Michael T. Battista
Douglas H. Clements
Educators continue to debate the relative emphasis that formal proof should play in high school geometry. Some argue that we should continue the traditional focus on axiomatic systems and proof. Some believe that we should abandon proof for a less formal investigation of geometric ideas. Others believe that students should move gradually from an informal investigation of geometry to a more proof-oriented focus.
Many of the discussions in this debate ignore what should be a crucial component of any serious discussion of curriculum issues -- research on learning and teaching. In this article, some of the research that is relevant to this debate will be discussed and a course of action recommended that is consistent with the research.
Establishing Truth in Geometry
No one would deny that establishing the validity of ideas is critical to mathematics, both for professional mathematicians and for students. But how do people establish "truth"; how can they prove things? According to Martin and Harel (1989), in everyday life, people consider "proof" to be "what convinces me." Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms.
But mathematicians most often "find" truth by methods that are intuitive or empirical in nature (Eves 1972). In fact, the process by which new mathematics is created is belied by the deductive format in which it is recorded (Lakatos 1976). In creating mathematics, problems are posed, examples analyzed, conjectures made, counterexamples offered, and conjectures revised; a theorem results when this refinement and validation of ideas answers a significant question. Hanna (1989) argues that because mathematical results are presented formally by mathematicians in the form of theorems and proofs, this rigorous practice is mistakenly seen by many as the core of mathematical practice. It is then assumed that "learning mathematics must involve training in the ability to create this form" (pp.22-23). The presentation obscures the mental activity that produced the results.
In fact, according to Bell (1976), personal conviction grows out of internal testing and forming a judgment about whether to accept or reject a conjecture. Later, one subjects this judgment to criticism by others, presenting not only the generalization formed but evidence for its validity in the form of a proof. For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument.
In sum, formally presenting the results of mathematical thought in terms of proofs is meaningful to mathematicians as a method for establishing the validity of ideas. However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?
Research On Students' Learning Of Proof
Several studies have reported that formal deduction among students who have studied secondary school geometry is nearly absent (Burger and Shaughnessy 1986; Usiskin 1982). (Note, however, that most students mentioned in these data would have taken a proof-oriented geometry course because these studies were conducted before informal treatments were widespread.) In fact, according to Schoenfeld (1986), most students who have had a year of high school geometry are "naive empiricists." For example, in one series of interviews (Schoenfeld 1987), college students who had taken one year of high school geometry were asked to use a straightedge and compass to show how to construct the circle that is tangent to two intersecting lines, with one point of tangency being a given point P on one of the lines. In the same problem session, the students had already proved that the center of a circle tangent to two given lines lies at the intersection of the angle bisector of the angle formed by the lines and the perpendiculars through the lines at the points of tangency. Despite this finding, 30 percent of students proposed that the center of the circle was the midpoint of the segment that joined P and its counterpart on the other line. See figure 1.
Evidently, either the students' proof activity did not help them understand or the knowledge itself was compartmentalized and was not accessible as a part of constructions. Similarly, Fischbein and Kedem (1982) found that high school students, after finding or learning a correct proof for a statement, still maintained that surprises were possible and that further checks were desirable. Galbraith (1981) reported that 18 percent of twelve- to fifteen-year old students believed that one counterexample was not sufficient to disprove a statement. For most geometry students, deduction and empirical methods are separate domains with different ways to establish correctness (Schoenfeld 1986)
Research on students' learning of proof in geometry is consistent with the foregoing findings. For instance, numerous attempts have been made to improve students' proof skills by teaching formal proof in novel ways, albeit largely unsuccessful ones (Harbeck 1973; Ireland 1974; Martin 1971; Summa 1982; Van Akin 1972). Even more telling is Senk's research (1985) on over 1500 students: only about 30 percent of students enrolled in full-year geometry courses that teach proof attained a 75 percent mastery level in proof writing.
Developing The Notion Of Proof
How do students develop the ability to prove ideas formally? Piaget described how this development occurs without considering curricula; van Hiele analyzed progress with curricula. Both views shed insight into how students can develop the ability to use proof and to judge where in this development our students might be.
According to Piaget, developing the ability to construct a proof as logical necessity passes through several stages (Clements and Battista 1992). At stage 1, the child's thinking is nonreflective, unsystematic, and illogical. Various pieces of data collected or examples examined are treated as separate, unrelated events. Exploration proceeds randomly without a plan. Conclusions may be contradictory. For instance, in putting the three angles of a triangle together so that they are adjacent, students were shown what happened for one triangle and asked to predict what would happen for others. Many stage-1 students fail to generalize the pattern to subsequent presentations of the task and are not sure if a straight angle would be formed if the order of the angles were changed. They ignore the size of angles and do not attempt to determine why the pattern occurs.
At stage 2, students not only use empirical results to make predictions but try to justify their predictions. They anticipate results in their searches for information and think logically only about premises that they believe in. In the angles task, students attempt to analyze the angles for each new example. But because they do not see the sizes of the three angles as being interdependent, they are often misled by the appearance of the angles. Gradually, however, they establish a relationship among the three angles. The induction that leads students to believe that the angles of any triangle yield a straight angle guides their thinking about the angles of new triangles.
Only at stage 3 does the student progress beyond a belief that something is simply always true to making a logical conclusion that it must necessarily be true. The student is capable of formal deductive reasoning based on any assumptions and so is capable of operating explicitly with a mathematical system. For example, students progress beyond an empirical generalization that the sum of the angles of a triangle is a straight angle to a belief, based on logical reasoning, that it must necessarily be so. They see this relationship as necessary because they understand that the angles of a triangle form complementary parts whose union is a straight angle. Furthermore, they can deduce that three angles that sum to more than 180 degrees cannot possibly belong to the same triangle.
How do students progress through the stages? At what point does the need for verification arise? "Surely it must be the shock of our thought coming into contact with that of others, which produces doubt and the desire to prove...Proof is the outcome of the argument" (Piaget 1928, 204). Owing to contact with others, the student becomes ever more conscious of his or her own though, becoming "conscious of his or her own thought, becoming "conscious of the definitions of the concepts he is using" and acquiring "a partial aptitude for introspecting his own mental experiments" (p.243). The student becomes increasingly able to take the perspective of others. Finally, with the onset of formal thought, experiments in which ideas are tested by mentally reproducing sequences of events as they occur or are imagined to occur are replaced by logical experiments in which the actual construction is reflected on and evaluated for inconsistencies.
Piaget's theory, on the one hand, describes how thinking in general progresses from being non-reflective and unsystematic, to empirical, and finally to logical-deductive. The theory of van Hiele, on the other hand, deals specifically with geometric thought as it develops through several levels of sophistication under the influence of a school curriculum (Clements and Battista 1992)
The levels of van Hiele
The van Hiele levels are numbered differently in various sources. However, all references to levels are specific to this article's numbering system.
Level 1 -- visual. Students reason about geometric shapes on the basis of their appearance and the visual transformations that they perform on images of these shapes. The identify such figures as square and triangles as visual gestalts, often after viewing prototypes. For instance, they might say that a given figure is a rectangle because "it looks like a door."
Level 2 -- descriptive/analytic. Students reason experimentally; they establish properties of shapes by observing, measuring, drawing, and making models. They identify shapes not as visual wholes but by their properties. For example, a student might think of a rhombus as a figure with four equal sides.
Level 3 -- abstract/relational. Students reason logically. They can form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand and sometimes even present logical arguments. They can classify figures hierarchically by analyzing their properties and give informal arguments to justify their classifications (e.g., identifying a square as a rhombus because "it's a rhombus with some extra properties").
Level 4 -- formal deduction. Students reason formally by logically interpreting geometric statements, such as axioms, definitions, and theorems. They are capable of constructing original proofs by producing a sequence of statements that logically justifies a conclusion as a consequence of givens.
Level 5 -- rigor/metamathematical. Students reason formally about mathematical systems rather than just within them. They can analyze the consequences of manipulating axioms and definitions.
The author de Villiers (1987) suggests that deductive reasoning in geometry first occurs at level 3 when the network of logical relations between properties of concepts is established. He claims that because students at levels 1 or 2 do not doubt the validity of their empirical observations, formal proof is meaningless to them -- they see it as justifying the obvious. Van Dormolen (1977) argues that at the visual level, single cases are justified, conclusions are restricted to the specific example for which the justification is given, say, for a particular rectangle. At the descriptive/analytic level, justifications and conclusions may be made for specific cases but refer to collections of similar objects, such as a class of rectangles. Only following level 3 can students justify statements by forming arguments that conform to accepted norms, that is, give formal proofs.
Research by Senk (1989) supports the notion that a proof-oriented geometry course requires thinking at least at level 3 in the van Hiele hierarchy. She found that less than 22 percent of students below level 3, but 57 percent, 85 percent, and 100 percent at levels 3, 4, 5, respectively, mastered proof writing. Thus, at van Hiele's level 4, students master proof, with level 3 being a transitional level. Unfortunately, over 70 percent of students begin high school geometry at level 1 or below, and only those students who enter at level 2 or higher become competent with proof by the end of the course (Senk 1989: Shaughnessy and Burger 1985).
In summary, both Piaget's and van Hiele's theories suggest that students must pass through lower levels of geometric thought before the can attain higher levels and that this passage takes a considerable amount of time. The van Hiele theory suggest that instruction should help students gradually progress through lower levels of geometric thought before they begin a proof-oriented study of geometry. Because students cannot bypass levels and achieve understanding, prematurely dealing with formal proof can lead students only to attempts at memorization and to confusion about the purpose of proof. Furthermore, both theories suggest that students can understand and explicitly work with axiomatic systems only after they have reached the highest levels in both hierarchies. Thus, the explicit study of axiomatic systems is unlikely to be productive for the vast majority of students in high school geometry.
Alternatives To Axiomatic Approaches
Research suggests that alternatives to axiomatic approaches can be successful in moving students toward meaningful justification of ideas (Bell 1976; Fawcett 1938; Human and Nel 1989). In these approaches, students worked cooperatively, making conjectures, resolving conflicts by presenting arguments and evidence, proving nonobvious statements, and formulating hypotheses to prove. Teachers attempted to involve students in the crucial elements of mathematical discovery and discourse -- conjecturing, careful reasoning, and building validating arguments that could be scrutinized by others.
Computer Construction Programs
Consistent with the alternatives to axiomatic approaches, the focus of computer construction programs such as The Geometric Supposer software series (Schwartz and Yerushalmy 1986) and The Geometer's Sketchpad (Jackiw 1991) is to facilitate students' making and testing conjectures. The Supposer programs allow students to choose a primitive shape, such as a triangle or quadrilateral, and perform measurement operations and geometric constructions on it. Both the Supposers and the Sketchpad record sequences of constructions performed on shapes and can repeat the action on other shapes. The Sketchpad software permits the user to construct a shape, change it, and then maintain any constructions that were made. Thus, both programs permit students to explore the generality of the consequences of constructions.
Research has demonstrated the effectiveness of such construction programs. In one evaluation, students using Supposer software performed as well as or better than their non-Supposer counterparts on geometry examinations (Yerushalmy, Chazan, and Gordon 1987). Students' learning went beyond standard geometry content as they invented definitions, made conjectures, posed and solved significant problems, and devised original proofs. Making conjectures was not easy for students, but eventually nearly all students made conjectures and justified their generalizations. Supposer-based activities helped students move away from considering measurement evidence as proof (Chazan 1989; Schoenfeld 1986; Wiske and Houde 1988). Unlike textbook theorems, which students can assume as true "because they are in the book," students believed that theorems generated with Supposer software needed to be proved before they could be accepted as true. Students using this software also made gains in understanding diagrams. They were flexible in their approach to diagrams, treated a single diagram as a model for a class of shapes, and were aware that this model for a class of shapes, and were aware that this model contained characteristics not representative of the class (Yerushalmy and Chazan 1988; Yerushalmy, Chazan, and Gordon 1987).
The secondary school geometry curriculum should be appropriate for the various thought levels through which students pass in a yearlong study of the topic. It should guide students to learn about significant and interesting concepts. It should permit students to use visual justification and empirical thinking because such thinking is the foundation for higher levels of geometric thought. The curriculum should require students to explain and justify their ideas. It should encourage students to refine their thinking, gradually leading them to understand the shortcomings of visual and empirical justifications so that they discover and begin to use some of the critical components of formal proof. But it is the meaningful justification of ideas -- establishing their validity while making sense in the mathematics classroom -- that should be a major goal of the geometry curriculum. Formal proof is appropriate only to the extent that students can use it as a way to justify ideas meaningfully. We shall describe two examples to illustrate how geometric ideas can be investigated at increasing levels of sophistication for both thinking about and justifying ideas.
Students can productively investigate the notion of similarity using the dilation transformation in The Geometer's Sketchpad. This activity might be preceded by having students perform dilations using pencil, paper, and rulers, and demonstrating shadows projected with a pointed light source. As students dilate polygons using various ratios, they can be asked to describe how the original and dilated figures compare. Their descriptions can at first be general and visual, for instance, "They are the same shape, but not the same size." Gradually, students can be encouraged to be more precise and to investigate properties, for example, "How are the parts of the shapes related?" This next phase leads to analyzing sizes of angles and lengths of sides and to a discovery about properties of similar figures. Students can be asked to characterize or define similar figures: How should we define similarity? Do alternative definitions exist? Could we say that two triangles are similar if they have two angles that are congruent? Students can empirically explore the latter problem in the Sketchpad software using the script in figure 2. From any two points, this script constructs a segment joining the points and a pair of sides that make given angles with the segment. It then measures the angles and sides of the triangle. Students find that as long as these two angles are kept constant, the triangles constructed are similar. Thus, students can devise an alternative characterization of similarity.
Next, we might pose a problem about similarity that puzzles many students: How are the areas of similar figures related? Students can first investigate this problem in the sketchpad program using the Measure Area command. See figure 3. Once they have a conjecture, they should encouraged to prove it. Using algebra, students can not only verify their conjecture but better understand why it is true.
By basing students' study of the concept of similarity on dilations instead of taking the axiomatic approach seen in most textbooks, we permit students to construct the visual knowledge and operations that will enable them to discover and comprehend the properties of similarity. They can also use an empirical approach that is appealing to, and necessary for students at this level. Using Sketch-pad software permits the students to use this empirical approach efficiently, helping them firs to find properties and second to convince themselves that their conjectures are valid. Through visual manipulations and empirical explorations, students move naturally from the intuitive idea of same shape to a property-based definition for similarity. Finally, students can use their Sketchpad explorations to decide what postulates to make about the basic characteristics of similarity.
Quadrilaterals inscribed in circles
Sketchpad explorations can not only encourage students to make conjectures, they can foster insight for constructing proofs. For instance, in the Sketchpad the user can construct a circle, pick four points at random on it, and consider the quadrilateral formed by joining consecutive points. If the lengths of the sides and the angles of the quadrilateral are measured as points are moved around on the circle, these measures are automatically updated as the shape of the quadrilateral changes. With enough practice, students will notice that the sum of the measures of the opposite angles seems to be 180 degrees. See figure 4. The Sketchpad demonstration is quite convincing because the size of the circle can be changed and the vertices moved at will. But is our conjecture always true? If so, why?
As one of the vertices of the quadrilateral is dragged around the circle, the angle measure at that vertex does not change, even though the lengths of the sides that include the angle and adjacent angles change. Why? The arc subtended by this angle does not change. Once this condition is noticed, students who have explored angels inscribed in circles can see a path for constructing a proof.
So exploration with a graphic representation for the class of quadrilaterals that can be inscribed in a circle not only helps us discover a property of these figures, it has given us the insight we need to prove and better understand the property.
Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work. By focusing instead on justifying ideas while helping students build the visual and empirical foundations for higher levels of geometric thought, we can lead students to appreciate the need for formal proof. Only then will they be able to use it meaningfully as a mechanism for justifying ideas.
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This material was partially funded by the National Science Foundadtion under grants no. MDF-9050210 and MDR-8954664. Any opinions, findings, and conclusiong or recommendations expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Michael Battista is professor of mathematics education at Kent State University, Kent, OH 44242. He is currently involved with research and curriculum development projects in geometry and is interested in how students construct personal meaning for mathematical ideas. Douglas Clements is professor of education at the State University of New York at Buffalo, Buffalo, NY 14260. He is interested in students' learning of geometric concepts in computer and noncomputer environments and learning of mathematics in Logo environments.
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