THE RULES

of the game of mathematics January 22, 2021

A student of mine, fan of fantasy novels and aspiring writer, is interested in geometry for its import into the visual arts. She recently made a very common error in geometry: to draw conclusions based on perceived measurements. For practical applications, perceived measurements are necessary. In geometry, perceived measurements are irrelevant. For example, I could play a cruel trick on any geometry student by directing them to locate the midpoint of a rod. The cruelty of this exercise is that it is impossible to locate the midpoint of anything. Therefore, I could logically dismiss any student's effort sight unseen! This, of course, is not good teaching for it would immediately result in exasperation. We all make assumptions. We make assumptions because we perceive that they simplify the problems of living and relationships. However, the wrong assumptions complicate our lives or unfairly frustrate the lives of others. This is just as true in mathematics as it is in life. An example of this is the widespread trope, "Time is money." Logically, this is an absurd statement. Practically, it leads to a conclusion that is fortunate for the contractor in the short run (not the long run), but unfortunate for the client. Where has quality craftsmanship gone? Quality has been sacrificed for quantity because "Time is money." In fact, mathematics and fantasy writing are the same process. In each, a world is imagined, rules are developed to provide a cognitive framework for exploration, and what is learned is recorded for posterity. The difference is the subject. Mathematicians write about numerical and spatial relations. Fantasy novelists write about human relations.

ARE WE HUMAN OR ARE WE DANCER?

mathematics vs. computation..., February 28, 2023

Calling all who dread teaching kids how to solve systems of linear equations. You worry with crossed fingers that they will never understand it. You worry they may take over the world with such knowledge... a terrifying thought because you know those kids! Worse yet, you worry whether you understand it well enough to dare teach it. I've got news for you. Chances are, you don't. Not to worry. You have until next year to prepare. And we're here to help! Initially, we need to properly situate the government sanctioned methods approved for children who have to solve 2 x 2 linear systems on state tests. Then, we need to develop activities that prioritize critical thinking and inquiry above practice in a way that doesn't sacrifice algorithmic competence. Finally, we need to steer our pedagogy toward guiding targeted discovery.

Mathematics began with computation, but didn't end there. So we begin with methods for solving linear systems. Mathematicians use computation to observe patterns. Students who can solve a 2 x 2 linear system algebraically are ready to derive Cramer's Rule and generalize it. Why not let them do this? Cramer's Rule is just the formalization of a pattern found in solutions of linear systems. Let the students search for this pattern until they find it. No time? Why reinvent the wheel? We only abuse and discard what we don't understand. Mathematicians use formalized patterns to gain insight. More than solving a linear system, Cramer's Rule also provides a test for the solvability of linear systems and characterizes their geometry in terms of determinants. This is precisely what algebra is designed to do: encode geometric propositions. Furthermore, Cramer's Rule provides the simplest motivations for both the definition of a determinant and the standard matrix multiplication.

Recently, I helped my daughter with solving systems of linear equations. In her algebra class, she was learning how to use substitution and elimination. That seemed to be the end of the story. I marveled at this because she was introduced to matrices two quarters prior with no connection to solving linear systems, and there is far more to the story. Far from proposing that all high school students take linear algebra if they are to learn how to solve 2 x 2 linear systems, I am questioning the educational value of stopping short of introducing matrix methods. Why not take the topic all the way to Cramer's Rule for 2 x 2 linear systems? Why not go further with those interested in doing so? Why are so few interested in going any further than what the teacher demands? The general experience with school mathematics is the culprit. Look, I get it. I was once in the trenches having to make sense of administrative nonsense about what to teach and how to teach it. There was one problem. I love mathematics. Let me explain. Passion for mathematics is the reason why mathematicians are rarely found in public schools. Link that with a passion for teaching and you've got someone incredible to prepare for high school classrooms, but rarer still. One could hand out a worksheet of systems to solve the prescribed way and follow it with a test. One could also ensure students gain competence with substitution and elimination, then ask them to derive a formula after challenging them to construct systems with specified geometric constraints. The inevitable formula provides fertile ground to introduce the concept of matrices and determinants. It also reveals a rather elegant representation of solutions for 2 x 2 linear systems that easily extends to the n x n case. Students who can solve a 2 x 2 linear system algebraically are ready to derive Cramer's Rule and generalize it. The point is not to listen and recite. The point of any mathematics class is to pursue to discovery of propositions about what computations suggest and then attempt to prove or invalidate what seems to be true. Verification of results is what situates mathematics among the sciences and its deductive fervor has distinguished it as the most peculiar science. If all that is going on in a classroom is computation, then the class is not worthy of the subject "mathematics".

Follow this link for further discussion about the linear algebra of systems of linear equations.

OF SQUARE PEGS AND ROUND HOLES

motivating irrationals..., June 2, 2021

Common Core Number Systems Standard for 8th grade reads "Know that numbers that are not rational are called irrational." What does this mean for teachers of 8th grade mathematics? Seems easy enough... just tell them "Today, you will learn that there are irrational numbers. A number is irrational if it is not rational. For example, √2 is irrational because it never ends and never repeats in decimal. Also, π is irrational. Let's try one. Is √3 rational? If we try to find its exact value, we run into trouble. So, we say it's irrational because we can only approximate it. What about √4? Yes, it is rational because √4 = 2. Now, open your books to page 123 and let's do 1-10 together."

Why is this a standard at all? Apart from being trained as a mathematician, there is no practical use for it. There is no job which pays for the identification of irrationals. This is not to suggest we shouldn't discuss this in classrooms with 8th graders. However, making it a college/career readiness standard is dishonest and just telling them is intellectually abusive. The matter is purely philosophical. Is the point of math class developing mathematicians or providing busy work to keep children out of trouble?

The Babylonians show in their calculations they were aware of some numbers that cause problems in certain contexts. The best way to motivate the existence of quantities which defy rational expression is to let students explore those contexts. One such context is trying to minimize rectangular perimeters which enclose a desired area. This is a very practical problem which hints at the existence of irrational magnitudes. Would it not be better to induce students to ask the question than to club them over the head with it and then answer it for them? Are we educating or inculcating? Only that student who is intellectually compelled to ask what you want to ask them is ready to begin the journey to understanding.

Follow this link for further discussion about the existence of irrational numbers.

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MATH SUCKS???

An explanation..., January 14, 2021

Why would a mathematics education firm title their blog, "MATH SUCKS!"? If you inquire into people's feelings about math, most will admit to fear or hatred of math. Math has been misrepresented, suffered orthodoxies, and been taught by dogmatists for too long in too many classrooms. The result is an American citizenry characterized by mathematical, and thus scientific, illiteracy. The attempts to force improvements in math student outcomes with carrot and stick -- standards and testing -- have failed the nation by missing the point of mathematics. Most math teachers are inadequately prepared, or disgusted with what has been done to students in the name of mathematics. Teachers can not meet students' needs while appeasing the pacing demands of the state passed down by their administrations. Parents, like teachers, don't know what to make of the standards because it's not how they were taught; they feel powerless to help their students. Students, like their parents and teachers, are overwhelmed by too much content and naturally develop a distaste for the dogma pushed by the current orthodoxy. We are here to help fix the problem, one pupil/parent/teacher at a time.

WHAT'S THE POINT?

Thinking about the decimal system..., January 13, 2021

Whenever we ask people about decimals, they nearly all haltingly respond with something like, "You mean, like 3.4?" They are never convicted in their response. Something akin to seeking explanation they were never given is betrayed. They desperately crave a deeper understanding, for they are suddenly embarassed at their recognition of their ignorance. The follow up question, "That's an example, but what is decimal?" rarely fails to invoke, "Is it the point?" Decimal is not the point. Then what's the point?

The reason for the widespread confusion and lack of understanding is the absence of any culturally promoted contrast to the decimal system. Ironically, the smart phones and laptops we use daily to stay connected, conduct business, and consume entertainment conceal the global dependence on a contrasting system! When we further consider the increasing demand for citizens to learn to code, the mathematical experience of American children being incarcerated by the decimal system appears tragic. Without comparison, there is no understanding. It's no wonder that most children graduate thinking that a decimal is the point sometimes placed between numbers; they are protected from suffering any understanding of decimal in a more general context. This is the outcome despite some attempts to do better. Why? Teachers teach the way they were taught and most teachers of math students were not themselves introduced to any of decimal's cousins.

CLOCK EARTH

Talking time and its measurment..., March 15, 2015

Does your child really understand the way time is measured? Do you? Have a discussion with your child about how to estimate time based on the position of the sun relative to their position on the earth. Draw some pictures of where the sun is relative to them at noon, six, and midnight. Fill in the gaps to include three and nine, and further. Talk about what time it will be other places when it is noon where you are. Geography and directions naturally become part of the discussion. You will be amazed at the discussion your child will have with you if you let them. And you may learn too!

WHY?

The most important question..., March 14, 2015

A proper understanding of mathematics starts with its beginnings. The modern formula for the area of a triangle is more than 5000 years old, according to the discovery of stone tablets unearthed in Iraq. Most of us may recall what that formula is. Here is a worthwhile question for every curious mind to wrestle with: Why is the area of a triangle given by half the product of its base and height? Please share your thoughts on this. You'll be glad you did!