Free Schooling Lesson Search! Enter Your Topic Below:

Chapter - 05

“… And Arithmetic”

One of the glorious attributes of the young is its ability to improve suddenly and almost mysteriously. The fact remains, however, that good habits of study, a sound foundation for future learning, and a pattern of maturity and responsibility are built with more permanency, and less drudgery in the building, during the years corresponding to the fourth grade through seventh than at any other time in the child's life. These are the choice-and-all-too-few-years of life when interest is so consuming that it has to be curbed, and energy for learning and improvement so abundant as to need only direction into the proper channels.

The world of arithmetic offers one such channel. And although in the field of arithmetic the parent will be moving to a large degree on new or forgotten paths, there are some practices which can profitably be encouraged at home. Of course, to most parents the whole field of mathematics narrows to knowing how to find out how many hens can be accommodated by a coop eight feet by ten, what the return will be on two thousand dollars at four and one-half per cent, how many square feet of floor space there are in the basement, or what the size of the mortgage will be when three children have finished their formal schooling. Be that as it may, the home can still provide direction in areas where careless mistakes are made, where word problems confuse, and where the first real work in thinking—which is what mathematics is—brings on dislike for and revolt against, and makes math "the hated subject" or produces the fatal attitude, "I just can't seem to get it."

Perhaps the subject of arithmetic, more than any other study in elementary school, introduces your child gradually and consistently to the habit of thinking. At the same time arithmetic is the subject which requires more individual attention than any other subject if a foundation of precision, thoroughness, and speed is to be built.

The beginning of precision, which will be of continued importance, can be emphasized in three simple areas. These require little more than suggestion and observation by the parent. They are: (1) writing numbers correctly, (2) copying problems correctly, and the correct problems (that is, the problems assigned), and (3) demanding exactness for the relative positions of numbers in problems. The correct form of the number in arithmetic is the beginning of neatness and a sense of exactness which all later mathematics will require. Numbers should be closed, angles definite, and stems straight. Check against exaggerated slants, and artistic-railed endings. Accuracy in copying problems is an important exercise and can be checked at home. There is something rather fatal and futile about having to pass back failing papers to bright eighth and ninth graders who fail because they copy the wrong problems and work them correctly, or copy the right problems incorrectly. Home practice in precision can also deal successfully in the (sometimes tragically overlooked) position of numbers in relationship to each other in problems. In the problems which follow, the same amount of energy was expended but different results achieved.

Thoroughness and speed, although at first sight appearing to counteract each other, must go together in the basic performances of handling numbers. Certain fundamentals must be learned so thoroughly that instant performance results. Four from six must leave two without counting, pecking with pencil, or moving fingers. Hundreds of operations must be so thoroughly memorized as to become second nature. For example, the possible combinations of the numbers 6 and 3 would require hours to write, and yet they must be ready for use with speed which is simple action rather than thought. Quick, short, easy-to-think-of exercises are better than long, hard ones. One mother taking 3, 6, and 9 worked out 80 combinations of multiplication, addition, and subtraction with her 10-year-old daughter, driving to and from music lessons and taught her daughter to enjoy arithmetic, make mental reaction to simple operations automatic; and most important of all — she brought imagination and simplicity to the most exacting of all sciences.

Exercises for thoroughness and speed in the basic performances are available everywhere for every interested parent. One of the brilliant naval officers who helped build the first atomic submarine amusingly says

Which of the papers below show that somebody really cares?

that his understanding and love of mathematics grew out of an imaginary field of cows. It seems that his father built all kinds of exciting arithmetic problems around cows seen on Sunday afternoon drives.

There are seven cows in the field. Two are lying down. How many footprints do the others make as they move across the field?

There are nine cows. How many ears do they have?

There are eleven cows. If each gives three gallons of milk a day, how much milk does the farmer get?

There are nine cows and they all have single calves except two, and those two have twins. How many calves and cows are there in the field?

Here are the practices for reviewing the tables of multiplication, learning the simple facts, mechanizing operations with a view to accuracy and speed, and beginning an alertness for key words in verbal problems. If cows are scarce, there are plenty of light poles, trees in rows, houses to be sold or rented, and traffic lights which change twenty times in sixty minutes and in three colors. For practice in fundamentals the field is unlimited, and one parent has gone so far as to ask, "What did I do with my children before?"

Of the many recent investigations and studies in the whole field of elementary school mathematics, one seems particularly applicable for giving the child help at home. Out of thousands of arithmetic papers, from fourth to approximately tenth grade level, the most commonly made errors in addition, subtraction, multiplication, and division have been compiled. The investigation also revealed that these same operations generally take a longer pause on the part of the student. They might be profitably used as a source for review at home.

The most errors in addition seem to occur in the following operations:
            *9 + 7                                      6 + 8                            8 + 7
            6 + 9                                        8+5                              9+5
            5 + 9                                        6 + 7                            5 + 6

In subtraction the following operations take the longest pauses for execution and present difficulty for the greatest number of students:
            15-9                                         15-8                             11-9
            9-9                                           15-6                             13-5
            16 — 7                                    1 — 0                          6 — 0
                                                            5-5
*AH troublesome combinations are used with permission of the Oxford University Press.
Attraction seems to be the fatal element in some number operations such as 5 — 5,9 — 9,15 — 15, etc. The answer given is frequently the same number repeated 5 — 5 = 5, 9 — 9 = 9, etc. In operations where o is subtracted, there is almost a universal desire to subtract something, consequently: 1 — o = o, 6 — o = o.

In multiplication, although the ninth table contains the most errors when the tables are written out, single operations with 7 and 8 involved prove to be equally as troublesome. Errors are frequent in the following operations:

In multiplication, as in subtraction, attraction seems to be the first reaction, and is frequently substituted for thought; thus: 3 × 3 = 6, 4 × 4 = 8, 5 × 5 = 10.

In division, most errors occur with the zero, and after the zero, nine appears as the second greatest trouble maker. Attraction is less frequent than in multiplication, and occurs most often in the cases of 9 -÷- 9, and 6÷6. The most common errors in division are:

Perhaps an interesting home project would be a test using these most commonly made errors.

Other mistakes which occur frequently are inversion of numbers—36 as 63, 24 as 42, 19 as 91—and change of operation where one step of addition slips into multiplication:

It is evident from the study above that 9 is the most troublesome number. The child can be warned to watch closely when nine appears. It was also learned that individuals sometimes have individual trouble makers. One boy could not handle five successfully; another made more mistakes with the use of three. It is within the power of any parent to observe a numbers-blindspot if one exists. And, of course, the cure for it is practiced attentiveness.

After the basic performances of addition, subtraction, multiplication, and division, interested parents should concern themselves with their child's reaction to word problems. Of course, the introduction to word problems can be so skillfully accomplished as to be one with the simple operations. The nine cows, the trees in rows, three people taking six grapes each from a bowl which contained 36 grapes—all these introduce word-problems, and will save the child from much trouble later.

For children do have a great deal of trouble with word-problems, and some are so completely lost that they attempt to save their pride by becoming sour on all arithmetic. Why is there more difficulty with word-problems than any other mathematical problems? If this question can be answered for the child, he will have been given some help. The answer lies in the fact that word problems involve more than arithmetical knowledge. The following statement puts it very well: "The pupil must not only be able to understand the vocabulary of the statements, but he must also be able to visualize the situation that is presented, and to sense the relationships among the quantitative elements that are involved. In addition to this he must be able to perform the necessary computations to find the answer to the problem."*

In language which the child can understand this means: (1) The meaning of the words used must be understood and intensively read. (2) The mind's eye must be used to picture the situation. (3) From the picture of the situation the operational process (addition, subtraction, division, or a combination) must be thought out. And (4) The operation must be done correctly and in the proper order.

In word problems there are no signs to indicate add or subtract, and the challenge of manipulating numbers and words sometimes proves too great an obstacle. Children become confused and afraid, and complaint replaces thought:

"I don't understand them.

I don't know what to do.

* Bueckner and Grossnickle, Making Arithmetic Meaningful, page 492.

It doesn't tell whether to add, subtract, multiply, or divide.

I don't know how many of the numbers in the sentences to use.

I don't know what some of the words mean."*

How can this bugbear of doubt and confusion be turned into a positive approach and real thought? For the five areas of confusion let us substitute five constructive steps of attack: (1) The problem does make sense if read properly. Intensive reading (locating information, noting key words, knowing the meaning of basic arithmetical words) will answer two important questions: What is given? What is needed? (2) Always reread the problem, and put it in your own words for clearer understanding and meaning. (3) Organize the information given into steps for finding the solution. What do I put down first? Next? etc.? (4) What operation is to be used? Addition and multiplication put groups together. Subtraction and division take groups apart. (5) Decide, after arriving at the answer, if the answer is reasonable. Check its correctness against previously known facts if possible.

The mathematical aptitude and achievement tests required of the pupils entering independent secondary schools and some public high schools are indicative of the amount of work and the standard of work desired in the field of mathematics. Besides the indications already mentioned, there are other signs which bid fair to increase the demands upon the elementary school child in the realm of mathematics. It seems reasonable to assume that the leisurely pace of six years for twelve lines of multiplication is doomed. The trend is clearly shown by a program for elementary school mathematics recommended by the Commission of Mathematics of the College Entrance Examination Board. Referring to the program to be completed in elementary school (by the end of the eighth grade), the Commission says: "We are convinced that it can be mastered by all college-bound students." Here is an implication also that has been too long overlooked in education—there is a significant element of college preparation in elementary school; it is not the last two years of high school as the American public would like to believe. Here then is the recommended program for elementary school mathematical goals:

"ARITHMETIC Fundamental operations and numeration:

Mastery of the four fundamental operations with whole numbers and fractions, written in decimal notation and in the common notation used for fractions.

*Milo K. Blecha, "Helping Children Understand Verbal Problems," The Arithmetic Teacher, Vol. VI, Number 2.

This includes skill in the operations at adult level (i.e., adequate for ordinary life situations) and an understanding of the rationale of the computational processes. The understanding of a place system of numeration with special reference to the decimal system and the study of other bases, particularly the binary system. Ability to handle very large numbers (greater than 1,000,000) and very small numbers (less than one ten-thousandth). The nature and use of an arithmetic mean. In addition, a knowledge of square root and the ability to find approximate values of square roots of whole numbers is desirable. (The process of division and averaging the divisor and quotient—often called Newton's Method—is suggested.) Ratio:

The understanding of ratio as used in comparing sizes of quantities of like kind, in proportions, and in making scale drawings. Per cent as an application of ratio. The understanding of the language of per cent (rate), percentage, and base. In particular, the ability to find any one of these three designated numbers given the other two. The ability to treat with confidence per cents less than 1 and greater than 100. Applications of per cent to business practices, interest, discount, and budgets should be given moderate treatment. GEOMETRY Measurement:

The ability to operate with and transform the several systems of measure, including the metric system of length, area, volume, and weight. Geometric measurements, including length of a line segment, perimeter of a polygon, and circumference of a circle, area of regions enclosed by polygons and circles, areas of solids, volumes enclosed by solids, interior of angles (by degrees). The use of a ruler and protractor. The pupil should know the difference between the process of measuring and the measure of the entity. The ability to apply measurement to practical situations. The use of measurement in drawing to scale and finding lengths indirectly.

Relationships among geometric elements;

These include the concepts of parallel, perpendicular, intersecting, and oblique lines (in a plane and in space); acute, right, obtuse, complementary, supplementary, and vertical angles; scalene, isosceles, and equilateral triangles; right triangles and the Pythagorean relation; sum of the interior angles of a triangle; sides and interior angles of a regular polygon with six or fewer sides. The use of instruments in constructing figures; ideas of symmetry about a point and a line.

ALGEBRA AND STATISTICS

Graphs and formulas:

The use of line segments and areas to represent numbers. The reading and construction of bar graphs, line graphs, pictograms, circle graphs, and continuous line graphs. The meaning of scale. Formulas for perimeters, areas, volumes, and per cents—introduced as generalizations as these concepts are studied. The use of symbols in formulas as place holders for numbers arising in measurement. Simple expressions and sentences involving Variables.' "*

This is indeed an ambitious program when compared with what is now required of the majority of American boys and girls in elementary school mathematics. It comes as a shock because parents realize that junior, in such a program will soon be beyond the ability of the average parent to give much actual mathematical aid. It comes as even a greater shock to parents who are familiar with the textbooks which now set the pace for what is required through the eighth grade. Such a program will necessitate the assigning of textbooks now used in high school to the elementary school and throwing many elementary school mathematics textbooks out the window.

Is the parent hopelessly lost in such a program because he has forgotten most of what is required of junior, or is there work in which parent and child can share? Doubtless, junior will have to work most of his problems and do most of the exercises without help, as he rightfully should, but his work can be made more meaningful by guidance and direction which give purpose and a sense of maturity to the whole study of mathematics.

The importance of neatness emphasized from the first day of school must be continued, and. there are habits in performance which if developed in elementary school will save the student from the monstrous penalties which many pay for careless and unreasonable procedure later. There are certain common sense approaches to the study of mathematics which, though they seem obvious, are constantly neglected. They might well be designated as basic practices for a successful pattern of study for mathematics. If learned at home and used at school by the fifth grader and those above, these simple practices can change not only attitude, but also grade:

* Quoted by permission of College Entrance Examination Board.

1. Be interested in means rather than ends; in most cases the method of solving is more important than the answer.


2. Copy the sample problem that the teacher works on the blackboard.


3. Listen closely to what the teacher says and ask questions as soon as a doubt arises.


4. As you listen to the teacher, be keenly aware of the value of the principle behind a problem, and the little value of the problem itself.


5. Do assignments in mathematics as soon after class as possible, while the explanations are still clear.


6. Never begin to work examples until all introductory material (rules, sample problems, etc.) which the book provides has been studied so thoroughly that it can be recalled and applied without a backward glance.


7. Make each paper as neat as is possible—numbers distinctly and orderly written, and problems spaced evenly on the page.


8. Put each step of the problem on the paper, down the page rather than across; no single part of a problem, however easy and obvious it seems, is unimportant enough to omit.


9. Make your paper attractive, leave space between problems, designate final answers by underlining 16 pounds or writing the word Answer after your results. Always use the same format or design. Write your name in the same place and fold each paper the same way.


10. If mathematics is your most difficult subject, have the courage to do your mathematics assignment first.

What will be the effect upon the future mathematics achievement of the fifth, sixth or seventh grader who takes these Ten Commandments for Better Mathematics to heart, posts them on the wall behind his desk, has them explained over and over again until there can be no mistaking the meaning, and then follows each religiously?

Interest is the heart of the whole matter, for with sufficient interest the commandments become part of the person rather than axioms tacked on a wall. The parent and child can follow the code together. The parent can repeatedly explain and review the rules, and thereby play a far more important role in junior's mathematical education than if an explanation of "Per cent as an application of ratio" were kept on the tip of his tongue ready for junior to use.

Indeed, there is much need for explanation, and even the fifth grader in many cases must be convinced. The errors of short-cuts, quick answers, and ignorance of methods become patterns for poor study habits almost as soon as the study of mathematics is begun. To be interested in means rather than ends, the first commandment is to appreciate the fact that mathematics is not a jumble of brain-twisting puzzles, but rather a process of thinking. Perhaps a comparison of the writing of a paragraph and the working of a problem is the best means of explaining the importance of development over end results. Both the paragraph and the problem develop an idea, the only difference being in the symbols used.   In the paragraph words are used and in the problem, numbers or letters (as in the case of algebra substituted for numbers) are used. Both the paragraph and the problem are ways of thinking about an idea, and the principles by which these ideas are developed lead to the proper ends. The end without the principle of development is mere conjecture and has no value. Note the two ideas below:

"The cowboy sat astride his horse and looked down the long street, seemingly trying to 'figure out' what was behind each closed door."

Immediately we are anxious to develop this idea: What is he looking for? Which door hides the thing that has brought him here? Why is he interested?

Now—"If 18 golf balls cost $30.00, how much are they a dozen?"

Here again there is immediate need for development: What is a dozen? 18 is what in relationship to a dozen? What is the cost of one golf ball?

To say that John Turner stood waiting behind the closed door of the Gold Nugget Saloon does not answer the questions raised in the idea of the cowboy scanning the street. Neither does the statement (golf balls are $30.00 a dozen) answer the questions raised in the problem. The principles of development must be followed in each. Both require thought processes that lead to satisfactory conclusions, but the conclusions without the processes prove nothing. The first commandment for a better understanding of mathematics (Be interested in means rather than ends.) pushes the question that always pops to the front of the mind (What's the answer?) into the background. The real and important question is always: "By what means do I arrive at the conclusive answer?"

The next three success commandments for mathematics deal with directing the student's attention toward the significant and important parts of classroom operations. (Copy the sample problem that the teacher works on the blackboard.) Why? It's just like one worked out in the book, or it is the one from the book. First of all, copying the whole problem trains away from the tragically attempted short-cuts revealed by extensive snooping into book-bags and desks—the copying of answers. One need only imagine the time consumed in baseless guesses and innumerable futile attempts to arrive at the methods for obtaining the answer in order to realize the importance of this admonition. To copy after the teacher is also a means of assuring attentiveness—"studying with a pencil in one's hand"—making the mind proceed in an orderly manner rather than by a skip and a hop to the meaningless answer. If commandment two is observed closely, three and four will automatically follow. The student will ask questions about parts he does not understand, and he will become keenly aware of the principle used in solving the problem.

The mathematics homework paper, like all other homework papers, is a self-portrait drawn by the student of his ability, his intention, and his working potentiality; yea . . . even his character. Imagine the effect upon our educational system if this important lesson could be stamped upon the heart of the fifth grader at home, and then be demanded by the school. Assignments would be done as soon after class as possible, before explanations were forgotten. Homework papers would be executed with a degree of neatness requiring well-sharpened pencils, distinctively formed symbols, neatly ordered development of steps, and carefully spaced problems. Thus it is easy for the uses of rules five, seven, nine, and ten to become habit through pride and a sense of obligation for the dignity of work.

Commandments six and eight, however, require tremendous effort of the will. There seems to be almost a universal tendency on the part of students to omit the study of introductory material that would make problems so much easier, and save the student from trial and error procedures which are costly both in time and grades. The second tragic habit which becomes ingrained early, and is difficult to change, is the tendency to scratch parts of the problem somewhere other than on the paper, or figure some steps orally; so that the answer might be arrived at as quickly as possible. Most mathematics teachers grade the paper for what is recorded on it, and not the student's guess. The answer has no value without the steps of thinking by which a conclusion was reached. Teachers and parents should guard against the use of mathematics workbooks which provide only space for answers. The student who uses them is bound to believe that the hop-scotch, skip-to-the-answer short-cut is permissible. Later he will probably pay dearly for careless mistakes. The time required to correct the poor habits of study caused by omitting introductory material and not putting all work on the paper can be saved for the older student for more profitable use, but if not corrected in the fifth or sixth grade, they will be somewhere in high school or in college.

The fifth or sixth grader need only be told of the wisdom of the practice. The eighth grader needs to be argued with and convinced with patience and endurance that can bring parent and teacher to the point of distraction. The high school junior will probably continue making careless mistakes and finding no pleasure in mathematics, because he hasn't "time to bother" with the two most important of all rules of better mathematics.

The home can indeed play a vital role in junior's success in mathematics. The mathematics might well be past the depths of many parents, but the methods are never beyond the realm of common sense. However, the task of neither parents nor teacher is to be underestimated. It is far easier to talk of proper methods, or use proper methods, than it is to teach proper methods. It is far easier for the Tasmanian native to count one, two, many (the extent of his arithmetic) on his fingers than for him to teach his child to count one, two, and many. There are no carbon copies of personalities and every child reacts differently to the same treatment. Response comes in varying degrees to stimuli and motives. Not one person in a thousand (starting at fifth grade, or earlier or later) uses the powers that are actually possessed to come anywhere near the limit of intellectual ability.

The main ingredients of success in any kind of study seem to be interest and confidence. Perhaps sometimes parents are too quick to exploit their own dislikes as excuses for lack of interest on the part of their children. This not only does nothing to help the child, but builds a barrier which the teacher must struggle to destroy before even an approach to interest can be made. With the correct approach a student may become interested in any subject, and interest changes the degree of response. Confidence grows from an attitude toward, and an understanding of, the thing being studied—that's why we show our greatest abilities and powers of sustained effort on our hardest and most time consuming work—our hobbies. Who can doubt that it lies within the power of the home to make school a hobby, and thereby provide the mathematics teacher and the grammar teacher and all other teachers with a child eager to learn—one who will respond because he has been fed from the heart at home.

The child's world of wonder begins to touch reality from the fourth grade forward, and some of the child's attitudes become very real, and take on a permanent character during this period. The energy, the imagination, and the common sense of the parents should be directed toward making the child's likes and dislikes take positive and constructive points of view. In this respect there is scarcely a sphere of influence which needs so much attention as the world of numbers and reasoning which for your child is arithmetic.

Home Helps For Study Of Arithmetic

1. Precision, thoroughness, and speed are necessary background elements for the carrying out of basic operations in arithmetic.


2. An awareness of numbers can be taught by practices involving the things we see around us.


3. A knowledge of troublesome combinations of numbers can prevent mistakes that sometimes persist even into high school.


4. Make a basic vocabulary for arithmetic. Know the meaning of key words.


5. Approach word problems by proper steps: (I) The problem makes sense. (2) Put it in your own words. (3) What information is given; what is needed? (4) Do I add, subtract, multiply, divide, or do I need a combination of these? (5) Is my answer reasonable; can I check it with known facts?


6. Be concerned with the methods for solving a problem rather than with the answer. If the proper methods are followed, the answer will take care of itself.


7. Study  all  introductory   material   (rules,   samples, notes, etc.) before starting the homework problems.


8. Put each step of the problem on the paper. Later one's own individual approach and solution will be respected, but for now, do not ignore this orderly training.


9. Make the paper attractive and neat. Space the problems evenly and designate answers by underlining 643 or Ans. 643.


10. If arithmetic is the most difficult subject, look for interesting and positive reasons for doing it.
    

Are You Ready To Move Onto The Next Lesson? Click Here...