Concluding the Project Final

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Well, it’s about over. Over the next two days, my students will submit their project finals for grading. I have been intimately involved in developing each project as a consultant, so I already know the work they have done very well. Some, inevitably, have slacked off and will pay for that — but most have engaged pretty deeply and the products in most cases are better than I expected, which is very exciting and inspiring.

Fears

When I started this process, I had no idea what to expect. I feared that students couldn’t tie the concrete system they built to the abstract model they had to create. After I wrote the project specifications, I feared they couldn’t learn to do what I required of them — to be honest, I was not sure how to do all of it myself. Most importantly, I feared that we would emerge from the process with a rather convincing feeling that the project had not been a good learning experience. I did not want to go back to the drawing board on this one, but I knew that was a possible outcome.

I am very pleased with the outcome of the process. I expect some really nice work and I look forward to showing it off to my students early in the year next year to get them fired up to take on this very same project (with some changes) in its second iteration. We have all learned a lot.

Blessings

It has been providential that I chose at the last moment to have all of my classes do this project. I modified the requirements so that my Pre-Calculus students would have to draw on their new knowledge of trigonometric functions — knowledge my Algebra 2 students don’t yet have — and so that my Calculus students would have to combine their differential calculus skills with linear algebra and polynomial skills they developed two years (or more) before. As a result, I have a well-defined sequence of projects, and it is clear just by viewing the graphs of the models which class each group represents. This gives the students a visual sense of how their mathematical power grows as they move through the advanced math sequence. I could not have devised a better teaser for those students who wonder just what awaits them in the advanced courses.

Another providential development was my requirement that students match their graphs to photographs of the course they created for their ball bearings (for my Algebra 2 and Pre- Calculus groups). This drove several of them to the web in search of graphing utilities, where, independently, they discovered and began mastering Desmos. We also took time to learn how to graph piecewise functions on the TI-8x series calculators we use in class. The students also had to learn how to integrate the Desmos product — a downloadable pdf file containing an exportable graphic — into a presentation application so that they could combine the graph and the photograph to make a new graphic. Many chose PowerPoint, but others took the time to discover how to do this using Prezi, Glogster, or other applications. One group projected their Desmos graphic on my Interwrite board (we turn our homework in on baked clay tablets in cuneiform, too) and traced it on a printout of their photograph. Low-tech? Sure — but the product meets specifications; the solution is ingenious in its own way.

“I love it when a plan comes together.”

The diverse methods and the determination that spread among the groups in most of my classes were a joy to see. The final satisfaction began to grow inside me as my more accomplished students, leaders in their groups, began to express personal pride not only in the product but in the simple fact that they managed to do the project in just a few weeks with no exemplars or specific experience. It is very satisfying for me to see and hear hard-working and capable students sit back after finishing a tough project and say, to anyone in particular, “I feel really good about doing this.”

Riding a Snowball?

I am already very excited about next year: we’ll do this project for the first semester final in all of my classes, and the projects I receive over the next two days will be “Exhibit A” for the argument “You can do this.” At the same time, I’m going to make the Algebra 2 and Pre-Calculus projects more demanding. The Calculus project has yet to be done in its entirety, so I will not adapt it yet. The Pre-Calculus survivors who take it on next year will have plenty of experience and confidence from this year to push them along.

“You gotta try this”

Teachers, I think we should all give this a shot. Try a project. Write up a crazy project — one that forces students to look for a connection between a concrete system and the abstractions that populate their advanced math courses — and play along. Don’t worry if parts of it seem impossible: encourage them to solve the problem, and take part in the process as a consultant. The one rule I had was this: I did not spread the word on any essential discovery until one group had made that discovery independently. Desmos is very, very popular in my classes now, but I told them not a word about it until one group discovered and mastered it on their own. The technique for overlaying the Desmos product on a photograph using PowerPoint is something all of my students understand now, but I did not guide any group to it until one group discovered it. The Prezi technique is still beyond me; I’ll have to learn that this summer.

Did they learn math? Oh, yeah.

That’s all very well, you might say, but what about the math? Well, that wasn’t as easy to pull off. We began with the simple process of turning a relative maximum or minimum and a point between that extremum and the next into a partial parabola; this mimicked the smooth curves that defined the course of the ball bearing through our vinyl tubing. This was the technique my Algebra 2 students chose to use: they had to solve ten to seventeen three-variable systems of three equations to generate their piecewise functions. They did not discover until very late that they could have used matrices and their graphing calculators to make the algebra much quicker and less prone to minor errors. That’s right: they solved each system using substitution and elimination. Had they discovered this earlier, several groups would have done much better, but, frankly, the Algebra 2 classes suffered not only for lack of mathematical knowledge but for maturity, and this will probably prove to be a benefit to them as they face projects in the future. They’ll be wiser for it.

For the Pre-Calculus students, the mathematical turning point in the project came when they realized they could abandon the parabola approach for trigonometric functions. These functions make their graphs look much “smoother” than the Algebra 2 graphs, though there are still obvious points where even those who do not understand calculus can see that the rates of change at intersection points are unequal.

My Calculus students were undermanned — three pairs of students — and lacked time because of the traditional expectations of seniors in their last semester, including early finals, graduation, and other additional duties. I exempted them from the graph-photograph overlay requirement, but the requirement that their graphs be differentiable (and therefore perfectly smooth) proved a stiff challenge. Our solution this year was to alternate quadratic and cubic graphs, which created systems of four equations in four unknowns that had solutions. One quirk in this process is that the numbers involved at the conclusion of the course, which we measured over twenty feet (610 cm) in centimeters, were so large that the TI-8x series calculators we use calculated that the resulting matrices were singular (the determinants were too close to zero). We had to find high-powered on-line linear algebra calculators to do the final calculations … but it worked out.

What did I learn? I learned we can trust this to work.

What, then, did I learn about project-based learning from this experience? I learned that it’s nice to hit on some intuitive blessings in your first run. I don’t believe in luck, so I am properly grateful for grace: the providential decisions and inclusions in the project plan brought out exactly the kinds of experiences that project-based learning should produce. Students had to do independent research without knowing that a solution even exists, and they had to improvise using the tools they discovered to rig a solution. In the process, they had to apply their mathematical skills to connect the abstract world of mathematical models to a concrete system — in this case, one of their own devising. I know from this experience that students can indeed learn new knowledge this way, and the activity, when led by capable students and guided by even a very inexperienced teacher of this kind of learning generates much academic leverage.

The students I taught this semester are leaving my class with confidence and knowledge that applies far beyond my classroom. They know they can seek, find, and master new resources. They understand how to learn to create unfamiliar forms of presentation from these new resources. Finally, they know they can build a bridge between the abstract and the concrete using these tools and their communicative talents. I’m not just looking forward to seeing what these students will create in their other classes — I’m looking forward to seeing them do all this for me again soon.

In closing …

A final reflection on the year fits here: I don’t do this job for the kids. I love them. I can’t work without them. Nevertheless, I do this for me. I do this because I want to be the best I can be at it, and I can’t get better without them. This attitude suits the activity well, because high school students particularly are looking for what is exceptional. Their emotional makeup and the society surrounding them drive them to seek out and emulate extremes — this is what drives so many adults to frustration when dealing with them. If teachers are serious about arresting that urge to shock and harnessing that reckless energy for positive purposes, then we’ve got to stand out as exemplars of ability and of teaching. We have to be authentically cool — our ideas have to work out and our ability has to be clear — to gain the authority and trust we need to help our students. Experienced teachers know the drill here: you can’t get away with faking it in front of a group of adolescents, because they’ll find you out and, once they do, you’ll never win them over.

If this project had flopped, I’d have faced a long road back to credibility with my students. That was the source of my fear. Why did I fear? Because I just don’t know how blessed I am and I don’t appreciate the effect I have on my students. I don’t know about the other teachers out there, but this year has opened my eyes to the impact I have. The little pitchers have big ears and big eyes. We have to give them an authentically good show. Did they know I was taking a risk? I think they did. Did they want it to work? For themselves, they wanted to get good grades and, as they realized that I intended to show others their work, they wanted to put their names on good products. Did they want it for me? Who cares? I teach students who want “it” — now I have to make the most of that both for me and for them. I am already very excited about pushing them more and discovering what other wonderful things we can do together, because this has gone better than I ever expected.

I have a lot to learn about this before I’m really in control of it. This project, for example, allowed students to fall back on methods they had learned in more traditional instruction. When I’m truly good at this, I’ll be able to use projects to drive students to learn new techniques and strategies. I’ll be striving for that kind of subtlety as I learn more, because once that’s in place, I’ll have a model for truly authentic life-long learning. In general, however, I am now sold: project-based learning is good for my students and is definitely going to be part of my quest to be the best teacher I can be for the foreseeable future.

Ora et labora ut in omnibus glorificetur Deus.

Common Core: Baby and Bath Water

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Maggie Gallagher has a piece about Indiana’s suspension of implementation of the Common Core here. You should read it. It’s okay, I’ll wait here.

I am implementing the Common Core standards myself, according to procedures I understand to be best practice. I have not even considered the possibility that the Common Core includes other guidelines — and that’s a good thing, as this article makes clear. I guess I should be proud and concerned that I have only read the Common Core standards and not the practical recommendations for Common Core implementation.

I’m a little concerned because I suppose if I’m going to implement something I should read all of it, but I’m proud because the confidence I have in the training I’ve received over the last few years has guided me to avoid the idiotic mistakes that launched the Indiana controversy.

I’m worried because the knee-jerk association of “watered-down” with “Common Core” could cause people to oppose what I’m doing without hearing me out. It’s the usual conservative self-defeat: what could a mere teacher know about education? We all went to school. We know what should happen there.

The Common Core is a new attempt at a national curriculum and thus it is, as the National Review article claims, trying to undermine local control of curriculum. That’s a fact, just as true of CC as it was of No Child Left Behind (NCLB). As a Catholic with conservative political leanings, I’m not a fan of that. I am a fan of rigorous, challenging curriculum that imparts skills and knowledge in a memorable and compelling fashion, and, in the practical and content standards of the Common Core, I see improvement over the standards my state used to use. The fact that a curriculum purports to be national is not enough to toss it all out as a Commie plot, as conservative education “reformers” are wont to do.

The trigger for the Indiana controversy, according to Ms. Gallagher’s article, was a series of third-grade math assignments that eschewed repetition, including only four items each evening, but attempted to draw analysis- and evaluation-level answers from 8-year-olds after they have left the classroom.

This should not have become a controversy. The school should simply have fired the teacher and the principal who backed the teacher up and the publishing company that provided support for such idiocy. This is not best practice and they should all know it, because that fact is well-documented in the research literature. No matter how many white papers the Gates Foundation or the Common Core folks put out, they cannot undo the good work done by Robert Marzano and many other serious researchers, including conservative darling E.D. Hirsch (one of my personal heroes), who have established that repetitive practice is important in skill acquisition and knowledge retention. In his book Classroom Instruction that Works, Marzano et al. tell us that 18-20 repetitions are necessary to drive retention to the 80% mastery level. For third grade math — the grade where I learned my basic math facts from 0 to 12 — the repetitions must be much more and the mastery level must be at or above 95%. Without those facts, math instruction is lost on students, and the Indiana parents were right to key on this. Where I fear they err is in romanticizing their cause: this will inspire others to chuck the Common Core as a whole when there is actually some good in it.

The article describes people who purport to be professional educators and education publishers defending the idea that repetition is not important to learning. Fire those people. I’m with anyone who wants to do that; however, don’t throw the baby out with the bath water. There are some really nice things in the Common Core that we can put to good use.

I’m not defending the Common Core in its entirety — because I haven’t read all of it. Now that I read Ms. Gallagher’s article, it’s clear that I’m not implementing Common Core as a whole — and, based on what’s happening in Indiana, that’s something of a relief. I just hope my own students’ parents will give me time to explain that to them, as they will surely challenge me.

National Review is one of the few information sources I generally trust for my political news. Unfortunately, as an educator, I see the coverage Ms. Gallagher gives to the Indiana story as sensationalistic and likely to produce overreactions among parents who are usually my natural allies. I’m talking about the parents of my students who actually want me to push them harder and make the mathematical element in their educations more rigorous and challenging. Those same parents may now retch and jerk their knees at the mere mention of Common Core, tainting some well-written practical and content standards with their appropriately negative reactions to this story (and probably others) of poor implementation.

I’m the last guy to defend the American education establishment, either. As a group, we’re not good, to redirect Chevy Chase’s assessment of Rodney Dangerfield’s golf skills in Caddyshack. I’ve not always been as good as I am now, and I am going to get better because I can and I want to improve. On too many occasions earlier in my career, I was, quite simply, bad. I am quite sure that over the last four years of my career, I’ve done the best job of teaching content that I could do at those points in time, and each of those four years has seen me improve and grow. I expect to continue that trend by implementing the enumerated Common Core standards according to the practices I’ve learned over the last four years (and not according to the notions that misguided its implementation in the case Ms. Gallagher documents).

Thanks to unions (yes, the NEA is a labor union, disgustingly so), tenure, and institutionalized arrogance, there are plenty of teachers who simply don’t even try to improve each year. They insulate themselves against legitimate concerns from parents and colleagues (administrators are colleagues, but the NEA treats them as enemies) and let kids down year after year. If there is one such teacher, then there are too many — and there appear to be many more than one.

The teachers have a few points in their side of the education “debate.” I put that it in quotes because what we call a “debate” has for years now regressed into two or three groups of kids yelling “nyah nyah nyah nyah nyah nyah!” at each other — and this is just how the NEA and progressives like it, because it strengthens them in their desire to produce a malleable, controllable populace that will identify its hopes and dreams with those of the government, making liberty expendable. Indeed, they have liberty on the ropes in America, and the prognosis is not good.

As a conservative, I see the last twenty years as proof of their success. It is absolutely true, as teachers claim, that higher salaries would attract teachers of greater quality into the profession and also support teachers in pursuing more professional growth, and it is absolutely true that education is too often a political football and not often enough a collective responsibility that all American adults take seriously. Benefiting as much as they do from such stagnation and paralysis, however, the NEA and progressives (but I repeat myself) have no desire to improve things. Apparently they have embedded that agenda in the part of the Common Core I haven’t read.

I guess I’ll read those implementation guides someday, but not soon. I’m too busy preparing to teach my students so that knowledge and skills will stick and so that they will learn to question and criticize assumptions and assertions of fact that lack the support of data and experience. Perhaps the first thing we should learn to do is question the wisdom of rejecting all of a complex thing when some parts of it can support the thing’s appropriate purpose.

I believe the practical and content standards of the Common Core are rigorous and comprehensive; they are certainly more so than the ridiculous standards I taught under NCLB for the last decade. The Indiana controversy which Ms. Gallagher wants to spread all over the nation is about bad educational practice — and people should be vigilant about that. The Common Core standards — the enumerated and organized statements of what is to be taught — are not the issue here: we all agree that we should have rigorous and comprehensive standards, and the Gates Foundation progressives have actually put together just such a set of standards, apparently to veil their desire to dumb down educational practice even more than we have seen in the last century. The issue is in the fine print of implementation guidance and in the mushy minds of people who shouldn’t ever be allowed to work in or around schools.

I don’t see a problem with implementing the enumerated Common Core standards rigorously and according to best practices supported by research, and I won’t be changing course. This controversy will simply force me to clarify what I am doing and defend it. I don’t resent that, but I will probably resent the fact that some parents will just assume that anything “Common Core” is bad. They will key on the flawed ideas about implementation and won’t countenance the enumerated standards, assuming that they simply repeat what appears elsewhere. As is usual with educators, I will wish that I could focus on serving students instead of winning over parents: I want to engage in pedagogy, not advocacy. Ms. Gallagher is trying to turn good parenting into saving the world and probably the conservative equivalent of a Pulitzer, but she will only succeed in making parents tilt at windmills. Thanks to her and others like her, and well-meaning but condescending parents, I will have to be an advocate as well as a teacher.

I hope that the knee-jerkers among my own conservative brethren will not insist that the entire Common Core — both baby (enumerated standards) and bath water (implementation drivel) — must go. Given the knack conservatives have these days for proudly and publicly shooting themselves in their feet and then inserting those same bloody feet firmly in their mouths, I’m preparing for the worst.

Ora et labora, ut in omnibus glorificetur Deus.

Douglas Adams, Reading Lists, and Project Finals

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A couple of posts ago I remarked at the schooling my project final was giving my students in the realities of the scientific method. I mentioned a student whose group had attempted to design a vinyl-tubing course for a gravity-powered ball-bearing without ever running a gravity-powered ball-bearing through vinyl tubing, and the joy he experienced when, upon embarking on a course of actual observation of the phenomenon in question, his group completed the design part of the project in less than one class period.

Please give me a few paragraphs to make a point. Thanks.

This student is a member of my “Scholars’ Bowl” team. Scholars’ Bowl is a watered-down, overly regulated, overly scripted version of Quiz Bowl, for those who are curious. I have a really good team — we got 5th in our state tournament this year featuring a team that had a single senior. In our last match, we defeated the state champions without that senior. Anyway, they’re good.

Because they’re good, I am pushing them. I wouldn’t do that if they weren’t good, as there would be no point to it. The students have a summer reading list consisting of ten books. Five are fun, in my opinion: I have assigned the five-book trilogy (yes, yes, I know) of comedic science fiction penned by the late Douglas Adams, known as The Ultimate Hitchhiker’s Guide, consisting of: The Hitchhiker’s Guide to the Galaxy, The Restaurant at the End of the Universe, Life, the Universe, and Everything, So Long, and Thanks for All the Fish, and Mostly Harmless.

Now for the point: I finished So Long, and Thanks for All the Fish, this evening. The book is generally the most awful of the five. For thirty chapters, we read Adams diving into inanities and obscenities he wisely avoids in the previous three novellas and in the sequel. Finally, in Chapter 31, he gets down to business and writes seven sentences that should form the premise of every science class in every school in the world. Here they are:

“… a scientist must … be exactly like a child. If he sees a thing, he must say that he sees it, whether it was what he thought he was going to see or not. See first, think later, then test. But always see first. Otherwise you will only see what you were expecting. Most scientists forget that.”

No, I do know that this is only six sentences. Here’s the other one:

“You can’t possibly be a scientist if you mind people thinking you’re a fool.”

(Douglas Adams, The Ultimate Hitchhiker’s Guide, 1996, Random House, p. 587)

We must forgive my student and Scholars’ Bowl player and every other kid we teach if they resist us on this matter, because our scientists have lived up to these statements a lot lately and very publicly. An excellent example is Michael Mann of Penn State, who has lived up to them so brilliantly that, if we really want to apply Adams’ criteria rigorously, we must conclude that Mr. Mann is not a scientist — and that would come as a shock to the government agencies and university officials who fund his so-called scientific research.

Of course, all of his “climate change” science friends are in the same boat. They created models and made predictions about what would happen when carbon dioxide concentrations were greater than they were. Now that carbon dioxide concentrations are greater than they were, we see that their predictions were false. The problem is simple and, if I may pat myself on the back a little, was obvious when these people made their predictions: these self-identified scientists had never seen an atmosphere with as much carbon dioxide as ours now has in it, so they could not know what the effect of that on temperatures would be. They made predictions anyway, based on models that merely encoded their own assumptions and opinions in mathematical equations. They also cast aspersions on anyone who pointed this out or discovered actual facts that seemed to contradict their assumptions, as the leaked e-mails from the University of East Anglia revealed.

No one will probably say all this in implementing better science education based on a scientific method that demands that we see a phenomenon before we think about it and test our hypotheses about it. Most of the educators we need to do this think Michael Mann really did win a Nobel Prize and believe he deserved it. I don’t care what they think about Michael Mann or climate change, as long as they teach their students to follow the scientific method rigorously and completely, and as long as they tell their students the truth about the risks of extrapolation. If they do this, of course, their students will not follow in Michael Mann’s footsteps — which is a fine outcome, in my opinion.

Douglas Adams was an arrogant, atheistic left-winger with a remarkable gift for original smart-alecky humor — a lot like me except for the atheistic left-wing bit and that no one is going to pay me to write anything — but in his Hitchhiker’s Guide books, I think he wrote a very insightful and, in its own eccentric way, spiritual treatise. I like to believe many things about him that I probably could never confirm. I do understand that he held Christmas-carol sing-alongs every year at his house, for instance. I like to believe that, had he lived to see the unraveling of the climate change hoax in the simple, innocent light provided by data, he would have berated scientists from his own vantage point on the left. Of course, he may have had to walk some things back in his own right, but I believe he would have been self-aware and (barely) humble enough to do it.

I do hope that, in his last days, he discovered time to consider God more seriously, and that he changed his heart about Him. I’ve often felt, after hearing a certain deep lyric in a rock song or reading particular passages by authors, that the Holy Spirit that inspired the Bible is still infecting both discerning and unsuspecting humans all around us, and, except for the first thirty chapters of So Long, and Thanks for All the Fish, I think Adams was among the latter group.

In any case, I love the convergence between the lessons my students are learning in my classes as the school year draws to a close and the books I’ve chosen to force my Scholars’ Bowl players to read over the summer. Even better, as I don’t believe in coincidences, I do enjoy believing that this is happening for a Purpose.

Ora et labora, ut in omnibus glorificetur Deus.

Project Final Just Keeps Getting Better!

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It’s been a great 48 hours for students attempting my Pre-Calculus project final.

As they took their last unit test Wednesday, I began trying to work out how my Pre-Calculus students could incorporate a trigonometric function into the piece-wise function formula they must produce as part of their final project in the class.

The problem comes down to discovering a formula for a sinusoid function that imitates the course the students constructed in the project’s first phase and passes through two specified points — one a relative minimum or maximum and the other a point of inflection.

Working together over two class periods in the last two days, we have discovered how to do this and most of my groups have immediately abandoned the polynomial method we started with (using quadratic functions) for the trigonometric method.

The courses my students constructed in the first project phase have four phases: descending from a maximum to an inflection point, descending from an inflection point to a minimum, ascending from a minimum to an inflection point, and ascending from an inflection point to a maximum. These correspond to the four phases of a sinusoid function, which in turn correspond to the quadrants of either the real plane or the complex plane.

When we used parabolas for this project — which is how we started — the students had to use the fact that a relative minimum or a relative maximum of their function could be the vertex of a parabola. We then discovered the equation of a parabola passing through an inflection point and a maximum or minimum point, employing the common knowledge that the x-coordinate of the vertex of a parabola f(x) = ax2 + bx + c always has an x-coordinate equal to -b/(2a).

That results in a system of three equations in three variables (a, b, and c), which is not terribly hard to solve, but the results feature rather obscure fractions or decimals that represent rational numbers beyond our calculators cannot identify. Students were bogging down in complicated numbers — which isn’t a bad thing. Textbook authors contrive nice, neat solutions and problems to help students develop confidence. One casualty of this is reality: in the real world the measurements are fraught with error and nothing can ever be exact beyond a given number of significant digits — a fact I’ll be incorporating into everything we do beginning next year.

The trigonometric approach to the problem, in contrast, is quick, deterministic, and easily reduced to formulas. I’m excited about this because we developed our formulas as we worked on the project. I had been rather skeptical that we could actually learn new information while we did projects. I had been fearful that students would resist diving into projects whose analytic phases I had not explicitly taught in advance. I am starting to see how a project can support specific and important analytic lessons, and even give plenty of repetition and drill.

The formulas are quite straightforward. We use a basic sinusoid function in the form
y = a•sin(b(x – c)) + d, and we work with two points: a point on the sinusoid mid-line (x1, y1) and a point that is either a maximum or a minimum (x2, y2).

The period of the sinusoid we use is p = 4 • | x2 – x1 |.

a = | x2 – x1 |

b = π / (2 • | x2 – x1| )

c = x1 if the function is in an ascending phase
c = x1 – (p / 2) if the function is in a descending phase.

d = y2

Another reason I’m excited about this is that I have given and want to continue to give all of my classes this project, but with appropriately different parameters. Algebra 2 and Pre-Calculus students must create continuous functions, but, as they have not learned differentiation, their curves do not have to be differentiable. My Algebra 2 students have not learned about trigonometric graphs yet, so they must use the polynomial approach with parabolas — which figure prominently in the course, as do the systems of linear equations required to discover their formulas. My Pre-Calculus students spend a lot of time on trigonometric functions, so using trigonometric graphs fits well with their course goals. My Calculus students have the additional burden of creating a differentiable function and calculating the area under the curve.

This project — closing my first year teaching these three courses — will be the first semester project next year. I am already excited to see the progress of my students as they repeat this process empowered with the new knowledge and skills they will acquire. I believe this process will enrich their high school math experience tremendously and help me to discover exciting new ways to challenge them and test them not only on their ability to use the skills and knowledge I teach them in units, but on their ability to learn new skills and knowledge within the project context. This is, of course, what companies and organizations pay problem-solvers and decision-makers to do. This is the sort of math that makes new things and makes old things better.

In this way, then, I can fulfill my promise to them that they would learn not only mathematics content but how to learn mathematics in my class. This is an exciting way to close the school year — and, like that birdie on the seventeenth hole that keeps a golfer coming back despite a triple-digit day on the links — it’s already pumping me up for next year.

Ora et labora ut in omnibus glorificetur Deus.

Final Project Update: The Scientific Method


We are one day from the deadline for the first phase of our project final. The broad objective is to arrange 915 cm of vinyl tubing along a straight path 610 cm long so that a ball bearing, powered only by the energy it gathers from gravity, completes a transit of the entire length of the tube. The math follow-up is to create a piecewise function formula that matches the course of the marble using measurements of the successful last effort.

A student came to me after school yesterday, concerned about his group. I asked him (knowing the answer) what had happened when his group had experimented by running the marble through the tube. He responded that his group had not yet done so, explaining that they had thought theorizing their way to a plan would be the right first step.

A lot of the kids I teach have this problem. The problem is easy to identify if you’ve ever worked with (or been) a good science teacher. A good science teacher is one that starts every unit with an opportunity for students to research or observe the phenomenon the class is about to study — one who has students doing things and gets them asking questions, wondering, reasoning, searching for explanations, and debating.

The problem is that my students do not understand the scientific method. It’s not that they weren’t lectured about it and tested on their knowledge of the information in a book or a lecture: it’s that they didn’t learn how to do the method. My students have believed that the first step in the scientific method is to create a hypothesis. They believe it no more. Creating a hypothesis is proposing an explanation. How can students explain anything if they haven’t observed it? They can’t.

The right first step in creating a system like that described above is to set up the tube so that its beginning and end are on vertical lines 610 cm apart and its configuration mimics a function ending in a path tangent to the floor, and then drop the ball into one end of the tube. The fact that my students don’t do this right away is evidence that they have not learned the scientific method correctly.

I think all teachers are going to have to teach students about systematic methods of learning in order to implement the Common Core. Students must not only learn about math in my classroom but they must learn how to learn. As a group, mine are just starting to learn how to learn, and it’s fun to be a part of that.

Whatever we end up teaching students concerning a systematic method of learning, we must inculcate a desire to “play,” or to observe, or to do some other activity that creates experience. Theorizing and hypothesizing before observing or playing is too often fruitless because students lack the experience to theorize correctly about the right aspects of a problem. Experience is not hard to get, once students realize they can create it themselves by playing, experimenting, and observing.

The larger lesson is obvious: experience is better knowledge. The tricky part for me is that I do not like to tell my students how to learn; when I can, I prefer to let them struggle for a while. When the student I mentioned above finally saw his group complete the project today, he actually danced in front of everyone with joy. The motivation that comes from success is greater when the struggle to succeed is significant and stressful.

It’s not always possible to get authentic experience, of course, but experience is the best kind of knowledge — one that provides true and memorable premises from which to reason and build hypotheses. If that is the only lesson that changes how we teach and how students learn in school, the entire Common Core transition will be worthwhile.

Ora et labora ut in omnibus Deus glorificetur.

Misconceptions about the Power of Mathematics


I recently read this article by Mike Adams at NaturalNews.com. It motivated me to write this post about intelligence, wisdom, mathematics, and libertarianism. As much as I love mathematics, it does not deserve the treatment Mr. Adams gives it; the error is very serious. To appreciate this, let’s take a moment to discuss intelligence and wisdom before tackling the place of mathematics in the intellectual pantheon and the current culture of American education and leadership.

Intelligence

Intelligence is the aptitude to learn without experience that which others have known before you, without experiencing the original learning of it or even the phenomenon in question directly.

Intelligence is a prized commodity to those who value intuition over learning by doing, original sources, and context. Why is a person who can “learn” without experience so intelligent? Because that person’s apparent intellectual growth faces no impediments from patient and rigorous observation, reasoning, and analysis. They “just know” something is true by reading about it or listening to someone speak about it and can immediately merge it into other intellectual activities. Such people can compound their learning quickly if they succeed in placing themselves among other intelligent people, whose pronouncements will have standing with them merely because of the credentials of the source, the cleverness of the presentation, or their own ambitions.

The mark of intelligence is intuition. Intelligent people can jump to correct conclusions based on an incomplete sequence of clues, often without appreciating the full set of premises and the arguments that support the conclusion. An intelligent person is not hard to identify: consider your circle of acquaintances for a moment. Is there a person among them who speaks often, says very interesting things that provoke responses and make conversations pick up pace, without seeming to decline in substance? Among the people in such a conversation, that person has gained standing as a source of information and insight: the others judge that one to be intelligent. This person is a valued source of information to the others for this reason.

Intelligence produces innovations in the solving of problems and the making of decisions. Wherever people act decisively to gain advantage in competition and then realize the advantage, and particularly wherever that habit compounds its own success by repeating itself in successive innovations, we see the fruits of intelligence.

It is not bad to be intelligent, but it is dangerous. Basing substantive action on intuition draws criticism from other intelligent people because of competing ambitions and the possibility that one’s intuition is leading to contradictions of others’ intuitions. Those who ignore this and then succeed accumulate wealth and power in our society at dizzying rates: intelligence often manifests itself in hubris.

Wisdom

Wisdom is the aptitude to learn by experience and to discern and trust sources of learning grounded in experience.

Wisdom is a prized commodity to those who appreciate the value of learning by doing, original sources, and context. Those who practice wisdom question and reevaluate every purported piece of “knowledge” that comes their way. If those whose prior knowledge has proven wise endorse it, or if one’s experience confirms it, then the new knowledge may gain standing, qualifying it for application, reevaluation, and additional questioning at a later time. To the wise, nothing is beyond question but those things that survive long questioning and consideration become preferred premises for arguments.

The mark of wisdom is discipline. Wise people reserve judgment on new ideas, asking incisive questions and listening very carefully to the answers, probing connections between the conclusions of arguments and their premises, and constructing their own arguments that might confirm or contradict the knowledge in question. A wise person is not hard to identify: consider your circle of acquaintances for a moment. Is there a person among them who speaks less often, but when that person speaks, everyone gives what they say extra weight and consideration? Implicitly, the others have acknowledged that this person is wiser. Among those people, this person has established a cautious and rigorous standard for evaluating new ideas: this person is a valued source of knowledge to the others.

Wisdom produces paradigms that restrain human action. Wherever people voluntarily restrain themselves from jumping to conclusions, hasty generalizations, shoddy reasoning, and bad habits of behavior and reflection, there is wisdom.

It is good to be wise, but it is not everything. When many people attempt a thing, such as competition to solve a problem through invention or innovation, it is often a person who dispenses with the reflective practices of the wise that leaps to a correct conclusion first and gains a decisive competitive advantage. In that event, a product or service may emerge of great benefit to others. The habits of wisdom restrain risk-taking, which is a necessary phenomenon.

I see wisdom and intelligence not as competitors but proper to different domains of human action. It seems to me that other conclusions have led to disastrous consequences.

Mathematics

In the era of secular modernity, mathematics has ascended to an intellectual pedestal previously reserved for theology. Mathematics is the theology of modern statist education in America, where, in the name of “freedom of religion,” the state has attempted to expurgate all evidence of the nation’s Christian foundations, thanks to an intentional revision of the meaning of “religious freedom.” For example, In the state of Kansas, where I teach, the new U.S. history curriculum calls for high school U.S. history courses to begin with the Theodore Roosevelt administration, a time at which statists were gaining the momentum that led ultimately to their final victory over American traditions of freedom through President Barack Obama’s election in 2008. (Kansas has also thus begun its transition from a solidly “red” state, meaning politically Republican, to “blue,” meaning politically Democrat.)

American history in Kansas will, from now on, be limited to the history of the movement that has sought to divorce citizens’ hearts and minds from God to marry them to the State. The premises and foundations of the noble American experiment are no longer worthy of examination and debate by our students as they grow in experience and intellect. I believe this is an intentional act by statists: if students engage in that debate as they grow into intellectual adults, they might reach conclusions contrary to statists’ conclusions about the Constitution, the Bill of Rights, the Declaration of Independence, the causes of the Civil War, the nature of our westward expansion, the oppression of the land’s previous occupants, and the extent to which America became an imperial power before 1900. Thanks largely but not exclusively to this sort of perversion of the noble role of education in forming our youth, America is no longer a Christian nation: in many ways, it is now a mathematical nation.

Why has math ascended thus? It has happened because mathematics is a way of thinking that is amenable to any set of premises. The methods of reasoning create valid arguments, so that the conclusions are true when the premises are true. Statists simply use mathematical reasoning in their arguments and choose the beliefs that give them the greatest advantage as premises. They do not tolerate anyone who questions the truth of those beliefs.

Mathematicians have based the modern presentation of geometry on four undefined terms: point, line, plane, and space; to this day philosophers consider this to be a mockery of philosophy (literally “love of wisdom”), a consideration supported nicely by subsequent events. This decision made mathematics less rigorous but made it seem more powerful, inviting the unscrupulous among the intelligent to advance it as superior to theology, which they have done to great and savage effect.

Leaving the founding terms of mathematical argument undefined made the system seem more powerful because it encouraged intuition through analogy. The results of this are morally ambiguous and yet powerful: there is a geometry of computer networks because the nodes or terminals of the network can be compared to points, their connections can be compared to lines, their networks can be compared to planes, and the network of networks (which we metaphorically call “cyberspace”) can be compared to space. This is morally ambiguous because the “good” or “evil” of a computer network, which is a tool, depends entirely on its use; the very fact that people around the world can read these words if they discover them, which many will do, is evidence of its power.

The statist thinker sees God as an intentionally undefined term with which we may dispense when we wish. Nobody who thinks seriously about God intends to leave God undefined; it remains, however, a fact that God is undefinable. Despite this, there is much spiritual, moral, and intellectual profit to gain by spending time and energy attempting to reason rigorously and patiently about God. To do so is to strive for perfection and learn that perfection is attainable only with God: thus the closest definition of God we can find is that God is that without which we cannot attain perfection. That way lies wisdom but it is not the sort of thinking that appeals to one who has made intelligence a prized commodity.

There is much to recommend using undefined terms when we are doing mathematics. We can build whole systems of thought without bogging ourselves down in the details of what the smallest elements mean. Mathematics is a science because we judge the merit of such a system by both the effectiveness of the solutions that emerge from it and their simplicity. We merge these two qualities into the mathematical virtue of elegance. Mathematics cannot be philosophy because philosophy demands that we define our terms.

The development of modern physics has taken a philosophical turn in the last century. Newton’s physics took stridently mathematical foundations that gave it great credibility and provided a foundation for much admirable and welcome progress; three centuries later, Einstein reexamined its conclusions and improved upon it with work that actually has more to do with philosophy in its method than mathematics. Einstein’s intuition revealed and corrected flaws in the details of physics; unfortunately, it also guided a turbulent and somewhat scandalous personal life. Mathematics, like intelligence, is appropriate to some but not all domains of human action. We will return to that lesson at the conclusion of this post.

The modern statists have found in mathematics the intellectual silly putty they need to expel theology from its traditional place as queen of the sciences. Just as the religious fervor of statism is the unspent spiritual momentum of the Puritans, so mathematics has ascended on the accomplishments of the pious men who discovered its secrets, to supplant the very faith that they embraced.

Mathematics is the closest thing to religion taught in American public schools, and those students who master it advance to the seminaries of the State: the service academies, the Ivy League institutions, and the cabal of “liberal arts” colleges that caters to the self-anointed elite that Angelo Codevilla has accurately dubbed “the ruling class” in America.

If I Had to Choose …

I do prefer wisdom to intelligence, and I consider mathematical skill to be intelligence and not wisdom. This is because while I am a very intelligent person, my experience has taught me not to trust my intuition in making the most important decisions in my life. My intuitive actions spring from irrational optimism and have led to unpleasant consequences in my personal life. I have failed many times to discover the true sources of wisdom around me, and, even when I have identified them, I have often failed to accept the wisdom flowing from them. I am not naturally very wise — perhaps none of us are — but I am learning to seek wisdom from the pain I experience when I do not use it. The sorrows of my many failures are teaching me wisdom, as my favorite author J.R.R. Tolkien taught me they would in The Lord of the Rings and The Silmarillion.

My intelligence is real, but it has a narrow band of application: when I use my skills of intuition and mathematics as a teacher, a quiz bowl coach, or a track-and-field coach (the work to which I am best suited), it serves me well, sometimes brilliantly. When I use it to govern my coaching of other sports, professional relationships, my work with my wife in leading my family, my family relationships, my political judgments, my involvement in my parish, or my finances, I fall far short of my goals and sometimes defeat myself utterly. In those areas, I desire ever more to govern my actions according to wisdom, which means that I wish to restrain my intuition and base my decisions on my experience or the experience shared with me by wise people.

Libertarianism

What has this to do with libertarianism? The current crop of leading American libertarians loves intelligence over wisdom, by which I mean that their views are not founded on reasoning from first principles but intuitive reactions to the very real weaknesses in the positions of the two dominant flavors of statism. One flavor is what I call the progressive version, whose high priests are Hegel and Darwin, which gave us eugenics, Planned Parenthood, the Nazis, the Communists, and all others who seek to foist death on those who do oppose or might oppose the chaining of the human spirit to the interests of the State. They claim to be humanitarian and patriotic in their words but are in fact statist in action.

The other flavor calls itself conservative in its current manifestation. It gave us the Bush family, Richard Nixon, UK Prime Minister David Cameron, and others who seem to profess limited government as a guiding principle, but in practice do not oppose the gathering of a nation’s power to its government as long as the levers of power happen to be in their own hands. They claim to favor personal liberty in their words but are statist in action and in their inaction.

The main difference between the two flavors of statism is that the progressive flavor is more activist, while the so-called conservative flavor does less and thus the progress to absolute state power is slower. Today’s libertarians will claim that this is a distinction without a difference, but, thanks to the American habit of voting every four years in elections, what we call it does not matter, but it matters. I prefer that my nation place itself on the ash-heap of history more slowly.

The so-called “conservative” flavor of statism claims George Washington, Calvin Coolidge, Margaret Thatcher, and Ronald Reagan as part of its legacy, but this is a claim that falls apart when comparing the conduct and ideas of these great leaders with the conduct of the rest of the group’s leadership. I consider these great people to be the best among many good world figures since 1787 who have stood for liberty, because they truly stood opposite statism. They are not libertarians, however, because they developed their views through the habits of wisdom, not the habits of intelligence.

As a movement based on intelligence, libertarianism is intuitive and reactionary and thus prone to reaching bad conclusions at times because of a lack of experience or rigorous reasoning; this intuition is not always wrong, of course. The problem is that political matters are those of real life, not abstract problem-solving and model-building. Such matters are better handled with the habits of wisdom than of intelligence. As a consequence, to apply either experience or rigorous reasoning to some of libertarians’ most cherished policy positions — isolationism and legalization of drugs, to name the two most obvious — is in many cases to destroy them.

The author of the article linked above, Mr. Mike Adams, presents libertarian traits of thought and advertises products associated with leaders of the libertarian movement on his website. According to this article, he believes that liberty springs from intelligence as expressed by mathematical skill. This is false: true liberty is a gift offered to every human being by God and God alone. Mr. Adams presents much that is true, of course: it is certainly true that mathematical knowledge can help us make better financial decisions and thus enhance our economic liberty.

The wrong conclusion Mr. Adams presents that is most relevant to this post is his thesis and title, which is that learning mathematics trumps oppression. This is not the case, and, sadly, I do not need to leave my experience to prove it. It is not true that ignorance of mathematics is the entirety of the problem and it is not true that learning mathematics is the entirety of the solution; Mr. Adams has made correct associations in seeing ignorance of mathematics on the “problem side” and acquiring mathematical knowledge on the “solution side,” but that is by no means the whole of the situation. Mathematics is wonderful and few love it more than I do, but it is by no means a source of liberty: that is a philosophical argument that ends quickly once engaged seriously.

It is my experience that libertarians believe with religious fervor in the power of the intuition that leads them to their political ideas. This leads them to invest their faith in the seductive idea that it is more often true that complex problems have simple solutions than is actually the case. I see that as evidence of a lack of critical thinking or at the very least flawed critical thinking. In this error, ironically and sadly, I believe libertarians have much in common with the statists they despise.

My Conclusions:

We should base our decisions on experience and wisdom, and use our intelligence, including mathematical skill where it applies, to work out details. Life is neither art nor science: it is a journey to wisdom and thus to salvation. Wisdom is both a means and an end; intelligence is only a means, and a treacherous weapon in the hands of those who do not value wisdom.

Mathematics is the process of creating elegant descriptions of experience based on abstract and undefined terms. Elegance is that rare quality in a creative work of maximizing both simplicity and effectiveness, particularly in the case of a mathematical model or proof.

To place mathematics before experience in seeking solutions to problems is to put the cart before the horse; more seriously, it is also a step toward perdition.

Ora et labora ut in omnibus glorificetur Deus.

Final Project: A Math Modeling Challenge

This is the final project in all my classes for the school year. In groups of 2 for Calculus and 3 or 4 for Algebra 2 or Pre-Calculus, I have challenged my students to do the following:

Final Project: Advanced Mathematics 2013

Introduction:

Math is about much more than numbers and formulas. The formulas we have in mathematics are the result of extensive research, often done by many different people over many years. This research includes innovative reasoning, substantial collaboration and peer review, and finally the test of time: do the conclusions of the research sustain many replications?

This project is about your ability to learn from doing mathematical research. The project has narrow limits so that you can complete the process in the time allotted using ideas we have studied together this year.

This kind of learning is less familiar to you than memorization, understanding, and application, which we teachers know as “lower-level” cognitive domains. “Higher-level” cognitive domains consist of analyzing, evaluating, and creating. You will do these things – in concert with your knowledge and ability to apply it – in completing this project.

Create:

Using provided materials and creativity, create a course for a rolling ball bearing. The course must consist of an initial phase where the ball bearing gathers kinetic energy from gravity and then a series of transitions from falling to climbing and climbing to falling, finishing at floor level. Here are the parameters for an acceptable course:

1. The ball bearing must pass from release at one end of the tube to the opposite end powered only by energy it gathers from the effects of gravity.

2. The ball bearing must not come to rest at any point in the course.

3. The course must occupy a single plane: no lateral curvature is allowed.

4. At no time may the course be vertical or horizontal over a measurable distance.

5. The course length must be 915 cm of vinyl tubing.

6. The course’s beginning and end must be on vertical lines that are no more than 610 cm apart, and the end of the course must be at floor level.

Evaluate:

The best course is the one that causes the ball bearing to take the greatest amount of time to travel from one end to the other under the specified conditions.

[There is a reward for creating the best course of all those created in a class and for creating the best course overall.]

Analyze:

Students must develop a mathematical model for the course they created.

Tasks:

1. Document the process of learning and lessons learned in developing the course.

2. Document detailed measurements of the course. Record all data for your final course, including the ball’s travel time. The final time trial must be recorded on video.

3. Devise a mathematical model that corresponds to your course as exactly as you can. Include its complete piecewise formula and a complete analysis of its graph, including intercepts, ranges where it is increasing and decreasing accurate to ±0.1 cm, and locations of local maxima and minima.

4. Create a technical memorandum:
Overlay an accurate photographic portrayal of your course with a graph of your model, so that a viewer may compare them visually.
Give a narrative of the process you used to create your course.
List all knowledge gained in the physical development of the course.
Give a narrative of the process you used to devise your model.
List all knowledge gained in the development of the model.
Note any direct connections between items on these two lists.

5. Calculus additional requirements:
The model must be continuous, differentiable, and integrable, and you must demonstrate that this is the case for every closed interval on the curve. Your technical memorandum must include the calculation of the area under the curve, a complete first derivative test, and a complete second derivative test.

6. Pre-Calculus additional requirements:
The model must include at least one portion of the curve modeled using a trigonometric function.

Acceptable media for the technical memorandum are:

paper, poster, Prezi, Glog, ShowMe, PowerPoint presentation, web page, web site, or video. If you wish to use another medium for your technical memorandum, please secure my approval first.

You may not post video of this project on the Internet without [our principal's] permission, my permission, and permission from all who appear in the video and, in the case of students, their parents or guardians.

I reserve the right to alter these as projects develop.
No changes will happen after 2 May.

I will issue the rubric for the model by 2 May.

I am evaluating each of you at all times for participation in the process of constructing the course and in the process of devising the model. If you decline to participate adequately after I prompt you to do so, then you will take a zero on your final, which is worth 10% of your semester grade.

My room, your materials, and I will be available to you most school days until at least 4:00, and later or on Saturdays by appointment.

[end project]

I hope to repeat the project next year for first semester and to develop different projects for each class for the second semester. I’m having my students do this in every class — Algebra 2, Pre-Calculus, and Calculus — so that I can gather more data and get more examples for next year’s students to improve.

Thoughts? Suggestions or ideas? Comments?

Ora et labora, ut in omnibus glorificetur Deus.

Video 1.3-3: Functions That Agree at All Points But One


As part of the process of flipping my classroom, I have embedded a new ShowMe video on my Teaching: Calculus page, explaining how to calculate the limit of a function with a point discontinuity by using a function that is equivalent to it at every point except the point where the discontinuity occurs.

As always, I appreciate any comments or feedback.

Ora et labora, ut in omnibus glorificetur Deus.

Video 1.3-2: Limits of Rational and Radical Functions


As part of the process of flipping my classroom, I have embedded a new ShowMe video on my Teaching: Calculus page, outlining the rules for calculating the limits of rational and radical functions.

As always, I appreciate any comments or feedback.

Ora et labora, ut in omnibus glorificetur Deus.

Video 1.3-1: Properties of Limits and the Limit of a Polynomial


As part of the process of flipping my classroom, I have embedded a new ShowMe video on my Teaching: Calculus page, listing some relatively obvious properties of limits (but not their proofs) and demonstrating their use in calculating the limit of a polynomial function.

As always, I appreciate any comments or feedback.

Ora et labora, ut in omnibus glorificetur Deus.