The Orthodox Christmas icon is a bit different from any depiction of the Nativity of Christ in Western art. It is a bit strange to our eyes, lacking the sentiments and techniques that we are used to seeing. It is even foreign, almost unrecognizable at first glance, but upon careful study, is deeply meaningful and beautiful. Like every icon, each translation of the image differs according to the iconographer's style and region, but they all have certain image elements in common.
I went on a brief rant on Facebook yesterday about an Op-Ed piece in the New York Times which I found remarkably bad. As with many rants, the author wished that it could have been longer… So, below you will find my thoughts a bit more fully expressed. Enjoy!
Aristotle wrote that “all men by nature desire to know.” And yet, too often, educators treat math students as if the pursuit of knowledge is irrelevant to them: truth is nice and all, but what we must give our students is economic benefit.
Sadly, just such an approach was advocated in an editorial in the New York Times this past Sunday. In “Who Says Math Has to Be Boring?”, the author advocates fixing math education in this country by implementing an approach known as STEM—which stands for Science, Technology, Engineering, and Mathematics—which treats these subjects as an integrated whole. The author would have math class be filled with technology and real-world examples, with the clear aim of preparing students for 21st century careers.
And yet, the reality is that math curricula in the United States have been geared towards this approach, or ones quite similar to it, for a very long time. Today, math education in this country continues to be as much of a failure as ever, despite our consistent push for the pragmatic and the relevant.
The problem is this: as long as we treat math as a tool, as a subject that should be pursued because of its practical, real-world benefits, we will fail at math education. We can only fix math education by showing students that math is first and foremost a pursuit of the true, the good, and the beautiful, through the use of logic and reason.
The STEM approach has a long pedigree in the American education system. We can properly trace it back to the Pragmatism of John Dewey, who said that an education is that “which increases ability to direct the course of subsequent experience”—as practical and career-oriented approach to education as ever you’ll find. This philosophy was infused into math and science classes with a vengeance in the wake of Sputnik: the Cold War made it so that Science, Technology, Engineering and Mathematics were the responsibility of every good American, and every math and science teacher’s duty was to prepare the next generation of Rocket Scientists who would defend us from Moscow.
Decades later, we still find ourselves belaboring the practical side of math, only now the menace we face is our economic competition with China. It is now the duty of every math teacher to prepare his students to increase GDP. Of course, this supposedly carries benefits for the math student as well: since, as the theory goes, a math student who is shown the practical benefits of math class will be more motivated to pursue a career in science or engineering, his likelihood of making a high wage will increase substantially. And so, it’s out with the abstract, in with the pragmatic: math education is only seen as being valuable as a loyal handmaiden of our technological enterprise.
But the reality is that the economic benefits of higher math studies are widely known by the general public, including children and teenagers. The fact that accountants and actuaries are well-compensated is so familiar to our students that to repeat it is to indulge in banality. Young people know that high achievement in math class can lead to a rewarding and remunerative career, and yet somehow, no matter how practical we make math class, no matter how subservient we make math to science, technology, and engineering, we cannot stifle the yawns of our students. Math achievement in this country has gone down in the decades since Sputnik; clearly the pragmatic approach is broken.
So in a world where math class is already supremely practical, and yet still overwhelmingly boring, what’s a math teacher to do?
I would have math class be less practical. As long as we treat math as simply another tool in the technological shed, students will find it as interesting as so many wrenches and screwdrivers. We must instead show our students that math is, first and foremost, a relentless pursuit of the true, the good, and the beautiful. We must orient math class around the use of logic and human reason.
Mathematics as a discipline has always been about using logic and reason to arrive at truth. The great mathematicians of history, from the days of Pythagoras and Euclid, found a supreme beauty in the pursuit of knowledge: they knew that they were proving timeless truths. Practical applications came along, of course, but math found its inspiration and its drive as a branch of philosophy, and as St. Thomas Aquinas said, philosophy arises from awe.
Today, we must give back to our students this beautiful pursuit of the truth. We must allow them to experience the joy of mathematical proof. We must help them train their minds in the use of logic and reason, to think abstractly, and to go after knowledge relentlessly. Anyone can look up a mathematical formula in a book and then punch it into a calculator; only a well-formed mind can find a good argument for why that formula works in the first place. It should be the priority of math teachers to make such well-formed minds.
So, enough with the real-world models and the overuse of technology. For too long, we have given students “real world scenarios” involving right triangles, and then asked them to plug numbers into the Pythagorean Theorem to have the length of the hypotenuse spat back out to them. I would rather spend my time training my students to make good arguments for why the Pythagorean Theorem is always true. Have them play with abstract arguments, have them fail and fail and then finally succeed, and have them always be encouraged to return to the logical pursuit of knowledge with greater vigor the next day. Let the science teachers and the engineering professors worry about the pragmatic side of things, and let the math teachers return to what math has always been about.
Ironically, the approach I am suggesting would actually have enormous practical benefits. Is it hard to imagine that a bridge or a building designed by an engineer who had been taught to love the truth and beauty of mathematics would be a superior product? Moreover, can it not be said that any democracy would be better off with a citizenry who had been immersed from childhood in Logic and Reason? We would have more economic utility from our math classes if we stopped treating math as a mere tool and instead saw it as truth.
The pragmatic side of mathematics will always be with us. But rather than following an instructional approach whose origins lie in a desire to aim missiles at the Soviet Union, let us craft classes designed to show our students how to pursue mathematical truth through Logic and Reason. We owe it to our students, because as Aristotle predicted, they desire to know.
The Link-Up in honor of the Sacred Heart is now live over at Catholic Cravings.
Check it out! Nos in Cor Tuum Trahe!
I am a Catholic Gentleman, and I approve of the entirety of the contents of the following post from THE Catholic Gentleman:
…and you know what that means: the First Friday Link-Up in honor of the Sacred Heart of Jesus is almost upon us!
We will be doing things a bit differently going forward. It’s proven to be a bit much to maintain a separate site, so First Friday Link-Ups will now be hosted at “Catholic Cravings“, the blog of Australia’s leading Catholic nerd, Laura.
Our writing prompt for this month:
December means one thing for many of us: Christmas. Every year we celebrate the birth of Our Lord, who was very God made flesh in the womb of Mary. This month is also the month of Our Lady’s Immaculate Conception, another miracle of life! So we’d love to know: how does the Sacred Heart helps you treasure life and understand the horror of abortion? Does the Sacred Heart have a special role to play in the Pro-Life movement?
So get writing, help us spread the word, and determine with us to love, honor, make reparation to, and adore the Sacred Heart of Jesus Christ!
It is truly right and just,
our duty and our salvation,
always and everywhere to give you thanks,
Father most holy,
through your beloved Son, Jesus Christ,
your Word through whom
you made all things,
whom you sent as our
Savior and Redeemer,
incarnate by the Holy Spirit
and born of the Virgin.
Tomorrow is Thanksgiving here in the United States. In an historical sense, it’s an ironic holiday for an American Catholic to celebrate: we look back to the Pilgrims, who left England because the established Anglicanism of the time wasn’t Protestant enough for them… But in another sense, Thanksgiving ought to be the most natural holiday in the world for Catholics to celebrate. Each and every time we gather for the Eucharist, we gather to give God thanks–”thanksgiving” is, after all, the literal Greek meaning of eucharistia.
2013 has been a year of tremendous blessings for me and my family. And so, I’d like to take this opportunity to just give thanks to God for some of the ones that stand out most in my mind. And so, I am thankful:
*For the birth of our daughter in January. She is beautiful and sweet and healthy!
*For our friends, Zach and Heather, who became our daughter’s godparents at her baptism. Zach, by the way, was also my sponsor when we entered the Catholic Church…
*That as of this past Easter, all of my siblings are now safely home on this side of the Tiber.
*For a new job that I am absolutely loving and that is providing well for our family. A special thanks for St. Joseph’s intercession is in order for this one!
*For a smooth and relatively easy move back to my home state of Florida, and for the help of my wonderful parents and siblings along the way.
*For the many friends we left behind up North. We miss you all terribly!
*For the many new friendships we are building here in Florida, as well as the old ones we are reviving.
*For getting to see my little brother marry the girl of his dreams this past weekend!
*That I was able to attend the funeral of my dear friend, Carl, in June. My principal gave me the day of the funeral off with only about 48 hours notice, and there were flights available for a price I could afford. Although it was incredibly painful to lose him, I was blessed to celebrate his life alongside old friends.
*For the constant love of my wife and children!
*That 2014 is already shaping up to be a year with a lot of professional opportunity for me: I’ve been invited to speak at a conference, to teach an evening class to adults, and to run several summer camps.
For this and everything else that’s happened in the last year, I give thanks to God through our Lord Jesus Christ!
I don’t know much about being a good math teacher… I mean, I’ve seen good math teachers at work. And some days, I think I’ve been a good math teacher myself. But it’s depressingly difficult to pin down what it is that makes for good math instruction, or to train someone to be a good math teacher. When it comes to good math teaching, I guess I’m like the famous judge who, when trying to explain what counted as pornography for the purposes of the law, said simply “I know it when I see it“.
But while I’m not very knowledgeable about what makes a good math teacher, I know plenty about being a bad math teacher. I think most of us do, as a matter of fact. We’ve all had that math teacher who muttered into the board, with his back turned to the class, making the problems as clear as mud. As the department chair at my old school, I even got to be in charge of that guy, and let me tell you, he’s not any more fun for the adults at the school than he is for the students…I could probably sit down and write a 300 pg. book, off the top of my head, no re-writing required, if the topic was “Being a Bad Math Teacher”, and the only problem would be that no one would buy it because you’re all already experts on the subject, thanks to the national pandemic of bad math teachers.
I do think, though, that over the last couple of years, I’ve gained a key insight about bad math teaching that the general public has not yet realized: bad math teachers usually begin with a bad philosophy of math…
I first learned to love math from an old man with a “Santa Claus” beard, who rode a Harley and cussed like a sailor in front of his students. His name was Mr. Atkinson, and he was my 10th grade Geometry teacher, and it’s mostly his fault that I don’t teach literature or history.
Mr. Atkinson had a way of explaining math as if he was that one uncle you have, who was in the Navy, and uses that as an excuse for being inappropriate at the family dinner table. Once, when explaining that Side-Side-Angle congruency doesn’t prove that you have congruent triangles, he looked out at the class and proclaimed: “Side-Side-Angle congruency is usually abbreviated SSA. A lot of students like to laugh at that, because it’s ass, backwards. I’m ok with that, because as you’ll discover when you’re an adult, most things in life are, in fact, ass-backwards.”
Once, when talking to my best friend and I after school, Mr. Atkinson told us: “look, I know that at your age, all you’ve got on your minds is girls, fast cars, and alcohol. All I’m asking is that you think about math every once in a while.” For a man trying to reach teenage boys with the Good News of Math, this was pure pedagogical gold.
Mr. Atkinson couldn’t have given less of a damn about teaching us the real world applications of the math we were learning. Sure, he had been an engineer before becoming a math teacher, and so he new a lot about applying math to the real world. But for him math was first and foremost epistemology. Math wasn’t just a tool for your toolbox, it was a path to truth.
I don’t recall doing a single word problem in his class. We did proofs. We started with assumptions–called “postulates” in Geometry–and used logic to work our way to the truth. Truth, mind you, that would always be true, that you knew for sure now because you had proven it.
We live in a technological age. I don’t mean merely that we live in an age driven by computers and gadgets, though that is certainly true. I mean it in the sense of the Greek root word, technos: we are obsessed with skills. In an age where knowledge is cheap and easy to get, your ability to wield tools becomes paramount.
As a society, we value functionality over truth. We’ve been that way for a while now. Maybe it’s the Pragmatists fault. Maybe it’s just that we’ve had technological progress and economic expansion for so long, we’ve given into the natural human tendency to become obsessive. But whatever it is, this I know: math and math education have been reduced to teaching skills.
And I think that lies at the heart of a lot of bad math teaching. The bad philosophy of math that we have adopted is one which says that math is purely a functional set of skills, not the use of reason and logic in order to know. In fact, we no longer believe that reason is a path to Truth at all…if we believe that Truth can be known at all, we usually believe only in empirical truth, or in a blind faith kind of religious truth. Reason is right out, and with it goes math…
Math certainly is a useful tool for the real world, but that isn’t what math is at its core. Math is a subset of logic. Its part of a rational epistemology. It’s man using reason to know something that he didn’t know before. And if I teach math as if it is just a set of skills, just tools for a tool box, then I’m not really teaching math. And if I’m not really teaching math–rather, I’m just showing someone how to use a numerical tool set that apparently just fell from the sky, and which I have no more reason to find truthful than I do Greek mythology or the headlines on the National Enquirer–then it’s pretty hard to be a good math teacher.
‘That’s NOT to say that if all schools started going back to this tomorrow, we would suddenly have a lot of good math teachers. But we might just eliminate some of the bad math teaching that goes on…
Incidentally, this is also my complaint about the Common Core State Standards, the much-maligned federal list-o’-stuff-you-should-know-by-the-end-of-this-grade. It fosters bad math teachers and bad math teaching by reducing math to a list of skills to be acquired, rather than portraying it as it really is: a relentless pursuit of the truth. The funny thing is, that the numerical tool set is fairly easy to learn and use if I’ve pursued math-as-epistemology, but *extremely* difficult to do if I have no idea why the math is the way it is. And so, Common Core will inevitably turn out bad math students, in addition to bad math teachers…
And there’s a deep spiritual problem that can go along with bad math teaching. With reducing math to a set of skills, we risk undermining the value of finding truth. And And if we undermine the value of finding truth, if we convey that Truth and Knowledge aren’t worth pursuing with all of our hearts, then we risk adding one more stumbling block to our students, who are told by everyone around them that truth is entirely subjective, or that the pursuit of Truth is ultimately fruitless… and that belief has eternal consequences…
Once, Mr. Atkinson alluded to a mystical experience he had had. He wouldn’t go into the details. He wasn’t a believer in any particular religion, but he told us that after the “experience”, he became completely convinced that there was more to the universe than just the material. And he kept pursuing the truth.
I kept in touch with Mr. Atkinson until he passed away. I was in college, majoring in math at the time, so Mr. Atkinson never knew I went on to be a math teacher. But my friend and I went to his funeral, and had a chance to tell his wife how grateful we were to have had him as a teacher.
In his encyclical letter, Spe Salvi, Pope Benedict wrote this about salvation:
Yet we know from experience that neither case is normal in human life. For the great majority of people—we may suppose—there remains in the depths of their being an ultimate interior openness to truth, to love, to God. In the concrete choices of life, however, it is covered over by ever new compromises with evil—much filth covers purity, but the thirst for purity remains and it still constantly re-emerges from all that is base and remains present in the soul. What happens to such individuals when they appear before the Judge? Will all the impurity they have amassed through life suddenly cease to matter? What else might occur? Saint Paul, in his First Letter to the Corinthians, gives us an idea of the differing impact of God’s judgement according to each person’s particular circumstances. He does this using images which in some way try to express the invisible, without it being possible for us to conceptualize these images—simply because we can neither see into the world beyond death nor do we have any experience of it. Paul begins by saying that Christian life is built upon a common foundation: Jesus Christ. This foundation endures. If we have stood firm on this foundation and built our life upon it, we know that it cannot be taken away from us even in death. Then Paul continues: “Now if any one builds on the foundation with gold, silver, precious stones, wood, hay, straw—each man’s work will become manifest; for the Day will disclose it, because it will be revealed with fire, and the fire will test what sort of work each one has done. If the work which any man has built on the foundation survives, he will receive a reward. If any man’s work is burned up, he will suffer loss, though he himself will be saved, but only as through fire” (1 Cor 3:12-15). In this text, it is in any case evident that our salvation can take different forms, that some of what is built may be burned down, that in order to be saved we personally have to pass through “fire” so as to become fully open to receiving God and able to take our place at the table of the eternal marriage-feast.
From time to time, I pray for Mr. Atkinson. I’m hopeful for his soul, because I do think that, at least from my experiences with him, he retained in the depths of his being an ultimate interior openness to truth. He would’ve been a bad math teacher if he hadn’t.