Thomas R. Post
University of Minnesota
The Rational Number Project was a multi-university cooperative research effort funded by the National Science Foundation Research in Science Education between 1979 and 1983. It developed instructional and evaluation materials concerned with the learning of rational number concepts, e.g., part-whole, measure, quotient, decimal, ration. These materials reflected the cognitive psychological principals suggested by Bruner (1966), Dienes (1967), Gagne (1877), and Piaget (1952).
A major focus of the project was an exploration of the role of physical models as facilitators of the acquisition and use of mathematical concepts as learners understanding of those concepts proceed from concrete to abstract. Psychological analyses show that physical models are only one component in the development of formal representational systems, and that verbal, pictorial, and symbolic modes of representation also play a role in the acquisition and use of concepts (Lesh, Landau, & Hamilton, 1980). A major hypothesis suggests that the ability to make translations among and between these modes of representation makes mathematical ideas meaningful to learners.
Rational number concepts are among the most complex and most important mathematical ideas children encounter during their presecondary years. The increased attention to research in children's acquisition of rational number concepts appropriately reflects this importance. Recent National Assessment results [Post. 1981a] have shown that children experience significant difficulty learning and applying rational number concepts.
What Makes a Rational Number Meaningful?
Is there a notion of "size" or "bigness" accompanying children's thinking about fractions which is similar to an accompanying notion in their thinking about whole numbers? Are children less inclined to estimate the result of operations on rational numbers? Do children have a quantitative notion of rational number? DO they see 3/5 as a single number having its own integrity or does the concept stay at the level of "3 of the5 parts shaded?"
The answer to each question appears to be in an undesirable direction. Of the nation's thirteen-year olds included in the sample, 30% added numerators and denominators to find the sum of 1/2 and 1/3, even though consideration of the size of the numbers 1/2 and 1/3 would indicate that it makes little sense to add these two numbers and obtain a sum less than either one of the original quantities. Only 24% of the thirteen-year olds were able to choose correctly an estimate for 12/13 + 7/8 among the distractors 1, 2, 19, 21, and "I don't know" (Post, 1981a). A recent Minnesota State Assessment in mathematics identified fractions as the area most in need of attention (Minnesota State Department of Education, 1976). This situation is not unique to Minnesota.
It appears that many students do not have a workable internal concept of rational number. Many times numerators and denominators are not considered in relation to one another but are handled as separate entities and are operated on independently. The fact that 28% of thirteen-year olds selected "19" ad 27% selected "21" as the estimate of 12/13 + 7/8 substantiates this conjecture (Post, 1981a). A quantitative notion of rational number appears to be lacking. What is this quantitative notion of rational number? How do we want pupils to think about rational numbers when they are encountered? Here are some candidates:
What is a Qantitiative notion of rational Number?
A quantitative notion of rational number included but is not limited to each of the following:
There may be, of course, other understandings which are prerequisite to a quantitative concept of rational number. This list indicates that a complex network of interrelated sub-concepts is required and that these subconcepts at times must subordinate previous learnings.
Observations Arising From Teaching Experiments
This paper refers to a teaching experiment in which children were taught many aspects of fractions. An ability to order fractions is certainly one very important part of any definition of quantitative understanding.
The following observations have evolved from an analysis of 132 interviews with fourth- and fifth-grade children; these interviews occurred over 18 instructional weeks. Extensive observational data substantiate each of the major points made. Some observations may be familiar while others will undoubtedly be new. We hope that they will encourage serious contemplation by teachers who are personally involved in teaching these very difficult concepts to children.
1. For all children there is an initial influence on their ability to order rational numbers which evolves from previous understandings of the ordering process for whole numbers. This adversely affects their ability to internalize the order relation on fractional numbers. For some of these children, this influence is persistent (Post, 1985, p. 23-33).
With whole numbers, children may deal with ordering questions in two different ways: first, they can compare the "bigness" of two given numbers by attempting to match elements of two finite sets representing the numbers. In this way the cardinal aspect of number is salient.
Children might also relate a question about the order relationship between the two whole numbers to the counting sequence, deciding that the larger number comes later. Here the ordinal aspect is salient.
These well-internalized approaches are inappropriate when children are to deal with fractions. In the first place, these new "numbers" can be considered to be an extension of the number system previously learned. This might suggest to the children that the symbol system used is also a continuation of the whole number symbol system. But these new symbols look quite different, with different interpretations assigned to the symbols employed. Moreover, with equivalent fractions children encounter symbols which look different but are said to denote the "same amount."
Even more important is that there does not exist an obvious ordinal aspect to fractions which allows one to exhaustively order them along a number line. But the symbols involved in 1/3 and ¼ surely do suggest such an order relation. Children may try to make use of an order relation which they incorrectly assume to be present. The fact that different strategies are necessary for ordering 2/3 and 2/4 than for ordering 3/9 and 4/9, for example, is something new to children and in fact causes considerable difficulty. For example, when asked to compare 3/9 to 4/9 some students argued that "3/9 is greater than 4/9 because in 4/9 the pieces are smaller and it would take more of then to equal the whole unit," or similarly, "because in 3/9 the pieces are larger and there are less of them." These arguments point out the obvious confusion between fraction type and appropriate strategy. This observation may, in fact, be and artifact of instruction, since in this investigation same numerator fractions were dealt with at an earlier time than same denominator fractions.
2. The words "more" and "greater" (and their counterparts, "less" and "fewer") cause difficulty for some children in dealing with the question of order.
Inherent in this difficulty is the fact that to many children "more" can mean more pieces in the partitioned whole, or more area covered by each part similarly, "greater" means a greater number of parts in the partitioned whole or a greater fraction size. So in order to determine which of two fractions is "greater" (i.e., denotes a greater amount - is "more"), it is of critical importance to clearly distinguish between the words "more" and "greater" (or "fewer" and "less"); that is, whether more pieces or greater sized pieces are in question. A similar situation with the words "size" and "amount" was encountered. One student when asked which of two fractions was greater would routinely ask, "Do you men on the size (area) or on the amount (number of subdivisions)?" Clearly it is important to carefully define for children the words to be used to identify rational number concepts. This is particularly important during the early stages of instruction.
The decision as to whether to focus on the size of each piece or the number of pieces is in reality dictated by whether one is dealing with like numerators (size of piece) or like denominators (number of pieces). The fact is that fractions would be ordered differently if one chooses the size of piece over the number of piece consideration. For example, 1/3 is greater than ¼ if we use as the criterion the size of the respective pieces, but ¼ > 1/3 if we consider the number of pieces into which the whole has been divided. Children initially do not understand this. It seems reasonable to hypothesize that this situation is further confounded by the language problem alluded to above. Words such as "more," "greater," "size," and "amount" are often used with less than the precise mathematical connotation by the teacher. The students probably continue to use them in their colloquial context, causing still further miscommunication.
3. An inability to accomplish translations between and within modes of representation retards abstraction of mathematical relations. Does a child realize that one-third of a pie is written 1/3 and vice versa? Can a child "convert" two-sixths of a pie into one third of a pie using appropriate physical and mental repartitioning strategies?
Associated with this is the ability of children to make meaningful associations between mathematical ideas expressed via manipulative aids and with accepted oral and written symbolism. It is important that children be able to translate a sentence comparing the sized of physical fractional parts to a sentence that symbolically states the order relationship between the represented fraction. We hypothesize that it is the ability to make translations between and within the various modes of representation (pictorial, manipulative, oral, and symbolic) which makes ideas meaningful for children. The oral mode may in fact be an indispensable intermediary in this process. The ability to progress from making single translations to situations which require multiple translations is likewise important.
4. Formal vs. concrete level thinking with respect to fraction concepts seems particularly related to overall success with fraction order and equivalence tasks.
In comparing children's work with manipulative aids, we frequently saw in some children the ability to demonstrate a preplanned procedure as compared to the step-by-step, trial-and-error process of others. Giving children the opportunity to explain verbally (without actually doing it) how a manipulative demonstration would be carried out required mental assimilation and synthesis. Ina sense it involves meta-cognitive processes since it requires thinking about thinking. It is hypothesized that such activity is related to the advancement from concrete to formal thinking.
5. Many children develop or invent strategies for dealing with fraction order and equivalence tasks which likely have origins in whole-number arithmetic or even more elementary experiences. The application of strategies, whether learned from instruction or invented by the child, is frequently problem-specific and interactive with problem type. Problems containing fractions with "more difficult" denominators frequently involve solution strategies different from identical problems with "simpler" denominators even though the problems should (could) be solved in precisely the same way.
This more general discussion of children's strategies may be considered in light of the fact that, with fractions, children explore a new realm of the numbers which in various ways reminds them of the whole numbers already familiar to them. Thus they may evolve strategies used in connection with whole number, being triggered by the whole-number symbols, and apply them incorrectly to fractional symbols. (1/6 is les than 1/7 because 6 is less than 7.)
It was observed that the idea of strategy is fairly fluid among many children. It appears, however, that children's strategies are frequently local strategies. That is, the strategy employed is often a function of the specific task, and does not necessarily persist through or transfer to different situations. This was found between children and within particular individuals. Variation in the numerical characteristics of a problem frequently will generate different solution strategies even within a single individual. It was illustrated above that the same child within a short period of time might employ two vastly different algorithms for similar questions, one algorithm referring to the physical aspect of a number of pieces and the other being based upon number relationships, dealing at a higher level of abstraction. This supports a hypothesis of interaction between solution strategies and the numerical characteristics of the task situation.
It has been observed that children often regress to additive strategies when dealing with fraction order and equivalence tasks. (i.e., 2/5 = 4/7 because 2 + 2 = 4 and 5 + 2 = 7.) It may be that fourth grade students are hesitant to use multiplicative strategies because the concept of multiplication has not yet been fully understood. One explanation could be that students retreat to more familiar approaches in their efforts to deal with newly emerging rational number situations. This observation has implications for revising instruction so as to exploit this tendency, e.g., to point out clearly that use of counting strategies is appropriate only for comparing fractions with like denominators.
We have elsewhere (Post. 1985) referred to this phenomenon as whole number dominance. Ways must be found to help students deal with this issue. In particular, we would like to suggest that strategies which help children bridge the conceptual gap between additive and multiplicative structures should be developed. It appears that some whole number reflexes based upon additive understandings become debilitating to the development of the newer, more complex multiplicative structure.
It is interesting to note that children employ strategies which have not been taught directly. The residual and transitive strategies are examples of this. Residual: 5/6 < 7/8 because they both have one piece left over (to make a whole) and since 1/6 is greater than 1/8, 5/6 (1 - 1/6) must be less. Transitive: 4/9 < 5/8 (because 5/8 < ½). Such strategies seem to be natural extensions of strategies previously used. For example, fourth-grade children would have been comparing objects by using a transitive schema for several years. It is not unusual for children to order several sticks by their lengths by comparing them to a third calibrated stick such as a ruler. Likewise, children have had previous experience with residual strategies; making change at a grocery store is an activity which can involve residual thinking; adding number facts to ten also involves residual thought processes. That is, 8 + 7 = (8 + 2) + 5 = 10 + 5. Adding 7 and 8 as 7 + 7 + 1 also involves a residual approach. Likewise a residual frame of reference is involved in the mental addition of two digit numbers where 37 + 46 becomes (30 + 7) + (40 + 6), which then becomes 30 + 40 + 7 + 6.
It should not be surprising then that strategies evolving from children's knowledge of whole numbers are also applied to new situations involving fractions. Improper fractions tend to increase the likelihood of transitive and transitive with residual strategies. If these thought patterns are natural for children, they could be accommodated in instruction.
It was also observed that a self-generated strategy was less likely in tasks with fractions less than 1. Perhaps this is because a proper fraction such as 3/5 can be dealt with more easily by imagining its "bigness" as part of physical unit than would be the case for 12/5.
Children often invent strategies, many of which are incorrect, when they are asked to compare two unequal fractions, neither of which have like numerators or like denominators. For example, 3/5 and 5/8 are difficult to order by any means than an abstract approach such as converting to a decimal, using a common denominator, or using a cross multiplication. Residual strategies would hardly be appropriate in this case. For such general case fractions, the number of available strategies not only limits the number of ways an individual can think about a problem situation: it also generally implies a reliance on a more abstract solution. We have concluded that the specific whole numbers contained in a faction, as well as the relationships which exist within and between numerators and denominators, influence the way in which children proceed to address the fraction ordering tasks.
The research from which these remarks were drawn demonstrates that fraction concepts in general and fraction ordering tasks in particular are quite complex for children probably because they involve the coordination of several variables. Order and equivalence questions concern both a number and size aspect of fractions. One might hypothesize that when we assess how children order fractions we also learn about their quantitative notion of fractions, that is, how fractions are perceived as numbers. However, this is only partially correct, since one can learn how to order fractions without having an understanding that they can be perceived as quantities. The situation appears to be bi-directional: as children learn to order fractions they acquire a quantitative concept of fraction: as they extend their concept of number to include fractions they also learn to order them.
A crucial point in the acquisition of order and equivalence concepts is reached when children's understanding of factions becomes detached from concrete embodiments and children are able to deal with fractions as numbers. Such a quantitative notion does imply the ability to order given fractions by comparing the amounts they denote, be they unequal or equivalent. This ability to order fractions is therefore a necessary but not a sufficient indicator of children's quantitative notion of fractions.
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* This research was supported in part by the National Science Foundation under Grants No. SED 79 20591 and No. SED 81-12643. Any opinions, findings, and conclusions are those of the authors and do not reflect the views of the National Science Foundation.