The Psychology of Arithmetic

Woody ['16] has constructed his well-known tests on this principle, though he uses only one example at each step of difficulty instead of eight or ten as suggested above. His test, so far as addition of integers goes, is:—

SERIES A.     ADDITION SCALE (in part)

By Clifford Woody

In his original report, Woody gives no scheme for scoring an individual, wisely assuming that, with so few samples at each degree of difficulty, a pupil's score would be too unreliable for individual diagnosis. The test is reliable for a class; and for a class Woody used the degree of difficulty such that a stated fraction of the class can do the work correctly, if twenty minutes is allowed for the thirty-eight examples of the entire test.

The measurement of even so simple a matter as the efficiency of a pupil's responses to these tests in adding integers is really rather complex. There is first of all the problem of combining speed and accuracy into some single estimate. Stone gives no credit for a column unless it is correctly added. Courtis evades the difficulty by reporting both number done and number correct. The author's scheme, which gives specified weights to speed and accuracy at each step of the series, involves a rather intricate computation.

This difficulty of equating speed and accuracy in adding means precisely that we have inadequate notions of what the ability is that the elementary school should improve. Until, for example, we have decided whether, for a given group of pupils, fifteen Courtis attempts with ten right, is or is not a better achievement than ten Courtis attempts with nine right, we have not decided just what the business of the teacher of addition is, in the case of that group of pupils.

There is also the difficulty of comparing results when short and long columns are used. Correctness with a short column, say of five figures, testifies to knowledge of the process and to the power to do four successive single additions without error. Correctness with a long column, say of ten digits, testifies to knowledge of the process and to the power to do nine successive single additions without error. Now if a pupil's precision was such that on the average he made one mistake in eight single additions, he would get about half of his five-digit columns right and almost none of his ten-digit columns right. (He would do this, that is, if he added in the customary way. If he were taught to check results by repeated addition, by adding in half-columns and the like, his percentages of accurate answers might be greatly increased in both cases and be made approximately equal.) Length of column in a test of addition under ordinary conditions thus automatically overweights precision in the single additions as compared with knowledge of the process, and ability at carrying.

Further, in the case of a column of whatever size, the result as ordinarily scored does not distinguish between one, two, three, or more (up to the limit) errors in the single additions. Yet, obviously, a pupil who, adding with ten-digit columns, has half of his answer-figures wrong, probably often makes two or more errors within a column, whereas a pupil who has only one column-answer in ten wrong, probably almost never makes more than one error within a column. A short-column test is then advisable as a means of interpreting the results of a long-column test.

Finally, the choice of a short-column or of a long-column test is indicative of the measurer's notion of the kind of efficiency the world properly demands of the school. Twenty years ago the author would have been readier to accept a long-column test than he now is. In the world at large, long-column addition is being more and more done by machine, though it persists still in great frequency in the bookkeeping of weekly and monthly accounts in local groceries, butcher shops, and the like.

The search for a measure of ability to add thus puts the problem of speed versus precision, and of short-column versus long-column additions clearly before us. The latter problem has hardly been realized at all by the ordinary definitions of ability to add.

It may be said further that the measurement of ability to add gives the scientific student a shock by the lack of precision found everywhere in schools. Of what value is it to a graduate of the elementary school to be able to add with examples like those of the Courtis test, getting only eight out of ten right? Nobody would pay a computer for that ability. The pupil could not keep his own accounts with it. The supposed disciplinary value of habits of precision runs the risk of turning negative in such a case. It appears, at least to the author, imperative that checking should be taught and required until a pupil can add single columns of ten digits with not over one wrong answer in twenty columns. Speed is useful, especially indirectly as an indication of control of the separate higher-decade additions, but the social demand for addition below a certain standard of precision is nil, and its disciplinary value is nil or negative. This will be made a matter of further study later.

MEASUREMENTS OF ABILITIES IN COMPUTATION

Measurements of these abilities may be of two sorts—(1) of the speed and accuracy shown in doing one same sort of task, as illustrated by the Courtis test for addition shown on page 28; and (2) of how hard a task can be done perfectly (or with some specified precision) within a certain assigned time or less, as illustrated by the author's rough test for addition shown on pages 28 and 29, and by the Woody tests, when extended to include alternative forms.

The Courtis tests, originated as an improvement on the Stone tests and since elaborated by the persistent devotion of their author, are a standard instrument of the first sort for measuring the so-called 'fundamental' arithmetical abilities with integers. They are shown on this and the following page.

Tests of the second sort are the Woody tests, which include operations with integers, common and decimal fractions, and denominate numbers, the Ballou test for common fractions ['16], and the "Ladder" exercises of the Thorndike arithmetics. Some of these are shown on pages 36 to 41.

Courtis Test

Arithmetic.     Test No. 1.     Addition

Series B

Courtis Test

Arithmetic.     Test No. 2.     Subtraction

Series B

Courtis Test

Arithmetic.     Test No. 3.     Multiplication

Series B

Courtis Test

Arithmetic.     Test No. 4.     Division

Series B

SERIES B.     MULTIPLICATION SCALE

By Clifford Woody

SERIES B. DIVISION SCALE
By Clifford Woody

Ballou Test

Addition of Fractions

An Addition Ladder [Thorndike, '17, III, 5]

Begin at the bottom of the ladder. See if you can climb to the top without making a mistake. Be sure to copy the numbers correctly.

A Subtraction Ladder [Thorndike, '17, III, 11]

An Average Ladder [Thorndike, '17, III, 132]

Find the average of the quantities on each line. Begin with Step 1. Climb to the top without making a mistake. Be sure to copy the numbers correctly. Extend the division to two decimal places if necessary.

As such tests are widened to cover the whole task of the elementary school in respect to arithmetic, and accepted by competent authorities as adequate measures of achievement in computing, they will give, as has been said, a working definition of the task. The reader will observe, for example, that work such as the following, though still found in many textbooks and classrooms, does not, in general, appear in the modern tests and scales.

Reduce the following improper fractions to mixed numbers:—

¹⁹⁄₁₃         ⁴³⁄₂₁         ¹⁷⁶⁄₂₅         ¹⁹⁸⁄₁₄

Reduce to integral or mixed numbers:—

⁶¹³⁸¹⁄₃₇         ²¹³⁴⁄₆₇         ⁴¹³⁄₄₁₃         ⁶⁹⁷⁄₂₂₅

Simplify:—

³⁄₄   of   ⁸⁄₉   of   ³⁄₅   of   ¹⁵⁄₂₂

Reduce to lowest terms:—

³⁵⁷⁄₅₂₇         ²⁶⁴⁄₃₁₂         ⁴⁹²⁄₇₇₉         ⁴¹⁸⁄₈₇₄         ⁸⁵⁴⁄₁₇₆₉         ³⁰⁄₇₃₅         ⁴⁴⁄₂₄₂         ⁷⁷⁄₈₄₇         ¹⁸⁄₂₄₃         ⁹⁶⁄₂₂₄

Find differences:—

Square:—

²⁄₃         ⁴⁄₅         ⁵⁄₇         ⁶⁄₉         ¹⁰⁄₁₁         ¹²⁄₁₃         ²⁄₇         ¹⁵⁄₁₆         ¹⁹⁄₂₀         ¹⁷⁄₁₈         ²⁵⁄₃₀         ⁴¹⁄₅₃

Multiply:—

²⁄₁₁ × 33         32 × ³⁄₁₄         39 × ²⁄₁₃         60 × ¹¹⁄₂₈         77 × ⁴⁄₁₁         63 × ²⁄₂₇
54 × ⁸⁄₄₅         65 × ³⁄₁₃         344¹⁶⁄₂₁    432²⁄₇

MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE SOLUTION OF PROBLEMS

Stone ['08] measured achievement with the following problems, fifteen minutes being the time allowed.

"Solve as many of the following problems as you have time for; work them in order as numbered:

The criteria he had in mind in selecting the problems were as follows:—

"The main purpose of the reasoning test is the determination of the ability of VI A children to reason in arithmetic. To this end, the problems, as selected and arranged, are meant to embody the following conditions:—

The test is purposely so long that only very rarely did any pupil fully complete it in the fifteen minute limit."

Credits were given of 1, for each of the first five problems, 1.4, 1.2, and 1.6 respectively for problems 6, 7, and 8, and of 2 for each of the others.

Courtis sought to improve the Stone test of problem-solving, replacing it by the two tests reproduced below.

ARITHMETIC—Test No. 6.       Speed Test—Reasoning

ARITHMETIC—Test No. 8.       Reasoning