UNIVERSITY
OF FLORIDA
LIBRARY
'cUsr
. tc
Ss.
ARITHMETIC in the
ELEMENTARY SCHOOL
Bulletin No. 26
September, 1942
Prepared at
THE CURRICULUM LABORATORY
FLORIDA STATE COLLEGE FOR WOMEN
MARGARET MCCURDIE o
T CoDirectors
THELMA TEW
w. T. EDWARDS, Consultant
f/cr d, STATE DEPARTMENT OF EDUCATION
TALLAHASSEE, FLORIDA
COLIN ENGLISH, State Superintendent of Schools
M. W. CAROTHERS, Director of Instruction
CONTENTS
Diagram of Bulletin Production ................... ................... ............. iv
Foreword .. ........ ............................... v
Chapter OneTHE PLACE OF ARITHMETIC IN PROBLEM SOLVING
Place of Problem Solving and Drill. ................. ........ ...................... 1
Measurement as a Process........................... 3
Outline of Mathematical Outcomes............... ...... 4
Chapter TwoDEVELOPMENT OF MATHEMATICAL CONCEPTS
Persistent Problems ......................................... 7
Function of Arithmetic .............................. 8
Analysis of Quantity ................ .... ........ ................... 9
Sequential Development of Concept of Time................................ 11
Sequential Development of Concept of Length.................................. 18
Sequential Development of Concept of Mass......................... 24
Sequential Development of Concept of Exchange............................... 27
Chapter ThreeDEVELOPMENT OF COMPUTATIONAL SKILLS
Discussion of Computational Skills................................. 38
Addition and Subtraction of Whole Numbers ................................ 43
Multiplication of Whole Numbers............................. 48
Division of Whole Numbers ...................... ...... ...... 51
Fractions .................. ...... ...... 55
Common Fractions ......... ...... ....... 56
Decimal Fractions ......................... 61
Percentage Fractions ..................... .............. 66
Roman Numerals ......................................... 67
Chapter FourORGANIZATION OF MATERIALS OF INSTRUCTION
Outline of Meaningful Learning Situations.................. ........ 69
Detailed Treatment of Primary and Intermediate
Grade Problems .................. ... 90
Specific Illustrations ...................................... 114
Chapter FiveEvALUATION
Instruction and Evaluation ............. ....... .................121
Evaluation Procedures .........................121
Testing Devices ....................  ................  123
149355
CONTINUING PRODUCTION OF CURRICULUM BULLETINS
FLORIDA PROGRAM FOR IMPROVEMENT OF SCHOOLS
SERIES BEGINNING 1938
No. 9
A Guide to
Improved
Practice in
Florida Ele
m e n t a r y
Schools (1940)
No. 2 Avenues of
W a y s t Understand
Better In ing, A Bulle
struction tin for Parents
in Florida a n d L a y
Schools (1930) Groups (1940)
No. 10
A Guide to
a Function
al Program
in the Sec
ondary School
(1940)
S No. 4.
SNo. 21.
No. 22K.
SNo. 26.
W No. 27.
P.
No. 1.
No. 4.
No. 5.
No. 11.
m No. 12.
No. 22K.
 No. 25.
E No. 27.
6 No. 2S.
5 No. 29.
, No. 40.
. No. 41.
5 No. 42.
Florida's School Health Program (1942, Revised)
Physical Education (1941)
Teaching Actions and Effects of Alcohol and Other Narcotics (1941)
Arithmetic in the Elementary School (1942)
State Adopted Library Books for Florida Schools (1942)
Guide to Exploratory Work* (1938)
Florida's School Health Program (1942, Revised)
Physical Education (1942, Revised)
Business Education (1940)
Industrial Arts (1940)
Teaching Actions and Effects of Alcohol and Other Narcotics (1941
Home Economics Books and Other Source Materials (1941)
State Adopted Library Books for Florida Schools (1942)
Social Studies in the Secondary School (1942)
Everyday Living, Grades 7 and 8 (1942)
Mathematics Essentials (Twelfth Grade Pupil Text) (1942)
Background Material in War Mathematics (for Teachers) (1942)
Teacher's Manual for War Emergency Physics Course
Technology Series:
Book 1. General Mechanics (1940)
Book 2. Engines (1942)
Book 3. Aeronautics 1942
* Now out of print.
Foreword
Following the publication of basic curriculum bulletins including
Ways to Better Instruction in Florida Schools and A Guide to
Improved Practice in Florida Elementary Schools the State Depart
ment has begun the issuance of more specific materials. The latter
constitute official courses of study in the areas treated and supersede
all statements concerning these fields which were made in the former
Course of Study (1933).
The material contained in the bulletin Arithmetic in the Ele
mentary School was developed by committees of Florida teachers
working at the Florida State College for Women during the last
two summers. Membership of these committees included: Katherine
Adams, Pensacola; Frances Belcher, Clearwater; Emily Brackman,
Pensacola; Doris Brownell, Sarasota; Mary Delamater, Largo; Har
riet Dyer, Panama City; Hazel Elliott, Winter Garden; Frances
Ellis, Fort White; Thelma Johnson, Walton; L. E. Jones, Bradenton;
Winona Jordon, West Palm Beach; Pauline Messer, Grand Ridge;
Jessie Moon, Tallahassee; Dorothy Oliver, Lake Wales; Rosalie
Powell, Jacksonville; Flossie Sharp, Ft. Lauderdale; Frances Sum
mers, Panama City; Nellie Swinney, Panama City; Rose Varlin,
Panama City.
Acknowledgment is hereby made for the splendid services of
the codirectors of bulletin, Miss Margaret McCurdie and Miss Thelma
Tew both of whom are critic teachers at the Demonstration School,
Florida State College for Women. To members of the elementary
faculty of the Demonstration School, to Dr. Robert C. Moon and
Mrs. Dora Skipper of the College of Education, Florida State
College for Women, and to many others who gave freely of their
time and effort in bringing the work to a successful conclusion,
grateful appreciation is also extended.
It is hoped that this material will be diligently studied and used
by all the elementary teachers of Florida. Faculty meetings in
which the entire group discuss the suggestions and problems together
will prove a valuable means toward this end.
State Superintendent of Public Instruction
Chapter One
THE PLACE OF ARITHMETIC IN PROBLEM
SOLVING
Your teaching of arithmetic . may merely train your class in a
number of processes, which will let them pass an examination at the end of
the term. That is "useful". It may also help them manage their savings
accounts better or get a job on graduation. That is usefuland this is
without quotation marks. But if you can develop in them an understanding
of number relations, if you can teach them to visualize distances and quan
tities, to appreciate imaginatively the meaning of "ten million" or of "one
thousandths of an inch", then you are training them culturally; they will
forever after be more sensitive, more appreciative, more understanding, even
though they may do no better on a formal examination.1
Before attempting to develop an arithmetic program teachers need
to consider many phases of the program in terms of pupil needs and
the function of arithmetic.
The conception of drill and problemsolving is wider in application
today than was formerly the case. No longer are teachers satisfied
to accept computational skill as the only outcome of drill, nor do they
limit problem solving to the solving of textbook problems. They have
come to understand that arithmetic has but one purposeto assist
in the solution of problems with quantitative aspects. The problems
may involve the comparing of amounts, the measuring or determining
of amounts, or the estimating of amounts. The solutions may depend
on counting, computing, measuring, estimating, and comparing, or a
combination of these processes.
Both incidental and "plannedfor" problems are faced by children
in the classroom. The incidental problem when utilized by the teacher
1Henry W. Simon, Preface to Teaching, (New York: Oxford University
Press, 1938), p. 34.
ARITHMETIC IN THE ELEMENTARY SCHOOL
may be the means of developing many worthwhile understandings
and skills. The following is illustrative of the use of an incidental
problem. One morning in February, elementary children may come
to school and report that their father and neighbors have been up all
night tending fires in local orange groves. The primary teacher
asks if anybody knows how cold it was during the night. If no one
can tell her, she may supply the information herself, or she may help
children understand differences in temperature through making a
comparison of thermometer readings inside and outside the Echool
room. She will use the terms freezing, above freezing, below freezing,
and will indicate these points on the classroom thermometer.
The intermediate grade teacher will do more. In addition to
helping children get information about the temperature during the
previous night, she may help them make comparisons between that
temperature and the temperature during the coldest night of February
of the previous year. She may help them make comparisons between
the warmest nights and the coldest, the warmest months and the cold
est, the Centigrade scale used to record temperature on airplanes and
the Fahrenheit scale used in the classroom. As a result of such
experiences, children will become more familiar with the terms aver
age, temperature, high ground, low ground, above freezing, freezing,
below freezing, Centigrade, Fahrenheit, and will become independent
in the use of a Fahrenheit thermometer for measuring heat within the
classroom. In addition, the children will discuss the reasons for keep
ing fires burning in orange groves, will learn why trees have a better
chance of survival during a frost when they are growing close to the
edge of a lake, and why trees sometimes freeze in the hollows but not
on the slopes.
The "plannedfor" problems are those from a unit of experience
about which children are concerned at the time, or textbook prob
lems. Chapter IV contains many examples of good problemsolving
situations. There is no limit to the number and variety of this type
of problem. They are a result of research and planning on the part
of the teacher who is as interested in developing meanings as in
developing computational skill. Such a teacher is aware of the
sequential development of understandings and skills and of the
necessity for looking into the past experiences of children for help
in interpreting present situations. Textbook problems are as suitable
ARITHMETIC IN PROBLEM SOLVING 3
as other types of problems provided they relate to the pupil's interests
and background of understandings.
The development of understandings and skills requires the repe
tition of experiences. This is another way of saying that drill must
be provided if understandings and skills are to be developed. Buck
ingham points to the importance of drill in the progressive organ
ization of knowledge when he says, "We must use the occasions of
the living present; we must recognize the ongoing character of ex
perience; we must invoke repetition as well as insight; we must look
toward a reorganizationwhich, in spite of the prefix, does mean
organizationof experience; and we must seek, if men are to rise
above the fortuitous and the occasional, the orderly arrangement of
knowledge.' '2
Drill should follow understanding but it should also increase
understanding. Written problems should follow oral problems and
drill with abstract numbers should come after the meaning of the
process has been established. Experiences which have led to the
drill should help the child understand what he is learning. He should
believe in what he is doing. If the drill seems worthwhile his attitude
will contribute to his success. In addition, he should be challenged
by the drill he is undertaking. It should never be entirely new but
the range of difficulty should be increasingly greater.
As was suggested earlier, drill is no longer used exclusively for
developing computational skill. Drill involves a great deal more than
computational skill. Teachers now recognize that it needs to be ap
plied to the whole field of problemsolving. Children should have
opportunities for practice in computing, comparing, estimating; in
manipulating standard instruments of measure; in interpreting fig
ures representing amount; in developing meanings for the number
system.
Whether the child compares, estimates, uses a standard instru
ment, counts or computes, he is measuring quantity, either accurately
or by means of descriptive terms.
7B. R. Buckingham, "What Becomes of Drill?", Arithmetic in General
Education, Sixteenth Yearbook of the National Council of Teachers of Mathe
matics, (New York: Bureau of Publications, Teachers College, Columbia
University, 1941) p. 204.
4 ARITHMETIC IN THE ELEMENTARY SCHOOL
In the primary grades, children compare and estimate without
the use of standard instruments, but they also have many experiences
measuring with rulers, cups, speedometers, clocks, and the like. They
first learn to tell the number of things (how many) by counting.
Later, they learn to find how many by adding, subtracting, multi
plying, or dividing and discover that computing is a more efficient
means of determining the amount than counting.
In the intermediate grades, a certain amount of real measuring
is continued, but children also deal with numbers representing moun
tain heights, population changes, value of property, and others.
While they have not actually held the instruments of measure in their
hands, children have a means of interpreting these numbers because
of former measuring experiences.
If the emphasis in an arithmetic program is on computation
children will have difficulty in recognizing and solving problems.
But if from the first grade they have been conscious of arithmetic
as a system by which measurement is expressed, and in addition, have
achieved skill in the use of the number system, they will be sensitive
to quantitative problems of the social and physical environment and
will have a certain amount of independence in solving and interpret
ing these problems.
If teachers are to help children become more sensitive, more ap
preciative, and more understanding of the quantitative aspects of their
environment, and at the same time help them develop skill in solving
problems, they need to plan the arithmetic program in terms of broad
outcomes.
Any statement of arithmetic outcomes should include understand
ings and skills involving the description and measurement of quantity
by means of the number system and a suitable vocabulary. The out
line below suggests the kind of outcomes appropriate for elementary
school children. These outcomes are treated fully in Chapters Two,
Three and Four.
Major Outcomes: The development of mathematical understandings
and skills which will contribute to the solution of problems of the
social and physical environment.
ARITHMETIC IN PROBLEM SOLVING 5
Understandings
1. Of the meaning of measure. (Treated throughout the bulletin. See
charts included in Chapter Two.)
2. Of measure as a process. (Treated throughout the bulletin. See Chapter
Two.)
3. Of the number system. (Treated throughout the bulletin. See Chapter
Three.)
Skills
1. In comparing. (Treated throughout the bulletin. See charts included
in Chapter Two.)
2. In measuring. (Treated throughout the bulletin. See charts included
in Chapter Two.)
3. In estimating. (Treated throughout the bulletin. See charts included
in Chapter Two.) 4
4. In counting and computing with whole numbers and fractions. (Treated
throughout the bulletin. See Chapter Three.)
Chapter Two
DEVELOPMENT OF MATHEMATICAL CONCEPTS
Arithmetic has been described as a system of quantitative thinking.
The need for quantitative thinking arises early in life when the child
is faced with the necessity for describing the size, weight, shape,
position, and amount of various things in some adequate fashion. In
the beginning, the description does not have to be accurate and the
child is satisfied with such expressions as "a long way off", "bigger
than John"' "before I came to school". However, it is not long
before activities at home and school help him realize the need for
describing things in terms of exact amount, exact weight. For
example, he is concerned over the number of pennies he has, how
much he weighs, how old he is. Later, after many experiences with
describing things in exact amounts he is able to become independent
of instruments of measure and can say, "nearly two dollars", "less
than five miles". In addition, there are other activities and ex
periences which are more or less new to the child. Therefore, while
he is gaining skill in expressing some of his quantitative ideas with
much more exactness or accuracy, he is attempting to express other
newer ideas in a more or less descriptive way.
These needs for describing and obtaining exact measurements are
a result of demands made on the child by his social and physical
environment. It is the responsibility of the school to help him
discover ways of meeting these needs, and in addition, provide ex
perience which will enable the child to make better interpretations
of quantitative situations.
In the past, children have been given rigorous training in the
manipulation of abstract numbers with little regard to their meaning
and significance, or to their function in interpreting the social and
physical environment. Actually, drill and repetition of skills within
the computational processes constituted the body of the arithmetic
MATHEMATICAL CONCEPTS
program. Children frequently did not know what they were doing
and even after mastering certain computational skills, were unable
to solve the simplest problems in which those skills were involved.
Only the more capable children came through this training without
unfavorable attitudes regarding mathematics.
CHANGING POINT OF VIEW
The Child and His Environment.Today, educators believe
arithmetic has a more significant role to play in the life of the
child. They are examining the child and his environment in an
effort to determine what are those understandings and skills (learn
ings) which should be developed in school. They are asking such
questions as, "What are the things adults and children do each
day?" They have identified such activities as eating, sleeping,.
adjusting to others, learning to use different kinds of equipment,
securing things they need and want, interpreting signs and symbols,
expressing themselves in various ways. As teachers study the prob
lems faced 'by both the child and the adult they have been able to
define important problems which seem to persist through life.
thus, persistent problems. The statement of persistent problems in
which mathematics is involved, and the approach and treatment of
these problems suitable for elementary school children is given
below.
I. A. The child experiences the need for expressing ideas
quantitatively as he makes adjustments to the social
and physical environment.
B. The child recognizes the need for expressing quantitative
ideas according to standards accepted by society.
II. The child experiences the need for selfexpression through
music, dancing, painting, oral and written language, and
becomes aware of the elements of designline, dark and
light, and colorand of such guiding principles as rhythm,
dominance, and balance. He learns that these elements
and guiding principles are dependent on proportion, or
fine relationships of line, mass, and color.
III. The child experiences needs which involve the use of
exchange, and comes to an understanding of the various.
media of exchange employed by society.
8 ARITHMETIC IN THE ELEMENTARY SCHOOL
Persistent Problem IA is an expression of that need which
all children and adults face in describing and interpreting the
physical and social environment. It concerns the need for express
ing how much, how many, how long, size, weight, shape, and
position. Persistent problem IB is an expression of the need to
express how much, how many, by means of a number system which
has been accepted by others of the same culture.
Persistent Problem II concerns the field of aesthetics, par
ticularly that part which deals with selfexpression. Probably very
little use. of actual numbers is made in this particular area but
the ability to compare, estimate, contrast, see relationship develops
gradually as a part of a good mathematics program. Children
respond to rhythm long before they learn that it can be described
mathematically. In planning proportions of various objects, they
learn that a 2 to 3, 3 to 5 relationship is more pleasing than a
2 to 2, or a 2 to 4. In doing a simple thing like arranging flowers,
they can appreciate the need for suitable vases. They soon learn
to float some flowers in a low bowl, and put tiny flowers in minia
ture vases. Later, they are able to estimate, to see that tall flowers
are about two and onehalf times the height of the vase.
Persistent Problem III concerns the use of exchange. Early in
life, the child discovers that money will obtain goods that he
wants, or that which his parents desire. He also gets some under
standing of the method of bartering through hearing parents talk
of trading fruit, vegetables, and the like for other necessities, and
in "swapping" articles with other children, himself.
The persistent problems then, indicate needs which face all
children and adults in adjusting to the social and physical environ
ment. An arithmetic program based on skill development of the
four computational processes, with relatively no emphasis on the
function of those processes in a significant social setting, cannot be
adequate for helping a child express ideas quantitatively by means
of an accepted number system, make the best possible use of our
system of exchange, or meet the needs of selfexpression in the field
of aesthetics. Teachers are discovering that society demands of us
the ability to compare distances, weights, ages, and to estimate how
much, how far, how deep, without the use of any number system
but by means of such descriptive terms as "just a little", "not very
MATHEMATICAL CONCEPTS 9
far", "a long way off". It is this same need for estimating more
accurately which helps the child see the necessity for gaining
control of a number system which will enable him to describe
amounts of time, distance, and the like in terms of exact amounts.
After many experiences comparing, measuring, estimating he will
be able to say "a little over three feet", "about two inches".
After examining arithmetic and questioning its place in the
elementary school program, teachers are beginning to see that its
function is to measure and describe quantity. What is quantity?
How does an understanding of quantity help Bill learn to subtract,
or Mary keep a record of war stamps, sales, or the teacher, herself,
make a yearly report on pupil attendance?
Quantity.From physics it is discovered that quantity is three
foldtime, length, mass.
An understanding of time develops as the child becomes aware
of the instruments for measuring time. Clocks measure short
periods of time, while calendars measure longer periods of time.
Speed, an interrelationship of time, motion, distance, can like
wise be measured by instruments.
Length, in its broadest sense, means distance, or height, or
depth, or widthanything of a linear nature falling between two
points. Area is a part of length in the sense that it may be ob
tained abstractly by multiplying the length of one side of a rec
tangle by the length of another side. Area is a length relationship.
The broad term mass may be applied to all solids, liquids,
gases. In analyzing the solids, liquids and gases with which the
elementary child frequently deals, it is found that he uses these
in a very concrete sense. In the place of thinking of solids, the
young child is likely to refer to specific objects which to him
seem solid, such as people, food, lumber or cloth. Water and milk
are to him common liquids, while air represents a gas. In using
the objects or things which surround him in an effective manner,
the child has need for a knowledge of quantity and of quantitative
relationships. The wise teacher will utilize to the fullest extent
the motivation thus generated.
The above analysis gives teachers a clue to the nature of the
problems which children face in expressing and interpreting quan
10 ARITHMETIC IN THE ELEMENTARY SCHOOL
titative ideas. They need to measure and describe time by means
of a clock or calendar. They need to be able to compare, esti
mate, and measure accurately various lengths, widths, depths,
heights. Likewise, they need to be able to find the amount and
weight of things, of themselves, their friends, food they buy, and
the like. They need to be able to interpret numbers representing
amount, size, weight, capacity, height, distance, and to understand
that different things are measured in different ways. In addition,
they need to have understandings regarding the use of money.
Continuous and Sequential Development.As teachers attempt
to identify needs, experiences, activities, of both the younger and
the older child, it becomes evident that a continuous growth in
understanding and in skill development occurs and that this
:growth begins before the child's entrance in school and extends
throughout life. This growth is sequential in its development,
beginning with the specific in the first and second grades (at
school) and culminating in the ability to generalize in later grades.
Many understandings and skills develop slowly and a child needs
repeated experiences over a long period of time. In some cases
it will be necessary to plan for reteaching and more thorough
development of particular skills and understandings throughout
the entire elementary school experience of the pupils. Other un
derstandings and skills develop more quickly.
Following the general Concepts Chart opposite this page are four
smaller charts containing the analysis of the sequential development
of understandings and skills involving different aspects of quantity
. .time, length, distance, height, width, depth, and volume. These
understandings and skills are interrelated and interdependent and do
not exist in real life as separate items, as they have been presented in
the charts. For example, in measuring length, a child is actually
dealing with time, or solids, or liquids, or gases, depending on the
problem at hand. To find the height of a column of mercury be is
dealing with a liquid. To find the length of a river he is dealing with
a liquid. But in finding the altitude (height) to which man can
ascend in a pressuresealed gondola, he is dealing with a gas.
'Teachers, then, do not try to develop understandings about each
separate aspect of quantity exclusively, but draw on all aspects
to give meaning to each.
MEANING AND USE OF STANDARD UNITS OF MEASUREELEMENTARY GRADES
GRADE ONE
1. Ability to use ruler or yardstick as a straight edge.
2. Ability to count the number of inches in a foot and in a
yard.
3. Ability to appreciate the necessity for placing ruler or yard
stick at edge or beginning point before measuring.
4. Ability to measure accurately an inch or a number of inches.
5. Ability to interpret statements of measurement in which the
fractional part of an inch is significant (heights of children).
6. Ability to appreciate nearness or "farness" from home to
school when distance is expressed as blocks, miles, or frac
tional part of a mile.
7. Ability to understand the length of a mile as the distance
between two well known land marks.
8. Ability to understand relationship between distance and time
as child walks or rides from home to a destination (school,
places within the community, places frequently visited by
the family).
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1. Knowledge of own weight.
2. Ability to read number which records child's weight.
3. Ability to determine gain or loss of own weight (by counting).
4. Some undertsanding of relationship of size and weight (com
paring various members of the class).
5. Some appreciation of the weight of adults known to the child.
6. Some understanding of the weight of one, two, or five
pounds through handling packages of commodities usually
sold by one, two, or five pounds (sugar, meal, meat, potatoes).
7. Some understanding of the weight of onehalf and onequar
ter pound through handling packages of commodities sold
by onehalf and onequarter pounds (cheese, butter).
GRADE TWO
1. Ability to place ruler or yardstick at edge or beginning
point before measuring.
2. Ability to measure inches and half inches.
3. Ability to interpret statements of measurement in which
the fractional part of an inch is significant (heights of chil
dren, widths of boxes).
4. Ability to express (through use of rulers) measurements of
more than twelve inches in terms of feet and inches (twenty
five inches the same as two feet and one inch).
5. Ability to appreciate nearness and "farness" from school to
many different places within the community.
6. Ability to gain some idea of the time it takes to walk a
mile.
7. Ability to understand the relationship between distance and
time as child (or group) walks or rides from home or school
to a destination (places within the community, places fre
quently visited by the family)
1. Ability to measure thirds, fourths, or a half of a cup.
i L. 
2. Ability to measure by teaspoon or tablespoon.
3. Ability to follow a recipe.
4. Ability to double a recipe.
5. Knowledge of a variety of containers used for selling a
variety of foods, the contents equivalent to or approximating
that of a pint, quart, or gallon.
6. Some understanding of the relationship of size to the number
of pieces of fruit or vegetables contained in the crate and in
various size baskets.
1. Knowledge of way to express or record weight of self or
others in class using fractional part of pound when necessary.
2. Ability to solve oral or written problems dealing with gain
or loss of weight (various members of the class, using ad
dition or subtraction process).
3. Ability to interpret statements of weight in which the
pound or fractional part of a pound is significant (butter,
meat, cheese).
4. Knowledge of the abbreviation for pound.
5. Acquaintance with commercial use of scales.
6. Some understanding of relationship of capacity to weight in
connection with observing truckloads of fruit, refrigerator
trucks, moving vans.
7. Acquaintance with use of word ton to mean a great deal of
weight.
GRADE THREE
1. Ability to measure inches, half inches, quarter inches (heights,
widths, depths).
2. Ability to measure an inch or a number of inches any place
on the ruler.
3. Ability to interpret statements of measurement in which
inches, half inch, and eighth inch are significant (heights,
widths, depths).
4. Knowledge of the relationship of inches to foot, feet to yard,
and inches to yard.
5. Ability to express measurements of more than twelve inches
in terms of feet and inches, or of yards and inches.
6. Ability to express 3 inches, 6 inches, and 9 inches as a frac
tional part of a foot; 12 inches and 18 inches as a frac
tional part of a yard.
7. Knowledge of abbreviations and symbols for inch, foot, yard.
8. Ability to estimate length of an inch or foot.
9. Knowledge of length of statute mile in feet and yards.
10. Ability to solve simple oral or written problems using linear
measurement (addition and subtraction).
11. Acquaintance with square measure by marking off square
inches.
1. Ability to estimate the amount contained in a cup, a quart,
a pint.
2. Knowledge of the relationship of amount measured by tea
spoon to that measured by a tablespoon, by a tablespoon to
that measured by a cup.
3. Knowledge of the number of tablespoons in a half cup and
quarter cup.
4. Knowledge of the amount of milk used by the family each
week; each month.
5. Knowledge of the amount of milk produced in a local dairy
in a day, in a week, in a month.
6. Knowledge of the use made of gallon, quart, pint, and half
pint measures in the sale of such products as milk, cream,
fruit juice, ice cream, oysters, syrup, gasoline, kerosene.
7. Ability to interpret statements of measurement in which the
gallon, quart, and pint, are significant.
8. Ability to change the larger units of liquid or dry measure
to the smaller units of liquid or dry measure (quarts and
pints).
9. Acquaintance with ounce as a unit of measure in connection
with reading labels on containers of various products.
10. Ability to systematize the following units of liquid measure
into a table, using abbreviations and symbolspint, quart,
gallon.
11. Ability to solve simple oral or written problems involving
use of gallon, quart, pint (addition and mutiplication).
12. Ability to double or half a recipe.
1. Some ability to estimate the weight of members of the class,
of adults, of various articles or equipment used in the home
and school.
2. Knowledge of amount measured by a pound, a half pound,
a quarter pound, expressed in cups or tablespoons.
3. Knowledge of relationship of ounces to pounds, of ounces
to half pound, of ounces to quarter pound.
4. Acquaintance with ounce as a unit of weight (sealed pack
ages of macaroni, crackers, cereals, cheese).
5. Ability to solve oral or written problems in which there is
a difference in weight.
6. Acquaintance with use made of pounds in connection with
air pressure in automobile and bicycle tires.
GRADE FOUR
1. Ability to measure inches, half inches, quarter inches,
eighth inches (heights, widths, depths).
2. Some ability to estimate height, width, or thickness to the
fractional part of an inch.
3. Ability to estimate the length of an inch, a foot, a yard.
4. Ability to interpret accurate to scale drawings, including
maps.
5. Ability to interpret statements of measurement in which
inches, half inch, quarter inch, and eighth inch are signi
ficant.
6. Ability to gauge or make fair estimates of distances used
while playing familiar games.
7. Ability to establish relationship between distances and time
through real or vicarious experiences with speed.
8. Ability to change larger linear units of measure to smaller
linear units of measure (multiplication).
9. Ability to solve oral or written problems using linear measure
ment (addition, subtraction, multiplication).
10. Knowledge of length in feet of statute mile, half mile, quar
ter mile, tenth of a mile.
11. Knowledge of abbreviations and symbols for inch, foot, yard,
mile.
12. Ability to determine the fractional part of a foot or yard
when a number of inches is given.
13. Acquaintance with meaning of square measure by marking
off square inches and square feet.
1. Ability to estimate the amount contained in a cup, a pint, a
quart, a gallon.
2. Knowledge of the way small amounts of liquid (less than
a pint) are measured.
3. Ability to make such comparisons as three pints is the
same as a quart and a pint.
4. Ability to appreciate the significance of the output of vari
ous Florida industries whose outputs are measured by the
gallon, quart, pint, and half pint (dairying, sugar industry).
5. Ability to systematize the following units of liquid measure
into a table, using abbreviations and symbolspint, quart,
gallon; number of cups in a pint, a quart; number of tea
spoons in a tablespoon, number of tablespoons in a cup,
in a half cup, in a quarter cup.
6. Ability to systematize the following units of dry measure
using abbreviations and symbols: pint, quart, pecks, bushel.
7. Ability to use information of the tables developed above
in the solution of oral and printed problems involving use
of liquid or dry measure.
8. Ability to increase or decrease a recipe to meet needs of
the group (under teacher guidance).
9. Knowledge of the relationship of a gallon of gasoline to
mileage and speed.
10. Ability to appreciate the significance of the output of various
Florida industries whose outputs are measured by the crate,
bushel, hamper, and various size baskets (citrus, berries,
celery).
1. Ability to estimate weight of familiar objects with some
degree of accuracy.
2. Some appreciation of the varying amounts contained with
in the same unit of weight (pound of spinach vs. a pound
of potatoes).
3. Ability to interpret the output of a Florida industry whose
output is recorded in tons and pounds (sugar, phosphate,
fish).
4. Ability to determine total weight when weight of one article
is given (addition or multiplication).
GRADE FIVE
1. Ability to measure inches, half inch, quarter inch, eighth
inch (heights, widths, depths).
2. Ability to estimate heights, width or thickness to the frac
tional part of an inch.
3. Ability to make and interpret accurate to scale drawing,
including maps.
4. Ability to construct articles involving use of linear measure
according to oral or written directions.
5. Ability to interpret statements, charts, or graphs which make
use of the inch, half inch, quarter inch, or eighth inch in
expressing an idea.
6. Ability to change from one unit of linear measure to an
other (multiplication or division).
7. Ability to estimate height of manmade structures within
the community.
8. Ability to express relative heights through art.
9. Ability to guage distances used while playing familiar games.
10. Ability to systematize units of linear measure into a table,
using abbreviations and symbols for inch, foot, yard, mile.
11. Knowledge of length of nautical mile and fathom.
12. Ability to determine arithmetical relationship between speed,
distance, time.
13. Ability to find area of a rectangle when dimensions are given
in inches, feet, or yards.
14. Acquaintance with distance of various solar bodies from the
earth.
15. Acquaintance with size of various solar bodies.
16. Ability to solve oral or written problems dealing with linear
or square measure.
1. Ability to estimate the output of significant industries when
daily or weekly output is given in units of liquid or dry
measure.
2. Acquaintance with the facts of the United States govern
ment standards.
3. Knowledge of the value of the imperial gallon as contrasted
with standard American gallon.
4. Understanding the relationship of amounts of gasoline and
horsepower of engines to speed and range of plane.
5. Ability to increase or decrease a recipe to meet needs of the
group (under teacher guidance).
6. Ability to solve problems involving more than one of the units
of liquid or dry measure by reducing measurements to the
smallest unit within the problem.
1. Some ability to estimate weight.
2. Some understanding of the relationship of increase in height
to gain in weight.
3. Knowledge of way to express small amounts of weight
(gram).
4. Some understanding of weight and capacity in connection
with limits placed on the use of certain roads and bridges.
5. Some understanding of the significance of weight in trans
portation (truck, boat, train, plane).
6. Ability to systematize the following units of weight into a
table using abbreviations and symbols.
7. Ability to determine the weekly, monthly or yearly output of
industries whose outputs are recorded in tons and pounds.
I
1. Ability to estimate one, two, five, and ten pounds of various
food articles.
2. Some understanding of the use of pound and ounces for
both liquid and dry measure.
3. Ability to appreciate the significance of net weight to gross
weight, of live weight to dead weight.
4. Acquaintance with government standards relative to weight.
5. Knowledge that water weighs 62.4 pounds per cubic foot
at sea level.
6. Knowledge of the number of articles contained in a gross,
half gross, and quarter gross (in wholesale buying) and the
relationship of these to carton and dozen.
i__
(SEE OTHER SIDE)
1. Ability to measure by a cup, a quart, a pint.
2. Knowledge of relationship of pints to quart, quarts to
gallon.
3. Knowledge of the size and shape of containers most com
monly used to measure a gallon, quart, pint, and half pint.
4. Knowledge of the number of cups contained in a quart, a
pint, a half pint.
5. Ability to appreciate the significance of the marks on a
measuring cup dividing it into thirds, fourths, halves.
6. Ability to follow a recipe calling for a cup or a fractional
part of a cup.
7. Some understanding of the relative amounts contained in
a hamper, field crate, crate, and different size baskets
(bushel, peck, two quarts, one quart, pint).
A
GRADE SIX
1. Ability to measure inches, half inch, quarter inch, eighth
inch (heights, widths, depths).
2. Ability to visualize, estimate, and think in terms of inch,
foot, yard, mile, and fractional parts of inch, foot, yard,
or mile.
3. Acquaintance with sixfoot carpenter's rule or steel tape,
twentyfive or fiftyfoot steel or cloth tape measure.
4. Knowledge of the length of a rod in feet and yards.
5. Ability to estimate height, width, or thickness to the frac
tional part of an inch.
6. Abiilty to make and interpret accurate to scale drawings,
including maps.
7. Ability to construct articles involving use of linear measure
according to oral or written directions.
8. Ability to interpret statements, charts, or graphs which
make use of the inch, half inch, quarter inch, or eighth inch
in expressing an idea.
9. Ability to change from one unit of linear measure to another
(multiplication or division).
10. Ability to estimate height, width, or depth of natural or
manmade structures.
11. Ability to express relative heights through art.
12. Ability to gauge or estimate long or short distances.
13. Ability to determine arithmetical relationship between speed,
distance, and time.
14. Ability to find area of large surface, expressing area in acres
or fractional part of an acre.
15. Ability to solve oral or written problems involving more than
one of the units of linear measure by reducing the measure
ments to the smallest unit mentioned in the problem.
16. Ability to solve problems in which astronomical figures are
used.
17. Knowledge of the length of a millimeter, a meter, and a
kilometer (metric system).
1. Ability to estimate the output of significant industries
throughout the world when monthly, seasonal, or yearly out
put is recorded in units of liquid or dry measure.
2. Knowledge of the facts of the United States government
standards.
3. Knowledge of units of measure used in wholesale trade and
how they differ from those used in retail trade.
4. Acquaintance with the use of the liter instead of the quart
(metric system).
5. Understanding the relationship of high octane gasoline to
speed and range of planes.
i A i I I I,
6. Ability to increase or decrease a recipe independently to
meet the needs of the group.
7. Ability to solve problems of liquid or dry measure involving
more than one of the units of measure by reducing the
measurements to the smallest unit mentioned within the
problem.
I I I
I
I 
MEANING AND USE OF STANDARD UNITS OF MEASURE (Continued)
GRADE ONE
1. Ability to adjust windows and doors for regulating tem
perature within a room (under teacher guidance).
2. Ability to find point on the thermometer which indicates
correct room temperature.
3. Ability to tell whether room temperature is too high or too
low by looking at the thermometer.
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1. Knowledge of own birth date.
2. Acquaintance with date of important holidays.
3. Ability to read date from the calendar.
4. Ability to read and write date in the form, September 15,
1942.
5. Knowledge of the number of days in a week.
6. Knowledge of the names of the days in the week.
7. Knowledge of the number and names of the days of the
school week.
8. Knowledge of the name of the months.
9. Ability to read the names of the days and months and their
abbreviated forms.
10. Knowledge that all months do not have same number of
days.
11. Ability to associate the position of hands on clock with
opening time, recess time, lunch time, and closing time.
12. Ability to count strikes of the clock.
13. Ability to recognize fire alarm signal by counting.
14. Acquaintance with various time patterns (in singing games,
simple dances, and other rhythmic interpretations).
15. Ability to interpret some rhythmic patterns as "1 clap,
2 shakes," "1 clap, 3 shakes".
1. Ability to recognize these coins: cent, nickel, dime.
2. Acquaintance with one and five dollar bills.
3. Knowledge of the relative value of cent, nickel, dime.
4. Knowledge of purchasing power of cent, nickel, and dime
for articles which child ordinarily purchases, (milk, candy,
picture show ticket).
5. Knowledge of the relationship of nickel to dime.
6. Knowledge of the special use of the nickel (using pay station
phone; securing candy, chewing gum, peanuts, or cold drinks
from a slot machine).
7. Ability to use tokens for bus or street car fare.
8. Ability to interpret and express small amounts of money
in these forms10 cents, 100, $.10.
9. Ability to interpret the decimal point as a cents point.
10. Knowledge of the price of stamps commonly used.
11. Ability to read price tags marked 5, 10, 25, etc.
12. Acquaintance with the many ways things are sold (by the
yard, pound, dozen, etc.).
13. Knowledge of relationship of quantity and quality in con
nection with articles commonly purchased by the child
(chewing gum, candy, pencils).
GRADE TWO
1. Ability to adjust windows and doors for regulating tem
perature within a room (under teacher guidance).
2. Ability to find the point on the thermometer which indicates
correct room temperature.
3. Ability to mark correct room temperature on a pictured
thermometer.
4. Ability to tell whether the room temperature is too high or
too low by looking at the thermometer.
5. Acquaintance with clinical thermometer for indicating fever.
6. Acquaintance with the means used for protecting plants,
radiators, water pipes against freezing.
7. Ability to read room temperatures expressed in written form.
1. Knowledge of birth date of brothers and sisters.
2. Ability to recognize relative ages of brothers, sisters, and
playmates.
3. Knowledge of birth date of some classmates.
4. Knowledge of date of important holidays.
5. Ability to read and write the date including its abbreviated
form.
6. Ability to write correctly the names of days in the week
and their abbreviations.
7. Ability to write correctly names of the months and their
abbreviations.
8. Knowledge of the names of the months in sequence.
9. Knowledge that February is the shortest month.
10. Ability to determine (through use of the calendar) the
months having 30 days and those having 31 days.
11. Knowledge that the space between numbers on a clock face
represents five minutes.
12. Ability to tell time accurately within five minutes, as "it's
almost 15 minutes after two".
13. Acquaintance with various time patterns (in singing games,
simple dance, other rhythmic interpretations).
14. Ability to interpret some rhythmic patterns as "1 clap, 2
shakes," "1 clap, three shakes."
1. Ability to recognize these coins: cent, nickel, dime, quarter,
half dollar.
2. Ability to recognize the one, five, and ten dollar bills.
3. Knowledge of the relative value of cent, nickel, dime, quar
ter, half dollar, dollar.
4. Knowledge of purchasing power of cent, nickel and dime
for articles purchased for the home or school (soap, milk,
writing paper).
5. Knowledge of relationship of quantity and quality in con
nection with purchase of food, home and school supplies.
6. Ability to determine amount of money needed in order to
make a purchase (within range of ten cents).
7. Ability to determine amount of change (within range of
ten cents).
8. Ability to make change, limited to use of nickel and dime.
9. Ability to interpret and express amounts of money up to
ten dollars in these forms: 65 cents, 820, $5.45.
LO. Knowledge of price of stamps commonly used.
11. Acquaintance with the many ways things are sold (by the
yard, pound, dozen, etc.)
L2. Ability to determine the cost of more than one unit when
the price for one unit is known.
L3. Ability to read price tags used in drug stores, ten cent stores,
grocery stores.
14. Acquaintance with relationship of price to size number of
can.
L5. Ability to interpret a sales slip as a record of items pur
chased.
GRADE THREE
1, Ability to assume responsibility for adjusting windows and
doors for regulating temperature within a room.
2. Ability to assume responsibility for checking temperature
at intervals.
3. Ability to mark points on pictured thermometer indicating
correct room temperature, and common deviations which
occur in the community.
4. Acquaintance with clinical thermometer for indicating fever.
5. Acquaintance with the means used for protecting plants,
radiators, pipes against freezing.
6. Ability to read temperatures above zero expressed in written
form (whole numbers only).
1. Acquaintance with the year of birth of people significant in
developing the local community.
2. Knowledge of the date of events significant in the develop
ment of the local community (big freeze, building of power
plant).
3. Knowledge of the birth date of a few outstanding national
leaders.
4. Ability to establish sequence of days of the week and months
of the year.
5. Ability to interpret date written in this form, 2/7/42.
6. Ability to express this sequence by means of ordinal num
bers.
7. Ability to determine number of days in a month using a
rhyme if necessary.
8. Ability to read the minutes involved in counting by 5's as
the teacher points to the numbers 1, 2, 3, and by l's for the
extra minute marks.
9. Ability to count to the right to see that the time is 40
minutes after ten; to count to the left to see that the time
is 20 minutes before eleven.
10. Ability to identify rhythmic patterns in 2/4, 3/4, 4/4 time.
11. Ability to recognize the value of a whole, half, and quarter
note.
1. Knowedge of relationship of cents to dime, of dimes to
dollars, of cents to dollar.
2. Ability to interpret amounts of money larger than nine cents
and less than a dollar in terms of the number of cents and
dimes contained in the amount.
3. Acquaintance with half fare rates for children.
4. Acquaintance with range of price for same article (sold
individually, by the yard, pound, dozen).
5. Ability to determine the cost of fractional part (%2, 4)
when amount can be determined by inspection (one dozen
oranges cost twenty cents; half dozen oranges costs ten
cents).
6. Ability to determine cost of several articles (sold by yard,
pound, dozen) when unit price is known.
7. Ability to use catalog for finding cost of equipment for home
and school.
8. Ability to determine amount of library fees.
9. Ability to interpret and express amounts of money needed
to interpret the social environment.
10. Ability to read the decimal point as "and" in reading an
amount of money.
11. Ability to solve addition and subtraction problems involving
use of large sums of money.
12. Ability to make change and to verify change made by clerk.
13. Ability to solve oral or printed problems involving the use
of money (addition, subtraction, multiplication).
GRADE FOUR
Ability to assume responsibility for maintaining correct
room temperatures.
Knowledge that warm air rises.
Ability to understand how this fact is related to proper
ventilation.
4. Ability to mark points on a pictured thermometer which
indicate the freezing point of water, critical temperatures
for certain Florida crops, correct room temperature, normal
body temperature, boiling point.
5. Acquaintance with the means used for protecting plants,
radiators, pipes against freezing.
6. Acquaintance with the point on the clinical thermometer
which indicates normal body temperature.
7. Knowledge that the normal body temperature is 98.6.
8. Ability to read temperatures above zero expressed in written
form (whole number and mixed number).
1. Ability to locate special information in which the date is
important (radio schedule, sports schedule).
2. Knowledge of certain dates of historical importance (dis
covery of America, Florida, landing of Pilgrims).
3. Acquaintance with time zones through travel, radio.
4. Ability to understand reasons for daylight saving time.
5. Ability to make necessary adjustment when daylight saving
time is in operation.
6. Ability to select appropriate task for time available.
7. Ability to tell the time and the date independently.
8. Ability to identify rhythmic patterns in 2/4, 3/4, 4/4 time.
9. Ability to recognize the value of a whole, half, quarter,
eighth note.
10. Acquaintance with the use of the dot to increase the value
of a musical note.
1. Ability to understand place value of number through inter
preting amount of money to find number of cents, dimes,
dollars contained in the amount.
2. Ability to find fractional part of cost by dividing.
3. Ability to find cost of a limited number of articles when the
unit price is known.
4. Acquaintance with commonly used rates of transportation
and communication services.
5. Ability to read and write amounts of money significant in
interpreting expanding environment.
6. Ability to understand relationship of banks to thrift and the
safe keeping of money.
7. Ability to budget a sum of money when cost of materials
is known (teacher guidance).
8. Ability to make purchases and check change.
4. Acquaintance with clinical thermometer for indicating fever.
5. Acquaintance with the means used for protecting plants,
radiators, water pipes against freezing.
6. Ability to read room temperatures expressed in written form.
GRADE SIX
1. Ability to assume responsibility for maintaining correct room
temperatures.
2. Ability to discover body temperature by using clinical
thermometer.
3. Ability to interpret graphs and chart indicating temperatures
(locally and throughout the world).
GRADE FIVE
1. Ability to assume responsibility for maintaining correct room
temperatures.
2. Ability to understand why warm air rises.
3. Ability to mark points on a pictured thermometer which
indicate freezing, critical temperatures for certain crops,
correct room temperature, normal body temperature, low
and high deviations of temperature for the United States,
boiling point, points below freezing point of water, and
below zero.
4. Ability to interpret graphs and charts indicating local and
regional temperatures (teacher guidance).
5. Ability to construct graphs and charts indicating temper
ature of local community for definite period of time (teacher
guidance).
6. Acquaintance with Centigrade thermometer.
7. Acquaintance with the freezing and melting points of butter,
alcohol, mercury, and iron.
8. Ability to understand significance of antifreeze for radiators.
9. Ability to associate specific temperatures with weather report.
10. Ability to read plus and minus temperatures expressed in
written form (whole numbers or mixed numbers).
1. Ability to locate special information in which the date is
important (items from magazines of previous years, items in
newspaper files).
2. Acquaintance with anthropological and geological evidence
that the earth is very old (fossils, Indian relics, Aztec cal
endar).
3. Ability to determine how long ago an event occurred.
4. Ability to identify the different time zones by using symbols
on a map.
5. Ability to interpret adjustments made for radio schedules
because of time zones.
6. Ability to understand the reason for time zones.
7. Ability to understand the significance of the prime meridian
and the International Date Line.
8. Understanding of the significance of latitude and longitude.
9. Ability to use these lines effectively in connection with a
map.
10. Ability to understand the meaning of degree when used to
express latitude and longitude.
11. Ability to estimate the position of local community with re
gard to latitude and longitude.
12. Acquaintance with the relationship of latitude and longitude
S to the astronomical basis of time.
13. Ability to accept some responsibility for budgeting time
independently.
14. Ability to identify rhythmic patterns in 2/4, 3/4, 4/4, 6/8 time.
15. Ability to recognize the value of a whole, half, quarter,
eighth, sixteenth note.
16. Ability to express and interpret the dot when used to in
crease the value of a musical note.
1. Acquaintance with millage tokens used in other states because
of sales tax.
2. Acquaintance with coins and currency used in neighboring
countries.
3. Knowledge of rate of exchange between the United States
and neighboring countries.
4. Ability to find fractional part of cost by dividing.
5. Ability to find the cost of any number of articles when the
unit price is known.
6. Acquaintance with rates of transportation and communi
cation services to points within the United States and to
neighboring countries.
7. Ability to read and write amounts of money significant in
interpreting expanding environment.
8. Ability to understand that interest is money paid for the
use of money.
9. Ability to read amounts expressed as interest in written form.
10. Ability to budget a sum of money when cost of material is
known (teacher guidance).
11. Ability to make purchases and check change.
1. Knowledge that one mill is onetenth of a cent.
2. Acquaintance with coins and currency used in various
parts of the world at different times.
3. Knowledge of rate of exchange between United States and
other countries of the world.
4. Acquaintance with the term, legal tender, and its significance.
5. Ability to find the average price when the total cost is
known.
6. Knowledge of rates of transportation and communication
services to places throughout the world from the United
States.
7. Acquaintance with different rates of postage for different
countries.
8. Ability to understand that interest is money paid for the
use of money.
9. Ability to read and write amounts expressed as interest in
written form.
10. Ability to read and write amounts of money independently.
11. Ability to budget a sum of money when cost of materials is
known.
12. Ability to make purchases and check change.
4. Ability to construct charts and graphs indicating temper
ature (locally and throughout the world).
5. Knowledge that 00 C. and 32 F. are the same and that
100 C. and 212 F. are the same.
6. Knowledge of the freezing, melting and boiling points (water,
butter, alcohol, mercury, and iron).
7. Ability to understand significance of antifreeze for radiators.
8. Ability to associate specific temperatures with weather
reports.
9. Ability to read plus and minus temperatures expressed
in written form (whole numbers and mixed numbers).
1. Ability to locate special information in which the date is
important.
2. Ability to understand the significance of the birth of Christ
as a point before which and after which time is recorded
differently.
3. Ability to use B. C. and A. D. significantly.
4. Acquaintance with the measurement of time in decades,
scores, and centuries.
5. Knowledge of the age of the earth based on anthropological
and geological evidence.
6. Ability to solve printed problems in which time and speed
are significant.
7. Knowledge of some of the different systems man has devised
for measuring time.
8. Ability to understand the relationship of latitude and longi
tude to the astronomical basis of time.
9. Ability to understand the meaning of degree, second, minute,
when used to express latitude and longitude.
10. Ability to budget time independently, usually with good
results. . \
11. Ability to interpret rhythmic patterns and express them in
written form.
I I I I
MATHEMATICAL CONCEPTS 11
A total picture of the development of understandings and skills
concerningg time, length, mass, and exchange would include all
distinguishing characteristics of each. The charts show only that
part of the picture which will be of interest to and within the
level of ability of elementary children. A development of the
concept of density, for example, has been omitted.
DEVELOPING THE CONCEPT OF TIME
AND TIME RELATIONSHIPS
Complete understandings and skills involving time cannot be
established at any one stage of the child's development. It seems
certain, however, that better understandings and skills can be
developed if each teacher recognizes the major problems involved.
As a result of experiences with clocks and calendars the child
should come to an appreciation of man's resourcefulness and an
understanding of the sequence of events in world development.
Likewise, he becomes aware of the part speed (an interrelation
ship of time and distance) plays in modern life.
In the beginning children have many experiences with telling
time, and make decisions regarding the use of time. As they
grow older, they become interested in the variety of clocks avail
able for telling time and are curious about their differences.
Children hear such words as Seth Thomas, Big Ben, Ingersoll,
and wonder about them. They notice the difference between tiny
watches and grandfather clocks. They become conscious of the
fact that some people have sundials in their gardens, and perhaps
they hear that the sundial on Bok Tower is one of the most ac
curate in the world. They read about candle clocks, water clocks,
and the like, and recall that an eggtimer is really a miniature
hourglass. It is at this point that they are ready for other in
formation about the historical development of time telling de
vices. This involves a partial understanding of the astronomical
basis of time. How much information children get and are able
to assimilate will depend on their level of maturity, and the
variety and number of previous experiences with time.
Learning to tell the date by means of a calendar is another
skill which should receive attention in the primary grades. Many
teachers capitalize on the need for finding the date as a means of
helping children identify numbers, and learn their sequence.
12 ARITHMETIC IN THE ELEMENTARY SCHOOL
In the upper grades children become curious about the origin of
the modern calendar when they read about calendars used by
other races and cultures. Information about the astronomical
basis of time is necessary if the child is to understand some of
the differences in calendars. This should lead to an appreciation
of man's resourcefulness in developing and improving ways of
measuring and recording time.
An understanding of the sequence of events in world history
develops gradually from the moment when children say, "When
Daddy was a little boy", or "Last Christmas", or "Before I was
five". It is not until they are in the intermediate grades, how
ever, that children are able to understand the significance of
B. C. and A. D. and the various periods of world development.
An understanding of the part speed and velocity play in our
daily life begins early but it is not until they are more mature
that children recognize speed and velocity as an interrelationship
of time and distance. Children become aware of the importance
of design as they make comparisons between slow and fast mov
ing animals, old model automobiles with new ones, fast moving
clipper ships with other ships of that period, old model airplanes
with those common today. They also become conscious of various
instruments for measuring the speed of different things and of
the several ways in which speed may be expressed.
It becomes increasingly evident that understandings and skills
involving time develop gradually and sequentially. It is the re
sponsibility of the teacher, then, to recognize the problems in
volved and to understand the sequential nature of their de
velopment.
Explanation of the Chart.The chart below contains six major
problems involving time and speed (a time relationship). In all
cases, beginnings of the understanding are indicated, and at least
one extension. No effort has been made to put the beginnings
and extensions into grade allocation. The vertical columns are
used for separating the extensions, and not to indicate on what
level of maturity the understandings and skills develop. The
value of the chart lies in the fact that it shows clearly that an
understanding of time is sequential in its development and can
be anticipated by the teacher.
DEVELOPING THE CONCEPT OF TIME AND TIME RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II [EXTENSION III VOCABULARY
PROBLEM ITo know the place of calendars in our society and to appreciate their utility.
Frequent reference to the Adult and children refer Children become inde Children are able to lo date, calendar,
date by adult, who gives to calendar to determine pendent in the use of the cate special information September 2, 1942
special notice to impor correct date. Teacher calendar. They can read in which date is impor Sept. 2, 1942
tant dates (birthdays, helps children determine and write the date in all tant. 9/2/42
holidays, special events). proper sequence of days forms. The form 2/7/41 today year
in the week. is introduced as the child yesterday
meets it on library card tomorrow
or school bank slip. month
names of months and
their abbreviations
days of week and their
abbreviations
PROBLEM IITo understand how man has been resourceful in developing and improving ways of measuring and recording time
by means of a calendar.
Children are aware of day Children learn that a cal Children begin to under night
and night, the rising and endar represents the days stand the significance of day
setting of the sun and of the week, the months a calendar for recording year
moon, changes in the size of the year. the passage of time and rotate
of the moon, seasonal learn something of its axis
changes. astronomical basis.They planet
read about primitive ancient
peoples and of their at modern
tempt to reckon time. solar system
In addition, they make season
simple comparisons be
DEVELOPING THE CONCEPT OF TIME AND TIME RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
tween the modern cal
endar and those of the
Babylonians, Egyptians,
Greeks, Romans, Az
tecs, Mayas, Incas.
PROBLEM IIITo understand something of the time sequence of events in world development.
Children hear frequent Children continue learn Children begin to sense primitive
references to events which ing meaning and signifi the chronological rela B.C.
took place before their cane of special days and tionship of events in A.D.
birth. They use such ex events in history. In ad world development, time line
pressions as, "Before I edition, they learn that They understand the
came to school," "Last the earth is very old and significance of B.C. and
Christmas," "When Dad that changes have oc A.D.
dy was a little boy." They curred on its surface
are conscious of their own from time to time.
birthdays, and begin to
understand the signifi
cance of outstanding holi
days such as Christmas,
Easter, Thanksgiving.
PROBLEM IVTo understand the place of clocks in our society and to appreciate the value of exact time measurement.
A. Children learn how the use of a clock helps them meet home and school responsibilities.
Children associate recess Children learn arrange Children learn to tell clock face
time, lunchtime, with cer ment of numbers on time independently, hands of clock
DEVELOPING THE CONCEPT OF TIME AND TIME RELATIONSHIPS
BEGINNING EXTENSION I
tain position of hands on clocks. They learn that
clock, the space between num
bers represents five min
utes.
VOCABULARY
hour hand
minute hand
A.M., M., P.M.
half hour, quarter hour
hr., min., second
B. Children begin to see the necessity for making worthy use of time through estimating and budgeting time.
Adult determines whether Adult guides children in Adult guides children in Children budget time budget
child has time to begin a selecting appropriate task evaluating choices they independently and may
task. for time available. (Indi have made, particularly be successful or unsuc
vidual or group in con in individual projects, cessfulin carrying it out,
nection with planning depending on how fam
program.) iliar the situation is.
Later they are able to
budget time independ
ently with good results.
PROBLEM VTo understand how man has been able to weave order out of chaos by developing various instruments for measur
ing time.
Through participation in Children learn ways of rhythm, pattern
rhythmic activities child expressing time patterns quarter note
ren learn that there are in written form in music whole note
definite time patterns (mu and poetry half note
sic, poetry, dancing, eighth note
games). metronome
repeat
____L__ _____________ _____________________________repetition

DEVELOPING THE CONCEPT OF TIME AND TIME RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
Through experiences with Toward the end of the hour glass
clocks, sundials, hour elementary grades child sundial
glasses, children begin to dren are aware of many water clock
see that man has been re devices for telling time candle clock
sourceful in developing through seeing them in
and improving devices for the home, reading about
telling time. them, and developing a
real appreciation of the
efforts of man to simpli
fy timetelling.
Children comment on or Through experiences with The teacher demon Children learn to inter time belt
ask about differences in radio, travel, globes, states with globes and pret time zone symbols time zone
time indicated by radio, maps, children learn that maps how it is possible on their maps. Investi latitude, longitude
They may ask about dif the world is divided into for darkness to come to gate reasons for time meridian
ferences in time observed time zones, the Atlantic Seaboard boundaries being placed Greenwich
while traveling from one before it comes to the where they are. Learn International Date Line
section of the country to Pacific. meaning and signifi degree
another. chance of the Interna
tional Date Line, Green
wich.
PROBLEM VITo understand the part speed and velocity (an interrelation of time, direction, and motion) play in modern life.
Children can distinguish Children learn the speed Children begin to under Children become inde velocity
between "slow"and"fast" of an automobile is indi stand the relation be pendent in the interpre mile per hour
when walking, riding, cated by a speedometer, tween design and speed station of numbers repre MPH
communicating. Hear Later, they learn that va through a comparison of senting speed of man speed
adults describe speed as rious instruments and old and new aeroplanes, made machines, and knot
DEVELOPING THE CONCEPT OF TIME AND TIME RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
40 miles per hour, 300 standards are used for clipper ships with other speed in nature. Like odometer
miles per hour. measuring the speed of ships. In addition, they wise, they begin to speedometer
automobiles, aeroplanes, are able to make com understand how modern chronometer
boats. prisons involving ani economic life is depend cyclometer
mals and birds with re ent to a great extent on
spect to their structure speed.
and ability to move
quickly from place to
place.
18 ARITHMETIC IN THE ELEMENTARY SCHOOL
DEVELOPING THE CONCEPT OF LENGTH
AND LENGTH RELATIONSHIPS
The major problems involving length have to do with distance,
height, width, depth, and area.
Length in the Primary Grades.In the primary grades, most
experiences with distance, height, and the like will be actual
ones within the classroom. The teacher will see that children
have ample opportunities for comparing, measuring, and esti
mating such things as their own heights, heights of furniture in
the classroom, paper for art work, materials for various kinds
of construction. The children will acquire meanings for such
terms as feet, inches, yards, and will recognize these terms when
they are written in their various forms.
Length in the Intermediate Grades.In the upper grades, chil
dren will continue measuring as they need to in the classroom.
In addition, they will be able to attack problems of a more com
plex nature. Activities in which finding area is necessary seem
appropriate at this stage, since children hear adults refer to area
and surface and find the terms frequently in their reading. Chil
dren may measure to find the area of the school garden in square
feet or yards, and of the playground in acres. They may estimate
to determine the area of their community in square miles. They
are usually interested in finding the area of their classroom and'
in determining the average amount of space per pupil. However,
while the manipulation of numbers for finding area may be neces
sary in the intermediate grades, children are not expected to be
come independent in the use of this skill until much later.
Likewise, it is in the intermediate grades that children first
feel the need for using large numbers in connection with heights,
altitudes, astronomical distances, depths, and the like. They will
need help in learning to express and interpret these figures. This
will not be difficult if the teacher exerts leadership in making
comparisons between the known and familiar, and the new ele
ments under consideration. For example, before Florida children
understand land elevation it is usually necessary for adults to
make comparisons by means of diagrams. Florida elevations are
slight. Therefore, children should compare (without the use of
numbers) the elevation of their own community with the eleva
MATHEMATICAL CONCEPTS 19
tion of the highest place they have ever been, or with pictures
of mountains. Then they should discover what is the highest
point in Florida and compare its elevation with that of their
own community. Along with this explanation of height should
go the explanation of sealevel and its significance in determin
ing altitude and elevation.
Explanation of the Chart.The chart below contains three
major problems involving length and length relationships. In
all cases, beginnings of the understandings are indicated, and at
least one extension. No effort has been made to put the beginnings
and extensions into grade allocations. The vertical columns are
used for separating the extensions and not to indicate on what
level of maturity the understandings and skills develop. The value
of the chart lies in the fact that it shows clearly that an under
standing of length is sequential in its development and can be
anticipated by the teacher.
DEVELOPING THE CONCEPTS RELATIVE TO LENGTH AND LENGTH RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
PROBLEM IRecognizing that there are differences in lengths and that these differences are expressed in terms of height, width,
depth, length, and distance.
A. Children learn use of yardstick, tapeline, and ruler for measuring in classroom.
Teacher estimates height Children experience need Childrenexperience need Children have many ex inches, inch, in., "
of children in relation to to express height in terms to express differences experiences in estimat feet, foot, ft., '
seats, chairs, tables, so of feet and inches and in height, using methods ing and measuring with yard, yd.
that each child uses furni parts of an inch. compatible with their emphasis on accuracy tapeline
ture of correct size. Teach maturity, and speed. yardstick
measures the heights of ruler
children. They learn to weight
read the numbers which height
tell their heights. Teacher measure
refers to height of adult scales
for comparison. pound, lbs., #
B. Children learn to appreciate heights of manmade structures.
Teacher and children sur Heights of manmade Teacher and children re Teacher and children height
vey community to find structures are interpreted fer to printed materials find ways of expressing high
structures of more than orally as teacher and for information about relative heights through low
one story (houses, barns, children refer to pictures, Florida cities, with em art, reading and writing tall
steeples, tanks, silos, busi movies, and other visual phasis on heights of numbers, reading and
ness buildings), aids to get feelings of rela buildings. Teacher and interpreting charts.
tive heights (buildings in children investigate rea
various parts of the coun sons for varying heights
try, monuments, dams) of buildings in various
cities, reasons for heights
of dams, etc. Makes
comparisons with heights
of monuments.
DEVELOPING THE CONCEPTS RELATIVE TO LENGTH AND LENGTH RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
C. Children learn to appreciate natural heights and depths.
Children have experiences Children have real or vi Children learn that land Children and teacher re elevations
observing the depth of carious experiences with is sometimes below the fer to elevations and depths
water in the gulf, ocean, natural elevations and level of the sea. (Death depths outside of Florida sea level
springs, sink holes, arte depths as they discover Valley, Dead Sea.) as means of understand fathoms
sian wells, Lake Okeecho continental shelf, that ing Florida elevations continental shelf
bee, limestone quarries, water is measured in fath and depths.
phosphate mines; as they oms, mountain heights
observe height of sand and how they vary in dif
dunes along coast and ferent parts of the coun
hills of Central and West try and in the world.
Florida; as they have ex
periences with caverns.
PROBLEM IIRecognizing that distance is how far one thing is from another and that various units may be used to measure
this distance.
Children accept adult de Children learn how to de Under adult guidance Children become inde direction
scription of distance from termine distance, at the children experience need pendent in use of road distance
one place to another until same time learning to give to read and use house maps and in interpret odometer
they have had various ex and follow directions in numbers, street names, ing maps in a city direc cyclometer
periences dealing with dif going from one place to road signs, road maps, tory. miles
ferent distances. the other, and to interpret dis fraction of miles
tances indicated on such
instruments as odom
eters, cyclometers.
DEVELOPING THE CONCEPTS RELATIVE TO LENGTH AND LENGTH RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
Children gain a feeling of Children learn to estimate Children learn that "far near
nearandfarbyexperiences distances and the need for ness" and nearness are
in physical activities such accurate estimation relative matters. An far
as reading, throwing or through participation in airplane seems near
kicking objects, and activities which require when only four hundred
through comparisons of them to gauge distance feet in the air.
distance from local cor (organized games, cross
munity to another place. ing street, riding a bi
cycle).
Children gain feeling of Through adult guidance Children become curious Children become inde solar body
astronomical distances by children learn that solar as to size and distance pendent in reading and diameter
looking into the sky, no bodies are different, have of moon, sun, and other writing of large num astronomical
ticing differences in sky at different names, solar bodies from the bers and in solving prob circumference
night and in the day time, earth. Through adult lems in which astronom Earth, Mars, Jupiter,
noticing such phenomena guidance, and reading, ical information is in etc.
as falling stars, meteors, their curiosity is partial volved (distance, diam planet, star
comets, eclipses, ly satisfied, eter, circumference), meteor, moon
PROBLEM IIIRecognizing that area is the amount of surface within specified limits and that it can be measured in different units.
Teacher helps children Teacher and children Children independently Teacher helps children scale
gain a feeling for space mark off area for games, measure any area (in become independent in area
through such activities as gardens. Later they mea games, of garden, of more accurate estimat compare
selecting paper of proper sure accuratelysuchareas farm, for construction). ing and comparing, estimate
size for drawing, clearing as portfolio covers, base acre
floor space for rhythmic ball diamonds, garden, square mile
activities, arranging ma city block (for scale map), square inch
trials on work tables, volleyball and tennis square yard
courts, farm or commu square foot
nity. (also abbreviations)
DEVELOPING THE CONCEPTS RELATIVE TO LENGTH AND LENGTH RELATIONSHIPS
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
Children learn that there Children learn that a map Children learn to inter Children begin to deal accurate to scale
are ways of representing is an accurate to scale pret distances on maps with figures represent rectangle
things graphically and a drawing, and globes by using ing areas and population circle
map is really a picture of scale of miles and ruler. (as these are affected by triangle
land and water bodies. They learn to draw to political and economic
scale such things as factors)
school garden plots, mu
rals, back drops, maps,
puppet show equipment,
geometric figures for
finding areas (rectangle,
circle, triangle).
ARITHMETIC IN THE ELEMENTARY SCHOOL
DEVELOPING THE CONCEPT OF MASS
AND MASS RELATIONSHIPS
The word mass has no place in the vocabulary of elementary
school children. It is used here because it is the only accurate
term which is broad enough to include solids, liquids, and gases.
Children apply the term, liquid, to such things as milk, water,
orange juice, and certain medicines, because of its distinguishing
characteristic. It is later that they recognize the existence of air
at all, and even later before they associate it with other gases.
The solids children know are people, chairs, tricycles, animals,
dishes. Their like elements are not apparent to them. For this
reason, an elementary teacher will not use the term mass. Neither
will she speak often of liquids, gases, solids, but will speak instead
of water, milk, fruit juice, air, helium, hydrogen, people, desks,
books, and the like.
Explanation of the Chart.The chart below contains six major
problems involving weight, capacity, pressure, buoyancy, and
selfexpression in the arrangement of mass. In all cases, be
ginnings of the understandings are indicated, and at least one
extension. No effort has been made to put the beginnings and
extensions into grade allocations. The vertical columns are used
for separating the extensions, and not to indicate on what level
of maturity the understandings and skills develop. The value
of the chart lies in the fact that it shows clearly that an under
standing of mass is sequential in its development and can be an
ticipated by the teacher.
IdEVELOPING CONCEPTS RELATIVE TO MASS AND MASS QUALITIES
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
PROBLEM IRecognizing that air is a gas and that gases occupy space, have weight, and exert pressure; and that man has de
veloped ways of controlling and using these factors.
Teacher and children ad
just windows for proper
ventilation.
Teacher helps child recog
nize the need for closing
windows as a fire precau
tion by means of con
trolled experiments show
ing various ways of ex
tinguishing fire through
eliminating oxygen sup
ply.
Child gains a partial un
derstanding of compres
sion through inflating bal
loons and footballs, using
peashooters, filling bi
cycle tires.
Child has experiences with
partial vacuums (medi
cine droppers, straws).
Teacher and children con
sider proper ventilation
in relation to tempera
ture.
Teacher guides childtoan
understanding that fresh
air and proper tempera
ture is necessary for main
taining health. Points out
reasons for cutting holes
in boxes where animals
are placed. Children read
about and discuss causes
of asphyxiation, and
drowning.
Child learns to discrimin
ate between objects that
are well inflated and those
that are not.
Child is able to assume
responsibility for proper
ventilation and tempera
ture.
The class visits the local
fire department to un
derstand community
service rendered by the
department. (Cost of
equipment, time to reach
fire certain distance
away, etc.)
Child learns the impor
tance of using gauges for
determining amount of
air in a tire.
As a result of adult guid
ance and many experi
ences the child begins to
understand that when
Child recognizes that
there are different kinds
of thermometers and
that they have various
uses.
Through reading, mov
ies, radio, child becomes
familiar with many uses
society has made of
compressed air.
Child begins to under
stand and appreciate the
many uses society has
made of vacuums and
thermometer
degree ()
Fahrenheit
temperature
Centigrade
oxygen
air
vocabulary of fire de
partment equipment
gauges
pressure
barometers
vacuum
atmospheric pressure
DEVELOPING CONCEPTS RELATIVE TO MASS AND MASS QUALITIES
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
Teacher helps child un
derstand certain uses of
air (gas) by directing his
observation as he plays
with air toyskites, pin
wheels, gliders noting
possible influence of de
sign, size, material.
Child learns to discrim
inate between materials
for constructing these
toys. Learns to distin
guish between types of
airplanes, and lighter
than air and heavier than
air craft. Learns that he
lium and hydrogen are
used for lighter than air
craft, and the advan
tages of each.
Children are conscious of Children learn that wind
changes in weather. is really air in motion.
air is removed from a
container a vacuum oc
curs and there is no pres
sure within it. (Ther
mometers, vacuum
cleaners, barometers.)
As a result of adult guid
ance, pictures, reading,
movies, and direct con
tacts, child begins to un
derstand the interrela
lationship of design, ma
terial, weather con
ditions in aeronautics.
Children learn about in
fluence of temperature
and atmospheric pres
sure on weather.
partial vacuums. (Boil
ing sugar under pressure
to avoid scorching, ther
mos bottle, instruments
of all kinds, airplanes).
Teacher and pupils ex
plore man's utilization
of all gases. (Aircraft,
fuel, in hospitals)
Children learn to appre
ciate significance of
weather forecasts, and
the use made of infor
mation about atmos
pheric pressure, visibili
ty, wind velocity, tem
perature, in his own
community and through
out the world.
size
helium
hydrogen
ether
natural gas
velocity
pressure
highlow pressure areas
weather map symbols a
weather signals
weather reports
DEVELOPING CONCEPTS RELATIVE TO MASS AND MASS QUALITIES
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
PROBLEM IIRecognizing that liquids have weight and exert pressure; and that man has developed ways of controlling and
using these factors.
Through experiences child
learns there are differ
ences in weight of objects
due to presence or absence
of liquids (towels, bathing
suits, flowers after a rain,
dried fruit).
Child first feels and sees
effect of pressure (in
swimming, walking up
stream, garden hose, ob
jects floating down
stream).
These experiences help
child understand that wa
ter has weight (62.4
pounds per cubic foot at
sea level). Later he is
able to understand that
liquids differ as to weight
(alcohol, mercury).
Out of these experiences
the child acquires limited
understanding of modifi
cations man has made to
use or overcome natural
force of pressure (how his
home is supplied with wa
ter, use of water wheels,
power plants).
Through experiences the
child learns that all
floating objects must be
constructed so that their
their total weight is less
than that of the water
they displace. Floating
depends on design, ma
terials, weight.
Through movies, read
ing, adult guidance, child
gets a better under
standing of effects and
control of air and water
pressure (submarines,
tunnel construction, div
ing suits).
Child learns use of Plim
soll mark and ballast in
proper loading of ships.
cubic foot, cu.ft.
Plimsoll mark
displace
ballast
sandhog
diving helmet
air chamber
air hose
periscope
submerge
emerge
bends
PROBLEM IIIRecognizing that solids have weight and that man has developed ways of controlling this factor.
Through experiences the Teacher weighs child. Child experiences need Child continuesestimat pounds
child learns that he can Child learns to read num to express differences in ing and weighing with fractional parts of a
lift some objects; that he ber which tells his own weight. emphasis on accuracy. pound
fers to weight of adults.
DEVELOPING CONCEPTS RELATIVE TO MASS AND MASS QUALITIES
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
As a result of many expe
riences child learns that
there is no direct relation
between size and weight
(cork, balsa, iron)
Child uses simple ma
chines for moving weight
(wagon, spade, ramp).
Later, the child experi
ences need for expressing
weight in terms of pounds
and fractional part of a
pound.
As a result of experiences
and with adult guidance
child acquires information
as to relationship of ma
terials and weight.
Child observes many ma
chines used by man for
liftingweight (steamshov
el, grease rack, jack).
Child learns how man
has invented machines
for moving weights more
conveniently and effec
tively (levers, pulleys,
inclined plane, magnets,
canal locks). Child also
learns than man has de
veloped agencies for
transporting goods (post
office, express agency,
moving vans, planes).
Adult helps child under
stand that weight is
merely the name we give
to the pull which exists
between the object and
the earth by helping him
notice that everything
tends to fall to the earth.
By observing aircraft
the child learns that man
has overcome the effects
of gravity to some ex
tent.
gravity
ascend
descend
lever
pulley
magnet
plane
ramp
DEVELOPING CONCEPTS RELATIVE TO MASS AND MASS QUALITIES
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
PROBLEM IVRecognizing that solids, liquids, and gases sometimes change weight and size as they change form and tempera
ture (hot air rises, cane juicehot syrup to coldbrown sugar to white, molten metal to solid, water to ice. Oral
level with little use of numbers).
PROBLEM VRecognizing that there are different systems of weight and that the weights are expressed in different units.
Child becomes aware of Child begin to see that Child learns that not all Child comes to realize avoirdupois
gross differences in weight liquids, gases, solids, each countries use the same that it is his responsi troy
(coal is weighed in tons, have their special way of standard of measure ability to know facts metric system
butter in pounds). being measured, ment as the United about government stan imperial gallon
States. Becomes curious ards and to be able to grain, gram
as to the reasons for use this information to pint, pack, bushel
these differences, good advantage. (also abbreviations)
PROBLEM VIRecognizing the place of line, dark and light, and color in arranging mass for selfexpression.
Child sees and is able to Child performs at this (1) Child is able to par
reproduce shape of ob level for some time. He ticipate in group pro
jects and learns to make begins to pay more atten jects (a) in which the
prominent or dominant tion to details. Pictures group strives for realism
things in the setting which of horses, dogs, cows, look and all objects are in
which are most important more natural. Pictures of correct proportion to
to him. The child associ lesser known animals are each other and the ar
ates characteristic shapes less accurately drawn. rangement is properly
with objects and the balanced, (b) where chil
teacher gives them a name. dren express a dominant
Child becomes acquainted idea through distortion proportion
with size, shape, color of arrangement
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
objects not in his immedi
ate environment through
use of pictures, before he
tries to draw them.
Child does simple model
ing of objects so that the
various parts are in pro
portion.
Child constructs toys, etc.
so that they look right to
him.
Child is directed to draw
picture to fill the page.
Teacher gives detailed
help and child learns that
any one object may be
made in a variety of sizes.
Teacher calls attention to
parts of child's work
where improvement is
needed in order that he
will be able to construct
an article in better pro
portion.
Child is given help in
placing objects in order
to give proper emphasis
to picture or setting.
of color, shape, propor
tion. (In a mural on Ha
waii the pineapple may
be over large).
(2) Child learns about
proportion in the human
figure through observa
tion (classmates used as
models).
Teacher presents an ob
ject to be modeled by
the class, emphasizing
proper techniques and
and proportion.
Child learns to estimate
or measure object for
which article is to be
made, and measure his
materials so that articles
conform to standards
and are usable (curtains,
bookshelves, stage prop
erties backdrop, fur
niture).
Child learns to leave a
margin to make his pic
ture more attractive.
Child receives continu
ous evaluation and guid
ance in all clay model
ing.
Child learns importance
of detail as he draws
maps to scale and con
tinues to make marion
ettes, backdrops.
Child learns that the
idea to be expressed
often determines mate
rials and techniques.
balance
dominant
distortion
margin
emphasis
setting
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
Child sees attractive bul Child learns to appreciate Child uses his initiative Teacher uses bulletin arrangement
letin board which teacher a pleasing arrangement in arranging attractive board to call child's at
arranges frequently. by observing and by plac bulletin board. tention to art principles
ing things on the bulletin balance, proportion,
board under teacher di rhythm.
reaction.
Child learns that a simple Child learns to appreci Child uses geometric fig triangle
drawing of a flower, bird, ate the repetition he figures in repeated de circle
boat, airplane, when re finds in art work of prim designs. square
peated makes a pleasing itive peoples as well as rectangle
design. in that of today. repetition
border
repeat design
32 ARITHMETIC IN THE ELEMENTARY SCHOOL
DEVELOPING CONCEPTS RELATIVE TO VALUE
AND VALUE RELATIONSHIPS
Many opportunities occur in the first year of school for
children to gain skill in using money, in reading and writing
numbers representing amounts of money, and in making de
cisions regarding appropriate exchanges without the use of coins
and currency. For example, the teacher may write on the board
the number of pennies brought by individual children for Junior
Red Cross. From this experience, children begin to associate the
cent sign with the number symbol representing amount. Later,
they have experiences with simple column addition as various
amounts of money are brought to school for charity, parties,
supplies. At all times the teacher is conscious of her responsi
bility for developing attitudes regarding the use of money, trad
ing, sharing. Perhaps the most important outcome in the ele
mentary school regarding exchange, aside from skill in manipu
lating, reading, writing, and interpreting numbers representing
sums of money, is the development of proper attitudes regarding
the use of money, trading, responsibility to individuals, the na
tion, and the world at large.
Explanation of the chart.In the chart below five major prob
lems are indicated. While beginnings are made in the solution
of these problems early in life, the extensions are farreaching
and in some cases, are hardly touched on in the elementary school.
The vertical columns are used for separating the extensions, and
not to indicate on what level of maturity the understandings and
skills develop. The value of the chart lies in the fact that it
shows clearly that an understanding of exchange is sequential in
its development and can be anticipated by the teacher.
DEVELOPING CONCEPTS RELATIVE TO VALUE AND VALUE RELATIONSHIPS (EXCHANGE)
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
PROBLEM IRecognizing that all things have value and that they may be obtained by various means of exchange.
A. Children learn proper care and expenditure of money (coins, currency, checks, money orders).
Through such experiences Through such activities Children have experi Children come to real penny, cent,
as buying candy, lunches, as buying lunches, the ences with trading, bar ize that it is their re nickel
school supplies, warbonds children have experiences tering, borrowing, and sponsibility to select dime
and stamps, tickets, car in finding the cost when learn the meaning and carefully when making quarter
fare, and making contri purchasing or selling and use of credit and trading purchases and to budget halfdollar
butions to organized char learn to receive and make in kind. money wisely. dollar, $
ity children learn the com correct change, decimal point (.)
parative value of a pen coins
ny, nickel, dime, quarter, currency
half dollar, and dollar, checks
They become familiar with money orders
amounts of money ex
pressed in written form, and
learn to be independent in
expressing these amounts.
PROBLEM IIRecognizing that there are agencies that restrict and promote exchange of goods.
A. Children begin to understand the advantages and disadvantages of various means of transportation.
Children become aware of Children begin to associ Children begin to under Child begins to appreci produce
trucks, trains, boats, ate these means of trans stand and appreciate va ate the fact that trans truck farm
planes, and their cargoes. portation with sale of lo rious means of transpor portation facilities in the transportation
They observe that at times cal produce in northern station, the use of rail United States are more delivery
fresh fruit, vegetables, markets. Inaddition, they roads in shipping large highly developed than facilities
meat from their own gar begin to associate these amounts long distances, those of other countries markets
den, nearby farms, are means of transportation use of trucks for lesser (camels, llamas, rick
DEVELOPING CONCEPTS RELATIVE TO VALUE AND VALUE RELATIONSHIPS (EXCHANGE)
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
available. They also ob with products sold in local amounts and conveni shas, canal boats pulled
serve that at certain time stores which are not pro ence of delivery, use of by manpower, ele
of year peaches, cherries, duced in the immediate cheaper boat transpor phants, small automo
etc., are available to them community (cherries, ba station where speed is biles of Europe).
although they are not nanas, certain fish, wool, not important, use of
grown i n their o w n coffee, silk), plane for small quanti
community). ties where speed is essen
tial, use of wagons, ox
carts for home to mar
ket and back transpor
tation.
B. Children recognize that communication plays a part in transporting, buying, and selling goods.
Children recognize impor Children begin to under Children begin to under Children begin to under stamp
tance of telephone in plac stand and appreciate ser stand place of telegraph stand and appreciate in air mail
ing local orders, vices available through in buying and selling. ternational services special delivery
postal system (mail order, Orders are frequently cablegrams, radiograms, parcel post
money orders, air mail, placed after information transoceanic telephone, registered mail
special delivery, parcel is received through radio C.O.D.
post, registered mail, and newspapers. mail order
C.O.D.)  cablegrams
radiograms
telegrams
long distance
telephone
zone
DEVELOPING CONCEPTS RELATIVE TO VALUE AND VALUE RELATIONSHIPS (EXCHANGE)
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
PROBLEM IIIChildren recognize part markets play in exchange of goods from producer to consumer.
Children share in the ef Children observe that Children begin to see Children begin to see taxes
fortoftheircommunityto market prices fluctuate place of middleman part overhead plays in overhead
put produce on the market from month to month and through comparison of fixing prices of commo expense
while prices are good. (A from year to year. prices on Georgia fruit ities. (A fruit peddler quart
Plant City child doesn't (Weather conditions, trucks with prices in lo can sell at lower prices pint
eat strawberries in De overproduction, labor calstores. Likewise they than canamerchantwho peck
cember if they bring $1.05 conditions, time of mar begin to see places of owns and operates his bushel
a quart as they did in keting, cost of materials, large buying, when they business from a build price
1925.) In January, 1938, one make wholesale pur ing.) income
quart of strawberries cost chases for school party wholesale
19 cents. By April, one or home use. retail
quart cost only 122z middle man
cents.)
PROBLEM IVChildren begin to understand that society has established laws for protecting and controlling exchange of goods;
they appreciate the need for these regulations and understand certain advantages and disadvantages which they entail.
Children identify as good Children observe name of Through adult supervi In addition to tariffs, tariff
or bad, cheap or expen country printed on back sion children learn that children learn that other sale
sive products of different of toy or other object. United States govern government regulations percent, %
brands and trademarks. May have some feeling ment charges taxes (tar have been made for pro percentage
(Shirley Temple dolls, (because of adult conver iff) on foreign goods as a testing production and import
DEVELOPING CONCEPTS RELATIVE TO VALUE AND VALUE RELATIONSHIPS (EXCHANGE)
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
Tinker Toys, electric
train versus spring oper
ated ones.)
station) about buying ar
ticles from certain coun
tries. Through adult su
pervision children learn
that labor is cheaper in
other countries than in
America.
protection to American
manufacturers and pro
ducers of raw materials.
sale of goods:
(a) Attempts to limit
production (ploughing
under, planting less acre
age).
(b) Attempts to improve
distribution, especially
since New Deal.
(c) Attempts to regulate
expense and elimin
ate unfair competition
in transportation (Inter
state Commerce Com
mission).
(d) Attempts to limit
production in order to
favor trade with U. S.
dependencies (sugar can
be produced in Florida
for 2.5556 cents a pound
which is less than cost in
Louisiana, Hawaii, Phil
ippines, Puerto Rico, but
laws restrict production
in U. S. to 29% of total
consumption.)
(e) Copyrights and pat
ents.
export
patent
copyright
raw products
manufactured goods
trademark
brand
DEVELOPING CONCEPTS RELATIVE TO VALUE AND VALUE RELATIONSHIPS (EXCHANGE)
BEGINNING EXTENSION I EXTENSION II EXTENSION III VOCABULARY
PROBLEM VChildren recognize that banks and credits are essential aids to exchange. Through experiences they learn to take
advantage of the services they offer.
Children observe and Children become aware of (a) Children become a (a) Children become a deposit
sometimes use various deposits in bank which ware of various services ware of local, national, check
ways of obtaining goods make it possible to use offered by banks (say and international credit charge account
coins and currency, check, a check. Children be ings department which system and some of their savings account
credit. (Children may pay come aware also of ne pays interest, checking farreaching effects (in interest
fees at school with check, cessity for paying bills at accounts, loans, safety eluding war stamps and deposit slip
buy groceries for adult and regular intervals, deposit boxes), bonds with interest), bank statement
charge them). (b) Children become a (b) Children become a loans
ware that in buying on ware of advantages and cash price
credit or in installment disadvantages in install installment
buying the cash price ment buying, see rela bonds
and credit price are often tionship of budgeting to stamps
different, this kind of buying, budget.
Chapter Three
DEVELOPMENT OF COMPUTATIONAL SKILLS
Chapters One and Two point to the desirability of helping
children become more sensitive to the quantitative problems of
their environment and of developing the necessary readiness for
participating in the solution of these problems. These chapters
also contain a discussion of measurement which, when used to
include comparing, measuring with standard instruments, or es
timating, may be considered the master processes. The processes
for which children have a need are describing and measuring
describing and measuring amounts of time, length, mass, and
money. This is done by means of a vocabulary (little, big, far,
near, like), and by means of the number system. Use of the
number system makes it possible for the description and measure
ment of amounts to be accurate.
In the beginning, it is possible for children to satisfy their need
for accurate description and measurement by means of counting.
Before long, however, they find themselves combining groups of
objects and through the teacher's leadership learn that there are more
economical ways of combination than by counting. It is at this
point that readiness for computation develops. Although children
may be ready for computation they may become lost in a maze of
abstraction if the teacher fails to help them bridge the gap between
the concrete and abstract.
Many times children go directly from counting to abstract drill in
the simple combinations. As a result, some of them became confused,
and frustrated, and lose sight of the real purpose of arithmetic.
What children need to do is follow counting of concrete objects with
grouping of concrete objects. Then they should learn to recognize
the number symbol which stands for the number of objects within the
group. After many experiences of this kind they will gradually
arrive at an understanding of the meaning of such combinations as
2 and 2 make 4, 3 and 2 make 5.
COMPUTATIONAL SKILLS 39
As teachers attempt to direct children in acquiring compu
tation skills they need to be aware of the real life problems faced
by children at home and at school. This includes a recognition
of the child's interest in his expanding environment as he gets
older and moves from grade to grade. But in addition, teachers
need to understand the stages of development in the computa
tional operations. For example, they need to analyze the prob
lems children face when they find it necessary to bridge a decade
in addition, borrow in subtraction, use a twoplace multiplier in
multiplication, or find the quotient in division when the divisor
ends in 8 or 9.
This chapter includes (1) recommendations as to techniques
and methods, (2) an analysis of the levels of difficulty of each
of the processes involving whole numbers'and fractions, and (3)
a grade allocation chart of the computational skills.
The recommendations concerning techniques and methods are
a result of research as to the most meaningful way of arriving
at the answer of a problem through computation. For example,
takeaway borrowing as the method for solving subtraction prob
lems is recommended because it can be understood if children
learn the place value of number, and because it is the method
best suited to subtraction of mixed numbers (whole numbers and
common fractions). Because an understanding of place value of
number is fairly difficult, and an understanding of the reason for
borrowing depends on an understanding of place value of num
bers, subtraction in which borrowing occurs (except in the ones
column) is recommended for fourth, fifth, and sixth grades, rather
than for the third grade.
Included in the analysis of the levels of difficulty in the com
putational processes is a description of each skill type, 'and sev
eral examples. The analysis should be useful in helping teachers
determine next steps of instruction for individual children and
for the group.
The skills allocation chart* includes recommendations regarding
the teaching of specific skills within each of the first six grades.
In setting up the skills allocation, an attempt has been made to,
distribute the arithmetic load throughout the elementary school
in relation to the increasing maturity of the learner. Through
*See Chart opposite page 42.
ARITHMETIC IN THE ELEMENTARY SCHOOL
an analysis of the levels of difficulty within the computational
processes, and an analysis of the problems faced by children as
they move from grade to grade, it appears that an emphasis on
specific skills can be made in each grade. Teachers recognize the
fact that beginnings of computational understandings with both
whole numbers and fractions are made before the child comes
to school. Since these understandings develop gradually teachers
should be unwilling to place the emphasis on any one process or
skill in a particular grade to the exclusion of all others. The
development of meanings for all whole number and fraction pro
cesses must continue from grade to grade. The more complex the
process, however, the later the emphasis is made. For example,
skills in adding, subtracting, multiplying, and dividing common
fractions are no longer treated exclusively in fifth grade. Drill
on subtraction of fractions where borrowing is necessary is recom
mended for sixth grade. Division of fractions is not recommended
at all. The development of meanings of fractional parts is recom
mended for the primary grades. Since meanings are developed
from the first grade, some drill on simple addition, subtraction,
and multiplication of fractions is recommended for fourth grade.
The allocation of skills to grades is suggestive; the maturity
of the child should determine grade placement rather than the
naming of a skill for a grade. Emphasis on the development of
meanings should precede emphasis on the development of skills.
Any adjustment or commission which will contribute to that end
should be made by the teacher. It is recommended that the allo
cation of skills presented here be followed unless an individual
school or countywide unit should see fit to develop a program
better suited to meet its needs.
READING WRITING AND COUNTING
"Teaching children to use number symbols as a means of ex
pressing ideas is comparable to teaching them to write words. If
children put together the symbols of the alphabet to name the
things with which they have direct experience, it would seem
that number symbols should be used in the same way. Thus the
symbol 8 may be recorded to indicate the number of windows in
a room. Even number combinations, such as 5 take away 2 leaves
3, may serve to describe what happens when 2 of 5 pennies were
COMPUTATIONAL SKILLS 41
spent, 2 of 5 cookies were eaten, or 2 of 5 marbles were lost.
The early study of number symbols is therefore more than an
exercise in hand writing and number readiness; more than the
repetition of number names, in a serial order; it is a study of
ideas and ways of recording them."
Before a child enters school, he has had many experiences
with counting. He has counted his fingers, counted for games
in playing with older children, and has heard rhymes and jingles
involving counting. He may be able to count orally to five or six
but be uncertain about seven, eight, nine, ten, eleven, or twelve.
The teacher should check her group for counting readiness as
indicated by their ability to count concrete objects (rational count
ing) and by their ability to count by rote. The contribution of
rote counting is twofold. It is the beginning of the use of num
bers in abstract form, and it helps children learn the sequence
of number names so that they will know with certainty what
number comes before and what number comes after a given number.
The need for telling the date, naming position of children in
line, giving a name to a certain row of chairs, will provide op
portunities for presenting ordinal numbers on an oral level.
There are many occasions for helping the child feel the need
for reading numbers. He needs to be able to read telephone num
bers, house numbers, dates, pages in a reader. He will learn to
associate the number symbol with the verbal sound and will be
come conscious of the fact that the way a number is read depends
on its use. If he is attempting to read 1942 as a telephone num
ber he will say, "One nine four two"; as the date, "Nineteen
fortytwo;" as a population number, "One thousand nine hundred
fortytwo"; as money, "Nineteen dollars and fortytwo cents".
The need for being able to read numbers is usually accom
panied by the desire to write numbers. During their first un
supervised attempts to write numbers, children often form fig
ures incorrectly. The teacher should, in the beginning, go through
the motion of making figures on the board, calling special at
3C. L. Thiele, "Arithmetic in the Early Grades". Arithmetic in General
Education, Sixteenth Yearbook of the National Council of Teachers of Mathe
matics, (New York: Bureau of Publications, Teachers College, Columbia
University, 1941), p. 48.
ARITHMETIC IN THE ELEMENTARY SCHOOL
tention to the beginning and ending points. Children should be
taught the formation of one figure at a time and the figures
should be of any convenient size without regard to position. This
will give them a chance to fix their attention on the correct
formation of the figures. Later, they may use ruled paper and
practice fitting their figures to the space between the lines. Chil
dren having difficulty in figure formation should be given indi
vidual attention. It should be the goal of the first grade teacher
to have children able to write large, wellformed figures repre
senting zero through nine. In addition, because of watching adults
write the date, weight, height, and the like, children will want to
learn to write larger numbers by themselves. They should be
encouraged to do this but they should be guided to the extent
that they do not write 17 for 71, 24 for 42.
As children learn to write numbers of more than one place,
they should begin to understand something of the place value
of number. If there are twentythree children in the class, the
teacher will help them understand that there are twenty and
three more. Then she will write the 20 and 3 on the board and
explain to them how the 20 and 3 may be put together to become
one number, 23. Another example, which may follow or precede
this one of class membership, is an example in which 2 dimes
and 3 pennies are used. Children can readily understand that
while there is a total of 23 cents, there are also 2 dimes and 3
pennies. They begin to understand the use of zero as a place
holder, and that in this connection zero means no number. Exam
ples of this kind should be repeated over and over until the
children know that the number in the extreme right column rep
resents ones, the next number represents tens, and so on. This
understanding develops gradually but is not difficult if teachers
use concrete objects for demonstration purposes.
Reading, writing, and interpreting large numbers is an im
portant skill in the upper grades, where the child's interest in
his expanding environment makes it necessary to read figures
representing large sums of money, long amounts of time, large
areas, population, and the like. Here an understanding of place
value of number plus an ability to use commas for grouping
numbers is necessary. When the figures are mixed numbers the
word, "and", should be used to join the fraction to the whole
COMPUTATIONAL SKILLS 43
number. The use of "and" prevents children being confused as
to where the whole number stops and the fraction begins; thus,
200.026 is read two hundred and twentysix thousandths, while
.126 is read one hundred twentysix thousandths.
ADDITION AND SUBTRACTION OF WHOLE NUMBERS
Children come to school with the ability to use such phrases
as some, more, less, as many as, how much, and what is left.
Primary teachers start with this readiness by capitalizing on
meaningful situations which make it possible for children, through
counting, to come to an understanding and use of the simple
combinations. By means of direct teaching children learn the
reverse of the combinations and how to express these quantita
tive ideas in both the addition and subtraction forms.
In the primary grades children have opportunities for using
numbers from one to thirtyone in connection with the calendar,
books, pencils, paper, number of children in the class. They will
have opportunities for using numbers greater than thirtyone in
activities involving weighing and measuring, scoring games, and
determining distances. Opportunities for using zero should not be
overlooked. From using zero to indicate no score in games, no
children absent, and the like, children come to understand that it
means "no thing" and that zero added or subtracted from a num
ber does not change the value of the number. Children learn that
zero is a place holder when they learn the significance of zero in
such numbers as 50 lbs., 20 cents. Later, they learn another
meaning of zero in connection with thermometers, sea level, time
(A. D. and B. C.).
Pupils will gain a knowledge of certain combinations through
counting objects in groups and then in combining the groups.
Which combinations they learn first will depend on the number
of experiences they have with certain combinations in real situa
tions, and not on the relative difficulty of the combinations. Ab
stract drill in all the combinations should be considered after
children have acquired meanings for each of the number symbols.
In other words, before a child learns that 5 and 2 are 7 he should
know five things and two things are seven things, whether they
are children, birds, dots, or desks.
44 ARITHMETIC IN THE ELEMENTARY SCHOOL
The language which conveys the idea of addition and sub
traction should be controlled by theteacher in verbal problems
so that children will be able to make intelligent decisions about
which process to use. "How much", "how many", "how much in
all", and "what is the sum" indicate the addition process. Some
of these same words"how much taller," "how much heavier",
"how much more will it cost"also indicate the subtraction pro
cess. Children may get the answer to subtraction or addition by
"counting on". While "counting on" enables children to get the
answer it does not help them understand subtraction. Therefore,
the teacher needs to help children associate what is left, loss, and
decrease in amounts with the subtraction process.
After children have had many experiences "counting on"
(counting on a calendar to find number of days remaining before
a holiday, counting on a tapeline to find difference in heights)
the teacher will introduce subtraction in its written form. She
will help them understand the significance of the positions of the
subtrahend and minuend through rearranging the numbers, and
having children observe the position of numbers on the calendar
or tapeline.
Because of the child's interest in his expanding environment,
he will need to subtract using two and three place numbers be
fore'he leaves the primary grades. Sometimes borrowing will be
necessary. The teacher will take the initiative in solving these
problems and will not attempt to teach children the takeaway,
borrowing method of subtraction before the third grade.
While the takeaway borrowing method is recommended, it
is important that teachers allow transfer children to continue using
any method they have learned if they are successful with it.
SKILLS ANALYSIS OF ADDITION AND SUBTRACTION
Addition
A. One place addends, column additions, sums less than ten
2 20 2 20
5 50 0 0o
1 1 5 5t
COMPUTATIONAL SKILLS 45
B. Two place upper addend
Two place lower addend, no carrying
45 450 $.45
32 32 .32
59
14
500
140
$.50
.14
C. Two place upper addend
One place loweraddend, no carrying
D. Two place upper addend
One place lower addend, carrying, bridging the decade
36 36 (No "$" or zero here.
5 5 Control situation.)
E. One place addends, column addition, sums more than ten
F. Two place upper addend (less than one hundred)
Two place lower addend, carrying in units column
65 650 $.65
16 161 .16
G. Two place upper addend
Two place lower addend, carrying
Sum more than one hundred
71 71l
49 490
$.71
.49
H. (1) Two place addends
$.27
.35
.14
ARITHMETIC IN THE ELEMENTARY SCHOOL
(2) Two place addends (irregular)
270
40
I. Three place upper addend
One, two, or three place lower addend
(1) Carrying from units to tens only
136 $1.36
237 2.37
(2) Carrying from tens to hundreds only
275
241
$2.75
2.41
(3) Carrying from units to tens and to hundreds
467 $4.67
274 2.74
(4) Carrying to blank in lower addend
$2.32
.85
(5) Carrying to two blanks in lower addend
594 $5.94
9 .09
J. Three place addends, column addition
No more than five addends
$1.25
2.41
2.50
1.25 miles
2.41 miles
2.50 miles
1.25 grams
2.41 grams
2.50 grams
COMPUTATIONAL SKILLS 47
Subtraction
A. Two place minuend
Two place subtrahend
No borrowing
34 340
12 120
B. Two place minuend
One place subtrahend
No borrowing
$.34
.12
34 340
10 100
$.38
.05
5
C. Two place minuend
One place subtrahend
Borrowing
32 320
4 40
D. Two place minuend
Two place subtrahend
Borrowing
Zero to left unexpressed
sents money.
42 42 $
29 290
E. Three place minuend
One place subtrahend
No borrowing
347
5
F. Three place minuend
Two place subtrahend
No borrowing
436 $4.3
$.32
.04
30 300
4 40
in difference except when difference repre
.42
.29
320 $.32
290 .29
$3.47
.05
6
355
5
$3.55
.05
$.34
.10
.12
12
ARITHMETIC IN THE ELEMENTARY SCHOOL
G. Three place minuend
One place subtrahend
Borrowing from tens
345 $3.45
7 .07
H. Three place minuend
Two or three place subtrahend
(1) Borrowing from tens and hundreds
423 $4.23 430 $4.30 403 $4.03 403 $4.03
156 1.56 156 1.56 56 .56 356 3.56
(2) Borrowing from hundreds only
423 $4.23 $4.23 $4.50
152 1.52 3.52 3.80
(3) Borrowing from tens or hundreds
423 $4.23 423 4.23
114 1.14 151 1.51
I. Three or four place minuend with two or three zeros
One, two, three, or four place subtrahend
Borrowing from tens, hundreds, or thousands
500 $5.00 $10.00 1,600 miles
198 1.98 5.95 540 miles
MULTIPLICATION OF WHOLE NUMBERS
Multiplication is an efficient means of adding like amounts.
Children come to this understanding slowly and only after many
experiences with adding amounts that are the same.
As children have repeated experiences adding like amounts
and are successful in determining the answer the teacher will
show the relationship between the addition and multiplication pro
cess and will introduce the multiplication form. She will help them
learn reverses through rearranging groupings from which they
will understand that a rearrangement of the multiplicand and
multiplier makes no difference in the product.
COMPUTATIONAL SKILLS
Systematic drill with the multiplication combinations should
hot occur before the third grade. In the third grade the drill on
facts should be limited to those having a product of no more than
36. This product was selected because most problems which chil
dren may be expected to solve independently in third grade will
include combinations no larger than 3 x 12, 4 x 8, 5 x 7, and their
reverses. Facts involving 12 are included because 12, when rep
resenting a dozen, or a foot in length, can be used in many so
cially significant problems. Facts involving 10 are included be
cause they are useful in solving problems about money. Facts
involving 11 are included because they are not difficult to learn,
and are of special interest to creative childrenchildren who
enjoy manipulating numbers and establishing their sequence.
Because of the child's interest in his expanding environment,
he will need to use multipliers of more than one place before he
leaves the primary grades. The teacher should take the initiative
in solving these problems but should capitalize on them to the
extent that children begin to understand the peculiar multiplica
tion form and the placing of partial products.
Throughout the elementary school the teacher will be concerned
about helping children understand the place value of numbers in
the ones, tens, hundreds columns. When children are ready to
multiply by twoplace numbers the teacher may continue develop
ing the understandings of place value, by showing children why
the partial products are placed as they are. For example, if the
class is interested in buying twentyfour boxes of crayons at
fifteen cents a box, they will need to multiply 24 by 150. The
problem will be written on the board thus: 24 boxes
x 15 cents
The teacher will then rewrite the problem after a discussion as
to the value of the five and one in fifteen cents, thus:
24 boxes 24 boxes
x 5 cents x 10 cents
The class, under the teacher's direction, will multiply each part
of the problem and find that the partial products are 120 and
240. When these products are added and the decimal point and
dollar sign placed, the result will be $3.06. The next step will
50 ARITHMETIC IN THE ELEMENTARY SCHOOL
be for the teacher and children to follow through the multipli
cation of 24 by 150 as it is usually done.
It is recommended that children accept the use of the zero
as they accept the use of any of the other number symbols through
learning the significance of the zero as they learn to use other
symbols. At first the zero will mean no score, no absences, no
tardies, and the like. As a result of many experiences of this
kind, children will be able to understand the importance of zero
as a place holder in any number. Since the emphasis in arith
metic is on the development of meanings, the further recommen
dation is made that shortcuts involving zero not be taught until the
teacher is certain that the multiplication form and the meaning
of zero have been learned.
SKILLS ANALYSIS OF MULTIPLICATION
Multiplication
A. Two place multiplicand
One place multiplier, no carrying
12 124 $.12 43
3 3 3 2
B. Two place multiplicand
One place multiplier, carrying
16 16t $.16
3 3 3
C. Two place multiplicand
Two place multiplier, no carrying
43 430 $.43 40
22 22 22 22
D. Two place multiplicand
Two place multiplier, carrying
25 250 $.25
23 23 23
COMPUTATIONAL SKILLS 51
E. Three place multiplicand
One place multiplier, no carrying
413 $4.13
3 3
F. Three place multiplicand
One place multiplier, carrying
114 $1.14 1.04 $1.04
6 6 6 6
G. Three or four place multiplicand
Two place multiplier
224 $2.24 4,241 $42.41
15 15 26 26
H. Three or four place multiplicand
Three or four place multiplier
345 $3.45 4,206 $42.06 4206
114 $1.14 514 $ 5.14 1643
I. Multiplicand or multiplier involving zeros with emphasis on multi
plier containing zeros.
4266 $42.66 $450.00 203 $5.00 60,000 $60,000.00
540 540 .02 104 30 250 200
DIVISION OF WHOLE NUMBERS
Young children have many experiences with sectioning, di
viding, and grouping. The use of these experiences in the pri
mary grades will enable the teacher to build an understanding
of division before the process is introduced.
"Recent investigation points out that children, when given
exercises to solve without being given instructions as to whether
to use long or short division almost universally use the long
division form; that children using the long division form are more
accurate than those using the short division form; and that chil
dren, when asked which method is easier for them usually choose
52 ARITHMETIC IN THE ELEMENTARY SCHOOL
the method of long division. The evidence thus seems to favor
emphasis on long division before short division. The writer, while
accepting the general thesis, believes that the problem of division
in teaching is complicated by thinking of short and long division
rather than thinking of division as a unitary process. He believes
that the simple division combinations may be learned as a reverse
of multiplication combinations and that the form and method
of long division should be practiced from the first in all exercises
calling for the use of pencil and paper and presented in the
form indicating division. . By constant use of this form, the
child will gain so much practice with this manipulation that the
more involved exercises usually thought of in connection with
long division will lose much of their difficulty." *
The introduction to division should occur in connection with
real problems involving concrete objects (sheets of paper, colored
pencils, food for a party). Through the use of concrete objects,
the division process can be demonstrated. Then, through recall
ing specific multiplication combinations the division combination
may be learned. For example, if the children already know that
five two's are ten, and two five's are ten, then they should have
little difficulty in discovering that there are five two's in ten,
or that there are two five's in ten.
An understanding of the remainder should be made clear to
children through the division of a group of concrete objects. A
bag may contain twelve pieces of candy. After five children
are given two pieces each, there are two pieces left. Problems
of this kind are solved orally and recorded on the blackboard so
that the children associate the division process with the written
form of division.
An introduction to the use of twoplace divisors should be
carefully controlled by the teacher, and should be made during
the solution of real problems in which a dozen or ten of some
thing is considered, and 10 or 12 would be the divisor. Children
should know the combinations in which 10 and 12 are involved
and should be able to use these numbers as divisors with little
difficulty. The next step in the use of twoplace divisors is one
4Clifford Woody. "Arithmetic", What Does Research Say, Bulletin No.
308, State of Michigan. Department of Public Instruction, 1937, p. 68.
COMPUTATIONAL SKILLS
which demands close supervision by the teacher. Easy divisors
like 21, 22, 23, 31, 44, should be used. Since the children will
not know division combinations containing these numbers, the
teacher must explain how multiplying between the quotient and
the divisor is accomplished. For example, in the problem:
2
21/ 42
42
the child may not know that two twentyones are fortytwo and
may need to multiply each number of the divisor by 2. Confusion
as to which number of 21 is multiplied first may result unless
the teacher calls attention to the fact that the multiplication
begins with the number in the one's column, just as in the case
of any other multiplication problem.
Throughout the intermediate grades, the teacher needs to con
trol the selection of divisors in practice material. Divisors ending
in a 0, 1, 2, or 3 are the easiest to manipulate since they rarely
call for a trial number in the quotient. Divisors ending in 9, 8,
or 7 are easier than divisors ending in 4, 5, or 6 since the quotient
number is apt to be the same as it would be if the divisor ended
in 0 and was in the next decade. Systematic drill in finding trial
divisors when the divisor ends in 4, 5, or 6 should not be engaged
in until the children have complete confidence in their ability
to divide, and have reached the stage where extra manipulation
of numbers to find the correct quotient becomes stimulating and
enjoyable.
SKILLS ANALYSIS OF DIVISION
Division (Long Division Method)
A. One place divisor
Two place dividend
No remainders, no carrying
No zeros
2/ 64 3/ 94
B. One place divisor
Three place dividend
Three place quotient
No remainders, no carrying
4/ 848 3 / 663
54 ARITHMETIC IN THE ELEMENTARY SCHOOL
C. One place divisor
Two and three place dividends
Remainder and no carrying
2/ 47 4/ 49 3/ 367 4/ 449
D. One place divisor
Two and three place dividends
Carrying, no remainder
4/ 96 3/ 72 3/ 423 4/ 524
E. One place divisor
Two and three place dividend
Carrying with remainder
3/ 47 4/ 59 2/ 511 3/ 742
F. One place divisor
Three place dividend
Two place quotient
No carrying, no remainders
2/ 126 4/ 128 6/ 126
G. Repeat A, B, C, with zero difficulties
2/ 60 4/ 804 4/ 120
H. Two place divisor, two or three place dividend
One place quotient, no remainders, with or without zero difficulties
12/ 24 21/ 105 32/ 192
I. Two place divisor
Two place dividend, remainder
Reverse of multiplication facts
32/ 69 24/ 49
J. Two place divisor
Two place dividend
One place quotient, easily estimated, with value greater than that of
multiplication facts, remainder
With or without zero difficulties
32/ 79 24/ 59
COMPUTATIONAL SKILLS
K. Two or three place divisor
Three or four or five place dividend
Two or three place quotient difficult to estimate
48/ 1546 49 / 3793
L. Three place divisor
Four or more place dividend
Quotient easy and difficult to estimate
195 / 4052 267 / 16,000
FRACTIONS
Children begin to use fractions very early in life. Studies have
been made which indicate that many children are able to dis
tinguish between onehalf, onethird, and onefourth before they
enter the first grade. Other first grade children, however, de
scribe fractional amounts less accurately and are content to speak
of three pieces (meaning thirds) or two pieces (meaning halves)
or the "big half" and the "little half" of an object cut into two
unequal parts. In dividing objects into halves, thirds, and fourths,
these children are less accurate in drawing and cutting and are
satisfied if the correct number of pieces is obtained.
Because of their immaturity and the nature of their home and
school experiences children become familiar with the written
symbol for common fractions before learning to recognize the
symbols for decimal and percentage fractions. As a result of
many experiences they learn that a fraction may represent part
of a group, or part of a single unit. Later, they learn that what
has been expressed in the common fraction form may also be
expressed in the decimal and percentage fraction forms.
Whenever possible, children should have opportunities for be
coming familiar with the three fraction forms on both the oral
and written levels. In the primary grades the emphasis will be
on the oral use of such amounts as onehalf, onethird, twothirds,
98.6 degrees, fifty per cent, one hundred per cent. The children
will become familiar with the written fraction forms as the teacher
writes amounts expressed in these forms on the board. These
figures may represent amounts in recipes, per cent of children
receiving innoculations and the like. In the intermediate grades
ARITHMETIC IN THE ELEMENTARY SCHOOL
children will continue developing meanings for the fraction forms
and will have systematic instruction in computating with common
and decimal fractions.
COMMON FRACTIONS
Although children begin to use fractions early in life, they
rarely have a need for solving a problem in which common frac
tions with large denominators are involved. For this reason, it
seems advisable that practice with common fractions be limited
to practice with fractions whose denominators are common to
problems of every day life. Research shows that halves, fourths,
fifths, tenths, thirds, sixths, twelfths are the most frequently
used fractional parts of a whole.
Addition and Subtraction. The processes of addition and sub
traction of fractions give meaning to each other and are usually
taught together. By the time children are in the intermediate
grades they will be able to solve orally many problems whose
solution depends on adding and subtracting fractions with like
denominators. Next steps will consist of learning to write prob
lems containing fractions in their proper form, to find a common
denominator, to reduce the answer, and to borrow (in subtrac
tion). If children continue to have experiences using a measuring
cup and ruler, cutting up fruit and paper, marking off play areas,
and the like, they will understand that any fractional part has
other common fraction equivalents. Many experiences pointing
to this fact should be shared with children before an attempt is
made to teach them to reduce the answer, find the common
denominator, or borrow.
SKILLS ANALYSES OF ADDITION AND SUBTRACTION
ADDITION INVOLVING COMMON FRACTIONS
A. Like denominators (use horizontal and vertical forms)
(1) Like denominators
Answer less than a whole
1/4 2/ + 1/8 =
2/4
56
COMPUTATIONAL SKILLS
(2) Like denominators
Answer a whole
1/2 2/3 + /3 =
/4 + 2/4 =
(3) Like denominators
Answer one whole and one fractional part
3/4 + 2/4 = 2/3
2/3
(4) Like denominators
Answer one whole and two or more fractional parts
3/4 5/6 + 3/6 =
(5) Like denominators
Reduction emphasized
(a) Answer equivalent to onehalf
1/4 + 1/4
(b) Answer has equivalent other than onehalf but less than
one whole
8 + 2/8
6/10 + 2/10 =
(c) Answer equivalent to one whole
% + 48 =
2/10 + /10 o 5/10 =
(d) Answer equivalent to one whole and one or more fractional
parts
2/10 + /o10 + 6/10 =
3/ + 78 =
B. Unlike denominators (use vertical form only)
(1) Unlike denominators
Common denominator equivalent to one of the expressed denomin
ators
Answer less than one whole
5/ 10 + 2/5 = /8 1/4 + '/2
112 + 1/4 =
58 ARITHMETIC IN THE ELEMENTARY SCHOOL
(2) Unlike denominators
Common denominator not one of the expressed denominators
Answer less than one whole
1/4 + /3 = 2/3 + /5 = 1/3 + 2/7 + /7 =
C. Mixed numbers
(1) Like denominator
No reduction
2 1/4 + 2 2/4 =
3 1/ + 2 3/5 =
21/6 + 3 2/ + 1 2/ =
(2) Unlike denominators, common denominator equivalent to one
of the expressed denominators. No reduction.
2 1/4 + 2 1/8 =
4 2/3 + 2 /6 =
4 1/2 + 2 1/4 + 3 /8 =
(3) Unlike denominators
Common denominator not one of the expressed denominators
No reduction
21/4 + 12/3 =
3 1/7 + 2 /3 =
21/ + 1 1/3 + 21/4 =
(4) Reduction type I (6 6/6 = 7)
2 2/ = 2 4/6
4 2/6 = 4 /6
6 6/6 = 7
(5) Reduction type II (6 /6 = 7 1/6)
2 2/3 = 2 4/6
4 /6e = 4 '/6
6 '/6 = 7 /6
(6) Reduction type III (6 /6 = 7 2/6 = 7 /3)
2 2/3 = 2 4/6
4 /6 = 4 4/
6 8/, = 7 '/6 = 7 1/
(7) Reduction type IV (9 12/6 = 11)
2 1/3 = 2 '/6
4 5/6 = 4 '/6
3 '/6 = 3 '/6
9 1/6 = 11
COMPUTATIONAL SKILLS 59
(s) Reduction type V (9 31/16 = 11 1/V5)
2 2/3 = 21/15
4 4/ = 4 2/15
3 9/ = 3 9/16
9 "/15 = 11 /15
(9) Reduction type VI (9 "/15 =11 5/15 = 11 1/s)
2 2/ = 2 10/15
4 4/ = 4 12/1
3 "/15 = 3 "/15
9 35/15 = 11 5/1 = 11 1/
SUBTRACTION Involving Common Fractions (Vertical form preferred)
A. Like denominators without emphasis on reduction
3/7 2/7 = 6/ 3/
3/8 2/5
B. Like denominators with emphasis on reduction
/6 2/ 8/]o 4/10 515
C. Unlike denominators with and without reduction
1/2 /4 2/3
1/5
D. Mixed numbers with like denominators. No borrowing
4 /3
2 1/
E. Fraction from a whole or a mixed number
Like denominators
(1) 1 1/3 2 1/3
(2) 1 /3 2/3 3 '/3 3
(3) 12/4 3 24 3/4
ARITHMETIC IN THE ELEMENTARY SCHOOL
F. Mixed number from mixed number. Unlike denominator. Borrowing.
4 1/ 4 '/
2 4/6 2 2/3
Multiplication. Multiplication of common fractions offers
few difficulties if the teacher selects the practice material with
care. For example, 7 1/2 X 6 3/4 looks simple but when it is changed
to 15 / X 27 /4 children are apt to lose sight of the process they
are learning in trying to multiply 27 by 15, and divide 405 by 8.
Better examples are 1/2 X 4, or 1/4 of 3 '/.
Cancellation should be introduced after the habit of multiplying
horizontally has been firmly established. Then it should be intro
duced as reduction. Reduction (cancellation) will eliminate un
wieldly improper fractions from the product and make division
of the numerator by the denominator less difficult.
Vertical multiplication is useful when a large whole number
and a mixed number are multiplied. However, elementary chil
dren are often confused by the form which includes division and
multiplication. For this reason, elementary teachers should con
trol the selection of drill material in which mixed numbers are
involved so that the horizontal form will be adequate. The vertical
form should be taught later when added maturity makes it desir
able for children to learn to use shortcuts.
SKILLS ANALYSIS OF MULTIPLICATION
MULTIPLICATION INVOLVING COMMON FRACTIONS (HORIZONTAL FORM)
A. (1) Fraction by a whole number
1/3 of 2 2/4 X 6
(2) Whole number by a fraction
2 X 1/a 6 X 2/4
B. Fraction by a fraction
1/3 of 6/8 5/e X /4
C. (1) Mixed number by a whole number
1 '/2 X 3
COMPUTATIONAL SKILLS 61
(2) Whole number by a mixed number
3 x 1 '/2
D. Fraction by a mixed number
(C) 2/ 1 /2 2of 2 1/2
(2) 1 /2 x 1/3
E. Mixed number by a mixed number
1 '/2 1 2/4
F. Mixed number by a large whole number
4 '/2 X 15
4 1/2 yards @ $ .15
2 dozen cups @ 3 '/20 a cup
G. Multiplying by three numbers
1 1/2 X 2 2/3
Division. There are few occasions when the division of com
mon fractions is socially significant. For this reason, it is recom
mended that when division of fractions is necessary, the common
fractions should be changed to their decimal equivalents. Through
out the entire elementary school the teacher should take the
initiative in solving problems of this type.
DECIMAL FRACTIONS
Experiences with decimal fractions should not be limited to the
sixth grade. Children in kindergarten and first grade regularly
use small sums of money which are expressed in the decimal form.
Teachers of young children can extend the use of the decimal
point and introduce the per cent sign through sharing attendance
records, and the like with members of the class. By the time
children are in the fourth grade they will have observed, either
independently or under adult guidance, the use of decimals in
connection with the odometer of an automobile, the rise and fall
of the alcohol in the classroom thermometer, the rise and fall of
mercury in a clinical thermometer, the rise and fall of mercury
ARITHMETIC IN THE ELEMENTARY SCHOOL
in a barometer, a record of the growth of rats in a nutrition
experiment. In addition, they will have used a twelve inch ruler,
a tape measure, a yard stick, and various scales in construction
and weighing activities. Many understandings and skills develop
as a result of the observations and the manipulations that follow.
Children learn that there are several ways of measuring objects,
distances, and the like, and that measurement may be expressed
in both the common or the decimal fraction form. They learn
that onetenth of a mile may be written .1 or 1/10 and on an auto
mobile is represented in still a different way. Likewise, .5, 5/10, 2
are the same whether used to express degrees of temperature or
grams of weight. 50% and are the same, also. Occasionally,
the children will hear someone say a number like 3.15. At this
time the teacher should explain that the other fractional form
is 3 15/100 and point out the simplicity of the decimal form.
It is at this time that children begin to understand some of the
advantages of each form. They learn that when the whole is
divided into few parts (when the denominator is small) the com
mon fraction form is usually adequate. When very fine differences
are to be measured, the decimal form is used. For example, the
diameter of a screw and the expansion of mercury is measured
in decimals. Scientific instruments usually measure in tenths.
hundredths, thousandths. Before children reach the sixth grade
they should have had many experiences with numbers expressed
as decimal and common fractions, and know that an amount ex
pressed in one form has its equivalent in the other form.
Addition and Subtraction. If children have had opportunities
to add and subtract small amounts of money, if they understand
the place value of numbers, and the use of zero as a place holder,
they should have little difficulty in adding and subtracting decimal
fractions and mixed numbers. Possibly the greatest difficulty
will be found when they try to write numbers expressed as decimals
from dictation. They will need to form the habit of listening
for the denominator as it is spoken so that they can visualize the
number of places beyond the decimal point. Subtraction of decimals
is more complex than addition when the minuend contains fewer
numbers than the subtrahend. In problems of this kind, it is
necessary to add zeros to the minuend and borrow. Again, it
is important that children understand the subtraction process.
COMPUTATIONAL SKILLS 63
and the place value of whole numbers and decimal fractions be
fore they engage in systematic drill on this form of subtraction.
SKILLS ANALYSIS OF ADDITION AND SUBTRACTION
ADDITION INVOLVING DECIMAL FRACTIONS
1. Mixed number with the fractional part expressed in tenths, with varied
whole numbers, no carrying.
(a) 2.5+3.1
(b) 2.5 + 31.4
(c) 4.3 + .5 + 24.1
2. Mixed number with the fractional part expressed in tenths with varied
whole numbers. Carrying.
(a) 2.8+3.4
(b) 2.7 +31.4
(c) 4.3 + .5+ 24.4
3. Mixed number with the fractional parts expressed in tenths and hun
dredths, varied whole numbers. No carrying.
(a) 2.5 +3.15
(b) 2.5 + 3.15 + .04
4. Mixed number with the fractional parts expressed in tenths and hun
dredths, varied whole numbers. Carrying.
(a) 2.5+3.65
(b) 2.5 +3.65 + .04
5. Mixed numbers with the fractional parts expressed in tenths, hun
dredths, thousandths, etc., whole numbers varied.
(a) 2.5+ .067
(b) 2.5 + 3.15 + .067
SUBTRACTION INVOLVING DECIMAL FRACTIONS
1. (a) Mixed numbers with fractional part expressed in tenths.
No borrowing
6.5 2.3
ARITHMETIC IN THE ELEMENTARY SCHOOL
(b) Fraction from mixed number. No borrowing.
Fractions expressed as tenths.
6.5 .3
2. Whole number from mixed number. Fraction expressed in tenths.
6.53
3. (a) Mixed number with fraction in minuend expressed in tenths, and
fraction in subtrahend expressed in hundredths. Borrowing oc
curs only once and at the extreme right.
6.7 2.51
(b) Mixed number with fraction in minuend expressed in tenths, and
fraction in subtrahend expressed in hundredths. Borrowing occurs
in both columns of fraction.
5.3 2.47
4. (a) Mixed number from whole number with remainder a mixed number.
6 3.5
(b) Mixed number from whole number with remainder a fraction.
4 3.5
5. Varied mixed number from varied whole numbers.
64 3.56 164 3.564
Multiplication. In multiplying decimal fractions it is important
that children learn how to place the decimal point. This some
times involves adding zeros. This need occurs very rarely and
is of little consequence in the elementary grades. However, the
proper placing of the zero requires a better than average under
standing of the decimal system and sixth grade children should
have opportunities for gaining in appreciation of the number
system and of the efforts of scientists who use this type skill
most frequently.
The most frequent need for multiplying decimals in the ele
mentary school seem to be in problems involving money. In order
to multiply, it is necessary to deal with two or more quantities of
like amounts. These like amounts occur less frequently in frac
COMPUTATIONAL SKILLS
tion or mixed form than in whole number form. Therefore, the
multiplicand will often be a number representing a sum of money
and the multiplier a number representing the number of children
in the class or other small group.
SKILLS ANALYSIS OF MULTIPLICATION
MULTIPLICATION INVOLVING DECIMAL FRACTIONS
1. (a) Multiplying sums of money by number of people, books, etc., with
product less than one dollar.
21 X 2 $ .22 X 2
(b) Multiplying sums of money by whole numbers with products more
than one dollar.
$2.53 x 2
2. Multiplying mixed nuniber by whole number.
4.5 miles by 2
3. (a) Multiplying mixed number by mixed number, varied frational
parts where it is not necessary to add a zero in the product before
putting in decimal point.
3.2 x 4.55 12.03 X 2.5
(b) Multiplication of numbers when zero must be added to product
before decimal point is placed.
.05 X .031
Division. Division of decimals is particularly useful in finding
per cent, averagesrainfall, grades, money, population, production
and in dividing money and financial responsibility among sev
eral people. In dividing money children learn to place the decimal
point in the quotient. In finding averages and per cent they fre
quently need to add zeros to make the fractional part of the
quotient more accurate, or add zeros when the quotient is less
than a whole (when a larger number is divided into a smaller).
Children rarely have a need for dividing by a mixed number. Be
cause of this, little emphasis in the elementary school is placed
on learning to put the decimal point in the quotient when the
divisor is a mixed number.
66 ARITHMETIC IN THE ELEMENTARY SCHOOL
SKILLS ANALYSIS OF DIVISION
DIVISION INVOLVING DECIMAL FRACTIONS
1. (a) Division of money, less than one dollar, into several parts.
750 + 5 5/ $ .75
(b) Division of money, more than one dollar, into several parts.
2/ $2.48 30/ $1.35
2. Dividend a fraction or a mixed number with fractional part expressed
in tenths, hundredths, or thousandths. Divisor a whole number.
.24 + 2 3.693 + 12 36.69 + 30
3. Dividend a fraction, whole number, or mixed number. Divisor a
whole number. Zeros added to dividend.
12/ .75 12/ 66 12/ 7.5
4. (a) Dividend a mixed number. Divisor a mixed number. Added zeros
not needed.
8.5/ 297.5
(b) Dividend a mixed number. Divisor a mixed number. Added zeros.
1.22/ 12.34
(c) Dividend a whole number. Divisor a mixed number. Zeros added.
1.5/ 68
(d) Fiftyfour dollars divided by one dollar seventyone cents.
PERCENTAGE FRACTIONS
In first, second, and third grades, children develop under
standings about percentage fractions if they hear the teacher speak
of half the class as fifty per cent. Likewise, if forty per cent of
the class are girls and sixty per cent are boys, they should hear
that, too, and come to understand that one hundred per cent
represents all the class. Since monthly attendance is described
in terms of per cent, children should have the attendance record
before them as they note the effect of illness, stormy weather, and
the like, on the class record. When they are older children learn
COMPUTATIONAL SKILLS 67
to use per cent in expressing such ideas as population decrease
and increase, production, and consumption.
While there seems to be no place in the elementary program
for independence in the solution of problems involving per cent,
teachers should take advantage of all suitable opportunities for
developing meanings about percentage fractions.
ROMAN NUMERALS
An interest in Roman numerals will develop after children
have had experiences using the HinduArabic numbers and begin
to wonder about the meaning of symbols they observe on certain
clock faces, in books, on sundials, or on the cornerstones of build
ings. This is the time to give them information which will lead
to an understanding that there are other number systems than
the one we commonly use and an appreciation of the way in which
all number systems evolved. If the explanation of Roman numeral
comes after children have learned to multiply they will be more
able to appreciate the fact that the Roman system was gradually
abandoned because of its cumbersome qualities, and that man is
constantly trying to find a better way of doing things. To
demonstrate this point, two numbers such as CXXV and V might
be added and multiplied in both Roman numeral and Hindu
Arabic numerals on the board. The economy of the HinduArabic
system will be apparent immediately.
Children like the story about the probable origin of Roman
numerals which tells how the fingers were first used for I, II, III
and possibly IV (IIII) and the index finger and thumb for V. A V
on top of a A(X) may account for the X we know today. Children
also like to contemplate a number system based on six or twelve
which might have evolved had man been born with six fingers and
six toes instead of five.
Children should have experiences with number systems other
than the HinduArabic and Roman systems from time to time
in order to become better acquainted with and learn to appreciate
more of their great social heritage.
Chapter Four
ORGANIZATION OF MATERIALS OF
INSTRUCTION
Chapter Two contains a discussion of the persistent problems
involving mathematics which face both adults and children. Briefly,
these persistent problems are: (1) The need for expressing ideas
quantitatively by means of an accepted number system, (2) the need
for using a system of exchange, and (3) the need for selfexpression.
In planning an arithmetic program based on persistent problems
the teacher needs to examine the environment of the child and
keep in mind his level of maturity. As a result, she should become
sensitive to many socially significant problems and should be able
to give children guidance in the solution of these problems.
One of the main objectives behind the production of this bulletin
has been that of helping Florida teachers become more sensitive to
the quantitative problems of daily living. This chapter contains
many examples of problems which should suggest others to teachers
interested in helping children develop understandings and skills
relative to the number system and its use.
Another objective has been that of helping teachers discover ways
of organizing materials for instruction in arithmetic in the various
grades.
Good organization calls for careful selection. In selecting
materials, teachers need to keep in mind the function of arithmetic,
and the needs, interests, and levels of ability of the children they
teach. Materials for much of the arithmetic program of the primary
grades are to be found within the classroom (the clock face, calendar,
small sums of money, thermometers, scales, yardsticks, midmorning
lunch, etc.). Materials for the intermediate grades are found in the
classroom and in textbooks (arithmetic, social studies, science, etc.).
MATERIALS OF INSTRUCTION
reference books, magazines, newspapers, radio, government bulletins.
Information from these courses will be useful in helping children
attack problems they face because of an increasing interest in their
expanding environment. In addition to using materials designed
for a particular grade, teachers also need to use materials designed
for other grades in order to meet the needs of individual children.
Illustrative Material. This chapter contains three sets of illus
trative material: (1) An outline of meaningful learning situations
suitable for teaching arithmetic in elementary grades, (2) a detailed
treatment of two typical primary and two typical intermediate grade
problems, and (3) two specific examples showing how a fifth grade
teacher and a second grade teacher, guided by the needs, interests,
and ability of the children in their classes, carried problemsolving
to the point of drill with abstract numbers.
The first set of illustrative material, An Outline of Meaningful
Learning Situations Suitable for Elementary Grade Children, shows
clearly how the problems become more complex as children grow
older. For example, Making a Reading Center has been suggested
for each grade. In the first grade, children count and group con
crete objects (mostly books) in their reading center. In the second
grade they continue to count and group objects but they also keep
simple records of their reading. In the third grade they group books
according to a simple system of classification, make chair covers
and table runners, and construct cards for use in filing reading
records. In the fourth, fifth, and sixth grades children may write
classification numbers on the books in their center, check out books
and return them on time independently, and construct such equip
ment as book cases, bookends, and the like.
Outline of Meaningful Learning Situations for Teaching
Arithmetic in Elementary Grades
GRADE I
Mathematical Understandings and
Meaningful Learning Situations Skills Involved
I. Living in the school
A. Making reading center A. 1. Selecting books and chairs for
1. Selecting books reading center
2. Selecting chairs 2. Estimating size of vase for
3. Arranging flowers on flowers
table
70 ARITHMETIC IN TIE ELEMENTARY SCHOOL
Meaningful Learning Situations
B. Making work center
1. Sorting blocks
2. Collecting tools
equipment
C. Making science center
1. Making a balanced aquar
ium
a. Preparing sand
b. Placing plants in sand
c. Selecting fish
d. Filling aquarium with
water
e. Collecting shells and
marbles to decorate
aquarium
D. Making a garden
E. Planning a party for mother
or another grade
1. Determining number com
ing to party
2. Selecting best time for
the party
3. Making invitations
4. Planning and preparing
refreshments
a. Planning amount of
food required
b. Buying materials
c. Making punch
5. Decorating for party
Mathematical Understandings and
Skills Involved
B. 1. Grouping smaller and larger
blocks
2. Counting smaller blocks and
larger blocks
3. Counting rulers, hammers.
saws, jars for paint, paint
brushes, and crayons
C. 1. Estimating and measuring
amount of sand needed (by
quarts)
2. Measuring water by pints and
quarts to determine number
of gallons to put into aquar
ium
3. a. Counting plants, fish, mar
bles, and shells
b. Adding different colored
marbles together (2 red, 3
blue)
c. Noticing differences in the
number and sizes of shells
and marbles put into
aquarium
D. See page 90
E. 1. a. Counting children coming
to party
b. Counting mothers coming
to party
2. a. Learning days of week
b. Selecting hour of the day
for party
c. Selecting day for party
d. Finding number of days
before the party will take
place
3. a. Measuring paper with
ruler
b. Folding paper in halves or
fourths as needed
c. Writing dates
4. a. Counting number of cook
ies needed; counting fruit
and sugar needed (under
standing of dozens and
pounds)
b. Finding cost of materials
c. (1) Cutting fruit in halves
(2) Measuring water and
juice by quart, pint,
half pint, cup
(3) Measuring sugar by
cups
5. a. Arranging bulletin board
attractively
MATERIALS OF INSTRUCTION 71
Meaningful Learning Situations
F. Using money in the class
room
1. Joining Junior Red Cross
G. Using the clock and calendar
in the classroom
II. Weighing and measuring
children
I. Playing games
1. Making formations
2. Choosing sides
3. Marching
4. Responding to music
II. Living in the home
A. Getting acquainted with the
home
B. Taking excursion to see dif
ferent kinds of homes
C. Making home booklet
1. Selecting pictures (kinds
of homes, rooms, members
of family)
2. Arranging pictures on
page
D. Making a piay house
1. Finding how much ma
terial is needed
2. Buying materials needed
3. Constructing the frame
work of the house
4. Making furniture
Mathematical Understandings and
Skills Involved
b. Arranging flowers attrac
tively
F. 1. Add total amount contributed
2. Add the number of pennies
3. Add the number of nickels
4. Add the number of dimes
5. Count by l's, 2's, 10's
G. See page 14
H. See page 96
I. 1. Becoming acquainted with cir
cles, squares, parallel lines
2. a. Becoming acquainted with
first, second, third, last,
etc.
b. Dividing into groups by
2's, 3's
3. Observing accented beats
A. 1. Counting members of the fam
ily, rooms in home, pets
2. Comparing sizes and ages of
brothers and sisters in the
family
3. Comparing sizes of homes
(stories, rooms)
B. 1. Counting different kinds of
homes
2. Counting number of blocks
walked
3. Gaining an understanding of
the term "mile"
4. Finding length of time con
sumed by trip
C. 1. Selecting and placing pictures
on page
2. Writing numbers on pages
3. Becoming acquainted with
such terms as: tall, taller,
high, low, wide, narrow, long
D. 1. Measuring with yardstick
and ruler to find how much
material is needed, and space
in room needed
2. a. Recognizing the value of
coins such as: pennies,
nickels, and dimes
b. Purchasing supplies and
making change
3. a. Comparing and estimating
sizes of furniture
b. Expressing quantitative
relationships such as lar
ger, smaller, longer, short
er. narrower, and higher
ARITHMETIC IN THE ELEMENTARY SCHOOL
GRADE II
Meaningful Learning Situations
I. Living in the school
A. Making reading center
1. Reading books
B. Making work center
C. Making science center
1. Making a balanced aquar
ium
2. Discovering reasons for
evaporation
3. Growing bulbs
4. Recording daily temper
ature
D. Making a garden
E. Planning a Halloween party
1. Determining number com
ing to party
2. Budgeting time in order
to be ready for party
3. Making favors (masks or
caps)
4. Buying materials for
party
5. Making or decorating
cookies
6. Serving refreshments
F. Money in the classroom
G. Using the clock and calendar
in the classroom
H. Weighing and measuring
children
Mathematical Understandings and
Skills Involved
A. 1. Keeping records of number of
books and pages read
B. 1. Finding cost of materials
C. 1. Measuring to find amount of
water evaporated from large
container
2. Finding cost of bulbs
3. Estimating correct size of
bowl in which to put bulbs
4. Measuring growth of plants
5. Recording growth of plants on
given dates (number of days
from planting to blossom)
6. Counting blossoms
7. a. Reading and interpreting
number on a thermometer
b. Subtracting to find differ
ences in temperature from
day to day
D. See page 96
E. 1. Adding the number of chil
dren and guests
2. Dividing day into hours or
part of hour
3. Measuring paper needed for
favors (inches and parts of
an inch on ruler)
4. Adding to find total cost of
materials
5. Buying by dozens, pounds, etc.
6. Making change
7. Adding to double recipe
8. Learning fractional parts by
measuring 1/ cup, 4 cup.
tablespoon
9. Timing the baking of the
cookies
10. Counting by 2's or 5's to de
termine number of cookies
needed.
F. 1. Adding to find cost of school
supplies
2. Adding to find cost of school
lunch for one week
G. See page 13
H. See page 96
MATERIALS OF INSTRUCTION
Meaningful Learning Situations
I. Playing gamessee Grade 1
II. Living in the neighborhood
A. Visiting post office
1. Walking to the post office
2. Buying different kinds
of stamps
3. Addressing letters
4. Mailing letters
5. Following route of letter
6. Examining the post office
schedule
7. Comparing post office
schedule with school
schedule
B. Taking a trip to the dairy
1. Determine the number of
children going
2. Planning date of trip
3. Transporting children
4. Going to the dairy
5. Exploring the dairy
Mathematical Understandings and
Skills Involved
H. 1. Adding in keeping scores
2. Subtracting to find differ
ences in scores and differ
ences in distances jumped
3. Dividing sides equally in
games
A. 1. a. Counting blocks in going
to post office in terms of
miles or parts of miles
b. Relating number of blocks
to mile
c. Figuring time spent in
getting to post office
2. a. Counting by l's, 2's, 3's,
5's, and 10's
b. Recognizing value of coins
c. Adding total number of
stamps bought
d. Adding cost of stamps
bought
e. Adding and subtracting
money in making change
3. a. Reading dates stamped on
letters
b. Reading box number, route
number, or house number
4. a. Comparing speed of de
livery
b. Understanding s 1 power
than, faster than, sooner
than, later than
5. Gaining an understanding of
distances
6. Reading and counting hours
each window remains open
7. Finding difference in post of
fice and school schedules
B. 1. Counting the number of chil
dren taking trip
2. Using the calendar
3. Assigning children to cars
4. a. Counting number of blocks
or miles in going to dairy
b. Finding amount of time
spent in getting to dairy
(understanding of hour or
parts of an hour)
5. a. Counting the buildings
b. Estimating and comparing
heights and sizes of build
ings (larger, smaller,
taller)
74 ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning Situations
6. Finding cost of milk per
child per week.
Mathematical Understandings and
Skills Involved
c. Counting cows in the pen
d. Counting calves in the pen
e. Learning the number of
times cows are milked
each day
f. Learning the time of day
for each milking (hour or
half hour)
g. Learning working hours of
different shifts of men
h. Finding total number of
men working in dairy:
milkers, men who prepare
milk for market, and de
livery men (rational count
ing, simple addition and
subtraction)
i. Adding to find amount
and cost of feed needed
for 1 cow, 2 cows, for 1
day or 2 days
j. Measuring quarts of milk
per cow per day
k. Reading temperatures nec
essary for pasteurization
of milk
6. a. Adding to find cost of
milk per week per child
b. Counting by 5's to find
how much milk may be
bought with 25c, etc.
GRADE III
Meaningful Learning Situations
i. Living in the school and com
munity
A. Making reading center
1. Classifying books
2. Making runner for li
brary table and making
chair covers
3. Making cards for filing
records of reading
B. Raising butterflies (or tad
poles, etc.)
Mathematical Understandings and
Skills Involved
A. 1. Grouping books according to
classification
2. a. Measuring material for
runner for library table
and covers for chair backs
b. Finding cost of material
3. a. Measuring with ruler in
making cards
b. Adding and subtracting in
keeping number of books
or pages read
1. 1. a. Keeping record of the
number of days that the
caterpillar was fed before
MATERIALS OF INSTRUCTION 7o
Meaningful Learning Situations
C. Making a garden
D. Roasting and selling peanuts
1. Buying peanuts
2. Roasting and
peanuts
3. Selling peanuts
bagging ,
E. Using the clock and calendar
in the classroom
F. Weighing and measuring
G. Planning health program
H. Playing games
I. Money in the classroom
J. Studying or visiting circus
or county fair
Mathematical Understandings and
Skills Involved
it began spinning its co
coon
b. Keeping record of the
number of days from co
ccon to caterpillar
C. See page 90
D. 1. a. Finding total cost of pea
nuts
b. Finding total cost of bags
c. Finding total cost of bags
and peanuts
2. Timing the roasting of pea
nuts
3. i. Dividing number of bags
among children
b. Multiplying to find gross
receipts
c. Adding and subtracting
money in making change
d. Adding gross receipts
e. Subtracting to find profit
E. See page 14
F. See page 96
G. 1. Budgeting time for work.
play, and rest
2. Adding to determine amount
of milk needed per child and
per family
3. Subtract to find difference in
cost of fresh and canned veg
etables
H. See page 7
I See page 8
J. 1. Adding to find total cost of
visit to circus
2. Measuring distances in terms
of miles to circus or fair
(measuring distance on map
to circus quarters in Sara
sota)
3. Making change in buying
tickets
4. Comparing heights and sizes
of tents or pens
5. Comparing sizes of animals
6. Securing information as to
weights of animals
7. Reading large numbers in
finding value of one animal
8. Adding or multiplying to find
total value of animals
76 ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning Situations
K. Making map of community
L. Study of school bus trans
portation
M. Studying or visiting a farm
1. Comparing relative val
ues of house and tractor
Mathematical Understandings and
Skills Involved
9. a. Adding to find how much
animals eat (pounds,
quarts, tons)
b. Adding or multiplying to
find cost of food
10. Securing information as to
days, weeks, or months baby
animals are cared for by their
mothers
11. Measuring one gallon of water
and discussing how many gal
lons a camel can carry (25
gallons)
12. Finding differences in the
speed of various animals
K. 1. Measuring the paper or space
on blackboard for maps (us
ing ruler and yardstick)
2. Locating and estimating dis
tances between different
buildings (children's homes.
post office, churches, police
station, fire department, etc.)
in order to indicate place of
buildings on map
3. Learning directions (2 miles
north, 3 miles south, etc.)
4. Expressing distances in terms
of miles or parts of a mile
L. 1. Adding total mileage the bus
driver covers each day
2. Adding or multiplying to find
mileage for two days, three
days, etc.
3. Adding or multiplying to find
cost of gas per day. per week,
or per month
4. Adding or multiplying to find
cost of tires for bus
5. Finding length of time each
child spends on bus
M. 1. a. Adding or multiplying to
find amount of feed needed
for horse per day, per week.
or per month (understanding
quarts. bale, bushel, peck)
b. Adding or multiplying to
find cost of feed for horse
by week or month
c. Adding or multiplying to
determine amount and cost
of gasoline used to run
tractor for day, week, or
month
MATERIALS OF INSTRUCTION 77
Meaningful Learning Situations
Mathematical Understandings and
Skills Involved
2. Studying poultry
3. Studying truckfarming
d. Subtracting to find time
needed to plow one acre of
ground by horse or by
tractor
e. Compare initial cost of
horse and tractor
2. a. Adding to find total num
ber of chickens
b. Adding to find total cost
of feeding chickens for a
month
c. Dividing to find cost per
week
d. Multiplying or adding to
find number of eggs gath
ered in 2 days, 4 days, a
week (understanding of
dozens or parts of a
dozen)
e. Subtracting to find differ
ence in number of eggs
gathered daily or weekly
f. Finding distance farmer
travels to market
g. Multiplying to find how
much farmer receives for
eggs
h. Subtracting to find differ
ence in cost of eggs at
store and amount received
for eggs by farmer
i. Adding and dividing to
find average cost and re
ceipts per month
3. a. Adding to find total ac
reage of vegetables (cab
bage, potatoes, beans,
etc.)
b. Determining length of
time between planting and
harvesting
c. Understanding relative
sizes of bushels, bunches.
hampers, crates, and
quarts through the study
of packing of vegetables
d. Finding distance farmer
travels in taking produce
to market
e. Comparing length of time
needed to deliver vege
tables now with length of
time needed when horse
and wagon were used
ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning Situations
Mathematical Understandings and
Skills Involved
f. Multiplying to find amount
of money received by
farmer for a load of cab
bage at a given price per
pound
g. Multiplying to find cost of
cabbage to customer
b. Comparing cost of fresh,
canned, frozen vegetables
GRADES IV. V, AND VI
Meaningful Learning Situations
A. Making reading center
1. Cataloging books
2. Checking out books
3. Making bookcases and
bookends
B. Keeping and checking at
tendance records
C. Money in the classroom
1. Selling tickets
Mathenmatical Understandings and
Skills Involved
A. 1. Writing numbers on books
(using decimal system)
2. Reading and writing dates
3. a. Measuring materials need
ed for book cases and book
ends (using foot, inch,
parts of inch)
b. Estimating amount of var
nish or paint needed (un
derstanding of pint and
quart)
c. Adding or multiplying to
find total cost of paint.
feet of lumber, and pounds
or parts of a pound of
nails
R. 1. Finding per cent of attend
ance and absences
2. Adding and dividing to find
average attendance per week,
per month
3. Finding per cent of bus chil
dren in room and in school
C. 1. a. Adding number of tickets
sold by children per day
and per week
b. Adding number of tickets
sold by group or by grade
per day or per week
c. Subtracting to find which
group sold more tickets per
day and per week
d. Adding to find total num
ber of tickets sold in school
e. Finding per cent of total
number of tickets sold by
each grade
MATERIALS OF INSTRUCTION
Meaningful Learning Situations
2. Studying cost of school
supplies and equipment
3. Finding cost of state
textbooks
4. Finding cost of furniture
and supplies
5. Studying cost of school
house and grounds
D. Studying birds
Mathematical Understandings and
Skills Involved
f. Multiplying to find total
receipts from tickets sold
per child and per grade
g. Multiplying to find total
receipts from tickets sold
by school
2. a. Adding to find cost per
child
b. Multiplying to find cost
per group
3. a. Adding to find cost of in
dividual sets of textbooks
b. Multiplying to find cost of
sets of books (readers,
science books, and others)
c. Adding to find total cost of
all state books in room
d. Subtracting to find the
difference in cost of sets
of books
4. a. Multiplying to find total
cost of pupils' desks
b. Adding to find total cost
of other equipment (maps,
teacher's desk, library
table)
c. Adding to find cost of sup
plies (chalk, erasers, pen
cil sharpener, paint,
paper)
d. Adding to find total cost
of all furniture and sup
plies in one room or in
several rooms
5. a. Reading the figures repre
senting the value of school
building
b. Reading the figures repre
senting the value of school
grounds
c.. Adding to find total value
of school building and
school grounds
D. 1. a. Adding kinds of birds
seen by group over certain
period of time
b. Measuring to find dis
tances traveled by differ
ent birds during migration
season (scale of miles)
c. Estimating length of time
birds spend in migrating
from one place to another
80 ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning Situations
E. Using thernmoeter
F. Making school garden
G. Studying the solar system
Mathematical Understandings and
Skills Involved
d. Reading figures represent
ing value of birds in insect
control
e. Measuring material for
bird houses (understand
ing and use of yard, foot.
and inch)
E. 1. Recording daily high and low
temperature
2. Adding and dividing to find
average temperature per day,
per week, and per month
3. Subtracting to find the differ
ence in temperature between
warmest and coldest day in
the month
4. Subtracting to find difference
in degrees between city hav
ing highest temperature and
city having lowest tempera
ture (in state and in country)
5. a. Comparing Fahrenheit and
Centigrade scales
b. Reading Fahrenheit and
Centigrade thermometers (un
derstanding meaning of zero
and plus and minus devia
tions)
6. Reading temperature to frac
tional parts of a degree
F. 1. Measuring the land for gar
den plot
2. Dividing the garden into plots
3. Dividing plots into rows
4. Dividing rows into ni!ls
5. Adding to find total number
of pounds and bunches of pro
duce gathered
6. Multiplying to find amount of
money received for vegetables
based on given price per
pound or bunch
7. Adding to find total receipts
from garden
8. Subtracting to find profit
G. 1. Developing meanings for lia
meter and circumference
2. Subtracting to find difference
in diameter of the different
planets
3. Subtracting to find difference
in distance of planet fr.'m
the earth and from the sun
MATERIALS OF INSTRUCTION 81
Mathematical Understandings and
Meaningful Learning Situations Skills Involved
4. Finding difference in number
of days in a year on the
planets
5. Reading the age of the earth
6. Reading the circumference of
the earth
7. Comparing the age of the sun
and the earth
8. Gaining understanding of
fractional parts through the
quarterly changes of the
moon
9. Reading the rate of speed
light travels per second
10. Reading the rate of speed
sound travels per second
11. Multiplying to find the speed
of light and the speed of
sound per minute and per
hour
12. Comparing the difference in
the speed of light and the
speed of sound
13. Reading temperature of the
sun
14. Reading cost of constructing
200inch telescope in Califor
nia
15. Subtracting to find how long
since the first telescope was
invented
H. Height, depth, distance H. See page 20
I. Studying citrus industry I. 1. Adding to find total amount
of oranges, grapefruit, and
tangerines, produced in one
grove
2. Dividing to find average num
ber of boxes picked per acre
(based on known number of
boxes picked on 20 acres, 10
acres)
3. Adding to find total amount
of oranges, grapefruit, and
tangerines produced in the
country
4. Multiplying to find cost of
operating several acres
(based on known cost per
acre)
5. Multiplying to find total re
ceipts from fruit (based on
known price per box)
6. Subtracting to find profit on
box of fruit
82 ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning. Situate
J. Sugar industry
K. Studying the local
munity
1. Age of community
places of interest in
munity
2. Making a list of ii
tant historical event
3. Studying city and c
government
ions Mathematical Understandings and
Skills Involved
7. Adding to find operating cost
of producing one box of fruit
from grower to consumer
when fruit is sent through a
packing house
8. Multiplying to find operating
cost of season's production
9. Multiplying to lind cost of
picking fruit (based on known
picking prices per box)
10. Reading the value and
amount of citrus fruit pro
duced in Florida
11. Reading the amount of grape
fruit and oranges produced in
in Florida
12. Reading the value of grape
fruit and oranges produced in
Florida
13. Subtracting to find difference
in amount of citrus fruit pro
duced in California, Florida,
Texas, and Louisiana
14. Determining the fractional
part of nation's citrus crop
produced in Florida
15. Reading figures representing
number of field boxes or
grapefruit canned in Florida
(19371938, six million boxes
canned)
16. Subtracting to find difference
in number of boxes of fruit
canned in different years
17. Using scale on map to find
distances fruit is shipped
J. See page 103
com
and K. 1. a. Subtracting to find how
com long it has been since com
munity was first settled
b. Subtracting to find age of
court house, post office,
and fire department
mpor 2. a. Reading and writing dates
s to gain understanding of
sequence of time
b. Subtracting to find how
long ago important events
took place
county 3. a. Adding to find operating
expenses including salaries
of city and county officers
MATERIALS OF INSTRUCTION 83
Meaningful Learning Situations
4. Studying population
5. Studying
in county
cost of school
6. Studying water plant in
community
L. Studying the state
1. Historical development
Mathematical Understandings and
Skills Involved
(judge, tax collector, clerk,
sheriff)
b. Dividing to find monthly
operating expenses
c. Dividing to find monthly
salary of city or county
officers based on yearly
salary
d. Multiplying to find 4year
salary
4. a. Reading population figures
in community and county
b. Comparing population fig
ures of community and
county at present time
with population 10. 20, or
50 years ago
c. Finding per cent of county
population living in local
community
5. a. Reading figures represent
ing cost of running schools
in county for 9 months
b. Dividing to find cost of
running schools for one
month
6. a. Reading amount of water
consumed per month in
terms of gallons
b. Dividing to find consump
tion per week and per day
c. Reading cost of operating
water plant for one month
d. Reading amount of money
received from water con
sumers for one month
e. Reading numbers repre
senting amount of chem
ical added to town water
supply (includes frac
tions)
L. 1. a. Tracing the routes of ex
plorers on map to find
their mileage
b. Dividing to find mileage
per day and per week
when total time for mak
ing voyage is known
c. Subtracting to find length
of time since first settle
ment was made
d. Reading dates of other
settlements
ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning Situations
2. Studying state govern
ment
3. Studying population
4. Studying education
M. ,Studying growth of our
country
1. Explorers and travel
Mathematical Understandings and
Skills Involved
e. Subtracting to find differ
ence in dates of settle
ments
2. a. Multiplying to find amount
of money received by sen
ators and representatives
based on known salary
per day
b. Comparing salary of gov
ernor and other officers
of our state with salary
of governors and officers
of federal government
3. a. Reading population figures
of the state at the present
time
b. Comparing population fig
ures of state at the present
time with population 10.
20, 50 years ago.
c. Comparing population of
Florida with other states
d. Reading percentage of
Negro population in the
state of Florida
e. Reading or finding per
cent of total country's
population living in Flor
ida
4. a. Reading figures represent
ing cost of education in
the state
b. Interpreting reports to
find the cost of education
per child in various states
c. Comparing cost of educa
tion per child in Florida
with cost per child in
other states
d. Subtracting to find how
long ago the first public
school was built in our
state and country
M. 1. a. Gaining an understanding
of time sequence through
reading of dates
b. Subtracting to find differ
ences in dates of settle
ments
c. Dividing to find rate of
speed per month, per
per week, and per day in
MATERIALS OF INSTRUCTION 85
Meaningful Learning Situations
Mathematical Understandings and
Skills Involved
2. Home life
3. Population
4. Government
making trips across the
ocean then and now
d. Dividing to find rate of
travel of modern airplanes
per day and per hour
e. Subtracting to compare
rate of travel in pioneer
days with present day
travel (covered wagon, au
tomobile, train, and air
plane)
f. Multiplying to find how
long it would take to
travel from Washington to
Plymouth based on known
rate of speed per hour
(covered wagon or bus)
2. a. Comparing cost of build
ing a home now with build
ing home in pioneer days
b. Measuring in constructing
and furnishing a miniature
log cabin (fractional parts
of an inch or foot)
c. Finding the cost of fur
nishing a modern kitchen
(addition, subtraction, per
centage, and discount)
d. Comparing cost of living
in California in 1849 with
that of today
e. Comparing value of land
today with value of land
10, 50. 100, 200 years ago
(in Florida, in California)
3. a. Reading population figures
of our country at the pres
ent time
b. Comparing population of
our country with the popu
lation 10, 20, 100 years
ago
c. Comparing population of
our country with other
countries
d. Reading populations of
different nationalities now
living in our country in
comparison with our total
population
4. a. Reading the salaries of
congressmen in our state
~
ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning Situations Mathematical Understandings and
Skills Involved
b. Multiplying to find salary
of a congressman for a
whole term of office
c. Reading the salary of the
president of the United
States
d. Multiplying to find salary
of the president for 4
years, 2 terms, 3 terms
N. Studying growth of British
Empire
1. Location and size
2. Studying early settlers
and settlements
British govern
4. Population
N. 1. a. Gaining understanding of
use of degrees of longitude
and latitude through find
ing correct location of
England
b. Gaining understanding of
the relation of time to the
primemeridian
c. Reading the area of Eng
land in terms of square
miles
2. a. Reading dates of settle
ments
b. Finding distances traveled
from home land to Eng
land
c. Reading dates of impor
tant buildings in England
d. Comparing dates of the
important buildings
e. Studying simple geometric
designs in architecture of
buildings
3. a. Reading Roman numerals
in connection with names
and kings
b. Comparing Parliament to
Congress (number of mem
bers, salaries)
4. a. Reading population figures
of England
b. Comparing population of
England with that of other
countries
c. Reading population figures
of London
d. Finding per cent of popu
lation of England living in
London
e. Comparing population of
London with that of New
York City
3. Studying
ment
MATERIALS OF INSTRUCTION
Meaningful Learning Situations
5. Possessions
6. Transportation to other
countries
O. Studying modern homes
1. Buying a home
2. Upkeep of home
3. Studying and interpret
ing blue prints of the
floor plan
Mathematical Understandings and
Skills Involved
5. a. Reading and comparing
sizes of different posses
sions of British Empire
b. Tracing on map the routes
from England to the pos
sessions through the Suez
Canal
c. Reading cost of Suez Canal
d. Reading time required to
build Suez Canal
e. Comparing cost and time
required for building Suez
Canal and Panama Canal
f. Measuring distance from
England to possession on
map
g. Reading amounts of goods
shipped to England from
other possessions for the
past five years
6. a. Tracing air routes from
London to other cities and
other countries
b. Estimating time and dis
tance traveled in making
these routes
c. Dividing to find miles cov
ered per hour
d. Multiply to find amount
of fuel needed for trips
e. Multiplying to find cost of
fuel for trip
0. 1. a. Reading sale price of the
home
b. Multiplying to find amount
of yearly payment on
home (based on known
monthly payment)
c. Dividing to find how long
it would take to pay for
the home
2. a. Adding city, county, and
state taxes on property or
house
b. Reading premium of insur
ance (fire, theft, and hur
ricane)
c. Adding to find total
amount of money needed
to pay taxes and insurance
3. a. Measuring and drawing a
floor plan to scale, show
ing arrangement of doors
and windows
88 ARITHMETIC IN THE ELEMENTARY SCHOOL
Meaningful Learning Situations
4. Planning and furnishing
one room in house
a. Studying the placing
of furniture in the
bedroom
b. Purchasing furniture
for bedroom
c. Buying curtains for
bedroom
Mathematical Understandings and
Skills Involved
4. a. (1) Measuring floor space
for bed, tables, chests
(2) Drawing the placing
of furniture
(3) Learning good propor
tion, balance and de
sign in placing furni
ture
b. (1) Subtracting to find dif
ference in price of fur
niture
(2) Multiplying to find
amount of discount
(% or 25% off)
(3) Subtracting amount of
discount from original
price to find sale price
(4) Adding to find total
cost of furniture for
the bedroom
c. (1) Measuring length of
windows to find
amount of material
needed for curtains
(2 Multiplying to find
cost of material (mul
tiplication of mixed
numbers)
The second set of illustrative material, A Detailed Treatment of
Typical Primary and Intermediate Grades Problems, has been in
cluded in order to demonstrate how it is possible for a teacher to
capitalize on significant problems for developing specific under
standings and skills. Each of these typical problems includes: (1)
a statement of the immediate problems which are of concern to the
children within a class, (2) a statement of the larger problem as
seen by the teacher, and, (3) a list of materials. In addition, each
typical problem also includes: (1) a group of smaller related prob
lems (incidental and "plannedfor"), (2) suggestions for solving
the problems and suitable extensions, and (3) a list of specific
outcomes.
No attempt has been made to allocate the related problems to
a particular grade, although it is apparent that most of them are more
appropriate for one grade level than for another. Since children
MATERIALS OF INSTRUCTION 89
of various degrees of maturity are apt to be in any one grade it is
desirable that teachers recognize and devise problems compatible with
their maturity. Many problems are as suitable for intermediate
grade children as for primary grade children. The difference lies
in the methods used in solving the problems. For example, fifth
grade children do not "count on" when finding the difference but
subtract, using the takeaway method. Since problems become more
complex as children grow older, and since the computational process
needed in solving these problems also become more complex, teachers
need to evaluate pupil progress regularly. It is through evaluation
that teachers are able to determine the point at which individual
children are ready to solve problems using abstract numbers and
specific skill types within the computational processes.
A Detailed Treatment of Typical Primary
and Intermediate Grade Problems
I. IMMEDIATE PROBLEMMaking a school garden.
TEACHER'S PROBLEMCapitalizing on children's experiences in planting a school garden for developing mathematical
understandings and skills involving measurement, arrangement, and exchange.
MATERIALSRulers, yardsticks, seeds, bulbs, coins, currency, a variety of vases.
Initial Problems
(From integrated phase)
1. Measuring to find size of gar
den.
2. Marking
garden.
off the rows in the
:. How many seeds are in Willie's
package of nasturtium seeds?
4. There are 34 seeds in Willie's
package. There are 32 seeds in
Tom's package. How many
seeds are there altogether?
Problem Solving
f Direct and individual instruction phase)
1. (a) Children lay rulers end to end along
width of garden. (b) Teacher and chil
dren count to find how wide the garden
is in feet. (c) Repeat performance for
finding length of garden. (d) If maturity
of the children permits, they may use
yardstick for measuring, and may find
area of the garden in square feet and
square yards.
2. (a) Children hear or read the number of
inches which should be left between the
plants. They use this information fo
laying out the rows of their garden
With teacher guidance they measure the
distance from one row to the next, and
from one plant site to another.
3. (a) Teacher and children count by l's
2's. and 5's, to determine number of
seeds in Willie's package.
4. (a) Children count to determine total
number of seeds. (b) If maturity of the
group permits, they may add 34 to 132
using the simple addition combinations
(no carrying necessary).
Outcomes
1. (a) Ability to measure length
using rulers and yardsticks.
(b) Ability to express length
with a mixed number if a frac
tional part of the foot or yard
is involved.
2. (a) Ability to use the ruler for
measuring distances more tluan
and less than a foot.
3. (a) Ability to do rational
counting by l's, 2's, and 5's.
4. (a) Ability to do rational
counting. (b) Ability to add
two place numbers where car
rying is not necessary.
