[Cp375.5r N87p \( l■^^''V^I^V|';V•.•:>1hv:;;!:■,■^^ ifliif Mathematics in the Public Schools Curricul-um Committee Cte JlilJtarp of tlft Slnitjet^itp of il3ortt) Carolina Collection of iRottS Catoliniana (gnBotDeS bp jlo^n ^ptunt l^ill of tt)e (Sla00 of 1889 Cc>ZlS.5\ UNIVERSITY OF N.C. AT CHAPEL HILL 00034036329 FOR USE ONLY IN THE NORTH CAROLINA COLLECTION 'orm Wo. A-368 ^^ATUCMATICS NORTH CAROLINA PUBLIC SCHOOLS -years i-u Publication No. 275 Mathematics in the Public Schools Prepared by The Mathematics Curriculum Committee with the help of Administrators and Teachers in North Carolina Issued by the State Superintendent of Public Instruction Raleigh, North Carolina FOREWORD Many children enter school with a well developed quan- titative language for their age levels, but somewhere in their school experiences they get the idea that mathematics is a body of facts to be memorized; they are plunged into processes beyond their range of comprehension ; and they develop an aversion to the subject. Aversion to mathematics and a lack of understanding go hand in hand. The lack of understanding of mathematics that is common to many chil- dren throughout the elementary and high school is a chal- lenge to teachers to find meaningful, effective ways of teach- ing the subject. This bulletin was written to help teachers meet that challenge. The suggestions in the bulletin were based on sound prac- tices recommended by superintendents, supervisors, prin- cipals, and teachers throughout the State. The illustrations were taken from reports of actual classroom situations in North Carolina. The pictures in the bulletin were made in North Carolina schools. The suggested sequence in develop- ing fundamental concepts, principles, and skills in mathe- matics was based on studies that have been made to show^ the levels at which children can understand certain mathemat- ical processes. The sequence was outlined originally by the 1942 mathematics curriculum committee. It is recommended that teachers get an overall view of the mathematics program from the first year through the twelfth. This publication will help teachers get such an overall view. State Superintendent of Public Instruction ^ May 22, 1950 111 "Arithmetic is a system of quantita- tive thinking which we teach children, so that, as children and later as adults, they may live more efficiently, more in- telligently, more richly, and more hap- pily in our culture." Dr. William A. Brownell. IV PREFACE Because of the importance of mathematics in the school cur- riculum this revision of the program in arithmetic and high school mathematics has been prepared. The work of the mathematics committee which prepared the outline for the 1942 publication, A Suggested Twelve Year Pro- gram for the Public Schools of North Caroliyia, has been care- fully studied by many teachers throughout the State. The pro- gram as originally planned has stood up well. In each adminis- trative unit a committee consisting of a primary, a grammar grade and a high school teacher was appointed and a number of conferences were held throughout the State. The conferences for the study of the program were arranged and conducted by Miss Mary Vann O'Briant and Mr. A. B. Combs. Mr. Henry A. Shannon rendered valuable assistance in the later stages of the work. In addition written comments were submitted by many teachers. The present revision seems to meet the approval of the vast majority of the teachers. The illustrations used throughout the bulletin were taken from actual classroom situations and should be helpful in suggesting good practices. In addition to the new material and to the careful check which was made of the outlines by years, additions have been made to the bibliographies for both elementary and secondary school teachers. From the bibliographies it should be possible to pro- vide a small collection of books, magazines and materials which will be of considerable assistance to teachers. This bulletin is an attempt to make available to every teacher in usable form suggestions which we believe will result in im- provement in the teaching of mathematics. J. Henry Highsmith, Director Division of Instructional Service ACKNOWLEDGMENTS The following curriculum committee wrote the section on mathematics in the publication, A Suggested Twelve Year Pro- gram for the Nor^th Carolina Public Schools: W. W- Rankin, Chahmian C. G. Mumford Grace Chappell N. C. Newbold Bonnie Cone W. C. Pressly Laura Efird A. M. Proctor G. H. Ferguson Mrs. Hannah McN. Stack W. M. Jenkins H. C. West Vilamae MacMillan Jane Williams H. D. Munch Carrie B. Wilson Dr. William Betz, Consultant A State-wide curriculum committee, made up of three people from every administrative unit, continued the work of the orig- inal committee. To facilitate travel and to make good working situations, this Mathematics Curriculum Committee was divided into sixteen groups, with sub-divisions when needed. This. bul- letin is a revision of the material originally published. The committee is indebted to the following administrative units for pictures, materials, and suggestions for the bulletin : Counties — Ashe, Beaufort, Buncombe, Carteret, Chatham, Cleve- land, Craven, Duplin, Durham, Edgecombe, Forsyth, Greene, Halifax, Harnett, Hoke, Hyde, Iredell, Jackson, Jones, Lenoir, Macon, Martin, Nash, Northampton, Perquimans, Robeson, Ruth- erford, Washington, Watauga, Wayne ; Cities — Albemarle, Ashe- boro, Asheville, Boone, Clinton, Durham, Elkin, Fayetteville, High Point, Lincolnton, Rocky Mount, Shelby, Statesville, Win- ston-Salem. We wish to express our thanks to the many teachers who at- tended the mathematics conferences and helped make decisions with reference to revisions. Special recognition is given to Henry A. Shannon, Adviser in Science and Mathematics, for the statement on Mathematics and Natural Science and work on the bibliography for the secondary school ; and to L. H. Jobe, Director of Publications, for his help- ful suggestions about the arrangement and his careful editorial work. Mary Vann O'Briant, Associate A. B. Combs, Assistant Director Division of Instructional Service vi CONTENTS .■■'■■ r . Arithmetic in the Elementary School Page I. Point of View 1 11. Suggestions for Helping Children Learn Arithmetic 2 A. Development of Meaning 2 B. Provision for Individual Differences 4 C. Problem Solving 4 D. Drill 5 E. Maintenance of Skills 6 F. Current Practices in Teaching Arithmetic 6 G. Homework 7 H. Games and Devices 7 III. Illustrations of How Children Have Used Arithmetic - 13 IV. Instructional Materials 17 A. General 18 B. Primary Years 18 C. Middle Years 19 D. Upper Years 19 E. Films and Filmstrips 19 V. Suggested Sequence in Developing Fundamental Con- cepts, Principles, and Skills in Arithmetic 20 VI. Measurement of Progress 46 VII. Bibliography 47 A. Books for Children 47 B. Books for Teachers 47 C. Magazine Articles 48 D. Professional Magazines 49 Mathematics in the Secondary School I. Ninth Year 50 Course A : General Mathematics 50 Course B: First Year Algebra 52 Vll Mathematics in the Secondary School Page II. Mathematics for Years X-XII ..___. 54 III. Tenth Year, Second Year Algebra 54 IV. Eleventh Year .1__.. 55 Course A : Plane Geometry „.. 55 Course B: Plane and Solid Geometry, Integrated 56 V. Twelfth Year ^„.___ 57 Course A : Business Arithmetic 57 Course B : Algebra 58 . Course C : Solid Geometry 58 Course D : Trigonometry 59 Course E: Basic Mathematics 59 VI. Mathematics and Natural Science 63 VII. Bibliography 67 A. Yearbooks 67 B. Magazines 67 C. Books on Various Phases of Mathematics 67 D. Booklets on Audio- Visual Aids 68 E. Mathematical Recreations 69 viu Arithmetic In the Elementary School POINT OF VIEW The curriculum should provide activities from the experiences of children which will promote understanding of the fundamental processes and principles of arithmetic. Concepts should be de- veloped through the use of concrete materials. Abstract quanti- tative situations should be introduced gradually. Attention should be centered upon relationships and principles of operation. Skills should be used to solve problems of everyday living. Mastery of any learning is not acquired all at once ; therefore, activities should be planned carefully, so that children will move toward mastery at the rates possible to them. Drills for mastery should be postponed until understanding has been thoroughly established. It is natural for children to work at different levels of accom- plishment and different rates of speed. For this reason, the arithmetic curriculum should be flexible enough to allow for many kinds of differentiation both as to the rates of learning and as to the content. The differences in needs and interests, as well as the differences in abilities, should be recognized and pro- visions should be made to take care of them. Arithmetic should be integrated with the entire curriculum. Consideration should be given to the many opportunities to teach arithmetic which arise in all classrooms and in all communities. SUGGESTIONS FOR HELPING CHILDREN LEARN ARITHMETIC In teaching children arithmetic there is a great need for atten- tion to be given to the development of meaning. Much of the lack of understanding of arithmetic and the dislike for the sub- ject stem from school programs that stress memorization and abstract drill. Children must see that arithmetic functions in their daily lives. They must see that it is a subject which con- tains information that they need in order to do many of the things which they want to do. Little children must do a great amount of work with concrete objects, then semi-concrete, before they work with abstract num- ^ , . bers. No abstract number, no arithmetical term, » ^ . no process should be introduced to children with- out sufficient background of activities of con- crete nature to insure meaning. Children of all ages need to 2 Mathematics in the Public Schools work with concrete materials and with actual life situations with- in their experiences in order to discover meanings. Schools should assemble concrete materials that can be used by classes to add meaning to their work in arithmetic. Some other suggestions of ways to develop meaning : 1. Teach 10 as a basic group. The teens will give more trouble than the 20's or 30's. Have each child put 10 sticks in a pile and put a band around the pile. Construct 11 by adding 1 stick to the pile. In like manner work through all the teens. When 20 is reached there will be 2 tens. Carry as far as nec- essary to build understanding. Prove that two tens and 2 are 22 by taking the bands off the two bundles of lO's and counting the lO's and the 2. Prove many of the numbers in the same way. Let the children know that teen means and ten. For example, thirteen means three a ?icZ iew. Eleven was once one-teen and twelve was two-teen. 2. Stress place value. A simple chart marked off into places for I's, lO's, and lOO's will help in the beginning of the work. Continue to stress place value in all of the fundamental opera- tions when the children are ready. Work, similar to the fol- lowing will help : Addition Read the following numbers and let the children figure out how they should be placed in a column: seven, one hundred, twelve, two, one hundred six. Subtraction -■ 237 Borrow 10 from. 30, add it to 7, making 17. Subtract — 118 8 from 17, etc. It is very misleading and inaccurate to borrow 1 from 3 to add to 7 to make 17. It just is not true. )Multiplicatio7i 23 2x3=6. Since 6 is a unit, put it , in the units' place. X 12 2x2 tens=4 tens. Put 4 tens in the lO's place. In 12 the 1 is ten. 1 ten X 3 = 3 tens. Put the 3 tens in the 46 lO's place. 1 ten X 2 tens = 2 hundreds (10 X 20 = '1^ 23 200). Put the 2 hundreds in the lOO's place. Show that the 23 tens equal 230. Avoid disregarding any rro step because of its simplicity. :UtiiB: Mathematics in the Public Schools 3 Division 20 16 tens ^8 = 2 tens. Place 2 tens in the lO's place. 8x2 16 tens. Show that 16 tens equal 160. Show that 8)160 tens 16 zero is the place holder. Stress the fact that zero is a place holder. Work many exam- ples with the children to show that zero's only function is to hold the place. Work with measures. Collect standard measuring vessels. Let the children measure liquids until they understand that two pints make a quart, etc. Let them work in a similar way with other liquid measures and with dry measures. Cut a linear inch, foot, and yard from construction paper or tag- board. Let the children estimate, compare, and measure lengths. Stress the importance of measuring accurately. Work in a similar way with square and cubic measures. Encourage the children to observe the geometric forms around them. Classrooms are filled with rectangles, angles, etc. Work tvitli concrete materials simplifies tables of measurements \ -*«s\ 1 7 y Wm^ ■-.i^y 4 Mathematics in the Public Schools 6. Avoid letting arithmetic become an isolated subject. There should be a close tie-up between arithmetic and everyday living. The wide range of abilities within any class, the difference in readiness and interests, the lack of success, and the danger of maladjustment when mass teaching is used m.ake it necessary to provide for individual differences. A study of the children through observations, tests, analysis of cumulative records, and interviews will disclose the differences and the needs. The needs should be met through a teaching procedure that will promote growth in each individual. Some ways of providing for individ- ual differences are: grouping children within the class accord- ing to their needs, keeping the grouping flexible in order to avoid „ . . „ , J. stigma; beginning where the children are Provision for Indi- ,. . -i . n , j, .-. i ■^^ . , , ^.^ realizing that all members of the class will Yidual Ditterences ^ , . .-, ^ • t not be at the same place ; providing mate- rials on many ability levels; enriching the program so that the work will be a challenge to the fast learners ; broadening the scope of the work so that the slow learners can achieve success ; and using as much individual instruction as possible. Units of experience that emphasize the social and economic phases of arithmetic can be used to a great advantage in bringing all mem- bers of the class together for some periods with each member working on his own level of ability and making some contribution of value to the class. Remedial work should be in the light of individual needs. It should consist of remedial instruction as well as remedial drill. Some of the major causes of children's difficulty in solving problems are : unfamiliar terminology, inability to read under- p , , Q 1 • standingly, inability to think through prob- lems, lack of knowledge of the fundamental operations, unfamiliar settings of problems, and use of careless procedures. From the beginning of a child's school life, he should be guided in building up his vocabulary of arithmetical terms. As early as is consistent with understanding, he should be guided away from using simplified terms, such as take away for subtract. Lack of ability to read arithmetic problems under- standingly is tied up often with tjie carry-over of simplified terms beyond the point of necessity. Early in the child's school life consideration must be given to training him to think through problems. This training should begin the first year of school, u^dng simple problems from everyday experiences, such as "How Mathematics in the Public Schools 5 many chairs do we need for Jim's reading group?" The prob- lems in the first year should be quite simple and should grow out of school experiences. The work should be oral. When children begin to work with written problems, a definite plan for solving problems should be started. The following steps should be included in the plan : Read the problem carefully ; decide what should be found; see what facts are given; decide what processes should be used ; see if the answer found answers the question the problem asked ; see if the answer is reasonable ; see if the work checks. In analyzing problems, children must see clearly what they want to find, what facts are given, and how to use these facts to arrive at a solution. Even then they are seriously handicapped in problem solving if they do not have a thorough command of the fundamental operations, since these are the tools for carrying out the solutions. In order to help children solve problems, it is necessary to follow their line of reasoning. It is more important to get children to tell why they worked problems in a certain way than it is to get them to tell how they worked them. To counteract the tendency of many children to use the trial and error method, exercises in solving problems without numbers should be used. Such work will cen- ter attention on principles. Frequently children can be helped in problem solving by working orally with simple problems in- volving the same principles as the ones which are causing trouble, using many problems that grow out of their experiences so that settings will be familiar, and by analyzing problems with the group. Children have less difficulty with problems if they feel the need for solving them ; therefore, the best problems are those that are tied with home, school, and community in the local sit- uation. Drill has two purposes: (1) to make responses to given stimuli automatic and (2) to increase retention. Therefore, it is very important that drill should follow — never precede — thorough un- 1^ ... derstanding of the ideas involved, and that drill periods should be frequent. If under- standing is not built before drill is undertaken, there is great danger of children practicing their errors, developing resistive attitudes, and memorizing instead of reasoning. Drill periods should be short and interesting. Accuracy should not be sacri- ficed to speed. Each child should be encouraged to keep a record of his own drill accomplishments in order to add interest and to build a desirable form of competition — that is, competi^-ion 6 Mathematics in the Public Schools Maintenance of Skills against cne's own record. The spirit of play may be used to bring- dull periods to life. If a skill is to be maintained, drill on it cannot be entirely aban- doned. Efficient drill should begin on a skill after understand- ing has been established and should continue throughout the school years on the level of each child's ability. Special attention should be given to drill in the four fundamental operations with whole numbers, n:;ixed numbers, and common and decimal fractions. Anything that gives meaning to that which is being learned aids in its retention. Helping children to generalize and to see that certain principles always hold true, helping them to see relationships, and encouraging them to use the skills they have learned to solve problems in their own experiences — these things will help them to maintain arithmetic skills. Current Practices in Teaching Arithmetic With the aim of making arithmetic meaningful to children, concrete materials are used to build arithmetical concepts ; arith- metic is tied closely to everyday living in school, home, and community ; emphasis is placed on the social and economic phases of arithmetic. The fact that many ability levels are found in every classroom is recognized, and instruction and materials are provided to meet the individual needs. Children are grouped within the class for work, and the grouping is kept flexible. The mental maturity of the children is taken into consideration in determining the placement of the work program. Teachers watch A daily chore is turned into a learning situation Mathematics in the Public Schools 7 p for undesirable practices which some children use in their arith- metic work; they help children to substitute desirable practices for undesirable ones. A few undesirable practices which some children use are: 1. Using inaccuracies, such as 41/2 = 2/4 for 41/2 = 4 2/4. 2. Using careless substitutions, such as clot for decimal. 3. Memorizing facts without understanding them. 4. Continuing to use simplified terms instead of building a vocabulary of arithmetical terms. 5. Juggling the numbers in problems in an effort to find answers instead of reading the problems carefully. 6. Holding on to crutches after they have served their purpose. 7. Using incorrect language forms. 8. Wasting time while one child struggles at the blackboard with a problem. 9. Working carelessly. 10. Regarding disabilities in arithmetic as permanent. In planning homework for children, teachers should consider such things as the following : the children's growth, the time chil- dren need for family living, rest, recreation, and home duties ; ,^ , the length of time some children spend on Homework , j .r, 1 1 ^ • . ^ buses ; and the lack of equipment lor study in some homes. The requirements for homework should be flexible in order to meet individual needs. Individual assign- ments are desirable. Much of the homework should be of the enrichment type; it should encourage independent study; it should provide opportunities for children to explore their inter- ests. Many of the activities may be self-chosen; many may be creative. Home and community resources should be utilized. If homework is of the drill type, care should be taken to see that the children know what to do and how to do it. A new process should never be assigned as homework. Games and devices can be very helpful in arithmetic work if care is taken in their selection and use. Some games do not con- tribute enough to the children's learning of arithmetic to justify ^ J -Tk • their use ; some are set up in such a way Games and Devices ,, , „ ' ,.,, . ,,^, , ,./ that a tew children do the learning while the others waste time. A thorough understanding of the processes involved should be established before drill games are played. The same holds true with the use of drill cards. Information about games and devices can be found in many of the books on 8 Mathematics in the Public Schools teaching arithmetic. Some of the following games and devices can be adapted to several ability levels. 1. Pictures to Talk About Select colorful, interesting pictures that are large enough for the children to see easily. Paste the pictures on cards made of tag board. Through questions about the pictures build understanding of first, last, tallest, over, etc. 2. Number Cards for Begi7i7iers Make the cards as large as necessary in order to arrange pictures on them. On the first card, paste the picture of one thing of interest to children. Put the number symbol 1 and i the word 07ie under the picture. Continue in the same way through 10. 3. Number Picture Books Make a number picture book with the title All About Four. Select pictures that show four things, such as four children or four dogs, and paste in the book with the number 4 and the word four printed or written in manuscript under each pic- ture. After a sufficient number of pictures that show four have been put into the book, select other pictures that show combinations that make four and paste in the book. Make other books about other numbers. The number picture books make interesting material for reading tables. 4. Materials for Countiyig Cut circles and squares from colored tag board. Make them large enough for beginners to handle easily. Use them for counting and grouping. Collect and sterilize ice cream sticks. Use them for counting and working with tens. Flat objects are better for counting than round ones, 5. Number Dictionary Make the number dictionary through tcTi. The information can be put in a scrapbook (num- ber dictionary), on cards, or on charts. Mathematics in the Public Schools 6. Number Puzzle 3 5 7 8 Turn some of the numbers on the cards toward the children ; turn some of them away. Let the children name the missing numbers. The cards can be drawn on the blackboard and the children can fill in the blank spaces. Many arithmetic concepts are gained from a simple measuring device How tolf are you ? 10 Mathematics in the Public Schools 7. Number Cards FRONT BACK OF CARDS OF CARDS 6 -2 2 x3 Make the cards about 8" x 5" if they are to be used in groui> work. Emphasize the gronpinjr of semi-concrete objects. At the beginning of the work with the cards, the fronts should be used more than the backs. After the concepts have been established, the backs should be used more; the fronts should be used then to verify the answers. Much in- dependent work for children can be built on these cards. Measuring Device Cut a piece of heavy cardboard about six inches wide and a few inches longer than the tallest child in the room. Use a yardstick to make the inches and feet on the cardboard. Mark the inches with short lines and the feet with long lines. Use color and pictures to make the measuring device at- tractive. Place the device where it will be convenient for the children to use to measure themselves.. 9. Nu7nher Disk Cut an eight-inch disk from heavy cardboard. Fasten a mark- er to the center with a nail. Put a cork on top of the nail to pre- vent accidents. Have the mark- er loose enough to spin easily. Collect and sterilize ice cream sticks to use in counting cores. Let each child spin the marker and take as many sticks as the Mathematics in the Public Schools 11 5^2 3+3 number of balls in the section in which the marker stopped. At the end of the game, let each child count his sticks to determine his score. For second and third year pupils, num- bers can be used and each child can add his score. 10. Climbing Stairs 4+6 Draw the stairs on the black- board. Let each child begin at 5+4 the first step and climb as far as he can by giving the correct 3+7 answer to the combination on each step. The child who reaches the top may become the teacher and may choose his pupil from the group. If his pupil reaches the top, he becomes the pupil- • teacher. This game can be used z_J for subtraction and multiplica- tion. Climbing the Ladder and Stepping Stones are similar games. 11. Ntunber Wheel 12. Magic Square 8 4 5 7 2 3 3 7 6 Draw a number wheel on the blackboard. Write the digits in random order near the rim of the wheel. Place the multiplier in the center. Let the children, one at a time, give the products, moving around the wheel in the order that the digits are given. This game can be used for addi- tion combinations. Fill in each empty square so that the sum of each row, ver- tically and horizontally, will be the same. The children may be interested in making magic squares for members of the group. 12 Mathematics in the Public Schools 13. Number Baseball 978 14. Draw a baseball diamond on the blackboard. Write three numbers at the homeplate so that the batter can have three strikes if necessary. Write one number at each of the other bases, and write the multiplier in the pitcher's box. Divide the group of children into two teams. Use baseball rules con- cerning outs, innings, etc. Count a score for each player who gives the product of the multiplier in the pitcher's box and the multiplicand at each base. The game can be speeded up by allowing only one strike. Filmstrip of Steps in Long Division Make a drawing of filmstrip on the blackboard. On the filmstrip write the explanation of what was done in each step. Leave the drawing and explanation on the blackboard until the children become familiar with the steps. If the blackboard space is needed, the filmstrip can be drawn on tag board. 15. Fraction Concepts Make two of each figure; cut one into fractional parts ; leave the other intact. Let the children place the fractional parts that have been cut upon the figure left intact to prove that 2/4 = 1/2, 2/6 = 1/3, etc. Carry as far as needed. Measure accurately. Make a chart showing many kinds of figures marked into fractional parts. 16. Meusurcments On the blackboard draw heavy lines of different lengths — 12", 18", 9", etc. Draw some of the lines vertically and some horizontally. Measure accuratelv. From colored corstruc- Mathematics in the Public Schools 13 tion paper cut strips the same lengths as the lines on the blackboard. While in his seat, let each child select a strip which he thinks is the same length as that of a designated line on the blackboard. Let the children measure the line with the strips they have selected. Work of this type will help children learn to estimate lengths and widths with a satisfactory degree of accuracy. Cut a linear inch, a linear foot, and a linear yard from colored construction paper. Mount on tag board and exhibit in the class- room. This should be the work of the children. Make each block a square inch and fold along lines to make a cubic inch. Follow the same plan to make a cubic foot, making each block a square foot. Exhibit in the classroom. This work should be done by the children. ILLUSTRATIONS OF HOW CHILDREN HAVE USED ARITHMETIC Many schools in North Carolina have reported activities that show how boys and girls have used arithmetic. The following activities which have been carried on successfully in many class- rooms may suggest some ways in which arithmetic experiences can be brought naturally and meaningfully into the school work of others. A first year group reported on a Christmas party : This is December 14, 1949. We went to town yesterday. We bought things for our part> The napkins cost 15 cents. The cups cost 90 cents. The cookies cost 40 cents. W^e spent $1.45. We still have $1.55. A second year group reported : "Numbers are fun ! We know, because every day we have fun buying and selling in our Corner Grocery Store. We made this store ourselves in the corner of our classroom. . . . We visited a nice, new grocery store. The next day we started our own 14 Mathematics in the Public Schools store. Mac, David, and Joe made the shelves. . . . The shelves measured 23 inches long and 7 inches wide. . . . Brenda brought her cash register, adding machine, and scales. She showed us how to use them. Mary brought her telephone. We learned how to dial " A child in a combination primary group reported : "The first thing that we do every morning is to count the cafe- teria money. On Mondays we take the dollars which pay for lunches all week. We collect twenty cents from the children who pay each day. We keep the record on the blackboard. On Friday we add the money collected during the w^ek." A P.T.A. committee chairman reported : "We have a school store that sells school supplies, and it is managed entirely by our third grade children One child writes on the board the amount of money collected for each item sold during the selling period, and the class totals the sales. . , . The use of zero is understood. . . . The use of the decimal point comes in naturally. . . . The difficult process of carrying and borrowing becomes meaningful." Teachers reported : "Beginners learned from the construction of a play house the meaning of many quantitative words." "There are situations arising many times each day which pre- sent opportunities for teaching arithmetic. For instance, the children and I work together in checking the attendance each morning." "We had a valentine shop in our room. . . . The children had practice in measuring, using money terms and signs, adding, subtracting, and multiplying." "The refreshment committee planned to make candy for the valentine party. The class figured out the amount of candy that v.'ould be needed for the party. It was necessary to increase the measures in the recipe. This called for multiplication and work with fractions. Finding the cost of the party called for much work with fundamental processes." "Before Christmas the class had a doll shop. Cowboy dolls, clowns, ringmasters, and teddy bears made the shop interesting to the boys as well as to the girls. In putting price tags on the dolls, the children learned to write the dollar sign and the cent sign. After Christmas they had a January clearance sale. All dolls were reduced 1/2, 1/3, or 1/5." "The children brought familiar vessels to school: milk bot- tles (half-pint, pint, and quart), fruit jars (pint, quart, half- gallon, and gallon) , a quart can, a gallon bucket, and a peck meas- ure. Using water that had been colored with cake coloring to Mathematics in the Public Schools 15 make it easily seen, the children proved the information given in the table of liquid measures. Using cracked grain, they did the same thing with dry measures. When they had finished the concrete work, one boy remarked, 'Well, seeing is believing'." 'The children measured the windows of their classroom to determine the amount of material needed for drapes. They fig- ured the cost of the drapes ; then they figured the total amount of other class expenses to see if they could afford to buy the drapes." "Each child weighed and measured himself and recorded his weight and height. He compared his weight and height with his former record, subtracting to find out how much hs had grown." "The fourth grade had an arithmetic contest for the purpose of creating interest and motivating drill work. Two captains were chosen, and they, in turn, chose their teams. . . . The daily scores were kept on a chart, and the winning team w^as the one that accumulated the greater number of points throughout the contest period. . . . The contest developed interest in arithmetic and an eagerness to learn, as each pupil wanted to be prepared to earn points for his team. . . . The children helped each other during the study and practice periods, thus a spirit of helpfulness and cooperation was developed." "Several of the fourth grade children had bought clothes during sales. . . . Sales were discussed. . . . Each pupil brought his problems to the class to be solved." (Problems were based on local sales.) "The children used geometric designs to make church windows for a Christmas pageant." "Each day the children would mark through the date on the calendar with a crayon and then count to see how many days be- fore our party." "The children came back from the Christmas holidays, walked into the room, and as usual announced something. This is the New Year. It's 1950.' I told them that a New Year is important and that this New Year is an especially important one. This year is a new half century. We talked about a century — that it is a hundred years — and that a half century is fifty years. . . . The cal- endar was used to look at 1950, to learn about the first month of the year, the first day of the month, the number of months in a year, the number of days in the months. Discussion went back to the century and half century. I wrote 1950 on the board again — 50 years gone by in this century. . . . Half a centurv is half of a hundred years — 50 years — the new year is 1950, the beginning of another half century. ... A child suggested that we make a mural for the bulletin board about the things that had happened in the last half century The group talked about the events and decided to name the mural 'Events of the First Half 16 Mathematics in the Public Schools Century'. Each child who had volunteered to draw a part of the mural took his turn while small groups worked with finding dates on the calendar and subtracting years to find out how many years between what happened then and now. The children wrote dates — the month, day, year — using the comma to separate day and year. They learned that the New Year is always 1 week — 7 days — after Christmas. No, it didn't take long. The morn- ing's discussion furnished creative work for the children. Dur- ing the day, while I was working with small groups in the reg- ular daily pattern, children by twos and threes looked for informa- tion in an encyclopedia, drew pictures, discussed the pictures, and planned for arrangement and spacing of the pictures and for the lettering of the mural. These children had a great thrill over the mural. They called it history and felt that they were doing important work in learning about happenings of the past. . . . There's a difference in helping children to create and in giving them busy work." Other activities reported which involve arithmetic work were: 1. Making graphs to show school attendance. 2. Working with classroom measurements for many practical purposes. 3. Finding the cost of the supplies furnished by the school; finding the total cost and the cost per pupil. 4. Reading the school electric meter and finding the cost of the electricity the school used. 5. Operating a school bank. 6. Making geometric designs and figures. 7. Reading thermometers and keeping records of the weather. 8. Planning the arrangement of materials on bulletin boards. 9. Planning trips. 10. Planning and issuing a school newspaper. 11. Making arithmetic exhibits. 12. Electing club officers. 13. Planning and making bird houses and feeding stations. 14. Making and using calendars. 15. Making Christmas cards and gifts. 16. Making budgets. 17. Finding the cost of school lunches. 18. Ordering from catalogues. • • ^ w • • ■ ■ 'I ^» <■ ■ ^ >»*».■, Seventh year puiJils put their -finishing touches to their exhibit of geometric, designs and figures 19. Decreasing and increasing measures in recipes. 20. Keeping scores in games. 21. Figuring batting averages. 22. Making maps of the classrooms, school grounds, and com- munities. Drawing to scale. 23. Making school gardens. 24. Planning patterns for weaving. 25. Planning an orderly arrangement of the classroom. 26. Making stages for puppet and marionette shows. 27. Keeping household accounts. INSTRUCTIONAL MATERIALS An abundance of concrete materials should be provided for instruction in arithmetic in order to establish meaning and to build arithmetical concepts. Many of the materials in the follow- ing lists can be made; some can be collected from homes and business places; some are parts of school equipment. It is rec- ommended that teachers add to the following materials as needs arise. 18 Mathematics in the Public Schools General: Library books Supplementary books Yardsticks Tape measures Rulers Scissors Paper (construction, drawing) Crayons, paints Scales Primary Years: Devices for measuring height Pictures with implications of arithmetic Charts Blackboards Bulletin boards Catalogues Thermometers Liquid measures Dry measures Tests (Use materials listed under Gen- eral) Blocks Sticks for counting and assem- bling into tens Squares and circles for counting and assembling into groups Spools Boxes for materials for count- ing Calendars with large numbers Rulers with inch and half-inch markings Number charts — Number cards Large thermometers Puzzles Toy dial telephones Tickets Clock faces Paper plates Textbooks for third year Number frames Children learn to tell time by trorking loith clock faces they have made from paper plates &^ Mathematics in the Public Schools 111 Middle Years: (Use materials listed un- der General) Reference books Globes Maps of countries Road maps Weather maps ^Measuring instruments Business forms (checks, deposit slips, money or- ders, etc.) Graph paper Graphs (picture, line, bar, circle) Upper Years: Blocks and circles cut into frac- tional parts Fraction charts Place value pockets Tag board cut into linear inches, feet, yards Square inch blocks Square foot blocks Cubic inch blocks Cubic foot blocks Newspapers Textbooks (Use materials listed under General and Middle Years) Protractors Compasses Board feet (sawed from lum- ber) Time tables Standard time charts Geometric figures House plans Government bulletins Films and Filmstrips: The Division of Instructional Service, State Department of Public Instruction, has prepared a list of films and filmstrips in the field of arithmetic. This list will be sent to schools on re- quest. It is recommended that school faculties preview films and filmstrips before buying them in order to be sure that they meet the needs of the school. Some films and filmstrips will be helpful in workshops for teachers, since from them teachers will get ideas of ways to work with children. Many of the professional books have chapters on using instruc- tional materials. Some of these are : Chapter VIII, "The Enrich- m.ent of the Arithmetic Course : Utilizing Supplementary Mate- rials and Devices," Arithmetic in General Education; Chapter 1, Part III, "Arithmetic Comes to Life," A2idio-Visual Methods in Teaching; Chapter XIII, "Materials of Instruction in Arithme- tic", How to Make Arithmetic Meaningful; Chapter 1, "Develop- ing an Understanding of Number", Teaching Arithmetic in the Elementary School, Volume I. See bibliography for full refer- ence list. 20 Mathematics in the Public Schools SUGGESTED SEQUENCE IN DEVELOPING FUNDAMENTAL CONCEPTS, PRINCIPLES, AND SKILLS IN ARITHMETIC The sequence outlined by school years in the following pages was planned for children who make average progress in the learning of arithmetic. Not all children make the same progress ; therefore, it is necessarj^ to adjust the arithmetic program to fit the children. It is necessary to find where each child is in his arithmetic learning and to begin there. It is recommended that a diagnosis be made of the children's arithmetic work at the beginning of each school year in order to discover strengths and weaknesses. The use of new experience will make the review interesting and helpful. Often during the year it may be necessary to review and reteach. It is recommended that children be trained to check their work for accuracy and neatness as one means of establishing confi- dence, developing habits of carefulness, emphasizing the im- portance of accuracy and giving them a basis for evaluating their work. It is also recommended that the arithmetic program be en- riched rather than accelerated in order to avoid plunging chil- dren into activities beyond their range of comprehension. First Year Children enter the first year of school with varying degrees of number readiness. Through the use of concrete materials and classroom situations, number readiness can be built and number concepts can be developed. The arithmetic program in the first year should not be left to chance. It is necessary to make definite plans for the work. Many situations in which quantitative relationships occupy prominent places may be used for stimulating the learning of needed skills. For example, a pet may be kept in some class- rooms. How to get the pet, how to build a pen or cage, how to provide food, when to feed the pet are a few of the questions which will arise. Many activities which call for work with numbers will grow out of such classroom situations. itwo JW-^! Jfv '# Om^breod CooKiti it cup molossej. 1 Tol»^poon buTtftr. 2 Tobleipoon \ord Ji TabiMjpcw ainaer. Si teaspoon sat! 4 teaspoon bakinq soda I ToUespoon worm woTer flour I ^Bm ■ v.holrgy, m the pedagogy of arithmetic. Its aim is to encourage scholti • riv i^i 'he -.ibject itself. It differs from an arithmetic for children ch ; • ih;. fact that it is addressed to more mature minds." Dale, Edgar. "Arithiiie. C'-'nes to Life." Audio-Visual Methods in Teach- ing. New York: The Di. dtn Press, 1946. 546 pp. Hildreth, Gertrude. Learning the Three R's. (Revised). Ch. 23, 24, 25, 26, 27, and 28. Nashville, Tennessee: Educational Publishers, Inc., 1947. 897 pp. Morton, Robert L. Teaching Arithmetic in the Elementary School. Volume 1, Primary Grades; Volume 2, Intermediate Grades; Volume 3, Upper Grades. New York: Silver Burdett Company, 1937, 1938, 1939. National Council Committee on Arithmetic. Arith^netic in General Educa- tion. Sixteenth Yearbook of National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia Univer- sity, 1941. 335 pp. 48 Mathematics in the Public Schools Spitzer, Herbert F. The Teaching of Arithmetic. Atlanta: Houghton Mif- flin Company, 1948. 397 pp. Stern, Catherine. Children Discover Arithmetic. New York: Harper and Brothers, 1949. 295 pp. $4.50. Stretch. Lorena B. "Number Experiences". Ch. X. The Elementary School of Today. Nashville, Tennessee: Educational Publishers, Inc., 1947. 488 pp. Van Engen, Henry. Arithmetic. No. 1, Developing the Fraction Concept in the Lower Elementary Grades; No. 2, For Developing an Understanding of Place Value; No. 3, Teaching Fractions in the Upper Elementary Grades. Educational Service Publications. Cedar Falls, Iowa: Iowa State Teachers College, 1948. 10^ each. . Magazine Articles Beatty, Leslie. "Re-Orienting to the Teaching of Arithmetic". Childhood Education. 26: 272-275. February 1950. Brownell, William A. "An Experiment on 'Borrowing' in Third Grade Arithmetic". Journal of Educational Research. 41: 161-171. November 1947. . "Frontiers in Educational Research: In Arithmetic". Journal of Educational Research. 40: 373-380. January 1947. . "The Place of Meaning in the Teaching of Arithmetic". Elementary School Journal. 47: 256-265. January 1947. Brueckner, Leo J. "Experience Units in Arithmetic". NEA Journal. 37: 20-21. January 1948. Carter, Paul D. "From a Mechanistic to a Meaningful Program of Arith- metic Instruction: A Suggested Approach". School Science and Math- ematics. 47: 604-608. October 1947. Coons, G. Edgar, and others. "Arithmetic at Work". Childhood Education. 26: 251-266. February 1950. Everett, Marcia A. "What About Homework?" Educational Leadership. 7: 331-334. February 1950. Gavian, Ruth Wood. "Helping Children to Grow in Economic Competence". Childhood Education. 20: 205-211. January 1944. Gray, Merle. "On Teaching Arithmetic". Childhood Education. 26: 242- 243. February 1950. Griggs, James H.. "Helping Children See Relationships in Developing Number Concepts". Childhood Education. 25: 115-118. November 1948. Hall, Jack V. "Color Clarifies Arithmetic Processes". Elementary School Journal. 50: 96-98. October 1949. Hartung, Maurice L. "Major Instructional Problems in Arithmetic in the Middle Grades". Elementary School Journal. 50: 86-91. October 1949. McSwain, E. T. "A Functional Program in Arithmetic". Elementary School Journal. 47: 380-384. March 1947. -. "Discovering Meanings in Arithmetic". Childhood Edu- cation. 26:267-271. February 1950. Morton, Robert L. "Arithmetic in the Changing Curriculum". NEA Jour- nal. 38: 430-431. September 1949. • . "Securing Better Results in Arithmetic". NEA Journal. 36:568-569. November 1947. Mathematics in the Public Schools 49 Moser, Harold E. "Advancing Arithmetic Readiness Through Meaningful Number Experiences". Childhood Education. 24: 322-326. March 1948. Shore, J. Harlan. "Why Children Dislike Arithmetic". Educational Admin- istration and Sxipervision. 32: 357-363. September 1946. Spache, George. "A Test of Abilities in Arithmetic Reasoning". Elementary School Journal. 47: 442-445. April 1947. Spitzer, Herbert J. "Techniques for Evaluating Outcomes of Instruction in Arithmetic". Elementary School Journal. 49: 23-31. September 1948. Sueltz. Ben A. "Measuring the Newer Aspects of Functional Arithmetic". Elementary School Journal. 47: 323-330. February 1947. Van Engen, Henry. "An Analysis of Meaning in Arithmetic". Elementary School Journal. 49: 321-329. February 1949. Wheat. Harry G. "Changes and Trends in Arithmetic Since 1910". Ele- mentary School Journal. 47: 134-144. November 1946. . "Teaching Thinking in Arithmetic". Mathematics Teach- er. 40: 217-220. May 1947. Wilburn, D. Banks. "Learning to Use a Ten in Subbtraction". Elementary School Journal. 47: 461-466. April 1947. Wimmer, George W. "The Dilemma of Homework". NEA Journal. 36: 633. December 1947. Young, May I. "We Must First Look at the Child". Childhood Education. 26: 244-246, 278. February 1950. Zirbes, Laura. "Materials for Instruction in Arithmetic". Childhood Edu- cation. 24: 372-373. April 1948. Professional Magazines The following magazines carry articles frequently on arithmetic in the elementary school: Childhood Education. Association for Childhood Education International, 1200 Fifteenth Street, N. W., Washington 5, D. C. $4.50. (Subscription is included in membership dues of $6.00.) Journal of Educational Research. Dembar Publications, Inc., 114 South Carroll Street, Madison 3, Wisconsin. $4.00. NEA Journal. National Education Association, 1201 Sixteenth Street, N. W.. Washington 6, D. C. (Subscription is included in membership dues.) The Elementary School Journal. The University of Chicago Press, 5750 Ellis Avenue, Chicago 37, Illinois. $4.50. Mathematics In the Secondary School Ninth Year (Course A or B Required) Course A: General Mathematics. This course will likely be taken by students who do not intend to pursue mathematics much further. Hence this course should give students a good check-up on what they have studied in mathematics. It should also give them some insight into the more powerful forms of analysis as found in algebra through the use of the equation and the formula, and in geometry through measuring and construction work establishing relationship of areas, volumes, etc. The following topics are suggested for work in this year ; teach- er and pupils should select topics and add new topics in keeping with needs and interests. 1. Inventory test on concepts, skills, and principles developed in years VII and VIII. Review and extension of concepts, skills, and principles relating to integers, decimals, frac- tions, fundamental operations, estimating and checking, denominate numbers, percentage, tables of squares, square roots, compound interest and other tables dealing with money matters. 2. Development and use of formulas for areas, perimeters, and volumes of such figures as pyramids and cones. Take a cone and a cylinder of the same base and altitude, fill the cylinder with sand and then see how many times this will fill the cone. Do the same for a pyramid and a prism. 3. Review of the basic concepts relating to the angle, triangle, and circle, making geometric constructions. 4. Constructions : triangle given sas, asa, sss ; triangle similar to a given triangle ; find the center of circle when an arc is given ; angles of 30, 45, and 60 degrees. 5. Informal proofs of elementary theorems by measurement and construction. Typical exercises : a. The sum of the angles of a triangle equals 180 degrees. Cut out paper or cardboard triangles and cut off angles and place together to form a straight line. Mathematics in the Public Schools 51 b. The Pythagorean theorem. Employ usual construction of right triangle with squares on hypotenuse and legs divided into units of equal measure. Multiply both sides of the equation c- = a- -f b- by pi over 4 and in- terpret the result: The circle on the hypotenuse as di- ameter is equal to the sum of the two circles on the legs as diameters. 6. Graphs: Draw circle graphs; read and construct more advanced bar and line graphs based on data of significance to pupils ; use graphs in study of linear equations.^ 7. Story of the development of geometry including indirect measurement.- 8. Use of home-made instruments for measuring, such as transit level, hypsometer, proportional compasses, adding machine. 9. Story of algebraic symbolism.^ 10. Evaluation of formulas. Example : The destructive power of an automobile is given by the formula K = WV-, where W = weight of car and V = speed of car. Let W = 3000 lbs. V = 30 miles per hour V = 60 miles per hour V = 90 miles per hour Note that in doubling the speed the destructive power is four times as great, and that trebling the speed makes the destructive power of the car nine times as great. 11. Review and extension of concepts of formula, equation, similar terms, ratio and proportion. Example: the distance a car will run after the brakes are applied is proportional to the square of the velocity, or as expressed in a formula D = .07V-, where D = distance in feet, and V = miles per hour. Plot this equation on graph- ical paper. 12. Develop concepts of low powers (a-, a^), positive and nega- tive numbers as found in current magazines. 13. Fundamental skills, techniques and principles for dealing with the equation and the formula : a. Four fundamental operations involving positive and negative numbers, using numerals and letters to repre- sent numbers. ^See Graphs. How to Make and Use Them, by Arkin. "See History of Mathematics by Sanford. ^See History of Mathematics Notation, ^'ol. 1 by Cajori. 52 Mathematics in the Public Schools b. Squaring a binomial: (a + b)-; (a + b) (a — b) ; etc. c. Equations with simple algebraic fractions: x+2__ Q — • f etc. 14. Establishment of relationships between table, formula, equation, and graph. See examples in 10 and 11 above. 15. Solution of first degree equations involving one unknown quantity using the following axioms : If equals are divided by equals the quotients are equal; if equals are multiplied by equals the products are equal; if equals are subtracted from equals the differences are equal; if equals are added to equals the sums are equal. 16. Translation of statements into formulas. Example: The arm strength of boys up to age 17 is determined by adding the pull-ups (chins) to the push-ups (as on parallel bars) and multiplying this sum by the result of taking 1/10 of the boy's weight in pounds plus the height in inches minus 60. In formula this is A = (P + p) (w/10 + h — 60). In physical education this affords a means of classifying boys as to athletic powers. 17. Use of equations in the solution of problems arising in home, business, shop, science, etc. 18. Problems involving equations applied to measurements of perimeters, areas of squares and rectangles, triangles, cir- cles, volumes and surfaces of the rectangular prism, tri- angular prism, cylinder, etc. 19. Scale drawing. Making maps of school grounds; making maps of the State showing by counties, population, pro- duction of agriculture, production in industry, and school population ; learning to read blueprints. . 20. Oral and written reports on special topics chosen by pupils because of individual interests. Course B: First Year Algebra. This course is designed for those students who should begin the study of algebra in the first year of high school. For those pupils who take Course A in the first year, Course B may be offer- ed as an elective in the second year. 1. Inventory test on skills and principles related to equations and formulas. Review and extension of concepts of co- efficient, equation, formula, similar terms. Using film striv as an aid to teaching 10. 11. story of the development of algebra.^ Practice in the expression of quantitative statements in algebraic shorthand. Evaluation of formulas and algebraic expressions. Development of concepts and skills in the use of base, ex- ponent, power, and laws of exponents. Interpretation and application of common formulas relat- ing to areas, volumes, etc. Making of formulas based on verbal statements. Making of formulas based on tables. Solution of simple linear equations involving only one un- known, using the following axioms : If equals are divided by equals the quotients are equal ; if equals are multiplied by equals the products are equal; if equals are subtracted from equals the differences are equal; if equals are added to equals the sums are equal. Use of equations to solve problems involving percent, simple interest, other business problems of discount, commission, profit and loss, taxation. Review and extension of ability to make and interpret sta- tistical graphs (bar, line, circle, picture, map etc.) ; graphs of formulas. iSee The Hisotry of Mathematics by Sanfnrd. 54 Mathematics in the Public Schools 12. Graphical and algebraic solutions of pairs of first degree equations in two unknowns. 13. Problems involving two first degree equations in two un- knowns. 14. Special products and factoring. 15. Fractions and fractional equations including decimals. 16. Relationships between table, formula, equation, and graph. 17. Oral and written reports on special topics. MATHEMATICS FOR YEARS X— XII (Elective) The courses in mathematics for Years X — XII are designed to meet two specific needs : 1. To equip students who are not going to college to meet prac- tical everyday problems of a quantitative type. For this group two courses are offered, (1) a course in business arithmetic with the emphasis upon arithmetical and graph- ical study of economic, social, and industrial problems, and (2) a course in basic mathematics covering the practical applications of arithmetic, informal geometry, algebra and trigonometry. 2. To prepare students to take mathematics in college and also for those who desire to continue the study of mathe- matics in high school, and therefore want the more formal type of courses. For this group any or all of the courses outlined may be offered. Students preparing for college should take at least second year algebra, plane geometry and if possible the course in advanced algebra. Students preparing to enter engineering or technical schools should add to their electives solid geometry and trigonometry. Tenth Year Second Year Algebra (Elective), 1 year, 1. Inventory test of skills and principles studied in first year algebra. Review of fundamental operations, with alge- braic expressions as needed. Emphasis on signed numbers. 2. Review the techniques necessary for the evaluation and solution of formulas and equations of the first degree. Em- phasis on checking. .. Mathematics in the Public Schools 55 3. Fractions and fractional equations, including decimals. Problems on motion, work, etc. 4. Ratio, proportion, and variation with application to indirect measurement, blue-prints, recipes, enlarging maps, etc. 5. The study of the four-fold relationships in connection with tables, formulas, graphs, and equations. Use squared paper. 6. Fundamental laws of exponents, square root, radicals, frac- tional exponents, negative exponents, imaginary numbers, equations involving fractional exponents, use of tables of logarithms. 7. Quadratic equations in one unknown. Graph of quadratic function, showing maximum and minimum values, with applications. 8. Simultaneous equations : both of first degree ; one first and one second degree. Problems related to motion, money, space, etc. 9. Arithmetic and geometric series with problems. 10. Story of algebra.^ 11. Oral and written reports on topics chosen by puipls because of special interests. Eleventh Year Course A: Plane Geometry (Elective), 1 year. 1. Inventory test for concepts, skills, and clarity of expres- sion, followed by review of construction work using ruler, compasses, and protractor; and re-establishment of con- cepts and principles relating to basic figures. 2. Establishment of the working principles consisting of defi- nitions, postulates, axioms, and undefined terms. 3. Development of an understanding of what is meant by a proof, different types of proof, both non-mathematical and mathematical. 4. Development of skill in demonstration of theorems. 5. Facts and principles related to such topics as congruence, similarity, concurrence, parallelism, indirect measurement, mensuration, loci, and construction. 6. A wide range of applications for the method of postula- tional thinking, as in athletic contests, court room, etc. iSee History of Mathematics by Sanford. Geometry is related to Safety Education 7. Construction and use of models and instruments, such as pantograph, proportional compasses, etc. 8. Applications of geometry to designing, art, architecture^ crafts, mechanical drawing, map making, etc. 9. Story of geometry ; builders of geometry ; discussion of some features of modern geometry, such as Theorem of Menelaus, nine point circle theorem, etc. 10. Oral and written reports on topics chosen by pupils because of special interests. Course B: Plane and Solid Geometry, Integrated (Elective) 1 year» 1. Basic propositions with lines, planes, and solids. (Standard lists of propositions as found in reports of Na- tional Commission on Mathematical Requirements, College Entrance Board, Regents of the University of the State of New York, etc.) 2. Free hand sketches of two- and three-dimensional figures and constructions using compass, ruler, and protractor. 3. Student-made models and instruments. 4. Correlation of work in geometry through mensuration to- arithmetic and algebra. 5. Development of greater skill in using deductive reasoning" as related to geometric ideas and as found in life situations. 6. Applications of geometry to designing, art, architecture,, crafts, mechanical drawing, map making, etc. Mathematics in the Public Schools 57 7. Story of geometry; builders of geometry; discussion of some features of modern geometry such as the nine point circle theorem, Theorem of Menelaus, etc. 8. Oral and written reports on topics chosen by pupils because of special interests. Twelfth Year Course A: Business Arithmetic (Elective) 1 year. (This course to be taken in either the Eleventh or Twelfth Year.) 1. Inventory test for concepts, skills and accuracy followed by a restudy of the fundamental processes with common and decimal fractions and per cents. Use problem situations. 2. Short-cut methods of computation. 3. Understanding and use of elementary statistical methods: a. "Gallup Poll," athletic records, reports of insurance companies, etc. b. Some technical skill in making accurate charts and graphs to express statistical data. Examples : Popula- tion (census figures), labor, production. State and county maps showing population, production, etc. c. Frequency tables, scatter diagrams, arithmetic average, median. 4. Use of tables : square root, logarithm, interest, investment, insurance. 5. Investments, such as building and loan, stocks, bonds, mort- gages, banking procedures, cost of owning a home, annui- ties, other forms of thrift. 6. Advantages and disadvantages of installment buying. 7. Family budgets. 8. Base of taxation and bond issues. Common forms of taxes. 9. Demonstration of use of computing machines. 10. Construction and use of index numbers (wages, cost of living, etc.) 11. Study of common forms of insurance. 12. National policies relating to price fixing, crop control, tar- iffs, social security, etc. 13. Measurement of lengths, surfaces, and volumes. 58 Mathematics in the Public Schools 14. Reports on special topics related to the work of the course. (There should be flexibility in this course both as to the content and the method of development of the course. See also Publication No. 267, Curriculum Guide and Courses of Study in Business Education for North Carolina, p. 54, "Business Mathematics".) Course B: Algebra (Elective) One-Half Year. 1. Review and extension of basic concepts, principles and skills. 2. Linear and quadratic functions and equations. Use of graphs. 3. Formulation and solution of problems of commerce, indus- try, science, home, and community. 4. Exponents, radicals, and radical equations, with prob- lems. 5. Arithmetic, geometric, and binomial series, with problems. 6. Solutions of simultaneous equations of first and second de- gree. 7. Establishment of the four-fold relationship between formu- la, graph, equation, and table. 8. Story of the development of algebra,^ reports on special topics. 9. Reports on symbols, fractions, computation, Vieta, Des- cartes, Newton, Records, etc- Course C: Solid Geometry (Elective) One-Half Year. 1. The basic propositions with the relation of lines and planes in space, and properties and mensuration of solids. 2. Free-hand sketching from student-made models, both solid and wire-outline ; tracing of shadows of models. 3. Basic formulas for mensuration: Lengths, areas, and vol- umes. 4. Correlation of solid geometry through mensuration with arithmetic and algebra. 5. Exhibit simple theorems from Modern Geometry. Exam- ples : Nine point circle theorem, Pascal's theorem on hexa- gons, etc.^ 6. Oral and written reports on topics chosen by pupils because of special interests. iSee The History of Mathematics by Sanford. ^Bell. Men of Mathematics; Sanfuid, (ii> cit. Daiitzig ; Xumber : Thte Language of Science. =Sanfnrd. op cit. : Cajoii. The History of Mathematics. Mathematics in the Public Schools 59 Course D: Trigonometry Elective) One-Half Year. 1. Six trigonometric functions of the general angle. 2. Reduction formulas. 3. Basic identities. 4. Addition formulas for sine, cosine, and tangent. 5. Radian measure. 6. Double-angle and half-angle formulas. 7. Solution of general triangle. 8. Practical problems on heights and distances, and elementary problems in air and sea navigation. 9. Field work using home-made transit and plane table. 10. Make and use slide rule. 11. Story of the development of Trigonometry. Course E: Basic Mathematics (Elective). (This course to be taken in either the Eleventh or Twelfth Year.) This course was devised during the war period to provide a refresher course for students prior to entrance into the Armed Services. It is believed that the course still has validity. The general purpose of the course is to provide, in the last year or two of secondary school, a study that will have the widest pos- sible usefulness. It should provide a review of knowledges and skills with special emphasis on the practical applications of math- ematics. It should include: Arithmetic A. Fundamental Operations. 1. Inventory test to determine which basic skills and tech- niques need emphasis. 2. Fundamental operations using whole numbers. 3. Reading and writing whole numbers. 4. Denominate numbers, 5. Making and solving problems related to every day affairs. B. Common Fractions. 1. Multiplication and division. 2. Addition and subtraction, mixed numbers. 3. Reduction of fractions, cancellation, 4. Making and solving problems involving fractions. 60 Mathematics in the Public Schools C. Decimal Fractions. 1. Reading and writing decimals. 2. Adding and subtracting. 3. Multiplying and dividing. 4. Changing common fractions to decimal fractions. Changing decimal fractions to common fractions. 5. Problems involving decimals, as money matters, measur- ing. D. Percentage. 1. Reading and writing per cents. 2. Finding a per cent of a number of quantity. 3. Finding what per cent one number is of another number. 4. Finding a number when a per cent of the number is given (Concrete illustrations of 2, 3, 4 above). 5. Rounding off numbers, approximations. 6. Problems involving the use of per cent and decimals as in : discount, commission, interest, other money matters, meas- uring. Informal Geometry A. Direct Measurement. 1. Instruments: ruler, compass, protractor, plane table, steel tape. 2. Measuring lengths, distances and heights. 3. Estimating and checking by measuring, using approxima- tion methods. B. Scale Drawing. 1. Making and reading maps, enlarging maps. 2. Reading blue prints, using plane table. C. Circular Measure. 1. Terms related to circle: radius, diameter, chord, circum- ference, tangent, etc. 2. Angle measures: degree, minute, radian, inscribed and circumscribed polygons, using protractor. 3. Formulas growing out of circular measure. Mathematics in the Public Schools 61 D. Triangle. Terms related to the triangle : base, altitude, median, right angle triangle, acute angle triangle, obtuse angle triangle, circumcenter, in-center. Construction of triangles. Angle sum, area, informal development of formulas. E. Area, Volumes, Weights. 1. Standard units of measure. 2. Relationships between units of measure; examples: ft. of water = 62.5 lbs., 231 cu. in. = 1 gallon. 3. Making estimates and checking by measuring. 4. Developing formulas which grow out of above work F. Symmetry, Congruence, and Similarity. 1. Drawing figures. 2. Collecting illustrations. Algebra A. Literal Numbers. 1. Fundamental operations. 1 cu. A class learns to make and use geometrical figures '^' OXQAIQO- i%*1 «ssiu^" 62 Mathematics in the Public Schools 2. Evaluation of formulas. Example : A = Trr-. Let r = 3 inches, then let r = 6 inches ; compare results. 3. Using monomial and binomial expressions with letters representing numbers. B. Graphs. 1. Read and make line, bar and circle graphs. 2. Using data related to motion, distance, areas, weight, money matters, etc. C. Equations. 1. Concrete illustrations of the equation concept. 2. Solving and evaluating equations. ^ _ 3. Permissible operations on the equation. 4. Special products, factors. 5. Signed numbers. 6. Short-cut methods. D. Exponents and Radicals. 1. Rules for operation. 2. Square and square root tables. Trigonometry 1. Instruments : ruler, compass, protractor. 2. Measuring angles, using proctractor, hysometer. 3. Learning and using the trigonometric ratios. 4. Fundamental identities, hypotenuse rule. 5. Computing heights and distances, using tables of loga- rithms, and natural functions. In the course outlined above, the approximate percentages of time to be devoted to different phases of mathematics are sug- gested as follows : Arithmetic 40% Algebra 30% Geometry 20% Trigonometry 10% The scope of the foregoing outline of content provides for a two-semester course. Adaptations should be made in the light of individual needs of the pupils enrolled. In the development of this course, it is recommended that major emphasis be placed upon industrial applications. Thoroughness and mastery should be major objectives rather than determination to have each pupil complete a pre-determined schedule of assignments. Mathematics in the Public Schools 63 Texts for the Course. There is no one basal book for this course, but there are several which could be used. The following are suggested : Hart. Basic Mathematics. (Brief edition). Heath. Hart. Basic Mathematics. (Complete edition). Heath. Betz. Basic Mathematics. Ginn. Although these books are not at the present time on the State adopted list, the Division of Textbooks has a number of copies on hand. Some of the schools probably have copies, since this course was offered during the war. Work in the course should be supplemented by the following books which are on the State supplementary list : Hausle, Braverman, Eisner, Peters. Mathematics You Need. Van Nostrand. Young. Rural Arithmetic. Bruce. Lasley and Mudd. The New Applied Mathematics. Prentice- Hall. Schorling, Clark and Langford. Mathematics for the Consum- er. World. Mathematics and Natural Science In the May 1950 issue of School Science and Mathematics there is an article entitled "A Unified and Continuous Program in Mathematics" by Harold P. Fawcett of Ohio State University. In one section of this article he says : "Mathematics is not a set of isolated and unrelated topics. It is a system of ideas unified by a number of fundamental concepts which grow in meaning and significance for the student as his study of mathematics con- tinues. To develop insight and understanding concerning the nature of these concepts is the major responsibility of the mathe- matics teachers on all levels, while it is also a part of the general responsibility of all teachers regardless of the area with which they may be associated." One of the fundamental concepts that he discussed was that of measurement. Since it is only by measurement that the forces of nature can be understood and brought under control, then this concept is particularly apropos to the work of science and mathematics teachers. To measure is to know and to have in- sight, and the story of measurement is the story of man search- ing for standards that do not vary. In any program of general 64 Mathematics in the Public Schools education we must provide experiences by means of which the students will develop insights and understandings related to this concept. In our natural science we can provide and do provide some of these experiences. Another concept which is discussed by Dr. Fawcett is that of relationship. As a student progresses from the first grade through the twelfth grade, he should become aware of the fact that he is dealing with well defined concepts and the nature of the relationships between them. The mathematics teacher then had the responsibility of providing experiences through which the student will develop understandings of this important con- cept. One of these understandings will be that formulas are generalizations of relationships. Again, in the natural science courses wonderful opportunities are offered for these experiences. According to Professor E. R. Hedrick, "'We seek in modern science to express as best we may the observed facts of nature in the form of the relationships between quantities." The following problems from the various science courses offer- ed in high school are given so that mathematics teachers will better understand the close relationship between these two fields of work. By seeing the fundamental operations involved in these problems and by observing the various forms of measurement and relationships, it is felt that not only will the mathematics teachers be benefitted but also the science teachers. Typical problems in the field of science : 1. At a faucet supplied from a reservoir the pressure is 26 pounds per square inch. How high is the water level in the reservoir above the faucet? Formula : Pressure = height times density P rr: h X d P = 26 pounds per square inch or 3744 pounds per square foot. D = 62.4 pounds per cubic foot To solve substitute the values in the above formula. 2. The small piston of a hydraulic press has an area of 2 square inches ; the large piston, an area of 50 square inches. How large a weight in the large cylinder can be held up by a force of 10 pounds acting on the small piston? Mathematics in the Public Schools 65 Formula : force on large piston area of large piston force on small piston area of small piston F A f = 10 pounds A = 50 sq. in. F = ? pounds a = 2 sq. in. To solve substitute the values in the above formula. A piece of metal weighs 100 grams in air and 88 grams in water. What is its specific gravity? Formula : Specific gravity = Weight in air weight in air — weight in water weight in air =100 grams weight in water = 88 grams To solve substitute the values in the above formula. Starting from rest, an object falls freely for 3 seconds. How far does it fall ? Formula : distance = one half the acceleration due to grav- ity times the time squared S = i/2gT- ^ =32 feet per second T =3 seconds To solve substitute the values in the above formula. An automobile weighing 2500 pounds is traveling at the rate of 30 miles per hour (44 feet per second). What is its kinetic energy? Kinetic energy = weight X velocity squared two X acceleration due to gravity K.E. = wv^ ~"2g w = 2500 pounds V =44 feet per second g =32 feet per second per second To solve substitute the values in the above formula. 66 Mathematics in the Public Schools 6. A certain weight of gas has a volume of 1200 cubic centi- meters at a pressure of 500 grams per square centimeter and a temperature of 21 degrees centigrade. What is its volume at a pressure of 1500 grams per square centimeter and a temperature of 315 degrees centigrade? Formula = Ps Vo Pj Vi T2 Ti Pi = 500 grams per square centimeter P2 = 1500 grams per square centimeter Ti = 21°C or 294°A T2 = 315°C or 588°A Vi = 1200 cubic centimeters To solve substitute the values in the above formula. What is the combined resistance of a 9-ohm coil and an 18- ohm coil connected in parallel? Formula: 1/R = 1/Ri + l/Ro Ri =9 ohms Ro = 18 ohms To solve substitute the values in the above formula. A paint manufacturer wishes to know how many pounds of lead nitrate, Pb(N03)25 he will have to add in a water solu- tion of sodium chromate, Na2Cr04, to make 25 pounds of the insoluble chrome-yellow pigment, lead chromate, PbCr04. X 25 Pb(N03)2 + Na,Cr04 — > 2NaN03 + PbCr04\ 331 323 X 25 331 323 A solution of the above equation will give the answer. Mathematics in the Public Schools 67 10. How many millimeters of 0.4 normal potassium hydroxide solution, KOH, will neutralize 100 millimeters of 0.05 nor- mal phosphoric acid, H3PO4? Formula : millimeters of acid normality of base millimeters of base normality of acid 100 0.4 X 0.05 A solution of the above equation will give the answer. BIBLIOGRAPHY A. Yearbooks of the National Council of Teachers of Mathematics. 15th Yearbook. The Place of Mathematics in Secondary Education. 1940. 16th Yearbook. Arithmetic in General Educatiori. 1941. 18th Yearbook. Multi-sensory Aids in the Teaching of Mathematics. 1945. 19th Yearbook. Surveying Instruments. 1947. 20th Yearbook. The Metric System of Weights and Measures. 1948. These books can be obtained from the Bureau of Publications, Teacliers College, Columbia University, New York City, New York, for $3.00 each. B. Mathematics Magazines. 1. '"The Mathematics Teacher", 525 West 120th Street, New York, New York. $2.00. 2. "School Science and Mathematics", 450 Ahnaip Street, Menasha, Wis- consin. $2.50. 3. "Balance Sheet", Southwestern Publishing Company, 245 Fifth Avenue, New York, New York. (Free) C. Books ox Vaeiot's Phases of Mathematics. 1. Anderson, R. W. Ro^nping Through Mathematics. A. Knopf. 1947. $2.50. 2. Arkin. Graphs: How to Make and Use Them. Harper. 1936. $3.00. 3. Bell, E. T. Men of Mathematics. Simon and Schuster. 1937. $5.00. 4. Bell, E. T. The Development of Mathematics. McGraw-Hill. 1940. $5.00. 5. Bendick, Jeanne. How Much and How Many: The Story of Weights and Measures. Whittlesey House, McGraw-Hill. 1947. $2.00. 6. Boyer, Lee Emerson. Introduction to Mathematics for Teachers. Holt. 1945. $3.40. 7. Breslich, E. R. Problems in Teaching Secondary School Mathematics. Chicago University Press. 1938. $3.00. 8. Breslich, E. R. Techniques of Teaching Secondary School Mathemat- ics. Chicago University Press. 1934. $2.00. 68 Mathematics in the Public Schools 9. Cajori, Florian. History of Mathematics. Macmillan. 1931. $4.00. 10. Cooley and others. Introduction to Mathematics. Houghton. 1937. $3.25. 11. Dantzig, T. Number, the Language of Science. Macmillan. 1939. $3.00. 12. Gamow, George. One, Tivo, Three . . . Infinity. Viking Press. 1948. $4.75. 13. Harris, Charles Overton. Slide Rule Simplified. American Technical Society, Chicago. 1943. $2.75. 14. Hassler and Smith. The Teaching of Secondary Mathematics. Mac- millan. 1936. $2.75. 15. Hogben, Lancelot. Mathematics for the Million. W. W. Norton. 1940. $4.50. 16. Hooper, A. Makers of Mathematics. Random House. 1947. $3.75. 17. Ivins, William. Art ajid Geometry. Harvard University Press. 1946. $3.00. 18. Kattsoff, L. O. A Philosophy of Mathematics. Iowa State College Press. 1948. $5.00. 19. Mathematics at Work,. A Report of the Mathematics Institute, W. W. Rankin, Duke University, Durham, North Carolina. 1948. Free. 20. Mathematics at Work. A Report of the Mathematics Institute, W. W. Rankin, Duke University, Durham, North Carolina. 1949. $2.50. 21. Progressive Education Association. Mathematics in General Educa- tion. Appleton. 1940. $2.75. 22. Sanford, Vera. Short History of Mathematics. Houghton. 1930. $3.25. 23. Schaff, W. L. (Ed.) Mathematics: Our Great Heritage. Harper. 1948. $3.50. 24. Schillinger, Joseph. ..The Mathematical Bases of the Arts. New York Philosophical Library. 1948. 25. Weyl, Herman. Philosophy of Mathematics and Natural Science. Princeton University Press. 1949. D. Booklets on Audio-Visl'al Aids. 1. Mathematics: Visual and Teaching Aids. New Jersey State Teachers College, Upper Montclair, New Jersey, 1947. Mimeographed (30 pages). 75(^. 2. Exploring the Teaching of Mathematics: Multi-setisory Aids to Learn- ing Mathematics. Cooperative Report of Third Summer Workshop, College of Education. Ohio State University. 1947. Mimeographed (53 pages). 3. Bihliography of Mathematics Films and Filmstrips. Compiled by D. A. Johnson and H. W. Syer. Spring 1949. Mimeographed (13 pages). Available from Henry W. Syer, Boston University, Boston, Massachu- setts. 4. BiUiography of Non-projected Multi-sensory Aids for the Teaching of Secondary Mathematics. Compiled by Bernard M. Singer. 1948. Mim- eographed (32 pages). Available from H. W. Syer, Boston University. Mathematics in the Public Schools 69 E. Mathematical Recrela.tions. 1. Ball, Walter. Mathematical Recreations and Essays. Macmillan. 1939. $2.50. 2. Freeman, Mae (Blacher) and Freeman, Ira M. Fun With Figures. Random House. 1946. $1.25. 3. Jones, Samuel I. Mathematical Wrinkles. Nashville, Tennessee, S. I. Jones. 1940. $3.00. 4. Jones, Samuel I. Mathematical Clubs and Recreations. Nashville, Tennessee. S. I. Jones. 1940. 5. Nott-Smith, Geoffrey. Mathematical Puzzles for Begimiers and En- thusiasts. Philadelphia. Blakiston. 1946. 6. Northrop, Eugene Purdy. Riddles in Mathematics, A Book of Para- doxes. Van Nostrand. 1944. $3.00. L