Today is Saturday, December 7, 2024 and this is William Michael of the classical liberal arts academy. Yesterday, I shared some thoughts on Facebook, a post about modern mathematics, because in my work in the classical liberal arts academy, I’m always amazed by how much time families choose to spend on modern mathematics. If I look around on the logs for the Academy Study Center, talk to parents, talk to students. There’s always so much attention being given to modern mathematics, so much so that much of classical Catholic studies can never even be attended to, can’t, can’t even be started, let alone made progress in or mastered. And there’s just this glaring cause of trouble in education, and it’s modern mathematics, and no one questions it, because we we sort of take it for granted that that no one can, no one can question that modern mathematics isn’t so important as to take first place in in our children’s studies. In fact, I would argue that in most educational circles, education becomes almost entirely consumed with attention to modern mathematics.
It really is the only criterion that’s used to judge the quality of an educational program. I’ve learned this even in in my experience with my own adult children applying to college, the only subject that there’s ever a question about, ever really any any clarification in from college admission boards, is modern mathematics. And as I look at this, I have to ask myself, are we sure? Are we sure that we’re not making a mistake? Are we sure that we’re right about this? Are we really sure that this attention and respect that we show to modern mathematics is reasonable. Is it actually right? Or have we made a great mistake in education? Have we made a great mistake because we really haven’t been honest with ourselves, or we haven’t really examined it carefully and honestly, but have just left mathematics alone as a subject that no one has any right to question or to examine.
I’ve begun, in my own spare time, to ask myself these questions, I’ve begun to ask myself, what are we actually learning? What are we actually doing in modern mathematics, are we actually doing what we pretend we’re doing? Are we actually learning what we what we think we’re learning, or have we been duped into a giant black hole of a distraction that is actually keeping us from the more important studies in our life? And one of the reasons why I’m especially concerned about this is because this was something that Pope Pius the 10th warned about back in the early 20th century. He said that these modern studies, and he referred specifically to the natural sciences, he said they have a tendency to crowd out more sublime studies, they have the ability or the tendency to crowd out more sublime studies. And I would argue that this is what has happened in modern education.
We have allowed modern mathematics courses, and we can list the names of these courses, from grade level arithmetic classes to pre algebra, Algebra One, high school geometry, algebra two, trigonometry, pre calculus, calculus. We’ve allowed these subjects to completely take over. For education, and I don’t think we’ve examined it. I don’t think it makes any sense, and I think that we’re being led to think that we’re doing something that we’re not actually doing. And I’d like to make what I’m sure will be a pretty radical argument. I’d like to make the argument that we need to stop studying modern mathematics, as we have been studying modern mathematics, I believe we need to stop studying modern mathematics because we’re not actually learning anything useful. We’re not actually learning anything that’s helping us in life. And I’d like to share my thoughts and just sort of argue this out in this talk today.
Now what normally happens, and I’d like to ask you, just to think back to when you were in school, think about what happens. You go through math classes, and math is pretty simple. You learn some basic facts. You learn the addition facts, you learn subtraction facts, and you take speed quizzes to test how quickly you can recall and answer addition and subtraction problems, and this is usually in first and second grade, and then we get to third grade and we move into multiplication and division, and we do the same thing. We have math facts that we memorize, and we have speed drills, and we see that we can quickly recall and answer questions and answers in addition, subtraction, multiplication, division, and then arithmetic goes on. We get into more complicated topics like fractions and percentages and decimals and so on. And what we’re doing in in these studies is learning lots and lots of facts, lots and lots of information. But what’s what’s interesting about these things is that it’s not it’s not new and advancing information. It’s not an increasingly complex information. What it really is is simply a system of shorthand. It’s a system of shorthand for expressing mathematical ideas. It’s a system of shorthand for expressing mathematical ideas.
In classical arithmetic, in classical arithmetic, we learned that quantity is that which can be increased or diminished. It can be increased or decreased. That’s what a quantity is, something that can be increased or diminished. So there are really only two possible operations in arithmetic. There are only two possible operations we can increase a quantity or we can diminish a quantity. This is what we learn when we talk about addition and subtraction. When we talk about addition, we’re talking about increasing a quantity. When we talk about subtraction, we’re talking about diminishing a quantity. This is what we can do with quantities. It’s what makes something a quantity. It can be increased or diminished. But when we move on to the next step, when we move on to multiplication and division, we’re not actually doing something different. We’re doing the same things. We’re just doing them in a peculiar way, and then establishing a shorthand way of expressing what we’re doing.
For example, when we talk about multiplication, if I say two times two equals four, all that I’m saying is that two plus two or two added twice makes four, and I express that with this shorthand called multiplication. I say two times two equals four, and that seems both simple and unnecessary. But if I have two times 12 or two times 23 then the multiplication shorthand allows me to express what would be very inconvenient to write on paper if i. Wanted to show that the number two was added 23 times, I would have to write two plus two plus two plus two plus two, and I’d have to continue that until I’d written 23 times. And so multiplication allows me to express that same operation in a shorthand way, by just writing two times 23 it’s really an expression of addition. It’s a shorthand expression of addition, and that’s what’s important for us to see. It’s not necessary. It’s not a concept that needs to be understood. It’s simply a shorthand way of expressing something that is already understood, mamely, addition.
Division does the same thing in Division, if I ask what is six divided by two. I’m asking, How many times is the quantity two contained in the quantity six? And what I’m basically asking is, how many times can two be subtracted from six? So if I say, How many times is two contained in six? I can just take two away and say, There I’ve taken two away once, take two more away and say, There I’ve taken two away twice, then take two more away and say, There I’ve taken two away three times, and there’s nothing left. So six contains two three times. It’s really just an expression of subtraction. And so in multiplication, we don’t actually have some new theoretical concept in arithmetic, we simply have a shorthand way of expressing addition and subtraction. Now why that’s important is because if we need to find the solution to some question of addition or subtraction, we don’t need to write it down on paper. We don’t need to think about it in this traditional way.
What I’d like to say is the development of modern mathematics has taken place basically from the 1600s until, I would say the computer age, when we’ve got calculators and computers being developed because mathematics, if you think about it, mathematics was done on paper. Mathematics was done on paper. When we think about men like leibonitz and Euler and the fathers of modern mathematics, algebra and so on, these men worked on paper and so all of the all of the ideas that needed to be expressed, or all of the problems that needed to be solved and worked out, they needed to be worked out in a way that could be done on paper with a pen or pencil. And so there had to be a way to express these ideas in writing. And what I would argue, and I’d like you to think about, I’d argue that modern mathematics is really nothing more than a shorthand way of expressing ideas in arithmetic or solving problems in arithmetic on paper. Modern mathematics, again, I would argue, is just a method for writing on paper and solving problems on paper. In arithmetic with the advent of the calculator and the computer, I would like to argue that that is no longer necessary. It’s not necessary.
Mathematics in our generation does not need to be solved on paper. Does not need to be expressed on paper. In fact, to take the system of mathematics that’s been established which which works well on paper and try to use it on computers actually doesn’t work very well. It’s actually very difficult to type using the modern written mathematical expressions. It takes, it’s very difficult. It’s actually, it’s actually very frustrating for students to have to try to type out expressions or solutions to quote, unquote, show the work in. Typing. Now there’s no question that mathematics and the scientific revolution have led us to the utilitarian developments that we enjoy in modern technology like smartphones and computers and calculators and so on. But what these tools do is free us from the need to do much of our intellectual work on paper. We don’t need to go and take a slice of a tree and get ink and pens to think and record and develop our thoughts on paper. We can do this digitally. We can do this in type written form. We can do this in drawn form and so on. We can just do it by means of calculations and expressions, not only written on a computer, but programmed into a computer.
So what is discovered or understood once, can be programmed to be to be done or performed an infinite number of times and and many times faster than can be done by human beings. And so we live in an age where modern mathematics can, and I would argue, should be done using calculators and computers rather than paper and the shorthand system that was developed to express many of the ideas that we just take for granted in modern mathematics courses are really no longer necessary. They’re not necessary so much of the time, nearly all of the time that’s spent in school studying mathematics, is really not studying mathematics at all. It’s really learning how to express mathematical ideas the way that men did it over the past three or 400 years, and we see this.
We see this conflict arise in modern classes, because students will have a calculator or a computer or a smartphone, and they’ll be given assignments in a modern math class that can easily be solved using a calculator or a computer or even a smartphone, the students will want to use the machine, and the teachers will tell them, No, you can’t use the machine. You need to show all of your work. And the teachers say this with an air of some kind of commitment to theoretical learning, to commitment to some real intellectual life, like we’re not going to let you just punch these numbers and symbols into a calculator and make the calculator do the work we want to see that you actually understand mathematics and and that that seems virtuous, and the teachers present themselves as these protectors of the virtue of learning and so on. But it’s actually false, because what the students are doing, even when they write out their work, even when they show their work, they’re not understanding mathematics at all. In fact, they’re doing what the calculators and computers do. They’re simply being made to do it themselves, rather than have the calculator do it. And I’ll explain what I mean.
Let’s just start with with a simple concept of squares in arithmetic. We take a number like two squared equals four, and the students learn to recite this, two squared equals four. And if I ask a student, okay, so I see you can write that down on paper, two with this exponent, two equals four. But I’d like you to explain to me what it is that you’re actually writing here, I want to make sure you’re not just being taught to work like a machine, but that you’re actually being given understanding of these mathematical concepts. So what does it mean when you write two squared equals four? What does that actually mean? And the student will likely say something like, Well, it means that we take two multiplied by itself, because square means that a number is multiplied by itself, and that gives us four for the number two. So two. Weird, means two times two, and it’s equal to four, and that that’s that seems like some kind of understanding, but it’s not. It’s not it’s simply reciting a multiplication fact. It’s simply reciting a multiplication fact. It’s not understanding. It’s just recalling from memory, a memory fact. Because if I were to go further and say, Okay, so, so two squared means two multiplied by two equals four. Well, what does that mean? What does two multiplied by two mean? What does two multiplied by two mean?
The student will likely, at that point, look at me confused and say it just means two times two. That’s just like basic math. Fact, we memorize our multiplication tables, two multiply. It just means two times two. That’s what it means two times two. And the answer is actually no. It doesn’t mean two times two. That’s just an expression. That’s just a shorthand expression. What that actually means is the number two is being added twice. That’s what two times two means. So two plus two, two times two and two squared are simply three different ways of expressing the exact same thing. Two plus two, two times two and two squared are three expressions that mean the same thing. So why would we do that? Well, we do that not because these are different concepts in mathematics, but because they’re simply shorthand forms, and when we’re dealing with small numbers like the number two and a square. It really doesn’t even make any sense why we would use these figures. And the key of what I’m trying to show is that addition, multiplication and squares are not different concepts. You’re not learning three different concepts. You’re simply learning different shorthand methods of expressing a simple concept, namely, addition.
But let’s, let’s go on and think further. Let’s say we have two cubed equals eight. Two cubed equals eight. And a student wrote that down, and I said, Okay, well, I’d like you to explain to me what that means. What does it mean when you say two cubed or two to the third power equals eight? What does that mean? The student would likely say something like cube means that the numbers multiplied three times. So two cubed is the same thing as two times two times two. And the student will say that with an attitude of understanding like, there, that’s what it means. Two cubed means two times two times two. Now my question is, is this actually mathematical understanding, or is this just recall from memory? Because I don’t think that most students really know what two times two times two means. What does it actually mean to say two times two times two? When we looked at two times two, we said, well, that’s the same thing as saying two plus two. We’re taking two and adding it two times that’s what two times two means. But what does two times two times two mean? It means that we’re taking two plus two and adding two plus two twice. That’s what two cubed means. It means two plus two and taking that twice. So taking the sum of two plus two, two times. That’s what it means. So again, we’re just back to addition. It’s a way of expressing, in a shorthand form, an addition that’s being made. Most students, I would argue, will not associate cubes or even squares with addition. They will associate them with multiplication. And as simple as this may be, they will not associate multiplication with addition. But because they’ve just memorized these multiplication facts, they’ll think of multiplication facts as the elements of arithmetic, when in fact they’re not. They’re simply shorthand expressions for additions. And the same thing is true of exponents of powers. They’re just shorthand expressions for additions. And what will get really difficult for the students is when I begin to ask them about higher exponents or higher quantities in arithmetic, because even at the level of two squared and two times two and two to the third power, they already don’t understand what they’re doing in mathematics, and so by the time we’re dealing with much higher numbers or complex expressions, they’ve never actually understood the concepts, and they don’t Understand that really, all they’re learning is a system of shorthand for addition and subtraction. That’s all that it is.
We then begin to talk with all of these modern mathematical expressions. The teachers will talk about things. The teachers will recite these math facts. The students will repeat the math facts. The textbooks will lead the students through all kinds of lessons and all kinds of exercises and things, and the teachers and the students will act like they’re learning to understand mathematics in some sort of conceptual way, when really all that they’re doing is memorizing the meaning of shorthand expressions. That’s really all that they’re doing. They’re memorizing the meanings of shorthand expressions and giving the answers by recalling that information from memory. They’re doing the work of machines. They’re not actually understanding they’re simply recalling information. And that’s what modern mathematics is. And over time, it becomes more and more complex. The system of so called shorthand becomes more and more complex, and to be perfectly honest, none of the students really have any idea what they’re doing. If you want to be a good math student, you simply say the right things. You simply write the expressions the way that you’re showed to write the expressions. And if you can write the expressions in the right way, you are said to understand mathematics, when in fact, you don’t understand what’s going on in those mathematics at all. You’re simply working like a machine. And the intelligent students are the ones who say, why can’t we just do this on a calculator, or why can’t I just do this on my smartphone, or why can’t I just do this on my computer? Why do I have to write this all out?
The teacher will come along as this defender of the virtue of theoretical learning and understanding things, and will say, Oh, no, no. We don’t want you using a calculator or a computer because we want you to understand it for yourself. Therefore we want you when, when you’re given a problem to solve on a test or a homework assignment, we want you to show all your work. Show all your work. And the question is, does showing all of the work prove understanding? And the answer is no, the work itself that’s being shown is nothing other than recall. The student is shown a certain kind of problem, and they’re shown the steps that need to be done to reach the solution to the problem, and then they are tested for whether or not they can repeat the steps.
It’s like giving someone directions saying, Okay, I’m going to give you directions to my mother’s house. There are five steps for you to get to my mother’s house. Step one, go out and make a right onto this street. Then go two miles, make a left, travel five miles down that street, make a right. Okay, now repeat to me the answer to the question, how can I get to your mother’s house? Repeat to me the steps. Tell me the answer to the question and the. Teacher is to so the person has to repeat the steps to me that’s not understanding, that’s simply recall, that’s simply recall. And this is what we’re doing in modern mathematics.
So when the teacher says, show me the work, the teacher is actually asking a student not to prove some understanding of mathematical ideas, but to simply show that the student is able to perform the operations like a machine, and show what, and again, what it means to show the work means to show the teacher that you didn’t just enter it into a calculator and hit enter. So you’re not allowed to just write the answer, even if it’s correct, you’re not allowed to just write the answer. You have to “show the work”. In other words, show that you are the calculator. Show that you are the machine that followed the steps that you have been programmed to follow, so that we can mark you correct and tell you that you understand mathematics, when in fact, you’re just acting like a calculator.
As I said, the smart students will ask, why can’t I just use the calculator which has already had these steps programmed into it, like that’s the whole benefit of the calculator is that these steps have been programmed into it, and we can just enter in the variables and hit enter and get the right answer. And it even protects us from from from typos or making some simple mistake and writing things out. We might not copy it accurately, or one of our numbers might not be clear. It actually makes it more accurate. Why would I not be allowed to just use the calculator and the teacher again?
The first answer is, well, if you use a calculator, that doesn’t necessarily show that you understand, but as I said, writing the work, showing the work, also doesn’t show that you understand anything. It just shows that you have been programmed to know the steps and that you’re working like a machine. But another answer that will be given is something like, Well, you know you’re not always going to have a calculator with you. You’re not always going to have a computer with you, and so it’s important for you to understand the concepts, because you can’t be dependent on machines.
But the problem with this is, is, when or where would I ever do these modern mathematical calculations? Where would I ever use trigonometry or calculus, or pre calculus, or the quadratic expressions or algebra. Where would I ever use these mathematics where there wasn’t a calculator available? Would I be designing a skyscraper with no computers available? Would I be solving some complicated calculation without a calculator available? When would I actually ever need to solve a problem like this, like the problems we’re learning about in in any course, from, let’s say, sixth grade on, when would I ever need to solve any of those problems or do any of that mathematical work where I didn’t have access to a calculator or to a computer or even a smartphone? The whole argument doesn’t make any sense.
These mathematical sciences, if you will (which I would argue really aren’t sciences) are for machines, they’re for calculators, they’re for computers, they’re for modern society. They’re for high tech society. So it really doesn’t make any sense to say that we’re going to learn algebra, and all that we’re going to learn in algebra can be handled by a calculator, but we’re not going to use the calculators. We’re going to be the calculators. And I’d like to just ask you in what other fields do we pursue mastery in this way? What other fields do we pursue mastery in this way?
If a student wants to learn computer programming, do we have the students take out a sheet of paper and write out. Uh, the code with a pencil on the paper, and say, Okay, here’s what we need to do. We need to we need to make a word move across a computer screen. We want the background to be blue and the word to be white, and we want the word to blink as it moves from left to right, across the paper, across the screen. How can we do that? Show your work? And so we’re now going to have students write out what the code would be that would be used in, let’s say, 1980 to make a computer screen blue with a white flashing word moving across the screen. Do we in a computer science course or programming course? Do we make students write out and show all their work? We don’t do that. That’s not even how programming works. We teach people to program, taking for granted that computers exist and that you’re going to be doing programming work on a computer.
We don’t teach any any trades like this. We don’t say, Well, if you’re going to learn how to be a carpenter in the 21st century, what you need to learn how to do is you need to learn how to start with raw timber, and you need to take the raw timber, and you need to learn how to, you know how to strip the bark from it. You need to learn how to run it through a sawmill. But we’re not going to let you use a powered sawmill. We’re going to make you do it by hand with a with a big old wood saw. So we’re going to make you cut all your own wood. Then we’re going to make you plan it, and then it’s got a cure. So we’ve got to give it times for the wood to to dry and straighten. So we’re going to start you with raw timber and hand tools. And this is how you learn to become a carpenter. No one would do that. No one would do that.
And if somebody said, Look, well, what happens if you know, what happens if there’s no electricity and you don’t have all of your fancy power tools? What if you don’t have, you know, lows down the road where you can go and buy whatever lumber you need already cut and measured. What happens if that’s not possible? And the answer is, well, you know what? Honestly, if that’s not possible, I’m probably not going to be building a bookshelf, because something’s wrong. You know? I’m learning, I’m learning carpentry, assuming that we’re not living in the midst of a nuclear holocaust or some kind of natural disaster where we’re pursuing the knowledge of trades and skills for use in regular everyday life. The trade schools don’t start kids with raw timber and hand tools, and yet we’ve allowed this to be done in modern mathematics. We’ve allowed this, this outdated approach to mathematical studies, to take over our children’s education.
Now, in recent times, we’ve begun to allow them to use calculators. So for example, the we’ll find that the on the SAT students will be told you’re allowed to use a calculator. Or in a certain course, let’s say a college physics course, there may be rules, and professors may say you’re allowed to use a calculator, but they’re going to limit which calculator you’re allowed to use. You’re only allowed to use a TI 84 calculator. You’re not allowed to use a smartphone or a computer. You’re only allowed to use this one calculator. And why are we doing this? I would argue that we’re doing this so that these modern mathematics courses, this whole this whole industry of modern mathematics can retain an outdated and obsolete, unproductive, even counterproductive, stranglehold on education. I’d like to say that again, I believe that we have allowed this modern mathematics industry to maintain a stranglehold on modern education, which really makes no sense.
If we were to think about what is actually necessary to learn for the practical use of modern mathematics, it would be very, very little, and it could be learned very quickly using a calculator or a computer. We would simply need to teach. Children a few simple concepts, and they would understand that these are just shorthand ways of doing addition and subtraction. These are just shorthand ways, and we can perform these operations very, very easily very, very quickly using calculators and computers. And if we would actually teach students to do that, we would blitz through mathematics in in no time, maybe one year, for all modern mathematics, but we allow the entire education of our children, from preschool all the way through college, to be subject to this stranglehold of obsolete so called modern mathematics, which are really not modern at all.
Now, as I said, even from the very earliest points of this education, the kids already abandon understanding and begin to just memorize and recall. Memorize and recall. Memorize and recall. I would say it starts as early as third grade, where kids start learning multiplication and division, and they no longer are really visualizing and understanding what’s going on. They’re just memorizing facts and reciting them, and it moves from multiplication and division to powers and roots and fractions and decimals, and it just continues from that point on. And there’s never again a moment where they’re really understanding the essential ideas of arithmetic, which really boil down to just two addition and subtraction.
Now, as they’re spending all of this time learning, to quote, unquote, show their work in what are practical mathematics, what they’re not doing is actually learning the sciences of mathematics. They’re not actually learning the mathematical sciences. They’re not actually learning the science of arithmetic. They’re not actually learning the science of geometry. And to be even more complicated, they’re not learning the arts of music or astronomy. If we think about the four arts of the ancient quadrivium, the kids are actually not learning theoretical mathematics. They’re simply being trained to show themselves to be machines that can, quote, unquote, show the work on their paper. They have no idea what they’re writing. They have no idea what it means. They simply know that if they write what the teacher shows them to write, they’ll get rewarded with good grades. They’ll be patted on the heads like good little calculators, because that’s really all that they’re doing. It doesn’t matter that they understand what’s going on.
For example, if the teacher teaches the students the properties of multiplication, and she says, Oh, well, you know, multiplication has these properties. Here’s the name of the first property. Here’s an example. Here’s a formula that shows what it is, and the kids will be all, you know, just, they just copy it down, and they look at it like, ooh, ah, okay. That’s the distributive property. This is the commutative property. This is the associative okay. And I’ve got this formula in my all they’re doing is they’re, they’re being told what to say, and then being tested to see if they say it. They’re being shown what to do, and then they’re tested to see if they can do it. It’s it’s simply the work of machines. They’re not understanding mathematics. They’re not studying the theory of the mathematical sciences. They’re doing the work of calculators, but they’re not allowed to use calculators to do that work.
So we take something that could take seconds, and we turn it into something that may take weeks, and we do this from the time the kids are in kindergarten all the way through their college years, where they continue to do this. Instead of showing them a concept and showing them how to perform it using a calculator or a computer, we say, no, no, no, no, no, we’re going to take six weeks to learn how to do this on paper, which you will never do in your entire life. But we’re going to do this so that we can, really, I would argue, just make a ton of money by continuing to hold these obsolete courses in schools and colleges and. That everyone has been persuaded are of the utmost importance in the future, and it’s just a big racket. It really doesn’t help the kids at all. In fact, it harms the kids because it takes them away from actually important subjects, which they completely neglect because they’re so busy studying math, math, math, math, math, math, for no reason.
What I would like to recommend is that students of all ages, and I’m not talking about what you need to make the kids do. I’m talking about what all of us as students need to do is we all need to recommit to the quadrivium, to classical mathematics, to the four mathematical arts of the ancient classical liberal arts curriculum, we need to go back to theoretical mathematics and realize that the practical mathematics are as simple as using calculators and computers. They really are that simple, and we can’t allow anyone to tell us otherwise what we what, what is valuable for us to study is the mathematical sciences, the four Mathematical Sciences of the quadrivium, and we can study them from their master texts, which, you know, we have in the classical liberal arts academy. But I think what we need to do is increase our focus in theoretical mathematics and make as efficient as possible our study of modern practical mathematics, making the use of calculators and computers, because it’s just a practical study the modern mathematics.
Now, these are the first times that I’ve talked about this. It’s something that I’ve had a feeling that there’s something that’s that’s really wrong with the the black hole of modern mathematics, in in modern studies. I haven’t taken time to really work out my thoughts on it before. I’ve just kind of lived with this nagging frustration in the back of my mind that I haven’t paid any attention to, but as I start to think through it, I start to realize we’re really allowing an obsolete approach to mathematics to be perpetuated for the benefit of those who perpetuated and at The great expense of our children and even of ourselves who need to learn the mathematical sciences, but the mathematical sciences are not what’s being taught in modern mathematics courses. And so what I’d like to encourage you to do is make an effort to turn away from modern mathematics and take up the study of classical arithmetic and classical geometry. And rather than asking, how does this course help me to satisfy modern school requirements, I’d like to ask you to think bigger picture than that, I’d like to ask you to say, how does this study help me to move forward in my pursuit of wisdom, wisdom that will actually help me in my life?
When we talk about grade level mathematics, one question that I have is, who actually requires these studies of us, who actually requires these studies of us, and who actually requires that we study modern mathematics in the way that they’re taught in modern mathematics courses, if we can demonstrate mastery of modern mathematics courses. Won’t we be credited with mastery of modern mathematics? We certainly will. And think about it. You can sit through your classes in fifth grade, seventh grade, ninth grade, 10th grade, being told that you’re not allowed to use calculators, you’re not allowed to use computers, you’re not allowed to do these things, and then you’re going to come to the sat where you’re going to be told, Oh, you know what, you actually can use a calculator.
So who’s going to do better? The student, who’s been made to spend all of this time, all of this energy, doing mathematics without a calculator, to ultimately be told that he can use a calculator, or one who’s actually taught to use the calculator to do his mathematics, I would argue that if we commit to abandoning this outdated approach to modern math, and actually get right to calculators and computers. Our students will actually do better on the testing and will prove to be better mathematicians, especially if they have a knowledge of theoretical mathematics, which modern students don’t. So I’d like to challenge parents, teachers and adult students to consider whether by going back to classical mathematics and using calculators and computers to master practical modern mathematics, we might actually end up ahead of the game in the future. I’d like to know your thoughts.
God bless your studies,
Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy
