by Barry Garelick
Chances are good that most students in high school today know that the sum of the measures of angles in a triangle equals 180 degrees. Unfortunately, chances are also good that most high school students today cannot prove that proposition.
Geometry as taught today is for the most part lacking in the most important aspect of the subject: Proofs. Prior to 1980, most if not all high school geometry classes were very much proof-based. While there are those who bemoan the teaching of K-12 math as being mired in “computational” and “procedural” aspects of math while ignoring the larger beauty of what mathematics is about, it is ironic that when it comes to geometry, the true mathematical nature of the subject is largely ignored.
A glance at the geometry textbooks that are typically used in high schools today reveals that the problems students are given in such courses require one or two proofs that are not very challenging in a set of problems devoted to the application of theorems rather than the proving of propositions. Most of the problems presented in these textbooks require students to apply various theorems and definitions to find the lengths of line segments and angles. Typical courses in geometry are lacking in proof-based problems; instead, they contain many problems in which missing angles or segments are indicated as algebraic expressions. For example, opposite sides of a quadrilateral that is identified as a parallelogram may be labeled x + 2 and 2x – 6; the student is asked to find the length of the segments. This problem requires knowledge of the properties of a parallelogram leading to the conclusion that the two segments of interest are congruent. The two sides, expressed as x + 2 and 2x – 6 then lead to the equation x + 2 = 2x – 6. Figure 1 shows another example of a problem that does not require formal proof.
This problem requires students to know and apply that the sum of the angles in a triangle equals 180 degrees, and to know what are linear pairs of angles, and that they sum to 180 degrees. From this, students can piece together information and compute angle R.
While the types of problem discussed above constitute a form of proof (requiring applications of theorems and definitions combined with deductive reasoning to justify the necessary computation), such problems do not fully develop the skills necessary to develop a logical series of statements that constitute proof. In contrast, consider a problem that requires a student prove a particular proposition, such as shown in Figure 2:
This problem does not require any numerical calculation. It requires knowledge of theorems of parallel lines in a plane and properties of isosceles triangles.
Defeating the Purpose of a Geometry Course
To limit the number of challenging and substantial proofs in a geometry text defeats the purpose of a course in geometry. Geometry differs from other math courses the student has had up through algebra. There is little disagreement that problem sets are the heart and soul of a mathematics course. Students learn the knowledge or skills of mathematics by solving problems that incorporate such knowledge The problems in a geometry course that require proofs of propositions are not only an application of the theory, but a part of it. If done right, the study of geometry offers students a first-rate and very accessible introduction to the nature and techniques of logical argument and proof which is central to the spirit of mathematics itself. As such, a proof-based geometry course offers to students—for the first time—an idea of what mathematics means to mathematicians, and how it is used. Also, unlike algebra and pre-calculus, since geometry deals with shapes, it is easier for students to visualize what it is that must be proven, as opposed to more abstract concepts in algebra.
To make matters worse, many of the modern day geometry textbooks are now in written in a disorganized manner: a hodgepodge of topics that do not follow in the heirarchy of postulates, definitions and theorems that make up a traditional geometry course. The theorem that the sum of the angles in a triangle is always 180 degrees is sometimes presented before it is actually proven. Also, it is not unusual to see that coordinate geometry (that is, graphing figures on a grid) is presented almost immediately and mixed in throughout the course with the traditional Euclidean geometry. Coordinate geometry is often used as the basis for proofs of theorems about triangles, using tools such as the distance formula, slope, and midpoint formula. This is done long before the Pythagorean Theorem is presented and (sometimes) proven. Coordinate geometry should be introduced only after presentation of the Pythagorean Theorem. This is because the coordinate grid is based on a network of horizontal and vertical lines that are perpendicular to each other; the formula for finding the distance between two points on the grid is based on the Pythagorean Theorem.
Today’s geometry textbooks are also written with an eye to being relevant to students, and therefore contain “real world” type problems which are, by and large, of a computational nature. This development is ironic considering the complaints of those who have advocated for reforms in math education. Such math education advocates claim that math as traditionally taught fails to teach true mathematical understanding because it is mired in computational and procedural aspects. They claim that such approaches ignore the larger beauty of what mathematics is about and, as evidenced in the Common Core Standards for Math, believe that students in lower grades (K-6) must “understand” the conceptual underpinnings of procedures.
A Topsy Turvy Approach to Mathematical Understanding
The focus on understanding in the lower grades, and the dearth of proofs in geometry seems to be a rather topsy-turvy approach. From a mathematical perspective, both understanding and procedural fluency are important. But in the early years, most students progress with procedural fluency to build up their level of problem solving efficiency and comfort, which in turn allows them to better understand the conceptual underpinnings. Some students learn these in the early grades, while most others gain the conceptual understandings later, particularly when they have the powerful tools of representing arithmetic operations in algebraic symbols. Taken to extremes, the emphasis of understanding over procedure in early grades can yield absurd results. Consider a student in 5th grade who is able to solve the problem of how many 2/3 oz servings of yogurt are in a 3/4 oz container of it. The Common Core standards’ embodiment of the “math is not just about computation” philosophy would judge such student to not “understand” fractional division if he/she can’t explain the invert and multiply rule. It is therefore perplexing if not frustrating that when it comes to geometry, the true mathematical nature of the subject is largely ignored.
One would think that the more rigorous treatment of geometry would be favored not just for the mathematical structure and logic, but also for the boost it gives to problem solving ability. A frequent criticism of the advocates for math reform of traditionally taught math courses are that students work problems for which they already know the procedure for solving. That is, the solution of a problem can be found by repeating a method that the student has learned, and thus using an “algorithmic procedure”. The criticism goes that students do not learn how to apply their prior knowledge to new and non-routine problems and for which they are not able to rely on worked examples. Yet, requiring students to prove geometric propositions would address this criticism. Proofs do not lend themselves to specific procedures. Rather, students must apply their knowledge of theorems, definitions and postulates to perform the proof successfully. They are forced to ask what needs to be shown for the proposition to be true? And from that they must work backwards to see the sequence of statements (referencing the appropriate theorems and definitions to verify their truth) that produces a logical demonstration of what it is that must be proven.
The Delay of Mathematical Maturity
I learned from a geometry book in the old SMSG series that prevailed as part of the 60’s new math. While the 60’s new math’s abstract and formal approach had disastrous results for the lower grades, the texts produced for high school were a different story. The SMSG Geometry book was written primarily by Edwin Moise (a first-rate mathematician) and Floyd Downs (a high school math teacher). The book eventually went into commercial production and is still available (Geometry, by Moise and Downs). The book is structured so that each theorem presented is proven using only theorems that have been proven previously. Thus, while some theorems presented could easily be proven using the theorem that the angles in a triangle always sum to 180 degrees, the proofs are presented only in terms of what came before. This highly structured approach taught me (and I assume many others) about the logical structure of mathematics and the nature of proof, which served as an important foundation for subsequent courses in math that I took as a mathematics major.
People roll their eyes when they hear about proofs because they may recall the “two column method” of proof: give a statement, and give a reason for every statement about the proof. The two-column method is used as an introduction to proofs to initiate students to the method of rigor and to force them to think about every statement made in a proof. The initial experience of the two-column proof is a basic training in how the heirarchy of the definitions, postulates and theorems that have been presented are used to prove something new. Students learn to ask “Can I make such a statement? How is this statement justified?” After a few weeks of such method, students are allowed to produce narrative types of proof. Having earned their “stripes” through a basic training of “rigor”, students are rewarded by not having to give a reason for things that have now become obvious (e.g., drawing in a diagonal in a quadrilateral no longer needs to be justified because “any two points determines a straight line segment”). Following this progression of learning, students are placed on a path to a new level of mathematical maturity which entails being able to tell the difference between what is obvious and what needs to be justified. It is a form of mathematical understanding, which entails the skills of logic and structured argument.
The level of mathematical maturity such understanding brings with it has been lost for two decades and counting. Whether it returns under the new Common Core math standards to be implemented in 45 states is an open question. From what I’m seeing so far, the implementation of the standards is turning out to be a matter of interpretation, and that interpretation appears to be the same emperor with the same wardrobe. My hope is that the standards allow those teachers who believe in the importance of the proof-based geometry course in high school to teach it, and that they have the proper textbooks with which to do it.