## The Modern Day High School Geometry Course: A Lesson in Illogic

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 [...]

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.

Figure 1:

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:

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.

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California.

Dear Barry, I am a geometry teacher and I think I am in love!! Brilliantly written. Can’t wait to share with my colleagues in the North Kansas City Schools, who, BTW, are interpreting CCSS to mean an increased implementation of proofs for geometry.

Thank you for sharing!!

2. SteveH

“Ironic”

Yup. Fractals, not proofs. Pie-chart understanding, not understanding that comes from application of fundamental identities and theorems.

Geometry is an interesting topic, and one can argue that not all students need the formality of proofs, but that’s not the point in a education world that gushes about the beauty of mathematics and about thinking like a real mathematician. And, one can see the same mismatch of conceptual and formal understanding going on with algebra. Going from algebra I to algebra II requires a lot of mathematical understanding, not conceptual understanding.

The problem is that few educators want to delve into the details of what understanding really means for math, and this goes well beyond whether proofs should be taught or not. Too many educators see math as some sort of top-down thinking process, when in reality, it’s based on a rigorous understanding of a large set of basic definitions, axioms, identities, and proofs. One can play at math with guess and check, but that’s not what a trained mathematician or engineer does.

My favorite examples have to do with basic identities like:

a/1 = a

Do students really understand what this means in all situations? Do they learn about why it’s good to see that:

a = a/1

a*b = b*a

What are ‘a’ and ‘b’ in

2(x-1)^3

Then there is

x^(-1) = 1/x

Do students know what this means for rational terms made up of many factors in the numerator and denominator? Do students even know what factors or substitution mean for complex expressions?

Some educators love to claim ownership of critical thinking and understanding, but as Barry points out, the reality is quite different. Are those “traditional” kids are just getting by using rote knowledge? They are the ones getting to STEM careers, so dig past pedagogical prejudices to see what’s going on. Don’t rely on rote ed school training.

“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.”

The top PLD level 5 (“disguinshed”) defined by PARCC just means being successful in a college algebra course. A PARCC curriculum does not define one that prepares kids for STEM careers. Educators will (continue to) see kids who get there, but they will never ask us parents how that happened. We know more than a little bit about what real mathematical understanding means. Our kids are successes not because we just want what we had when we were growing up. What many schools offer now is worse; talk of critical thinking and understanding that hides low expectations and avoids real mathematics.

• Josh

I don’t think anyone is arguing the point that these details are important. There’s no way to get past knowing what a/1 means.

However, for many students such as myself, it is important not to lose sight of the big picture. I constantly need to be making connections with what I previously learned and how it ties to the endless possibilities of mathematical language.

The idea behind math as a “top down” thinking process is correct in many ways. But you can’t teach it that way. Start with the basics, let kids learn what mathematicians are saying, how they are saying it, and what it means. Then, once they have grasped what the language means, they can begin to understand the more interesting and complex aspects.

I advocate keeping kids interested not through “Real world!” Applications that are completely irrelevant and recognized as bull by kids AND instructors – but through the many interesting things within the subject. I’ve found that most of my instructors, while computing very well and knowing what they were doing had trouble reaching the majority of the class because they themselves were only teaching as a way to make a living, and weren’t truly interested in the subject. After having explored mathematics myself a bit, I don’t understand how someone can lecture to a class without getting excited about what’s being taught or the possibilities.

That’s just me…someone who used to do poorly in math because of a lack of computational skills. Learning about the bigger web and how all the rules played into the bigger picture, what I call true understanding – was the deciding factor.

Forgive the haphazard organization of my reply, it’s late.

3. A Mom

Thank you, Barry, for your clear explanation. We need this.

Monday

February 18th, 2013

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