STEMing the Tide

Or: How I Learned To Stop Worrying and Try Again at Math

Let me tell you the story of two classrooms.  One was about ten years ago in a nontraditional high school with startling purple walls.  The class was Algebra II, although this particular curriculum integrated algebra and geometry.  I was a sophomore in high school, and a little advanced beyond my peers in the material.  It had been some time since math had been intuitive to me while English, history, and civics continued to be quite second-nature.  I didn't exactly struggle in math.  I could follow the procedures laid out for me, but I had less and less idea of what I was actually doing.  And I can still remember the concept that shut the door and turned me into a math-hater.

Sine, cosine, and tangent.

I remember my math teacher Ms. Gordon* going over the concept.  Neither I nor the other students really got it.  Her explanation of the process was straightforward enough (SOHCAHTOA and all that).  We kept asking questions of what sine, cosine, and tangent even were.  What are they used for?  What's this number even mean?  Why do we need this?  Ms. Gordon got impatient.  "Don't make it more complicated than it needs to be.  Divide opposite over hypotenuse and punch the 'sine' button," she said.  Maybe she thought we were being difficult.

Math is just a thing you have to do.  It's useful for everything.  That was what I'd heard for years.  I'd heard it that time in second grade that we visited the firehouse on a field trip; here it was again in this classroom in 2004.  Math was a thing you just had to do.  And here, as I really and truly tried to figure out what on earth these bizarre trigonometric terms were telling me, I heard it again.  Just punch the numbers in.  It's just a thing you have to do.

"What does this even mean?  It's just nonsense.," I pressed.  I got some laughs from some equally-confused classmates.  This emboldened me to push in more.  I loaded a silver bullet I'd been sitting on for a few months.  "For that matter, why are there 360 degrees in a circle?  Why not 400?  Why not 12 like a clock?  And why does every triangle have to be 180?"

"Because it's half a rectangle."

"But why?  Why can't a triangle exist by itself?  Isn't that like saying that no triangle is happy until it's married to another triangle?"  The chuckles continued as the teacher got a little more flustered.  Now I was a burgeoning comedian on a roll.  "Triangles can find fulfillment by themselves!  They don't need no man to make 'em happy!"

It was immature.  It was simplistic.  It was a distraction from finishing the lesson.  Most teenagers probably aren't asking for the Euclidean proofs underlying the calculations.  And yet behind the thin layers of teenage insecurity masked as comedy lay a very real struggle: What is this thing?  What does it mean?  And before I even started playing amateur hour at the Laugh Factory I had already checked out when I heard that deadly phrase: "Just mash the button."  Sine, cosine, tangent; even the degrees in a figure were just steps to memorize to get an answer.  Just like formulas in geometry.  Just schlock you have to vomit out on the SAT.  Just another thing you have to do so you can get into the college you want.

Years of pragmatism and lowest-common-denominator thinking wrought a dichotomy in my thinking.  Literature, history, and civics were classes where you got to do the fun stuff, wrestling through a dilemma or uncovering the mysteries of what it means to be human.  Math was a chore by contrast.  The humanities became like taking flight for the first time, each rush of discovery like the rush of a loop or a dive.  Math became the canvas bag I relieved my airsickness into when I came down.  Mind you, this was the same year I was reading The Count of Monte Cristo in World Literature and Korematsu v. United States in Civics.  In literature I kept wondering, How can Dantes see himself as a hero while he makes these people miserable?  And even worse, why am I rooting for him?  As I read Korematsu, I shuddered when I realized that the presidential power of imperium has only revolution to check it.  As I learned the formula for the surface area of a sphere, I wondered when I'd ever have to tile the outside of a sphere or why anyone made such a reckless purchase to begin with.

So I did what everyone told me.  I learned enough math to get into a liberal arts college, majored in a humanity, and graduated with Elementary Statistics and Personal Finance as my only math courses.

In another classroom ten years later, I find myself a somewhat reluctant science teacher.  I was always pretty decent at chemistry, but I never touched physics since I'd heard it was math-heavy.  I took the class primarily so I could work with the same students I did last year.  It's a physical science class, which ends up being an introduction to chemistry and physics.  The material is more challenging than anything I faced in the eighth grade.  I also find that middle school curiosity goes far beyond what one can prepare (even if you know the material very, very well).

Yet I had an enlightening experience last month.  We were going over chemical reactions for the third time, specifically combustion reactions.  A hydrocarbon plus O2 gas yields CO2 and H2O (and heat).  A student asked an off-handed question about light.  "What elements is light made out of?  How do we account for it in a chemical equation?"  I was sorta surprised because this wasn't a question I expected.  I answered, "Light isn't a compound.  It's a form of energy."

"Huh?" James asked.

"It's a wave and/or a particle that is produced by exothermic reactions like combustion."

"Combustion?"

"Yeah, like setting things on fire or explosions."

The boy-heavy classroom erupted into excitement.

"So... what is fire?" asked Jack.

"It's... I guess technically what you're seeing and feeling are the compounds turning into super-hot gas and particles in the smoke.  And lots of energy in the form of heat."

James was taken aback.  "Whoa... that's what all those symbols on the board mean?"

In that moment, I saw something I had never witnessed in this context.  It clicked.  I was used to seeing clicking all the time in my history and government lectures.  I hadn't seen (or experienced) it in a STEM subject (science, technology, engineering, mathematics) in years.  For at least half the class, two weeks of puzzling and confusion had given way to a breakthrough.  These weren't just runes tossed up on a board arbitrarily.  It was a meaningful statement of real process we observe all the time.  It was a really real explanation of a common phenomenon--all the way down to the molecular level--that proved that matter was neither created nor destroyed but changing form.

In a piece entitled "Understanding the Current Condition", Andrew Elizalde from the Veritas School in Richmond, Virginia, discusses the ineffective reforms pushed through curricula since the Sputnik launch.  Attempts are consistently made to "teacher-proof" curricula, especially in math and science where America is perceived to be falling behind international competitors (be they China today or the Soviets in 1957).   He notes that these attempts at reform have ironically made math education worse, not better.  Because publishers are trying to market their books as widely as possible, they cover much more material than necessary.  Really, it's more than any teacher really could cover in a year.  This means that math teachers must emphasize certain topics and minimize or ignore others.  This makes the transition between courses rough.  He writes that this minimizes or eliminates "experiences of investigation, adaptation, discovery, contemplation, collaboration, perseverance and creative problem-solving."  If a student doesn't get a concept, he will ask the teacher to do it on the board for him.  Math teachers may be rushed to cover everything demanded them in a given year--or worse, to achieve a certain pass-rate on a standardized test--without devoting time to the struggle students need to learn it.

He continues, "This fast-paced learning experience leaves little time for exploration, experimentation, discovery, and argumentation--the processes that were necessary for the development of mathematics across history and are still at the very core of what real mathematicians do."  So students need to be able to ask questions, make mistakes, try and try again.  They need to fight to understand a problem, learn a plan of attack, and really wrestle with a concept in order to have any hope of understanding any material--be it STEM or humanities.  A truncated pace that deprives students of the struggle is short-sighted; it gives them procedures but not real understanding.

Struggling is easier in the humanities because 1) it's harder to objectively manage and 2) it lends itself to discussion.  Students are encouraged to throw out lots of answers both individually and as a group until they find the explanation or solution that sticks.  They can wrestle with a text both individually and corporately, chew it over, try and fail and eventually succeed in synthesizing and solving a dilemma without easy resignation.  Teachers are less likely to feel a pressure to fast-forward through material because few are anxious about how our poetry compares with Japan's or China's.

I think there's also a necessary component of pragmatism preferred over theory.  Alexis de Tocuqeville said our American addiction is toward practice rather than theoretical science.  Notice, for example, that theoretical physics is not in the STEM acronym.  Giving students formulas might get them the right answer, but it doesn't inspire them.  It can't capture their attention or give them insight.  It tells them how to complete a process, but not why that process works or even necessarily what all the symbols on the page mean.  That comes only through the arduous process of struggle.

In spite of disparaging, contrary remarks I have made in the past, STEM subjects are not boring, rote chores or the refuges of uninspired pragmatists.  They only become that when they are procedures for profit or application rather than for the joy of the discovery.  The best way to learn something is to let it capture you, perhaps even to briefly obsess you.  Now not everything can or will catch you.  It isn't necessary for learning something; I never recall shoe-tying holding me under its thrall.  Nevertheless, leaders and innovators in any field should be (and almost always are) sorta quirky and weirdly obsessed about their subject.  I want my neurosurgeon to have a bedside manner and have memorized the surgical procedure; I want the neuroscientist who taught him to have brains in jars all over his creepy science lair.

I also don't mean to say that practical application of theoretical concepts is bad.  It is only a problem when the practical dominates the theoretical.  Science and math are best studied for their own sake; if you make money using them, fine.  Just don't try to inspire a generation of students with the promise of dollar signs.  Show them instead that there's something intrinsically fascinating about quantifying the amount of energy they burn running a 5k or how to isolate variables in an equation.

So maybe after all these years, it's time for me to reexamine sine, cosine, and tangent.  Maybe the problem-solving skills presented there really do correspond with reality.  Maybe Euclid does have an explanation as to why all rectangles are really triangles who finally decided to settle down.  It could even be that a 361st degree in a circle is tantamount to dividing by zero.  Guess it's time for me to find out.  It should be at least as interesting as Jack Burden's character progression in All the King's Men.  And if not to me, then maybe I'll at least see why someone else gets stoked over it.


*Name changed to protect the guilty.

To Grow in Wisdom: Jesus as a Model of Liberal Arts (Part 2)

            Perhaps he could, some may argue.  “Of course he knew how to refute them; he was God and God knows everything.  Jesus didn’t need to learn the Bible because he wrote it!”  Yet this ignores what Luke told us earlier in his gospel.  “Jesus grew in wisdom and stature, and in favor with God and man” (Luke 2:52).  This means that the four-year-old Jesus did not have a supernatural knowledge of reality.  He was not discoursing in Christian metaphysics while his kindergarten playmates ate mud pies.  He had to learn things just as we all must.  He knew the Bible not because he had a supernatural, divine memory of it but because he applied himself to it.  He learned it.  The Book of Hebrews tells us that he lived just as we did and felt the full human experience.  He cannot be held to be fully human if he didn’t live the human experience—including long days of note-taking and memorization.

            However, Jesus not only demonstrates intellectual knowledge but also earthy, tangible service, though always to serve the greater purpose of his ministry.  His sermons were not designed for academic audiences but popular ones.  His audiences were primarily farmers and fishermen.  His parables frequently use blue-collar metaphors familiar to them.  One of the most explicit stories of Jesus’s concern for the practical comes in his healing of the paralytic.  Matthew, Mark, and Luke all tell us that a crippled man is brought to Jesus.  He is lowered through the roof of the house in which Jesus is teaching.  When the man is lowered to him, he announces that the man’s sins are forgiven.  The Pharisees balk, murmuring to themselves that this is blasphemy.  Jesus knows their hearts and announces that he will prove his authority to forgive sin by healing the man.  We see in this that Jesus’s tangible ministry is spurred by a spiritual imperative.  Time would fail me to explore other examples like the feeding of the multitudes, his social discourse in the Sermon on the Mount, his affirmation of paying taxes to Caesar, even raising the dead.  Nevertheless, it is clear that Jesus believes that a bodily life must be grounded in a spiritual foundation.

            Most striking for our purposes, however, is how Jesus is passionately concerned with the hearts of his hearers over the particularities of how they make a living.  We read in Matthew, Mark, and Luke that a rich young ruler came to Jesus and asked how he may inherit eternal life.  He appeared to have all the correct practices, was obviously wealthy, had a promising political career ahead, and showed a real desire to follow Jesus.  To use Berry College language, his “head” and his “hands” appeared to be in order.  Then Jesus, loving him, drops a dramatic challenge upon the young man.  “Go, sell what you possess and give to the poor, and you will have treasure in heaven; and come, follow me.”  Rather than follow, this pious, intelligent, practical young man goes away sad.  In saying this, Jesus was not saying that all who follow him must be poor and destitute.  We see numerous people in the Bible who are wealthy Christians.  Yet Jesus knows something about this man; Mammon is his god and he will never be truly fulfilled until he casts his idol into the dust.  He must drop everything and follow this itinerant preacher for as long as he has left on Earth.  This man must leave his doubtlessly promising political career, all the financial gain he has achieved through his job, and follow the Master, an unwise career move in any century.  In saying this, Jesus illustrates what he taught in the Sermon on the Mount: “No one can serve two masters, for either he will hate the one and love the other, or he will be devoted to the one and despise the other.  You cannot serve God and money.”  For Jesus, the heart of what we do cannot be devoted to profit.  It must be devoted to God.

            The greatest challenge to the mechanical/employability view of education is the very core of Jesus’s life and teaching.  Bill Gates told the National Governors Association in February 2011 that “everybody should have a sense of which of the colleges—both community and four-year institutions—are doing very well.  You can even break that down by the departments. […] The amount of subsidation is not that well-correlated to the areas that actually create jobs in the state—that create income in the state.”  And surely this is common sense, right?  Pour taxpayer resources into departments which create jobs, take the humanities off of life support, and you will maximize state investment in education.  Let the English majors be baristas and let the hard sciences do the real work in the economy.  Isn’t that what China and Japan do?  But the very premise of the Gospel Jesus preaches, the very premise of the entire Bible, is that truth must be revealed from the outside.  It is not enough that students graduate, make money, create jobs, and figure out who they are on their own.  The Bible says that truth must be revealed from without, not from within.  Jesus tells us that he came that we might have life and have it more abundantly.  He tells us that the Scriptures have been fulfilled within our hearing.  He tells us that we are in far more trouble than we ever realized in the Sermon on the Mount.  Most importantly of all, he tells us that greater love than we can possibly imagine has come in the body of the modest carpenter calling us home.

            This is why Christianity cannot affirm a libertarian view of education.  Education cannot consist solely in acquiring skills for a marketplace.  Christians throughout the centuries have followed Christ in affirming that education must be holistic.  We cannot be only concerned with right belief, lest we become like the teachers of the law who believed many of the right things yet were “whitewashed tombs”, clean on the outside yet full of death.  Nor may we be concerned only with the pragmatics and measurable outcomes (or incomes) of education lest we become like the rich young ruler who was successful and affluent, yet in the end worshiped money and comfort.  Christian education must be fully embodied, recognizing that Jesus Christ, the Truth, is concerned with the whole person.  Jesus’s final charge to the newly-formed Church is to “Go and make disciples of all nations, teaching them to obey everything I have commanded you.  And surely I am with you always, even until the end of the age.”  Disciples cannot make themselves no matter how much money or job fulfillment they find.  They must be taught, head, heart, and hands.

To Grow in Wisdom: Jesus as a Model of Liberal Arts (Part 1)

When it comes to educational philosophy, the liberal arts are falling out of vogue.  Organizations like the Bill and Melinda Gates Foundation or the Casey Research Institute issue dire warnings about the future if America does not devote more resources to mathematics and engineering.  Alex Daley with the Casey Institute even went so far as to write a piece entitled, “Friends Don’t Let Friends Major in the Liberal Arts”, an unsubtle title to say the least.  There he argued that American students are “graduating less prepared for the technological future than their peers from around the world – a trend that is already having serious economic effects as jobs like biotech research now head overseas as quickly as manufacturing once did.”  No doubt Mr. Daley has in mind countries like Japan, Germany, or China, nations which excel in engineering and research.  Or perhaps he has in mind numerous media reports about fledgling American science and math scores or the dire predictions of our economic future if we don’t shed our obsession with the arts.  As any reasonable libertarian like Doug Casey, founder of the Casey Research Institute, will tell you, history, literature, dead languages, and theology are really nothing but diversion or a hobby at best.  Maybe as well the ironically unscientific anecdote of freeloading Occupy protesters with loads of debt and useless Medieval Literature degrees still rings true in the ears of many hearers.  So off to war march the pragmatists and the artists, engineers citing the indisputability of common sense.

            As a Christian, however, I am left wondering how Jesus informs this discussion.  While I no longer sport a knit bracelet advertising it, I still have to ask “What Would Jesus Do?”  (My Sunday School teacher said so.)  More aptly in this discussion, what can we learn from Jesus’s pedagogy here in 2013?  What can we glean from the famous teacher Jesus Christ, the most influential man who ever lived?  I believe that we can see from Jesus a holistic view of education, one which engages the mind with the arts, values the practical life of work and career, and transforms the human heart.  Thus, Jesus the liberal artist sees and educates the whole person.

Jesus demonstrates a holistic liberal education in his knowledge not of only of the Scriptures (a decidedly unscientific book, despite our loudest fundamentalists’ yelling).  In Luke 4:16-22, Jesus attends the synagogue in his hometown of Nazareth.  In this passage, we read that Jesus opens the scroll of Isaiah and announces the fulfillment of Isaiah’s prophecy in the presence of the assembly—namely, that the Messiah had come and was now standing before them.  We may note two very significant facts here which may not be apparent to the reader.  The first is that Jesus is familiar with the Hebrew Bible.  This story takes place in a time before the conventional chapter and verse designations with which we are familiar.  This was also in a time before the codex, meaning that the entire text of Isaiah was on a single scroll.  Jesus would have studied the synagogue’s copy of the book; it was too expensive to produce individual copies.  Further, Jesus was familiar with Hebrew, a language which was essentially dead at the time.  Modern Hebrew spoken in Israel today is the only example in history of a revived language.  Jesus probably spoke Aramaic as his native language; spoken Hebrew was almost dead by this time.  In fact, most young Jews would not have begun any kind of practical job training before learning Hebrew from the local rabbi.  Jesus was well-versed enough in the Hebrew faith that he was even able to astound the rabbis in Jerusalem… at the age of twelve!

            However, beyond simply knowing the Bible, Jesus the blue-collar carpenter was also familiar with the theological ideas in vogue at the time.  In Luke 20, we read an account of Jesus’s time in Jerusalem during Holy Week.  The Sadducees approach Jesus and pose a hypothetical example concerning the resurrection of the dead.  “Teacher, Moses wrote for us that if a man's brother dies, having a wife but no children, the man must take the widow and raise up offspring for his brother. Now there were seven brothers. The first took a wife, and died without children. And the second and the third took her, and likewise all seven left no children and died. Afterward the woman also died. In the resurrection, therefore, whose wife will the woman be? For the seven had her as wife.”  The Bible tells us that the Sadducees denied resurrection and they also held only the Torah—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—as canonical Scripture.  The most explicit references to the resurrection of the dead in the Old Testament come in the writings of the Prophets (such as Isaiah and Ezekiel), books denied by the Sadducees.

            Jesus is familiar with their theological opinions and meets them on their own terms.  He answers their question by refuting their silly example (there is no marriage in the resurrection) and then citing Exodus.  He says that God is the god of the living, not the dead.  How can he be (present tense) the God of Abraham, Isaac, and Jacob if they are dead?  While modern-day Jews may dispute Jesus’s interpretation of this verse, we are told by Luke that this completely dumbfounded the Sadducees.  Jesus the carpenter quoted passages his opponents considered authoritative, something he could not have done if he hadn’t studied their theology.