Prisms & Cubes
David Wheeler
Educational Solutions Worldwide Inc.
First published in 1974. Reprinted in 2009. Copyright Š 1974-2009 Educational Solutions Worldwide Inc. Author: David Wheeler All rights reserved ISBN 978-0-87825-029-5 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com
Table of Contents Presenting The Gattegno Prisms And Cubes ................ 1 Some Of The Topics That Can Be Taught Through Activities With Prisms And Cubes.............................. 19 Arithmetic ..............................................................................19 Algebra .................................................................................. 20 Geometry............................................................................... 20
Presenting The Gattegno Prisms And Cubes
David Wheeler The set of Prisms and Cubes is a natural extension of the set of Algebricks and gives students access to a range of new topics in the fields of arithmetic, algebra and geometry. Some activities with the Prisms and Cubes are suitable for students in kindergarten classes; others can be challenging to advanced students in high school and beyond. In this introduction the characteristics of the material and its special contribution to mathematical education are discussed. At the end of the leaflet is a list of some of the mathematical topics that can be reached through activities with the Prisms and Cubes.
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Prisms & Cubes
The reader who has not already used the material will find it helpful to have a set near him as he reads and use it to illustrate the points that are made here only in words. This leaflet tries to create a context — the right frame of mind, as it were — for work with students; it is not a manual. Subsequent leaflets will spell out in more detail the ways that students can use the Prisms and Cubes. 1 The set of 1,407 pieces is packed in a wooden box. A number of the larger pieces are stacked in recessed trays in the two drawers; the rest are in polyethylene bags in the compartment below the lid. Anyone coming to the set for the first time is immediately aware that the pieces are colored. If he is familiar with the set of Algebricks he will recognize that the same set of colors is used, but he may be struck by the fact that the colors and shapes are not in one-to-one correspondence as in the case of the rods — pieces colored red are not all the same shape, for instance. Closer inspection will also reveal that some, but not all, of the pieces with the same shape and size appear in different colors. The coloring system is not random and it can readily be discovered. The teacher has the choice of inviting the students to crack the color-code straight away, or of delaying the question until the students have more experience in handling the pieces and are more able to be articulate about their properties. In the latter case, because the coloring system has a physical reality, the colors still carry visual messages to the user that he can understand even if he cannot verbalize them.
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Presenting The Gattegno Prisms And Cubes
Older students may recognize and be able to express the fact that a common characteristic of the pieces is that they are all right prisms having a square cross-section (or alternatively, that they are all rectangular parallelepipeds), but neither the technical language, nor the immediate awareness of the common property, is necessary for work to begin. 2 Many students will recognize some of the pieces as cubes. In practice it can be helpful to put into circulation several labels, such as rods, tiles, cubes, slabs, pillars. Although these are not mathematical terms, with the exception of cube, when used with adjectives denoting color they make it easy to construct verbal pointers which identify particular shapes — for instance, orange cube, black rod, dark green tile. The set of Prisms and Cubes only contains one piece that can be denoted by “orange cube,” but several that can be called “dark green tile.” It is a property of the set and not the labels alone that ensures that these particular words can be used unambiguously. Visual inspection of the pieces that can be called dark green tiles, perhaps supported by physical juxtaposition, gives the necessary evidence that for all practical purposes they are equivalent to each other and that it therefore does not matter which particular one is indicated. A similar verbal construction — “yellow pillar,” for instance — turns out to be potentially ambiguous since the pieces that share this label are not all perceptually equivalent to each other. The words “tile,” “slab” and “pillar” bring enough meaningful associations to the teacher so that he will be in no doubt which
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set of pieces is covered by each label. But some of the students may not be at all sure at first, or may give the words a different range of application. As when they meet unfamiliar words at any time they have to discover the criteria for their use, and if they cannot find anything in their previous experience that offers clues they will have to search for them in the present situation. The teacher can become aware that this uncertainty is common enough when students are learning mathematics, particularly geometry, and observe himself dealing with it, perhaps more easily than when the words involved are standard mathematical terms. Is uncertainty best dealt with by giving a verbal definition? Or by immediately correcting mistakes? Or by pointing out further examples? Or by allowing students to use any word they like? etc. Sooner or later the students will become aware that some of the classes covered by the color-shape vocabulary overlap. The smallest yellow pillar may be called a yellow rod; the smallest black slab may be called a black tile. In particular, the small white pieces are cubical in shape, and so belong to the class of cubes, but they can also be “the smallest rods” and “the smallest tiles,” and therefore belong to the class of rods and the class of tiles. Whether a white piece is called a rod or a tile or a cube will depend on the context, depending on the other pieces it is being related to at that time. Changing the labels clearly does not alter the physical properties of the pieces, but it usually indicates that these relationships rather than those are being stressed at the time. (Overlapping classes are fairly common in geometry: squares are also rectangles, parallelograms are also quadrilaterals, hexagons are also polygons, and circles are also ellipses. Only the introduction of quantifiers into the language 4
Presenting The Gattegno Prisms And Cubes
will remove any mysteriousness about the fact that “squares are rectangles” is a true statement, whereas “rectangles are squares” is regarded as false. In fact, the corresponding statements should be: “all squares are rectangles” and “some rectangles are squares.” The teacher can make many opportunities to introduce these verbal distinctions.) 3 When students manipulate pieces from the set of Prisms and Cubes their actions and perceptions give them direct experience of certain aspects of space and spatial relationships. The pieces are tangible, having particular form and substance, occupying space but freely movable. They can be stacked, juxtaposed, balanced, and arranged in a multitude of positions relative to each other. (They can also be sawed, painted, dropped into water and thrown.) But the perception of spatial properties does not of itself inevitably lead to mathematics. What must happen before activities with the prisms will give entries into mathematics? We have already noted in the previous section that the dark green tiles, for instance, can be said to be equivalent to each other. As soon as such a statement is made and acted upon, the user of the material is no longer operating simply at the “concrete” or physical level even though he has based his statement on features of the physical situation. He is saying that he will regard any dark green tile as having the same function as any other in the manipulations he performs with it — that one dark green tile can be substituted for any other without it making any difference.
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It is clear that he is not taking all the features of the physical situation into account since it would certainly make sense to ask such questions as: Does this tile actually have uniform thickness? What is the range of thicknesses in a set of orange tiles? What is the average weight of the dark green tiles? Questions that would be important for quality control purposes, for instance, take into account physical properties of the pieces that the statement of equivalence ignores. How many red tiles will exactly cover a dark green tile? The same words can make an answerable or an unanswerable question. The question cannot be answered if all the physical features of the pieces are taken into account: the naked eye can easily detect discrepancies in the “fit” of nine red tiles on top of a dark green tile. But since manipulation and perception indicate that a “square” of nine red tiles is much more nearly the shape and size of a dark green tile than anything else in the set, one can decide to overlook the physical discrepancies and say that nine red tiles “exactly cover” a dark green tile. (The decision is not difficult to make, of course, and the teacher will notice how questions like the one at the beginning of the paragraph urge the students to make it.) Awareness of this process brings out a number of significant elements. First we notice that the vast majority of the attributes of the pieces that could be attended to are cavalierly ignored. Secondly, a particular physical relationship is idealized into a precise statement of equivalence: the precision is not an artifact of the material but of what is said about it. Thirdly, the idealized relationship taken together with other idealizations provide a supply of mental material that can be operated on. 6
Presenting The Gattegno Prisms And Cubes
Fourthly, the physical material generally behaves in a way which is compatible with operations that can be carried out on the mental material. As a simple illustration of the last two remarks we can consider two students finding out how many red tiles will cover a brown tile. One may manipulate the material and make his statement when he sees how the physical situation behaves. The other, already having at his disposal several idealizations — that the red tiles are all equivalent, that nine of them will cover a dark green tile, and that the difference between the edge of a brown tile and the edge of a dark green tile is equivalent to the edge of a red tile — can argue that the brown tile will need seven more red tiles; two more rows of three each and another to fill the remaining corner. Both students will arrive at the same answer; the second is certainly operating mathematically. In general, the role of the physical material is to trigger and, where necessary, to confirm the idealizations of relationship which can lead to mathematics when they are operated on by the mind. The idealizations are held and communicated by language and notation and can, in principle, be changed when the user wants to stress a different aspect of relationship. 4 There are other, less obvious, ways in which mathematical thought can “take off” from the physical situations created with the material. For example, eight white cubes can be assembled into a shape equivalent to an orange cube. At the level of the physical material, the latter equivalence is one out of a large number of possible ways of constructing a shape equivalent to the orange cube, whereas there are only a small number of different ways of constructing a shape equivalent to the red cube with the available material. Nevertheless, at the 7
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level of mathematical thought one can become aware that the scale of the physical model could be entirely ignored. Immediately this brings the awareness that everything that can be obtained by direct manipulation and perception of the equivalent transformations of the orange cube is available to one’s inner perception when one looks at a red cube — or, indeed, at any cube, including the white one. One is then able to “see” transformations of the white cube which are quite unrealizable in physical terms. As another example, consider a cube equivalent to an orange cube constructed from horizontal layers of orange tiles. It can be subjected to a physical transformation which pushes the tiles at right angles to a vertical face so that each overhangs the one below by a small amount. The shape is no longer cubical, but since no wood has been added nor taken away, it can still be regarded as in some sense equivalent to an orange cube. The transformation leaves some properties unchanged — mass, weight, volume, density, color, height, number of components, for instance — but changes others. What has the transformation done to the external surface area? A changing situation has been created by an action and a question about it focuses on one attribute of the situation. One’s perception may or may not yield immediately the awareness that the surface area is now greater than before. If it does not, how can one be sure? Calculation can be used as a tool, though a rather cumbersome one, possibly by supposing that each tile overhangs a distance of a centimeter, say, and working out the new surface area and comparing it with the surface area of the orange cube. Alternatively, the perception of an increase may be 8
Presenting The Gattegno Prisms And Cubes
forced if the tiles are pushed further in the same direction (supporting them so that the edifice does not topple). If this still does not bring awareness because of the complexity of the movement, further questions can direct attention to particular features — does the area of the base change, or the area of a side, for instance? In fact, the only change in area occurs on the sloping “staircase-like” faces and it increases as the stairs become wider. Once this certainty has been reached there is usually no doubt that there is a continuous relationship between the physical action and the increase in area, and that therefore any similar distortion, however small, from the form of a cube will have increased the surface area a small amount. The situation may now be abandoned as having produced a correct response. But the chance exists, and can be seized, of taking the consequences further. For example, can one conceive of tiles with half the thickness of the orange tiles being available? And can one follow through, in one’s mind, the effect of using these tiles instead in a similar experiment? Once the question has been raised, most people find no difficulty in entertaining it; but the question now becomes a “thought experiment,” physical action is replaced by virtual action, and perception of the physical material by perception of one’s store of images. Enough similarities remain for one to be sure that the same kind of “imaginary” physical transformation would also increase the area for the same kind of reason. It will be necessary to be more precise or to probe more deeply to find whether the use of thinner tiles brings anything new to the situation.
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It may come to us to consider whether the surface area of the solid made with the tiles can be increased by the transformation to any amount; the transformation certainly increases the area, but is there a point beyond which it cannot continue to increase it? Again, once the question has been formulated it brings the answer with it. The area obviously cannot be increased beyond the total surface area of all the individual tiles used to make the cube. And it can be seen that this maximum is somewhat greater in the case of the thinner tiles. The mathematician within each of us, having listened so far, will now feel an irresistible urge to pursue the intellectual implications of these discoveries. What if the thickness of the tiles is halved again? And again? We have, in fact, produced a new transformation, quite different from the first, which also generates increasing areas — i.e., the successive maximal areas as the tiles are successively halved in thickness. Will this transformation produce increases only up to a certain value or can it produce increases of any size? This question may be more difficult to answer than the earlier questions and our unsupported intuition may be unreliable (as the classical paradox of Zeno reminds us). However, whether we succeed in resolving it to everyone’s satisfaction or not, we have constructed a question that is both challenging and quintessentially mathematical out of a situation that was simple enough for anyone to come to terms with. 5 Almost everyone with access to a set of Prisms and Cubes can quickly become familiar with the properties and the
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Presenting The Gattegno Prisms And Cubes
particular decisions that the inventor made in designing the forms of the material. Almost everyone brings to it the capacities for action and perception that provide raw material for mathematical activity. But most people do not know when they begin how to “enter into a dialogue” with the material so that the mental activity the interaction stimulates is as deep and wideranging as possible. It may at first seem unreasonable to use the word “dialogue” since the prisms are inanimate a-mathematical entities by themselves. Nevertheless the observation that almost everyone given access to a set spontaneously begins to handle the pieces and “make something” with them shows that the material can force someone into acting upon it. At a more subtle level, “doodling” with the material can often trigger unexpected thoughts in someone who is sensitive to the potential significance of a newly-perceived relationship. But the word “dialogue” is more than a metaphor in the sense that anyone working with the prisms is conscious of an activity in his own mind that is directly influenced by what the material brings to him as well as by what he brings to his encounter with the material. That the dialogue is entirely within his own mind does not, indeed, distinguish this kind of mental activity from that which might arise from his encounter with another person. A key element in establishing fruitful dialogues with the material is knowing how to ask oneself mathematical questions. The characteristic on-going quality of mathematical inquiry can perhaps be suggested by the tone of the question, Would it make any difference if . . .?, which can be thought of as a crude pro forma for generating fresh initiatives. The effect of this kind of 11
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question is to press one to be clear about the class of situations to which a mathematical statement refers and, by emphasizing the idea of difference, it asks one to consider again the equivalences involved. For example, suppose that we have discovered that a black tile can be covered with a pink tile and a light green tile and two congruent rectangles made either of pink or light green rods. As it stands, this statement remains at the level of the particular. Asking oneself “Would it make any difference if tiles of other colors were used instead?” opens up a number of possibilities. One might (1) retain the black and alter the pieces covering it, finding out that if one of the tiles is replaced by a larger one, the other has to be replaced with a smaller one and the rectangles modified correspondingly (in this case discovering that instead of “pink” and “light green” in the above statement we could equally well say “yellow” and “red” or “dark green” and “white”); or one might (2) try covering other tiles than the black, finding out whether the same pattern of two squares and two rectangles will work in other cases (so that, for instance, if we replace “black” by “dark green” and “light green” by “red,” the statement at the beginning of the paragraph remains true). The reader who is following this description with the pieces at hand will see that this activity can be put into the language of algebra — e.g., the last statement can be written as d2 ~ (p + r)2 ~ p2 + 2pr + r2.* In these terms it gives an understanding of how “the square of a sum” differs from “the sum of two squares,” *
The sign ~ is read “is equivalent to.”
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Presenting The Gattegno Prisms And Cubes
and how the standard identity for (a + b)2 can be arrived at. We can say that this direction of investigation is equivalent to attempting to generalize the original discovery. (The actual moment of awareness of a generalization can vary from activity to activity and from person to person; an observer can rarely be sure. Someone who is in the act of covering a black tile may already know when he picks up, say, a pink tile that he is choosing from a number of equally valid alternatives and follow through his choice, knowing that he could deal with any of the other possibilities open to him in a similar way. For this person the general is already implicit in the particular. Another may require seeing that others have chosen differently and yet obtained a pattern that is in some sense “the same” as his. Yet another may find the generalization in the language and the notation that someone applies to the situation. And still another may need to repeat the activity several times, varying what can be varied, until his perception “discovers” the constancy of structure — i.e., he finds that something can be said which holds over all the instances. Of these possibilities the paradigm of mathematical generalization is the first — where the mind takes in the freedom available in a situation even while it is working out the details of the particular instance.) But if the tile to be covered is dark green or orange, say, it is possible to make the two covering tiles the same color, and in this case the rectangles also become squares. In fact, the covering can then be composed of four congruent tiles — so that d2 ~ 4g2; or o2 ~ 4y2. If we investigate other instances in which this can occur we can collect all together in the statement (2x)2 ~ 4x2. In taking this path we have specialized the original situation 13
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and then generalized from the specialization. It is an important characteristic of mathematical thinking that it moves back and forth between these two modes. The purpose of asking the question, Would it make any difference if . . .? or What would happen if . . .?, is to prompt the generation of new questions from old ones. The development of “lines of inquiry,” in which the exploration of any question can be made the starting point of others, gives the learner the authentic experience of mathematics as unfenced territory crisscrossed by innumerable roads and tracks. Any learner at any level can benefit by knowing how to transform solved problems into new unsolved problems. It often requires little more than the awareness that it can be done. For example, the material allows one to explore “the cube of a sum.” Association with the word “sum” brings the word “difference” to mind. Can the material be used to express “the cube of a difference” and can an algebraic identity be developed? (The answer is not necessarily “Yes”; although there is a standard identity for “the difference of two squares,” there is no corresponding identity for “the sum of two squares.”) Again, in Section 4, we examined the transformation of a cube into shapes with the same volume but greater surface area. Turning the problem around in the mind may prompt the question, Does the cube have the least surface area of all the solids having the same volume? Or, focusing on the fact that one quantity stays the same while the other changes, the mind can play with the interchange of the two quantities and come up with the question, Suppose the surface area is kept constant, is it possible for the volume to change and, if so, how? Or again, Can 14
Presenting The Gattegno Prisms And Cubes
there be different shapes with equal volumes and equal surface areas? 6 So far in this introduction we have been concerned with the way in which mathematical activity can arise from actions with the prisms through systems of statements that idealize relationships that the material suggests to the perception of the user. Now although the material is specially designed to give access to mathematics, its tangible physical existence makes it available for other kinds of activity. For example, anyone who manipulates the material may find his attention caught by the aesthetic characteristics of his constructions and explore, say, properties of pattern, proportion and balance which it would be stretching a point to call conventionally mathematical (although the fact that the same words are shared by architecture and mathematics is an indication of some common ground). Further, there are questions one can ask which belong more to other sciences than to mathematics. The question Are all the pieces cut from the same kind of wood? may suggest a study of the grain or, if this fails to be conclusive, an experiment to determine the densities of various pieces. The set of Prisms and Cubes will not produce very many significant questions in the areas of, say, art or botany or physics since it was not intended for this purpose. Nevertheless a teacher who is aware of some of these possibilities can suggest inquiries to the students which are interesting in their own right and instructive in giving a more precise awareness of the distinctive quality of mathematical thinking through the direct contrast with other modes. Why are some faces rougher than others? Can a way be found to measure the relative roughness 15
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of two faces? Given a tower of three cubes, how far can the middle one or the bottom one be displaced horizontally before the tower falls? How much water is displaced if an orange cube is lowered gently into a full container? Finally we note some of the consequences of paying attention to discrepancies between the physical behavior of the material and the behavior of the mathematics we apply to it. For example, an orange rod is almost the same length as a train of ten white rods and we idealize this relationship in the statement o ~ 10w, or some statement equivalent to this. So a length of 10 orange rods is equivalent to a train of 100 white rods (since 100 is another name for ten 10s). But if we make a train of ten orange rods and lay a train of white rods alongside, it is possible that we shall find, say, that 102 white rods are required to get as close as possible to the length of the orange train. Do we therefore lose confidence in our earlier statement? We may informally patch up the disagreement by saying that a train of ten orange rods “should” be the same length as a train of 100 white ones, just as we may say that the three altitudes of a triangle “should” have a single common point even though our attempts to construct them don’t bear this out. But the phenomenon sheds further light on the idealization that takes place when we make mathematical statements: our mathematical statement is not really about the orange and white rods in the set but about idealized rods which are related in precisely the way we said they were. Most of the time we remain happily unaware of the distinction because we get the same statements whether we are referring to our observations of the physical relationships of the material or to our deductions from 16
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our idealizations; only blatant discrepancies like the one discussed make us open our eyes. Noting these discrepancies introduces a new possibility — that instead of being neglected they can be mathematized, and this can yield a study of estimation, approximation and error. New kinds of questions can be asked: for instance, Estimate how many white rods would be required to make a length of 100 meters. If 102 white rods makes a train almost equal in length to 1 meter, how many will almost cover a surface with an area of 1 square meter? What is the average percentage error in the area of a face of a white cube? If it is known that each piece is cut with an error of not more than 0.5mm, between what limits does the area of the top surface, or the volume, of a light green cube lie? 7 The preceding discussion will have made it clear that the set of Prisms and Cubes is much more than an attractive construction toy. But it is also much more than a mathematical teaching aid. First of all, it is “multivalent” — that is, it can be used to create a variety of different situations that yield a wide variety of results. Secondly, and fundamentally, it is an “instrument of mathematical awareness.” Because the situations that can be constructed with it are immediately available to anyone’s perceptions, the learner can follow all the stages in his actions, perceptions, images and thoughts that lead him to be able to formulate publicly acceptable mathematical statements and conclusions. Its ability to initiate this process, and sustain the awareness of the process, constitutes the principal value of the material. The mathematical end-products are important to the learner as signs that he has worked well and earned his 17
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reward, but the lasting benefit he can get from his contact with the material is an increased awareness of how he functions as a mathematical thinker and a mathematizer. Activities with the material give him a chance to know in specific, concrete terms: how his perceptions become sensitized to mathematical relationships; how and when he makes a shift from the level of actual actions to the level of virtual actions; how he controls and increases his stock of mental images; how and when he checks the validity of an intuition; how he reaches the conviction that a discovery is universal; how he utilizes speech and notation; how he constructs good mathematical questions for himself; and so on. The principal function of the teacher here — which he cannot fulfill unless he has at least begun to work on these awarenesses in himself — is to use his resources, expressed through his choice of activities and his technique of questioning the students, to make clear to them all the resources they have that can be turned to good mathematical account if they choose to utilize them.
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Some Of The Topics That Can Be Taught Through Activities With Prisms And Cubes
For convenience the topics are listed under the conventional headings, but many of the categories overlap and can be related and combined in many different ways.
Arithmetic Areas of squares and rectangles Square numbers and square roots Rectangular (composite) numbers; factors Surface areas of simple and compound rectangular solids (i.e., those that can be constructed with the prisms) Volumes of simple and compound rectangular solids Cubes and cube roots Metric system units of length, area and volume Fractions; fraction of a fraction 19
Prisms & Cubes
Ratios of two or more quantities; proportion Representation of numbers in different bases Approximations; errors
Algebra The distributive law for multiplication over addition Difference of two squares Square of a sum; square of a difference Sum of two cubes; difference of two cubes Cube of a sum; cube of a difference Use of these identities in calculations
Geometry Vertices, edges and faces; dimensions Parallel and perpendicular edges and faces Symmetry axes of a cube and a square prism Cross-sections of a cube and a square prism Ratios of lengths, areas, or volumes of similar shapes
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