Frank Swenton - Middlebury College
The function is well known as one of the most fundamental concepts in all of mathematics, and it is equally well known as one of the most misunderstood by students in mathematics courses. Some would say that this is simply the nature of the student, and others would blame the advent of the graphing calculator. But the root cause of this problem lies much deeper, its seeds planted more than 300 years ago: it is the graph itself, which through the primacy it has attained as a means of dealing with functions, has obscured—and even supplanted—in the minds of students the very concept it was meant to illuminate.
In the Beginning, there were the Domain and the Range, each a distinct set in its own right, and there was the Function sending each element of the domain to one element of the range. Domain was domain, range was range, and never the twain did meet. Fast-forward to the 17th century: René Descartes rotates the range by 90 degrees and clanks it down right on top of the domain; upon these axes he draws a function’s graph: a construction both revolutionary and visually seductive—in fact, enough so to eclipse the function concept itself.
Considering functions via their graphs is a practice so deeply ingrained in the mathematics classroom that it escapes the critical eye given to all other aspects of mathematical pedagogy. Graphs readily provide easy answers (in some cases), but the answers are all too ready, too easy, and—most importantly—quite often incomplete or deceptively facile. X has become synonymous with “input,” and y with “output”; the derivative means “slope”; the integral means “area”; continuity means that a graph can be drawn “without lifting the pencil.” Even the very heart of being a function has sadly devolved into the “vertical line test.”
The fundamental problem is that these rough and ready geometric answers often serve to distract the learner from well-warranted further consideration of the true nature of each concept; they provide just enough of a simple, soothing “answer” with which to be content and to quiet any call for further thought. Moreover, when a learner is presented time and time again with graphs, it becomes far too easy to see only the graphs. When applying a function f to a value, the eyes quickly move directly to the point of the graph at the correct horizontal location, and there they stop. Seldom does the eye ever stray to the domain, at which the action starts, or the range, at which the action ends—the function’s meaning as the link from one to the other is lost, all attention drawn by the immense visual pull of its graph.
The effects of this graph-centric perspective on the function are felt far beyond the topic of the function concept itself. Inverse functions have students immediately flipping the plane or wrenching their necks; that inverses are simply rules sending elements of the range back whence they came becomes secondary. Differential approximation becomes the circuitous unraveling of a triangle tangent to a graph, finally arriving at a formula that is a direct consequence of the definition of derivative. The epsilon-delta definitions of limits and continuity become a crisscrossed clutter of horizontal and vertical lines on a graph, yielding a similar tangle in the learner’s mind, when the domain and range have no business intersecting in the first place. Careful inspection across the gamut of function-related concepts shows that graphs very often complicate or confound in their effort to simplify and explain.
Is the graph logically inconsistent? No. But we must reply equally in the negative to the question of whether a graph properly expresses the function concept in its entirety. One might rebut that the modern definition of the function from X to Y is as a particular subset of the Cartesian product X x Y. However, consider the logical definition of the implication “if P, then Q”: do we properly conceptualize it as “Q or not P,” or is this merely the logical statement that defines its formal meaning? A function, while encoded as a subset of X x Y, is conceptually no more a subset of X x Y than a poem is ink on a page; the difference is that no one forgets that a poem must be read aloud (at least in the mind) to be fully appreciated or understood.
A function acts: it sends each element of its domain to some element of its range; it maps subsets of the domain to their images in the range; it even pulls back sets in the range to their preimages in the domain. A function acts—and a graph simply sits, dead. We must view that static graph as a machine ready to act, or else the graph only serves to gloss over the concept; we must make the crucial effort not to be content with the limited perspective on functions afforded us by graphs alone. The graph is only a valid tool for studying a function when viewed not as the function itself, but as a representation of the function—only in vigilance of this crucial distinction will the concept retain its integrity; and with a healthy independence of graphs, the function concept can properly grow and flourish within the minds of students.
Very interesting (and important) food for thought.
ReplyDeleteAs I began to read, I envisioned teaching real functions using two number lines, labeled domain and range, running parallel to one another and a function described and depicted as acting in a way that sends points from the domain to points in the range. I think you could get some interesting pictures of functions and have some great discussions with students around this representation -- then at some point maybe move the range line into its typical perpendicular position, which leads to the notion of the graph as another representation of the function (not the function itself).
You sound like an old-timer pining over the good ol' days when you could see two matinees down at the nickelodeon for the price of a quarter. Ah, but then "the talkies" came and ruined the imagination of our youth...
ReplyDeleteI understand functions perfectly well and their graphical representation certainly never stood in my way.
Anonymous, there's no need for personal (and vapid, albeit florid) attacks here. Points of contention or questions with intellectual content would certainly be welcome, on the other hand---much is to be gained through discussion! We're all very happy for you that you understand functions "perfectly well." But any dependence on graphs (rise, run, slope, area, etc.) in the understanding or explanation of any function-related concept belies a fundamental gap in that understanding---which is just what a great many students of mathematics end up with.
ReplyDeleteThis is not to say that graphs _prevent_ a solid understanding of functions; it is certainly possible, with effort, to push through the nonsense and come to a proper understanding of the function concept, graphs notwithstanding. However, for many students, graphs do pose an _obstacle_ to such a full understanding, and thus there is good reason to question the graph's place as a centerpiece in the mathematics classroom.
This is an interesting point and I agree with Frank that graphs may pose an obstacle to a full understanding. There are cases where graphs are not necessarily useful; take for instance normal groups in algebra. Using graphs to visualize this may be a bit challenging. On the other hand, intuition through graphs comes in handy when first learning real analysis.
ReplyDeleteAgreed! For concepts like the integral or derivative (much less so, limits and continuity, though uniform convergence is back on the good side again), the graph is a great intuitive starting point, as long as it doesn't become the end of the story---which by the time real analysis comes around, it generally can't even pretend to be, so the threat is lessened. I can't reasonably argue with the use of graphs in moderation, but I'm always a bit worried when things get too graphy in introductory (or pre-calc) courses.
ReplyDeleteWhen teaching real analysis or measure theory, I end up using graphs (and thinking in terms of them) in many spots---sometimes even without any lingering dissatisfaction or guilt! There are definitely some places that graphs simply seem to be the best entree to one's intuition. Now, whether this is due to the graph-heavy instruction in my early education, that's something that I try to be introspective about, and I'm honestly still not sure. If it's possible to build a better understanding of concepts in my students than I obtained at their stage, I'm all for it!
The book "Limits and Continuity" by William K. Smith, uses graphs very effectively in illustrating what an epsilon-delta proof is all about. After showing how the definition works using a graph, Smith then proceeds to the algebraic form of proof with inequalities, but the intro certainly didn't stand in my way of understanding what was going on.
ReplyDeleteThe history sounds wrong to me. I don't think the idea of function (in the modern sense) was developed until well after Descartes. And the modern definition of function as a set of ordered pairs is implicated here as well, whether or not you decide to visualize the ordered pairs on a Cartesian plane. So it's difficult to argue that graphs are not a central object in modern mathematics. It's true that students often see them as static pictures, without a dynamic understanding of how they related two covarying quantities along each axis. But surely that is a failure of instruction. It seems very odd to me to blame a mathematical object.
ReplyDeleteThe account of Descartes is not literally correct. A glance at Descartes' Geometry---a Dover edition is available---shows he did not use anything closely resembling our modern-day coordinate axes. Moreover, he also did not think about functions explicitly. In parts of his book, was concerned with relationships among variables that denoted the lengths of segements in moving diagrams---almost a "geometer's sketchpad" in the imagination.
ReplyDeleteI believe a large part of your criticism on the over reliance on graphs and their potential to lead to incomplete knowledge of what a function actually is, is that conventional graphs are merely static illustrations or simplistic conceptual representations whereas functions by their very nature are dynamic and do not necessarily have useful graphical representations. Furthermore we can all agree it is definitely the case that even classes of functions that do have quite useful graphs do not necessarily expose their full meaning.
ReplyDeleteWhile still flawed in some ways, I believe that, using (well chosen examples and) modern computer generated animations of such graphs (interactively controllable whenever practical), preferably combined with corresponding tables or other representations of the actual domain and image) would somewhat alleviate the chance of such misunderstanding.
Indeed I believe that there is much to be gained whenever possible in from moving from static to dynamic visualizations when it comes to epsilon, delta, limiting process, or similar proofs. These were either impractical or impossible in the past but with todays tools (Maple, Mathematica, Powerpoint, Keynote, etc...) this kind of presentation can be prepared, or done on the fly, and even interactively during a class. (or even a website associated with a textbook perhaps)
I agree however with the gist of the article, that over reliance on any one technique, especially a technique that does not, and indeed can not encapsulate the full breadth or subtlety of the definitions, is likely to harm understanding in the long term if exposure to the deeper underlying concepts is prolonged to long or simple avoided entirely.
Personally I would suspect that you would very much prefer that more formal and mathematically rigorous methods be more widely employed in introductory mathematics so as to build a much more solid mathematical foundation to build on going forward. I think the main problems with realizing this better foundation is finding a better way to treading the fine line between the level of abstraction necessary, the level of mathematical maturity of incoming students, and most of all, the ability to build a motivation for the average student as to why the handwaiving approach currently employed is insufficient.
Exposing students early to the type of general functions that they now typically first meet only in a real analysis or Fourier analysis course could definitely help with this but the question becomes how to motivate or justify this level of generality in a concrete fashion at a grade school level.
For reasons of a similar ilk there has been some call for teaching the Gauge integral right out of the box rather than the Riemman integral and avoiding the many difficulties that result, but as best I can tell this approach is not getting any traction despite its merits.
Mathematical academic tradition is such that mathematicians hide or obscure the original motivations, misadventures, and failures that eventually lead to theorems and instead present ever more compact formalized proofs of the latest and greatest result of the most generality.
I would humbly submit that no one, no matter how genius, can truly understand anything in a deep way if they (try to) begin at a level of abstraction that is beyond their ability to adequately conceptualize or motivate, so in that regard and with an eye toward history I have to agree with Bill McCallum's take on the history of the graphs vs the most generalized modern definition of a function. Furthermore I agree that the main problem is a failure of instruction and that of course content not meeting the students needs, rather than the problems your seeing being truly fundamental to one or more conceptual tools such as graphing.
Glad to see some good intellectual discussion here (after a bit of a rocky start). Let me offer a mea culpa for the reference to Descartes---the use of Cartesian coordinates in most graphing plus things I was taught decades ago must have led me to that misattribution (I'm admittedly a poor math historian)---at any rate, I needed _someone_ to blame! :).
ReplyDeleteThe "animated graphs" note above, I think, is right on point. The entire motivation for the development of the graph (in my view) was the need to exhibit a function visually, using---and this is the key---the limited technology available. Only within, say, the past 30 years has it really become feasible to do better; perhaps this would have been a good (and more positive) thing to say in the article, but there was a bit of a word-count issue as it was. At any rate, now---with Java, Flash, or whatever---is precisely the time that we (as teachers) should be breaking the mold and moving beyond teaching tools (such as, in my opinion, the graph) that have inherent limits and shortcomings due simply to the technological limitations of the time of their development.
For instance, an interactive animation showing simply two _parallel_ axes (one each for the domain and codomain) and a point or interval in the domain that can be moved (with a corresponding image drawn in the codomain) would much better express a wide variety of calculus concepts, from limits and continuity to the derivative (doing particularly well with differential approximation, the chain rule, and inverse functions). But my main point is that I think the correct next move does breaks apart the two axes, because we've finally found ourselves largely absent the limitations of static print, which means it's time to rethink some of these constructions from scratch.
For a nice example of how mathematical animations can be used very effectively to convey an entirely different level of understanding and elucidation that static presentations just cannot compete with see any of the episodes of the excellent physics series "Beyond The Mechanical Universe". ("The Mechanical Universe" was an earlier series that was also good but not on the level of its sequel as far as animation quality is concerned.)
ReplyDeleteBy todays standards the computer generated graphics in it are somewhat quaint but my reference to it is more about content than visual fidelity -- this was actually a very high budget/high production value series (a UC Berkely? prof hosted the video course, it was full of historical reenactments and anecdotes and fully animated equations and mathematical (mainly calculus) manipulations, 2D & 3D graphics were used throughout, produced in the early 1990's if I recall correctly).
It would be very nice to see a full curriculum worth of mathematics courses get this level of attention and funding - perhaps as a free resource to be used nationally to augment standard instruction.
Also with iTunes U being freely available nowadays it is starting to be possible for students and others to access decent qaulity video of some very good instructors (from some fairly prestigious and expensive institutions no less) giving their full lectures. I can only imagine if we had this archiving and distribution ability earlier how inspiring it would be to be able to see the original lectures of the great mathematicians and scientist of past generations.