Tuesday, July 2, 2024

Geometry & How to Think

I know that anyone that comes across this will most likely find it a bit on the boring side. I get that. I'm not trying to build an audience here, I gave up on that dream long ago. At the same time, what I am going to say, I believe has relevance. I believe it is important. I believe that it borders on essential. And so I write it, not thinking that it is a statement that will sweep the nation or the educational system or even the 5 people I know online. I write it instead, as an actual exercise in thinking, which is the core topic of this post. So I encourage you to read it, to think about it, and to communicate with me about it, not for social media hype, but for community and dialogue and for thinking... 


The longer I teach Geometry, the more I believe that the real purpose of Geometry is not how to find angles and identify shapes and compare congruent triangles... No. All of those things are secondary to, what I believe is, or ought to be, the real purpose of a Geometry Course. Don't get me wrong, all of those things, along with slopes of lines and right triangle trigonometry and the surface area of spheres is all important: amazingly, exceptionally, actually important. The more technical our world gets, the more these things are drastically important. And even if I go another way with Geometry... the "Geometry in Construction" way, which is another course that I teach, these things are still really, really important. Houses still need to be built with the proper pitch of the roof. Rafters need to be cut, doorways need to be square, and floors need to be level. Again... all important stuff... but... But the longer I teach Geometry, the more I believe that the real purpose of Geometry is not necessarily any of these things. The real purpose may be... ought to be... probably should be or at least could be... how to think. 

I guess that if I am going to talk about this topic in this format, I should probably attempt to accomplish a few things in this "talk". I most likely need to say something about the need for our youth to be taught how to think. You wouldn't think that this would need an explanation, but it might. From there I should try to show you why I think that the Geometry classroom is the proper delivery vessel for students to be taught how to think... I think. Let's start with the need for thinking by thinking about thinking, you think?

Let's think about this. 

Most today would assume that thinking is simply a natural response. That thinking is something that we do internally and naturally. On the one hand, this is true, but on the other hand, there exist things like bad thinking, flawed thinking, thinking based on a deception, critical thinking, non-critical thinking, random thoughts, intrusive thoughts, prolonged thought, and educated thought. Thinking can be lazy thinking or it can be exercised thinking. It is possible to think about things in a better way than you used to think about things. Just consider for a moment the thoughts of a baby. They are without words initially and know of a limited number of responses. Which is really weird if you think about it. Nearly all of my thought is accompanied by language, but before you have language, is there thought? Yeah. I'm feeling something in my lower body... but I don't have words for any of that, but I know that the lady, which I also don't have a word for, makes wonderful tasting sustenance. A baby knows all of this and thinks it in some way, but without language? Fascinating. Something to really think about. But steer your thoughts toward how your thinking has changed over your lifetime.

Your thinking over the course of your life has not simply become more complex, though that is part of it. Your thinking over the course of your life has not simply become more informed, though that is also a part of it. There have been times in your life where the very way that you think has been altered. Sometimes it is just a natural part of the development of the brain, most notably recognized when someone has a trauma or genetic condition that keeps the brain from continuing to develop. Sometimes the alteration in thinking has come from a specific moment. An eye-opening moment. A moment that alters your perception. This can go horribly wrong when that altering moment is based on a falsehood or when a person's natural brain development is inhibited by limited viewpoints being introduced in that person's life. 

I could go on from here and we could really develop some interesting explanations as to the crazy way people think in our world, and why thoughts can be so completely diverse. This could also (potentially) help to explain the seeming inability for many in our respective social circles to think outside of their own thought-boxes, and the ensuing emotional outbursts that can happen when one voices thinking outside of that safely validated thought box. I could really, but instead I would ask you to consider whether or not one has the ability to improve their thinking... to think better? 

Thinking Better

One of the things that good teachers do is bridge the gap between their student's current scope of knowledge and connect that to additional, and usually beneficial knowledge. Not to branch off into educational psychology, but one of the few things that I distinctly remember learning and then experiencing over and over again for the last 24 years as a math teacher, is the way a person learns. There must be previous knowledge, there must be a question... a question developed in that person's mind, and then there is the answer, attaching the new knowledge to the old knowledge through a question attached to an answer. It isn't that the student needs to ask the right question, they just need to feel the "brink" of their knowledge and be curious about what lies beyond that. Without that curiosity, there cannot be any real learning. (Unless, of course we consider "learning things the hard way" by bumps, bruises, and broken things. Even then I would still say that there needs to be a measure of curiosity about what brought the bump, bruise, or breaking: in order to actually learn what caused it.)

I have noticed, not only within myself, but also with my students that when the new piece of knowledge I am introducing them to is filled with more complexity, the question in the student's brain must ask more complex questions for them to acquire that knowledge. For example, at a young age, when a student is learning 1+1, they are simply asking the question, "how many?"  They are learning numbers, how to say them and how to write them. That knowledge is built on previous knowledge as they quite naturally began to learn more, as in "I want more" or "do it some more!" or "why do they have more?"  When a student moves to multiplication, they are essentially asking the same questions, but combined: "how much more?" and "how many?" are combined into a multiplication of numbers. Division then builds on that, but what we need to see is that the questions are becoming more complex.  No longer is the learning accomplished by the simple question of "how many?" but there is now an additional layer placed on top of the "many" that will result in a student learning how to multiply and divide. 

Now this is critical right here. At this stage, this learning can happen in a couple of ways. Sometimes a student will simply learn the math facts. I don't have a problem with this, which I will explain in a moment, but other students do something else beyond the facts.  Some students, instead of only learning the facts and the techniques and the methods of getting to the answer, begin to understand what is actually happening with those numbers. This becomes evident when a student hits fractions and percents, because they can "see" that they are essentially doing the same things. Common Core math attempts to solve this issue by attempting to demonstrate math facts, but it ends up coming across like trying to teach someone how to walk by telling them which muscles to flex and in what order. All you would do is confuse the child. Even if you didn't use the technical terms of those muscles, walking is not learned in the details of movement... but you can teach an athlete how to run faster by exploring what they already know and thinking about what they are actually doing when they are running.  

The same can be true of math, and I would suppose other educational topics as well. When learning math, one ought to expect the child to add and subtract, if necessary, by simple fact memorization and a few basic processes. Most of those common core math methods are things that a child will do naturally, if left to their own learning. Attempting to get technical on how addition works with one who can't add, makes it too complex... we are trying to make them think complex questions for simple answers. What is 1+1? It is 2. That is all that needs to be said initially. When a young student has mastered those basics, their brains, incorporating their imaginations into their learning, will begin to naturally find ways to categorize and store that information.  From this categorization and mental filing, their brains will be more equipped for new information or for more complex information. Without ever being told what addition is, multiplication will "feel like" an extension of addition, and the multitude of numbers and their corresponding, purely imaginative mental renderings, will be properly equipped for more complex answers to more complex questions. 

The complex questions begin to appear quite naturally for a child in the form of "how?" questions and "why?" questions. For a young child, those are usually easily answered, from the cliche' "Because I told you so." and the less helpful, "because I said so." to the "that's just how it is."  But as anyone who has been around children knows, those answers don't always satisfy. But the complexity of our answers will usually depend on their ability to hear and comprehend those answer or on whether or not those answers will unduly burden the young child with too much information that they are not yet equipped to handle. Quite often we give foundational, simple answers to those questions in their youth, that we will be able to build on as their brains mature. But many questions require one to do more than just hear an answer. Sometimes the mind has to be taught how to hear the answer. This is where I believe that a properly taught Geometry course can come in handy. 


No matter what you may think you know about Geometry, allow me to correct your view, for just a moment. This will require some thinking, so some of you may not be able to grapple with what I am about to say. Geometry is usually thought of in terms of shapes, but that is not what is happening in Geometry. In Geometry what one is doing is thinking about the real world... without the real world. Let's think about the world, but let's ignore the actual molecules and chemical interactions and the laws of physics. Let's not consider the substance or the composition of actual physical objects without any of its purely physical attributes. Let's consider distances and locations without any of the locales included. When one does this, one can cut away all of the extra stuff and simply think about those shapes. They can think about a point, not as a dot, but as a location. The question, "how do you get from point A to point B, stripped of roads and hills and valleys is now a line and lines are infinite collections of points. Stripping away all extra information about an object or a place or direction or location or quantity is to simplify it down to its geometry. Calculations can then be made on the basic problem at hand. Of course, when bringing your answer back into the real world, there are quite often adjustments that need to be made... just ask any good carpenter. But quite often, the essential answer can only be found in the process of thinking about a problem in its purest sense, and this is what is, or ought to be happening, in a Geometry Course. Which is why most Geometry courses included an introduction to logic, which led to the writing Geometric proofs. 

Unfortunately, many Geometry courses no longer include the teaching of logic and proofs. There are even some Geometry teachers that don't miss this, and in one sense I understand why. But I think that we are missing out on a golden opportunity to teach our youth how to think. How to clear away all the rubble around a problem, not just the problems physical attributes, but all of its emotional connections that play into the answers that are given. As a people, we must be able to do this, and I find that more and more adults have either lost this ability or never learned it to begin with. And I can't help but wonder if they could have benefited from a good, proficient Geometry teacher, that understood the Geometry's real purpose. How to think about the world in simple terms. In terms we can wrap our minds around. Sure, when taking these answers back into the real world, there will need to be adjustments. Like an old house that has settled, nothing is longer level, squared up, or straight. A good carpenter may aim for the perfect and make some slight adjustments, but a really good carpenter will sometimes tear out the whole wall because it has become so warped it is no longer safe. 

To whatever degree I can, this is always my goal when teaching Geometry. I don't simply think that I want to meet state standards and get students to pass, those things come naturally when you have a student who can actually think. 

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