On Mathematical Creativity II

Continued from last week….

Content is subordinate to engagement. Again, a few paragraphs will not convince you to favor this position if you do not already — but given my own experience as an educator, I stand by it. I am clearly at my best when both my students and myself are thoroughly engaged in the work at hand…those occasional days when students say, “I can’t believe class is over already!” I wish I had more of them.

The waters muddy. Comparatively speaking, it is easy to teach content to pre-service teachers. But teaching them how to engage their students is a challenge.

Of course this is misleading — is there really a “how” when it comes to engagement? There may be many techniques and methods for drawing students in to learning mathematics. But engagement is about relationship. And here we confront a fundamental of the human condition — our profound inability to relate to one another.

Perhaps this is an exaggeration, though I might cite any number of large-scale wars as evidence. In the classroom, the student-teacher relationship is the scaffolding of the learning situation. But I am rather at a loss at what more to say.

Do my students laugh in class? What about the student who spent much of the last exam in tears? And what about the student at the table in the corner who never talks to anyone else? Why won’t that student come to visit my in my office? Why does this particular student always seem angry? depressed? tired? lonely?

We each handle such situations differently. Teaching is idiosyncratic. But how we relate to our students as human beings ultimately creates our classroom. Imagine, if you can, walking into your classroom and being able to instantly capture the individual responses of your students seeing you walk in. How would you feel?

I maintain that it is quite important that students like me as a teacher. I enjoy some moderate success here; I do not think that I am the most popular teacher in my department, but nor am I the least. Students are more likely to be engaged if they enjoy my being in front of the classroom. Of course this is common sense, but a point which I find is downplayed in discussions of curriculum.

Curriculum, pedagogy, content, engagement, relationship. Curriculum can be successfully standardized only to a degree — purposefully vague, but unavoidably so. Here in the US, more colleagues than not (at least among my acquaintances, both at my current and former institutions) are constrained by the curriculum they teach rather than inspired by it. Is it truly a mystery why our students are not engaged? Currently, a curriculum is seen as a sequential list of topics — complete with learning goals and outcomes — together with a nominally meaningful way to assess whether the outcomes have been met. As this list grows, students become superficially exposed to a breadth of topics, but are never given the opportunity to think deeply about any of them. Perhaps this is because it is difficult to measure depth of thought.

Measurement drives curriculum. I need hardly mention the situation in the United States and the infamous No Child Left Behind Act.  Accountability drives assessment. Of course measurement and assessment need not be the same, but in practice, there is little difference. Simply put, the analysis of the results of standardized assessments is currently the means by which we decide whether our teachers and schools are doing their jobs.

Thus has assessment become political. Parents must be appeased, administrators validated, and legislators satisfied. Of course it is always the children who suffer. By any number of indicators, our educational system is becoming less and less effective. Reasons given for this decline are legion, but there is no need for finger-pointing here.

We imagine that the solution to this dilemma is the ideal curriculum, packaged so that teachers everywhere can deliver the necessary content, with the end result being a sufficiently pleasing number. It matters little what that arbitrary number represents, but that is still what is being sought — a sufficiently high number.

It is as though we were training would-be artists by selecting a certain number of classical works of art, turning them into paint-by-number exercises, and then counting the number of times students cross over the lines. At the very least, a prospective artist should be able to color within the lines! And so, charcoal in hand (due to limited resources, all work is done in shades of gray), artists of the future are ushered out into an unfriendly world.

At university, everything changes. Colored pencils! Perhaps the student of art is amazed for a brief moment. But only until it is time to learn how to teach younger children how to color within the lines. And, of course, create their own paint-by-number exercises for their own students. Now if I just make the lines a little thicker, then more of my students will be able to color within them….

Allow me this poor analogy. Suffice it to say that our educational system does not foster mathematical creativity. The teaching of creativity cannot be standardized, nor can creativity be easily measured (by those who feel so inclined). Thus it has no place in a “curriculum.”

What is required is that we cease to think of education as delivering a curriculum.

So how must we think about education?

I shall certainly disappoint the reader by having no ready answer to this question. Or perhaps not, for any pithy answer would necessarily be glib and certainly be suspicious.

But we might say at least this:  Our classrooms should foster mathematical creativity.  It is a sobering thought to realize that most individuals go through their entire lives without appreciating mathematics as a creative endeavor. I would go further to speculate that most of these think mathematics is nothing more than advanced arithmetic.

The reader will surely be able to supply any number of reasons for why this is the case. Unfortunately, the current legalistic approach to educational reform — an approach centered around standardization, assessment, equity, etc. — only worsens the problem. Such trends essentially serve one purpose: to insulate students from poor teachers. We can no longer guarantee that a student graduating with a teaching degree is competent. Our standards — especially in mathematics — are too low.

Thus the teacher is put on the defensive. Innovation is now suspect, and the impulse toward creativity is dampened. Teach the standard curriculum and have your students pass the standardized tests — or else suffer the very real consequences.

We must get students excited about learning mathematics.  Force-feeding content to unmotivated students simply doesn’t work.

We must get teachers excited about teaching mathematics. The enthusiasm a teacher has for teaching mathematics is communicated to her students.

We must foster creativity in our classrooms. This is not an answer to a particular question, but rather a focusing point for conversation about pedagogy.

We need a paradigm shift in the way we think about curriculum.  As technology develops, the ways students learn and students’ attitudes toward learning change much more rapidly than our teaching strategies do. Yet the current approach toward curriculum emphasizes standardization and homogeneity, when in fact more flexibility is needed. Technology develops more quickly than standards change, so that much of what we teach students to be able to do by hand can be accomplished with a few keystrokes. It may be the case that most students, after they graduate, will rarely perform a mathematical calculation by hand. We simply cannot ignore this sobering fact.

So there is much work to be done. A teacher whose primary focus is to be creative, spontaneous, and engaging in the classroom is a very different teacher than one whose primary focus is to prepare students for a standardized exam. We must radically change the way we train teachers, and we must make teaching a more attractive profession for our especially talented students. We must acknowledge that our current way of thinking about curriculum and pedagogy is not adequate in our technologically advancing world — and find alternate, workable perspectives.

I shall not end with a few hopeful platitudes — frankly, the situation is not really hopeful at all. Education might be about empowering students to create their own Starry Nights, or teaching them to color within the lines of paint-by-number imitations. Which shall it be?

On Mathematical Creativity I

What follows is an essay I wrote about five years ago.  I’ll let it speak for itself — but it’s quite a bit longer than my usual posts, so I decided to separate it into two installments.  Here is the first part….

(Note:  References are to when the essay was written, so that “I currently teach…..” refers to what I was teaching when I wrote the essay.)

What is mathematical creativity?

Forgive me for not answering this question. Better minds have attempted to do so, but no consensus has been reached. I am not confident that a definitive answer will be forthcoming any time soon.

So why ask?

Now this is an interesting question! Perhaps even answerable.

There are those who seek to quantitatively measure creativity in some limited way — but I am not among them. Nor am I convinced that this is a worthwhile endeavor. Of course you will agree with me or not — and I am fairly certain I will not sway you with a few hastily written paragraphs.

We might instead attempt to qualitatively describe mathematical creativity. To what end? Perhaps we might arrange for a team of educators to individually write condensed paragraphs about creativity, but then what is to be done with all the diverse responses? Certainly many such paragraphs have been written already. Consensus is still lacking.

Should I withdraw the question?

Allow me a tentative rewrite. Perhaps, “How might we foster mathematical creativity?”

Much better! But why? We could find an answer potentially useful. Knowing what one teacher did successfully in his classroom could give a colleague an idea which she can adapt for use in her classroom.

Well, this seems to be a promising beginning! A fruitful exchange of ideas, followed by a suitable adaptation, then finally an enthusiastic implementation, and oops! What went wrong?

Learning is situational; teaching is idiosyncratic. From this there is no escape.

Many of us are familiar with the situation where we have two sections of the same class, and what seemed to work wonders in the earlier section is, somehow, not so wonderful in the later section. Perhaps one section was right before lunch, one after. Or a particularly energetic student in one section was sick that day. Maybe a desperate email from a parent just before the later section is lingering heavily on our mind. Rather more likely, however, is just the fact that there are different students in the sections.

Now add to this inescapable fact — that no two classes have the same students — the additional inescapable fact that you are not your colleague. You bring very different backgrounds to your classrooms. Moreover, in creating the lesson, your colleague likely thought through many potential difficulties, then arrived at something he could truly be excited about — and communicated this enthusiasm and confidence to his students in a way which you could not quite match in your classroom.

Nothing went wrong — unless you expected your experience to be the same as your colleague’s. Fortunately, often times it is sufficiently close, but more frequently than we would like, it is not.

This is simply the usual give-and-take we as teachers experience when we are ourselves creative in the classroom. That a new idea is implemented flawlessly is rare; often many revisions are necessary before we are satisfied with the result. An artist may make several sketches before deciding on a particular composition for a painting. A similar patience is required for artistry in teaching.

This suggests that there is no such thing as a successful curriculum. For success is not derived from the structure of a lesson, no matter how cleverly devised. It should be obvious that teachers must be sufficiently well prepared; but sadly this is often not the case. As I have found from interacting with colleagues from around the world, teachers — especially those working with younger children — have meager backgrounds in mathematics. There is an uneasy tension between insisting that teacher candidates have adequate mathematical experience and the real necessity of having them be certified to teach.

For those of us teaching older students, issues of training in both mathematics and pedagogy are significant. I currently teach at a secondary school for students especially talented in mathematics and science. Some of my colleagues (myself included) had previously taught mathematics at university, while others’ careers primarily involved teaching at the secondary level. It should not come as a surprise that such diverse backgrounds result in different views on mathematical creativity — and what is needed to foster it.

As an example, I currently teach a course entitled Advanced Problem Solving. My approach to fostering creativity? Among other things, I have students write an original problem each week on a topic of their choice.

Now given the nature of the students in this course and the course content, students write problems involving logic, geometry, number theory, probability, recurrence relations, generating functions, and geometrical inversion, among others. I give them relatively little guidance, so that they are free to explore and create. I am moderately successful with this approach.

Would I recommend this approach for a new teacher just out of college? With these topics, I would be hesitant except for the most mathematically proficient teacher.

Does that mean new teachers should forego teaching problem posing until they have more experience? Certainly not. I hope to suggest that my style of fostering creativity in the classroom is intimately related to my background and experience — different teachers will take different approaches. Perhaps more importantly, this particular approach plays to my strengths. And — dare I confess? — I get excited about it.

I suspect that every educator knows precisely what I mean. There are courses you teach, and there are courses you are excited to teach. Likely there is no need to wonder in which courses your students are more receptive.

Content is subordinate to engagement. Again, a few paragraphs will not convince you to favor this position if you do not already — but given my own experience as an educator, I stand by it. I am clearly at my best when both my students and myself are thoroughly engaged in the work at hand…those occasional days when students say, “I can’t believe class is over already!” I wish I had more of them.

To be continued….

Writing Original Problems

How do students view mathematics?

Not surprisingly, many (if not most) students see mathematics as a set of known problems to be solved — changing a few numbers here and there, perhaps — but essentially, all of mathematics is known.

Mathematicians have rather the opposite view — we’re just scratching the surface.  There is so much more underneath.

As I mentioned in my first post, mathematics is creative.  What makes this difficult for students to appreciate is that the artistic medium is that of abstraction, and without a real understanding of abstraction in mathematics, the creative aspect is hard to see.

But there is a way to help students experience the creative side of doing mathematics — and that is by having them write their own Original Problems.  I began thinking about this while I was teaching a course in problem solving at a magnet STEM high school — and being an avid problem writer myself, I imagined that having students write problems would help them solve problems.  Whether this is true or not is difficult to determine.  Regardless, an assignment was born….

What really got me interested in this assignment was the student comments at the end of the semester.  One student wrote,

Anyone can write tedious, difficult problems that review core math subjects, but to write problems in a novel, challenging, and refreshing manner, one must be imaginative. I feel that this creative side of math is an often overlooked aspect of the field as many believe math to be an extremely black-and- white, rigid, and boring subject.

I was intrigued by the fact that even though students were not prompted to address creativity in writing their course evaluations, some spontaneously did so.  As a teacher, I was delighted — an unanticipated side effect of an assignment designed for another purpose was somehow more significant to me than the intended outcome.

Fueled by this success, I introduced the assignment in an Honors Calculus section I taught, and students responded positively again.  Then I incorporated writing Original Problems into a traditional calculus classroom, then precalculus, then algebra — and students kept getting it.  Posing problems was no longer an assignment just for advanced students.

What does the assignment look like?  I break it down into four sections.  First, Motivation.  Where did the problem come from?  For some students, they might start looking in their textbook at interesting problems.  For others, they take inspiration from their daily life.  One calculus student said he came up with his problem because he dropped his backpack down the stairs and had to retrieve it — and he immediately thought of this as a displacement/velocity problem.  Another student was doodling figure eights, and created a problem about ice skating on a figure-eight shaped rink.  It is remarkable what students can create, given the opportunity!

Second, the Problem Statement.  This is actually quite difficult for some students.  And we teachers know the challenge of writing a test whose problems can be interpreted in only one way.  Now that I’ve moved on from the STEM high school to teaching university again — and work with a different set of students — I now assign the Motivation and Problem Statement as a separate assignment.  That way, I can give written and verbal feedback to students and help them craft a well-stated, manageable problem.  This has been very helpful for the students, and the quality of their final submissions has improved.

Third is the Problem Solution.  This is fairly self-explanatory, but a few comments are in order.  I like to give students wide latitude in selecting a problem of interest to them — sometimes they want to challenge themselves with a difficult problem.  In this case, a partial solution is fine.  The point is to get them writing mathematics — and a partial solution to a difficult problem often involves more mathematics than a complete solution to a more routine problem.

Finally, there is the Reflection.  I only ask for a few sentences or a paragraph — enough to give me a sense of how students are responding to the assignment.  These can be very revealing, and you sometimes get students who really appreciate the assignment and understand its purpose.  All four sections are to be included in the final submission.

You might be interested in a recent Original Problem prompt.  This assignment is highly adaptable.  I’ve had colleagues who wanted to narrow the focus because the assignment seemed to broad.  Suggesting a specific application — such as the Pythagorean theorem — will give students a starting place.  In my mind, the assignment is about creativity, writing, and self-determination.  Let students choose a topic to create a scenario and write about it, and they start to get a handle on what creativity in mathematics is all about.  There is no one way to accomplish this.

I should say a few words about grading these assignments.  At their broadest, these assignments read like short essays.  But they’re all different, so you can’t really develop a rhythm in the grading process.  So Original Problems take more time to grade — this semester I’m just giving two assignments, so it’s more manageable.  I do think it’s important to give at least two assignment, so students have a chance to improve.  Generally, I’m more lenient when I grade the first assignment, since often this is the first time students will have encountered such an assignment.

To encourage creativity, I tell students that if they just do the assignment  — and get their mathematics correct — they won’t earn lower than a B.  I don’t want them worrying about grades (and we’re stuck with them for a while!), although some inevitably do.  I rarely give a C, unless it’s evident a student waited until the last minute, or a student worked below their potential.  I do believe that for an assignment like this, you should evaluate students relative to themselves, not their peers.  More able students should be pushed — and frankly, most of them appreciate it when you do push them.

Many students really do begin to understand the creative aspect of mathematics after doing these assignments.  They really do enjoy getting to choose their own problem — and though it is sometimes challenging to come up with a way for them to develop a particular idea, I rarely tell them to just choose another topic.  I try to find some avenue they can pursue.

So I encourage you to give Original Problems a try!  Let me know how it goes.  For additional reading, you can find an article about writing Original Problems in Publication 10 on my website.  There is also a discussion of several student problems in Chapter 6 of Mathematical Problem Posing.  It really is time to have all students experience creativity in mathematics.  This is one of the main purposes of writing this blog, after all.

The Problem with Calculus Textbooks

Simply put, most calculus textbooks are written in the wrong order.

Unfortunately, this includes the most popular textbooks used in colleges and universities today.

This problem has a long history, and will not be quickly solved for a variety of reasons. I think the solution lies ultimately with high quality, open source e-modules (that is, stand-alone tutorials on all calculus-related topics), but that discussion is for another time. Today, I want to address a more pressing issue: since many of us (including myself) must teach from such textbooks — now, long before the publishing revolution — how might we provide students a more engaging, productive calculus experience?

To be specific, I’ll describe some strategies I’ve used in calculus over the past several years. Once you get the idea, you’ll be able to look through your syllabus and find ways to make similar adaptations. There are so many different versions of calculus taught, there is no “one size fits all” solution. So here goes.

1. I now teach differentiation before limits. The reason is that very little intuition about limits is needed to differentiate quadratics, for example — but the idea of limits is naturally introduced in terms of slopes of secant lines. Once students have the general idea, I give them a list of the usual functions to differentiate. Now they generate the limits we need to study — completely opposite of introducing various limits out of context that “they will need later.”

Students routinely ask, “When am I ever going to use this?” At one time, I dismissed the question as irrelevant — surely students should know that the learning process is not one of immediate gratification. But when I really understood what they were asking — “How do I make sense of what you’re telling me when I have nothing to relate it to except the promise of some unknown future problem?” — I started to rethink how I presented concepts in calculus.

I also didn’t want to write my own calculus textbook from scratch — so I looked for ways to use the resources I already had. Simply doing the introductory section on differentiation before the chapter on limits takes no additional time in the classroom, and not much preparation on the part of the teacher. This point is crucial for the typical teacher — time is precious. What I’m advocating is just a reshuffling of the topics we (have to) teach anyway.

2. I no longer teach the chapter on techniques of integration as a “chapter.” In the typical textbook, nothing in this chapter is sufficiently motivated. So here’s what I do.

I teach the section on integration by parts when I discuss volumes. Finding volumes using cylindrical shells naturally gives rise to using integration by parts, so why wait? Incidentally, I also bring center of mass and Pappus’ theorem into play, as they also fit naturally here. The one-variable formulation of the center of mass gives rise to squares of functions, so I introduce integrating powers of trigonometric functions here. (Though I omit topics such as using integration by parts to integrate unfriendly powers of tangent and secant — I do not feel this is necessary given any mathematician I know would jump to Mathematica or similar software to evaluate such integrals.)

I teach trigonometric substitution (hyperbolic as well — that for another blog post) when I cover arc length and surface area — again, since integrals involving square roots arise naturally here.

Partial fractions can either be introduced when covering telescoping series, or when solving the logistic equation. (A colleague recommended doing series in the middle of the course rather then the end (where it would have naturally have fallen given the order of chapters in our text), since she found that students’ minds were fresher then — so I introduced partial fractions when doing telescoping series. I found this rearrangement to be a good suggestion, by the way. Thanks, Cornelia!)

3. I no longer begin Taylor series by introducing sequences and series in the conventional way. First, I motivate the idea by considering limits like

\displaystyle\lim_{x\to0}\dfrac{\sin x-x}{x^3}=-\dfrac16.

This essentially means that near 0, we can approximate \sin(x) by the cubic polynomial

\sin(x)\approx x-\dfrac{x^3}6.

In other words, the limits we often encounter while studying L’Hopital’s rule provide a good motivation for polynomial approximations. Once the idea is introduced, higher-order — eventually “infinite-order” — approximations can be brought in. Some algorithms approximate transcendental functions with polynomials — this provides food for thought as well. Natural questions arise: How far do we need to go to get a given desired accuracy? Will the process always work?

I won’t say more about this approach here, since I’ve written up a complete set of Taylor series notes.  They were written for an Honors-level class, so some sections won’t be appropriate for a typical calculus course. They were also intended for use in an inquiry-based learning environment, and so are not in the usual “text, examples, exercise” order. But I hope they at least convey an approach to the subject, which I have adapted to a more traditional university setting as well. For the interested instructor, I also have compiled a complete Solutions Manual.

I think this is enough to give you the idea of my approach to using a traditional textbook. Every calculus teacher has their own way of thinking about the subject — as it should be. There is no reason to think that every teacher should teach calculus in the same way — but there is every reason to think that calculus teachers should be contemplating how to make this beautiful subject more accessible to their students.

Continue reading The Problem with Calculus Textbooks

What Is Mathematics?

Mathematics is creative.

Unfortunately, this is lost upon many — if not most — students of mathematics, in large part because their teachers may not understand mathematical creativity, either.  One way to address this issue is to have students write and solve their own original mathematics problems.  This seems daunting at first, until students realize they are more creative than they were led to believe.  (I’ll discuss this more in a later post.)

The difficulty is that the creative dimension of mathematics is a bit elusive.  Give a child crayons and ask her to draw a picture, sure — but give a student some ideas and ask him to create a new one?  To appreciate mathematical creativity, you need some understanding of the abstract nature of mathematics itself.  To create mathematics, you need imagination much like you do in any of the arts — or other sciences, for that matter.

Over the years, I’ve created my fair share of mathematics.  How much of it is really new is hard to determine — how do you know if any of the billions of other people in the world already created something you did?  (Proof by internet search notwithstanding.)

This blog is about sharing some of my ideas, problems, and puzzles.  Some were created years ago, some are new — and I will consider myself lucky if some are entirely original.  I truly did have fun creating them, and I enjoy writing about them now.

I’m hoping to convey an enthusiasm for mathematics and its related fields — in other words, all human knowledge — and to share something of the creative process as well.  The creation of mathematics is not a mystical process, and needs no explanation to a mathematician.  But we can surely do more to make this enlivening process accessible to all in a time when it is certainly necessary.

As you follow, you’ll notice a heavy emphasis on programming.  Every student should learn to program — and in more than one language.  Perhaps this should be an axiom in the 21st century, but we’re not even close.  So many of the tools I use are virtual — the ability to write code to perform various tasks is essential to my creative process, as you’ll see.  In fact, many posts will have links to Python programs in the Sage platform (don’t worry if you don’t know what these are yet).  These tools are all open source, and available to anyone with internet access.

Finally, blog posts will usually have a “Continue reading…” section.  Some posts (like this one) will be essays on teaching, creativity, or a related topic.  Since not everyone may be so philosophically minded, the “Continue reading…” sections of these essays will be a puzzle or game.  Enjoy!

Continue reading What Is Mathematics?