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Word Problems: Applications vs. Mental Manipulatives Andr´ e Toom University of S˜ ao Paulo E-mail [email protected]

Dear Editor: Thank you for giving place to articles on word problems [1, 2]. I am concerned with word problems very much and would like to say something about them also. First of all, in my opinion, any discussion should include a definition of the subject and in such situations it is better to keep as close as possible to the exact meaning of the words. I suggest that a non-word problem is a problem, which is formulated using mainly mathematical symbols and only a few special words like “Solve the equation...” Correspondingly, a word problem is a problem which uses non-mathematical words to convey mathematical meaning. At the K-12 level non-word problems tend to be technical exercises, which are necessary, but not exciting. It is only natural that most interesting and non-standard problems are word problems. It does not mean that all word problems are difficult, but all of them need understanding of the natural language and ability to translate between different modes of representation: in words, in symbols, in images. This is similar to Thomas’ main idea [2], although I do not quite agree with his treatment of solving equations. In America word problems are often considered difficult and unpleasant. There is a well-known cartoon of the ‘Far Side’ series, which shows Hell’s library, consist-

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ing completely of story problem books. Many Americans seem to have experiences like the following: ‘The high school algebra coulrse I took (honors, too) was nothing but imposed charts and imposed algorithms. It was boring boring boring and if anyone had told me I’d grow up to be a mathematician I’d have laughed at them.’[3] I attribute these unpleasant experiences to the manner of teaching widely used in America, rather than to word problems themselves, but some American educators disagree with me. In Russia, where I grew up, word problems have been used for decades and always were considered interesting and enjoyable. According to my experience, word problems are exciting and important at all age levels and all levels of difficulty from dissemination of mathematical literacy to challenging the most gifted children. If you open a Russian elementary, middle or high school problem book, or a collection of recreational or olympiad problems, you will see a lot of word problems. For example, Perelman’s famous book [4] contains many excellent word problems, including the following: A team of mowers had to mow two meadows, one twice as large as the other. The team spent half-a-day mowing the bigger meadow. After that the team split. One half of it remained at the big meadow and finished it by the evening. The other half worked on the smaller meadow, but did not finish it at that day. The remaining part was mowed by one mower in one day. How many mowers were there? [4, p. 39] This problem is more than a hundred years old. We know that Leo Tolstoi,

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who was very interested in public education, liked it, especially because it can be solved in a visual way, without algebra. I do not explain this solution in detail, it is clear from the picture.

done by all the team in 1/2 day

done by 1 man in 1 day

done by

done by

1/2 team

1/2 team

in 1/2 day

in 1/2 day

big meadow

small meadow

Another example. When I was in elementary school, a Russian writer Nikolay Nosov published a story [5], which immediately became very popular. The main character, Vitya Maleyev, is in fourth grade. Last year he failed in math and promised the teacher that he would catch up. So he solves several problems at the third grade level. This is one of these problems (p. 624): A boy and a girl collected 120 nuts. The boy collected twice as many nuts as the girl. How many nuts did each collect? First Vitya does not know what to do. He draws a picture of a boy and a girl. He draws two pockets on the boy’s coat and one pocket on the girl’s apron to express the fact that the boy collected twice as many nuts as the girl. Then he looks at his picture and sees three pockets. Then suddenly a thought “like a lightning”

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comes to him: to divide the total number of nuts into three parts! He divides 120 by 3 and gets 40 which is the number of nuts in a pocket. Thus the girl has 40 nuts. The boy has twice of 40, that is 80. Vitya is very excited: first time in his life he solved a problem completely on his own.

In my opinion this is

a piece of good mathematical education. One should not think that Vitya is an exceptionally gifted or priviledged child. On the contrary, this episode is similar to what has been typically going on in Russian schools for decades. My main intention is to address the following important question asked by Gerofsky [1, p. 41]: ‘All this leads me to a question for which I have no answer as yet, the question of the purposes of word problems as a genre.’ Although Russian educators have been using word problems very productively for a long time, they, as far as I know, never cared to explain rationally why are word problems so useful, because nobody questioned this in their presence. They were and still remain guided by tradition, experience, intuition and esthetical criteria, all of which should not be ignored. However, I agree with Gerofsky that in the present situation we need to discuss the purposes of word problems. Gerofsky observes: ‘The claim that word problems are for practicing real-life problem solving skills is a weak one, considering that their stories are hypothetical, their referential value is nonexistent, and unlike real-life situational problems, no extraneous information may be introduced. Nonetheless, they have a long and continuous tradition in mathematical education, and that tradition does seem to matter.’ (p. 41). I believe that Gerofsky’s question has no single answer: like many other cultural

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phenomena (fables, for example), word problems have several purposes. Let me concentrate on two of them and compare them with each other: word problems as applications vs. word problems as mental manipulatives. Word problems as applications.

In this case a word problem is an ap-

plication of mathematics to some situation which can occur in everyday life. An example: ‘A grocery store sells eight oranges for a dollar. A customer wants to buy seven ones. How much should he be charged?’ This problem is based on a real event from my life. I was buying food in a grocery store, where eight oranges were sold for a dollar. I put (as I thought) eight oranges in a plastic bag and went to the cashier, who counted my oranges and said that there were only seven. I asked her to prorate the price. She took out her calculator, but did not know what to calculate. I easily calculated it mentally, but kept silent to see what she would do. She called anothet clerk with a bigger calculator, but he also could not figure it out. He counted the oranges again, found that there were eight and this settled the matter. I know that stories about mathematical incompetence of young people are told in quantities. My question is, what should we conclude from them? Some educators suggest to increase attention to ‘real-world problems’, perhaps similar to this one. I maintain that ‘real-world problems’ should not constitute the only or even the main kind of problems used in classroom and I shall present two reasons for this. One reason is that study of mathematics should be systematic: if study of some topic is undertaken at all, it should result in a complete understanding, that is an ability to solve all problems within a certain range of difficulty, most of

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which, of course, have no counterpart in everyday life. The other reason, which is still more important in my opinion, will be presented below. Word problems as mental manipulatives.

These problems deal with

imaginary situations, which don’t need to be met in everyday life. Numerical data don’t need to be taken from reality. Quantities asked do not need to be unknown or needed in real situations and quantities given do not need to be easily accessible in everyday life. What matters in this case is intrinsic consistency and interesting mathematical structure rather than consistency with or importance for everyday life. Some people would say that such problems are useful as mental gymnastics, and this may be true, but I believe that this reason is insufficient and shall present a more important reason to use such problems. Of course, these two kinds of problems do not exclude each other. Many problems, actually used in schools and included in textbooks, are mixtures of these two kinds. However, many of the best and most useful problems clearly belong to the second kind: they certainly are not ‘real-world’. Their purpose is to convey mathematical meaning, that is to use suitable concrete objects to represent or reify abstract mathematical notions. Like animals in fables, ‘real objects’ in these problems should not be taken literally. They are allegories or mental manipulatives or reifications, which pave children’s way to abstractions. For example, coins, nuts and buttons are clearly distinct and countable and for this reason are convenient to represent relations between integer numbers. The youngest children need some real, tangible tokens, while older ones (like Vitya Maleyev) can imagine them,

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which is a further step of intellectual development. That is why coin problems are so appropriate in elementary school. Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be (and in Russia are) common already in middle school. Trains, cars and ships are so widely used in textbooks not because all students are expected to go into transportation business, but for another, much more sound reason: these objects are easy to imagine moving at constant speeds and because of this are appropriate as reifications of the idea of uniform movement, which, in its turn, can serve as a reification of linear function. Thus, we can move children further and further on the way of dereification, that is development of abstract thinking. Many problems used at mathematical olympiads and circles can serve as good illustrations of these ideas. In fact, these problems implicitly introduce children into substantial mathematics, such as number theory, graph theory or combinatorics, but without heavy professional terminology. This is closely connected with the practice of mathematical circles, where interesting and substantial mathematics is taught in such a manner that children in a relatively short time, without any pomp, become proficient and creative.

For example, if you want to introduce

children into graph theory, you don’t need to start with cumbersome terminology and definitions. Instead you can give them a problem: 2n knights came to King Arthur’s court, each having not more than n − 1 enemies among the others. Prove that Merlin (Arthur’s advisor) can place the knights at a round table in such a manner that nobody will sit beside his enemy. [6, p. 89]

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This problem was proposed for the 27th Moscow Mathematical Olympiad, but it was unusable in its original form: “A graph has 2n vertices, each vertex being incident with at least n edges. Prove that this graph has a hamiltonian cycle.” I proposed to represent the hamiltonian cycle by the legendary round table, and in this form the problem was accepted. After that it was discussed at mathematical circles, where knights were represented by circles and friendship relations by lines connecting them. Thus discussion of a “jocular” problem smoothly turned into a study of graph theory, which was non-trivial from the very beginning. The two preceding paragraphs actually present my second and more important reason for inclusion of non-applied, non-real-world word problems into curriculum. Let us remember that formal and abstract thinking, which is essential for success in the modern civilized technological society, does not come as a straightforward result of physiological maturation or social adjustement. Children do not develop abstract or formal thinking as naturally as they learn to run or jump or speak their mother’s tongue. We know this from several expeditions into regions populated by people belonging to so-called ‘traditional’ cultures. The scientific experiments and observations showed that these people do not solve even simple word problems if these problems go beyond their experience, even when they can perform arithmetical operations. This is a telling report: Luria: “Subjects who lived in remote villages and had not been influenced by school instruction were incapable of solving even the simplest problems. The reason did not involve difficulties in direct computation (the subjects handled these fairly easily, using special

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procedures to make them more specific). The basic difficulty lay in abstracting the conditions of the problem from extraneous practical experience, in reasoning within the limits of a closed logical system, and in deriving the appropriate answer from a system of reasoning determined by the logic of the problem rather than graphic practical experience.” [7, p. 120].

Members of ‘traditional’ cultures do

not perform even single syllogisms if the data go beyond their experience: Cole & Scribner: Experimenter: Spider and black deer always eat together. Spider is eating. Is black deer eating ? Subject: But I was not there. How can I answer such a question ? [8, p. 162]. I completely agree with the following conclusion: Tulviste: “Knowing how to solve ‘school’ problems is, of course, not an end in itself. In school, pupils are taught primarily scientific information and scientific thinking. It would be impossible to create, confirm and use scientific information if every separate deduction had to be compared each time with reality or with available information on reality.” [9, p. 122]. Observations of this sort make me conclude that training students in solving non-real-world problems, where data should be accepted at face value, as abstract hypotheses, rather than statements about reality, and conclusions should be deduced from these data, is very important for developing the ability of formal reasoning, which in its turn is essential for success in modern society. The necessity to apply deliberate and organized efforts to develop abstract thinking of children is especially visible in countries like Brazil and Russia, where many people are not or were not educated. For example, most Russians were illit-

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erate at the beginning of this century and I believe that Vygotsky’s ideas should be viewed in connection with a powerful trend towards national education, which took place in Russia in the end of 19 and beginning of 20 century. This pathos of enlightement is also felt in writings of Perelman, who published several excellent popularizations, including [4]. I am writing this article in Brazil, a wondeful country with a great future, but still full of considerable social inequalities. Some people are rich, some are well-to-do, some are poor, some are very poor. It is abundantly evident that income and social position depend on the level of education. Well-educated people, like me, are needed and well-paid, while there are illiterate and semi-literate people who work hard and gain little. There are many unemployed people who barely make ends meet and sometimes beg in the streets. How should we educate children in Brazil and elsewhere to make them employable? In this connection let me quote Gerofsky again: ‘studies in ethnomathematics [Lave, 1988; Lave, 1992; Nunes, Schliemann & Carraher, 1993] have revealed that people who are successful and efficient mathematical problem-solvers in “real-life” (i.e., life outside of school) may be unable to solve school word problems with pencil and paper, even when these word problems appear to be similar to “real-life” problems that the person is quite capable of solving’ [1, p. 36]. This important observation needs, however, the following clarification: those ‘successful and efficient mathematical problem-solvers’ are poor because that ‘real life’, where they are successful, is not only outside school, it is also outside modern industry, banks, government and business offices, laboratories, design and computing

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centers, in short all places where salaries are high. In fact, many of these ‘successful and efficient mathematical problem-solvers’ are street peddlers, belonging to the lowest strata of society. What all children in the world need most urgently is well-organized schooling, that is systematic development of an ability to deal with rigorous scientific formalisms and abstractions. For most students the study of mathematics does not occur spontaneously and needs all the typical academic attributes: regular classes, curriculum, textbooks, exercises, tests, homeworks etc. Since its formation, the United States of America for a long time have been a symbol of democracy and liberty for the whole world. If this image is going to remain, we may expect American educators to promote teaching all children to master formalisms and abstractions. Regretfully, there are opposite tendencies in American education.

When I came to America nine years ago and started to

teach, I found that many university students had very poor experience of solving word problems. When I started to read American educational literature, I found a strange approach to word problems, which was quite different from that to which I had got used in Russia. Some American educators seem to think that problems solved in classrooms should be as close as possible to everyday life. Now I believe that this approach can be traced back to the well-known American psychologist and educator Thorndike, whose influential book [10] contains a chapter called “Unreal and Useless Problems”, which starts as follows: “In a previous chapter it was shown that about half of the verbal problems given in standard courses were not genuine, since in real life the answer would not be needed. Obviously

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we should not, except for reasons of weight, thus connect algebraic work with futility”(p. 258). Although Thorndike tried to apply a strict scientific approach, it should be noted that such words as ‘genuine’ on one hand and ‘unreal and useless’ on the other have strong evaluational connotations; in other words, these are strong good and bad words, similar to those used (and abused) in political propaganda. Let us apply Thorndike’s approach to the problems quoted at the beginning. Leo Tolstoi’s problem: Is it possible for several men mowing one meadow not to know how many of them are there? Of course, not. In dealing with Nosov’s problem, Thorndike would ask: Is it possible to know that the boy has twice as many nuts as the girl without counting the nuts? Of course, not. Thus, according to Thorndike’s theory, both problems are unreal and useless and if given to children, will produce a sense of futility. I can assure you that Thorndike’s theory does not work in Russia, where these and similar problems were always liked. Does it, did it ever work in America? I doubt this. Did Thorndike ever watch children play? Did he notice that all life of children is a continuous play of imagination? There is an important similarity between children’s play and mathematics: in both cases creative imagination is essential. This idea is not new. For example, Martin Gardner wrote in one of his wonderful books: “Perhaps this need for play is behind even pure mathematics.” [11, p. IX] However, some modern American educators seem to follow Thorndike rather than Gardner. For example, Usiskin’s article, reprinted by the journal ‘Mathe-

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matics Teacher’ on the 75-th anniversary of NCTM, says: “Algebra has so many real applications that the phony traditional word problems are not needed.”[12] Whoever uses such a pejorative term as ‘phony’, puts the burden of proof upon himself. I have never seen any proof of the statement that traditional word problems are phony. On the contrary, I have seen them to be enormously productive in Russia. How could such a pejorative term be used without being immediately criticized by the professional community? The explanation, as I see it, is that there is a strange, but widespread, theory in American mathematical education, which I shall call no-transfer theory. According to this theory, children cannot transfer their skills and knowledge from classroom to life outside school and therefore the curriculum should be filled with those problems which people solve in everyday life. According to this theory, children cannot be interested in anything that is not related to everyday life.

When I came to America and became aware of

arguments based on this theory, I first ignored them as absurdist jokes, so crazy they seemed to me. It took me several years or communication and reading educational literature to accustom myself to the strange fact that arguments of this sort are taken seriously and really influence curriculum and manner of teaching. Just one example. The following problem may be used almost everywhere around the globe without objections: Sally is five years older than her brother Bill. Four years from now, she will be twice as old as Bill will be then. How old is Sally now? In America it is declared unfit for the following reason: “First of all, who would ask such a question! Who would want to know this? If Bill and Sally

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can’t figure it out, then this is some dumb family.”[13, p. 85] As an example of an opposite, much more productive approach, let me quote [4], where the second chapter, called “The language of algebra”, consists of 25 sections, each devoted to a problem. One of them, called “An equation thinks for us”, starts as follows: “If you doubt that an equation is sometimes more prudential than we are, solve the following problem: The father is 32 years old, the son is 5 years old. How many years later will the father’s age be ten times the son’s age?” An equation is made and solved, but the answer is negative: −2. What does this mean? Perelman explains: “When we made the equation, we did not think that the father’s age will never be ten times the son’s age in the future - this relation could take place only in the past. The equation turned out more thoughtful and reminded us of our omission.” I believe that this comment is really instructive, and constitutes a sufficient reason to discuss such a problem. Imagine that prospective teachers of literature in a certain country are made believe through their professional preparation that all fairy tales, fables, fantastic stories are useless. When told a fable, where animals speak to each other, they cannot comprehend and enjoy it in a normal way, as all children do, but exhaust their imagination in figuring out how could it happen in real life: perhaps, animals were especially trained to speak? perhaps, they were made some operation? perhaps, it were disguised people? etc. This is similar to the approach of some American educators towards word problems: they insist that it should be possible for the situation and for the question asked to take place in reality. Actually

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these educators suffer from some sort of mental deficiency which is not innate but artificially created by their professional preparation.

America has a strong tra-

dition of inventing interesting problems. The Fifteen game (S. Loyd) and the Life game (J. Conway) are among valuable recreations invented in America. However, this great tradition seems to be pushed away now by those educators who are indoctrinated in the “real-world” spirit. All this does not mean that children need or are interested in cumbersome or far-fetched plots. I do not propose to use arbitrary or chaotic problems. On the contrary, my point is that word problems are mathematical problems presented in the form accessible for children and their quality depends first of all on the quality of their intrinsic mathematical structure and also on their elegance and accessibility. This means that they should not be overburdened with arbitrary or irrelevant details. A good word problem should be as aesthetic as a piece of art. Take Aesop’s fable “The Crow and the Fox”. On one hand it uses images known to every child. On the other hand, it is cleaned of all irrelevant details. But, from the strange viewpoint, which is widespread now, Aesop’s fable is useful only for those who have a chance in some future to perch on a tree branch with a cheeseburger in their mouths. We are dealing here with some of the most fundamental laws of culture: human culture never describes reality one-to-one. It condenses, simplifies, idealizes. For example, geographical maps are not, can not and should not be equal to those landscapes which they represent. Creations of the human mind are subject to

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the drastic law of economy: redundancy should be avoided, every detail must serve the purpose. A good problem shares all the same attributes which Bentley mentioned in connection with one of his excellent computer programs: “General Chuck Yeager (the first person to fly faster than sound) praised an airplane’s engine system with the words ”simple, few parts, easy to maintain, very strong”; this program shares those attributes.”[14, p. 6] Many traditional word problems share these attributes. Many so-called “real-world problems” are cumbersome and loose. The real world is full of waste, redundancy, confusion and boredom, all of which should be excluded from classroom.

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References [1] Susan Gerofsky. A Linguistic and Narrative View of Word Problems in Mathematical Education. For the Learning of Mathematics, 16, 2 (June 1996), pp. 36-45. [2] Robert Thomas, Susan Gerofsky. An Exchange about Word Problems. For the Learning of Mathematics, 17, 2 (June 1997), pp. 21-23. [3] Judy Roitman. E-mail communication. [4] Ya. I. Perelman. Recreative algebra. “Nauka”, Moscow, 1976. [5] Nikolay Nosov. Vitya Maleyev at school and at home. Library of the world literature for children. I Vasilenko, A. Neverov, I Likstanov, Aleksy Musatov, Nikolay Nosov. ‘Destkaya literatura’, Moscow, 1983. [6] G. A. Galperin and A. K. Tolpygo. Moscow mathematical olympiads. “Prosveschenie”, Moscow, 1986. [7] A. R. Luria. Cognitive Development. Its Cultural and Social Foundations. Translated by Martin Lopez-Morillas and Lynn Solotaroff. Edited by Michael Cole. Harvard University Press, 1976. [8] Michael Cole & Sylvia Scribner. Culture & Thought. A psychological introduction. Wiley & Sons, 1974. [9] Peeter Tulviste. The Cultural-Historical Development of Verbal Thinking. Translated by Marie Jaroszewska Hal. Nova Science Publishers, 1991. [10] Edward L. Thorndike a.o. The psychology of algebra. “The Macmillan Company”, New York, 1926. [11] Martin Gardner. The Scientific American Book of Mathematical Puzzles & Diversions. “Simon and Schuster”, New York, 1959. [12] Zalman Usiskin. What Should Not Be in the Algebra and Geometry Curricula of Average College-Bound Students ? Mathematics Teacher, v. 88, n. 2, February 1995, pp. 156-164. [13] Michael K. Smith. Humble Pi. “Prometeus Books”, 1994. [14] Jon Bentley. Programming Pearls. “Addison-Wesley”, 1989.