Proportional reasoning and the linguistic abilities required for hypothetico-deductive reasoning

JOURNAL OF RESEARCH IN SCIENCE TEACHING VOL. 21, NO. 2, PP. 119-131 (1984)

PROPORTIONAL REASONING AND THE LINGUISTIC ABILITIES REQUIRED FOR HYPOTHETICO-DEDUCTIVE REASONING

ANTON E. LAWSON

Department of Physics, Arizona State University, Tempe, Arizona 85287

DAVID I. LAWSON

Department of Mathematics and Computer Science Stetson University, DeLand, Florida 32 724

CHESTER A. LAWSON

Lawrence Hall o f Science, University of California, Berkeley, California 94 720

Abstract

The hypothesis is advanced that a necessary, though not sufficient, condition for the acquisition of proportional reasoning during adolescence is the prior internalization of key linguistic elements of argumentation, essentially those used in hypothetico-deductive reasoning. This hypothesized internalization, which does not occur in all individuals, results in some who have acquired the ability to reflect upon the correctness of self-generated answers in a hypothetico-deductive manner, and others who have not. As an initial test of the hypothesis, 46 subjects (Ss) (mean age = 21.03 years) were classified into additive, transitional, or proportional reasoning categories based upon responses to a proportions task. Group differences were found in which proportional Ss performed better than transitional Ss who in turn performed better than additive Ss on a number of items testing Ss’ abilities to identify, generate, and use the linguistic elements of argumentation. Further it was found that some Ss who were successful on the linguistic items failed the proportions task, but no Ss who were successful on the proportions task failed the linguistic items. This result supports the hypothesis that the internalization of linguistic elements of argumentation is a prerequisite for proportional reasoning and by inference other advanced reasoning schemata as well. Implications for science instruction are drawn.

A considerable body of evidence indicates that a number of cognitive abilities increase during adolescence (e.g., Garfinkel & Thorndike, 1976; Keating, 1979; Lawson, Karplus, and Adi, 1978). One such ability is the ability to solve problems that involve quantitative proportional relationships. For example, Lawson, Karplus, and Adi (1978) found only 10% of a sample of 13-year-olds used proportions to solve a task involving pouring water from a wide cylinder to a narrow one in which the ratios of water levels was 2 to 3. Older subjects (Ss) from the same area, however, performed progressively better. The following percentages of correct responses were obtained: 20% for the 14-year-olds, 69% for the 16-year-olds, 76% for the 18-year-olds, and 80% for the 20-year-olds. According to Piaget (e.g., Hall, 1970; Inhelder and Piaget, 1958) the acquisition of abilities such as these do not arise solely as a consequence

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120 LAWSON, LAWSON, AND LAWSON

of the school curriculum. Data such as those cited above support this position as proportions are part of the normal curriculum only in the seventh and eighth grade. What then accounts for the development of proportional reasoning ability during adolescence?

The purpose of the present study is to test the hypothesis that successful performance on specific proportions problems such as the pouring water task used by Lawson, Karplus, and Adi (1 978) is dependent upon the prior acquisition of general hypothetico-deductive reasoning abilities which are acquired as the result of the internalization of key linguistic elements of argumentation: specifically the abilities to recognize, generate, and use sentences which repre- sent hypotheses, predictions, results, and conclusions. In specific task situations this general ability manifests itself as the S asks him or herself the following question: Is the answer I have generated to solve the problem correct? and his or her ability to answer this self-imposed question by generating predictions based upon its assumed correctness and by comparing those predictions with known empirical facts.

Let us consider the pouring water task in greater detail. Ss are shown that four units of water in the wide cylinder rise to six units when poured into the narrow cylinder. They are then asked to predict how high six units of water in the wide cylinder will rise if poured into the narrow cylinder. The most salient response, and that given by most children, is eight. “It rose two before, so it will rise two again.” Ss who initially guess eight, but think further and ask themselves if that answer is correct, may discover that it indeed cannot be. A constant difference of two units between the water levels would lead one to predict that zero units in the wide would rise to two units in the narrow. Since the creation of two units of water from nothing is clearly impossible, an S following such a hypothetico-deductive line of reasoning would be left with the conclusion that eight cannot be correct and would feel the need to try an alternative strategy, perhaps one in which the ratio, not difference is constant. Thus it would seem that the pouring water task requires subjects to hypothesize an initial answer, deduce its implications, and compare those with other known facts, such as the fact that water can not be created from nothing.

In short, what we are hypothesizing is that the central ability which develops during adolescence is the ability to ask questions, not of others, but of oneself and to reflect on the correctness or incorrectness of answers to those questions in a hypothetico-deductive manner. This hypothetico-deductive question asking and answering behavior involves the acquisition of linguistic skills associated with hypothesis testing and leads ultimately to the development of hypothesis testing schemes. In other words, prior to adolescence the child raises questions, generates answers, yet has no systematic means of asking himself if his answers are correct or not. He simply generates them and for the most part uses them for better or for worse. Without such an ability Ss confronted with the pouring water task simply choose the most salient response of eight and conclude that it is correct without further thought.

We believe the position that the ability to reflect on the correctness of ones thought arises as a consequence of the internalization of patterns of argumentation is in essential agreement with Piaget’s earlier thinking. Piaget (1928) advanced the hypothesis that the development of advanced logical reasoning occurred as a consequence of “the shock of our thoughts coming into contact with others, which produces doubt and the desire to prove” (p. 204). Piaget went on to state:

The social need to share the thought of others and to communicate our own with success is at the root of our need for verification. Proof is the outcome of argument. . . Argument is therefore, the backbone of verification. Logical reasoning is an argument which we have with ourselves, and which produces internally the features of a real argument. (p.204)

HYPOTHETICO-DEDUCTIVE REASONING 121

In other words, the development of hypothetico-deductive thought during adolescence, as defined the ability to ask questions of oneself, generate tentative answers, deduce predictions based upon those answers, and then sort through the available evidence to verify or reject those tentative answers, all inside one’s own head, occurs as a consequence of attempting to engage in discussions of the same sort with other persons and listening to arguments of others in which propositions are put forward and accepted or rejected as the basis of evidence and reason as opposed to authority or emotion.

This position also seems consistent with that of Vygotsky (1962) who views speech as social in origin and only with time does it come to have self-directive properties that eventually result in internalized verbalized thought. This position is also similar to that of Luria. According to Luria (1961) the progressive differentiation of language to regulate behavior occurs in four steps. First, the child learns the meaning of words; second, language can serve to activate behavior but not limit it; third, language can control behavior through activation or inhibition via communication from an external source; and fourth, the internalization of language can serve a self-regulating function through instructions t o oneself.

If our hypothesis with regard t o the central importance of self-questioning behavior is correct and if our hypothesis with regard to the hypothetico-deductive requirements of the pouring water task is also correct, then adult subjects who respond to the pouring water task with a prediction of 8 (an additive response presumably indicating a lack of development of the general linguistic abilities of the hypothetico-deductive thinker) ought to possess demonstrable linguistic deficiencies, while adult subjects who respond with a prediction of 9 (a proportional response suggesting the development of the general linguistic abilities of the hypothetico- deductive thinker) ought to demonstrate linguistic abilities superior to those of his child-like counterpart. On the other hand, if the ability to use proportions to solve the pouring water task is not linked to the more general linguistic abilities of hypothesis testing, then subjects classified into different groups based upon their responses to the pouring water task would not necessarily be expected to exhibit significantly different linguistic abilities.

Further, since our hypothesis assumes that the acquisition of the language of argumentation is a necessary, although not sufficient, condition to success on the proportions task we predict (1) some subjects who possess the linguistic abilities will also succeed on the proportions task, (2) some subjects who possess the linguistic abilities will fail on the proportions task, and (3) subjects who do not have the linguistic abilities will fail on the proportions task. To the extent that subjects who do not have the linguistic abilities succeed on the proportions task the hypothesis will have been falsified, Thus the comparison of linguistic performance of adult subjects who have been classified into different categories on the pouring water task with categories of successful and unsuccessful use of the linguistic elements of hypothesis testing will provide an initial test of the hypothesis. Needless to say positive results will in no sense prove the hypothesis. Such proof is beyond the realm of scientific investigation as alternative explanations can always be offered.

Method

Subjects

Ss were 46 students (6 males and 40 females) enrolled in a nonmajors biological science course at a major southwestern university. Ss ranged in age from 18.75 years t o 26.83 years, mean age = 21.03 years, SD = 1.94 years.

Procedure: Assessing Proportional Reasoning Ability Ss were initially classified into additive, transitional, or proportional reasoning categories

based upon their responses to the pouring water task. The task was administered on the first


day of class by showing Ss two glass cylinders of equal height but different diameters with iden- tical equally spaced marks from top to bottom. Ss were shown that water which rose to the fourth mark in the wide cylinder rose to the sixth mark when poured into the narrow cylinder. This water was then poured out and the Ss were asked to predict how high water which rose to the sixth mark in the wide cylinder would rise if poured into the narrow one.

Ss who responded with “8” and justified this by stating that it rose 2 marks before (from 4 to 6) so it will rise 2 marks again (from 6 to 8) were classified as additive reasoners. It was assumed that these Ss responded with the first and most salient response available and failed to ask themselves if their initial answer was correct.

Ss who responded with the correct answer of 9 and justified this by noting the constant ratio of 2 to 3, 4 to 6, or 1 to 1$ or correctly used a cross multiplication algorithm, were classified as proportional reasoners. It was assumed that these Ss considered the additive strategy either during the course of the testing or at some time in the past when considering quantitative proportional relationships and via a self question asking and testing process rejected it in favor of a correct strategy which they either invented themselves or had been taught.

Ss who responded with answers other than 8 or 9 (e.g., 7:, 10, 14) were classified as transitional reasoners. It was assumed that such responses indicated that the Ss considered the most salient response of 8 but rejected it as inappropriate, yet in the time allotted for the testing were unable to generate an appropriate strategy. These responses were assumed to show some ability to reflect upon the correctness of ones answers but its development was either too little or too late to allow assimilation of a correct proportions strategy.

Instruction

Following pretesting Ss received approximately 30 hours of classroom instruction involving laboratory experiments, lectures, and readings designed to teach biological concepts and hypothesis testing procedures. Piaget (1 972) advanced the hypothesis that persons who were other- wise formal operational may fail to use formal reasoning in specific task situations because they lack familiarity with the phenomenon inquestion. Thus instruction ensured, as much as possible, that lack of familiarity would not be the cause of poor performance by our Ss. Further the instruction ensured that all Ss were introduced to the terms involved in the hypothetico- deductive reasoning process and had been given ample opportunity for their use in designing and carrying out experiments.

The pattern of instruction, which followed the exploration-invention-discovery learning cycle (Science Curriculum Improvement Study, 1974), involved raising questions, generating hypotheses, deducing predictions, and designing and carrying out experiments to test the hypotheses advanced. Such a pattern led to the introduction of biological concepts concerning the phenomena that had been investigated as well as discussions of the reasoning used during the various phases of the experimental process. During class time the questions, hypotheses, predictions, and results were explicitly mentioned and written on the board. Terms such as hypothesis, theory, prediction, independent variables, dependent variable, and control of variables were introduced. Learning cycles 1-6, 8-11, and 14 from the class study guide (Lawson, Munch, and Tillery, 1978) formed the basis of instruction. The study guide also included six “Reasoning Modules” designed to accompany the classroom instruction to further highlight the reasoning and terms involved in testing hypotheses. Out-of-class textbook reading (Baker & Allen, 1977), included chapter one “The Nature and Logic of Science,” and chapter two “Test- ing Hypotheses and Predictions,” in which the hypothetico-deductive reasoning process is discussed in some detail along with examples of its use in laboratory and field situations.

HY POTHETICO-DEDUCTIVE REASONING 123

Assessing the Linguistic Abilities of Hypothesis Testing

A written examination consisting of three sets of questions was designed to assess Ss’ ability to recognize, generate, and use questions, hypotheses, predictions, and experimental results to establish the truth or falsity of the hypotheses advanced. The examination, which required approximately one hour for completion was administered during classtime. Although previous research suggests that adolescents have somewhat better developed linguistic abilities than children (e.g., O’Donnell, Griffin, & Norris, 1967; Hunt, 1970; Davelaar, 1977; Lawson & Sheppard, 1979) to date it is not clear if or in what way or ways these differences relate to advances in problem solving and hypothesis testing ability. The present study represents a novel approach in that we are not looking for the use of longer and/or more complex sentences, or the increased frequency of use of specific terms such as if, then, should, therefore, and so on. Renner, for example, found little correlation between the frequency of use of such terms and level of intellectual development (Renner, Prickett, & Renner, 1977; Renner, 1979). Thus, rather than looking at the frequency of use of key terms, we are looking for the correct, as opposed to nonsensical, use of such terms, embedded in sentences the students are asked to generate to refer to different aspects of the hypothesis testing process. In other words, the working hypothesis is that when essentially empirico-inductive adults are confronted with instruction in the language of the hypothetico-deductive thinker they will attempt to use that language but will do so incorrectly. On the other hand, when hypothetico-deductive adults are given the same instruction they will respond appropriately and will demonstrate correct use of the language of hypothesis testing.

The examination items were as follows: (1) The Trees. Imagine that you are on a walk through the park and observe the following: (a) What question is raised by your observations? (b) Generate three hypotheses to tenta-

tively answer your question. (c) What predictions can be derived from your hypotheses? (d) Describe an experiment to test your hypotheses.

(2) The Salmon. In an experiment to determine the way salmon find their way to their home river, A. D. Hasler captured fish from the East Fork and Issaquah Rivers. He then blocked the olfactory sacs of some of the fish from each river (the experimental group) and did nothing to the remaining fish (the control group). He then tagged all the captured fish and released them below the fork where the two rivers join. He then recaptured the fish as they swam back up into the East Fork or Issaquah Rivers. For each statement below indicate whether it represents a question, hypothesis, prediction, or result.

(a) Salmon find their way to their home river by a sense of odor. (b) Blindfolding salmon should prevent them from finding their home river. (c).Most of the salmon with blocked olfactory sacs did not find their way home. (d) How do salmon find their way to the river of their birth? (e) Salmon find their way back by the sense of sight.

A 8


(f) What causes salmon to return to their home river? (g) Blocking olfactory sacs should prevent the salmon from finding their home river. (3) Hasler’s Data. (a) What hypothesis is Hasler testing in the experiment mentioned in

question 2? (b) What is the independent variable in this experiment? (c) What is the dependent variable? Table I shows Hasler’s data for the control and experimental groups. (d) Do the data support his hypothesis? Explain (be specific).

Scoring. The Trees: Each part of the Trees Item was scored separately. Part (a) responses were simply scored correct or incorrect depending upon on whether or

not the response was indeed a question and pertained in some way to the differences in grass growth under the two trees. For example, the response “Why is grass growing so close to A’s trunk and not Bs?” was considered correct, while the response “Does a tree grow better in the sunlight or shaded areas?” was not. Our position suggests that both empirico-inductive and hypothetico-deductive persons are able to observe phenomena and raise appropriate questions; therefore, significant group differences are not expected on this item.

On part (b) Ss were given one point for each hypothesis they generated which represented a possible explanation for the observed differences in grass growth. For example, the following two responses were considered correct: “The soil is different by tree B.” “No sun hits tree B underneath the tree.” But these two responses were considered incorrect: “Grass is dying.” “Weather is changing.” Significant group differences are expected on this item as it calls for Ss to generate multiple hypotheses which, in and of itself, admits of more than one possible explanation and clearly raises the issue of which, if any, are correct. Although empirico-inductive Ss presumably are able to generate an answer to a question, presumably they seldom, if ever, generate multiple answers. Therefore, this item should prove difficult for them.

On part (c) Ss were awarded one point for each reasonable prediction that could be derived from their hypotheses based upon an if.. .and. . .then form of reasoning. For example, the following responses were considered correct: Hypothesis: “Tree B creates more shade.” Predic- tion: “Trimming tree B would allow for more sunlight and growth.” Hypothesis: “The grass grew fuller on A than B because around one tree there was more fertilizer.” Prediction: “That if each A and B had the same fertilizer they would probably grow the same.” Incorrect responses were placed into one of three categories: (1) restatement of part or all of the hypothesis (e.g.,

TABLE I Hasler’s Data

Control Group Experimental Group

Capture S i t e

Recapture S i t e

Issaquah East Fork

~~~ ~~~

Recapture Site

Iasaquah Eas t Fork

I ss aquah

Capture S i t e

Easr Fork


Hypothesis: If leaves fall from the tree then the grass dies.” Prediction: “Grass can’t grow if something is on top of it such as leaves.”); (2) restatement of the hypothesis with the addition of an if. . .then. . .(e.g., Hypothesis: “Water gets under A but not B.” Prediction: “If A gets watered all the time under the bark and B doesn’t, then B won’t grow.”); and (3) statement of the then portion of an if. . .then. . .(e.g., Hypothesis: “In B the tree used up the water closer to the trunk.” Prediction: “Grass will not grow near the trunk.”) Significant group differences are expected on this item in that empirico-inductive Ss presumably have not internalized the language of hypothesis testing and would not, therefore, be able to effectively generate predictions based upon the assumed truth of the hypotheses while hypothetico-deductive Ss would. Given the fact that empirico-inductive Ss have received instruction in hypothesis testing, they may show partial success but most likely will give one of the types of incorrect response as mentioned above.

Part (d) responses were awarded 0-3 points. One point was awarded for each suggested procedure that, if done, would lead to results that would test one of the hypotheses in a controlled fashion. In other words, suppose the S hypothesized that more fertilizer under tree A was the cause of differential grass growth. An experiment in which two containers of soil were planted with grass seeds, one with fertilizer and the other without, was considered a satisfactory test. It was assumed (although no doubt incorrectly in some cases) that the Ss had previously estab- lished that tree A actually had more fertilizer around it than tree B. For example, the following responses were considered correct: “In two containers exactly the same we could put half as much fertilizer as in the other.” “Water the area around the trunk of tree B, as rain water falls on area around trunk of tree A.” But the following response was considered incorrect as the variables were confounded: “Put grass in small container provide one container with fresh soil, give it plenty of water and sun. Put grass in another container and give it drier, older soil and not as much sun or water.” Again significant group difference are expected in that the scheme of controlling variables required for a successful test is presumably lacking or poorly developed in empirico-inductive Ss.

Each of the seven parts of The Salmon Item was simply scored correct or incorrect. For re- porting purposes Ss scores on parts (a) and (e) were summed as both statements represented hypotheses. Scores on parts (b) and (8) were also summed as these statements represented predictions. Likewise scores on parts (d) and (f) were summed as they both represented questions. With inclusion of part (c), which represented an experimental result, seven points were possible on this item. With the exception of parts (d) and (f) (the questions) significant group differences are expected on this item as it requires Ss to identify and separate hypotheses, predictions and results from one another. Differences may not be as large as on some of the other items, however, in that the identification and separation of these elements of hypothesis testing in a matching format as used here is presumably easier than identifying or generating those elements from more embedded contexts.

Each part of Item 3-Hasler’s Data was scored separately. For part (a) Ss either named Hasler’s hypothesis correctly (Salmon find their way upstream by a sense of odor) or incorrectly and were scored as such. The incorrect responses were most instructive and were classified into one of three types: (1) Those that were stated as questions (e.g., “Do the salmon use their sense of smell to find their way back home.” “How do salmon find their way up the river.”); ( 2 ) those that predicted what will happen if the olfactory sacs are blocked (e.g., “If the olfactory sac is blocked the fish will not be able to find their way home.” “By blocking olfactory sacs; this will prevent fish from finding way home.”); and (3) those that in effect are answers to the question: What is Hasler trying to find out? (e.g., “Whether the blockage of the olfactory sacs have anything to do with the way salmon find their way home.” “If the salmon from the East Fork and the salmon from the Issaquah River could find their way home.”) Again significant


group differences are expected on this item as empirico-inductive Ss are expected to have difficulty in separating stated hypotheses from questions, predictions and results.

One might question whether it is fair to score responses such as “Do the salmon use their sense of smell to find their way back home,” or “Whether the blockage of the olfactory sacs have anything to do with the way salmon find their way home” as incorrect. After all, both indicate that the S has grasped the notion that it is the sense of smell that enables the salmon to return home. Yet considering such responses correct would miss the essence of the scoring scheme. No doubt the Ss have the general idea that the sense of smell is crucial, but they have failed to place that thought in its proper linguistic form which is, of course, a straightforward declarative sentence. The point is that these Ss have mixed questions and hypotheses which suggests that they have failed to differentiate between these linguistic elements in the process of hypothetico-deductive thought.

For part (b) Ss either named the independent variable correctly (i.e., the ability to smell, blockage of olfactory sacs) or incorrectly (e.g., kind of fish, number of fish, the environment) and were scored as such. Also for part c Ss either named the dependent variable correctly (i.e., the place where the fish end up, the number of fish returning to their respective rivers, direction of fish movement) or incorrectly (e.g., water conditions, sense of smell, number of fish) and were scored as such. Although parts (b) and (c) do not directly involve the linguistic elements of hypothesis testing, the identification of independent and dependent variables is most certainly a part of the hypothesis testing process, therefore, significant group differences are expected.

Part (d) responses were scored either correct or incorrect. Correct responses were awarded one point. For a response to be considered correct the S must have been able to recognize that the data supported Hasler’s hypothesis and justified it by stating that a greater portion of the control group was able to find their home river than the experimental group. Some Ss noted that while 65/73 (89%) of the control group were successful only 42/70 (60%) of the experimental group was. For example, one S replied: “Yes, of the fish in the control group approximately 10% did not find their way home as compared to 40% in the experimental group.” Like parts (b) and (c), part (d) does not directly involve the linguistic elements of hypothesis testing. Rather it requires utilization of a correlational reasoning schema (cf. Inhelder & Piaget, 1958, pps. 232-242; Karplus, Lawson, & Adi, 1980). Nonetheless, significant group differences would be expected as the development of such a data analysis scheme presumably has not occured in empirico-inductive Ss who have not as yet begun to consider means of testing multiple hypotheses. The items with somewhat subjective responses were scored independently by two raters. Agreement was reached on 85% of the items. Discussion resolved the remaining differences.

Results and Discussion

Twenty-four of the 46 Ss (52.2%) gave additive responses of 8 to the pouring water task (mean age = 20.8 years, SD = 1.3 years). Seven Ss (15.2%) gave responses of 7, 7$ 10, or 14 which were considered transitional (mean age = 21.2 years, SD = 2.5 years), while 15 Ss (32.6%) gave responses of 9 and were classified as proportional reasoners (mean age = 21.2 years, SD = 2.5 years).

Table I1 shows scores and percentages of correct responses to the individual parts of the examination as well as subtotal and grand totals for the three levels of Ss. The table reveals that the proportional Ss performed better than the additive Ss on all parts of the examination. Further, the transitional Ss performed at an intermediate level on nearly all parts. The overall differences in group performance, as shown in the last row of the table, were highly significant.


TABLE I1

Mean Scores and Percentages of Correct Responses for Subjects at Each Level of Reasoning ~~ ~

Item Reasoning Level F R a t i o F Prob

(Poss ib l e p o i n t s ) Add i t ive T r a n s i t i o n a l P r o p o r t i o n a l

The Trees

Generate Quest ions ( 1 p o i n t )

Generate Hypotheses ( 3 p o i n t s )

Generate P r e d i c t i o n s ( 3 p o i n t s )

Tes t Hypotheses (5 p o i n t s )

The T r e e s Sub to ta l (10 p o i n t s )

The Salmon

I d e n t i f y Quest ions (2 p o i n t s )

I d e n t i f y Hypotheses ( 2 p o i n t s )

I d e n t i f y P r e d i c t i o n s (2 p o i n t s )

l d e i i t i f y R e s u l t s (1 p o i n t )

The Salmon S u b t o t a l (7 p o i n t s )

H a s l e r ’ s Data

Naming Hypotheses (2 p o i n t s )

Naming Independent Var i ab le s (2 p o i n t s )

Naming Dependent Var i ab le s ( 2 p o i n t s )

Analyzing Data ( 2 p o i n t s )

1 H a s l e r ’ s Data S u b t o t a l (8 p o i n t s )

GRAND TOTAL

(25 p o i n t s )

.92 92%

2.13 71%

0.63 21%

0.79 26.3%

4.46 44.6%

1.95 97.5%

1.33 66.5%

1.29 64.5%

U.83 83%

5.42 77.4%

1.00 50%

1.00 50%

0.75 37.5%

0.41 20.5%

3.16 39.5%

13.16 I

52.6%

1 .00 1 . 0 0 100% 100%

2 .71 2 .87 90.3% 95.6%

1 .00 1 . 9 3 33.3% 64.3%

1 .29 2 . 2 0 43% 73.3%

5.86 8.00 58.6% 80%

2.00 2.00 100% 100%

1.86 1 . m 93% 90%

0.85 1.00 85% 100%

6 . 4 3 6 .60 91 . Q % . 3%

1.14 1 .73 57% 86.5%

1.14 1.60 57% 80%

0.86 1 .86 43% 93%

0.86 1 .60 43% 80%

4.00 6.80 50% 85%

16.28 21.40

65.1% 85.6%

0 . 9 3 0,400

2.89 0.066

4 .84 0.012

7.60 0.001

8.74 0.001

0.44 0.642

2 .83 0.069

2.16 0.128

1.36 0.266

2.75 0.074

2.89

1.79

7 .87

8.60

11.99

0.066

0.179

0.001

0.001

0.0001

13.61 0.0001

Group differences o n the Trees Item and on the Hasler’s Data Item subtotals were also highly significant, however, group differences on the Salmon Item subtotal failed to reach significance at the 0.05 level.

Two of the four parts of the Trees Item showed significant group differences. As expected, the groups did not demonstrate significant differences in their ability to observe a phenomenon and generate questions. Differences in their ability to generate hypotheses failed to reach significance at the 0.05 level. However, as expected, the groups did differ significantly in their


ability to generate predictions and test hypotheses. Of the 29 Ss who were unable to generate satisfactory predictions, the unsatisfactory responses were categorized as follows: 7 Ss (6 additive, and 1 transitional) gave responses that restated part or all of the hypothesis; 16 SS (8 additive, 4 transitional, and 4 proportional) gave responses that were restatements of their hypotheses with the addition of the words if . . .then, and 6 Ss (5 additive and 1 proportional) gave responses that stated the “then” portion of an if . . .then construction. As expected group differences were not significant on parts (d) and (f) of the Salmon Item which required Ss to identify questions. Although group differences on the other parts of the Salmon Item were in generally the expected direction, they also failed to reach statistical significance.

Significant group differences fp < 0.05) were found on two of the four parts of the Hasler’s Data Item. Group differences reached significance on the naming the dependent variable part and analyzing data part. Although group differences failed to reach significance on the naming Hasler’s hypothesis part and the naming the independent variable part, the trends were in the expected direction. Only 50% of the additive Ss were able to correctly state the hypothesis Hasler was testing even though it was stated for them in the previous item. Of the 17 SS who responded incorrectly, 6 Ss misstated Hasler’s hypothesis as a question (5 additive Ss and 1 transitional S); 4 Ss misstated his hypothesis as a prediction (2 additive, 1 transitional, and 1 proportional); 6 Ss misstated his hypothesis as an answer to the question “What is Hasler trying to find out?” (4 additive, 1 transitional and 1 proportional), and 1 additive S responded “Fish don’t use smell to find their way to the river and place of their birth.” This response was considered to be in a class of its own.

In summary the data in Table 11 provide evidence in support of the hypothesis that proportional thinkers as categorized by responses to the pouring water problem have better developed abilities to generate, name, and identify the linguistic elements of hypothetico-deductive thought than transitional and additive thinkers of the same age. The table also provides support for the hypothesis that proportional thinkers are better able to utilize schemes of hypothesis testing (the control of variables) and data analysis (correlational reasoning) than are transitional and additive thinkers. Thus support has been obtained for the hypothesis that the development of proportional reasoning ability is related to the acquisition of elements of argumentation and the development of hypothesis testing schemes.

Table 111 compares linguistic ability as reflected by performance on the written test with proportional reasoning ability. Ss were categorized into low (scores of 0-6), medium (scores of 7-13), and high (scores of 14-20) linguistic ability groups. A 20 point scale was used (rather than the 25 points for the entire test) in that the two test items which required use of the control of variables and the correlational reasoning schemes were omitted from consideration. They were omitted because they required more than linguistic ability as defined. Additive, transitional, and proportional groups were formed for proportional reasoning ability. The pattern of responses fits the expected pattern as none of the low or medium linguistic ability Ss were successful on the proportions task. A number of Ss demonstrated medium and high linguistic ability but failed to utilize these abilities to solve the proportions task.

The results shown in Table 111 are of particular importance for they support the hypothesis that acquisition of the linguistic elements of hypothesis testing is a necessary (but not sufficient) condition for solving the pouring water task and by inference for other advanced reasoning tasks as well. In essence, the linguistic ability score can be viewed as a measure of the presence or absence of specific linguistic elements that must be correctly assembled and utilized in a hypothetico-deductive manner to solve the pouring water task. If they have not been internalized Ss fail to utilize them to solve the pouring water task. Therefore it would seem that successful performance on the pouring water task requires more than schooling on how to use


TABLE 111

Contingency Table Comparing Linguistic Ability with Proportional Reasoning Ability

Propor t iona l Reasoning A b i l i t y

Addit ive T r a n s i t i o n a l P r o p o r t i o n a l

L i n g u i s t i c A b i l i t y

LOW

Medium

High

proportions. It also requires the internalization of a hypothetico-deductive mode of problem solving.

In conclusion, the data are consistent with a three level theory of knowledge development similar t o that proposed by Piaget (1976). The first level is one in which language plays little or no role as it has yet to be acquired. The child learns primarily through sensory-motor activity and knowledge is that of action. The second level is characterized by the acquisition of language. The child is able to respond to spoken language and acquire knowledge transmitted from adults who speak the same language. To learn, the child is able to raise questions and have adults respond verbally t o those question. Of course this is not to say that all adult responses are understood; nonetheless, a new and powerful mode of learning is available to the child. The essential limitation of this level is that the use of language as a tool for reflection and as a guide to behavior has not yet occurred. Reasoning a t this level is essentially empirico-inductive. The final level begins at the moment at which the individual begins to ask questions, not of others, but of himself, and through the gradual internalization of elements of the language of argumentation acquires the ability to “talk to himself” which constitutes the essence of hypothetico- deductive thought and allows one t o test hypothetical explanations and arrive at internally reasoned decisions to solve problems such as the pouring water task. Contrary to Piaget’s position we see no need to invoke his system of 16 binary operations and INRC group to account for this internalization.

No distinct age norms are suggested for the passing from one level to the next, yet we see no biological or psychological reason why a child as young as say six years old could not begin to reflect upon his own thoughts given an environment in which such reflective behavior was strongly encouraged. Of course this represents just a beginning and one would still require con- siderably more time and experience to internalize the language of argumentation and develop the associated hypothesis testing schemes. On the other hand, a dogmatic environment in which the relative merits of ideas are not discussed and rules are strictly and unthinkingly enforced could most likely lead to the failure of hypothetico-deductive thought to be acquired.


Educational Implications

The results of the present study imply that important components of instruction in the science classroom which aim to promote formal reasoning are discussions and activities in which students are actively engaged in argumentation with one another over the truth or falsity of hypotheses advanced and the meaning of data gathered. It is not enough to teach students isolated aspects of scientific reasoning such as controlling variables, proportional reasoning, or correlational reasoning as though there is no glue that binds them together. Rather, it would seem that the various schemata of formal reasoning need to be treated as subcomponents of a more general mode of reasoning which has hypothesis testing and argumentation as central themes and students are provoked to rethink their ideas and arguments. Further, according to the theory, only via the gradual internalization of the linguistic elements of hypothesis testing and argumentation will students come to comprehend the meaning and usefulness of each schema and will successfully employ them in the solution of novel problems. This gradual internalization is seen as no short-term process. Even after 30 hours of classroom instruction, the additive Ss in present study displayed considerable difficulty in the generation and use of sentences which represented questions, hypotheses, predictions, results, and conclusions.

Implications for the teaching of proportions can also be drawn. Clearly, the results suggest that teaching algorithmic solutions to proportions problems, such as the cross-multiplication algorithm, is not sufficient to effect its successful use among students who have not developed the ability to reason hypothetico-deductively. In addition to algorithmic procedures these students need some means of deciding when various procedures should be applied. Although it is not clear how this ability can be developed, it would seem that hypothetico-deductive reasoning may be essential as successful students try various procedures and then reflect upon their consequences before deciding which procedure is most likely correct.

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Manuscript accepted April 12,1983

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Proportional reasoning and the linguistic abilities required for hypothetico-deductive reasoning