DOCUMENT RESUME - ERIC · ed 036 426. author title. institution. report no pub date. note. edrs price descriptors. identifiers. document resume. se 007 530. suepes, patrick; ihrke,

ED 036 426

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INSTITUTION

REPORT NOPUB DATENOTE

EDRS PRICEDESCRIPTORS

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DOCUMENT RESUME

SE 007 530

SUEPES, PATRICK; IHRKE, CONSTANCEACCELERATED PROGRAM IN ELEMENTARY-SCHOOLMATHEMATICS--THE FOURTH YEAR.STANFORD UNIV., CALIF. INST. FOR MATHEMATICALSTUDIES IN SOCIAL SCIENCE.TR-148AUG 6942P.

EBBS PRICE MF-$0.25 HC-$2.20*COMPUTER ASSISTED INSTRUCTION, *CURRICULUM,*ELEMENTARY SCHCOL MATHEMATICS, GRADE 4,*INSTRUCTION, *MATHEMATICS, PROGRAM DESCRIPTIONSNATIONAL SCIENCE FOUNDATION

ABSTRACTREPORTED IS THE DESCRIPTION OF THE FOURTH YEAR

(1966-67) CF A LONGITUDINAL STUDY OF THE ACCELERATED PROGRAMINVOLVING 30 BRIGHT STUDENTS IN ELEMENTARY SCHOOL MATHEMATICS. THISREPORT WAS WRITTEN TO BE AS HOMOGENEOUS AS POSSIBLE WITH EARLIERREPORTS BY SUPPES AID HANSEN (1965), SUPPES (1966), AND SUPPES ANDIHRKE (1967). TABLES AND FIGURES IN THIS REPORT REFLECT THE EXPANSIONOE THE CURRICULUM AND ADDITIONAL DATA AVAILABLE FROM THE USE OFCOMPUTER ASSISTED INSTRUCTION. THE SECOND SECTION OF THIS REPORTCONTAINS A DESCRIPTION OF THE CURRICULUM FOR THE CURRENT YEAR.. THETHIRD AND FOURTH SECTIONS DESCRIBE THE GROUP COMPOSITION AND CLASSPROCEDURE. TEE FIFTH SECTION PRESENTS THE SYSTEMATIC BEHAVIORAL DATACOLLECTED. FINALLY, THE APPENDIX CONTAINS A BRIEF DESCRIPTION ANDDATA CLASSROCM PROBLEM SETS. (RP)

,

Alex*

ACCELERATED PROGRAM IN ELEMENTARY-SCHOOL

MATHEMATICS--THE FOURTH YEAR

BY

PATRICK SUPPES AND CONSTANCE IHRKE

TECHNICAL REPORT NO. 148

AUGUST 7, 1969

PSYCHOLOGY SERIES

INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCES

; (.STANFORD UNIVERSITY

1 1\-) STANFORD, CALIFORNIAIn,i

lo i

;I.Llt 1

iv)

TECHNICAL REPORTS

PSYCHOLOGY SERIES


(Place of publication shown in parentheses; If published title is different from title of Technical Report,this is also shown in parentheses.)

(For reports no. I - 44, see Technical Report no. I25.1

0 R. C. Atkinson and R. C. Mathematical learning theory. January 2, 1462. (In B. B. WcIman (Ed.), Scientific Psycholrgy. New York:-Basic Books, Inc."; 1965, ,.pp. 254 -275), -

I P. "Supdes,- E. Crotheri, and R, Weir., Application of mathematical learning theory and linguistic analysis is vowel phoneme matchirg inRussian wadi:, DeceMber 28, 1962,. ,

- , - - '- ~' R. C' Atkinson, 0:-Calfee, D. Sommer, W. Jeffrey andA. Shoemalcer. A test of tine models for stimulus compounding with cluiLleen.

January,29,-1963. (i.' exp,,OsyChol., 1964,--67, 52-58) . ., : . _= -

_. .E....Crothers.:General Maskov-models for-leaning With inter-blal'forgetting. April 8,1963:j'."1, Myers and,R,,C.,Atkinton: Choice behavior andieviad structure. May 24, 1963. (Journal math. Psychol., 1964, I 1 170-203)R.: E.' Robinson: A Set-theoretical'apprOach to empirical meaningfulness of measurement statements. June 10,1963.

-E."'CrOtheii; k: Weir and 0, Palmer. The role of transcription the learning of the or thographic representations of Russia' sounds. June,17,. 1963.-.,

P. itippeS.-" Problems of optimization in learning-a-Hit of simple Items. July 22,190. (In Maynard W. Shelly, 11 and Clenn L. Bryan (Eds.), _

HumanJiidgments andOptimilitY.`New York: Wiley., (44.- Pp. 11(426) _-

R. tAtkinson and Ei J; Crafters.- Theoretic:al note: all -or -none (earning and intertrial forgetting. July 24, 1963.= _ . ,

R. C. Calfee,'Long-term of.rati'under Probabilistic reinforcement schedules._ October 1, 1963. .- ,

"R.-C.: Atkinson and E. 4.- Crother3,. Testi of acquisition and retention, axioms,for paired-associate learning. October 25, 1963. (A comparison-= 'of-paired-associate learning models having different acquisitiOn and %.2tention axioms, .J. math. Psychol., 1964, I, 285-315)

and Gibbon. The general-gamma distribution and reaction times. November20,1963. (J. math. Psychol., 1965, 2, 1-18)62 NOrnien, Incremental learning- on random trials December 9, 1963. (J. math. Psychol., 1964, 1:331;551)

' 63 p:SuPpes;,. ,The'developinent of mathematicalconcepti hs children. February 25, 1964. (On the behavioral foundations of mathematical concepts.Monographs of the'Society for Research tn Child Development, 1965, 30, 60-96)

Suptes: Matiemiticai concept formation in children.April 10, 1964. (Amer. Psychologist, 1966, 21,139-150)- <

R: C. Calfee, R. C. Atkinson, and T. Shelton, Jr. Mathematical models for verbal learning. August 21, 1964. (In N. Wiener and J., P. Schoda- ".(Eits.);"CybeinetiCs of tie Nervous-System: Progress In Brain Research. Amsterdam, The Netherlands. Elsevier Publishing Co.,,1965.

6- J; Burke;and W. K. Estei. Paired associate learning with differential rewards. August 20, 1964. (Reward and; :Information values-of trtapidcoMei in paired asseciatelearning. (Psychol. Monogr., '1965, 79, I.21)

.67; F:41iinrien: A probabilistic- model fcr:free-responding. DeCember 14, 1964.---,

iv. K. EsteiApikli: A. Taylor.: Visual detection In relation to display size and redundancy ofcritir,e1 elements. January 25, f965,,Revised7+65. (perceptionAnd PsyClioph)isicS, 1966, 9 =16)

p....Sappes_ and J. Dimio; Foundations of stimulus-sampling theory for continuous-time processes. February 9,1965. (A. math. Psychol., 1967, .

4;62027225). - _ -.C'.:AtIclimOnand Kinchla. A learning model for forced-cLice detection experiments. February 10, 1965. (Br. .1. math stat. Psycisoi.t1965,18;184-206)

E. Crothers. --Preientition orders for items fronidifferenkcategories. March 10, 1965.. I , ,

P. 1.3uppe4 ; G. GrOen, and N. Schlag-Rey. Some models fcir response latency in paired-associates leaning. May 5,1965. (J, math. Psychol.,:1966,3, 99!-128),'

M,y;-,tevIne.'_-1:fili generalization function in the probability learning experiment. June 3,1965.b:,:Hansen and I., t; Itodgeri:: An exploratiOnOf_psyCholinguistic units in Initial, reading. July 6, 1965.B. C,Arnold.%=ivcarelated urn- scheme for a continuum of responses, July 20, 1905. .

-76- C. Izawa and VV.-IC: Estes. Reinforcement -test sequences in paired-associate learning. August I, 1965. (Psychof Reports, 196 6, IS, 879-919)L:Alehart. Pattern dissevindnation learning with Rhesus monkeys: September 1, 1965. (Psychol. Reports, 1966, 19, 311-324)

J; L., Phillips and 12:-C:2 Atkinson. The effects oldisplay size on short-term memory. August 31, 1965..79 .."R: C,-.- Atkinson and R. M. Shiffrin. Mathematical 'models fa memory and learning. September 20, 1965.

P.P. Suppei. y foundations df mathematics. October 25, 1965. (Colloques Internationaux du Centre National de to RechercheScientlfique.: Editions du Centre National de 14 Recherche Scientiflque, Paris:1967. Pp, 213-242)

P. Suppes "Computer-assisted instruction In the schools: potentialities, problems, prospects, October 29, 1965.R. A. Kincina, J. ToWnsend,I.J. Mellott, Jr:, and R., C. Atkinson. Influence of correlated visual cues on auditory signal detection.

November 2, 1965. (Perception and PsYchophysics,I966, I, 67-73)83 P. Suppes, M.- Jerman, and G. Groen. Arithmetic drills and review on a computer-based teletype., November 5, 1965. (Arithmetic 'Teacher,

April 1966, 303-309.84 P., Suppesand L, Hyman. Concept learning with non- vernal geometrical stimuli_ November 15, 1966.85 P. Holland.' A variation on the minimum chi-square test. (J. math. Psychol., 1967; 3, 377-413).86 P., Suppes. Accelerated prcxpam in eleinentary-school mathematics -- the second year. November 22, 1965. (Psychology in the Schools, 1966,

3, 2914-307)87' Lo enzel and F. Binford. Logic as a dialogical game.' November 29, 1965.88 L. Keller, W. J. Thomson, J. R. Tweedy, and R. C. Atkinson. The effects of reinforcement interval on the acquisitioi of paired-associate

responses.- December 10, 1965. (J, exp. Psychol., 1967,J3, 268 -277)89 J. 1.Yellott, Jr. Some effects on noncontingent success in human probability learning. December 15, 1965.

".' 90 P. Suppes and G. Groen. Some counting models for first-grade performance dataon simple addition facts. January 14, 1966. (In J. M. Scandura(Ed.), Research in Mathematics Education. Washington, D.- C.: NCTM, 1967. Pp. 35-43.

+ _

(Continued on inside back cover)

U.S. DEPARTMENT OF HEALTH, EDUCATION& WELFARE

OFFICE OF EDUCATIONTHIS COCUMENT HAS BEEN REPRODUCEDEXACTLY AS RECEIVED FROM THE PERSON ORORGANIZATION ORIGINATING IT. POINTS OFVIEW OR OPINIONS STATED DO NOT NECES-SARILY REPRESENT OFFICIAL OFFICE OF EDU-CATION POSITION OR POLICY.


MATHEMATICS- -THE FOURTH YEAR

by

Patrick Suppes and Constance Ihrke

TECHNICAL REPORT NO. 148

August 7, 1969

PSYCHOLOGY SERIES

Reproduction in Whole or in Part is Permitted for

any Purposr' of the United States Government


STANFORD UNIVERSITY

STANFORD, CALIFORNIA

R


MATHEMATICS- -THE FOURTH,YEAR1

Patrick Suppes and Constance Ihrke

Institute for Mathematical Studies in

the Social Sciences

Stanford University

This report describes the fourth year (1966-67) of a longitudinal study of

the accelerated program in elementary-school mathematics conducted by the

Institute. The first year of the study, including details of the procedures by

which the students were selected, was reported in Suppes and Hansen (1965). The

second year of the study was reported in Suppes (1966) and the third year was

reported in Suppes and Ihrke (1967). This report was written to be as homogeneous

as possible with the earlier ones. Results are presented in formats similar to

those used in previous reports. Additional tables and figures reflect the

expansion of the curriculum and additional data available from the use of computer -

assisted instruction. The second section of this report contains a description

of the curriculum for the current year. The third and fourth sections describe

the group composition and class procedure. The fifth section presents the

systematic behavioral data collected.

The 30 students who participated in the 1966-67 program were bright fourth

graders in the fourth year of an accelerated program in elementary-school

mathematics. This fact should be considered when reading the description of

the curriculum.

Sets and Numbers

CURRICULUM DESCRIPTION

The students continued to use the Sets and Numbers textbook series for

their basic classroom instruction. Each student proceeded from the last problem

he had completed the previous year to as far as he could progress at his

individual rate when the fourth school year ended. During the school year

students worked in Books 3B, 4A, 11.B, 5, and 6 (Books 3B, 4Aland 4B were preliminary

editions of Books 3 and Ii). Chapter descriptions are included in Table 2. The

following topics are presented in the Sets and Numbers texts: numbers and

numerals, place value, addition, subtraction, multiplication, division, sets,

fractions, measurements, geometry, problem-solving, equations and inequalities,

graphs and charts, laws of arithmetic, and logic.

The student proceeded through his Sets and Numbers work by reading examples

when a new concept was introduced and by a.king the teacher for any additiOnal

instruction he desired. The student was allowed to skip as many as half of the

('-v

problems on a text page if he felt he understood the material; but, if after

checking his work the teacher decided the student's error rate was too g)eat,

he was asked to solve additional problems on the page after he had corrected

his errors.

Probabilitz

Although the fourth- and fifth-grade Sets and Numbers z texts each contain

a chapter on probability, it was decided to give this special group of students

a more extensive introduction to probability. During the month of February

the first author taught nine sessions with each of the two classroom groups.

Session 1 dealt with an intuitive consideration of the possible outcomes

that could be expected in flipping a coin. The students were asked to estimate

how many heads they would expect in a hundred flips and in a thousand flips.

A simple experiment with 10 flips was carried out in class. The children were

assigned the homework problem of flipping a coin a hundred times and observings

how many heads were found.

2

Session 2 began with a discussion of the results of the homework

experiment. The students were encouraged to conceptualize why a certain

variability in result was obtained and also to explain why the variability

was limited. This session then turned to the problem of representing the

set of possible experimental outcomes and how events could be thought of as

sets. The particular example dealt with in detail was the set of possible

outcomes of flipping a coin twice, the event of at least one head, the event

of exactly one tail, the event of a head on the first-flip, and the event

of at most one head. The students were given the homework assignment of

finding the set of possible outcomes for three flips of a coin and descriling

the following events as subsets of this set of

of at least one head, the event of exactly two

possible outcomes:

tails, the event of

one head and at least one tail, and the event of at most two heads.

the event

at least

Session 3 discussed the homework from Session 2 and then also turned to

the discussion of the chapter on probability in Sets and Numbers,Book 5 with

an emphasis on the various types of spinners discussed in that chapter. These

simple spinners, familiar in children's games, were used throughout as an

alternative to the flipping of coins and the distribution of sex of children.

in a family. Conceptually an emphasis Was put on distinguishing between a

symmettical and a nonsymmetrical spinner. There was considerable discussion

of how one could estimate probabilities in a nonsymmetrical spinner. The

session also dealt with the problem of representing the set of possible outcomes

from spinning a two-color or three-color spinner twice.

Session 4 continued the discussion of some of the exercises in

Sets and Numbers Book 5 on probability and then turned to an extensive

discussion of the distribution of boys and girls in families of three children.

3

As before, the first problem was to work out explicitly the set of possible

outcomes, then to deal with various events and their probabilities, for

example, the probability of the event of at least two boys and the probability

H3

of the event of exactly one girl.

Session 5 dealt once again with the flipping of a coin three times but

with an emphasis on determining the probability of a number of different events

and a first introduction to the union of two events and the probability of

the union when the events are disjoint.

No. emphasis Was placed in the initial discussion on the necessity of

mutual exclusiveness holding, in order for the law of addition to hold,

but at the end of the session an example was given to show that the*

addition of probabilities will not always work, and a discussion was

begun of the conditions under which additivity would hold. The session

ended with a simple statement of the ordinary law of addition for

exclusive events.,

Session 6 concentrated on determining the number of possible outcomes

when a coin, is flipped 10 tithes. The objective was to come up with the

standard fbnailation that with n flips there are 2n possible outcomes.

A similar disaussion and a similar final result was obtained in the

case of a three-color spinner with of course the base 2 being replaced

by base 3.

Session 7 continued the discussion of the number of possible

outcomes. Spinners with various numbers of colors were considered.

The question was posed of how many flips of a coin would be needed- to have

4

about 1,000 possible outcomes. The discussion went on to consider the

number of possible outcomes for different numbers of flips, running up

to 20, and there was a discussion of how we could estimate approximately

the number of possible outcomes for number of flips above 10.

Session 8 returned to the example of three flips of a coin and

consideration of the additivity of events. The law of addition was

approached in a 'slightly different way, but the same systematic

statement was obtained as in the case of the earlier session on this

same topic.

Session 9 reviewed the various topics discussed in the previous

sessions and once again stressed the conditions under which the law of.

addition holds, with the consideration of counterexamples when the two

events being considered have a nonempty intersection.

its

5

Logic,

The logic program initiated during the summer session of 1965 and continued

during the 1965-66 school year was expanded during the 1966-67 school year.

Since the teletype machines connected to the PDP-1 computer at Stanford University

were to be used by these students for an arithmetic drill-and-practice program,

an earlier plan of presenting logic via computer-assisted instruction was

revived. Presenting the logic curriculum in a programmed format allowed students

to proceed at an individual rate through the linear structure of the program.

Lessons were prepared for two courses of study, sentential logic and elementary

algebra. Both courses used the same logic program, but had separate introductory

bracks.for'rule names and applications. For most of the year the sentential

logic stressed derivations using symbols, and the algebra emphasized numerical

derivations; however, rules from both were required for some proofs near the

end of the year. Each child alternated his course of study from one day to the

next; logic one day, algebra the next.

The fourth year's logic program was intended to be self-contained as tutorial

computer-assisted instruction at a teletype terminal, but students were able to

question a staff member who was available in the teletype room when the logic

program was running. Although a considerable amount of individual instruction

was given to some students while they were working at the terminals, very little

group instruction occurred.

The format used for the logic problems was similar to that used for

computer - assisted, instruction during the summer of 1965 and for the classroom

materials used during the school year 1965-66.

Lesson 1 of the sentential logic contained 19 problems that were written

in symbolic format with two or three premises and that required one-step proofs

6

applying modus ponendo ponens, a rule of inference familiar to all the students.

The rule was abbreviated AA for Affirm the Antecedent. The students needed to

knoW' that 'R --)S' meant iif_R then S' that 'R -,8' was a conditional sentence

whose antecedent was R and consequent was S, that 'P' was the abbreviation for

'premise'', and that the use ,`of AA required two line numbers with the line

number-of-the conditional sentence followed by the line number of its antecedent.

A period separated the line numbers. After the teletype had printed, what

the student was to derive and the given premises, the typewheel positioned

itself for the student's instructions. The student then typed the abbreviation

for the rule and the line numbers reqUired for its application. The next

information printed by the teletype was either a valid step based on the -student's

input or an error message if the student had given instructions for an invalid

step. The i;eletype proceeded to the next problem when the student had completed

the desired-derivation. :An example from LessonA is the following:

Derive: -I,'

P.

(1) K

P (2) M

P (3) K

AA1.3 (4) L.

The underlined phrase indicates what the student typed for this problem. The

remainder .of the typing was performed automatically under computer control.

Lesson 2 contained 8 more problems that had either two or three premises

and that required only a one-step proof. Mathematical sentences were included,

as-well as the usual symbols of sentential logic. Each of the 7 problems in

Lesson 3 had three premises and used modus Emends ponens. Two-step problems

were presented for the first time in this lesson.

7

The Rule of Conjunction was introduced in Lesson 4 as the rule that would

Form a Conjunction CFO. The 17 problems in this lesson involved one-step,

two - step,, and three-step derivations using modus ponendo ponens and the Rule of

Conjunction.

In Lesson 5 the Rule of Simplification was presented as two separate

commands for the student to give the computer: to derive the Left Conjunct

he typed LC, or.to derive the Right Conjunct he typed RC with a designated line

number to complete the instruction. For. example:

Derive: R

P (1) S & Q

(2) S

AA1.2 (3) R & Q

Lc3 (4) R.

The underlined sections of the problem indicate the student's input for the

derivation. There,were 21 problems in Lesson 5 that involved one-step, two-step,

and three-step derivations that used from one to five premises.

In Lesson 6 there were 20 problems that contained two, three, or four

premises using all the rules introduced up to that place in the curriculum.

The problems required from one-step to four-step derivations. Another new rule,

modus tollendo ponens, was introduced as the rule that would Deny a Disjunct

(DD). For example:

Derive: D

P, (1) A v (B & C)

P (2) D v -B

P (3) -A

DD1.3 (4) B ac; C

LC4 (5) B

DD2.5 (6) D.

8

As before, the underlined sections indicate the student's typed work, and the

teletype printed the remainder of the problem.

Lesson 7 contained 21 problems that required from one to four lines to

solve problems based on two or three premises. Another new rule, modus tollendo

tollens, was introduced as Deny the Consequent (DC). The underlined statements

represent the student's work in the following problem:

Derive: R

(1) N

P (2) -R -) -S

P (3) N -) S

AA3.1 (4) s

Dc2.4 (5) R.

At approximately this stage in the: curriculum (depending on each's:mdent's

individual rate of progress), .a multiple-choice mode was available for use at

the teletype terminals. Two inserted lessons used this multiple-choice mode

for review and practice on logical vocabulary. One new rule, Double Negation

(DN), was introduced by using the multiple-choice mode for direct instruction.

The first inserted lesson contained 20 problems and the second lesson 19 problems.

Lesson 8 contained 18 problems having from one to three premises and

required one-step through four-step derivations to derive the conclusions.

Practice in applying the rouble Negation Rule was emphasized; For example:

Derive: B

(1) -1-4A

P (2) A

DN1 (3) A -,B

AA3.2. (11) B.

The problems in Lesson 9 featured another new rule, Hypothetical Syllogism

(HS). There were 21 problems in this lesson that required from one-step through

five-step derivations. From one to three premises were provided for each problem.

9

One problem required the use of an algebraic rule in its derivation. The rule

of the Hypothetical Syllogism was applied in the following typical problem:

rerive: A, -) D

P (1) A -) B

P (2) B -)C

P (3) C D

BZ1.2 (4) A -) C

BZ4.3 (5) A -4 D.

Lesson 10 contained 27 problems with from one to five premises that

required from one-step to twelve-step proofs for solution. Many applications

of the algebra rules were necessary for the problems in this lesson. Also, the

Law of Addition, Form a Disjunction (ET), was-presented. This rule permitted

the student to type the second part of a disjunction formula. The underlined

sections indicate work typed by-the student. For example:

Derive: -S

P (1) S -(R )

P (2) R

FD2 (3) R v (T)

DC1.3 (4) -S.

In Lesson 11 some of the 17 problems required derivations and some of the

problems were presented in the multiple-choice mode. Those problems of the

multiple-choice type reviewed the vocabulary and required the student to

identify a certain type or part of a formula. The derivations contained from

one to six premises with from two to twelve lines of rule applications for the

solutions.

Lesson 12 combined both derivations and multiple-choice problems for the

introduction of two new rules that applied the Commutative Laws. The first

rule was called Commute Disjunction (CD), and the second rule was called

Commute Conjunction (CC). There were 18 problems in this lesson; the nine

10

derivations had either one or two premises and were one-step or two-steps in

length. The Rule CD was applied as follows:

Derive: A v B

P_ B

FD1 (2) B v (A)

0D2 (3) A v B.

Lesson 13 emphasized the combined use of algebra rules and logic rules.

The 27 problems included both multiple-choice problems and derivations having

one to three premises with as many as six lines of rule applications. The 16

problems in Lesson 14 followed the same format of combining multiple-choice

problems with derivations that included the use of algebra rules.

Algebra

The algebra curriculum was presented in much the same format to the students

as the logic curriculum, with the exception that rules were introduced in a

notebook written in a programmed format. This approach was initiated because

theiewai-no multiple- choice mode available when the algebra program started.

Directions written into the program instructed the student when to read

the introduction and when to solve the problems for a new rule in his-notebook.

The student then used the answer section in his notebook to check his work.

The first tWO pages of the notebook included the rule names for both the logic

and algebra programs and examples of their application. Each student had his

notebook at the teletype terminal available for reference each day.

Lesson 1 contained 10 probems in which the student practiced the rule

Number Definition (ND). (Each positive integer greater than 1 is defined as

its predecessor plus 1. Thus 2 = 1 + 1, 3 = 2 + 1, etc.) This rule was printed

with a prefix that indicated which number the machine was to present (Ind define.

For example:

11

Derive: 6 = 5 + 1

6ND (1) 6 = 5 + 1.

The underlined section shows the student's command to the computer.

Lesson 2 presented 15 problems that required the student to apply the

Rule of Number Definition and then the new rule, Definition ED), that allowed

the definition of a particular number to be substituted for (the name of) the

number in a given number sentence. A prefix number in front of the rule

abbreviation indicated the number that was to be replaced by its definition, and

a postfix number indicated which occurrence of the number in the given sentence

was to be defined. For example:

Derive: 8 = ((5 + 1) + 1) + 1

8 N D (1) 8 = 7 + 1

7D1 (2) 8 = (6 + 1) + 1

6D1 (3) 8 ((5 + 1) + 1) + 1.

Lesson 3 contained 20 problems using both the Rule of Number Definition and

the Rule of Definition for two-step to four-step derivations. Lesson 4 provided

further practice using the same rules for 15 problems that required three or four

lines of proof.

In Lesson 5 a new rule, Commute Addition (CA), was introduced. To apply

this rule to the previous line of the problem, a postfix number indicated which

occurrence of the plus sign was used for the commutation. For example:

Derive: 7 = 1 + 6

7ND (1) 7 = 6 + 1

CA1 (2) 7 = 1 + 6.

For the 20 problems in this lesson, both the Rule of Number Definition and the

Rule of Definition were used continuously.

12

Lesson ,6 contained 23 problems that required as many as four steps of proof.

The three rules available for algebra proofs were used. Lesson 7 provided

further practice with the same rules. The 22 problems required as many as seven

steps for a solution. Lesson 8 extended the use ,of the same three rules. The

13 problems needed as many as eight lines of proof for the derivation.

In. Lesson 9 a new rule, Associate Addition to the Right (AR), was introduced.

The student typed a postfix number to indicate which plus sign was to be dominant

after applying Associate Addition to the Right. For example:

I' (1) (4 + 3) + 1 = (4 + 3) + 1

AR2 (2) 4 + (3 + 1) (4 + 3) + 1.

There were 20 problems in this lesson that needed as many as five steps of proof

for solution.

Lesson 10 provided practice with all rules that had been presented. There

were 21 problems that required as many as seven steps of proof. Lesson 11

contained 11 problems that provided further practice with the same rules.

Lesson 12 contained a new rule, Inverse Definition (ID). This rule put a

number in place of its definition. A, postfix number was required to indicate

which occurrence of a number's definition was to be replaced by the number.

For example:

P (1) 5 +.1 = 5 + 1

61is2` (2) 5 + 1 = 6,

The postfix 2 indicates that the second occurrence of the definition of 6 is

to be replaced. 'There were 20 problems in this lesson and some required as

many as `seven steps of proof for a solution.

In Lesson-13 there were 17 problems that needed as many as six steps for

a derivation. Lesson 14 introduced a new rule, Associate Addition to the Left

(AL); This rule allowed the students to reassociate numbers to the left using

13

the same format as Associate Addition to the Right. There were 17 problems in

this lesson.

Thus, in these 17 algebra lessons a total of six algebraic rules of

inference were introduced. The introduction of these rules gave the students

experience with the sort of mathematical inferences that are widely used in

elementary algebra and that are rather different from the rules of sentential

inference.

Computer - assisted Arithmetic Drill and Practice

A computer-assisted drill-and-practice program for elementary-school

arithmetic instruction was included in the curriculum for this class during

1966-67. The program is described in detail in Suppes, Jerman, and Brian (1968)

These students drilled on fourth-grade problems that were prepared at five

different levels of difficulty. This curriculum was written in concept blocks,

like addition or subtraction, that required seven days of work. The first

lesson of the concept block contained a pretest for the material included. in

that concept block with problems representing all five levels of difficulty.

The student's performance on this pretest determined the level of difficulty

of his next drill. Each day's performance within the concept block determined

the level of difficulty of his following drill. On the seventh day of the

concept block a posttest designed like the pretest completed the concept block.

The next concept block started with a pretest for the new material. In

addition to the particular concept block the student was studying, approximately

five problems were reviewed according to his individual need determined by his

lowest posttest score on previous concept blocks. Each time he reviewed a

concept block his new posttest score for that block was determined by his

latest performance. A description of the concept blocks is given in Table 4.

Daily Problem Sets

During the previous school year these children drilled on arithmetic problems

for a few minutes each day in the classroom, -The same procedure was followed

this year until the teletype drill-and-practice program began operating at the

end of November. There were usually 10 to 15 problems to be Solved each day.

After the individualized teletype program started, a problem set with seldom

more than five problems was presented daily.for the remainder of the year. The

short problem sets were used for group discussion emphasizing different possible

solutions to the given problems. Special formats required for responding to

the teletype drill-and-practice program were given during these discussion

periods;

The problem sets included review, of all arithmetic operations presented in

various formats in the Sets and Numbers texts and other current texts. Number

relations, units of measure, fractions, word problems, and dictations were

included"inithese problem sets. Detailed descriptions and results of these

problem sets are in the Appendix of this report. MathematiCal games and puzzles

were also brought to the class by the, students and the teacher..

GROUP COMPOSITION

In September, 1966, the group was composed of 32 children, 15 girls and

17 boys, who were distributed among for schools in subgroups of 5, -6, 9, and 12.

The oldest child was 9.5 years old and the youngest child was 8.5 years old.

With the exception of one student who had been promoted from third grade to

fifth grade, "the students were in 'fourth grade.

In -june, 1967, 30 students, 14 girls and 16 boys, remained in the group

after '2 students moved from the school district.

15

CLASS PROCEDURE

The students attended four neighboring elementary schools, but all of the

students met at one of those schools for their mathematics instruction this year.

School buses provided by the school district transported the students from

three of the schools to the fourth school. None of the neighboring schools was

more than a 10-minute bus ride from the school used for the classes. Children

from two schools comprised each of the two class sessions that were taught by

the same teacher (Mrs. Ihrke).

In September, 1966, there were 17 members in one class session and 15 members

in the other; in June, 1967, there were 16 members in one class session and 14

members in the other. The students' total mathematics instruction was contained

in five daily class sessions that lasted from 45 to 55 minutes.

Two adjoining schoolrooms were used for each session. One of the rooms was

used as a classroom and the other room housed the 8 teletype machines used for

the computer-assisted instruction. Each day in the classroom the students

were asked to solve a set of problems that were appropriate for fourth graders.

Again this year each student worked individually through the Sets and Numbers

textbooks. Each student corrected all errors in his previous day's work

before he began any new problems. Occasionally the classroom setting was used

for group activities that included problem-solving and drill presented as games

or puzzles. Games and puzzles for individual use were available during designated

class time and for use at home.

The room adjacent to the classroom contained 8 teletype machines connected

by telephone lines to a PDP-1 computer at Stanford University. The teletypes

were spaced several feet apart against three of the walls in the room. This

arrangement allowed the students to work independently with little distraction

from each other.16

Two separate programs were presented as computer-assisted instruction.

First was the drill-and-practice program for arithmetic skills that started

during the last week of November, 1966. Each drill provided from four to ten

minutes-of practice in basi-c arithmetic skills. Often students were allowed to

do more than one arithmetic diill when class time permitted.

Second was the sentential logic which started the second week in December,

1966. Each student worked from six to ten minutes per day solving logic.

problems. In conjunction with the logic program, the, algebra program started

the fourth week of January, 1967. After that date the students automatically

alternated between the logic program and the algebra program each day and,

continued with the separate arithmetic drill program. Each student worked

4

approximately fifteen to 'twenty minutes in the teletype room daily. The

classroom teacher assigned additional arithmetic drills to students who had been

absent or needed supplementary practice. When the logic program was initiated

a staff member was available in the teletype room to answer the students'

logic questions and to direct machine problems to the Institute's computer staff.

As machine problems became less frequent and improvements were made in the

logic program, it was not necessary for someone to be in the teletype room

with the students every day. The classroom teacher supervised the students in

the teletype room with the aid of a one-way window between the two rooms.

RESULTS

Sets and Numbers Texts

For the fourth year the results show considerable differences in the number

of problems solved and the error rates for individual students. Table 1 presents

Insert Table l'about here

TABLE 1

Number of Problems Completed and Error Rate forSets and Numbers Text Material, Top Four and

Bottom Four Students

Student BooksNumber of

problems completedPercentage of

problems in error

1 5, 6 9196 5.86

2 4A, 4B, 5 8836 3.97

3 4A, 4B, 5 8655 6.15

4 4A, 4B, 5 7206 5.37

27 4A, 4B 3133 3.5728 4B, 5 2921 6.71

29 4B 2117 7.46

30 4B 157 8 7.29

°Mean (30 students) 5031.70

18

5.88

the results of the four students who completed the greatest number of problems

and the four,students who solved the least number of problems.in the Sets and

Numbers texts. Only one of the students who completed the most number of problems

in 1965-66 completed the most number of problems in 1966-67. One of the students

who completed the least number of,problems in 1965-66 completed the most number

of problems; in 1966-67. The 30 students who participated during the entire

school year-were considered,for the results in the Sets and Numbers work. The

mean percentage of errors was 5.88 for the 1966-67 year, which was an increase

of .44 per cent over 1965-66. In addition to Sets and Numbers problems, arithmetic

drills, logic and algebra problems, and daily problem sets were salved by each

student.

Table 2 shows detailed results of the students' work in the Sets and Numbers

Insert Table 2 about here

On

texts. Included are chapter descriptions, the mean number of problems completed,

range of the number of problems completed, the mean percentage of problems in

error, range of the error rates, and the number of students who solved problems

in each section. Differences in the number of problems completed occurred when

students started or finished the 1966-67 school year within a given book section

or when the teacher suggested that a stildent.,301-4e a certain portion of a given

section. Approximately two years of the Sets and Numbers curriculum separated

the fastest and the slowest students at the end of 1966-67. This is a greater

spread than existed at the end of 1965-66. Table 3 shows each student's class

-


TABLE 2

Detailed Results of Students' Work

in Sets and Numbers Texts

Chapter Book section

Mean no.problems

completed

Range ofproblemscompleted

Mean %problems

in errorRange of% error N

Book 3B

4 Commutative 73.00 73 0.00 0 1

5 Associative laws 342.00 276-408 1.59 1.45-1.72 2

6 Shapes and sizes 8.00 8 0.00 0 1

7 Distributive law formultiplication

193.67 116-340 14.90 3.45-33.60 3

8 Lines of symmetry 70.00 18-122 0.82 0-1.64 2

9 Word problems 129.67 115-137 16.51 8.70-9.45 3

Book 4A

1 Review of sets, addition,and subtraction

195.43 59-219 5.97 1.38-12.39 7

2 Review of geometry 44.00 44 12.66 2.27-22.73 7

4

Addition and subtractionGeometry

363.0084.18

64-47537-10e

8.9812.86

3.36-13.474.90-23.08

911

5 Review of multiplicationand division

280.09 238-309 3.95 1.26-11.94 n.

6 Laws of arithmetic 797.41 403 -896 2.99 1.12-7.47 12

7 Shapes and sizes 45.54 16-48 8.97 0-31.25 13

8 Distributive law for

multiplication

669.12 97-910 3.91 0.93-11.34 17

9 Applications 380.59 285-367 - 6.84 3.10 -17.92 17

10 Distributive law fordivision

466.0 200-540 5.26 1.3E:1.13.55 19

Book 4B

1 Circles and lines 85.81 44-90 3.69 0 -25.56 21

2 Subsets and less than 58.79 15-62 8.46 0-22.58 19

3 Fractions 350.05 125-315 4.75 0.40 614.67 19

4 Division 27.00 27 0.00 0 1

5 Average 54.37 26-60 15.93 3.85-29.31 19

6 The number line 272.00 93-333 10.29 1.83-33.60 19

7 More about geometry 109.35 36-127 2.99 0-9.45 17

8 Review of the commutative,associative, anddistributive laws

496.20 94-584 4.80 1.71-9.62 15

9 Word problems 3.41.50 125-146 12.66 6.16-26.03 14

10 More about sets 162.64 130-187 8.24 2.53-22.78 14

11 Logic 94.00 94 0.00 0 1

Introduction to long

division 1,176.67 987-1474 4.52 0.75-9.53

Book 5

1 Sets 179.05 66-195 7.24 2.06-13.64 20

2 Laws of arithmetic 455.94 180 -488 5.55 1.84-10.86 18

3 Multiplication and division 498.94 111-70e 10.84 2.26-19.54 17

4 Geometry 167.08 6-209 3.76 0-7.18 12

5 Fractions 515.25 444-529 4.69 1.91-7.75 8

6 More divisions 297.25 80-528 7.35 0-13.13 8

7 Measuring 320.50 235-345 14.00 8.41-17.70 6

8 More fractions 459.50 144-569 6.47 2.99 -10.37 6

9 Space figures 150.00 150 4.17 2.00 -10.67 4

10 Systems of numeration 432.75 420-437 6.39 3.81-9.38 4

11 Integers 209.33 1 134-246 4.91 2.99-5.28 3

12 A coordinate 213.00 9-319 4.37 0-9.97 3

13

_system

Graphs 156.0o 156 5.77 5.13-6.41 2

14 Mathematical sentences 240.50 145-336 6.37 4.46-8.28 2

15 Using fractions 436.00 436 12.39 0 1

16 Decimal fractions 396 396 6.82 0 1

17 More geometry 212 212 4.25 0 1

18 Logic 176 176 2.27 0 1

19 Probability 57.76 26-94 7.42 o-46.81 29

20 Graphs and functions 129 129 12.40 0 1

21 More about sets 119.50 42-l97 6.3.3. 5.08-7.14 2

Book 6

1 Sets 240 4.17 0 1

2 Laws of arithmetic 438 3.65 0 1

3 Fractions 331 5.44 0 1

4 Geometry 201 3.98 0 1

5 Factors and multiples 160 2.50 0 1

20

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s

ranking for the Sets and Numbers text work and other areas of the curriculum.

Logic and Algebra

The,number of logic problems solved by the students ranged from 162 to 285

problems. The mean number solved was 220.5 problems. From 130 to 244 algebra

problems were solved with 194.7 the mean. Some further details about the results

of the logic program 'are reported in Suppes and Morningstar (in press).

Computer - assisted Drill and Practice-

Table 4 shows the mean class performance on pretests and posttests for the


aI

various concept blocks. The data are based on the performances of the 30 students

who participated in the class throughout the year. Block 16 was the last concept

block for most of the students, and some of the students were not able to finish

that block before the end of the year. For certain concept blocks, complete data

for 30 students were not available. Concept Blocks 22, 24, 25, and 26 were special

blocks inserted into the curriculum during the school year. Figure 1 shows the

NO

Insert Figure 1 about here

students' mean performance, on pretests and posttests. One student showed an

average decline on posttests compared with pretest performances. All other students

showed definite improvement for the combined performances of tests.

Figure 2 -shows the spread in performances on pretests and posttests. Most of

Insert Figure 2 about here

the scores are clustered above 8o per cent correct. Again, the improvement on

posttests is a general rule.

22

TABLE

Drill-and-practice Pretests and Posttests

Conceptblock Description

Mean % correctPretest N

Mean % correctPosttest N

1 Addition 78.9 30 87:3 30

2 Subtraction 84;4 30 94.7 30

3 Subtrabtion 87.2 30 90.7 30

4 Addition 94 8 3o 96.0 30

5 Addition, Subtraction 8oio 30 87.3 30

6 Wasures 81.5 30 89'.4 29

8 Mixed drill 89.3 29 89.8 29

9 Laws of arithmetic 78.3 29 91.5 30

10 Division 91.1 30 95:8 30

11 Multiplication 92.0 30 96.7 30

12 Fractions 60.7 30 72.2 30

13 Mixed drill 74.6 30 89.2 29

15 Laws of arithmetic 85.4 30 87.7 29

16 Fractions 76.1 29 87.0 12

22 Mixed drill 88.6 29 92.8 29

24 Division 75.4 30 89.6 30

25 Mixed drill 96.1 30 96,8 30

26 Mixed drill 96.5 29 97.1 30

23

100

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I

90

80

70

60

Z3t 50a.

40

303040 50 60 70 80 90 100

Percent Correct Pretest Scores

$ 1 * Xao

$ IF IF -41

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=1111M

Fig. 2. Pretest and Posttest performance for each student

on each concept block.

25

Standardized Tests

Table 5 shows the results of the standardized tests administered during the


school year. _The.achool and College Ability Test (SCAT) measured the students'

verbal and mathematical abilities and the Stanford Achievement Tests (SAT) measured

the students' progress in mathematics. A wide range occurred for each section of

each test. Percentile ranks were based upon the grade level of the students at

the time the tests were administered.

The comparison group which is one year younger and was selected by the same

instruments as the accelerated group completed its third year of mathematics

(see Sears, Katzjand Soderstrum (1966) for details about the comparison group).

In the spring of 1967, these students were given the Stanford Achievement Test,

Arithmetic Primary II. Some grade-level results for the accelerated class and the

comparison group, based on the different forms of the Stanford Achievement Tests

for the spring of 1967, follow, The accelerated group had a mean score of 6.4

years for computation (1.5 years above grade level) and a mean score of 7.7 years

for concepts (2.8 years above grade level). The comparison group had a mean score

of 3.9 years for computation (at grade level) and a mean score of 5.5 years for

concepts (1,6 years above-grade level). One student in the accelerated class

was below grade level in computation (-0.6 year). -In the comparison group 14

students were below grade level in computation (range -1.6 to -0.2 years) and one

student was below grade level in concepts (-0.8 years).

A study of the personality and social development of both groups of students

was continued under the direction of Dr, Pauline S. Sears, and the results will

be published elsewhere.

26

TABLE 5

Standardized Tests

Test SectionMean

percentilePercentilerange Date

School and Verbal 92.7 31.0-99.9 Fall, 1966College Ability Quantitative 97.5 70.6-99.9 Fall, 1966Test (total) 97.0 65.8-99.9 Fall, 1966

Stanford Computation 65.9 34-98 Fall, 1966Achievement Concepts 96.2 66-99 Fall, 1966Test, Form W Applications 92.5 58-99 Fall, 1966

Stanford Computation 82.4 28-99 Spring, 1967Achievement Concepts 96.8 74-99 Spring, 1967Test, Form X Applications 94.8 77-99 Spring, 1967

27

REFERENCES

Suppes, P. Sets and numbers, Book 3B. Syracuse: L. W. Singer, 1963.

Suppes, P. S- ets and numbers, Books 4A and 4B. Syracuse: L. W. Singer, 1964.

Suppes, P. & Hansen, D. Accelerated program in elementary-school mathematics- -

The first year. Psychology in the Schools, 1965, 2, 195-203.

Suppes, P. Accelerated program in elementary-school mathematics--The second year.

Psychology in the Schools, 1966, 3, 294-307.

Suppes, P. Sets and .numbers, Book 5. Syracuse: L. W. Singer, 1966.

Suppes, P. & Ihrke, C. Accelerated program in elementary-school mathematics- -

The third year. Psychology in the Schools, 1967, 4, 293-309.

Suppes, P., Jerman, M., & Brian, D. Computer-assisted instruction: Stanford's

1965-66 arithmetic program. New York: Academic Press, 1968.

Suppes, P. & Morningstar, M. Computer-assisted instruction: Stanford's 1966 -67 -

elementary-mathematics programs. New York: Academic Press, in press.

Cooperative School and College Ability Tests; School Ability Test, Form 5A.

Princeton: Educational Testing Service, 1956.

Stanford Achievement Test, Intermediate I Arithmetic Tests, Forms W and X.

New York: Harcourt, Brace & World, Inc., 1964.

28

Footnote

1The work reported here has been supported by the National Science

Foundation (Grant G-18709). Participating members of the staff in addition

to the authors were Frederick Binford, Dow Brian, Jamesine Friend, John Gwinn,

Max Jerman, and Phyllis Sears.

29

1

APPENDIX

Data on Daily Classroom Problem Sets 1966-67

Drill Possible Mean Number ofnumber Description score correct students

1 Mixed drill; all operations 15 62.5 32

2 Subtraction, check by addition 5 95.6 32

3 Mixed addition, subtraction;check subtraction by addition 8 93.3 28

li. Mixed addition, subtraction;check sub - traction by addition 8 93.5 31

5 Mixed addition, subtraction;check subtraction by addition 8 91.9 31

6 Multiplication by 1 digit 12 95.0 30



9 Mixed drill; addition,subtraction, multiplication 10 92.4 29











20 Nixed drill; multiplication,subtraction, addition 12 94.8 27

21 Nixed drill; multiplication,subtraction, addition 12 92.8 31

22 Mixed drill; multiplicationsubtraction, addition 12 94.6 31



30

Drill Possible Mean % Number of

number Description score correct students

25 Mixed drill; multiplication,subtraction, addition 12 96.8 23



28 Mixed drill; multiplication,subtraction, addition 12 90,8 29

29 Mixed drill; multiplication,addition, subtraction 12 90.1 32


31 Mixed drill; multiplication,addition, subtraction 12 90.3 32

32 Multiplidation; units of measure 12 97.0 25

33 Mixed drill; multiplication,addition, subtraction, units

of measure 12 86.6 31

34 Mixed drill; multiplication,subtraction, addition, units


35 Mixed drill; multiplication,division, addition 12 86.6 31

36 Mixed drill; multiplication,subtraction, addition, units


37 Multiplication by 1 digit; units


38 Mixed drill; multiplication by 1and 2 digits, addition,subtraction, units of measure 12 81.8 28

39 Mixed drill; multiplication,subtraction 12 98.4 31

1,0 Mixed drill; multiplication,Subtraction, addition, dictation 14 95.9 28

41 Inequalities; dictation: addition,

subtraction 95.0 30

42 Mixed drill: all operations;

dictation: addition, subtraction 14 87.6 19

31

Drill Possible Mean Number of


43 Inequalities: multiplication,division; dictation: addition,

subtraction 14

44 Inequalities: multiplication,division; dictation:subtraction, addition 14

45 Inequalities: multiplication,division; dictation:subtraction, addition 14

46 Inequalities: multiplication,

division; dictation: addition,

subtraction 14

47 Multiplication: missing factors;

dictation: subtraction,

multiplication 14

48 Inequalities: multiplication;

dictation: sums of money,

multiplication 14 93.1

49 Mixed drill: addition,

subtraction; dictation:multiplication by 1 and 2

digits 14 86.9 32

50 Inequalities: multiplicationand division; dictation:differences of money, addition 14 85.1 32

51 inequalities: multiplication;dictation: differences of

money, multiplication by 1

digit 14 90.4 31

52 Mixed drill: addition,

subtraction, multiplication;dictation: multiplication 14 89.1 31

53 Inequalities: multiplication;dictation: differences of

money, multiplication 14 91.2 30

54 Inequalities: multiplication;dictation: multiplication 14 91.2 30

55 Inequalities: multiplication 12 94.9 31

56 Inequalities: multiplication;

dictation: differences of

money, addition 14 93.6 31

57 Multiplication by 1 and 2

digits 12 86.8 31

87.7 29

91.2 30

87.2 28

92.1 31

93.6 31

32

Drillnumber Descriptiom

Possible Mean % Number of

score correct students

58 Multiplication by 1 and 2digits 12 93.1

59 Multiplication by 1 and 2digits 12 80.9 28

60 Mixed drill : addition,subtraction, logic 12 62.3

61 Logic; multiplication by1 digit 10 86.5 31

62 Inequalities: division;commutative laws for'

addition and multiplication 1k 85.9 32

63 Mixed drill: sums anddifferences of money;multiplication by 1 digit 80.1

64 Logic; multiplication by 1digit 9 61.7 29

65 Multiplication: 3 and 4digits by 1 digit 8 89.,6 29

66 Multiplication: 3 and 4

digits by 1 digit 8 85.4 30

67 Division by,1 digit 10 97.1 31

68 Multiplication: 4 digitsby 1 and 2 digits 4 75.-8 31

69 Multiplication by 1 and2 digits 8 71.4 31

70 Multiplication: 3 and 4digits by 1 and 2 digits 4 85.5 31

71 Multiplication: 3 and 4digits by 1 and 2 digits 4 81.8 30

72 Mixed drill: addition and

multiplication 4 89.3 30

73 Multiplication: 4 digitsby 1 and 2 digits 4 93.5 27

74 Multiplication: 4 digits

by 1 digit 4 78.8 27

75 Multiplication: 4 digits by1, 2, and 3 digits 71.5 29

76 Mixed drill: subtraction,multiplication by 1 digit 60.3 27

77 Mixed drill: addition, ,

subtraction, multiplication 4 75.0 25

33

Drill Possible Mean Number of



digits 4 75.o 31

79 Mixed drill: multiplicationand subtraction (some

money) 4 74.3 29

80 Mixed drill: multiplication;subtraction and division of

money 4 83.5 29

81 Mixed drill: multiplication,subtraction (some money) 4 86.0 25

82 Mixed drill: multiplication,subtraction (some money) 5 80.0 25

83 Mixed drill: multiplication,subtraction, addition(some money) 5 75.8 28

84 Multiplication by 2 digits 1 65.5 29

85 Dictation: addition of money,

multiplication 2 76.0 23

86 Mixed drill: all operations 5 81.8 22

87 Mixed drill: multiplication,addition, subtraction 5 89.6 29

88 Dictation: subtraction ofmoney; multiplication by2 digits 2 69.5 28

89 Addition of money;multiplication by 2 digits 2 70.0 30


digits 2 70.5 17


digits 2 72.0 27

92 Mixed drill: subtraction of

money; multiplication by2 digits 2 65,0 27

93 Addition of money, multiplicationby 1 digit 2 84.5 28

94 Multiplication by 2 digits 2 614- 29



digits 2 65.5 26

97 Multiplication by 1 digit;subtraction of money 2 85.5 17

34-

Drillnumber Description!

98 Addition of money;multiplication by 2 digits

99 Money: subtraction,multiplication by 1 digit

100 Multiplication by 1 digit(money) and 2 digits

101 Multiplication by 1 digit(money) and 2 digits

102 Mhltiplication by 1 digit(money) and 2 digits

103 Multiplication by 2 digits;division; checked bymultiplication

104 Division, checked bymultiplication; multiplicationby 2 digits

105 Division, checked bymultiplication; multiplicationby 2 digits

106 Mhltiplication of money by 1digit; division checked bymultiplication

Multiplication of money by 1digit; division checked bymultiplication

108 Addition of money; multiplicationby 2 digits

109 Division checked by multiplication;multiplication by 2 digits

110 Addition; multiplication by2 digits

111 Multiplication of money by 2digits; division, checked bymultiplication

Possible Mean Number ofscore correct students

112 Additicin; division, checked bymultiplication

113 Multiplication by 1 digit (money)and 2 digits

114 Division, Checked by multiplication;multiplication by 2 digits

115 Dictation: addition; multiplicationby 2 digits

35

2 70.5 29

78.5 30

70.5 29

2 59.5 26

65.5 29

2 65.0 30

2 65.5 29

2 85.0 30

92.5. 26

2 81.0 29

65.5 29

2 77.0 28

2 75.0 30

70.5 29

89.0 27

2 80.5 28

2 67.0 29

2 75.0 30

Drill Possible Mean % Number of


116 Division checked bymultiplication; multiplicationby 1 digit

117 Multiplication by 2 digits;division, checked bymultiplication

118 Multiplication of money by2 digits; division checked bymultiplication

119 Division checked bymultiplication; multiplication

by 2 digits

120 Multiplication by 2 digits;division checked by

multiplic.ation

121 Multiplication of money by 2digits; division

122 Multiplication of money by 2digits; division checked by

multiplication

123 Multiplication by 2 digits

2 79.5 29

2 65.0 27

2 611.5 24

2 75.0 28

2 60.5 28

2 72.0 27

2 57.0 28

(money) and 3 digits 2 39.5 28


125 Multiplication by 3 digits;

division 2 81,0 21

126 Multiplication by 3 digits;

division 2 76.0 27

127 Division; multiplication by 3digits 2 84.o 28

128 Division; multiplication by 3digits 2 78.0 25

129 Division; multiplication by 3digits; evivalent fractions 3 68.0 27

130 Division; multiplication by 3digits; equivalent fractions 3 70.3 28

131 Division; multiplication 1y 3

digits; equivalent fractions 3 71.8 27

132 Division; multiplication by 3digits; equivalent fractions 3 76.8 30

133 Division; multiplication of moneyby 2 digits; equivalentfractions 3 76.7 30

36

Ii

Drillnumber Description;

134 Subtraction; division of money;equivalent fractions 3 80.0 30

135 Dictation: mixed drill; unitsof measure 10 76.2 29

136 Dictation: mixed drill; unitsof measure 10 65.7 30

137 Dictation: .mixed drill; unitsof measure 10 76.3 30

138 Dictation: mixed drill; unitsof measure, division 10 65.5 20

139 Dictation: mixed drill; unitsof measure, division 10 64.3 28

140 Dictation: _equivalent fractions 5 76.2 26

141 Multiplication by 2 digits;division; equivalent fractions 3. 85.0 29

142 Mixed drill; units of measure,division 5 74.7 19

143 Multiplication by 2 digits;division; equivalent fractions 3 72.2 18

144 Division by 2 digits; multiplicationof money by 2 digits, equivalentfractions 3 56.7 30

145 Division by 2 digits; multiplicationby 2 digits; equivalent fractions 3 75.0 28

146 Division by 2 digits; checked bymultiplication 2 55.5 27

147 Division, checked by multiplication,multiplication of money by 2digits; equivalent fractions 3 60.3 27

148 Division, checked by multiplication;equivjalent fractions 3 82.7 27

149 Division by 1 and 2 digits 2 72.0 27

150 Division checked by multiplication;equivalent fractions 3 75.3 27

151 Division checked by multiplication 2 63.8 29

152 Division checked by multiplication;equivalent fractions 3 81.1 23

153 Division by 2 digits;multiplication by 3 digits 2 62.1 29

154 Mixed drill; subtraction, additionof money, multiplication with afraction as 1 factor 3 74.3 26

Possible Mean %, Number of

score correct students

37

Drill Possible Mean % Number ofnumber Description score correct students

155 Subtraction of mixed numbers;division checked bymultiplication 2 69.3 26

156 Multiplication by 2 digits;subtraction of mixed numbers 2 78.3 23

157 Subtraction of mixed numbers;multiplication by 3 digits 2 67.4 23

158 Multiplication with a fractionas 1 factor; division by 2digits, checked bymultiplication 2 77.8 27

159 Division by 2 digits, checkedby multiplication;

multiplication with a fractionas 1 factor 2 77.1

160 Multiplication of money by 2digits; division by 2 digits,checked by multiplication 2 65.2 23

27

Lr

LI

IJ

38

(Continued from inside front cover)

91 P. Suppes. Information processing and choice behavior. January 31,1966.

92 G. Groen and R. C. Atkinson. Models for optimizing the learning process. February 11,1966. (Psychol. Bulletin, 1966, 66, 309-320)

93 R. C. Atkinson and D. Hansen. Computer-assisted instruction in initial reading: Stanford project. March 17, 1966. (Reading Research

Quarterly, 1966, 2, 5-25)94 P. Suppes. Probabilistic inference and the concept of total evidence. March 23, 1966. (In J. Hintikka and P. Suppes (Eds.), Aspects of

Inductive Logic. Amsterdam: North-Holland Publishing Co., 1966. Pp. 49-65.95 P. Suppes. The axiomatic method in high-school mathematics. April 12, 1966. (The Role of Axiomatics and Problem Solving in Mathematics.

The Conference Board of the Mathematical Sciences, Washington, D. C. Ginn and Co. , 1966. Pp. 69-76.

96 R. C. Atkinson, J. W. Brelsford, and R. M. Shiffrin. Multi- process models for memory with applications to a continuous presentation task.

April )3. 1966. W. math. Psychol., 1967, 4, 277-300).97 P. Suppes and E. Crothers. Some remarks on stimulus-response theories of language learning. June 12, 1966.

98 R. 8jork. All-or-none subprocesses in the learning of complex sequences. al. math. Psychol., 1968,1, 182-195).

99 E. Ga nmon. The statistical determination of linguistic units. July 1, 1966.

100 P. Suppes, L. Hyman, and M. Jerman. Linear structural models for response and latency performance in arithmetic. In J. P. Hill (ed.),

Minnesota symposia on Child Psychology. Minneapolis, Minn.:I967. Pp. 160-200).

I 01 J. L. Young. Effects of intervals between reinforcements and test trials in paired-associate learning. August I, 1966.

102 H. A, Wilson, An investigation of linguistic unit size in memory processes. August 3,1966.

103 J. T. Townsend. Choice behavior in a cued-recognition task. August 8,1966.

104 W. H. Batchelder. A mathematical analysis of multi-level verbal learning. August 9,1966.

105 H. A. Taylor. The observing response in a cued psychophysical task. August 10, 1966.

106 R. A. Blot. Learning and short-term retention of paired associates in relation to specific sequences of interpresentation intervals.

August II, 1966.

107 R. C. Atkinson and R. M. Shiffrin. Some Two-process models for memory. September 30,1966,

108 P. Suppes and C. Ihrke. Accelerated program in elementary-school mathematics- -the third year. January 30, 1967.

109 P. Suppes and I. Rosenthal-Hill. Concept formation by kindergarten children in a card-sorting task. February 27, 1967.

110 R. C. Atkinson and R. M. Shiffrin. Human memory: a proposed system and its control processes. March 21,1967.

111 Theodore S. Rodgers. Llnyuistic considerations in the design of the Stanford computer-based curriculum in initial reading. June 1,1967.

112 Jack M. Knutson. Spelling drills using a computer-assisted instructional system. June 30,1967.

113 R. C. Atkinson. Instruction in initial reading under computer control: the Stanford Project. July14, 1967.

114 J. W. Brelsford, Jr. and R. C. Atkinson. Recall of paired-associates as a function of overt and covert rehearsal procedures. July 21,1967.

115 J. H. Stelzer. Some results concerning subjective probability structures with semiorders. August 1,1967

116 D. E. Rumelhart. The effects of interpresentation intervals on performance in a continuous paired-associate task. August 11,1967.

117 E. J. Fishman, L. Keller, and R. E. Atkinson. Massed vs. distributed practice in computerized spelling drills. August 18,1967.

118 G. J. Groen. An investigation of some counting algorithms for simple addition problems. August 21,1967.

119 H. A. Wilson and R. C. Atkinson. Computer-based instruction in initial reading: a progress report on the Stanford Project. August 25,1967.

120 F. S. Roberts and P. Suppes. Some problems in the geometry of visual perception. Augast 31,1967. (Synthese, 1967, 17, 173 -201)

12 I D. Jamison. Bayesian decisions under total and partial ignorance. D. Jamison and J. Kozielecki. Subjective probabilities under total

uncertainty. September 4,1967,

122 R. C. Atkinson. Computerized instruction and the learning process. September 15, 1967.

123 W. K. Estes. Outline of a theory of punishment. October 1,1967.

124 T. S. Rodgers. Measuring vocabulary difficulty: An analysis of item variables in learn.ng Russian - English and Japanese-English vocabulary

parts. December 18,1967.

125 W. K. Estes. Reinforcement in human learning. December 20,1967.

126 G. L. Wolcird, D. L. Wessel, W. K. Estes. Further evidence concerning scanning and sampling assumptions of visual detection

models. January 31,1968,

127 R. C. Atkinson and R. M. Shiffrin. Some speculations on storage and retrieval processes in long -term memory. February 2, 1968.

128 John Holmgren. Visual detection with imperfect recognition. March 29,1968. .

129 Lucille B. MI odnosky. The Frostig and the Bender Gestalt as predictors of reading achievement. April 12,1963.

130 P. Suppes. Some theoretical models for mathematics learning. April 15, 1968. (Journal of Research and Development in Education.

1967, 1 , 5-22)131 G. M. Olson. Learning and retention in a continuous recognition task. May 15, 1968.

132 Ruth Norene Hartley. An investigation of list types and cues to facilitate initial reading vocabulary acoaisition. May 29, 1968.133 P. Suppes. Stimulus-response theory of finite automata. June 19, 1968.

134 N. Moler and P. Suppes. Quantifier-free axioms for constructive plane geometry. June 20, 1968. (In J. C. H. Gertetsea and

F. Oort (Eds.), Compositio Mathematica. Vol. 20. Groningen, The Netherlands: Wolters.Noardhoff, 1968. Pp. 143 -152.)

135 W. K. Estes and D. P. Horst. Latency as a function of number or response alternatives in paired-associate learning. July I, 1968.136 M. SchlagRey and P. Suppes. High-order dimensions in concept identification. July 2, 1968. (Psycho). Sci., 1968, II, 141-142)

137 R. M. Shiffrin. Search and retrieval processes in long-term memory. August 15, 1968.138 R. D. Freund, G. R. Loftus, and R.C. Atkinson. Applications of multiprocess models for memory to continuous recognition tasks.

December 18, 1968.139 R. C. Atkinson. Information delay in human learning. December 18, 1968.

140 R. C. Atkinson, J. E. Holmgren, and J. F. Juola. Processing time as influenced by the number of elements in the visual display.

March14 , 1969.

141 P. Suppes, E. F. Loftus, and M. Jerman. Problem-solving on a computer-based teletype. March 25,1969.

142 P. Suppes and Mona Morningstar. Evaluation of three computer-assisted instruction programs, May 2,1969.

143 P. Suppes. On the problems of using mathematics in the development of the social sciences. May12, 1969.

144 Z. Cantor. Probabilistic relational structures and their applications. May 14 , 1969.

145 R. C. Atkinson and T. C. Wickens. Human memory and the concept of reinforcement. May 20, 1969.

146 R. J. Titiev, Some model-theoretic results in measthement theory. May 22, 1969.

147 P. Suppes. Measurement: Problems of theory and application. JuneI2, 1969,

148 P, Suppes and C. fluke. Accelerated prog,am in elementary-school mathematics--the fourth year. August 7, 1969.

Documents

DOCUMENT RESUME - ERIC · ed 036 426. author title. institution. report no pub date. note. edrs price descriptors. identifiers. document resume. se 007 530. suepes, patrick; ihrke,