J. VAN DORMOLEN THE USELESSNESS OF VENN DIAGRAMS* Attempts at introducing notions like intersection, subset, element of a set to highschool students by means of Venn diagrams turned out to be less successful than I expected. This is partly due to my lack of experience, but also implied by the subject itself. First I tried it as follows. I drew an oval on the blackboard and said Fig. 1. something like 'This represents a set and every element of the set is put into the oval'. If for instance I would suggest the set of positive integers less than or equal to 20 and divisible by 3, I marked some points within and some outside the oval, wrote numbers beside them, and finally said: This point represents the number 6, which belongs to the set; that point represents the number 7, Fig. 2. which does not belong to the set. Then I asked the students which numbers had to be written inside and,,which not. There are a few disadvantages of this method: (a) The attention is drawn to the oval rather than to the characteristic which causes the elements of the set to belong to each other. (b) Not all points of the drawing plane are used to represent a number. 402 Educational Studies in Mathematics 1 (1968-69) 402-407; 9 D. Reidel, Dordrecht-Holland
THE USELESSNESS OF VENN DIAGRAMS This drawback is much more serious than I initially supposed. The pupil needs time to understand that it is a drawback. He feels there is something wrong but he does not realize what is the matter. It is a handicap if he finally succeeds in taking the fence, and he has to make efforts to catch up. Of course I will adjust my teaching to this experience and give students the time to overcome this difficulty though it would be better to avoid it entirely by another teaching method. (c) Theelementsofmysethavenotbeenneatlyarranged. Even the sloppiest student appreciates some order. Actually there is a certain order. I need not tell my students that the number sets they know are ordered. They know it, and the only thing I have to teach them is the word, and the fact that this order may be useful. The numbers on the blackboard are not ordered. This is not a serious drawback, but in any case one the students have to overcome and I have to account for. (d) It may happen that my oval remains void. Then I have drawn something that is useless and, actually, does not exist. (e) If somewhat later I have to draw the set of points of the drawing plane which lie at a given distance from a given point, 1 get again such a figure. But now the boundary itself forms the set rather than the things inside. Fig. 3. It is even worse if I turn to the set of points at a distance from a given point, larger than a given distance. There are two advantages of this method: (a) The conclusive force I can find in such diagrams if I have to prove theorems like AnB=BnA (AnB) uc=(auc)n(buc) A~B--,AnB=A. If I would deal with such theorems, it would certainly pay to use diagrams, to have the students to meet the difficulties mentioned above, and to teach the students to overcome them. If I succeed, I would have helped them to understand that pictures are no concepts, but rather representations of con- 403
J. VAN DORMOLEN cepts, which can serve to guide our reasonings. Then, of course, to prove theorems I should adopt conventions on how the ovals have to be drawn. In the case of two sets, for instance, I have to account for the fact that parts can be void. From Vredenduin's paper 1 it appears that then I have still to draw the general figure, and only to indicate which parts are void. But leaving parts void leads to didactical absurdities, which have probably never been intended by old Venn. My students would consider it a new proof for the saying that mathematics is to make easy things difficult. I have to add that I do not need at all this kind of theorems. Sets and related concepts serve to simplify the language whenever it is useful, and to make things clearer and more perspicuous. Theorems like the ones mentioned are Fig. 4. appropriate in an introduction into set theory, but set theory is not a school matter. Moreover, the first and third theorem are so evident for high school students that dealing with them explicitly can only create confusion. (b) Reviewing some teaching matter I can explain the students the connection of concepts by means of diagrams, for instance to show them that the properties of parallelograms are shared by rhombs, and the properties of rectangles by squares (Figure 4), and how the most usual number sets are related to each other (Figure 5). The earlier mentioned drawbacks are still palpable in this application (a point in the oval 'rhombs' means a rhomb rather than a point). Even then I would use such a method at most in a recapitulation, not when I start speaking about quadrangles or numbers. My results in the classroom improved when I organised my questions a bit differently. I took a finite universe, for instance the positive integers below 13. I put the elements upon the blackboard. Not as points with labels, but the labels themselves. I chose the places such that my questions could be easily answered (Figure 6). Then I asked: Draw a closed curve around the even numbers, around the multiples of 3. What is the intersection of both sets? Disadvantages: 404
THE USELESSNESS OF VENN DIAGRAMS (a) Again only part of the points of the plane is used, but this is not serious since the curves are drawn after the children have learned what is the universe. (b) Again the numbers are not ordered. Fig. 5. (c) I cannot draw a void set. Once I saw an attempt to explain the void set by first drawing a curve around a pile of popcorn and then eating the popcorn. An original idea though I do not know how to eat numbers. (d) Geometric sets cannot be drawn in this fashion. (e) The set must be finite (and not too large because otherwise I have to write too much before I can start working). Advantages: (a) The attention is not diverted from the essentials, that is the characteristics by which the elements belong together. The dosed curves now function as a fence around the elements. There is no danger any more that the set is identified with the oval. I once heard somebody compare the curve with a rope that binds the elements together. My first reaction was: the rope is not tight, the elements will drop out. Of course this is a childish reaction which, however, will occur in our students. It is not so serious; they can think about it, but meanwhile the attention is diverted from the gist of the exposition. This I like to avoid. (See the earlier remarks on overcoming handicaps.) (b) The students feel that whenever they need it, they can draw a fence in another set. (c) I cannot draw a void set. I thought this was an disadvantage, but in fact it is not. If there is nothing, I would draw nothing. Every representation of the void set is wrong. An empty matchbox is a wrong representation. It suggests that a set be a box where you may or may not stuff in things. A class of students is not the classroom where they sit. Nobody can point to a class of students, but one can point to its elements. The classroom is at most the fence around. We use the void set because it simplifies the language; 405
J. VAN DORMOLEN often it allows us to drop casual distinctions. The same happens if we introduce points at infinity: it makes things easier if we can always speak of the intersection of two lines, rather than distinguishing special cases of parallels. Summarizing I would like to remark that Venn diagrams can be successfully used in the just described situations with finite sets. But these are so exceptional that if it happens I would draw a fence without explaining it explicitly. /"7 ".... -zl ; 11 Fig. 6. In general I would draw a picture of a set such as it fits most appropriately in a given situation, and then it is almost never a Venn diagram. I do not use Venn diagrams: if drawing the set of points at the same distance from two given points; if answering the question on the intersection of the sets {xl x > 2} and {xl x<4}; if counting the numbers of lattice points inside or on the boundary of the quadrangle with the corners r0, 07, r3, 07, r0, 5 -~, r2, 4-1; if determining the common solutions of the equations x2+y2=25 and x+y=l; if determining the set of lines that form equal angles with two given interseeting lines and pass through their intersection point. I can continue this list indefinitely. There is an argument which may be used by champions for Venn diagrams at school: the students learn abstract thinking. I seriously doubt it, and even if it should be true, I could point to a few subjects which were more suited to this purpose and which, nevertheless, nobody would advocate in the high school (at least in the first cycle). There are enough opportunities in the existing curriculum to teach abstract thinking - whatever this may be - without Venn diagrams. 406
THE USELESSNESS OF VENN DIAGRAMS REFERENCES * Translated from Euclides 43 (1967-68), 273-278. 1 This issue, p. 394. Note that Vredenduin's paper is not a plea for Venn diagrams, but a successful attempt to show teachers how conclusive force can be derived from Venn diagrams. 407