PROOF AND JUSTIFICATIONS IN MATHEMATICS EDUCATION OMAR HERNÁNDEZ RODRIGUEZ, JORGE M. LÓPEZ, AND ANA HELVIA QUINTERO
A BSTRACT. Simulation programs for geometry exploration developed using Dynamic Geometry software (DG) can significantly enrich the idea of proof or justification in geometry education. Such programs are almost exclusively used to illustrate curious and interesting behaviors of artificially constructed environments for Geometry exploration. However, these programs can also be used to foster student discovery of geometric “truths”, to promote conjecturing, and to construct proofs or justifications of geometric statements. We present two examples. The first one includes our reflections on the use of the Geometric Sketchpad to develop the idea of area in the plane and we make connections to the unit Reallotment of the Mathematics in Context series for teaching the concept of area; the second one presents the proof of the existence of a circumference inscribe in a triangle.
1. I NTRODUCTION The purpose of these notes is to subsume the central points discussed in our class session on Wednesday the 23rd of January, 2019, of the Seminar Mate 8985: Dynamic Geometry. These notes are written in English as a deference of our followers who are not cognizant of the Spanish language. The central theme of the class revolved around what constitutes a proof in mathematics, in general, and, in geometry in particular. It should be remarked at the onset that, perhaps, one of the notable contributions of Euclid’s Elements to mathematics is the establishment of a rather coherent guidelines regarding what constitutes a formal argument in Geometry. It comes as Date: January 24, 2019. 1
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F IGURE 1. First Book: Euclid’s Elements no surprise however, that historically there have been a great latitude regarding the amount of rigor that one requires of mathematical “proofs” and, in fact, rigor may show great variations from one branch of mathematics to the next, depending also on the conventions in the history of mathematics regarding proofs. Arguments that have been held as completely acceptable as justifications or proofs of mathematical statements in certain mathematical epochs, plainly do not give the grade as rigorous justifications in other mathematical times. For instance, the first proposition of the Elements in which it is required to construct an equilateral triangle on a segment AB has been severely criticized for its lack of rigor. According to Joyce 1 [8, 2010] “It is surprising that such a short, clear, and understandable proof can be so full of holes. These are logical gaps where statements are made with insufficient justification. Since the first proof in the Elements is the one in this proposition, it has received more criticism over the centuries than any other.” The criticism voiced by Joyce (ibid) are anachronistic and unfair to the extent that it presents objections that took mathematics almost two millenniums after Euclid’s Elements to sort out. Let’s see what the real objection is. Euclid proposes to draw two circles, one with center A and radius |AB| (the length of segment AB) and another one with center B 1
We urge the reader to examine this reference since it contains a rather complete account of all books of Euclid with commentaries and clarifications.
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F IGURE 2. Figure of Proposition 1 in Euclid’s Elements linked with Postulate 1. and radius |AB|. Then if one chooses the point C in Figure 2 it is clear that ∆ABC is equilateral. Similarly one can also take the other intersection (C 0 ) of the two circles. The point is that in all truthfulness today that argument would require the justification that point C exists, that is that C corresponds to some point of the plane. If the reader does not realize the problem being posed we suggest that he/ she tries to solve the following problem. 1.1. Exercise. In this exercise we denote the set of all rational numbers by Q, and P = Q × Q, the corresponding rational plane, that is the Cartesian product of Q with itself. x2 + y 2 = 1 (x − 1)2 + y 2 = 1 do not intersect on that plane P. [Hint: Find the points of intersection in the usual plane R × R (which contains the rational plane P) and show the solutions are not points of P.] The criticism is, as remarked previously, anachronistic in that if fabricates an objection that was not in the mind frame of the mathematicians of the time of Euclid, and even though the existence of irrational numbers were known, it was unimaginable that this required any further justification. Nonetheless the fact remains that if the argument is written in our time, it requires a justification of the existence of the points of intersection of the two circles.
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2. P ROOFS
AND THEIR RELATION TO THE COGNITIVE DEVELOPMENT OF STUDENTS OF MATHEMATICS
The DG programs are clearly helpful to illustrate interesting animations of challenging geometric environments. As this course continues you will no doubt have more examples of the kinds of “created” environments for geometric exploration and conjecturing. In principle there is a promising future in the use of geometric simulation programs for the purpose of discovering the geometric “truths” that deserve proofs or justifications and the distinction of being elevated to the category of geometric truths or theorems. But the justifications that a student or a geometry teacher can give to “explain” a certain geometric behavior can have quite different cognitive levels, and that is not always good since a justification or an argument of too high a mathematical sophistication is often not understandable by students. Teachers must be able to place themselves at the level of understanding of the students, that is at the right cognitive level, for students to be able to take the maximum possible benefit of such justifications or “proofs”. For the purposes of showing a simulation of a geometric exploration was given to the seminar’s participants. The environment showed a triangle with its interior shaded, a parallel line to its base, two fields indicating the perimeter and the area of the triangle, and an instruction asking the user to move the point along the parallel line; see Figure 3. In the figure there is a segment AB of fixed length that is a side to ∆ABC and the vertex C of the triangle moves freely on a parallel line having a fixed or unchanging distance to the line that extends segment AB. The resources of the Geometer’s Sketchpad readily indicate that while the perimeter is constantly changing in value, the area remains constant. In the discussion some of the students gave various justifications to the observed behavior, and they revolved around the observation that the area of a triangle is given as half the length of the base times its height; in the present case both the base of ∆ABC as well as the height of the triangle remain unchanged during the exploration. It was remarked that this, after all, might not be a good justification to the observed behavior, since it relays on the validity of a formula unknown to many students at the time they
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F IGURE 3. Exploration: Area versus perimeter do the exploration. This is an important situation to consider since often teachers offer justifications that are not manageable by students in terms of their actual cognitive capabilities, although they are, certainly, adequate for the teachers who think of ways to clarify the behavior observed in the exploration. Such a high level explanation of the observed invariance of the area in the exploration would be an example of “didactical inversion”. Didactical inversion is a term introduced by Freudenthal 2[3, 1983] and it describes the fact that some teachers prefer to present the final results of mathematical thinking and not the way these results were developed. It is characterized when "a sophisticated logical approach is preferred to a phenomenological one" (p. 305). In our case, emphasis may be on the triangle’s formula to compute the area and not on the redistribution of the areas. In Freudenthal words, "the final result of the developmental process is chosen as the starting point for the logical structure in order to finish deductively at the start of the development" (p. 305). In pursuing this while keeping within the framework and guidelines of Realistic Mathematics Education (RME) students must be provided with mathematical models that are amenable to their customary and everyday mathematical informal knowledge. For instance, in Figure 4, there is a representation of 2
Hans Freudenthal is a Dutch mathematician considered to be the founder of the Realistic Mathematics Education
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the unit square for the measurement of area (the unit square). The Figure is obtained by drawing two perpendicular lines, one at A and another one at B with intersections of C 0 and C on the line parallel to the segment AB. It is clear that each of the colored triangles have the same area as they are formed by dividing the square along the diagonal. Furthermore, the area of each triangle is half of that of the unit of area measurement, that is 1/2 unit of area. If we now translate the upper
F IGURE 4. Exploration: Unit of measurement of area triangle horizontally one unit to the right, it is clear that we are only “redistributing” the area of the unit square without increasing or diminishing it, and thus we get a parallelogram as in Figure 5 Redistributing other portions of the figure in
F IGURE 5. Exploration: Unit of measurement of area the same fashion leads to a very convincing arguments that all such “deformations” of the unit square lead to an infinite number of parallelograms of the same unit area.
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We include as an appendix the Mathematics in Context Unit Reallotment with the teacher guide with a Realistic Mathematics approach to areas and geometry. The problem of the adequacy of geometric arguments for different stages of the student’s cognitive development, is somewhat harder to sort out and it has been accomplished to very adequate levels of detail in Usiskin [12, 1982], Van Hiele [14, 1986] and Van Hiele [13, 1999]. 3. T HE
NOTION OF A HEURISTIC ARGUMENT IN DYNAMIC GEOMETRY
As proposed in Plato’s famous dialogues, Platonic idealism has it that knowing is actually an educated way of remembering. In the dialogues of Socrates and Meno, Socrates by means of poignant questions get his slave Meno to discover some “truths” of geometry such as the Pythagorean Theorem. So in a very real way, knowing is remembering to the extent that it entails the recalling known knowledge with some savvy guidance. A twist on the Socrates Meno dialogues incorporates Polya’s Heuristics, in so far as it brings about heuristic resources as those described in Polya [9, 1971] and [10, 1990]. These efforts tried, at different times, include some dialogues on locus problems and Ceva’s Theorem; see Campistrous et al. [?, 2012], Hernández Rodríguez and López Fernández [5, 2011], [6, 2014], and [7, 2016]. There are somewhat opposing views related to the epistemology of mathematics knowledge. The platonic view is to be contrasted with other views like those of King Solomon son of David and King of Israel. As stated by Bacon in his essay LVIII. “Salomon saith, There is no new thing upon the earth. So that as Plato had an imagination, that all knowledge was but remembrance; so Solomon gave his sentence, that all novelty is but oblivion.” Bacon [1, 1680, Essay LVIII] proposes interestingly, that knowing is possible because our forgetfulness and not our remembrance as Plato will have it. In any case, both points of view are idealistic to the extent that all knowledge is denied and then pursued as in Plato (it exists in Plato’s cave) or given and then forgotten as in king Solomon’s sentence. In any case, the logic of Geometry has built in the
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Peano Axioms as well as the predicate calculus (statements with quantifiers). With some axioms for set theory (like the Zermelo Frankael) we include also the real numbers and make impossible the criticism of Proposition 1 of The Elements; see, for instance, Stoll [11, 1961]. 4. R EPHRASING
GEOMETRIC STATEMENTS : T HE CONCURRENCE OF THE INTERNAL ANGLE BISECTORS OF A TRIANGLE
Regarding the logic of Geometry we would like to comment on another activity discussed in the same class. We remark that the purpose of this part of this essay is to use the exploration to write up a proof of justification for the existence of an inscribed circumference on a triangle. We should remark that the resources in line for Euclid [8] there is an illuminating discussion of the definitions, postulates, common notions (called axioms by Euclid) and the propositions (results) of Euclidean Geometry (https://mathcs.clarku.edu/~djoyce/java/elements/bookI/ .html#cns). Any person planning on making a systematic reading of the Elements is advised to examine this virtual link. We are not planning on doing any systematic reading of the Elements but we will pursue, nonetheless proofs for the discovered or conjectured geometrical statements. In this section we shall use Figure 6. In the Figure there is a triangle ABC in which the segments bisecting angles A and B have been drawn; The intersection of these segments with the opposite sides are A0 and B 0 . The intersection of segments AA0 and BB 0 is the point I; and IX, IY and IZ are the perpendiculars from I to the sides of the triangle as shown in Figure 7. In Figure 7 the triangles IY A and IXA are congruent (criterion side − angle for right triangles). Thus, segments IX and IY are congruent. Similarly triangles IY B and IZB are congruent (same criterion) and thus segments IY and IZ are congruent. Thus all three segments IX, IY and IZ are congruent taken in pairs. Thus, the segments have the same lengths and I is the center of a circumference containing all three points X, Y, Z. In Figure 8 we see the inscribed circumference. par If we now construct the line segment CI and extended to the
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F IGURE 6. Exploration: The incenter of a triangle
F IGURE 7. Exploration: I is the incenter of ∆ABC. .
side opposite to C to point C 0 , then the right triangles IXC and IZC are congruent (criterion side − angle for right triangles). In particular CC 0 is the angle bisector of angle C. It is clear that all three bisectors of the internal angles of ∆ABC are concurrent, that is, have a point in common.
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F IGURE 8. Exploration: The bisectors of the internal angles of any triangle contain a common point which is the center of the inscribed circle of the triangle. . 4.1. Exercise. Explain why the circle of Figure 8 is tangent to each side of ∆ABC. Begin by defining the notion of a tangent line or segment to a circle. 4.2. Exercise. A cevian of a triangle is any segment with one vertex as and endpoint of the segment while the other endpoint is an interior point of the opposite side of the triangle. Explain why we proved the following important and interesting theorem. 4.1. Theorem. The bisectors of the interior angles of any triangle are concurrent cevians at a point (called the incenter) which is the center of the inscribed circle to the given triangle (circumference tangent to all sides of the triangle). 5. C ONCLUSIONS Dynamic geometry programs are useful for creating experimental environments that foster mathematical creation and conjecturing. With a carefully planned design, DG can be used to help students to write justifications of geometric results based on the observed dynamic exploration. In the planning, teachers must have in consideration how the DG software can be used as an instrument to develop the expected
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competencies. We should beware as teachers to avoid the error of didactic inversion as we try to justify geometrical behavior based on principles and observations of too high a cognitive levels for the students participating in the exploratory activities designed for them.
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R EFERENCES [1] F. Bacon. The Essays or Counsels, Civil and Moral, of Sir Francis Bacon Lord Verulum Viscount St. Alban: with a Table of the Colours of Good & Evil: Whereunto Is Added The Wisdom of the Ancients .. London. Printed by M. Clark for Samuel Mearne ... John Martyn ... and Henry Herringman ..., 1680. [2] L. A. Campistrous et al. La dimensión “dinámica” del problema de la determinación de los lugares geométricos en la geometría. Epsilon: Revista de Educación Matemática, 29 (1), 9-22, 2012. [3] H. Freudenthal. Didactical phenomenology of mathematical structures. D. Reidel Publishing Company, Dordrecht, Holland, 1983. [4] K. Gravemeijer, et al. Reallotment. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.). Mathematics in context. Chicago: Encyclopædia Britannica, Inc. 2010. [5] O. Hernández Rodríguez & J. M. López Fernández. Geometría Dinámica: Una opción novel para la capacitación de maestros. Uno, Revista de Didáctica de la Matemática, 57, 93-106, 2011. [6] O. Hernández Rodríguez & J. M. López Fernández. Mate 8980: Taller de Geometría Dinámica Conversaciones sobre el Teorema de Ceva. Technical Report, ResearchGate. December, 2014. [7] O. Hernández Rodríguez & J. M. López Fernández. Heuristic Conversations on Ceva’s Theorem. ResearchGate. October, 2016. [8] D. E. Joyce. Euclid’s Elements https://mathcs.clarku.edu/ ~ djoyce/elements/ bookI/bookI.html , Department of Mathematics and Computer Science, Clark University, Worcester, MA. [9] G. Pólya. How to Solve It. Princeton University Press. November, 1971. [10] G. Pólya. Mathematics and Plausible Reasoning, 2 Vols. Princeton University Press. August, 1990. [11] R. Stoll. Sets Logic and Axiomatic Theories. W. H. Freeman. NY. 1961. [12] Z. Usiskin. Van Hiele Levels and Achievement in Secondary School Geometry. University of Chicago, 1982. [13] P. M. Van Hiele. Developing Geometric Thinking through Activities that Begin with Play. Teaching Children Mathematics, Vol. 5, 310-316, 1999. [14] P. M. Van Hiele. Structure and Insight. A theory of Mathematics Education. Academic press Inc, 1986.
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