# 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.

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