Mathematical understandings of elementary school teachers

Mathematical understandings of elementary school teachersAuthor(s): RUSSELL A. KENNEYSource: The Arithmetic Teacher, Vol. 12, No. 6 (OCTOBER 1965), pp. 431-442Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186964 .

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Mathematical understandings of elementary school teachers

RUSSELL A. KENNEY Kern County Schools, Bakersfield, California Mr. Kenney is curriculum coordinator in the office of the Kern County superintendent of schools. His duties include any and all activities

for the improvement of instruction in generalf and mathematics in the elementary school in particular.

L his article is a condensation of a study made by the author with the assistance of the Committee on Teacher Education of the San Joaquín Section of the California Association for Supervision and Curricu- lum Development, and with the cooper- ation of many teachers, including those who took the preliminary test and the 356 teachers who took the test in its final form.

The purpose of the study was to assist in the improvement of teacher education for the California state-adopted arithmetic series by finding the strengths and weaknesses of elementary school teachers in arithmetical understanding. Emphasis was placed on the understandings presented in the texts and in the manuals.

Procedure used ¡n the study The writer found no adequate instru-

ment in the literature to measure these understandings. Some tests covered certain areas but not others. Glennon1 had developed and used one of these tests, and with his permission parts of his test were used in our instrument. The preliminary draft of a fifty-item test prepared by the author was submitted for suggestions to the Committee on Teacher Education mentioned above and to other associates. The instrument was revised in light of the resulting suggestions, and was adminis- tered on a trial basis to two classes of

students in teacher education. The test was then revised on the basis of these results, and again submitted to the mem- bers of the Committee on Teacher Educa- tion for final approval.

The approved test contained fifty items in six categories.2 Most of the items were of a multiple-choice type. Other items were of a word-answer or completion form. Copies of the test, with a sheet of instruc- tions for administering it, were distributed to school supervisors and administrators who had agreed to cooperate in the study. A total of 356 usable responses was re- ceived. The results were recorded on punched key-sorting cards, which were used in the preparation of tables and charts.

Findings A study was made of the results for the

total group and for selected subgroups de- scribed below.

The total number of correct responses on the fifty-item test ranged from 5 to 44. Thirty-six individuals had scores of 40 to 44, while only four had scores below 10. The median was 29.7, the 75th percentile was 35.4, and the 25th percentile was 23.0.3

The range of percent of correct responses by category as shown in Chart 1 was 49 to 69 percent. The highest rating

i Glennon, Vincent J.f "A Test of Basic Mathematical Understandings. ' '

'See pages 436-442. * These scores were calculated by the short method, using a grouped frequency distribution.

October I960 431



20 40 60 80

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Chart 1. Percent of correct responses by category

was in the category of Number Systems, the lowest in the categories of Whole Numbers and Per Cent.

The range for individual items within each of the categories is shown in Table 1. The range of correct responses may be seen to vary from 3 percent for one item in decimal fractions to 98 percent for one item in common fractions.

In the category of Number Systems only 35 percent of the individuals could answer the following question (1.7) correctly: "In the above example [adding feet and inches] what base is used to determine the amount carried in adding the number of feet?" /Ninety-three percent answered item 1.2 correctly: "Rearrange the digits in the number 435,852 to make the largest possible number."

In the category of Measures, Graphs, and Scales, only 6 percent were able to identify the most accurate or precise measurement from a given list (2.1). Ninety- two percent, however, were able to identi-

fy the year of greatest production from a line graph (2.5).

In the Whole Numbers section, the poorest response was shown on item 3.4: "The three types of situation in which we use subtraction are Чаке-away/ and ." The strongest showing (89 percent) was made on 3.2: "In the problem at the right, why is the '1' carried to the second column and added to the '9'?"

307 95

402

The Common Fractions category had the highest percent of correct responses for a single item on the whole test (98 percent). This item was 4.1, "Which of the following fractions is smallest? A |, В %, Ci, Dì, E%." The lowest percent of correct response was made to 4.2: "Which of these statements best tells why we cannot

Table 1 Range of percent of correct responses to items ¡n each category

Category Number of items Range ( %)

Number Systems 10 35-93 Measures, Graphs, and Scales 8 6-92 Whole Numbers 8 9-89 Common Fractions 8 14-98 Decimal Fractions 8 3-89 Per Cent 8 29-85

432 The Arithmetic Teacher



say that the shaded parts of this picture represent!?'7

The section on Decimal Fractions had the item (5.7) with the poorest response. This item dealt with the placement of the decimal point in the multiplication of decimal fractions. The best response in this section was made to item 5.5, which dealt with subtraction of decimal fractions.

The percent of correct responses to the items in the category on Per Cent had the narrowest range. The low items were 6.5 and 6.6, which dealt with finding the whole when the percent and percentage were given. The high response item re- quired the finding of a percent of a number.

In an effort to find possible causes for these results, the scores were grouped and compared. First, scores were grouped according to the teaching preferences of the respondent. Each of these groups was then divided into subgroups, according to whether or not the respondents had had any coursés in arithmetic methods or content since 1054. The number having had such courses made 21.4 percent of the total. The number in each of the subgroups was too small to be significant, but there seemed to be no significant difference in median scores (69, 62, 62) between the subgroups. The median percent of correct responses for the total group having had courses was 59 percent (148 cases), while those having no courses was 59 percent (208 cases) (from Table 2 not shown).

If we compare the percent of correct responses of the teachers teaching at different levels shown in the three columns under "Totals" in Table 1, we find a con-

sistent difference, whether it be in the top or bottom quartile or in the middle group. The medians for the middle 50 percent, as shown, are primary teachers, 51 percent; intermediate teachers, 62 percent, and upper grade teachers, 72 percent.

If the results are grouped according to number of years of teaching experience, slightly different results are seen. Re- spondents with no experience scored low, as may be expected (53 percent). Teachers with one to four years experience scored slightly higher (60 percent) than those with five or more years (58 percent) .

Conclusions The test upon which the results for this

study were based was not standardized. The author made a sincere attempt to confine the concepts and understandings measured to those directly taught or im- plied in the California State Arithmetic Series to assure a high degree of validity. Samples were selected in an effort to find typical strengths and weaknesses which might shed some light on how the teaching of arithmetic in the elementary school might be improved. That it measures the respondent's knowledge of a basic arithmetical concept necessary for the effective instruction of arithmetic in the elementary school seems unquestionable in light of the process used in developing the instrument. Evidence seemed to indicate that errors were largely due: (1) to inability to understand the language or vocabulary used in the test, or (2) to a lack of understanding of the concepts, relationships, ot mathematical generalizations involved. The test was designed to measure the latter, as far as possible. The extent to which the lack of adequate vocabulary or understanding of the language interfered with this primary purpose is not known. In spite of this, certain strengths and weaknesses can be discerned. These findings should have significance in teacher education in this area.

The relatively high median score for the section on the Number System indicated

Oùtóber 1966 433



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that there is considerable understanding in this area. The answers to two of the questions (1.1 and 1.7) indicate a weakness which may be due to a language difficulty, but the low scores in response to two other questions (1.3 and 1.10) are in keeping with the observations of the writer made in many classrooms.

The relatively low median for the section on Whole Numbers points out a need for more education in the rationale of this subject. The weakness revealed in the answers to two questions (3.3 and 3.4) points up a need for a better understanding of the function of subtraction. The response to another question (3.1) would appear to indicate not only a lack of understanding of the difficulty of the unseen addend, but also a superficial understanding of other difficulties in addition. The answers to other questions (3.7 and 3.8) reveal weaknesses in the understanding and use of zero, which are known to exist, and point to the need for greater emphasis on the understanding of the processes and algorisms used in multiplication and division.

The wide range of correct responses in the section on Common Fractions would seem to indicate poor teaching of this subject or the difficulty of this subject. Uni- tary fractions seem to be well understood, but the weakness shown in one question (4.2) seems to indicate that there has been too great a reliance on a few stereotyped illustrations in teaching fraction concepts, resulting in a less than adequate understanding. The responses to question 4.3 appear to indicate a rather superficial and mechanical treatment given to the chang- ing of terms of a fraction or to the finding of a convenient equivalent fraction.

The extremely poor showing on one item (5.7) in the Decimal Fractions category may be due to a lack of understanding of the language used rather than a lack of understanding of the principle involved. The answers to another question (5.4), however, seem to indicate a need for more emphasis on teaching an understanding of

our number system and how it works. Decimal points in division (5.8) is quite generally recognized as a difficult subject which needs further study. Finding the decimal equivalent for a common fraction, where the fraction is sufficiently difficult to require the understanding of the principle, proved to be difficult for many. This would seem to point to the need for work in this area.

The findings in the section on Per Cent appear to indicate a need for a further analysis not only of the difficulties involved, but also of the methods used in teaching the concepts involved.

The relatively high median for upper- grade teachers, as shown in Table 2, as compared with that of primary teachers indicates stronger foundations for teaching arithmetic by this group. We may conclude that, since upper-grade teachers deal with many more aspects of the arithmetic program, they are more familiar with the material in some sections of the test. This would not necessarily mean that primary and intermediate teachers do not need this degree of understanding. More- over, this lack of understanding by primary teachers was evident in the phases ordinarily taught by these teachers in the section on the Number System. The intermediate teachers were weak in the processes of whole numbers at their level.

The fact that those who had had recent courses in arithmetic did no better than those who did not have such courses would appear to indicate that either these courses do not have the same objectives as the state texts and the test or that they have not been functional for some reason.

It is quite evident that understanding of arithmetic concepts and processes does not increase with teaching experience.

Recommendations 1 There should be serious consideration of

the optimum amount and kind of pre- service and in-service education in arithmetic for elementary school teachers.

October 1965 435



2 Support should be added to the efforts of the National Council of Teachers of Mathematics and the California Coun- cil of Teachers of Mathematics to require prospective teachers to be better prepared for the teaching of elementary achool arithmetic.

3 The type of understandings measured in the test and treated in our state texts should be an important part of any such preparation.

4 The degree of mastery of each of these areas needs to be on a higher level of achievement than is revealed in this study.

5 It is, therefore, imperative that every effort be made in both pre-service and in-service education to improve the teacher background in, and understandings of, the mathematics essential to the most effective teaching and use of arithmetic.

rak 261 p. 1

Mathematical Understandings of Teachers

Number Systems

1. 1 Write in words: 20, 104.

1.2 Rearrange the digits in the number 435, 852 to make the largest possible number.

1.3 In the number 7, 433, what is the largest possible number of tens?

1. 4 What would be 6, 542, rounded to the nearest thousand? _^

1.5 What is the place value or base of the "5" in the above number?

1.6 In the following example: 14 ft. 6 in. what base is used to 11 ft. 8 in. determine the amount 6 ft. 2 in. carried from the inch

32 ft. 4 in. column to the foot column?

1.7 In the above example what base is used to determine the amount carried in adding the number of feet?

1. 8 What base is used in converting quarts to gallons?

1. 9 Which of the following statements best tells why we write a zero in the number 4, 039 when we want it to say "four thousand thirty-nine"? (A) Because the number would say "four hundred thirty-nine" if we

did not write the zero, (B) Writing the zero helps us to read the number. (C) Writing the zero tells us not to read the hundreds figure. (D) Because the number would be wrong if we left out the zero, (E) Because we use the zero as a place-holder to show that

there is no number to record in that place.

1.10 Which of the following methods ig best for determining the value of a figure in a number? (Example: The value of "7" in 3748, ) (A) Its position in the number. (B) Its value when compared to the other figures in the number. (C) Its size in the order from 1 to 9. (D) Its position in the number and its sise. (E) Its value when compared with the whole of the number. _____




rak 261 p. 2 Mathematical Understandings of Teachers

Measurements, Graphs, and Scales.

2. 1 Which of the following indicates the most accurate measurement? (A) 1«, (B) 21 1", (C) 7« 3-1/4", (D) 6' 4-3/8", (E) 5» 7-7/1611

2.2 Which of the following is the most accurate statement? (A) A ratio is a fraction. (B) A ratio is the difference between two numbers. (C) A ratio is expressed by placing one number over another. (D) A ratio is an expression of the size of one quantity in terms of another. (E) A ratio is two numbers separated by a colon.

2. 3 The Dodgers had won 48 games and lost 29. What was their standing expressed to three decimal places?

Yearly Production

I 1953 1 1954 1 1955 1 1956 ' I 1957 1 1958 | 1959 | I960 50

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10 0 [ I I I 1 I 1 I 2.4 During what year does the graph show the greatest gain?

2. 5 During what year does it show the greatest production?

2.6 A line representing 3 feet 6 inches, using a scale of one-half inch to one foot, should be how long?

2.7 Select the best completion statement for the following phrase and place its identification letter in the space provided. In measuring something, (A) We must use a measuring device made for that purpose. (B) We must use a standardized unit of some kind. (C) We may use nothing but the above. (D) We may use anything but the thing itself. (E) We may use the thing itself.

2.8 Which of the following indicates the most accurate measurement? (A) 4, (B) 5. 1000, (C)6.26, (D)8.9, (E) 7.265.

October I960 437




Whole Numbers - Processes

3. 1 What new difficulty is introduced in addition when a third addend is involved?

3.2 In the problem at the right, why is the "Iй carried to the 307 second column and added to the "9"? 95

402

3. 3 - 3.4 The three types of situations in which we use subtraction are "take-away11, (and) 3. 3

and 3 . 4

3.5 In the example shown, 36 why is the "2" in the second 24 partial product placed where 144 it is? 72_

864

3. 6 In the example shown, why 56 do we "bring down" the 13 )728 "8" from the dividend? 65

78 78

3. 7 What would be the effect on the answer if you annexed 439 two zeros to 439 and took away the zero from 450? x 450 (A) The answer would be ten times as large. (B) The answer would be one hundred times as large. (C) The answer would remain the same. (D) The answer would be one -tenth as large. (E) The answer would be one -hundredth as large.

3. 8 What would be the effect on the answer if you added or 92) 4500 annexed two zeros to the 92 and changed 4500 to 450? (A) The answer would be ten times as large. (B) The answer would be one -tenth as large. (C) The answer would be one hundred times as large. (D) The answer would be one -hundredth as large. (E) The answer would be one -thousandth as large.





Common Fractions

4. 1 Which of the following fractions is the smallest? (A) 1/9, (B) 1/5, (C) 1/2, (D) 1/7, (E) 1/3

4.2 Which of these statements best tells why we cannot ̂

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4. 3 Which picture shows how the result would look if you divided the numerator and denominator of 10/8 by 2? (A) _ IB) ^ lCs'.

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4. 5 Which picture best shows the sample 4x2/3?

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October 1965 439



rak 261 p. 5

4. 6 When a number is divided by a common proper fraction how does the answer compare with the original number? (A) larger, (B) same, (C) smaller, (D) sometimes larger and sometimes smaller, (E) cannot tell _^_____

4.7 Which of the pictures best shows this example? 3 j- 1/2

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4. 8 Which sentence best tells why thé answer 5 I 3/4 is larger than the 5 in this example? (A) Because inverting the divisor turned the 3/4 Upside down. (B) Because multiplying always makes the number larger. (C) Because the divisor, 3/4 is less than 1. (D) Because dividing by a fraction always makes the answer

larger than the number divided* (E) Inverting a fraction put the larger number on top.





Decimal Fractions

5. 1 What is the decimal equivalent of 1/250? (A) .250, (B) .25, (C) .04, (D) .004, (E) .0004

5.2 How should you write the decimal "eighty and eight hundredths"? (A). 8008, (B) 80. 800, (C) 80. 08, (D) 80. 008, (E) 8008. 08

5. 3 Which decimal tells how long line Y is when compared with line X? Line X Line Y (A) .5, (B) .625, (C) 1.25, (D) 75, (E) 33

5.4 How many hundredths are there in . 634? (A) .634, (B) 6.34, (C) 63.4, (D) 634, (E) 6340

5. 5 Which of the following is the best reason for placing the decimal points in a vertical line in a subtraction problem? (A) The rule tells us to do so. (B) It is neater and makes a better impression. (C) The decimal point is the base in our number system. (D) The decimal is the central point in our system of numbers. (E) This places numbers with like bases or place value in the

same column.

5. 6 What would be the effect on the answer if you changed 368 to 3680 and 24 to 2.4? 368 (A) The answer would be smaller. x 24 (B) It would not change the answer. (C) It would be the same as annexing (adding) a zero to the answer. (D) The answer would be one -tenth as large. (E) Cannot tell without working the example both ways.

5. 7 Which of the following is the best statement on the placement of the decimal point in the multiplication of decimal fractions? (A) The decimal points in the multiplicand and multiplier must

be placed in a vertical line. (B) The decimal point in the product must be placed by rule. (C) The placement of the decimal point in the product must be

developed inductively. (D) The placement of the decimal point in the product may be

done by estimation. (E) The decimal point should be placed directly below the one in

the multiplier.

5. 8 How would the answer be changed .1. 47) 34. 75 if you changed 1. 47 to 147? (A) The answer would be the same. (B) The answer would be ten times as large. (C) The answer would be one -tenth as large. (D) The answer would be one hundred times as large. (E) The answer would be one -hundredth as large.

October 1965 441



Per Cent

6. 1 If the sales tax is 4%, what is the tax on 28. 50.

6. 2 If milk contains 3. 6% butte rfat, find the amount of butterfat in a shipment of 1850 pounds.

6. 3 Goods costing $70. 00 were sold for $87. 50. What was the percent of profit over the cost?

6. 4 If an article cost $40. 00 and sold for a profit of $24. 00, what was the percent of profit of the selling price?

6.5 Experiences of a benevolent organization indicated that 15% of the money collected was needed for the expenses of the drive. What should be their goal in order to have $34, 000 left for their primary- purpose?

6. 6 If a merchant wishes to make a profit of 10% of the selling price and the total cost was $17. 91, what must be the selling p.rice?

6. 7 In order to find a percent, (A) We must multiply the base number by the other number. (B) We must divide the base number by the other number. (C) We must find the difference between the two numbers and

divide the larger number by the difference. (D) We must find the difference between the two numbers and

divide the difference by the base number. (E) We must divide the number representing the amount of

which we want the per cent by the base number.

6.8 Which of the following is the correct answer to the question, 327 is what percent of 218? (A) 66.7%, (B) 33.3%, (C) 150%, (D) 15%, (E) 1.5%

"This is the way we count our trains " (The following narrative has been recorded with far more subjectivity than is usually asso- ciated with a description of the teaching and learning processes in elementary arithmetic. It describes the use of Cuisenaire rods in one of a series of sessions with a kindergarten child of low average mentality. The current efforts towards helping children to "see" relationships between numbers is the focus of this interview between a little girl and a person serving as an elementary school district consultant in arithmetic.)

Пег little fingers roamed lightly over the rods, sometimes pausing to stroke the length of a particular one, then moving restlessly on their quest. Her big brown eyes were staring off across the schoolroom, looking at neither the rods, the teacher, nor the other pupils. Suddenly a joyous flash crossed her face and with a triumphant

"I've found it!" she selected the last yellow rod from the pile and lined it up carefully with the other yellow rods. Encompassing the group with her fingertips she said, "This is a square shape - I know it is because I can feel it - and besides each rod measures the same as five little ones, and there are five ends across the top and so it's five across and five down, and five is five! And five of the fives are twenty-five!"

The visiting teacher handed her a red rod and said, "Tell me about this one." The child rolled it delicately on her fingertips, almost balancing it as she turned it over and over, seemingly doing so just for the fun of feeling it. "Well, it's two of the little ones so it's the red; I can count by twos - do you want to hear me?"

"How did you learn to do that?" asked the visiting teacher.

(Continued on page 449)




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Mathematical understandings of elementary school teachers