although skills are essential in the list, understanding the
concepts is also essential. This is an attempt to set priorities for emphasis in an
unambiguous fashion.

The amount of high school and college level mathematics that today’s workers
require varies dramatically, but more and more careers are dependent on some
college level mathematics. Early elementary school mathematics is the same for all
students because these students should have all career options open to them. The
system must not fail to offer opportunity to all students. Basics are for everyone.

It is perhaps very difficult for many fourth grade teachers to see the connection
between what they need to teach and why it is necessary for the future engineer,
doctor or architect. Those who regularly teach college mathematics to these
students understand well what is needed from their students coming into college if
they hope to fulfill the necessary mathematics requirements. A strong elementary
school mathematics foundation cannot be overemphasized. In light of this, the
work of our elementary school teachers is extraordinarily important.

There are five basic building blocks of elementary mathematics. Keep in mind,
throughout, that mathematics is precise. There are no ambiguous statements or hidden
assumptions. Definitions must be precise and are essential. Logical reasoning holds
everything together and problem solving is what mathematics allows us to do. These
essential building blocks are not just the foundation that algebra rests on, but, done
properly, prepare the student for algebra and the mathematics beyond algebra.

Precision, lack of ambiguity and hidden assumptions, and mathematical reasoning
are the fundamental defining principles of mathematics and it is difficult to
adequately emphasize their importance. If a problem is not well defined with a
unique set of solutions, it is not a mathematics problem. There can be no hidden
assumptions in a real mathematics problem. Terms, operations, and symbols must
be defined precisely. Otherwise ambiguity creeps in and we are no longer dealing
with mathematics.

Although it is easy enough to say that mathematics is logical, it is more difficult to
describe mathematical reasoning. Mathematical reasoning is what builds the
structure of mathematics. So, in order to understand a mathematical concept, a
student must absorb the mathematical reasoning that develops that concept.
Dividing fractions (invert and multiply) is an important skill in mathematics but it
is mathematical reasoning that explains why invert and multiply is the correct way
to divide fractions. It is important that mathematical reasoning of this sort be
taught and that new skills be understood through mathematical reasoning. This
understanding is important and assumed throughout this discussion. Skills without
understanding have little value, likewise for understanding with no skill. Each is
essential.

Ultimately, solving problems is what mathematics is all about. The content of
mathematics is all designed and built to solve specific types of problems. Our basic
mathematics is fundamental to this enterprise because almost all other mathematics
is built on it. Each new piece of mathematics allows a student to solve a new kind
of problem. To be sure, at the elementary school level, some of these problems
could be solved without the new mathematics being introduced, but that
mathematics becomes more and more necessary as problems get more and more
sophisticated. It is best to practice new mathematics on the easy problems first. It
sometimes appears that learning the new mathematics in order to solve a problem is
harder than just solving the problem directly, but once the newly learned
mathematics feels natural it usually becomes clear that it solves the problems much
more efficiently, and, importantly, can be used later to solve even harder problems
that cannot be solved without the new techniques.

The five building blocks

Numbers: Numbers are the foundation of mathematics and students must learn
counting and acquire instant recall of the single digit number facts for addition and
multiplication (and the related facts for subtraction and division ). Instant recall allows
the student to concentrate on new concepts and problem solving. It is of fundamental
importance in later mathematics.

This heading covers a lot of material in the earliest grades. Students must acquire
some number sense. First, of course, they must learn the numbers, both to speak
them and to write them. This comes along with counting and a working familiarity
with and understanding of commutativity, associativity, and distributivity.
Students will learn how to add (and then multiply) single digit numbers before they
learn instant recall of these facts. They must have an understanding of addition,
subtraction, multiplication and division that underlies the ability to instantly recall
these elementary number facts. Mathematics is built level by level. Multi-digit
addition and multiplication are built up from single digit operations using the place
value system and the basic properties of numbers such as distributivity. The
general operations reduce to the single digit number facts. Whatever their level of
understanding, students without instant recall of these foundational single digit
number facts are severely handicapped as they attempt to pursue the next levels of
mathematics. In later courses, the student who has to quickly do the single digit
computations, even if in their head, rather than just recall the answers, will find
they are unable to focus completely on learning and understanding the new
mathematics in their course.

Place value system: The place value system is a highly sophisticated method for
writing whole numbers efficiently. It is the organizing and unifying principle for our five
essential building blocks. Although its importance is often overlooked, it is the
foundation of our numbering system, and, as such, deserves much more attention than it
usually gets. It is much more than just hundreds, tens and ones. Arithmetic and algebra
are the foundation for college level mathematics. A solid understanding of the place
value system, and how it is used, is the foundation for both arithmetic and algebra.
Arithmetic algorithms can only be understood in the context of the place value system.
Since understanding is crucial, it begins with the place value system. Elementary school
mathematics must prepare students for algebra. Working with polynomials in algebra is
just a slight generalization of the place value system. The place value system is essential
algebra preparation.

The place value system is the foundation of our numbering system. The efficiency
of the arithmetic algorithms are based on it. A real understanding of the basic four
algorithms rests on a firm grasp of the place value system. Multiplication, for
example, is little more than the combination of the place value system,
distributivity, and single digit math facts for multiplication. This combination is
the mathematical reasoning that makes the multiplication algorithm work.

The algebra of polynomials is just a generalization of the place value system. The
place value system is based on 1, 10, 10 squared, 10 cubed, etc., and polynomials
are based on 1, x, x squared, x cubed, etc. A solid understanding of the place value
system naturally prepares students for the algebra of polynomials.

Without an understanding of the place value system and how it can be used there
can be no real understanding of the rest of elementary school mathematics and all
of the higher mathematics that rests on this. The place value system is learned in
the early grades precisely because everything else depends on it so it must be
taught first. Just because it is taught in the early grades does not mean that it is
either simple or unimportant . On the contrary, it is a deep concept and
understanding it makes all the difference. This puts a heavy burden on the teachers
in these early grades and it is important that they be aware of this.

Whole number operations: Addition, subtraction, multiplication and division of
whole numbers represent the basic operations of mathematics. Much of mathematics is a
generalization of these operations and rests on an understanding of these procedures.
They must be learned with fluency using standard algorithms. The standard algorithms
are among the few deep mathematical theorems that can be taught to elementary school
students. They give students power over numbers and, by learning them, give students
and teachers a common language .

The case for the importance of the standard algorithms for whole number
operations cannot be overstated. They are amazingly powerful. They take the ad
hoc out of arithmetic. They give the operations structure. The theorems that are
the standard algorithms solve the age-old problem of how to do basic calculations
without having to use different strategies for different numbers. They completely
demystify whole number arithmetic. As an elegant, stand alone solution to an
ancient problem they justify themselves.

There is more to the standard algorithms than just a very satisfying solution to a
major problem. As students progress in their study of mathematics they will be
confronted with more and more algorithms. They must start somewhere to learn
about algorithms and these are the easy basic algorithms that prepare students for
learning more difficult, complicated algorithms later on.

In high school and college mathematics these very same algorithms get slightly
modified and generalized and used in different settings with new mathematics.
This happens many times over and a mastery of the original algorithms makes this
process an incremental one. The standard algorithms put all students and teachers
on the same page when they make these transitions.

The standard algorithms are useful in other ways as well. The long division
algorithm is probably the most important in this sense. With it, for example, it is
quite easy to see that all rational numbers give rise to repeating decimals (any
repeating decimal is also a rational number). It, by its very nature, also teaches
estimation and begins to prepare students to understand convergence, a basic step
towards calculus.

More operations than just these four come into mathematics, these are just the first
four. These operations teach about operations. New operations fit into a pattern
first developed with these basic four. They form a firm foundation for the
conceptual development of future mathematics for the student such as the extension
of these operations to rational numbers and complex numbers as well as the
extension to polynomials and rational functions in algebra.

Fractions and decimals : The skills and understanding for the four basic arithmetic
operations with whole numbers must be extended to fractions and decimals, and fractions
and decimals must be seen as an extension of whole numbers. Students must become
proficient with these operations for fractions and decimals if they are to pursue additional
mathematics. Again, understanding fractions is a critical ingredient for algebra
preparation. A solid grounding in fractions is a necessary prerequisite for understanding
ratios, which show up everywhere including business.

Whole numbers are just not enough. Our number system must be extended to
include fractions (and decimals, which are really just fractions too) in order to solve
a wide variety of problems. Fractions are everywhere in mathematics and in day to
day life so the ability to manipulate them with fluency is essential. They are
seriously intertwined with algebra as well. First, you need them to solve simple
equations like 2x=1, and, second, in algebra, students must learn how to manipulate
fractions involving polynomials, i.e. rational functions. This is, again, an
incremental transition if students can operate with numerical fractions with fluency
and understand and work with their definitions.

Problem solving: Single step, two step, and multi-step problems (i.e. problems that
require this many steps to solve ), especially word or story problems, should be taught
throughout a student’s mathematical education. Each new concept and skill learned can,
and should be, incorporated into a series of problems of more and more complexity. The
translation of words into mathematics and the skill of solving multi-step problems are
crucial, elementary, forms of critical thinking. Developing critical thinking is an
essential goal of mathematics education.

Mathematics is an activity. It is not enough to believe you understand something in
mathematics. You must be able to do something with it. For example,
multiplication is not understood if you cannot do it. Problem solving is what you
do with mathematics. Problem solving at the elementary school level is a well-understood
process that can be taught. Going from one step to two step to multi-step
problems gradually increases the level of critical thinking.

By solving problems using new mathematics skills a student can confirm their
understanding of this mathematics by doing. New skills allow students to solve
problems that old skills did not suffice for. This reinforces the value of the new
skills.

The difficult process of extracting a mathematics problem from a word problem
requires a high level of critical thinking. However, such problems can start with
great simplicity and gradually work up to immense complexity. Mathematical
problem solving is a great place to hone logical critical thinking skills.

In normal daily life people are constantly being called upon to solve very complex
problems that are usually not very well posed. The logical thinking and
mathematical reasoning used to solve multi-step mathematics problems develops
the critical thinking necessary to face life’s more complex situations.