Part a: Paul likes to play dominoes, but he never puts them back in their ...... P. and Hill, S. (1964) First Course in Mathematical Losic, New York: Blaisdell.
NITSA
CHILDREN'S
HADAR
CONDITIONAL
REASONING
An Investigation o f Fifth Graders' Ability to Distinguish Between Valid and Fallacious Inferences* ABSTRACT. This study stemmed from a desire to redress the distorted view of mathematics in the elementary curriculum, created by the current imbalanced emphasis on computational rules and some applications, but very little logical analysis. The study is intended to show that fifth-grade students can significantly improve their use of logical analysis through a suitable instructional unit taught under ordinary classroom conditions. Concrete teaching materials were developed, through several trials and revisions, to familiarize students with the distinction between the valid inference patterns - Modus Ponendo Ponens and Modus Tollendo Tollens (AA, DC), and the fallacious ones - AffLrming the Consequent and Denying the Antecedent (AC, DA). No formal rules were taught. The experimental unit was implemented four to five times a week for 23-25 sessions, by 4 ffffth-grade teachers in their ordinary classes. The teachers participated in a twelvehour pretraining workshop. A pretest/posttest treatment/no-treatment design was applied to assess resulting improvement in students' conditional reasoning ability. The sample consisted of 210 fifth graders in a suburban area, 104 in 4 experimental classes and 106 in 4 control classes. A written group test was developed, through trials and revisions. Test items are formulated with a reasonable hypothetical content. Each item includes two premises: the first a conditional sentence, and the second either its antecedent, its consequent, or the negation of one of these, thus determining the logical form: AA, DC, AC, or DA. The question following the premises is stated positively. AA and DC are answered correctly by 'yes' or 'no': AC and DA by 'not enough clues' (NEC). The test contains 32 randomly-ordered three-choice items, eight in each logical form (two of the eight in each of the four possible modes in which negation may or may not occur in the antecedent or consequen0. No sentential connective other than negation and conditional appears in the premises. Test/retest reliability was 0.79. Experimental and control group pretest performance levels did not differ (a = 0.05). More than 78% of the answers on AA and DC, and fewer than 33.1% on AC and DA, were correct. Overall pretest mean scores were 54.3% and 53.8% for the experimental and control groups respectively. There was a significant difference (a = 0.01) between the experimental and control groups' posttest overall performance - 74.7% and 55.4%, respectively. There was no significant change in the control group's pretest and posttest performance levels on any logical form, or for the experimental group's on AA and DC. However, on AC and DA the two groups' gain scores were found significantly different. Negation mode, unlike logical form, was not found to be independently influential in analyzing test scores, but interacted with logical form. There was a pretest/posttest increase of 3.5 in experimental group frequency (percentaged) of incorrect NEC answers (AA and DC). As NEC appeared infrequently on the pretest, this increase was interpreted as learning that NEC is an acceptable answer. Separaling out this effect from the percentaged frequency of correct NEC answers (AC and DA) left a pretest/posttest average increase of 37.8. This increase was attributed to learning when NEC is correct. * This study was supported by NSF grants Nos. GW 7659 and PES 74-018450 and by an AAUW International Fellowship. It was carried out at the University of California, Berkeley, in 1974/75 under supervision of Professor Leon Henkin.
Educational Studies in Mathematics 8 (1977) 413-438. All Rights Reserved Copyright © 1977 by D. Reidel Publishing Company, Dordreeht-Holland
414
NITSA HADAR
Teachers were excited at the beginning, frustrated in the middle, and felt competent and involved in the project at the end. They felt the teaching should be less condensed. The majority of the students reacted positively to most parts of the experimental unit. However, some thought the unit as a whole was too repetitive and boring. No correlation was found between learning logic through the experimental unit and standard school achievement as measured by the Stanford Achievement Test (SAT). High, average, and low SAT achievers of the experimental group did not differ significantiy in their pretest and posttest gain scores. Results of the study call for further investigation of the value and usefulness of teaching various parts of logic as an ordinary part of the elementary mathematics curriculum.
PART I AN INTUITIVE APPROA CH TO THE LOGIC OF IMPLICATION* 1. G O A L S OF T H E S T U D Y Reasoning from conditional premises is widely recognized as basic for logical analysis. However, extensive psychological research indicates a substantial need for improvement in young children's, as well as in adults', conditional reasoning ability. O'Brien and Shapiro (1968) found that children aged six to eight experienced great difficulty when challenged to discriminate between a necessary conclusion and one that does n o t necessarily follow from conditional premises. When no logically-necessary conclusion existed, these students were rarely able to perform above the chance level. Further, growth in this ability over the three-year span studied was negligible. In a later study (1970), O'Brien tested a sample of upper-middle-class children of ages six through thirteen. He found that in recognizing logical necessity, i.e., in the ability to apply the rules of inference known as 'Modus Ponendo Ponens' and 'Modus Tolendo Tollens'**, subjects had little difficulty this ability leveling off at six to eight years of age. However, students aged 6 - 1 3 exhibited considerable difficulty in testing for logical necessity, i.e., in
* This is a revised paper. The author wishes to thank Professor L. Henkin, Professor H. Freudenthal and Professor S. Avital for their remarks, upon the basis of which some revisions were made in this paper. The author wishes also to thank Dr. M. Finegold for editing this sequence of papers. ** a) A conditional sentence is a declarative (English) sentence of the form 'If p, then q', where p, q stand for two declarative sentences. The p-part, which starts right after the word 'if', is called the antecedent. The q-paxt, which starts right after the word 'then' is called the consequent. (Footnote continues on page 415.)
CHILDREN'S CONDITIONAL
REASONING
415
avoiding the fallacies known as 'Affirming the Consequent' and 'Denying the Antecedent'. Many other studies show similar findings from a variety of age-groups and population-samples. (Ennis and Paulus 1965, Paulus 1967, Roberge 1970, Evans 1972, Wason and Johnson-Laird 1972, O'Brien 1973, Osherson 1974, Eisenberg and McGinty 1974). The study described in this sequence o f three papers was set up to find out whether elementary school children's hypothetical-deductive ability can be altered by systematic intervention. The three major stages of the study were as follows: 1. A unit in conditional reasoning for the intermediate elementary grades was developed. 2. Pretrained elementary school teachers implemented the unit in their ordinary classes as a regular part o f their curriculum; and 3. The effectiveness o f that implementation in improving students' performance in conditional reasoning was examined. The present paper describes the experimental unit, its development, and implementation. The second paper discusses the measurement instrument and considers problems o f developing a reliable test in conditional reasoning. The Final paper presents analyzes and discusses the experimental fmdings.
1. T H E E X P E R I M E N T A L
UNIT
2.1. The Approach The purpose o f the present study was to teach intermediate-elementary-grade students the process of making valid deductions by providing experience in making valid deductions and by providing practice in avoiding nonvalid b) 'Modus Ponendo Ponens' is the rule by which one infers 'q' from 'if p, then q' and ~', or in other words, from a conditional sentence and a sentence affirming its antecedents. (This rule will be abbreviated by AA for Affirmation of the Antecedent). c) 'Modus Tollendo Tollens' is the rule by which one infers 'not-p' from 'if p, then q' and 'not-q'. (Abbreviated DC for Denial of the Consequent). d) 'Aff'Lrmingthe Conseqnent' (Abbreviated AC) is the fallacy by which one invalidly infers 'p' (or 'not-p') from 'ffp then q' and 'q'. e) 'Denying the Antecedent' (Abbreviated DA) is the fallacy by which one invalidly infers 'not~/' (or 'q3 from 'ifp then q' and 'not-p'. f) The Contrapositive of 'Ifp then q' is 'If not-q, then not-p'. Contrapositive sentences always convey exactly the same information. g) The converse of 'If p then q' is 'If q then p'. Two converse sentences are logically independent.
416
NITSA HADAR
deductions. It was felt that the formal analysis should be postponed to a later stage, after students had been exposed to a variety of concrete examples and had applied logic on an intuitive level. It was n o t the purpose of the present study to teach formal logic, nor to analyze the nature o f the deductive process itself. Rather, it was intended to increase student awareness of the existence of non-valid as well as valid conclusions, and to change student initial inability to separate out those cases in which the given information was insufficient to necessitate a certain conclusion. Several didactial ideas guided the development of the experimental unit: (i) The experimental unit should familiarize the student with the distinction between valid and non-valid inference from simple conditional premises w i t h o u t teaching formal rules or any algorithm by means of which such distinction may be achieved. Even though symbolic logic provides such an algorithm, algorithms were avoided from the direct teaching process in view of the fact that application of the purely syntactical inferences established by symbolic logic is a mechanical process which saves the 'trouble' of logical analysis rather than enhances it. Algorithms were eliminated because symbolic logic does not provide an ideal model for the thinking process o f human reasoning, which usually involves semantic as well as other considerations. Because symbolic logic provides an objective criterion from judging the validity of the outcomes of the thinking process, it might be used by the teachers perhaps as a judgement tool. 0i) The experimental unit should lead the students to recognize the syntactical and the semantical differences between a conditional sentence 'p ~ q' and its converse 'q ~ p ' . Namely: p ~ q ~ q -+p. Without this recognition one could hardly hope that a student would distinguish an AA case (p ~ q, p =~ q) from AC case (p ~ q, q ~ p). (iii) The experimental unit should lead the student to recognize the syntactical differences between, and the logical-semantical identity of a conditional sentence 'p ~ q' and its contrapositive 'not-q ~ not-p'. Namely: p ~ q ~* not-q --> not-p. Without this recognition it is difficult to believe that a student would be able to base a DC case (p-+ q, not-q =* not-p) upon logical considerations. Indeed, in the course of the study a new hypothesis emerged explaining a high rate o f success in DC cases upon a fortunate mistake achieved by a mechanism which has very little to do with logical reasoning. (A conjecture concerning the possible nature of such a mechanism will be developed in a future paper). (iv) The experimental unit should lead the student to recognize the syntactical and semantical differences between a conditional sentence 'p ~ q" and its inverse: 'not-p ~ not~/'. Namely: p ~ q ~ not-p -~ not-q. Otherwise it is hard to see how a student would distinguish a DC case (p -~ q, not-q =~ not-p) from a DA case (p -+ q, not-p ~ not-q).
CHILDREN'S CONDITIONAL REASONING
417
(v) The experimental unit should lead the student to recognize the logical equivalence between a conditional sentence 'p--' q' and the compound sentence: 'p-and-q or p-and-not-q or not-p-and-not-q'. This is equivalent to the recognition that the sentence ' p - - , q ' negates the sentence 'p and not-q'. Recognizing that two as logical equivalents of a conditional sentence is necessary for sorting out the undecidable cases - AC, DA - from the deciable ones - AA, DC. (vi) The experimental unit should bring the four cases - AA, DC, AC, DA into confrontation. Dealing with one case at a time proved unchallenging, and therefore inefficient in the pilot study. (Not reported here.) (vii) The experimental unit should be designed to improve logical reasoning on an intuitive level, therefore it should enrich student's individual experience in drawing valid conclusions within reasonable contents. Reasonable content does not necessarily mean factual one, but whenever it is hypothetical it must not contradict everyday experience and it should make sense to intermediate grade students. (viii) In order to obtain and maintain student's interest the experimental unit should make provision for student experience through autonomous activities such as games and manipulation of a variety of concrete materials. Oral discussions should be based upon such experiences. Generalization should be left to individuals reaching this level spontaneously. 2.2. The Structure of the Experimental Unit The unit consists of seven chapters: Electric Cards;Domino Activities;Pictorial Activity; Numbers and Their Properties; Playing Cards; Colored Light Switch Box; Prepare a Quiz. Each chapter is a set of small-group activities introduced by a teacher/whole-class activity. The first set, Electric Cards, is a motivational activity for the whole unit and it also leads to the final project: Prepare a Quiz. Between the first introduction and the final project, the unit is subdivided into three parts according to the objectives to which each set of activities addresses itself. The first part is designed to demonstrate the implications of a conditional sentence; the second part shows how conditional sentences ate derived from other sentences; and the third part integrates the first two in order to produce some generalizations.
2.3. Motivation and Final Project The first set, Electric Cards, sets the motivation for learning the whole unit through a self-conducted group activity of problem solving. There are about 300 problems, each typed on a separate card. Each subset of four problems
418
NITSA HADAR
has an identical conditional premise and the second premise is its antecedent, its consequent or the negation o f one of those. An immediate checking and feedback system is available through a simple electric tester. (See Figure 1). The tester lights a bulb when, and only when, its terminals are attached to the metal contacts (made out o f paper fasteners) connected to the right answer.
battery
bulb
term~aaal
_~:
'
tez'ni~
{I
=
Fig. 1.
'
The tester.
If Ann is a secretary, then Ann knows how to type Ann knows how to type I s Ann a secretary?
"0 NO, certainly not
Yes, for sure
•
represents
a paper
- - -
represents
the
that
students
Not enough clues
fastener.
wiring could
Fig. 2.
behind neither
the see
card nor
[covered feel
the
by cardboard
so
wiring)
One of the electric cards.
A group o f 6 - 8 students gets a set o f cards. The leader reads the problem out loud and each group member independently writes an answer. The leader counts the number o f yes no - not-enough-clues answers and leads a discussion in case o f disagreement among group members. After all the group agrees upon one answer, the leader checks that answer with the tester. The final project at the end o f the unit - Prepare a Quiz - involves students in the whole process o f producing self-made electric cards. This process starts -
CHILDREN'S CONDITIONAL REASONING
419
in inventing a true, or a reasonable conditional sentence*, and ends in wiring the card and challenging a friend with it. A group of students could go to the electric cards at any stage of the learning period. The teacher would assign them a set of cards having something in com.rnon, e.g., (i) Sets of four questions for which the conditional (first) clue is identical. (These cards had four successive numbers 1-4, 5-9, etc.) (ii) Specific mode of negation occurrence in the conditional premise i.e. negation occurs in the antecedent, or in the consequent, or in both, or in neither part of the conditional premise. ('fii) Cards that carry questions which have indefinite answers (DA or AC). (iv) Cards which have definite answers (AA or DC).
2.4. The First Part o f the Unit - Implications o f a Conditional Sentence The first part of the unit consists of the Domino Activities and the Pictorial Activity. These activities demonstrate by a variety of examples the idea that whenever a conditional sentence (if p, then q) is true or assumed, it excludes the possibility of p and not 35, then it is > 15. What sentence is that? (A conditional one.) How do you know? (It starts with 'If.') Is that a true conditional * To remove any doubts it should be noted again that the symbolic language used above
is not the language used in the classroom.
CHILDREN'S CONDITIONAL (i)
REASONING
P u t each aumber llsted b e l o v In t h e a p p r o p r i n t e
chart:
427
box in the matrix
(cross It off the lint as soon as you put ~t In the matrix).
100, 45, 37, 29, 76, 12, 30, 1, 4, 42, 26, 15, 28, 35, 94, 56, 49, 13, 3, 24, 18, 6, 79, 52, 64, 19, 2, 91, 85, 22, 7, 33, 82, 77, 14, 34, 5, 11, 21, 36, 71, 84, 46, 51, 55.
Greater t h a n 20
Not greater than 20
100, 76, 94, 79, 64, 91, 85, 82, 77, 71, 84,
Greater than 60
Not greater than 60
(2)
45, 37, 29, ~0, 42, 26, 20, 35, 56, 49, 24, 52, 22, 33, 34, 21~ 36, 46, 51j 55,
I n t h e c h a r t you made i n q u e s t i o n
(1),
12, I, 4, 15, 13, 3, 18, 6, 19, 2, 7, 14, 5, II,
i s t h e r e a n y empty box?
Can you f i n d a number t o p u t i n t h e empty box?
Y~s
No
Why? Becauee ttz~.re i s no number that i s a t the ~ame time ~l~,eater than
60, and not ~ r e a t e r than 20.
Fig. 5. Numbers and Their Properties - a sample worksheet (with answers).
sentence? (Yes.) Why? (Students will explain: no number can be greater than 35 without being greater than 15 because 35 is greater than 15, and so on.) T: Here is a matrix like the one we have in your worksheets.
>35 not > 35
>15
not > 15
A
B
C
D
(Draw it on the Blackboard).
428
NITSA H A D A R Choose as your secret number(~rite it and hid~ I.
Jimmy, can you read this? What kind of a sentence is it. How do you know? I choose a number. I'ii not tell you what my number is but I can tell you it is greater than 60.
65 (bide it)
Is my number greater than 20? (Yes)
Write on the (big) blackboard If X ? 60, then X ) 20
X ? 60
Is X ,20? Show your hidden number
2.
If X > 6 0 ,
Here is the same sentence. I choose two numbers now! My new numbers both are greater than 20. Are they greater than 60? (NEC)~ Why? (Children give examples like: you may have chosen 22 and 23~or 22 and 6 2 o r 62 and 63. In all cases X 720, but we can't tell whether X ) 60 or not.)
then X > 2 0
25 and 70 ( h i d e both) X? 20
Is X> 60?
Show your numbers 3.
Here I choose another number.
30
My secret number this time is not greoter than 60. ]s it greater than 20? (NEC) Why?
If X) 60, then X > ~-0 X is not > 60
Is X > 20? Show it
4.
Last number I choose.
15
This time it is not greater than 20. Is it greater than 60? (No[) Why?
If X > 6 0 ,
then X , 2 0
X is not) 20 Is X 7 6 0 ? Show it
This ~ m e
may be repeated by students.
~% ~EC ~s an abbreviation for Not-enough-c]ues
Fig. 6.
I think of a number. I'll call my number X. My number X is greater than 35. (Write on the board X > 35.) Which box does it belong to? S: The upper left, box A. T: Why? O.K., come put X there. Now I think of another number. I~l call it Y. My number Y is greater than 15. (Write 'Y > 15', under 'X > 35'.) Which box does it belong to? S: Give us another clue. We can't tell yet. It may belong to either box of the left column, either A or C.
CHILDREN'S CONDITIONAL REASONING
429
T: Come, put Y in both. My new number is Z. Z is not > 35. (Write it under the previous two statement.) Where does Z belong? S: You fooled us again: you didn't give us enough information. If Z is > 15 it belongs in C, the bottom left box; if it is not > 15, it belongs in D, the bottom right box (read ~ ' as 'greater than'). T: Come put Z in both. I have another number now in mind. I'll call it T. T is not > 15 (write it under the previous three statements). Can you tell which box T belongs in? S: D. T: Why? How come you don't need some more information? S: T is not > 15, so it is certainly not > 35. T: Does anybody have a number in mind? Andy, call your number a name. Tell us something about your number and we'll try to see if we can put it in our matrix, etc. Another teacher vs. whole-class guessing game goes like this: Each student needs paper and pencil. Teacher needs a small blackboard on which he can write his secret numbers. Choose any true conditional sentence about numbers and their relations; for example, let's take the one the students worked on in their worksheets. Write it on the big blackboard in front of the class (preferably using symbols like: If X is > 60, then X is > 20.) Teacher will choose a number, will write it on the small blackboard, and hide it. Then the teacher will give the class a clue, which will be written on the big blackboard under the conditional sentence, and will ask a question about his hidden number. Students will be asked to answer, then the teacher will show his number. All four questions will remain on the blackboard one next to the other. Figure 6 illustrates the course of the dialogue. Later on the teacher could try to invite a student to choose a secret number. The student will whisper his number to the teacher who will give partial information to the class in the following manner: T: Come here, Tom. Whisper your number to me. Oh, Tom's number is greater than 20. Is it greater than 607 (Or, according to the case: Tom's number is greater than 60, is it greater than 20? or: Tom's number is not greater than 60, is it greater than 20? or: Tom's number is not greater than 20, is it greater than 60?) Children write their answers first, and then carry on a vote along with their reasoning. At the end, the child that chose the number says his number. If there's any student who has trouble answering these questions, a number line may help ham visualise the problems. For our example, if x > 60, then x > 20, given that the unknown number is somewhere here (shaded area),
430
0
NITSA HADAR
,
,
20
40
///////////////H/I/!/lll
60
80
it is certainly to the fight of 20, too. But given that the unknown number ~eshere(shaded area), ,
0
///I/1111111111111111111111111111111111111111111111
20
40
60
80
it may or may not be to the right of 60. We can't tell. The given information is not enough to reach a decision. For the Playing-Card activities each group o f students receives a deck of regular cards and one o f 8 different charts like the one in Figure 7. Students take turns in putting cards in place, explaining each time why they do so, e.g., for 7 of spades they are supposed to say: it is not red and not a heart, so it belongs here (bottom right box). Eventually, usually before all the cards are distributed, students realize that there would be an empty box, in this case the upper right one. This box will stay empty because there is no heart card that is not-red. Rephrasing it in two ways as a conditional sentence caUs again for the use o f the contrapositive. It is particularly easy to see that the same chart can be described in two contrapositive ways: if one starts the conditional sentence once from the left marginal titles: 'If a card shows a heart, then it is red,' and once from the upper row titles: 'If a card is not-red, then it does not show a heart'. All the 8 different charts and their corresponding sentences are illustrated in Figure 8. Teachers were advised to use other decks of two attributes cards which would be less familar to the children.
Put each card in the right place. Red
What do you discover? Not-red
not- ( ~ Phrase your discovery as a conditional sentence in two ways.
Fig. 7.
A sample chart for playing card activity.
CHILDREN'S
CONDITIONAL
REASONING
Dimcovery
2x2 Natrix
No b l a c k diamonds
Co~iltloaal S e n t e n c e s
1.
I f a c a r d shows ~ , it i s r e d .
2.
If a card is not red, it i s n o t - 0
•
O
A l l dJ~unonds a r e r e d
sOt-
NO red spade
431
I,
then
A I £ a card shows t ~ , it's b l a c k .
then
then
All spades are black 2.
If a card is not black, t h e n i t does n o t show a spade.
I.
If a card Is~, is black.
2.
If s card is not black, then it 18 n o t - ~- j
1.
If a card is ~, is red.
2.
If a card is not red, then it is not-~
1.
If a card is ~ Is not black.
2.
If a card is black, IC i s n o t - 0
I.
If a card is~, is not red.
2.
If a card is red, it is not- ;
i.
I f a c a r d I s C~, t h e n it is not black.
2.
If a card is black, then it is not-~
i.
If a card is~. is not red.
2.
If a card is red, is not- e
not-
NO r e d c l u b
then i t
All clubs are black not-~,
No b l a c k h e a r t s
then it
A l l h e a r t s a r e red
not-(~
No b l a c k diamonds
0 not-
No r e d c l u b s
not-~
I
No b l a c k h e a r t s
not-(~
No red s p a d e s
not-~ )
, then it
then
then it
then
then i t
then it
Fig. 8. Charts for playing card activities. (R = red; B = black.) The following is an example o f a working sheet for the students. The words typed in italic are for the teacher.
432 1.
2.
3.
4.
NITSA
HADAR
Write T for true or F for false next to each of the following sentences. a.
If a card shows ~ , then it's red.
T
b.
If a card is red, then it shows ~ .
F
c.
If a card is not red, then it doesn't show ~ .
T
d.
If a card does not show ~ , then it's not red.
F
Write T for true or F for false next to each of the following sentences. a.
If a card shows ~ , then it's black.
b.
If a card is black, then it shows *
T
c.
If a card is not black, then it doesn't show ~ .
T
d.
If a card doesn't show ~ , then it's not black.
F
.
F
Write T for true or F for false next to each of the following sentences. a. If a card shows ~ , then it is not black.
T
b. If a card is not black, then it shows ~ .
F
c. If a card is black, then it doesn't show ~ .
T
d. If a card doesn't show ~ , then it's black.
F
Complete the sentences. a. If a card shows ~ , b. If a card is
red
then it is not-
red
, then it doesn't show ~ .
T
T
e. If a card is not- black , then it shows ~ .
F
d. If a card shows ~ ,
F
then it is not black .
The above questions should lead to an intuitive feeling o f the generalizations: 1. The truth o f a conditional sentence does not imply the truth o f its "flipped over" one (its converse). 2. A conditional sentence and its contrapositive are either both true or both false.
2.6. Third Part o f the Unit: Towards Generalization The third and last part o f the e x p e r i m e n t a l unit includes activities w i t h a colored-light switch b o x , m o r e paper and pencil w o r k , and preparations for the final project o f the Electric Cards p r o d u c t i o n . This part a t t e m p t s to lead the s t u d e n t towards generalizations t h r o u g h sorting o u t the p r o b l e m s dealt w i t h previously, and b y analyzing similarities and differences a m o n g the f o u r kinds o f inference patterns. T h r o u g h o u t the second and third part o f the unit, m a n y abbreviations are used. Students like t h e m because t h e y save writing. What t h e y m a y n o t k n o w is that these abbreviations also emphasize the s y n t a x o f the various logical forms. T h e y stand for constants in the logical sense, and thus serve as an intermediate step before variables take their place in the general pattern o f
CHILDREN'S CONDITIONAL REASONING
433
the four relevant logical forms. There is no direct teaching of any algorithmic distinction between valid and fallacious inferences. However a sense of its existence is expected to emerge from these activities, even though it is never forced. The colored light switch-box (see Figure 9) is operated by eight pushbuttons controlling three colored bulbs. Each push-button turns on one of the eight possible combinations of the three bulbs. Many true conditional sentences can be stated. Some of them are simple ones e.g. If switch No. 1 is pushed, then the yellow light comes on. Others are more complex, e.g. If Switch No. 1 is pushed, then the yellow and the green light come on.
-6-
-0-
01
02
-6-
Yellov
Green
03
04
0$
Red
06
07
08
Fig. 9. The colored light switch-box. Students use abbreviated notation for purposes discussed above: G.L., Y.L., R.L., for green, yellow and red light respectively, and $1, $2, etc. for switch-numbers. For instance, Ss -" R.L.O. represents the sentence: If switch No. 8 is pushed then the red light is on. The fact that each light is turned on by more than one push-button is the basis for sets of AA, DC, AC and DA questions based upon true conditional sentences about the switch-box. The following description should exemplify a way of using 'Ss --" R.L.O.' to phrase four questions: Question ] : Teacher: Cover the lights only, push switch # 8. Say Write on the blackboard If I push switch # 8 , then the red light is on. Ss ~ R.L.O. I pushed switch # 8 . . . . . . . . . . . . . . . . . . . . . . . . . . ,$s Did the red light come on? . . . . . . . . . . . . . . . . . . . . . (Right answer: Yes)
R.L.O.?
Question 2. Cover lights and switches. Push switch # 6 or any other switch that turns on the red, but not switch # 8. Say Write on the blackboard Ss "-" R.L.O.
434
NITSA HADAR
I pushed a switch. It's not switch # 8 . . . . . . . . . . . . . not $8 Is the red light on? . . . . . . . . . . . . . . . . . . . . . . . . . . . R.L.O.? (Right answer: NEC. During the discussion, teacher shows that some switches do and some don't turn on the red light, by pushing all switches one at a time). Question 3. Cover lights and switches. Push switch # 6 or any other switch
that turns on the red light, e.g., 2, 4 Say
Write on the blackboard
Sa ~ R.L.O. I pushed a switch. The red light came on. (Show the red light now) . . . . . . . . . . . . . . . . . . . . . . . . . . . R.L.O. Did I push switch # 8? . . . . . . . . . . . . . . . . . . . . . . . . Ss 9. (Right answer: NEC. Discussion: I t could be switch # 8, but it does not have to be switch # 8. Many other switches turn on the red light. Teacher can show it by pushing switches 2 , 4 , 6) Question 4. Cover switches and lights. Push switch # 1 or any other switch which does not turn on the red light, e.g., 3, 5, 7. Say Write on the blackboard
Ss -" R.L.O. I pushed a switch. The red light did not come on. (Show the red bulb only.) . . . . . . . . . . . . . . . . . . . . . not-R.L.O. Did I push switch # 8? . . . . . . . . . . . . . . . . . . . . . . . . Ss ? (Right answer: No, because switch # 8 turns on the red light, but the red light isn't on.) The following is the last activity before the final project of making electric cards. Its description should clarify the point of using letters for constants as a mean of preparation for their future use as variables.
Coding-Decoding Game - Part 1
First the teacher discusses the structure of a conditional sentence. The notion of a conditional sentence is by now familiar to the students, so the discussion may move inductively from examples used throughout the unit, to the general form of a conditional sentence: 'If . . . . t h e n . . . ' . Alternatively a teacher may wish to try the following procedure: T: Here are some sentences. (Post a list, see Figure 10). Now, here is my conditional sentence (write on the blackboard and say:) ' I f (a), then (e)'. Can you decode it?
C H I L D R E N ' S C O N D I T I O N A L REASONING a) b) c) d) e) f) g) h) i) J) k) 1) m) n) o) p) q) r) s) t)
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It is dark. Fred wears his shoes. His feet will hurt. He g o e s o u t . It is scary. Fred reads a book. The sun i s up in t h e s k i e s . It is raining. He f e e l s good. It is vacation time. He w o r k s h a r d . The s t r e e t s a r e s l i p p e r y . Fred wears his coat. He s l e e p s l a t e . He n e e d s some q u i t e a r o u n d . It is snowing. He t a k e s a s h o w e r . It is hot. Fred d r i v e s s l o w l y Fred is tired.
F~. 10. A sample for construction of conditional sentences. S: T: S: T: $1 : T: $2 : T:
If it is dark, then it is scary. If(g), then n o t (a). (Write it down under the previous one). If the sun is up in the sky, then it is not dark. Do any of you want to try to invent a conditional sentence? If (p), then (s). Who knows what this sentence says? If it is snowing, then Fred drives slowly. Very good. Anybody else want to invent a conditional sentence out of these? Students will suggest their sentences, and write them one under the other. Students may suggest conditional sentences including negations, e.g.: If (a), then not (f), which means: If it is dark, then Fred does not read a book. Students may also suggest ridiculous sentences like 'If (In), then ( g ) . . . ' When there are enough examples on the blackboard, the teacher will lead to the generalization of the form of a conditional sentence. Coding-Decoding Game - Part 2
Using the above sample of sentences, the teacher writes four questions, one at a time, on the blackboard, using shortcuts. For example: If (j), then (i) (j) (i)?
If (j), then (i) not (j)
If G), then (i) (i)
If (j), then (i) not (i)
(i)?
(j)?
(j)?
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Children will decode and answer each question before the next one is presented. The next four questions are written down below these, each one under a question of the same logical type. After a few examples, a student is asked to invent a question based on a conditional sentence from the list and to write it on the blackboard in the proper column. After one student's question is answered, the class is challenged to invent another question with the same conditional sentence. Each student is expected to discover the four possible relations between the conditional sentence (first clue) and the second clue which is the antecedent, or the consequent, or any of their negations. They are not expected, however, to be able to express their discovery verbally. Their discovery will be expressed by their ability to construct all four kinds for a particular conditional sentence. The above activity can also be conducted as a group activity in which students take turns in presenting their coded questions. At the end of this activity students start on the final project - Prepare a Quiz - as previously described.
3. THE UNIT D E V E L O P M E N T AND I M P L E M E N T A T I O N The experimental unit was developed in four cycles of teaching-revisionreteaching, with a changing role on the part of the investigator: (i) investigator's work with individual students, (ii) investigator's work with three small groups of students in succession. Ctii) teachers' implementation pilot study, in which the investigator was present in each and every class period and, (iv) teachers' implementation main study, where the investigator paid only occasional short visits to each class. Only the last part, the main study, is expanded upon here. Before the main study started, four elementary school teachers took a twelve-hour pretraining workshop given by the investigator. They attended weekly meetings during the instruction period. During the workshop teachers were taught the logic of conditional reasoning using the experimental unit materials. The experimenter presented the activities to the teachers in the way the teachers were expected to present the activities to their classes. Because the teachers had little or no background in logic, they easily played the student role with no discernible difficulty. The four 5-th grade classes of the pretrained teachers constituted the experimental group of the main study. Teaching took place for 30-40 minutes a session, 4-5 times a week for 23-25 sessions. Most of the class-sessions started in a short teacher-class discussion which led to one of the activities.
CHILDREN'S CONDITIONAL REASONING
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4. C O N C L U S I O N : AN I N T U I T I V E V E R S U S AN A L G O R I T H M I C APPROACH A well known approach to acquainting students with ideas of mathematical logic is to teach the construction of truth tables for sentences built with propositional connectives (e.g., Suppes and Hill, 1964). This task is essentially computational and algorithmic. The approach in the present study is different in that it is directed to leading students towards the intuitive construction of a deductive argument. Truth tables may be incorporated, in such an approach, as a technique for verification of the logical validity of a conclusion, which in turn will indicate the provability of the conclusion. But truth tables alone cannot serve the purpose o f putting the students into a thinking process o f distinguishing valid from nonvalid inferences. A truth table is an automatic algorithmic process that does not call for thinking. Furthermore, truth tables have operational limitations. For example, it is impossible to construct a truth table for sentences of the general form, 'For all x, x i s . . . ' since there may be models in which such a sentence is true and others in which it is false. Teaching the technique of constructing truth tables was not used in the experimental unit. Instead, the main aim was to lead students to an awareness of the existence of non-valid conclusions, and to do it without adding another algorithm to their repertoire, but through individual experience accumulated in an autonomous atmosphere.
REFERENCES Eisenberg, T.A. and McGinty, R.L. (1974) 'On Comparing error Patterns and the Effect of Maturation in a unit on Sentenfial Logic'. Journal of Research in Mathematics Education 5, No. 4,225-237. Ennis, R.H. and Paulus, DJ-I. 1965. Critical Thinking Readiness in Grade 1-12, Phase I: Deductive Logic in Adolesence, Itacha: Cornell University. Evans, J. St. Bt. B.T. (1972) 'Deductive Reasoning and Linguistic Usage (With special Reference to Negation)' Unpublished Ph.D. dissertation, University of London. Hadar, N.B. 'Children's Conditional Reasoning: An Investigation of Fifth Graders' Ability to Learn to Distinguish between Valid and Fallacious Inferences', Ph.D. dissertation, University of California, Berkeley, 1975. Eric accession number: ED 118359. Announced in Resources in Education, June 1976. O'Brien, T.C. and Shapiro, B.J. (1968) Whe Development of Logical Thinking in Children'. American Educational Research Journal. 531-542. O'Brien, T.C. (1970) 'Logical Thinking in Children Ages 6-13', Child Development 41, 823-829. O'Brien, T.C. (1973) 'Logical Thinking in College Students', Educational Studies in Mathematics 5, 71-79. Osberson, D.N., (1974) Logical Abilities in Children, Vol. 2. Logical Inference: Under. lying Operations, Potomac, Maryland: Lawrence Erlbaum Association,Publishers.
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Paulus, D.H. (1967) 'A Study of Children's Ability to Deduce and to Judge Deductions', unpublished Ph.D. dissertation, Corneil University. Roberge, J.J. (1970) 'A Study of Children's Abilities to Reason with Basic Principles of Deductive Reasoning', American Educational Research Journal, 583-596. Suppes, P. and Hill, S. (1964) First Course in Mathematical Losic, New York: Blaisdell Publishing Company.
