What is truth?
Andrew Farah tweeted this the other day, stating “we have infinite access to information and we can’t agree on truth”.
Andrew is right. We do have a lot of information. But having information doesn’t mean we know what to do with it. Just because information is readily available, doesn’t mean it’s parsed to be objectively truthful or reliable or accurate.
What Andrew has unintentionally posed here is also the main issue that we have with the way that we are taught mathematics: we have all the math information that we could possibly need, yet, 40% of adults can’t make a straightforward calculation based on simple math skills.
We know math. We know that it exists for 18 years of our life, in some form of Algebra 1 or Geometry / Trigonometry, but we never get taught the art of math.
Caitlin Flanagan, a very prolific author, called algebra stupid in her tweet from September 25th. A common sentiment among most of us.
Eric Weinstein (a Math Guy!) responded with “sounds like you got a bad batch of math”, and then listed out all the cool things that math encompasses: snake lemmas, lie groups, and Kac-Moody algebras.
Eric is right. Math is incredible. So why do most of us hate it?
The Story of George Boole
George Boole published The Laws of Thought in the 1850s (yes, Boole of boolean search). This was a book about the “symbolic language of Calculus”. It detailed how we can use operations, such as addition and multiplication, with symbols, such as x and y and z, to create logical arguments.
The algebra of thought.
This was from the opening section of his 1847 pamphlet, Mathematical Analysis of Logic:
They who are acquainted with the present state of the theory of Symbolic Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same processes may, under one scheme of interpretation, represent the solution of a question on the properties of number, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. … It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic
Boole wanted to convey logical thought in equations. He wanted to quantify Aristotle’s work on human reasoning and the Stoics’ logic of propositions, so x, y, and z became collections of objects, such as a collection of trees, a collection of carriages, a collection of people, etc.
For multiplication, if x were the collection of college students, and y were the collection of newsletter-writers, then xy would be all college students who are also newsletter-writers — the collection common to both x and y. [AND]
For addition, if x were all Business students and y were all Pre-med students, then x + y would be all students that are either Business or Pre-med. [OR]
Boole’s system looked like below— two commutative laws, two associative laws, and the distributive law (pretty similar to ordinary arithmetic). Here, Boole didn’t solve for x, but rather displayed patterns of logical thought through whatever the collections were — a collection of students, politicians, trees, etc.
If we wanted represent something that didn’t exist, we would write that collection equal to zero. So, (depending on your beliefs), let x = a collection of Santa Clauses, and thus x = 0.
Boole’s algebra was a bit different than ordinary algebra because x + x could equal x (a collection of men + a collection of men = a collection of men, but 2 + 2 ≠ 2) and y * y could equal y. Boole tied reasoning into algebra, and created a platform for other mathematicians to build on. Boole’s algebra of thought is key to our technological progress today.
What is math?
Boole’s story is incredible. But most of us are never taught of Boole and his attempt to quantify logic. We are simply taught the rules, and told to apply them. Imagine knowing this connection between the unknown and the known, the fact that in some worlds, x + x can equal x (this bothered me to no end when I was younger).
In fact a Tik Tok video from gracie.ham asked the question that most of us have been thinking our whole lives — how did all this math begin? Why? Why do we need math? Who came up with this, and how? She’s asking about the true meaning of math, something that gets skipped over in most educational systems.
Why is math, math? What is the truth of math?
What is Math to Me?
I am a public school kid. Kindergarten to high school to college, I attended public institutions. I loved my school experience, and am grateful to the Kentucky educational system and all the incredible teachers I had.
When I was in the 3rd grade, I moved classrooms 4 weeks into the school year. I don’t know what happened. But the class that I had moved into was moving at a faster pace than the class that I moved from. In my previous class, we hadn’t discussed multiplication.
But in my new class, they had already moved into long division.
I was lost.
The First Realization
I lost a rung in the mathematics ladder (a bad batch of math, as Eric would call it) and thus, I was scrambling for the next 10 years of my life, trying to keep up. I lost the learning. I lost the understanding of the principles, because I was trying not to drown — I didn’t care about swimming, I just wanted to survive.
I never learned how to multiply or divide. I still don’t really know how, honestly. Because of the way that school is taught — a sequence of subjects that advance as you age- if you miss out on one step, you fall.
In the 4th grade, I remember crying because I had gotten a bad grade on a multiplication test. I had to get a tutor. I never tested well. There was always a semi-permanent gap in knowledge, simply because I had missed out on 4 weeks of a different level of education. Those 4 weeks made a huge difference in my entire school career.
I went into middle school afraid of math, lacking the foundation that I needed for algebra. I had finally learned how to multiply, but I was slow. Long division was a mystery.
High school was the same. 9th grade, 10th grade, I did what I could but I spent a lot of time questioning my ability and my intelligence. I always felt like I was ‘bad’ at math, so I just believed it.
The Second Realization
It wasn’t until 12th grade, my senior year of high school, that I realized that I liked math. I was in AP Statistics. It was fun, it was challenging, and I wasn’t bad at it. So I went into college, took my first calculus class, muddled through with my peers-but still got an A (the discussion of grading is another article entirely). More importantly, I understood it.
In fact, I rather liked it. Everything I had thought about my math ability was wrong.I wasn’t bad at math.
I was just deathly afraid of it.
Melodramatically, it felt a bit like meeting the love of your life at the wrong time and the wrong place. “Where have you been all my life?” I would whisper to my Mathematics textbook, wishing that things could have been different for the first 18 years of our time together.
I have spent ~80% of my life living in fear of something that I really enjoyed.
I was trying to climb a ladder with missing rungs.
Would I say that I am good at math now?
Probably not. In our culture of speed and execution, I wouldn’t be considered a ‘math wizard’ by any means. I am still slow at adding and multiplying and dividing. But I know the theories, and know when to apply them. Most importantly, I enjoy it. I know what it means to me. I have found the ‘truth’ in my mathematics journey.
The Third Realization
Most of my friends were in the same boat as me, circumnavigating the choppy waters of arithmetic with fear of the great white whale of Math.
Our education can shape our worldview of ourselves. A lot of us live in fear of mathematics. We think that it’s something that we aren’t able to do, and thus live in a world where we avoid it completely. Why is this?
Math is a Process
Math has an element of specificity to it that other subjects don’t really have. You can fuddle your way through history if you have a general understanding of everything that happened With math, you have to know the specifics in order to feel comfortable — the answer is usually either Right or Wrong (sometimes).
Also, math is interconnected. If you can solve for the area of a triangle, you should be able to solve for the area of a rectangle. But we are often taught things in a disjoint fashion, so the connections between everything gets lost, so everything can feel like a bunch of floating objects that sometimes bump into each other.
This is partially because school systems try to cram so much into students that there is never time for synthesis. This isn’t the fault of the teachers — it’s the fault of the testing culture that we exist in. Get the kids to the test, get a good test score, rinse, repeat.
You memorize, and then you move forward.Math is more than a set of rules to be memorized or a methodology that can only be used in one place. It’s an adaptation to a set of problems, a process that can be used outside of the classroom.
Math is Art
Math has patterns, and ultimately can be a route to creativity and expression. Paul Lockhart wrote the very famous “A Mathematician’s Lament” about the importance of math as art.
Lockhart brings us into a world where musical education is taught like math is today, showing us through comparison how sad our math system sounds.
He tells us of a world of worksheets and memorization, preparing for standardized musical tests, and the despair of the students — the students are “bored in class, their skills are terrible, and their homework is barely legible”.
The world is “Paint-by-numbers”, in which students are tracked by ability and get a “good foundation” for college. The students don’t actually “paint” until high school- everything before that is simply learning colors and applicators, how to wipe and dip the brush.
It’s terribly clinical.
Lockhart explains that math is art — and the fact that it’s being taught as a prescribed ritual of memorization and testing is the reason that “math class is stupid and boring”.
Lockhart highlights that we miss out on figuring out WHY the area of a triangle is 1/2 x b x h. We are just expected to know that it is, and apply it moving forward. This is Paint-by-Numbers- no creative freedom, no application to the unknown, just rote and bland rinse and repeat.
We discount the importance of creativity and independence during the mathematical process, and instead place value on the outcome — who can test the quickest, who can solve the fastest? Along the way, we lose the art form.
We get to plug-and-chug. We don’t get ownership over math like we do with arts and science. Math should be fun.
Math is Fun
My brother is excellent at math. When he was little, he would take the sports section of the newspaper and memorize statistics. His brain turned math into something that was fun for him.
He was a whiz all throughout our school years, and aced most of his math classes (which I had struggled through two years prior). That’s because his mindset was different than mine. It was something that he could use to analyze basketball, and determine which player was going to do what based on their previous metrics.
For me, it wasn’t until I got to college that I began to have fun with math. I was using math without realizing it.
In economics, my professors were more interested in how I got there, rather than my final answer. In economics, the price elasticity is the slope of the line (algebra) or a first order derivative (calculus). The only difference between the two is the application.
But I had to understand the basics first. Only then was I able to apply it.
Data analysis is the language of math, taking those concepts that “we will never use again” and building them into narratives and stories.
We all do that. We all subconsciously calculate probabilities and count occurrences, building out our world view in our mind. We keep friends close, because on some level, we are quantifying our relationship with them. We conduct cost-benefit analysis every single day. We think about tradeoffs.
Math might be fun, but it’s not free of struggle. That’s one of the pitfalls with the process — we give up too early because it can get hard. Nobody ever said art was easy.
Math is Failure
Math is tough. It requires work. And if you miss a step along the way, you are moving against the current, which only gets more powerful the more that you miss.
We need to be told WHY math is important. Only then we can build mental models, understand processes, and continue to grow. One missed step can lead to a long time of frustration. Not understanding the ‘why’ leads to disinterest and disengagement.
When ten or more years instruction fails to leave people having even the faintest idea what something is, why it is done, or what it is used for, then something is seriously wrong.
Even Pascal and Fermat struggled, as noted in their famous letters to one another. Math is an art, a science, an approach to life. It is not free of struggle, and we shouldn’t give up on it.
The struggle is a part of the story.
But we have to make room for the art too.
“I was made to learn by heart: ‘The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.’ I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way.” — Bertrand Russell
Math should not be paint-by-numbers. Imagine a world where Bertrand Russell, a famous mathematician (amongst many other things) gave up on math. But how many young mathematicians have we lost due to what Russell describes above?
Math as Truth
Math is the art of thought and ideas. Our teachers used to say, “in the real world, you won’t carry around a calculator with you” (modern technology makes a mockery of predictions). So now that we have our calculators and Wolfram Alpha, what’s left to learn?
Keith Devlin explains —
A sufficiently deep understanding of all those procedures, and the underlying concepts they are built on, in order to know when, and how, to use those digitally-implemented tools effectively, productively, and safely.
No longer are computation skills of utmost importance, but rather, it’s the understanding of the concepts and the procedures, the application of the process that mathematics requires now. It’s called “number sense” —
Fluidity and flexibility with numbers, a sense of what numbers mean, and an ability to use mental mathematics to negotiate the world and make comparisons.
Computers can’t understand this yet — they can simply compute. But we, humans, can “think and reason flexibly with numbers, use numbers to solve problems, spot unreasonable answers, understand how numbers can be taken apart and put together in different ways, see connections among operations, figure mentally, and make reasonable estimates”, as described by Marilyn Burns in About Teaching Mathematics.
Mathematics is more than the algebra that Caitlin Flanagan described. It’s the science of discovering patterns, the science of developing logical thought around the processes, and ultimately using it in how we shape our world. Everything is a data point, and if we understand how to process and implement these data in our lives, we are that much more successful.
But we should also recognize math for what it is- an art form, digging deep into the reasoning of ideas and questions, building fantasies. Math is meant to be fun. As Lockhart wrote, “A good problem is something you don’t know how to solve.”
The world is meant to be a place to explore curiosity. A place to test ideas, try new things, explore concepts and questions. But we have placed so much rigidity on the ‘effectiveness’ of curriculum in order to build ‘ideal candidates’ that we have lost our students along the way.
We all want to know ‘why’ — why do we solve for the area of triangles, and what does that mean? It doesn’t need to apply to real life. Not everything needs to be a building block towards the ‘successful future’.
Math as art is probably part of a more idealistic world where students aren’t standardized and creativity is valued over letter grades. It’s not so much the pursuit of one great truth, pieced together from infinite pieces of information, but rather our own individual mathematical truth that teaches us to think and process. A world where we do not live in fear of math, but rather, embrace it for the beauty it holds.
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