Lecturer's Précis - Olckers (1951)

"A Factorial Study of Arithmetical Ability"

Copyright Notice: This material was written and published in Wales by Derek J. Smith (Chartered Engineer). It forms part of a multifile e-learning resource, and subject only to acknowledging Derek J. Smith's rights under international copyright law to be identified as author may be freely downloaded and printed off in single complete copies solely for the purposes of private study and/or review. Commercial exploitation rights are reserved. The remote hyperlinks have been selected for the academic appropriacy of their contents; they were free of offensive and litigious content when selected, and will be periodically checked to have remained so. Copyright © 2010, High Tower Consultants Limited.

 

First published online 12:50 GMT 3rd March 2005, Copyright Derek J. Smith (Chartered Engineer). This version [HT.1 - transfer of copyright] dated 18:00 14th January 2010

 

1 - Introduction

Olckers began by briefly reviewing the history of the notion of “mental factors” in psychological explanation, explaining how there had been a flurry of interest in psychometric testing in the early decades of the 20th century, and singling out the pioneering work of Sir Cyril Burt [Research Methods Glossary] and Charles Spearman [Research Methods Glossary] for special mention.

ASIDE: Both Burt (1917) and Spearman (1904, 1927) had resorted to mathematical procedures to help them look for subtle explanatory patterns in the accumulated results of their intelligence testing. At issue was whether there was really such a thing as “general intelligence”. If there was, then a given person would be expected to perform approximately equally on tests of mathematics, geography, and English, say, provided (a) that they had put in equal amounts of study, and (b) that the tests had been cross-matched for difficulty. The intelligent ones would do well on all the tests, and the unintelligent ones would do poorly on all of them. On the other hand, if there were several fundamentally different types of intelligence, then people would perform well at (and presumably find it easier to learn) the subjects Nature had best equipped them for, and less well at the others. Both authors concluded that intelligence was a combination of both specific and general abilities [known as s- and g-factors respectively], thus ushering in a period of scientific controversy which continues in some respects to this day. A good example of a modern theory of multiple specific intelligences is provided by Howard Gardner’s Theory of Multiple Intelligences (Gardner, 1983).

As for mathematical abilities in particular, Olckers quoted Spearman's (1927) claim that "the inter-correlations of arithmetical abilities are traceable to a single factor over and above g" (Olckers, p2). He also identified a number of other pre-1951 studies, including .....

·         Thorndike (1922): Olckers cited this monograph by the influential American learning theorist Edward Lee.Thorndike (1874-1949) as typical of the s-factor school of thought. Thorndike, he noted, “conceived the mind to consist of a large number of particularized and independent abilities” (Olckers, p2). There were, for example, “at least seven processes or minor functions involved in two-place column addition, each of which is psychologically distinct and requires distinct educational treatment” (Thorndike, 1922).

·         Alexander (1935): This study indicated two major factors in mathematical abilities, namely "a general factor and a specific factor common to all the arithmetical processes" (Olckers, p2).

ASIDE: In fact, Alexander introduced the potentially valuable notion of the "functional cluster" [alternatively "functional unit", "functional ability", or "unitary trait"] to describe bundles [our term] of abilities which "function almost exactly alike" (Alexander, 1935, p116). This allows such terms as "verbal ability" to remain useful and valid, despite the fact that more detailed investigation can find different sub-skills within them. Moreover, when one of those sub-skills is common to several abilities, it will naturally be reported as a common factor upon statistical analysis; but this does not automatically equate it to Spearman's g-factor. "The real problem," Alexander argued, was to find "the smallest number of such factors which will account for all the variation" (Ibid., p117), but this was not the same as the number of variables measured in the test battery used.

·         Thurstone (1938): Louis Leon Thurstone (1887-1955) used a battery of 56 various tests, and, like Thorndike, identified "a number of independent primary factors and no general common factor" (Olckers, p2). The primary factors included one for "number ability", another for reasoning, and another for induction.

·         Cunningham and Price (1934): Olckers cited this nationwide survey of arithmetical ability in Australia as having demonstrated that arithmetical calculation and mathematical problem solving [presumably Thurstone's "number ability" and "reasoning", respectively] were "almost, though not quite, distinct from one another" (Olckers, p2).

So the problem facing Olckers in 1951 was how to improve upon the factor analytic conclusions already reached. He carried out three separate studies of his own, one a preliminary screening study, and two more precisely targeted follow-ups, as now summarised .....

2 - The Research

STUDY #1 - PRELIMINARY: In the first study, he subjected 112 secondary school pupils (46 girls, 66 boys, average age 14;4 years) to a battery of 15 different types of arithmetical test, as follows .....

Test 1 - Addition: This was a 4 minute test of columnar addition.

Test 2 - Subtraction: This was a 4 minute test of columnar subtraction.

Test 3 - Multiplication: This was a 4 minute test of multiplication, starting with single digits and increasing in difficulty to five digits by two digits.

Test 4 - Division: This was a 4 minute test of division, starting with single digits and increasing in difficulty to five digits by two digits.

Test 5 - Convert Roman Numerals: This was a 4 minute 25-item test of converting Roman numerals into everyday notation [e.g. LVI into 56]. [Perhaps not surprisingly, this test correlated poorly with the four arithmetical tests above - see the Row 5 correlation coefficients in the table below.]

Test 6 - Dot Counting: This was a 3 minute test to count the number of dots in an irregular 20-row array.

Test 7 - Repetition of Digits: This was a 15-item test of the ability to repeat orally presented digit sequences.

Test 8 - Fractions: This was a 7 minute 24-item test "of the four rules applied to ordinary and decimal fractions" (Olckers, p3).

Test 9 - Problem Solving: This was a 10 minute 28-item test of [unspecified] problem solving.

The remaining six tests were taken from the South African Group Test of Intelligence. The tests were named as follows, but no details were given .....

Test 10 - Classification, Test 11 - Analysis, Test 12 - Number Series, Test 13 - Letter Test, Test 14 - Same-Opposite Test, Test 15 - Figure Test.

Initial analysis of the accumulated test scores involved correlating all possible pairs of test scores. Thus the scores on Test 1 were correlated with Tests 2 to 15, the scores on Test 2 were correlated with Tests 3 to 15, the scores on Test 3 were correlated with Tests 4 to 15, and so on until every possible pairing had been exhausted. Here are the correlation coefficients which were reported. To find any particular pairing simply locate the first test as a column and the second test as a row, and read off the value in the cell of intersection of column and row .....

 

1

2

3

4

5

6

7

8

9

10

11

12

13

14

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

2

.49

-

-

-

-

-

-

-

-

-

-

-

-

-

3

.47

.57

-

-

-

-

-

-

-

-

-

-

-

-

4

.40

.59

.55

-

-

-

-

-

-

-

-

-

-

-

5

.09

.08

-.04

.05

-

-

-

-

-

-

-

-

-

-

6

.34

.32

.39

.24

-.04

-

-

-

-

-

-

-

-

-

7

.11

.03

.07

.24

.09

.07

-

-

-

-

-

-

-

-

8

.14

.40

.29

.39

.34

.00

.20

-

-

-

-

-

-

-

9

.18

.27

.12

.36

.35

.02

.32

.47

-

-

-

-

-

-

10

.21

.20

.13

.11

.28

-.04

.18

.28

.20

-

-

-

-

-

11

.05

.12

.08

.13

.41

.07

.26

.27

.24

.28

-

-

-

-

12

.27

.21

.25

.17

.20

.11

.05

.47

.51

.36

.28

-

-

-

13

.13

.04

.14

.16

.22

.10

.20

.28

.17

.51

.33

.25

-

-

14

.02

-.02

-.02

.06

.22

-.20

.15

.31

.43

.31

.28

.11

.20

-

15

.15

.22

.18

.31

.19

.12

.04

.42

.35

.36

.18

.37

.32

.18

NB: Note the colour-coding we have added to Olcker's black-and-white original. Where a correlation is high [we have bold green highlighted all correlations of .40 and above; there were none of -.40 or below], the argument is that both tests are, to that extent, measuring the same thing. Thus people who score highly on Test 1 (Addition) also score highly on Test 2 (Subtraction) [the correlation being a healthy (but far from absolute) .49]. Where a correlation is low [we have bold red highlighted all correlations between -.10 and .10], the argument is that both tests are, to that extent, NOT measuring the same thing. Thus people who score highly on Test 3 (Multiplication) are not automatically scoring highly on Test 14 (Same-Opposite) [the correlation being -.02]. One of the tests which correlates poorly with its neighbours is Test 7 (Repetition of Digits), a popular test of short-term memory, presumably because that type of resource was not heavily tapped by the other tests.

The correlation matrix was then factored down to five major factors, using Thurstone’s (1934/2005 online) “centroid method with guessed communalities” (Olckers, p4). Here are the five suggested factors …..

Factor I - "Concentration and Retentive Power": This is the "still obscure" (Olckers, p11) ability for a combination of retentiveness, alertness, and sustained attention to the problem at hand.

Factor II - Reasoning: This is the ability "to select essentials and to see relationships" (Olckers, p6).

Factor III - Speed of Performance: This is the ability to associate ideas quickly in "performing simple well-practised tasks of a continuous nature" (Olckers, p6). 

Factor IV - Directed Relational Thinking: This is the ability to manipulate symbols "according to well-known rules" (Olckers, p6).

Factor V - Residual: This factor was left undefined pending further investigation.

STUDY #2 - FIRST FOLLOW-UP: In the second study, Olckers subjected a new sample of schoolchildren to a different test battery. This time the battery contained 13 different types of arithmetic test, as follows …..

Test 1 - Four Rules A: This was a 4 minute test of "alternate rows of two-digit addition, subtraction, multiplication, and division sums with whole numbers, involving not more than one step each [which] would define the nature of the speed factor involved in computation" (Olckers, p7).

Test 2 - Simple Opposites: This was a 2˝ minute 80-item test in which subjects had to write down the first letter of the opposite of each given word [e.g. given "hot" answer "c" (for "cold")].

Test 3 - Alphabet: This was a 2˝ minute 80-item test in which subjects had to write down the letter following a given letter pair [e.g. given "MN" answer "O"].

Test 4 - Four Rules B: As Test 1, but with more complex four-step questions and two extra minutes.

Test 5 - Cancellation A: This was a 3 minute test of simple digit cancellation. Presented with a block of randomly sequenced digits, subjects had to make a line through every 9 they could find.

Test 6 - Four Rules C: As Test 4, but using a different question set.

Test 7 - Fractions: This was a 4 minute 20-item test of addition, subtraction, multiplication, and division of ordinary fractions.

Test 8 - Change: This was a 2 minute 20-item test of giving change.

Test 9 - Cancellation B: This was a 4 minute test of rule-driven digit cancellation, similar to Test 5, but more complicated. The specific task was to cancel Os, but only if preceded by 3s, UNLESS followed by a 7, in which case the 7 had to be cancelled!! The test was preceded by a number of untimed practice trials to ensure that subjects understood what was required of them.

Test 10 - Problems: This was a 5 minute 20-item test "covering the main principles involved in arithmetic" (Olckers, p8).

Test 11 - Number Series: This was a 5 minute 25-item test of series completion [e.g. given "1, 4, 9" the subject should detect that the rule was increasing squares, and continue "16, 25"].

Test 12 - Definitions: This was a 3 minute 30-item test in which subjects had to select the first letter of the word being defined from an array of five alternative possibilities [e.g. given "used for washing with" the subject should answer "s" (for "soap")].

Test 13 - Repetition of Digits: This was a 30-item test of the ability to repeat orally presented digit sequences.

The test scores were then correlated as before .....

 

1

2

3

4

5

6

7

8

9

10

11

12

1

-

-

-

-

-

-

-

-

-

-

-

-

2

.34

-

-

-

-

-

-

-

-

-

-

-

3

.30

.36

-

-

-

-

-

-

-

-

-

-

4

.67

.21

.31

-

-

-

-

-

-

-

-

-

5

.49

.40

.39

.30

-

-

-

-

-

-

-

-

6

.58

.07

.18

.64

.08

-

-

-

-

-

-

-

7

.45

.05

-.01

.42

.08

.51

-

-

-

-

-

-

8

.27

.17

-.06

.38

.02

.31

.44

-

-

-

-

-

9

.05

.12

.11

.15

.46

.08

.09

.11

-

-

-

-

10

.05

.06

.08

.13

-.01

.07

.33

.50

.02

-

-

-

11

.20

.19

.00

.28

.05

.29

.53

.56

.24

.49

-

-

12

.27

.25

.03

.18

.16

.17

.42

.21

.08

.37

.21

-

13

-.05

.12

.15

.10

.03

.14

.07

.10

.00

.16

.16

.16

NB: Colour-coding as for Study #1.

 The correlation matrix was then factored down to four major factors, as follows …..

Factor I - "Concentration and Retentive Power": Same as Study #1, Factor I.

Factor II - Speed of Performance: Same as Study #1, Factor III.

Factor III - Manipulation: Same as Study #1, Factor IV. 

Factor IV - Reasoning and Residual: Same as Study #1, Factor II, plus residual.

STUDY #3 - SECOND FOLLOW-UP: In the third study, Olckers subjected a new sample of schoolchildren to a different test battery. This time the battery contained 21 different types of arithmetic and other test, as follows …..

Test 1 - Speed of Movement: This was a test of how many short lines a subject could draw relatively accurately on a piece of paper in 15 seconds, averaged over three attempts.

Test 2 - Repetition of Digits: Same as Study #2.

Test 3 - Four Rules A: Same as Study #2.

Test 4 - Dividing Geometrical Figures: This was a 4 minute 20-item test of spatial recognition and analysis. In each item, a complex geometric figure was positioned alongside two component figures, and the subject had to trace the cutting line on the original needed to produce the two components.

Test 5 - Definitions: Same as Study #2.

Test 6 - Fractions: Same as Study #2.

Test 7 - Opposites: Same as Study #2.

Test 8 - Four Rules B: Same as Study #2.

Test 9 - Change: Same as Study #2.

Test 10 - Number Series: Same as Study #2.

Test 11 - Alphabet: Same as Study #2.

Test 12 - Three Digit Addition: This was a 4 minute test of 3-digit addition.

Test 13 - Maze: This was a 3 minute non-verbal reasoning test requiring subjects to trace their exit route on a series of 30 mazes of increasing difficulty.

Test 14 - Problems: Same as Study #2.

Test 15 - Reading: This was a 3 minute test of silent reading in which comprehension was periodically assessed by multiple-choice Cloze questions.

Test 16 - Reasoning: This was an 8 minute 15-item test "which could only be solved by eliminating the wrong answers", and which was intended to test "a different type of reasoning from that involved in arithmetical problems"  (Olckers, p13) [no further details given].

Test 17 - Part-Whole: This was a two part test, with 2 minutes allowed for each part. Firstly subjects had to select from lists of five options the names of parts of named objects, and secondly they had to select the objects which matched named parts.

Test 18 - Cancellation of 9s: Same as Study #2.

Test 19 - Compound Words: This was a 5 minute 90-item test in which given words had to be extended to form compound words [e.g. given "table" add "cloth" to make "tablecloth"].

Test 20 - Spelling: A spelling test taken from the Cape Departmental Standard VI Examination, with words either presented alone or in sentence context.

Test 21 - Arithmetic: A total of 2 hours arithmetic testing taken from the Cape Departmental Standard VI Examination.

The test scores were then correlated as before .....

 

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

2

.22

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

3

.32

.20

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

4

.09

.11

.19

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

5

.11

.08

.24

.20

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

6

.13

.13

.35

.20

.29

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

7

.19

.19

.31

.23

.25

.28

-

-

-

-

-

-

-

-

-

-

-

-

-

-

8

.11

.09

.50

.16

.23

.45

.30

-

-

-

-

-

-

-

-

-

-

-

-

-

9

.05

.12

.41

.20

.32

.48

.18

.44

-

-

-

-

-

-

-

-

-

-

-

-

10

.02

.13

.07

.40

.31

.41

.20

.26

.37

-

-

-

-

-

-

-

-

-

-

-

11

.12

.24

.24

.26

.18

.29

.45

.23

.20

.23

-

-

-

-

-

-

-

-

-

-

12

.04

.08

.53

.10

.15

.36

.28

.68

.44

.15

.25

-

-

-

-

-

-

-

-

-

13

.06

.05

.07

.45

.14

.11

.15

.13

.16

.32

.20

.14

-

-

-

-

-

-

-

-

14

.15

.16

.19

.26

.40

.38

.16

.29

.55

.49

.23

.18

.18

-

-

-

-

-

-

-

15

.10

.19

.20

.15

.23

.25

.38

.21

.17

.18

.27

.24

.23

.16

-

-

-

-

-

-

16

.05

.17

-.12

.14

.23

.21

.20

.15

.28

.29

.19

.05

.18

.31

.21

-

-

-

-

-

17

-.04

.09

.18

.25

.41

.20

.43

.23

.22

.28

.30

.15

.32

.12

.45

.30

-

-

-

-

18

.16

-.01

.31

.23

.01

.13

.13

.28

.01

.00

.11

.24

.14

.19

.06

.01

.15

-

-

-

19

.09

.09

.22

.21

.30

.19

.35

.20

.18

.19

.24

.12

.24

.15

.32

.24

.45

.14

-

-

20

.21

.23

.22

.04

.24

.25

.25

.28

.21

.18

.32

.26

.00

.26

.31

.11

.13

-.09

.16

-

21

.08

.18

.26

.19

.37

.52

.14

.34

.51

.46

.20

.27

.21

.62

.24

.22

.16

-.14

.10

.37

NB: Colour-coding as for Study #1.

As before, the correlation matrix was then factored down, and on this occasion suggested the following six factors .....

Factor I - "Concentration and Retentive Power": As in Studies #1 and #2.

Factor II - Speed of Performance: As in Studies #1 and #2.

Factor III - Manipulation: As in Study #2, Factor III.

Factor IV - Reasoning: As in Study #2, Factor IV.

Factor V - "Fluency of Verbal Association Under Restriction": This is the ability for precise use of verbal material such as reading, vocabulary, definitions and opposites. 

Factor VI - Visual Imagery and Residual: This is the ability to see the spatial characteristics of the problem at hand, and to use visuo-spatial intelligence to manipulate them.

 

3 - The Author's Conclusions 

The first conclusion was that "similar factor patterns were revealed in all three [of] the analyses" (Olckers, p19). The four most consistently seen factors were Concentration, Speed, Manipulation, and Reasoning. Olckers' second conclusion was that "there is no evidence of a general common factor in Spearman's sense" (Olckers, p20). In other words, there was no need to postulate a g-factor for mathematics, because the s-factors listed above account for the majority of the variance [although Olckers allowed for the possibility that this may not be the case in slightly younger children].

 4 - References

See the Master References List

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