tag:blogger.com,1999:blog-77755136174068888112015-07-05T04:20:34.522-04:00Math Horizon's AftermathAftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA.
Contact information available <a href="http://www.maa.org/mathhorizons/feedback.html">here </a>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger26125tag:blogger.com,1999:blog-7775513617406888811.post-29054529218538406152015-04-01T00:00:00.000-04:002015-04-10T15:30:51.964-04:00History Helps Math Make Sense <i>By Daniel E. Otero</i><br /><br />Has it ever bothered you that many mathematics textbooks begin with a number of strangely crafted definitions?<br /><br />These definitions invariably turn out to be extremely valuable for the development of the theory in question, but it may be days, months, or years—with many rereadings—before this is apparent. <br /><br />I remember wondering as a college sophomore why the definition of the linear independence of vectors <b>v<sub>1</sub>, v<sub>2</sub>, …, v<sub>n</sub></b> had to be so complicated and, in particular, what the algebraic condition<br /><br />The only scalars a<sub>1</sub>, a<sub>2</sub>, …, a<sub>n</sub> that satisfy a<sub>1</sub><b>v</b><sub>1</sub> + a<sub>2</sub><b>v</b><sub>2</sub> + … + a<sub>n</sub><b>v</b><sub>n</sub> = 0 are a<sub>1</sub> = a<sub>2</sub> = … = a<sub>n</sub> = 0.<br /><br />had to do with the ability of the vectors to fill out n-dimensional space. It was years before I figured out that connection.<br /><br />Anyway, after the cryptic definitions, the textbook author embarks on proving a series of theorems whose purpose is hidden until quite late in the theory’s development, if ever. <br /><br />The most important results, so identified because they are called The Fundamental Theorem of Something or Other, appear at the end of section 3.3 as a corollary to some other impenetrably technical theorem, apparently as an afterthought!<br /><br />If the intrepid reader has lasted this far, the author throws a bone late in chapter 4 in the guise of an application of the theory to some problem that may have helped someone at some time. <br />No wonder so many people think that mathematics is only for nonhumans!<br /><br />To be fair, more and more mathematics textbooks are far better written than this caricature I have painted for you, but sadly, plenty of examples of expositional writing in mathematics fit this mold. <br /><br />You can thank Euclid for this penchant professional mathematicians have of organizing their writing in axiomatic form: definitions, axioms, propositions, theorems, and corollaries. Indeed, Euclid didn’t bother adding applications to such expositions. You can thank Archimedes, Ptolemy, and Galileo for including them (although Descartes, Gauss, and Cauchy usually did not).<br /><br />I won’t argue that the traditional axiomatic style lacks value—mostly because I don’t believe this at all! I will suggest that it is not the best vehicle for learning mathematics. My contention here is that the best antidote for students who struggle with traditional forms of axiomatic exposition is to investigate the history of the subject.<br /><br />That definition of linear independence at the start of this article? I finally started figuring out the link between it and the geometry of space when I read about linear algebra’s history (specifically the work of Herman Grassmann in the mid-1800s and the later formalism of linear algebra under Giuseppe Peano and others later in that century). And this is not the only occasion when learning how a mathematical subject developed helped me make sense of what was going on.<br /><br />Studying the history of mathematics has much to offer the mathematics student:<br /><ul><li>Context (the conceptual and cultural circumstances for the underlying problems); </li><li>Motivation (the rationale or even the value of wanting to know the answers to the central questions involved); and </li><li>Connections (how the mathematicians thought in terms of other ideas that were already established at those times and places). </li></ul> What’s more, the history of mathematics humanizes the subject in a way that no formal presentation can, reminding us that it’s people who do mathematics, not textbooks. And that, after all, just makes sense.<br /><br />To purchase at JSTOR: <a href="http://dx.doi.org/10.4169/mathhorizons.22.4.34">http://dx.doi.org/10.4169/mathhorizons.22.4.34</a><br /><div class="MsoNormal"><o:p></o:p></div><br /><i>Danny Otero is an associate professor of mathematics at Xavier University, president of the MAA Ohio Section, and chair of the History of Mathematics Special Interest Group of the MAA. He still digs the Power Puff Girls. Email: <a href="mailto:otero@xavier.edu">otero@xavier.edu</a></i>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-75765265354496129372015-02-01T00:00:00.000-05:002015-02-01T00:00:04.904-05:00I Love Math And I Hate The Fields Medal<em>By Cathy O'Neill</em><br /><br />I’ve loved math for as long as I can remember. When I was five I played with Spirographs and learned about prime numbers, and in high school I <a href="http://mathbabe.org/2012/07/18/hcssim-workshop-day-14/" target="_blank">solved the Rubik’s cube with group theory</a>. Gorgeous stuff! Inspiring!<br /><br />In college, I was privileged to learn algebra (and later, Galois theory) from <a href="http://math.berkeley.edu/~ribet/" target="_blank">Ken Ribet</a>, who became my friend. He brought me to dinner with all sorts of amazing mathematicians: Serge Lang, J. P. Serre, Barry Mazur, John Tate, his Berkeley colleagues Hendrik Lenstra and Robert Coleman, of course, and many others. <br /><br />Many of the characters behind the story of solving <a href="http://en.wikipedia.org/wiki/Fermat's_Last_Theorem" target="_blank">Fermat’s Last Theorem</a> were people I had met at dinner parties, including Ken himself. Math was discussed in between slices of Cheese Board Pizza and fresh salad mixes from the Berkeley Bowl.<br /><br />The best thing about these wonderful people was how joyful they were about the serious business of doing math. It was a pleasure to them, and it made them smile and even appear wistful if I’d mention <a href="http://mathbabe.org/2011/07/20/what-tensor-products-taught-me-about-living-my-life/" target="_blank">my difficulties with tensor products</a>, say. They were incredibly generous to me, and honestly I was spoiled. I had been invited into this society because I loved math and was devoting myself to it, and that was enough for them. Math is, after all, not an individual act; it is a community effort, and progress is to be celebrated and adored. And it wasn’t just any community. It was an exceptionally nice group of people who loved what they did for a living and wanted other cool, smart people to join them.<br /><br />I mention all this because I want to clarify that in such a community, where math is so revered and celebrated, it is its own reward to prove a theorem and tell your friends about it.<br /><br />Now that I’ve explained how much I love math, let me explain why I hate the <a href="http://en.wikipedia.org/wiki/Fields_Medal" target="_blank">Fields Medal</a>. Through the filter of that award, the group effort I’ve just described is utterly lost, is replaced with a synthetic and false myth of the individual genius working in isolation.<br /><br />You see, journalism has rules about writing stories that don’t work for math. When journalists are told to “put a face on the story,” they end up with all face and no story. After all, how else is a journalist going to write about progress in some esoteric field? <br /><br />The mathematics is naturally not within arm’s reach: It is by nature deep and uses multiple layers of metaphor and notation that even trained mathematicians grapple with, never mind a journalist, and never mind a new result on the far edge of what is known. Too often the story becomes about what the mathematician had for breakfast the day of his or her discovery rather than what the discovery itself means.<br /><br />The Fields Medal, which is easy to understand (“it’s the Nobel Prize for math!”), is thus incredibly and dangerously misleading. It gives the impression that we have these superstars who “have it” and then we have a bunch of wandering nerds who “don’t really have it.” That stereotype is a bad advertisement for mathematics and for mathematicians. Plus, the 40-year-old age limit for the award is just terrible and obviously works against certain people, especially women or men who take parenting seriously. And while the fact that a woman has won the Fields Medal is a good thing, it’s a silver lining on in otherwise big old rain cloud, which I do my best to personally blow away.<br /><br />Lest I seem somehow mean to the Fields Medal winners, of course they are great mathematicians, all of them. To be sure, there are many other great mathematicians who never get awards, and awards tend to be given to people who already have a lot of resources and don’t need more. Even so, I’m not saying they shouldn’t be celebrated, because they’re awesome, no question about it. <br /><br />I’m just asking for more celebrations. I would love to see some serious outward-facing science journalism celebrating the incredible collaborative effort that is modern mathematics.<br /><br /><hr /><em>Cathy O’Neil is a mathematician and a data nerd. She wrote</em> <a href="http://shop.oreilly.com/product/0636920028529.do" target="_blank">Doing Data Scienc</a>e <i>and is working on a book called</i> <a href="http://mathbabe.org/2014/01/13/im-writing-a-book-called-weapons-of-math-destruction/" target="_blank">Weapons of Math Destruction</a>. <em>She writes regularly at</em> <a href="http://mathbabe.org/">mathbabe.org</a>.<br /><br />Email: <a href="mailto:cathy.oneil@gmail.com">cathy.oneil@gmail.com</a><br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-74620561264229737472014-11-11T13:44:00.000-05:002014-11-11T13:45:13.924-05:00When Will I Use This?<div>Douglas Corey—<i>Brigham Young University</i></div><div><br /></div>The top search engine completions for "when will I use . . ." are all related to school mathematics. Some students ask such questions as a challenge to the teacher, but others sincerely want to know. Like me, you may have a hard time remembering the last time you used multiplication for something other than schoolwork, yet we use multiplication all the time. It is so ingrained in our thought processes that we don't notice it.<br /><br />When someone asked when I had last used it, I eventually remembered that it was to calculate the area of my raspberry patch. But this example probably wouldn't convince a skeptical student that math is useful. Examples about baseball statistics, recipe conversions, or grocery store price comparisons may also be unconvincing. Why? Because if an application is outside the interest of a student, he or she discounts it. In truth, a teacher doesn't know when a student will use the math being taught (except on the exam). It is fraudulent to pretend otherwise.<br /><br /><h4>Connecting the Dots</h4><div><br /></div>Typically we don't know what we don't know. This makes it very difficult to predict what kind of knowledge we will need. It is also very hard to see how we could use knowledge that we don't have.<br /><br />In Steve Jobs's commencement address at Stanford University, he describes taking a calligraphy class in college. In the class he learned about typography: the technical aspects, the terminology, its history, and the characteristics of a beautiful typeface. This class, which at the time had no practical applications for him, led to his inclusion of multiple typefaces and proportionally spaced fonts in the first Macintosh computer. Jobs said, "You can't connect the dots looking forward; you can only connect them looking backwards. So you have to trust that the dots will somehow connect in your future."<br /><br /><h4>The Eye of the Mind</h4><div><br /></div>Knowledge enables us to see what others can't. When one of my sons puts his shirt on inside out and backwards, I think of the symmetry group generated by the actions on his shirt. When I watch the balls on my kids' trampoline roll around, I think about how their paths are modeled by hyperbolic geometry and how the model also governs the path of light through space.<br /><br />It doesn't go the other way. People don't stand on the trampoline and ask themselves what connection it has to hyperbolic geometry or to Einstein's theory of general relativity. They don't think about abstract algebra when they see a shirt turned inside out. They can't see these connections, so the connections don't exist to them.<br /><br /><h4>Just Look It Up</h4><div><br /></div>Students argue that it is a waste of time to memorize formulas, definitions, theorems, and proofs because they can always look up such things. But we look up things only when we know we don't know about them, and we need to know fairly specifically what we don't know in order to search for it.<br /><br />As an experiment, find a meaningful quote, such as this favorite of mine attributed to Thomas Edison, "We often miss opportunity because it is dressed in overalls and looks like work." Repeat it every morning and evening for two weeks. During this period you'll find that the quote comes to mind as relevant multiple times. Without having memorized it, you would not have stopped and thought, "I wonder if there is a quote by Edison that I could use right now?" You did not know enough to see any connection to Edison's ideas.<br /><br /><h4>Conclusion</h4><div><br /></div>No one knows when you will use the math you are learning in your classes. Most knowledge gets applied to situations we never anticipate. It pays to learn all you can about all you can. You will be able to see how you benefit from it only by connecting the dots backwards.<br /><br />An expanded version of these arguments, which I give to my students, is available at <a href="http://maa.org/mathhorizons/supplemental.htm" target="_blank">maa.org/mathhorizons/supplemental.htm</a>.<br /><br /><hr /><i>Douglas Corey is an associate professor in the mathematics </i><i>education department at Brigham Young University. </i><i>He stays busy with his eight kids, all of whom are </i><i>girls but seven.</i><br /><em><br /></em><em>This article was published in the November 2014 issue of </em><a href="http://www.maa.org/mathhorizons/" style="color: #005bab; text-decoration: none;" target="_blank">Math Horizons</a>.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-24793793672118795062014-09-01T05:00:00.000-04:002014-09-01T05:00:06.854-04:00Bad at Math Is a LieMatt Waite—<em>University of Nebraska-Lincoln</em><br /><br />All I had to do was test into college algebra. That was what the College of Journalism and Mass Communications at the University of Nebraska required in 1993. If you could score well enough on the math placement exam for incoming freshmen to get into college algebra—not take it, just get into it—you were done with math. <br /><br />I very nearly didn’t graduate from high school on time because of math, so this made the test a five-alarm panic attack. I struggled in every math class I took in high school. I needed tutors and small miracles to pass.<br /><br />But all of this was OK because I was Bad at Math. It was a thing. People I knew were Bad at Math. My mom was too. So it was probably genetic. I was born to agonize over math, my friends weren’t.<br /><br />So when I tested into college algebra, you would have thought I hit a home run in the bottom of the ninth in the seventh game of the World Series and scored the goal that won the World Cup all at the same time. I went running across campus jumping and pumping my fists like a lunatic.<br /><br />Fast forward almost 20 years, and there I was, taking that same math placement exam. I wanted to get an MBA, and calculus was required to get in.<br /><br />You could say I bombed, but that’s not true. I tested exactly where I should have: remedial algebra. I was going to have to take two math classes before taking calculus.<br /><br />That’s how I—a 37-year-old father of two, a professor with a résumé that includes reporting from a war zone, stargazing through the eye of a hurricane, starting my own software company, and building a website that was the first to get a Pulitzer Prize—ended up in a remedial algebra class with students half my age.<br /><br />And I was terrified.<br /><br />See, I was Bad at Math. I knew I was going to have to sit in the front row, ask millions of questions, and work harder than anyone in there if I had any hope of passing. Forget about getting a good grade.<br /><br />And that’s what I did. I sat in the front row. I raised my hand so much they asked me to stop. I was that student. The one who did extra homework. The one who started studying for tests a week ahead of time.<br /><br />And I learned something: Bad at Math is a lie. It’s a lie I believed to make struggling at math hurt less.<br /><br />I worked harder in that one math class than I had in whole years of schooling. And I got an A in math for the first time since the fifth grade. And I did the same thing in College Algebra—and got the same grade.<br /><br />When it came time to take calculus, I was beyond scared. I struggled. I went in for extra help. I used online videos. I did twice as much extra homework. And I lay awake at night, worrying, going over problems, doing them in my head.<br /><br />I might get the A in calculus tattooed somewhere. It means that much to me.<br /><br />Somewhere in the early years, I missed something. I daydreamed through a lecture, something. Something didn’t click, I didn’t ask, and I started struggling. The Bad At Math lie was born. And I believed it.<br /><br />But the truth is anyone can get math. Some of us just have to work harder. Some of us didn’t get the message that you have to practice. We didn’t get that math is really explaining how to solve a problem, not just solving the problem in front of you.<br /><br />If you’re reading a magazine about math, none of this might make any sense to you because you get it. But trust me, someone sitting near you is in agony, panicking that he or she doesn’t get it.<br /><br />Someone near you still believes the lie. <br /><hr /><em>Matt Waite is a professor of practice at the College of Journalism and Mass Communications at the University of Nebraska–Lincoln, teaching reporting and digital media development. He was the principal developer of the website PolitiFact.</em><br /><em><br /></em><em>This article was published in the September 2014 issue of </em><a href="http://www.maa.org/mathhorizons/" style="color: #005bab; text-decoration: none;">Math Horizons</a>. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-31584896624222775312014-04-02T09:52:00.000-04:002014-04-02T09:53:12.610-04:00Every Math Major Should Take a Public-Speaking CourseRachel Levy—<i>Harvey Mudd College</i><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">In mathematics courses we practice deep thinking, </span><span style="font-family: inherit;">clear writing, and effective problem solving. </span><span style="font-family: inherit;">Excellent public speaking complements these skills. </span><span style="font-family: inherit;">As one of my students put it:</span><br /><i></i><br /><blockquote class="tr_bq"><i><span style="font-family: inherit;">No matter what we all do after college . . . [we] </span><span style="font-family: inherit;">will have to speak to people. Every one of us will have </span><span style="font-family: inherit;">a limited amount of time that we can convince someone </span><span style="font-family: inherit;">else to see our point of view.</span></i></blockquote><br /><span style="font-family: inherit;">A public-speaking course can help you develop a superpower: </span><span style="font-family: inherit;">the ability to communicate to a live audience </span><span style="font-family: inherit;">in a clear, compelling manner. Every mathematics major </span><span style="font-family: inherit;">should take such a course. Comments in italics are </span><span style="font-family: inherit;">from my students in Math Forum, our required public speaking course at Harvey Mudd College.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Unfortunately, standing up in front of a group makes </span><span style="font-family: inherit;">us nervous. Our hearts beat faster; this throws off our </span><span style="font-family: inherit;">body chemistry and can make us feel ill. We fidget, rock </span><span style="font-family: inherit;">back and forth, make awkward hand gestures, or stand </span><span style="font-family: inherit;">unnaturally still. Our body language, voice inflection, </span><span style="font-family: inherit;">and gestures reveal our discomfort.</span><br /><blockquote class="tr_bq"><i><span style="font-family: inherit;">There’s something fundamentally nerve-racking about </span><span style="font-family: inherit;">giving a presentation. The first day of Math Forum, we </span><span style="font-family: inherit;">all attempted to describe that near-indescribable feeling </span><span style="font-family: inherit;">of speaking in front of an audience . . . Here I was, in a </span><span style="font-family: inherit;">class that I had dreaded taking since hearing about it my </span><span style="font-family: inherit;">freshman year, thinking I was the only person that had </span><span style="font-family: inherit;">these feelings, . . . and yet the dozen other people in the </span><span style="font-family: inherit;">class shared this same feeling.</span></i></blockquote><br /><span style="font-family: inherit;">Practice is key to taming our nervousness and to making </span><span style="font-family: inherit;">a successful presentation.</span><br /><blockquote class="tr_bq"><i><span style="font-family: inherit;">With each talk I delivered throughout the semester, </span><span style="font-family: inherit;">my confidence only increased. For my first talk, I was a </span><span style="font-family: inherit;">nervous speaker that feared the whole ordeal, unable to </span><span style="font-family: inherit;">deliver my opinions with sincere confidence. In contrast, </span><span style="font-family: inherit;">for my last 10-minute talk, I was completely comfortable </span><span style="font-family: inherit;">and calm. I had even begun to enjoy interacting with the </span><span style="font-family: inherit;">audience during the presentation.</span></i></blockquote><br /><span style="font-family: inherit;">We can learn a lot from watching other speakers—</span><span style="font-family: inherit;">professors, renowned lecturers, and classmates. If possible, </span><span style="font-family: inherit;">watch yourself giving a presentation.</span><br /><blockquote class="tr_bq"><i><span style="font-family: inherit;">In re-watching the video of my second talk, . . . I noticed </span><span style="font-family: inherit;">I sometimes shifted my body weight from one side </span><span style="font-family: inherit;">to the other. . . . In my [later] talk, I felt at ease, and </span><span style="font-family: inherit;">this was evident in my posture.</span></i></blockquote><br /><span style="font-family: inherit;">Careful preparation is essential to a first-rate lecture. </span><span style="font-family: inherit;">Speaking tasks </span><span style="font-family: inherit;">often have a fixed, </span><span style="font-family: inherit;">typically short, </span><span style="font-family: inherit;">time allotment. In </span><span style="font-family: inherit;">a public-speaking </span><span style="font-family: inherit;">course, you learn to </span><span style="font-family: inherit;">deliver a message </span><span style="font-family: inherit;">within a given time </span><span style="font-family: inherit;">and to pare your </span><span style="font-family: inherit;">talk down to its essence, so that there is no wasted moment. </span><span style="font-family: inherit;">Although there are many ways to construct a successful </span><span style="font-family: inherit;">presentation, you’ll learn how to write a strong </span><span style="font-family: inherit;">introduction and conclusion, and how to connect them </span><span style="font-family: inherit;">with a logical flow of ideas.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Written mathematics can be expressed elegantly and </span><span style="font-family: inherit;">efficiently with words and symbols, but in a presentation, </span><span style="font-family: inherit;">written words and complicated mathematical notation </span><span style="font-family: inherit;">are difficult to follow. A lecture benefits from an </span><span style="font-family: inherit;">effective use of images, clever uses of color, and careful </span><span style="font-family: inherit;">placement of choice information for each slide. Generally: </span><span style="font-family: inherit;">Less is more.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">These are a small sampling of the many skills you will </span><span style="font-family: inherit;">learn in public-speaking class. If no such class is available, </span><span style="font-family: inherit;">consider other options, such as a theater class or a </span><span style="font-family: inherit;">Toastmasters club; research groups and math clubs can </span><span style="font-family: inherit;">also provide opportunities for you to give a presentation. </span><span style="font-family: inherit;">These experiences can help you deliver confident, </span><span style="font-family: inherit;">compelling communication about any topic, including </span><span style="font-family: inherit;">mathematics.</span><br /><blockquote class="tr_bq"><i><span style="font-family: inherit;">I got a glimpse at the sorts of strategies I’ll need to </span><span style="font-family: inherit;">give good talks: approaching a topic from the eyes of </span><span style="font-family: inherit;">someone who is unfamiliar with it, eschewing notation </span><span style="font-family: inherit;">unless it is particularly elucidating, leaving out ideas </span><span style="font-family: inherit;">that don’t support whatever central message I want to </span><span style="font-family: inherit;">present, and many more. . . . I am happy I was able </span><span style="font-family: inherit;">to share the ideas I find interesting with the rest of my </span><span style="font-family: inherit;">Forum class. It was unbelievably satisfying to finally give </span><span style="font-family: inherit;">a talk I could be proud of.</span></i></blockquote><br /><hr /><i><span style="font-family: inherit;">Rachel Levy is an associate professor of mathematics </span><span style="font-family: inherit;">at Harvey Mudd College and editor-in-chief of SIAM </span></i><i style="font-family: inherit;">Undergraduate Research Online (SIURO).</i><br /><span style="font-family: inherit;"><br /></span><span style="font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 13px; font-style: italic; line-height: 18px; text-indent: 19.200000762939453px; vertical-align: baseline; white-space: pre-wrap;">This article was published in the April 2014 issue of </span><span style="font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 13px; line-height: 18px; text-indent: 19.200000762939453px; vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.maa.org/mathhorizons/" style="color: #005bab; text-decoration: none;">Math Horizons</a>. </span>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7775513617406888811.post-91809008054006745292014-02-23T10:12:00.001-05:002014-02-23T10:12:46.357-05:00Steven Strogatz on Mathematics Education<div><br /></div><b>Patrick Honner:</b> What are your thoughts about the state of math education right now?<br /><br /><b>Steven Strogatz:</b> My thoughts are mostly based on my own instincts as a teacher and what I’ve seen of teachers I admire. I don’t know much about the constraints that practicing teachers face in high schools right now, so my opinions are fairly uninformed. But I do worry about math communication and teaching in general.<br /><br />Can I give you my “I have a dream” speech?<br /><br /><b>PH:</b> By all means!<br /><br /><div><b>SS:</b> In my dream world, everyone would have the chance to be a teacher the way Mr. Joffray [Strogatz’s high school calculus teacher and the subject of his book <i>The Calculus of Friendship</i>] was a teacher. His job was to teach us calculus, but he had his own vision of how to teach it and he followed that vision. He was creative, and he put his personal stamp on the course for us. He trusted his judgment, and the school trusted him. He could teach us the way he wanted to teach us, and he was a great teacher.<br /><br />This is a profession that should be revered. What’s more important than teaching? Why not let teachers teach creatively and inventively? So that’s my dream: a world in which teachers are given the freedom to teach the subject they’re supposed to teach, the way that makes sense to them.<br /><br /><b>PH:</b> You have two daughters in school right now. Do you think they are being exposed to math in a positive way?<br /><br /><b>SS:</b> No, I don’t. I worry that my kids are not falling in love with math because it’s being presented as lots of procedures that they need to learn.<br /><br />It’s too fast. My eighth-grade daughter is taking algebra, and one day she’s doing word problems, like “find three consecutive odd numbers that add up to 123,” and the next day she was doing something I’d never heard of—literal equations.<br /><br />It just struck me as unbelievable that we’re doing word problems in one night’s homework. Students should spend at least two to three weeks on word problems. They’re hard! Every old-fashioned word problem is being thrown at her in one night.<br /><br /><b>PH:</b> And then it’s off to literal equations the next day.<br /><br /><b>SS:</b> I can’t imagine what any kid is doing who doesn’t have a math professor as a parent. The whole thing looks crazy to me. I’m sure even my daughter’s teacher doesn’t want to do it this way. Something is really messed up.<br /><br /><b>PH:</b> Should math be a mandatory subject for kids?<br /><br /><b>SS:</b> I’m conflicted about it—I don’t know what to think. There are a lot of students out there who would love math but don’t know that. So they have to be exposed, or maybe even forced, to take math to realize they like it. But after a certain amount of that, it becomes clear to a student that they don’t want to take more math. We as a profession should think about this again.<br /><br /><b>PH:</b> What math do you think all people should know?<br /><br /><b>SS:</b> Some amount of number sense is essential—for example, to know what it means when the store says certain items are 20 percent off. If you don’t know what that means, to me, you’re not educated. I feel comfortable saying that every person should understand fractions. But after that, what? Does a person need to know what a polynomial is? That’s not clear to me.<br /><br />What should a person learn, if anything, after arithmetic? That seems like a pretty interesting pedagogical question, and I don’t believe our current curriculum is the optimal answer. Algebra I and II are good subjects, but so is network theory. It would be nice if people could understand how Google works, for example; it’s not that hard.<br /><br />There’s a lot of fun in math. Do we really have to teach such dead material? If we could get a cadre of<br />people who love math and who get it the way you get it or the way I get it—people who know what math is about—you don’t need to tell them how to teach. You just leave them alone, and it’ll be okay.<br /><br /><hr /><i><br />Patrick Honner is an award-winning math teacher at Brooklyn Technical High School. He writes about math and teaching at MrHonner.com and is active on Twitter as @MrHonner.</i><br /><br /><div style="text-indent: 0px;"><i><br /></i></div><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px; text-indent: 14.4pt;"><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">This article was published in the February 2014 issue of </span><span style="vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.maa.org/mathhorizons/">Math Horizons</a><i>, along with more of Patrick Honner's interview with mathematician and author Steven Strogatz. Yet more of the interview is available online as a supplement. </i></span></span></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-40649621745654525412013-11-01T01:00:00.000-04:002013-11-01T01:00:00.840-04:00Limits to GrowthPriscilla Bremser—<i>Middlebury College</i><br /><br /><div>For your next mathematical modeling project, download “AP Program Size and Increments” from <a href="http://research.collegeboard.org/programs/ap/data/participation/2013" target="_blank">collegeboard.org</a>. Using the number of Advanced Placement exams given annually from 1989 (463,644 exams) to 2013 (3,938,100), develop a model that describes the growth of the program. In your analysis, discuss possible adverse consequences of such growth. I can suggest one or two.<br /><br />The College Board tells students that AP courses will help them “stand out in college admissions.” Guidance counselors, along with college admissions officers, advise students to take the most challenging courses at their schools. Dutifully heeding this advice, high school students rush through the mathematics sequence to get to calculus, often taking as many as six other AP courses before they graduate.<br /><br />At the end of this frenzy, a number of bright, hardworking students have weak algebra skills, effectively neutralizing any advantage they might have earned. They may have placed out of Calculus I, but they are only marginally prepared for Calculus II.<br /><br />Over time the AP program has shifted from being a way to meet the needs of a few students who are ready for a challenge to a de facto admissions requirement for many who may not be. Having used their AP credit to get into Middlebury, a number of our students try to take calculus again, saying “I know I got a 5 on the exam, but I didn’t really understand it.” If placement into advanced college classes is truly the main objective, then something is amiss.<br /><br /><h3>Breadth over Depth</h3><div><br /></div>Mathematics majors have told me that they didn’t see an ε or δ until junior year of college. Their AP Calculus courses did not include the precise definition of a limit, upon which calculus stands. The College Board’s course description calls only for “an intuitive understanding of the limiting process,” followed by a list of topics so exhaustive that I’ve never seen a single college course cover them all. Apparently “rigor” and “challenge” lie in breadth, not depth.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-4P4Wd5N6T4U/UmEs2nWuP4I/AAAAAAAAIno/I-FxCQksywE/s1600/graph.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-4P4Wd5N6T4U/UmEs2nWuP4I/AAAAAAAAIno/I-FxCQksywE/s1600/graph.JPG" height="234" width="400" /></a></div><br />Students in high school AP programs who love mathematics may end up with a weak conceptual understanding of their favorite subject. Meanwhile, students better suited to a different math course feel compelled to take AP Calculus to enhance their transcripts. Once they all get to college, their math professors have some students who earned 5s on the exam as well as others who scored 3 or lower (58 percent of those who took the AB exam in 2013). At a conference I heard one mathematician say to another, “We’re trying to figure out how to deal with students who have taken the AP.” Join the club.<br /><br /><h3>Figure the Expenses</h3><div><br /></div>Why has this happened? At $89 per exam, some grumble that it’s all about money. Defenders of the program would point out that the College Board is “a not-for-profit membership organization.” Still, nonprofits exist to perpetuate themselves and seem to be taking a grow-or-die approach. The majority of students who took the Human Geography AP exam in 2013— 67,070 of them—were in ninth grade. Are that many 14-yearolds truly mature enough to take a college-level course?<br /><br />There’s no going back to the time when the AP program was simply a way for well-prepared students to get advanced placement. Indeed, at my own institution, the faculty voted down a proposal to do away with giving course credit for high AP scores, choosing instead to limit each student to five such credits. Meanwhile, the College Board advertises the program as a way to “save on college expenses.” College may be too expensive, but this purported remedy blithely disregards the significant differences between high school and college.<br /><br />For extra credit on the modeling assignment, use demographic data to estimate the carrying capacity of this system. What will growth rates look like in the coming years? At what costs?<br /><br /><hr /><i>Priscilla Bremser is a professor of mathematics at Middlebury </i><i>College. Her interests include number theory, </i><i>mathematics education at all levels, and appreciating the </i><i>Vermont landscape on foot, bicycle, and skis.</i><br /><i><br /></i><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px; text-indent: 14.4pt;"><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">This article was published in the November 2013 issue of </span><span style="vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.maa.org/mathhorizons/">Math Horizons</a>. </span></span></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-78584616640768268812013-09-12T11:53:00.001-04:002013-09-12T14:25:46.516-04:00i Can't Stand It AnymoreTravis Kowalski—<i>South Dakota School of Mines and Technology</i><br /> <br />Complex numbers are essential tools of mathematics, providing beautiful connections between arithmetic and geometry, algebra and trigonometry, number theory and analysis. Unfortunately, few people outside the cloister of trained mathematicians know this. I teach a course on complex analysis, and each time I am dismayed to find that, even after 15 weeks of demonstrating how the use of complex numbers fundamentally unifies most mathematical concepts learned in undergraduate studies, there is still a nontrivial subset of students who say, “That’s nice and all, Dr. K, but they aren’t real. They’re still imaginary numbers.”<br /><br />By its very definition the lamentable word “imaginary” describes something that does not exist or is utterly useless. Of course, these derogatory implications were just what Descartes had in mind when he coined the term “imaginary number” in 1637. Two centuries later, Gauss advocated the term “complex number,” but Euler’s introduction of the symbol <i>i</i> means that, no matter whatever else we may choose to call it, the adjective “imaginary” will always be associated with the root of –1. Students know what the letter <i>i</i> stands for. To them, it is a number that is imaginary and therefore irrelevant.<br /><br />It is a self–fulfilling—and sadly self–defeating—prophecy. And so, it is time to retire <i>i</i>.<br /><br />If the previous plea of “pejorative prejudice” is a bit of a stretch (or at least, needlessly alliterative), allow me to strengthen it with a bone fide mathematical argument for retiring Euler’s chosen notation. In a standard presentation, the complex number \(a+b\sqrt{-1}\)<i><span style="font-size: 19px; white-space: nowrap;"> </span></i>is identified with the point (<i>a</i>,<i>b</i>) in the plane. The way complex multiplication is defined, the effect on the plane of multiplying by \(a+b\sqrt{-1}\) is <i>exactly the same</i> as left multiplying each point (written as a column vector) by the real matrix<br /><table style="color: black; text-align: center;"><tbody><tr><td><table border="0" cellpadding="0" cellspacing="0px" style="border-left: 1px solid #000; border-right: 1px solid #000; color: black;"><tbody><tr><td style="border-bottom: 1px solid #000; border-top: 1px solid #000;"></td><td><table border="0" cellpadding="0" cellspacing="0" style="color: black;"><tbody><tr><td style="text-align: center;" valign="center" width="30">a</td><td style="text-align: center;" valign="center" width="30">-b</td></tr><tr><td style="text-align: center;" valign="center" width="30">b</td><td style="text-align: center;" valign="center" width="30">a</td></tr></tbody></table></td><td style="border-bottom: 1px solid #000; border-top: 1px solid #000;"></td></tr></tbody></table></td></tr></tbody></table><!-- matrix expression end -->and so multiplying by the complex unit \(0+1\sqrt{-1}\)<span style="font-size: larger; white-space: nowrap;"> </span>is the 90° rotation<br /><table style="color: black;"><tbody><tr><td><table border="0" cellpadding="0" cellspacing="0px" style="border-left: 1px solid #000; border-right: 1px solid #000; color: black;"><tbody><tr><td style="border-bottom: 1px solid #000; border-top: 1px solid #000;"></td><td><table border="0" cellpadding="0" cellspacing="0" style="color: black;"><tbody><tr><td align="center" valign="center" width="30">0</td><td align="center" valign="center" width="30">-1</td></tr><tr><td align="center" valign="center" width="30">1</td><td align="center" valign="center" width="30">0</td></tr></tbody></table></td><td style="border-bottom: 1px solid #000; border-top: 1px solid #000;"></td></tr></tbody></table></td></tr></tbody></table><br /><!-- matrix expression end --> Whatever one wishes to label this matrix, the letter <i>I</i> is off limits because <i>I</i> <i>always</i> refers to the multiplicative “identity” matrix. This suggests that <i>I</i> should not be used for the complex unit. In keeping with a consistent lettering scheme, <i>I</i> should represent the complex number whose multiplication coincides with that of <i>I</i>, but that’s just the multiplicative identity—that is, <i>I</i> <i>should</i> denote the real number 1.<br /><br />What would be a better symbol? Why not just dust off the old $\sqrt{-1}$ notation and use that? Unfortunately, this is a choice fraught with peril. Following the traditional algebraic “rules of radicals,” we end up with paradoxes like \[-1=\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1\] A better proposal comes from multivariable calculus. In the notation of ordered pairs, complex multiplication takes the form (<i>a</i>,<i>b</i>)(<i>c</i>,<i>d</i>) = (<i>ac</i> –<i> bd</i>, <i>ad</i>+<i>bc</i>). Using this and vector algebra allows us to write any point in the plane as<br /><br />(<i>a</i>,<i>b</i>) = (<i>a</i>,0)(1,0)+(<i>b</i>,0)(0,1)<br />=<i>a</i>(1,0)+<i>b</i>(0,1)<br />=<i>a</i><b>i</b><i>+b</i><b>j</b><br /><b><br /></b>where <b>i</b>=(1,0) and <b>j</b>=(0,1) are the standard basis vectors for the plane. Not only does this provide one more piece of evidence to support the claim that the symbol <i>i</i> <i>ought</i> to refer to 1, it also means (<i>a</i>,<i>b</i>)=<i>a</i>+<i>b</i><b>j</b>. This looks exactly like the standard form of a complex number with the vector <b>j</b> standing in for \(\sqrt{-1}\). Even more compelling, note that<br /><br /><div style="text-align: center;"><b>j</b><sup>2 </sup>= <b>jj </b>= (0,1)(0,1) = (-1,0) = -1,</div><br />so <b>j</b> is indeed a square root of –1.<br /><br />Consequently, we should denote the complex unit by <b>j</b> or, if we want to emphasize its role as a complex <i>number</i> rather than a plane <i>vector</i>, the italicized letter <i>j</i>. In fact, electrical engineers already use <i>exactly this same letter j</i>, although their prime motivation is that <i>i</i> is already reserved for <i>current</i>.<br /><br />Our main motivation is that the letter <i>j</i> doesn’t stand for anything in particular, and it most certainly doesn’t stand for “imaginary.” The symbol <i>j </i>simply denotes a <i>complex unit</i>, a number that multiplies against itself to yield –1. It is a blank canvas on which to paint the utility of the complex number system, effectively banishing the confusion and distrust of that other letter, which shall no longer be named.<br /><br /><hr /><i>Travis Kowalski teaches mathematics at the South Dakota School of Mines and Technology.</i><br /><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px; text-indent: 14.4pt;"><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;"><br /></span></span></span><span style="font-family: inherit;"><span style="background-color: white; line-height: 18px; text-indent: 14.4pt;"><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">This article was published in the September 2013 issue of </span><span style="vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.maa.org/mathhorizons/" style="text-decoration: initial;">Math Horizons</a>. </span></span></span>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-73982249633482439012013-04-01T04:00:00.000-04:002013-04-04T10:18:47.186-04:00Mathematical Habits of MindKaren King<span style="background-color: white; line-height: 17.765625px;">—</span><i>National Science Foundation </i><br /><br /><br />When I look back on my own mathematical education, I have many people to thank for helping me develop productive mathematical habits of mind. I remember walking to the car with my dad on a bitter cold day on the way home from kindergarten, and I just had to understand how you could do subtraction with regrouping. Instead of brushing off my pesky questioning (and I was pesky), he explained it to me, writing in the frost on the car window to illustrate the ideas. Some years later, Linda Agreen, my Advanced Placement calculus teacher, made sure that I understood why the fundamental theorem of calculus was fundamental, even though that was not going to be on the AP test. These habits of seeking real understanding were solidified in the mathematics department at Spelman College, under Etta Z. Falconer and her colleagues.<br /><br />Building on the foundation laid by my father and my other mathematics teachers, I learned the mathematical habit of doggedly pursuing a complete understanding of ideas. I also learned how to recognize when my understanding was not complete and the reasoning skills to address the situation.<br /><br />Unfortunately, too many students of mathematics, whether in college algebra or abstract algebra, do not possess these productive mathematical habits of mind. Instead, they have picked up some bad habits along the way: a tendency to look for the quick answer, a lack of persistence when the answer is not obvious, memorization over understanding.<br /><br />Why do I keep referring to reasoning skills as “mathematical habits of mind”? Because I believe that if we start thinking about these unproductive practices as habits of mind, it opens up a different set of strategies for addressing the problem. When Al Cuoco, Paul Goldenberg, and June Mark introduced the concept of mathematical habits of mind (<i>The Journal of Mathematical Behavior</i> 15, no. 4 [1996]), it was a powerful concept for rethinking K-12 students’ learning of mathematics.<br /><br />Habits are behaviors we engage in unconsciously, but they are the result of a long evolution of choices we make at a young age. Habits of mind evolve from the choices that we make about how to think about ideas. Thus, my dad’s early intervention was important. At 5 years old, I was still making choices about how to learn. So were my teachers—in elementary school, high school, and beyond.<br /><br />But too few students develop the habits of mind needed for more advanced mathematical learning. Presented with a problem with no obvious example to follow, a poorly trained student might start writing things down or try some calculations with no real strategy in mind. Faced with the task of learning to write proofs, a person without sound mathematical habits usually attempts to memorize various arguments instead of re-creating them from their internal logic. These habits may have served them well previously, but no longer.<br /><br />Habits reflect what a person is likely to do in a given situation, especially a stressful one such as taking a test, and habits are notoriously hard to break. Smokers know that continuing to smoke has a high likelihood of leading to cancer and other diseases, but that knowledge alone is rarely sufficient for those who are trying to quit.<br /><br />With this in mind, we need to ask whether the way mathematics is currently taught reinforces bad habits of mind. Is it too easy to get by for too long using bad mathematical habits? And where did these bad habits come from in the first place? The likely answer is that there are some entrenched teaching habits in need of attention.<br /><br />Thinking in terms of habitual behaviors conjures up powerful analogies. How might we change our approach to learning—and teaching—math if we labeled as “unproductive habits of mind” those methods that serve us poorly? Just like the person who finally replaces smoking with a healthier habit—or better yet, who never starts in the first place—we will all be better served with healthier mathematical habits of mind.<br /><br /><br /><div><hr /><i>Karen King is the former director of </i><i>research for the National Council of </i><i>Teachers of Mathematics. She has </i><i>been a member of the mathematics </i><i>education faculty at New York University, </i><i>Michigan State University, </i><i>and San Diego State University.</i><span style="font-family: inherit;"><i> </i></span><br /><span style="font-family: inherit;"><b style="background-color: white; font-weight: normal; line-height: 18px; text-indent: 14.4pt;"><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;"><br /></span></b></span><span style="font-family: inherit;"><b style="background-color: white; font-weight: normal; line-height: 18px; text-indent: 14.4pt;"><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">This article was published in the April 2013 issue of </span><span style="vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.maa.org/mathhorizons/" style="text-decoration: initial;">Math Horizons</a>. </span></b></span></div><div><span style="font-family: inherit;"><b style="background-color: white; font-weight: normal; line-height: 18px; text-indent: 14.4pt;"><span style="vertical-align: baseline; white-space: pre-wrap;"><br /></span></b></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com4tag:blogger.com,1999:blog-7775513617406888811.post-65689924743880167922013-02-01T09:00:00.000-05:002013-02-11T17:44:41.222-05:00What to Expect When You’re Electing<span style="font-family: inherit;">Stephen Abbott<span style="background-color: white; line-height: 17.765625px;">—</span><i>Middlebury College </i></span><br /><span style="font-family: inherit;"><i><br /></i></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://imgs.xkcd.com/comics/math.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="color: black; font-family: inherit;"><img border="0" height="140" src="http://imgs.xkcd.com/comics/math.png" width="400" /></span></a></div><span style="font-family: inherit;"><i><br /></i> When the national election finally came to a merciful end in November, there was one universally recognized winner whose name did not appear on any ballot. In a stunning denouement, political blogger Nate Silver may have permanently altered the way elections are reported—and run for that matter—and he did so by staking his claim to the veracity of Bayesian statistics.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Like everything else in an election year, Silver’s story is nearly impossible to separate from its heated political overtones, but in this case it is worth a try. Not only was mathematics well served, but its objectivity emerged as a potential means for making headway into the political storms that lie ahead.</span><br /><span style="font-family: inherit;"><br />Nate Silver’s first statistical love was analyzing baseball, which he did successfully for a sports media company after college, but in the run-up to the 2008 presidential election Silver began applying his mathematical tools to political forecasting. In March of that year he started a blog called <i>FiveThirtyEight</i> and made a name for himself by correctly predicting the outcome of every state except for Indiana in the Obama-McCain race. With its star on the rise, <i>FiveThirtyEight</i> was picked up by <i>The New York Times</i>, just before the 2010 midterm elections. In anticipation of 2012, the <i>Times </i>signed Silver to a multiyear contract.</span><br /><span style="font-family: inherit;"><br />And this is where the plot thickens. In addition to being a first-rate statistician, Silver is also a self-professed progressive with ties to the Obama campaign. Thus, when Silver’s blog showed Obama with a comfortable polling edge going into the final weeks of the election, attacks from conservative pundits began to fly. Denigrating the messenger is standard procedure in elections, but Silver’s methods—i.e., his mathematics—also became fair game. An <i>L.A. Times</i> editorial characterized the <i>FiveThirtyEight </i>model as a “numbers racket.”</span><br /><span style="font-family: inherit;"><br />Referring to Silver, MSNBC’s Joe Scarborough proclaimed that “anybody that thinks that this race is anything but a toss-up right now is such an ideologue [that] they should be kept away from typewriters, computers, laptops, and microphones for the next ten days, because they’re jokes.”</span><br /><span style="font-family: inherit;"><br />Silver’s series of responses make for some pedagogically compelling reading. “There were twenty-two poles of swing states published Friday,” he wrote in a November 2, 2012, post. “Of these, Mr. Obama led in nineteen polls, and two showed a tie. Mitt Romney led in just one . . . a ‘toss-up’ race isn’t likely to produce [these results] any more than a fair coin is likely to come up heads nineteen times and tails just once in twenty tosses. Instead, Mr. Romney will have to hope that the coin isn’t fair.” Silver then goes on to give a razor-sharp explanation of the difference between statistical bias and sampling error and how one accounts for each in assessing uncertainty.</span><br /><span style="font-family: inherit;"><br />The <i>FiveThirtyEight </i>author’s mathematical rejoinders only agitated his antagonists, who vowed to make him a “one-term political blogger.” But on Election Day Silver’s model was correct for all 49 state results that were announced that evening. And what about Florida, which was too close to call for several days? Silver had rated it a virtual tie.</span><br /><span style="font-family: inherit;"><br />Predictably, this “victory for arithmetic” was quickly employed as weaponry in the red versus blue debate. This is as unfortunate as it is counterproductive, and here is why. If we can agree on anything in today’s political climate, it is the need for a more productive means of public discourse. If we ignore Silver’s political orientation for a moment, what we have is an illustration of how mathematics, in the proper hands, can provide an objective foothold when the partisan winds start to blow.</span><br /><span style="font-family: inherit;"><br />What could mathematics, and a mathematical approach that prioritized proof over punditry, contribute to our ongoing debates about climate change? The national debt? The relationship of gun laws to violent crime? What are the chances that some disciplined mathematical analysis might provide an objective first step in bridging at least some of our philosophical differences?</span><br /><span style="font-family: inherit;"><br />I’d rate it a toss-up. </span><br /><div><hr /><span style="font-family: inherit;"><i>Stephen Abbott is a professor of mathematics at Middlebury College and coeditor of </i>Math Horizons<i>. </i><b style="background-color: white; font-weight: normal; line-height: 18px; text-indent: 14.4pt;"><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">This article was published in the February 2013 issue of </span><span style="vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.maa.org/mathhorizons/" style="text-decoration: initial;">Math Horizons</a>. </span></b></span><br /><b style="background-color: white; font-weight: normal; line-height: 18px; text-indent: 14.4pt;"><span style="font-family: inherit; vertical-align: baseline; white-space: pre-wrap;"><br /></span></b><span style="font-family: inherit;"><span style="background-color: white; line-height: 17.98611068725586px;">Image by Randall Munroe </span><span style="background-color: white; color: #333333; line-height: 17.98611068725586px;">(</span><a href="http://xkcd.com/1131/" rel="nofollow nofollow" style="background-color: white; color: #3b5998; cursor: pointer; line-height: 17.98611068725586px; text-decoration: initial;" target="_blank">http://xkcd.com/1131/</a><span style="background-color: white; color: #333333; line-height: 17.98611068725586px;">)</span></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-83461716126565671502012-11-01T07:00:00.000-04:002012-11-02T15:21:48.500-04:00Necessary Algebra<span style="font-family: inherit;"><span id="internal-source-marker_0.9733413818757981"><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">Paul Zorn</span></span><span style="background-color: white; color: #222222; line-height: 17.77777862548828px;">—</span><b id="internal-source-marker_0.9733413818757981" style="font-weight: normal;"><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"><i>Saint Olaf College</i></span></b></span><br /><span style="color: #221e1f; font-weight: bold; vertical-align: baseline; white-space: pre-wrap;"><span style="font-family: inherit;"><br /></span></span><b style="font-weight: normal;"><span style="font-family: inherit;"><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">I remember vividly the moment—and the room decor, the time of night, and the LP on the stereo—when my cousin Jon taught me algebra.</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">He and I, then seventh-graders, enjoyed those hoary old story problems (Al is twice as old as Betty; in seven years . . .) that once appeared in magazines such as </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Life </span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">and </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Look</span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">. I had concocted a simple strategy that one might charitably call iterative: Make any old integer guesses and tweak them as the errors suggest. What Jon first saw, and memorably pointed out, was that an </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">unknown</span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">, say </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">A</span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">, for Al’s age, can be manipulated as though it were a </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">known </span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">quantity like one of my guesses. </span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">What thrilled me then was the prospect of zipping through an entire genre of contrived puzzles. What amazes me still is the power of one simple idea: You can manipulate unknowns and knowns to solve equations.</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">That prescription seems a decent nine-word summary of what algebra does, even beyond the seventh grade. Jon and I got a preview, however dim, of an idea bigger and better than we could have suspected. Every student should encounter, and eventually own, an idea so simple and powerful. I’m convinced that almost every student has a fighting chance.</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span></span></b><br /><h3><b style="font-weight: normal;"><span style="font-family: inherit;"><span style="color: #598fb0; font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">Is algebra necessary? </span></span></b></h3><br /><b style="font-weight: normal;"><span style="font-family: inherit;"><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">So asked a provocative </span><a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?pagewanted=all&_r=0"><span style="color: #1155cc; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">New York Times </span><span style="color: #1155cc; vertical-align: baseline; white-space: pre-wrap;">op-ed</span></a><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"> last July. In fact, the title is slightly misleading. Author Andrew Hacker, professor emeritus of political science at Queens College, doesn’t question algebra’s larger importance. He notes cheerfully that “mathematics, both pure and applied, is integral [Hacker’s good word] to our civilization, whether the realm is aesthetic or electronic.”</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">Hacker’s different but equally provocative question is how much “algebra,” that “onerous stumbling block for all kinds of students,” should be required in high school and college. His answer: Much less. And less of other mathematics, too.</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">Here “algebra” is in quotes because Hacker’s beef is not really with that subject in particular. Indeed, Hacker sees both “algebra” and existing curricula idiosyncratically. His examples of supposedly superfluous material—“vectorial angles” and “discontinuous functions”—are unlikely examples of “algebra” and even less representative of what is typically taught. And Hacker’s </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">en passant </span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">endorsement of teaching long division (right up there with reading and writing) surprised me. He doesn’t acknowledge, or seem aware of, creative efforts to improve school teaching of “algebra” by teachers like those supported and mentored by, say, Math for America.</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">Hacker’s real curricular concern is broader than algebra: It’s the curricular complement of quantitative literacy (QL). He refers generally to “the toll </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">mathematics </span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">takes” (my emphasis), not just to difficulties posed by algebra. In this sense Hacker’s three Rs proposal—require QR, but not “mathematics”—is more radical, and Philistine, than the article’s title suggests. But let’s concentrate on algebra.</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span></span></b><br /><h3><b style="font-weight: normal;"><span style="font-family: inherit;"><span style="color: #598fb0; font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">Where he’s right, and wrong. </span></span></b></h3><div><b style="font-weight: normal;"><span style="font-family: inherit;"><span style="color: #598fb0; font-weight: bold; vertical-align: baseline; white-space: pre-wrap;"><br /></span></span></b></div><b style="font-weight: normal;"><span style="font-family: inherit;"><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">Some of Hacker’s rhetorical targets are legitimate. Algebra can indeed be taught rigidly and applied ineffectively. (I remember the joy of solving algebra puzzles but also tedious hours of FOIL-ing quadratics.) Hustling high school students toward calculus sometimes pushes them too rapidly for effective mastery through prerequisite courses—including algebra. And Hacker, keen to avoid “dumbing down,” suggests some interesting applications of QL methods to such topics as the Affordable Care Act, cost/benefit analysis of environmental regulation, and climate change. (Whether such topics can really be approached without algebra is another question.)</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">As Hacker observes, few workers use algebra explicitly in daily life. (We </span><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">all </span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">use it implicitly.) To infer that algebra can therefore vanish from required curricula is mistaken. Similar arguments might be made against history, the humanities, and the sciences generally, none of which is widely practiced in daily life. More important in curricular design than eventual daily use are broader intellectual values, which algebra clearly serves: learning to learn, detecting and exploiting structure, exposure to the best human ideas, and—the educational Holy Grail—transferability to novel contexts.</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">Transferability is undeniably difficult, as Hacker duly notes. The National Research Council agrees (see </span><a href="http://www7.national-academies.org/BOTA/Education_for_Life_and_Work_report_brief.pdf">Education for Life and Work Report (pdf)</a>)<span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;"> </span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">and indeed stresses the value of “deeper learning,” of which a key element is the detection of structure. </span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">“Transfer is supported,” says the NRC, when learners master general principles that underlie techniques and operations. </span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;">Algebra is a poster child for deeper instruction. We should teach it. Students can learn it. </span></span></b><br /><b style="font-weight: normal; text-indent: 14.4pt;"><span style="font-family: inherit;"><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;"><br /></span></span></b><br /><hr /><b style="font-weight: normal; text-indent: 14.4pt;"><span style="font-family: inherit;"><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Paul Zorn is a professor of mathematics at Saint Olaf College and currently serving as president of the Mathematical Association of America.</span></span></b><br /><b style="font-weight: normal; text-indent: 14.4pt;"><span style="font-family: inherit;"><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;"><br /></span></span></b><b style="font-weight: normal; text-indent: 14.4pt;"><span style="font-family: inherit;"><span style="color: #221e1f; font-style: italic; vertical-align: baseline; white-space: pre-wrap;">This article was published in the November 2012 issue of </span><span style="color: #221e1f; vertical-align: baseline; white-space: pre-wrap;"><a href="http://www.maa.org/mathhorizons/">Math Horizons</a>. </span></span></b><br /><div dir="ltr" style="margin-bottom: 0pt; margin-top: 0pt; text-indent: 14.4pt;"><b style="font-weight: normal;"><span style="font-family: Calibri; font-size: 16px; vertical-align: baseline; white-space: pre-wrap;"></span></b></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7775513617406888811.post-21433513159261506752012-09-01T08:00:00.000-04:002012-08-31T11:21:43.073-04:00Measuring Women’s Progress in Mathematics <span class="s1"><span style="font-family: inherit;">Linda Becerra and Ron Barnes</span></span><span style="background-color: white; color: #666666; font-family: 'Trebuchet MS', Trebuchet, Verdana, sans-serif; font-size: 13px; line-height: 18px;">—</span><i>University of Houston</i>–<i>Downtown</i><br /><span class="s2"><span style="font-family: inherit;"><br /></span></span><span style="font-family: inherit;"><span class="s2">M</span>any believe that residual effects of past hindrances and discrimination against </span><span style="font-family: inherit;">women in mathematics are being overcome. Studies by the American Mathematical Society and National Science Foundation on women in mathematics appear to reinforce this belief. Conventional wisdom suggests it is only a matter of time before women achieve parity. Julia Robinson (instrumental in the solution of Hilbert’s tenth problem) suggested that one measure of parity would be when male mathematicians no longer consider female mathematicians to be unusual. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Unfortunately, a close reading of AMS and NSF data suggests that significant progress is not being made. One can be deceived by looking only at raw numbers without considering the related percentages.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Among entering students at U.S. institutions, data for the years 2000 to 2008 indicate the number of female and male freshmen expressing interest in a major in mathematics went from 44,500 and 49,500 in 2000, to 66,000 and 66,600 in 2008. These figures indicate the gap between female and male interest in majoring in math narrowed from 5,000 in the year 2000 to 600 in 2008. However, among all undergraduates, the percentage of females and males interested in a math major went from 0.6 percent (female) and 0.8 percent (male) in 2000, to 0.7 percent and 1 percent, respectively, in 2008. Hence, the percentage gap between the sexes </span><i style="font-family: inherit;">increased </i><span style="font-family: inherit;">during this time. This is because there was considerably higher growth in the overall female undergraduate population during this period.</span><br /><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">Total graduate enrollment in the mathematical sciences increased from about 9,600 in 2000 to 22,200 in 2009 (131 percent), while female graduate enrollment increased from 3,670 to 7,979 (117 percent). However, the percentage of female graduate enrollment in the mathematical sciences remained relatively static in the decade—38 percent in 2000 and 36 percent in 2009.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">AMS mathematics data noted that the number of Ph.D.s awarded to U.S. citizens in the mathematical sciences increased from 494 in 2000 to 669 in 2008, and the number of Ph.D.s awarded to women grew from 148 to 200. However, the percentage of Ph.D.s earned by women in 2000 and 2008 were both approximately 30 percent, with some variation in the intervening years. Also, the percentage of bachelor’s degrees awarded to females during this time varied little from 41 percent.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">The NSF used a different data set, and the conclusions are even less encouraging. It indicates that women’s percentage of bachelor’s degrees in mathematics from 2002 to 2009 steadily decreased from 48 percent to 43 percent.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">In mathematics, the number of doctoral full-time tenure/tenure-track (T/TT) positions held by women at U.S. institutions increased from about 2,850 in 2001 to 4,000 in 2009 (a 40 percent increase). However, the percentage of T/TT positions held by females increased only from 18 percent to 23 percent during this time. This smaller difference is explained by the fact that significantly more males also obtained T/TT positions in this period.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">There are reasons to believe that women’s progress in mathematics should be much better by now. Since 1982, considering all fields, women have annually earned more bachelor’s degrees than men. By 2011, more women than men had earned advanced degrees. Yet, the statistics cited show that in mathematics, women’s participation at advanced levels is still unusually low and either improving slowly or, in some cases, making no progress whatsoever.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">The real question is: How can meaningful progress be effected? Evidently the present strategies are not working.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">A few ideas for consideration:</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">• Engage in a rigorous, sustained intervention with girls throughout school-level mathematics and in universities—not a few small programs, but a broad, concentrated, and sustained effort to integrate girls into mathematics, its culture, and its relevance. This effort must involve all the professional mathematical societies.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">• Reengineer the culture in the mathematics professoriate with an eye toward more flexibility in the tenure and promotion process. The standards need not be watered down in any way, but the process should allow for a variety of pathways to meet them.</span></div><div class="p3"><span style="font-family: inherit;"><br /></span></div><div class="p3"><span style="font-family: inherit;">As Julia Robinson observed, “If we don’t change anything, then nothing will change.”</span><br /><span style="font-family: inherit;"><br /></span></div><div class="p3"><hr /><span style="font-family: inherit;"><i>Linda Becerra and Ron Barnes are professors of </i><i>mathematics at the University of Houston</i>–<i>Downtown.</i></span></div><div class="p3"><br /></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7775513617406888811.post-68456786749568330702012-04-02T08:00:00.000-04:002012-04-05T15:31:08.033-04:00My Conversion to Tauism<span style="font-family: inherit;">Stephen Abbott—<i>Middlebury College, </i>Math Horizons<i> Co-Editor</i></span><br /><span style="font-family: inherit;">There was no identifiable moment when I said, yes, I believe. My conversion must have come on silently and unexpectedly. I do, however, remember the moment when I realized something had inalterably changed...</span><br /><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;">Read the full article <a href="http://www.maa.org/Mathhorizons/apr12_aftermath.pdf">PDF</a></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: inherit;"><br /></span></div><div style="text-align: center;"><div style="text-align: left;"></div><hr style="background-color: white; color: #666666; line-height: 18px; text-align: -webkit-auto;" /><div style="text-align: left;"><span style="font-family: inherit;"><i style="background-color: white; color: #666666; line-height: 18px; text-align: -webkit-auto;">About the Author: </i><span style="color: #666666;"><span style="line-height: 18px;"><i>Stephen Abbott is a professor </i></span></span><i style="color: #666666; line-height: 18px; text-align: -webkit-auto;">of mathematics at Middlebury </i><i style="color: #666666; line-height: 18px; text-align: -webkit-auto;">College and currently co-editor of </i></span><span style="color: #666666; font-family: inherit;"><span style="line-height: 18px;">Math Horizons<i>. </i></span></span><i style="background-color: white; color: #666666; font-family: inherit; line-height: 18px; text-align: -webkit-auto;">Email: <a href="mailto:abbott@middlebury.edu" style="color: #005bab; text-decoration: none;">abbott@middlebury.edu</a></i><span style="background-color: white; color: #666666; font-family: inherit; line-height: 18px; text-align: -webkit-auto;"> </span></div><div style="background-color: white; color: #666666; line-height: 18px; text-align: -webkit-auto;"><i><span style="font-family: inherit;"><br /></span></i></div><div style="background-color: white; color: #666666; line-height: 18px; text-align: -webkit-auto;"><i><span style="font-family: inherit;">Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA.</span></i></div></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com2tag:blogger.com,1999:blog-7775513617406888811.post-64399319135314380392012-02-01T09:00:00.000-05:002012-02-23T17:12:44.735-05:00Unduly Noted<span style="font-family: inherit;">Tommy Ratliff—<i>Wheaton College </i></span><br /><span style="font-family: inherit;">When I opened the MathFest program in Lexington last summer, I took one look at the first page and nearly yelled out loud “NO! NO! NO!” The inside cover to the program contained an advertisement for an online homework system with the following example:</span><br /><span style="font-family: inherit;"><br /></span><br /><span style="font-family: inherit;">Find the derivative of<em> y</em> = 2 cos(3<em>x </em>− π) with respect to <em>x</em>. </span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-uvxJ9wQicrY/Tyr-1HMpMEI/AAAAAAAAECI/j-1ZrSLbu3c/s1600/1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-uvxJ9wQicrY/Tyr-1HMpMEI/AAAAAAAAECI/j-1ZrSLbu3c/s1600/1.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: inherit;">The assertion is that a competing system marked this answer as wrong, but the advertised system identified the expression as correct, demonstrating its superiority. I assume that the intent is to show that the system can recognize equivalent, but not identical, algebraic expressions. What caused me to react so strongly, however, is that I would have also marked the given answer as wrong. The answer should have been</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-L75-uK3OkP0/Tyr-45gwPXI/AAAAAAAAECQ/Dvd8o-gB4Vo/s1600/2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-L75-uK3OkP0/Tyr-45gwPXI/AAAAAAAAECQ/Dvd8o-gB4Vo/s1600/2.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: inherit;">The parentheses matter! The expression sin(<i>x</i>) represents a <em>function</em>!</span><br /><span style="font-family: inherit;"><br /></span><br /><span style="font-family: inherit;">I should be clear: My irritation is not directed at this particular homework system as much as at the entire mathematics community for the sloppiness in notation that we tolerate, and even encourage, when dealing with trigonometric functions. You can pick up almost any calculus text, peek into almost any math classroom, or attend any number of talks at various MAA events to find a plethora of examples of trig functions lacking their parentheses.</span><br /><span style="font-family: inherit;"><br /></span><br /><span style="font-family: inherit;">Why do I think the parentheses matter so much? This is not just a pedantic preference on my part. The lack of parentheses represents an irregularity in notation that obscures the meaning of the mathematics. We often use a space to indicate multiplication, as in </span><a href="http://4.bp.blogspot.com/-E2OduLTB9JA/Tyr-8tpsfXI/AAAAAAAAECY/wFAm6Ms0qfw/s1600/7.gif" imageanchor="1" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-E2OduLTB9JA/Tyr-8tpsfXI/AAAAAAAAECY/wFAm6Ms0qfw/s1600/7.gif" /></a><span style="font-family: inherit;">or 3 sin(</span><em style="font-family: inherit;">x</em><span style="font-family: inherit;">), so leaving off the parentheses hides the fact that we are using a trigonometric function. The confusion is compounded when we say that the derivative of “sine” is “cosine.” If we were to be consistent, this would lead to applying a distorted product rule to get</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-KNZOGikYSq8/Tyr_IhQS9KI/AAAAAAAAECg/KTji-Jh_a0Y/s1600/3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-KNZOGikYSq8/Tyr_IhQS9KI/AAAAAAAAECg/KTji-Jh_a0Y/s1600/3.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: inherit;">I have seen students struggle with this, even when they understand the intent of the original notation. They correctly apply the chain rule only to confuse the order of operations at the end because they did not put the constant multiple of 3 at the beginning of the expression:</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-e-1R7tlsCKQ/Tyr_P2wuf-I/AAAAAAAAECo/kc4E46RCyjU/s1600/4.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-e-1R7tlsCKQ/Tyr_P2wuf-I/AAAAAAAAECo/kc4E46RCyjU/s1600/4.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: inherit;">After all, why should you apply the cosine function to the first 3 in the 3<em>x</em> but not to the trailing 3? If we always used parentheses to enclose a function’s argument, then there would be no confusion.</span><br /><span style="font-family: inherit;"><br /></span><br /><span style="font-family: inherit;">An even worse abuse of notation occurs in the location of the exponent when a trig function is raised to a power. I will never write sin<sup>2</sup>(<em>x</em>) for sin(<em>x</em>)<sup>2</sup> because the first notation leads to ambiguity when discussing the inverse trig functions. Since <em>f</em><sup>− 1</sup> (<em>x</em>) is the standard, consistent notation for the inverse function of <em>f</em>(<em>x</em>) , we also use sin<sup>− 1</sup>(<em>x</em>) for arcsin(<em>x</em>). If we were consistent with notation, a perfectly reasonable calculation would be</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-2P2lJxRsMwM/Tyr_Tyu88sI/AAAAAAAAECw/h5ae5eh-yGg/s1600/5.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-2P2lJxRsMwM/Tyr_Tyu88sI/AAAAAAAAECw/h5ae5eh-yGg/s1600/5.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: inherit;">Notice that I had to make a choice about the meaning of sin<sup>−2</sup> (<em>x</em>) in simplifying the expression—an impossible task! Should it be sin<sup>2</sup>(<em>x</em>)<sup>-1 </sup>= 1/sin<sup>2</sup>(<em>x</em>) or sin <sup>-1</sup>(<em>x</em>)<sup>2</sup> = arcsin (<em>x</em>)<sup>2</sup>? The point is that we should never have to make this choice! We <em>should</em> be taking the derivative of the inverse sine function. This is horrific. The bad notation allows at least three different interpretations of the expression</span><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-pQpx_RdzTl4/Tyr_XB3qahI/AAAAAAAAEC4/qxPcQAyGgJY/s1600/6.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-pQpx_RdzTl4/Tyr_XB3qahI/AAAAAAAAEC4/qxPcQAyGgJY/s1600/6.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: inherit;">We in the mathematics community pride ourselves on the consistency and deterministic nature of our discipline. I think we do ourselves a genuine disservice when we use sloppy notation that requires another layer of interpretation to understand the intended meaning. The purpose of mathematical notation is to provide clarity and, ideally, to provide insight into the mathematics being notated.</span><br /><span style="font-family: inherit;"><br /></span><br /><span style="font-family: inherit;">Therefore, I implore you: The next time you use a trig function, please remember the parentheses, put the exponent on the outside, and never, ever write anything like sin3<em>x</em><sup>2 </sup>cos<sup>-2</sup>5<em>x</em>. </span><br /><span style="font-family: inherit;"><br /></span><br /><hr /><span style="font-family: inherit;">Tommy Ratliff is a professor of mathematics at Wheaton College in Massachusetts where he enjoys thinking about voting theory, building new science centers, and being precise in his notation. </span><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com4tag:blogger.com,1999:blog-7775513617406888811.post-24785300727315767002011-11-01T08:00:00.005-04:002011-11-15T11:43:39.359-05:00Statistics à la Mode<div style="text-align: left;"><span class="Apple-style-span" style="color: rgb(51, 51, 51); background-color: rgb(255, 255, 255); ">Meg Dillon—<i>Southern Polytechnic State University</i></span></div><p>The last time I taught introductory probability and statistics, I turned in my grades and asked my department chair to take me off the course permanently. I’d spent some time working on a committee to update the course and we’d modernized it roughly to my taste, so my chair was puzzled. The best I could offer by way of explanation was, “I just hate it.” Then I went to France and taught their version of the same course.</p><p> My stint in France lasted three weeks. Essentially, I was substitute teaching and not looking for more than an excuse to be in the country for a while. My students were second-year engineering students, pretty much like my students at home. And like my students, the French students were a few notches below elite. While the similarities between my home university and my French university were comforting, the contrasts in the probability/statistics courses could not have been more jarring.</p><p> Anyone who has taught or learned in a U.S. mathematics department recently knows the typical introductory probability and statistics course. It involves an expensive, gassy textbook with lots of color pictures, word problems involving industrial applications, and charts to help students navigate problems. American students purchase the textbook and far too often, the ancillaries the bookstore peddles alongside the text.</p><p> At my home university, the chair has some difficulty finding mathematics faculty willing to teach the course. While I can’t speak for my colleagues, to me the course seems oddly estranged from mathematics. There is a section on probability, and we love that: the probability laws, the counting. It’s possible to trick out that section and get a chewier piece of mathematics into the act, but, by and large, the course is a hodgepodge of recipes, motivated by problems involving IQ testing, rhesus monkeys, salamanders, and the like. Regardless of the text, there is almost invariably a peculiar pair of caveats presented as from on high: Never accept the alternative hypothesis, and never say the probability is 0.95 that the mean lies in a 95% confidence interval for the mean. I dreaded teaching it in France.</p><p> The French course, though, was a different kettle of fish. No one expects French students to shell out money for books, so the course was based on notes produced by the instructor of record. The notes were spare and lacked attribution. They started with simple examples involving coins, dice, and lifetimes of electronic gadgets, what one would expect. The definition of sample space appeared on page one. (<em>That was fast</em>.) The definitions of sigma-algebra (<em>Gasp! Are they joking?</em>) and probability space (<em>Is this a grad course?</em>) appeared on page two. The course spooled out from there. Yes, it assumed more calculus than we do but mostly in the more interesting problems, and it treated testing and interval estimates in much the same way we do. No one was joking, and this was not a grad course: it was introductory prob/stats, in an unapologetically mathematical setting. </p><p>Statistics is possibly the most important course we teach in mathematics: for life and for cultural literacy, a basic understanding of it is essential. The high schools teach it, yet I’ve heard excellent high school math teachers express fear, if not loathing, of the subject.</p><p> An introductory probability and statistics course based on mathematics is missing, not just from the math education curricula, but from American soil altogether, as far as I can tell. While we teach these courses from bloated texts that avoid mathematics, we might seize the opportunity to teach a critical life skill—understanding statistics—through an exposition that glorifies its foundation in mathematics.</p><p> A big chunk of statistics courses in the United States are taught by non-mathematicians, outside math and statistics departments. By the looks of things, students can often get by on facility with software and a foggy understanding of principles. We still see many of these students in the introductory course, though. Could we do better there? Could we rope these students in with mathematical ideas, and could this happen anytime soon?</p><p>I don’t know, but I’m hoping to go back to teach in France next year.</p><p><a href="http://3.bp.blogspot.com/-5wGUICnbU6g/TsKWcrTQwaI/AAAAAAAADoY/PwesNKlfGcU/s1600/aftermath-11-11.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img src="http://3.bp.blogspot.com/-5wGUICnbU6g/TsKWcrTQwaI/AAAAAAAADoY/PwesNKlfGcU/s320/aftermath-11-11.gif" border="0" alt="" id="BLOGGER_PHOTO_ID_5675263899844264354" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 142px; " /></a></p><hr /><p>About the author: Meg Dillon is a professor of mathematics at Southern Polytechnic State University in Marietta, Georgia.</p><p><em>Aftermath</em> essays are intended to be editorials and do not necessarily reflect the views of the MAA.</p>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com5tag:blogger.com,1999:blog-7775513617406888811.post-71032917787949081692011-09-01T00:00:00.002-04:002011-09-01T00:00:03.916-04:00If You Think You Know It, Try to Teach It<div style="text-align: left;">Maggie Cummings—<i>University of Utah</i></div><p></p><p>I am involved with a Math for America (MfA) project at the University of Utah that helps individuals with strong backgrounds in mathematics (typically a baccalaureate degree in mathematics) become secondary math teachers in high-needs schools. What has been extraordinary to me in this work is the gap between general mathematical knowledge and mathematical knowledge for teaching (MKT). This disparity has received significant attention in teacher education circles. (See, for example, Ball, Thames, and Phelps’s article, “Content Knowledge for Teaching: What Makes It Special?” in the Journal of Teacher Education 59[5], 2008.) The general theme of research in this area is that there is a difference between “doing” and “teaching” mathematics and that while teacher content knowledge is necessary for pedagogical knowledge and skill, the former does not guarantee the latter.</p><p>At the University of Utah, we are trying to develop a conceptual understanding of MKT at the secondary level and a means by which we might measure it. In particular, we are interested in identifying knowledge and skills that secondary mathematics teachers need but that are not necessarily possessed by those with degrees in mathematics. It may seem ridiculous to think that individuals with degrees in mathematics don’t know all the math they need for teaching secondary school students, but here are some concrete examples of where we see a gap:</p><p><span class="Apple-style-span" style="color: rgb(0, 0, 238); -webkit-text-decorations-in-effect: underline; "><img src="http://2.bp.blogspot.com/-3mHmH0K9bsA/TlgDBGdHZqI/AAAAAAAADOw/Hp6pQAtwpaU/s400/aftermathimage8-11.gif" border="0" alt="" id="BLOGGER_PHOTO_ID_5645265450356401826" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 400px; height: 115px; " /></span></p><div>How can you help a seventh-grade student mentally compute 35% of 80?</div><p>Why is the area of a trapezoid ½h(b1+b2)?</p><p>Is there a geometric reason that the slopes of perpendicular lines are negative reciprocals of each other?</p><p>How might you explain why 5 minus -7 equals 12, or why the product of two negative numbers is positive?</p><p>Why is anything to the 0 power equal to 1?</p><p>When our MfA fellows begin the program, most can provide the algorithms or rules related to the above topics, but when pressed, they generally are not able to give student-friendly explanations that connect a tangible model to the algorithmic procedure. For example, fellows usually set up a proportion to solve the problem of finding 35% of 80 (x/80 = 35/100), but do not know of any way to make this problem simple enough for students to compute the answer in their head. (One approach is to understand that 35% is three and a half groups of 10% portions. Ten percent of 80 is 8, so three of them would be 24 and another half [4] would give 28.)</p><p>Indeed, once they examine the conceptualization, the models are not just intuitive—they actually enhance prior understandings of our fellows. The issue is that (a) these conceptualizations are vital to the work of teaching mathematics and (b) they do not seem to be developed in conjunction with typical preparation in mathematics.</p><p>It is not enough for a secondary teacher to say a negative times a negative is a positive—she must also be able to engage students in understanding why this is the case and then how this logic can be applied to other situations. In a similar vein, it is not enough that a teacher knows that a student made a mistake in simplifying an algebraic fraction; he must also be able to identify what the student was thinking in the erroneous simplification process. That way, the teacher has a better chance of helping the student connect his or her understanding of numeric fractions to algebraic fractions.</p><p>As we prepare individuals with strong backgrounds in math to become teachers, what we have learned is that advanced content knowledge in mathematics must be deliberately linked to content-specific pedagogical knowledge and skills. If that linkage is not made, advanced content knowledge stays “siloed” in the instructor, where it doesn’t do the instructor or the students much good.</p><p>Individuals wishing to teach mathematics at the secondary level need more than a strong background in advanced mathematics; they need a strong foundation in the mathematics they are going to teach. So, while it is essential that secondary math teachers understand abstract algebra, it doesn’t necessarily translate into the ability to teach basic algebra. If the truest test of understanding an idea is being able to teach it to someone else, then even some of the strongest graduating mathematics majors still have much to learn about the foundations of their chosen subject. The more they are willing to learn, the more their future students will be likely to follow suit.</p><hr /><i><span class="Apple-style-span">About the Author: Maggie Cummings is an instructor with the Center of Science and Mathematics Education at the University of Utah. Email: <a href="mailto:margaritacummings@gmail.com">margaritacummings@gmail.com</a></span></i>
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<br /></i></div><div><i><span class="Apple-style-span" style="color: rgb(102, 102, 102); line-height: 18px; background-color: rgb(255, 255, 255); "><span class="Apple-style-span">Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. </span></span></i></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com7tag:blogger.com,1999:blog-7775513617406888811.post-78738363379320075512011-04-01T00:00:00.002-04:002011-04-01T00:00:03.770-04:00The Problem with Problem Solving<div style="text-align: left;"><b>Andy Liu</b><span class="Apple-style-span" style="color: rgb(51, 51, 51); font-size: 13px; line-height: 20px; "><strong>—</strong></span><i>University of Alberta</i></div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-OjD6g_s-xUk/TYiq0pZXYzI/AAAAAAAACp4/WezTo997_-Q/s1600/Pages%2Bfrom%2BAftermath-Liu.jpg"></a><div style="text-align: center;"><br /></div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-9sDeCadBR-U/TYiqqmF_tKI/AAAAAAAACpw/XUC46_wHSQo/s1600/mh-4_2011-1.gif"></a><div>There are two stages in solving a problem. The first stage is to convince yourself that you have solved the problem. The second stage is to convince other people that you have solved the problem. The first stage is the creative one and is indicative of how mathematics is actually done. The second stage is more formal and often has little in common with the first stage, but ironically it is how mathematics is usually communicated and taught.</div><div><br /></div><div>Let us illustrate with a simple geometry problem. </div><div><br /></div><div><b>Problem</b>. </div><div><br /></div><div>P is any point inside an equilateral triangle ABC. Perpendiculars are dropped from P to BC at D, CA at E, and AB at F. Which has the greater total area: triangles PAF, PBD, and PCE, or triangles PAE, PBF, and PCD?</div><div><br /></div><div>The symmetry of this problem compels us to jump in with both feet and say, “They are the same!” However, the proposer of the problem may be having fun with us, so let’s test our hypothesis with some special positions for P. Putting P at the center of ABC, then at the midpoint of BC, and then coincident with A, we see that in each case our hypothesis holds true. (See figure 1.) So we are confident that our conclusion is correct.</div><div><br /></div><div><span class="Apple-style-span" style="color: rgb(0, 0, 238); -webkit-text-decorations-in-effect: underline; "><img src="http://3.bp.blogspot.com/-w1Ty_tNUmW4/TYi7BvwybxI/AAAAAAAACqQ/jzDfwmJ2Gzg/s400/fig1new.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5586920976428461842" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 400px; height: 118px; " /></span></div><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: underline; " ><i>Figure 1 </i></span></div><div><br /></div><div>A good way to solve problems is to make use of special cases. The first two attempts at incorporating the special positions into the general diagram are not particularly fruitful. The shaded regions do not correspond exactly. (See figure 2.)</div><div><br /></div><div><span class="Apple-style-span" style="color: rgb(0, 0, 238); -webkit-text-decorations-in-effect: underline; "><img src="http://1.bp.blogspot.com/-u3Ob7LY5Z7o/TYi7HvagV2I/AAAAAAAACqY/XK3lC8Q2cg4/s400/fig2new.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5586921079414216546" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 400px; height: 177px; " /></span></div><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: underline; "><i><span class="Apple-style-span" >Figure 2 </span></i></span></div><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: underline; "><i><span class="Apple-style-span" ><br /></span></i></span></div><div>The third special position—the simplest one—delivers! The shaded regions correspond exactly. Moreover, instead of having P coincide with A, we can have it coincide with B and C as well, yielding the diagram in figure 3. </div><div><br /></div><div><span class="Apple-style-span" style="color: rgb(0, 0, 238); -webkit-text-decorations-in-effect: underline; "><img src="http://3.bp.blogspot.com/-_JsEyyMdjbM/TYi7bsZig8I/AAAAAAAACqg/P1uZC2bU6Ic/s400/fig3new.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5586921422202241986" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 400px; height: 177px; " /></span></div><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: underline; "><i><span class="Apple-style-span" >Figure 3</span></i></span></div><div><br /></div><div>Since PUV, PWX, and PYZ are all equilateral triangles, PZ and PW are both parallel to BC, so that P lies on ZW. Similarly, P lies on UX and VY. Moreover, PXAY, PZBU, and PVCW are parallelograms, and a diagonal of a parallelogram bisects its area. Everything falls into place!</div><div><br /></div><div>We are now at the end of the first stage—which is where the trouble begins. If you read the solution to this problem in some problem book, you are likely to get the following: </div><div><br /></div><div>Construct lines ZW, UX, and VY through P, parallel to BC, CA, and AB respectively. Then PUV, PWX, and PYZ are equilateral triangles while PXAY, PZBU, and PVCW are parallelograms. Denoting the area of a polygon Q by [Q], we have</div><div><br /></div><div><span class="Apple-style-span" style="color: rgb(0, 0, 238); -webkit-text-decorations-in-effect: underline; "><img src="http://4.bp.blogspot.com/-9sDeCadBR-U/TYiqqmF_tKI/AAAAAAAACpw/XUC46_wHSQo/s400/mh-4_2011-1.gif" border="0" alt="" id="BLOGGER_PHOTO_ID_5586902986510021794" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 400px; height: 119px; " /></span></div><div><span class="Apple-style-span"><br /></span></div><div>This is very neat and very impressive. The key step is in the first line—the construction of ZW, UX, and VY. But notice that in our work on the first stage, this construction emerges as practically the final step. The two stages run in essentially opposite directions!</div><div><br /></div><div>Emphasizing the second stage over the first has significantly negative consequences. Performing an excess of exercises may lead students to look at a problem and say, “This one I can do,” and start their write-up before thinking through their own reasoning. Alternatively, they may say, “I have never seen anything like this before,” and move on for lack of any ideas.</div><div><br /></div><div>The most difficult thing about solving a problem is getting started. Where do ideas come from? There are many techniques, but no sure-fire method because problem solving is not a science. It is an art. Progress comes only with regular practice and sustained effort on the first stage of the process. Unfortunately, only reading solutions by other people is ineffective because other people tend to write only about the second stage of problem solving.</div><div><br /></div><div>Perhaps this is to be expected. In the world of research mathematics, only the second stage is required to get a paper published. The paradoxical result of this is that we disseminate new theorems of mathematics in a fashion that offers little insight into how they were created.</div><div><br /></div><div>Bringing this back to the world of the classroom, the upshot is that more attention needs to be paid to the first stage of problem solving. Perfecting the expository skills is certainly important; however, there will not be anything to write about if students cannot conduct informal explorations. They must learn to make something out of nothing, which is the essence of research.</div><div><br /></div><div>About the author: Andy Liu teaches at the University of Alberta, from which he obtained a Ph.D. in mathematics and a graduate diploma in elementary education, thus becoming perhaps the only person officially qualified to teach mathematics from kindergarten to graduate school. Email: aliumath@telus.net</div><div style="text-align: left;"><br /></div><div>Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. To respond, go to Aftermath at www.maa.org/mathhorizons.</div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7775513617406888811.post-38826666687447565792011-02-11T14:05:00.003-05:002011-02-14T13:29:38.678-05:00The Lure of the Dark Side<div style="text-align: left;"><strong>Doug Szajda—</strong><em>University of Richmond</em></div><p>In the interest of full disclosure, I must confess that though I was once a mathematician, I have since gone over to the Dark Side—computer science. And like any self-respecting Dark Sider, part of my job is to entice others to follow my path. For an undergraduate math major, this translates simply—if you truly want to experience the power of mathematics, then, while there’s still time, take as many applied mathematics, computer science, and statistics courses as you can.</p><img src="http://3.bp.blogspot.com/-nZvcnNwwzBw/TVWKVIE3CuI/AAAAAAAACig/bOf_CEkJYNE/s400/Aftermath2-11.gif" style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 220px; height: 260px;" border="0" alt="" id="BLOGGER_PHOTO_ID_5572512209490610914" /><p>I know, your major doesn’t require you to take any statistics or computer science courses. Sadly, such programs still exist. And I understand that it’s comforting to live in the cocoon of pure math. Theory is clean. It is elegant. Yes, pure mathematics can be beautiful in the same way that great literature, art, and music are beautiful. Real-world math, on the other hand, is messy. Hypotheses are not always clear. Boundary conditions and transition phases complicate analysis. Models have to be carefully balanced between being simple enough to be tractable, yet sufficiently detailed that they accurately model phenomena. Dealing with this can be nasty business. But it’s what is required if you want to really use mathematics.</p><p>And there are at least two good reasons why you should explore real-world applications of math. First, you like math, and applied areas are where you’ll get to see some really amazing mathematics. In the corridors of my department (a combined math and computer science department), there are 45 AMS “Mathematical Moments” posters. These fliers, which in some math departments should be considered false advertising, depict problems or research areas where math plays a fundamental role. Topics covered include robotics, speech recognition, cell biology, protein folding, and even crime solving. Of this (admittedly unscientific) sample, only three posters discuss problems that might be worked on by a pure mathematician—and one of these is solving sudoku. On the other hand, the topics mentioned on the other 42 posters are most likely examined by experts in the techniques of applied mathematics, statistics, or computer science.</p><p> The mathematical techniques most often mentioned on these posters include statistics, dynamical systems, graph theory, mathematical models, pattern recognition, image analysis, differential and partial differential equations, linear algebra, combinatorics, and optimization. As a mathematics major, you’re not likely to see most of these techniques, even if you pursue a math Ph.D., although ironically, your non-math friends might very well be introduced to the basics of very useful topics in linear programming, graph theory, probability, combinatorics, and game theory in the non-major courses they take to fulfill their math requirements for graduation.</p><p> The second reason you should take more applied courses is that you likely have an interest in technology, and you live in a technological society. You use a computer and cell phone, probably own an iPod (if not several), and are surrounded by devices that are controlled by microprocessors. And let’s be honest: you probably couldn’t exist without them. Do you want to graduate without having even a basic understanding of how these work? Moreover, you live in a world in which you are bombarded by statistics. It thus behooves you, as a more technologically inclined citizen, to understand enough statistics to be able to see what statistical results really tell us—and also how they can be used in misleading ways.</p><p> In case you are inclined to dismiss the opinions of a Dark Sider, then perhaps you will listen to the Mathematical Association of America Committee on the Undergraduate Program in Mathematics, which recommends in its 2004 curriculum guide that mathematics programs should promote learning that helps students better understand the uses of mathematics. This is a refreshing change from the historical norm where applied mathematics was often viewed as a debasement of the Platonic ideals of pure math, and undergraduate programs were designed for the less than 10 percent of students who might have the desire and talent to continue their studies at the graduate level.</p><p> So, if you are fortunate enough to be a part of a program that has opportunities for engaging the applied side of math, you’d do well to take advantage. I can assure you, it’s more fun on the Dark Side.</p><p> The money isn’t bad either.</p><p><strong><em>About the author: </em></strong><em><a href="mailto:dszajda@richmond.edu">Doug Szajda</a> is an associate professor of computer science at the University of Richmond. He is currently general chair of the Internet Society Network and Distributed System Security Symposium.</em></p><p align="center"><span class="Apple-style-span"><em>Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. </em></span></p>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com2tag:blogger.com,1999:blog-7775513617406888811.post-50258091954659684702010-10-28T11:11:00.001-04:002010-11-01T13:56:36.878-04:00Does the Master’s Degree in Mathematics Get Too Little Respect?<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_Kom-mudcX5I/TM7_PcjKhaI/AAAAAAAACOY/OU1eGZfp2eI/s1600/diploma.jpg"></a><p class="MsoNormal"><b>Carl Cowen - Actuarial Science Program at Indiana University–Purdue University</b></p><p class="MsoNormal">If you think about the history of science, mathematics sits in a unique position: everything that has ever been true in mathematics is still true! We no longer believe that the elements are Earth, Air, Fire, and Water, for example, but Euclid’s description of geometry in the plane is still correct. Modern physics rests on developments from the late 19th century onward, with recognition that Newton’s discoveries provide a working foundation. Modern chemistry is largely a 20th-century science, and molecular biology starts with the discovery of the role of DNA in the mid-20th century. A fundamental difference between undergraduate education in mathematics and that of the other sciences is that we (mostly) take students to the early 20th century or so while the other sciences take students to the research forefront.</p> <p class="MsoNormal">As an example, a few years ago I taught a course on computational neuroscience for juniors and seniors with a mathematical background or a biological background (prerequisites: two semesters of calculus for biology students; differential equations for math students; and at least junior standing in a mathematics, statistics, engineering, or biological sciences major. Note that no biology prerequisites were asked of the math students). During the semester, we read a research paper from 1988. The math students were astonished: they mostly had never seen a research paper, or if they had, they had never seen one that new! The biology students were also astonished: they had seen many research papers, but they had never seen one that <i style="mso-bidi-font-style:normal">old</i>!</p> <p class="MsoNormal">Thus, our science colleagues have a quite different perspective on undergraduate and graduate education than we do. A Ph.D. in chemistry at Purdue University requires two (two!) classroom courses, and the rest is research. A Ph.D. in mathematics usually includes 10 to 15 classroom courses! My own opinion is that the study for the Master of Science degree is the most intensive learning experience in the mathematical sciences. Much more mathematics is learned than at the undergraduate level because the study is so much deeper, and more is learned than at the Ph.D. level because there the learning is specialized and research focused. Thus, first and foremost, I regard the M.S. as the time when students acquire a broad and deep understanding of mathematics.</p> <p class="MsoNormal">Further, most of the master’s program is devoted to studying late 19th-, 20th-, and 21st-century mathematics. Indeed, an M.S. program should put a student close (say 1950s–1970s era) to the research forefront in at least one area. Most M.S. programs include Ph.D. qualifier courses. This is fundamental, broad, and deep material in comparison to undergraduate work.</p><p class="MsoNormal"><a href="http://4.bp.blogspot.com/_Kom-mudcX5I/TM7_PcjKhaI/AAAAAAAACOY/OU1eGZfp2eI/s1600/diploma.jpg"><img src="http://4.bp.blogspot.com/_Kom-mudcX5I/TM7_PcjKhaI/AAAAAAAACOY/OU1eGZfp2eI/s320/diploma.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5534641632911721890" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 320px; height: 171px; " /></a></p><p class="MsoNormal">As a profession, we put too little emphasis on the M.S. and give it too little respect. We should be encouraging many more of our undergraduate students to go to graduate school and get an M.S. degree. Mathematics faculty are good at encouraging the “best” students to go to graduate school, but we should be encouraging the top third of our students to go on—they are surely qualified for the experience and would benefit greatly from the added education.</p> <p class="MsoNormal">Moreover, the job surveys I’m familiar with suggest that the M.S. is the most marketable degree in the mathematical sciences. This is a consequence, I believe, of the fact that M.S. students know much more mathematics than undergraduates and are less likely than Ph.D.s to be “distracted” by research interests (in the minds of those who are looking for mathematical expertise in filling job openings).</p> <p class="MsoNormal"><o:p> </o:p></p> <p class="MsoNormal">There are several important career paths for M.S. degrees. The M.S. in statistics is the professional degree for a statistician. As I understand it, except for specialized areas such as the pharmaceutical industry where the Ph.D. is preferred, most “working” statisticians have an M.S. in applied statistics or biostatistics. The two-year college faculty member in mathematics is usually expected to have a “plain vanilla” M.S. in mathematics with enough statistics background to be able to teach beginning statistics courses. Both of these career paths are full of opportunities!</p> <p class="MsoNormal">Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. To respond, go to Aftermath at <a href="http://www.maa.org/mathhorizons">www.maa.org/mathhorizons</a>. </p> <p class="MsoNormal">About the author: Carl Cowen is professor of mathematical sciences and director of the Actuarial Science Program at Indiana University–Purdue University at Indianapolis. He is a former president of the MAA. Email: <a href="mailto:ccowen@math.iupui.edu">ccowen@math.iupui.edu</a> <span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span></p> <p class="MsoNormal"><o:p> </o:p></p>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com2tag:blogger.com,1999:blog-7775513617406888811.post-21566081751342350622010-09-01T00:00:00.004-04:002010-09-02T09:29:32.984-04:00Facebook and Texting vs. Textbooks and Faces<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_Kom-mudcX5I/THf99A_akiI/AAAAAAAACBo/s-y_eQUIlro/s1600/laptop-Fisher.jpg"></a><p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><b><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">Susan D’Agostino - Southern New Hampshire University</span></span></b></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">Last semester, my business statistics students were not exactly thrilled when I announced an in-class ban on electronic devices, including laptops, phones, and digital music devices. </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">“But I use my cell phone as a calculator!” one student protested.<o:p></o:p></span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">“Can’t I use my MP3 player to help focus during exams?” another pleaded.</span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">“I found a cool app that gives p-values for the standard normal distribution!” another offered hopefully, as if using statistical jargon would entice me to cave.</span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">“Humor me,” I responded. “Let this class be the one hour and fifteen minutes of your day in which you are completely unplugged.” I felt like a counselor at an outpatient program for recovering addicts.</span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">Halfway through the semester, I did what any self-respecting statistics instructor would have done: I surveyed my 67 students and used the tools I was teaching—confidence intervals for means and proportions—to compile the data. My results provide estimates—with a 95 percent confidence level—for the in-class, electronic multitasking habits of business majors at midsized, regional universities. Every student in this category has, at some point, used a laptop, phone, or digital music device in class. In a seventy-five-minute class that permits students to be “plugged in,” a student with an open laptop takes electronic notes just as much as he social networks: 34 minutes with a margin of error of 5 minutes. Looking at websites that are relevant to class is only slightly more common than looking at websites that are irrelevant to class: 36 as opposed to 32 minutes. A student with an open laptop spends, on average, 27 minutes sending and receiving email and 11 minutes reading an electronic newspaper. That these numbers sum to more than the seventy-five class minutes hints at the prevalence of in-class, electronic multitasking.</span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';">Overall, when electronic devices are permitted in class, a majority of students using the devices—58 percent— multitask at least half the time. Students self-reported on the number of multitasking activities they engaged in beyond listening to the lecture or participating in class discussion: 52 percent of the examples involved one activity, including social networking or texting. Forty-six percent of the examples cited two, three, or four activities, including social networking, emailing, and doing homework. An intrepid 2 percent of the examples involved five multitasking activities: social networking, instant messaging, searching online, playing games, and texting.</span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:'times new roman';"> </span></span></p><p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"></p><div style="text-align: center;"><span class="Apple-style-span" style=" ;font-family:'times new roman';"><span class="Apple-style-span" style=" ;font-family:Georgia, serif;"><a href="http://4.bp.blogspot.com/_Kom-mudcX5I/THf99A_akiI/AAAAAAAACBo/s-y_eQUIlro/s1600/laptop-Fisher.jpg"><img src="http://4.bp.blogspot.com/_Kom-mudcX5I/THf99A_akiI/AAAAAAAACBo/s-y_eQUIlro/s400/laptop-Fisher.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5510151893790921250" style="display: block; margin-top: 0px; margin-right: auto; margin-bottom: 10px; margin-left: auto; text-align: center; cursor: pointer; width: 288px; height: 288px; " /></a></span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">To my surprise, the vast majority of students—94 percent—expressed either a favorable or neutral opinion of my policy. Were these the same students who originally made me feel like a counselor for substance abusers?</span></span></div><p></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">“Knowing I can’t text allows me to pay better attention,” wrote one student.</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">“Not having my computer out means that I can’t find myself on Facebook,” wrote another student.</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">“I like the reduced noise distractions from [the absence of] electronic devices,” wrote a third.</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">“It’s a good policy. I always see the students with laptops looking at Facebook or playing games,” another offered.</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">So what about the responses from students who did not appreciate my policy? One commented that he “miss[ed] the unlimited amount of information that a computer has.” Another was put off by having to “carry notebooks and pens for note taking.” Another mentioned his concern about being unreachable in an emergency. Of course, I had informed my students that the university’s security office would deliver an emergency message to a student in class if needed.</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">The Kaiser Family Foundation recently reported that the average 18-year-old spends over seven hours daily using electronic media devices for recreational purposes outside of the classroom. Based on my study, this statistic would likely increase dramatically if recreational use of electronics inside of the classroom were counted.</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">College students should not sell their in-class time short. Class should be a time and place devoted to wrestling with ambiguity, not deferring to online encyclopedias edited by anyone with an inclination to blog. Currently, this assistant professor of math is wrestling with whether the anonymous student who wrote the following comment on my survey intended to be ironic: “I think [the in-class ban on electronics] is a good policy.... In this age of technology, people need to stay connected at all times. It absolutely gets in the way during class. Unfortunately, I really do not know how to fix the issue. I guess you could Google it?”</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal;mso-pagination:none;tab-stops:.5in 1.0in 1.5in 2.0in 2.5in 3.0in 3.5in 4.0in 4.5in 5.0in 5.5in 6.0in; mso-layout-grid-align:none;text-autospace:none"><b><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">About the author:</span></span></b><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> Susan D’Agostino is an assistant professor of mathematics at Southern New Hampshire University.</span></span><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;"> </span></span></p> <p class="MsoNormal" style="text-align: center;margin-bottom: 0.0001pt; line-height: normal; "><i><span class="Apple-style-span" style=" color: rgb(102, 102, 102); line-height: 20px; "><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. Contact information is available </span></span><a href="http://www.maa.org/mathhorizons/feedback.html" style="color: rgb(85, 136, 170); text-decoration: none; font-weight: bold; "><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">here</span></span></a><span class="Apple-style-span" style="font-family:'times new roman';"><span class="Apple-style-span" style="font-size: medium;">.</span></span></span></i></p>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com4tag:blogger.com,1999:blog-7775513617406888811.post-76269885189017522942010-04-08T14:47:00.006-04:002010-04-08T14:55:41.564-04:00The Intermediate Under-Valued Theorem<span style="font-weight: bold;">Bruce Peterson - Middlebury College</span><p>“Well, duh.”<br /><br />A familiar student reaction to the Intermediate Value Theorem. After all, if a function is “continuous,” it can’t jump from place to place without stopping in between. Or,<br /></p><p> “Real functions are like <em>x</em><sup>2</sup>or sin(<em>x</em>). Those step thingies don’t really matter.”<br /></p><p>This cherished theorem usually falls flat in beginning calculus because, I would argue, students see it as so obvious as not to merit discussion. And it’s not their fault; the theorem that justifies the word “continuous” strikes most students as unimportant because they rarely see it do anything other than confirm their long-held intuition about what “continuous” ought to mean. If a continuous function is positive somewhere and negative somewhere else then, sure, it has a root in between. But this familiar “application” is of course just a restatement—or a special case—of the original result.<br /></p><p>So what kinds of applications are there? For starters, how do you cut a cake in half? What you don’t do is find the center and cut through it. Rather you mentally move a knife across the top until the area on the left of the knife looks to be about the same as the area on the right—a simple application of our old friend. Does the cake have to be round you ask? Nope. If <em>S</em> is any closed figure in the plane, then there is a line in any given direction that bisects the area of <em>S.</em> (A “closed figure” is a set bounded by a simple closed curve.)<br /></p><p>To prove this claim, we can construct a standard coordinate system with the <em>y</em>-axis parallel to the chosen direction. For each <em>x</em>, let <em>l</em>(<em>x</em>) be the line through <em>x</em> and perpendicular to the <em>x</em>-axis. <em>L</em>(<em>x</em>), the area of <em>S</em> to the left of <em>l</em>(<em>x</em>), is a continuous function of <em>x</em> as is <em>R</em>(<em>x</em>), the area to the right of <em>l</em>(<em>x</em>). Hence <em>D</em>(<em>x</em>) = <em>R</em>(<em>x</em>) – <em>L</em>(<em>x</em>) is continuous. For a line to the left of <em>S, D</em>(<em>x</em>) = Area of <em>S</em>, and for a line to the right of <em>S</em>, D(<em>x</em>) = – (Area of <em>S</em>). By the Intermediate Value Theorem there is an intermediate line for which <em>D</em>(<em>x</em>) = 0 and <em>L</em>(<em>x</em>) = <em>R</em>(<em>x</em>).<br /></p><p>If that were the whole story there would be no story. After all, we’ve really just beaten a simple theorem to a pulp and not learned much except that the Intermediate Value Theorem may be part of our DNA. Let’s look a bit further.</p><p>If <em>S</em> is a closed figure in the plane, then in fact there are two perpendicular lines that divide the figure into four “quadrants” of equal area. To see why this is so, let <em>l</em>(α) be a line that makes an angle α with the <em>x</em>-axis and, appealing to the previous argument, assume it bisects the area of S. Clearly<em> l</em>(α) and <em>l</em>(α + π/2) cut <em>S</em> into four quadrants. We’ll label them in the usual counterclockwise fashion and designate their areas <i>A</i><sub>1</sub>(α), <i>A</i><sub>2</sub>(α), <i>A</i><sub>3</sub>(α) and <i>A</i><sub>4</sub>(α). Since <i>A</i><sub>1</sub>(α) + <i>A</i><sub>2</sub>(α) = <i>A</i><sub>3</sub>(α) + <i>A</i><sub>4</sub>(α) and <i>A</i><sub>1</sub>(α) + <i>A</i><sub>4</sub>(α) = <i>A</i><sub>2</sub>(α) +<i> A</i><sub>3</sub>(α), we have at once that <i>A</i><sub>1</sub>(α) = <i>A</i><sub>3</sub>(α) and <i>A</i><sub>2</sub>(α) = <i>A</i><sub>4</sub>(α).<br /></p><p>The difference D(α) = <i>A</i><sub>2</sub>(α) – <i>A</i><sub>1</sub>(α) is continuous, because each component is, and <i>A</i><sub>1</sub>(α + π/2) = <i>A</i><sub>2</sub>(α) and <i>A</i><sub>2</sub>(α + π/2)= <i>A</i><sub>3</sub>(α) = <i>A</i><sub>1</sub>(α). Therefore <em>D</em>(α) changes sign between α and α + π/2, and there is an angle for which <i>A</i><sub>2</sub> = <i>A</i><sub>1</sub> (=<i>A</i><sub>3</sub> = <i>A</i><sub>4</sub>).<br /></p><p>A better known example is the “Ham Sandwich” Theorem: Given a piece of ham and a piece of bread (in the plane), it is always possible to cut both in half with one slice of a knife. Intuitive? Obvious? The proof combines the ideas explored in the previous arguments—give it a try.<br /></p><p>Here is a less familiar example: There is a square (not just a rectangle) that circumscribes any figure <em>S</em> in the plane in the sense that <em>S</em> lies inside the square and each side of the square contains a boundary point of <em>S</em> (possibly a vertex). To prove this one, let<em> l</em>(α) be a line tangent to <em>S</em> in direction α and with <em>S</em> on the left of l(α).The lines <em>l</em>(α), <em>l</em>(α + π/2), <em>l</em>(α + π) and <em>l</em>(α + 3π/2) define a rectangle <em>R</em>(α) circumscribing <em>S</em>. Let <em>L</em>(α) be the “length” of <em>R</em>(α), the dimension parallel to <em>l</em>(α), and <em>W</em>(α) the “width” of <em>R</em>(α), the dimension perpendicular to <em>l</em>(α). Since <em>W</em>(α) = <em>L</em>(α + π/2), applying the Intermediate Value Theorem to <em>L</em>(α) – <em>W</em>(α) proves the theorem. As you visualize the rectangle <em>R</em>(α) changing dimension, you can “see” the sought-after square.<br /></p><p>The Intermediate Value Theorem won’t matter unless the instructor makes it matter, so here’s a final problem to ponder: Consider a planar set where the maximum distance between any two points is 1. Find the side length of the smallest regular hexagon that is guaranteed to contain any such set. (And be sure to check out the Zip-line section of The Playground in this issue.)</p><p><span style="font-weight: bold;">About the author: </span>Bruce Peterson is Charles A Dana Professor of Mathematics and College Professor Emeritus at Middlebury College. His fondness for the Intermediate Value Theorem stems from a lifelong advocacy of geometry in general. He also has an avid interest in ornithology.<br /></p><p style="text-align: center;"><span style="font-style: italic; color: rgb(102, 102, 102);">Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. Contact information is available <a style="font-weight: bold;" href="http://www.maa.org/mathhorizons/feedback.html">here</a>. </span></p>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-57587334706100465892010-02-16T14:53:00.005-05:002010-02-16T15:04:18.890-05:00Thinking Inside the Box<span style="font-weight: bold;">Nathan Carter - Bentley University</span><br /><br />I love computers and related gadgets, but have been wary about integrating technology into my classroom. Calculators are not allowed on most of my exams, my students and I use plenty of chalk, and PowerPoint rarely shows up. I graph by computer only if a hand sketch would be messy. But as the math world gazes with interest on a shiny, new WolframAlpha, formerly dormant debates over technology begin anew. And they got me thinking.<br /><br />At first, I leaned right back on my old favorite argument, shared by many, that can be applied to many different pieces of high-tech math tools: “Technology is a black box that can actually get in the way of real learning when pushing buttons replaces a more rooted understanding of what’s going on below the surface. When used as a teaching tool, students may come away able to produce a few impressive answers, but they do so without real comprehension or the ability to apply their knowledge in any context other than the basic setting of the problems they’ve encountered in the assignment.”<br /><br />There’s a lot of truth to this argument. Sure, some instructors might use it to justify a pre-existing preference—not wanting to rework the whole curriculum in response to a shiny, new WolframAlpha!—but that doesn’t mean the argument isn’t correct.<br /><br />And I still think it is compelling, but I recently made an important realization. It’s also irrelevant. To see what I mean, let’s apply the same argument to a piece of mathematical technology that’s a little older than WolframAlpha, even older than the calculator—yes, even older than the slide rule. Let’s apply it to…algebra! (I’m talking quadratic formula and completing the square, not groups and rings.)<br /><br />But is algebra a technology? Merriam-Webster defines technology as “the practical application of knowledge especially in a particular area.” The American Heritage Dictionary is less brief, but allows any “technical means” even if only from “pure science.” No sprockets or circuits are required! Algebra is a technology.<br /><br />Why compare algebra, which takes so much thinking, to using a calculator or computer, which (often) takes comparatively less thinking? I suppose I could stave off this question by saying that in each case you must learn a technical skill, or you’ll make an error and thus get wrong answers. This is true, but there is a better answer.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_Kom-mudcX5I/S3r5TYizKVI/AAAAAAAABOk/H3TiAZpknAY/s1600-h/aftermath.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 319px; height: 189px;" src="http://1.bp.blogspot.com/_Kom-mudcX5I/S3r5TYizKVI/AAAAAAAABOk/H3TiAZpknAY/s400/aftermath.jpg" alt="" id="BLOGGER_PHOTO_ID_5438933611404536146" border="0" /></a><br />Our old, faithful friend algebra has just as much potential to be a “black box” as calculators and computers do. This includes not only the too-common example of using algebra to derive just as ridiculously incorrect an answer as you might with a calculator, but it includes much more sophisticated missteps as well. Consider the mathematician who attacks a problem or a proof with all the metaphorical levers, buttons, and knobs in the algebra arsenal and comes out the other side victorious. Then the referee’s report points out a far-more-elegant, two-sentence argument. In such a case, the referee’s report might very well say, “The author clearly doesn’t understand what is going on in this argument.” Oh, what we miss by fleeing to the trusted algebra crutch too soon!<br /><br />But isn't algebra useful precisely because it works even at times when we either don't know why or at least don't care to focus on why? Surely not every algebraic argument can be turned into elegant prose—at least not in short order. And more importantly, haven't we as instructors justified students' study of algebra for this exact reason—its utility?<br /><br />If the power of algebra, when used rightly, to churn out correct answers from correct inputs is the reason that students should become proficient at it, then shouldn’t that same reasoning justify their becoming proficient at even more powerful tools? In fact, if we use that reasoning to justify requiring students to be proficient with algebra, how can we do anything but require them to be proficient with the likes of WolframAlpha? (Software engineers may now cackle and/or cheer.)<br /><br />So this is how I saw the light. I am a new mathematics professor and I say that a mathematician who wants students to learn algebra should also want them to learn any similarly powerful mathematical invention, even if it has sprockets or circuits! Wait. This means that I have to rework my curriculum, doesn’t it?<br /><br /><span style="font-weight: bold;">About the author</span>: Nathan Carter is Assistant Professor of Mathematics at Bentley University in Massachusettes and author of the acclaimed new book Visual Group Theory, which employs the graphic power of computers to explore abstract algebra.<br /><br /><div style="text-align: center;"><span style="font-style: italic; color: rgb(102, 102, 102);">Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. Contact information is available <a style="font-weight: bold;" href="http://www.maa.org/mathhorizons/feedback.html">here</a>. </span><br /></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7775513617406888811.post-51718941114065346362009-11-11T17:04:00.024-05:002009-11-24T12:44:27.420-05:00A Politician’s Apology<span style="color:#333333;"><span style="color:#000000;"><strong>Steve Kennedy - Carleton College</strong><br /></span><br /></span><span style="font-family:lucida grande;color:#333333;">On September 10, 2009 British Prime Minister Gordon Brown issued a public apology, on behalf of the British nation, to Alan Turing for that country’s treatment of him. Alan Turing was one of the great mathematicians of the twentieth century. He worked on fundamental problems in mathematical logic in the 1930s in the process inventing some of the seminal concepts of computer science, including the basics of the theory of computation. During the Second World War he served at the British code-breaking school at Bletchley Park and was largely responsible for breaking the German U-boat code. After the war he worked on the design and construction of two first-generation computers and was an early advocate for, and theorist of, the future of artificial intelligence. Turing was also gay and in 1952 he was convicted of “gross indecency” and sentenced to a course of treatment with female hormones designed to eliminate his sex drive. Not surprisingly, this chemical castration had profound physical and psychological consequences. In 1954, at the age of 41, Turing took his own life.<br /><br />The extraordinary drama and tragedy of Turing’s life—war hero and mathematical genius driven to suicide by persecution for being gay—has provided inspiration for several artistic interpretations of his life. (Turing did his best to increase the drama by choosing a method of suicide inspired by Disney’s Snow White, a favorite of his—he ate a cyanide-laced apple.) There have been at least three plays, one novel, and two movies based on his story, as well as an extraordinary biography by mathematician Andrew Hodges. That biography reveals Turing as a good and decent man more or less bewildered by the barbaric treatment he received from his countrymen. These retellings of the Turing story have contributed to Turing’s status as something of an icon in the gay community. His profoundly original scientific contributions—in computer science he is memorialized in Turing Machines, the Turing Test, and computer science’s Nobel Prize, the Turing Award—have similarly preserved his iconic status among mathematicians and computer scientists. In fact, the Brown apology was provoked by a petition drive organized by a British computer scientist, John Graham-Cumming, inspired by admiration of Turing and dismay at his treatment.<br /><br />Once Brown issued the apology the news, and reactions to it, zipped around the Internet. Most folks believed that the apology was long overdue and constituted a genuinely positive development, especially as a small contribution that might help chip away at still existing homophobia. And, to Gordon Brown’s credit, he understood that this should not be just about Alan Turing. Turing is just one particularly egregious and notorious example; Brown explicitly expresses regret over the “many thousands of other gay men who were …treated terribly” and even recognizes the “millions more who lived in fear.” Most fans of Turing seemed pleased; one friend told me he felt “elated.”<br /><br />I didn’t feel elated and I wondered if there was something wrong with me. Oh sure, I recognized that this was a good and necessary step, but I couldn’t help but feel that it was not proportionate. The British government, in the name of the British people, tortured this good and decent man (and thousands of others) because they disapproved of his sexual habits. Now, half a century later, they offer only words of regret. Maybe I’d feel better if Gordon Brown vowed not to rest until gay marriage was legal in Britain. Of course in Britain today the legal status of gays is a thousand times better than it is in the US, so maybe Brown could work on educating America? How about passing a heavy tax—the Turing Tariff—on all computing hardware and software imported into the UK from countries, like the US, that still discriminate against homosexuals by banning gay marriage? The proceeds of the tariff could be donated, in the name of Alan Mathison Turing, to the leading gay rights organizations in the exporting country.<br /><br />I know that’s not going to happen. Nothing really dramatic is going to happen. Still, I admire John Graham-Cumming and the thousands who signed his petition and am grateful for their efforts. I’m also appreciative of Gordon Brown’s understanding and grace. I do feel badly that I’m unable to celebrate; a terrible maltreatment of one of my intellectual heroes is being publicly recognized as such. But it just seems that this goodness and benevolence—and maybe all goodness and benevolence—are slow and discreet and progress in tiny incremental steps while hatred and injustice bash and roar and wreak enormities. And we don’t seem to ever learn. Elated? No, I’m not elated; I’m just very sad.<br /></span><div align="center"><span style="font-family:lucida grande;"><a href="http://3.bp.blogspot.com/_Kom-mudcX5I/Svs4yiPTepI/AAAAAAAAA7c/04aei87ovvw/s1600-h/Aftermath+AppleImage.jpeg"><span style="font-family:lucida grande;color:#333333;"><img style="WIDTH: 317px; HEIGHT: 240px; CURSOR: hand" id="BLOGGER_PHOTO_ID_5402974618796653202" border="0" alt="" src="http://3.bp.blogspot.com/_Kom-mudcX5I/Svs4yiPTepI/AAAAAAAAA7c/04aei87ovvw/s400/Aftermath+AppleImage.jpeg" /></span></a></div></span>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com5tag:blogger.com,1999:blog-7775513617406888811.post-34634636675997928822009-09-23T12:46:00.010-04:002009-11-24T12:35:49.071-05:00Dispassionate Mathematics<span style="font-size:180%;"><strong><span style="font-size:78%;">Rick Cleary - Bentley University</span></strong><br /></span><br />I am coming clean. I do not have a passion for mathematics. And I don’t think that’s a bad thing. Here’s why.<br /><br />I was recently reading an article about a friend who solved an interesting open problem and was rewarded with some well deserved publicity. This mathematician is quoted as saying, “I get a problem like this and I don’t sleep at night.” Articles about mathematics and interviews with mathematicians seem to always include comments like this. There is an indication of a passion for problem solving and a related inability to function in the rest of the world while the math question remains open. I do not recall ever reading an article where a mathematician discusses a recently solved problem and simply says, “Yes, that was a nice result. A good day at the office. That’s what I’m supposed to do. It’s my job.”<br /><br /><img style="TEXT-ALIGN: center; MARGIN: 0px auto 10px; WIDTH: 250px; DISPLAY: block; HEIGHT: 215px; CURSOR: hand" id="BLOGGER_PHOTO_ID_5384706244774143682" border="0" alt="" src="http://4.bp.blogspot.com/_Kom-mudcX5I/SrpRzNuXJsI/AAAAAAAAAuk/_deQ_G_rEMs/s400/mh1.jpg" /><br />Does a person have to possess an extreme level of commitment to be a good mathematician? To choose mathematics as a major in college, should one feel that it is a calling? How about for success in graduate school in one of the mathematical sciences? Can a person forge a happy career in our subject without passion, or is something approaching devotion a vital ingredient?<br /><br />I think in many circumstances we have, sometimes almost unwittingly, made passion a pre-requisite to entering the major, much to the detriment of the field. There are some of us who have had successful and rewarding careers in mathematics and related fields without having been on the high school math team, without taking the Putnam exam as undergraduates, and without losing sleep over problems. I enjoy mathematics, and I think a day at a math meeting with interesting talks is a great way to spend my time. But at the end of that day when I’m at dinner and a colleague grabs a pencil and a napkin and says, “Here’s a cute problem,” count me among those who try to change the subject.<br /><br />Of course passion for mathematics can be a wonderful thing and it may be a necessity for the giant steps that move the field forward. The truly great mathematicians have that trait and I recognize its value. I enjoy reading mathematical history and I delight in accounts of the single minded tenacity shown in solving hard problems, and the euphoric feeling of triumph when successful. Anyone who has watched the wonderful Nova episode about Andrew Wiles and his solution of Fermat’s last theorem can see the great interaction between a scientific breakthrough and a personal victory. But tremendous devotion is demonstrably neither a necessary nor sufficient condition for building a successful career in the mathematical sciences. To use a baseball analogy, it might be necessary for a “Hall of Fame” career, but those of us making contributions as utility infielders have a place too.<br /><br />Making passion a requisite part of our culture has costs. Here are a few of them:<br /><br /><em>Requiring passion discourages talented students from studying mathematics.<br /></em>Recruiting students to mathematics is like a political party trying to decide whether to appeal to a committed base or expand a point of view to broaden participation. A common theme among colleges that have large and successful math majors is that they have put up the ‘big tent’ and allowed lots of students in. Does this hurt the quality of their top majors? It doesn’t appear to. For one thing it gives them enough students to offer required and elective courses more than once every other year!<br /><br /><em>Passion makes for poor advising.</em><br />Advisors who want passion as a pre-requisite often see graduate school as the only winning outcome for students with talent in mathematics. This runs counter to one basic goal of a liberal arts education, namely to encourage people to be thoughtful, adaptable and open-minded in career choices. Mathematics majors can go on to be wonderful accountants, machinists or landscapers and perhaps make unexpected contributions in those areas thanks to their point of view.<br /><br /><em>Faculty may quit research sooner than they should.<br /></em>When faculty members insist that their research must be elegant, ground breaking and the result of a deep commitment, a common mid-career result is to give up completely. But there are myriad open problems in other disciplines where we can be of great help. There are plenty of opportunities for pedagogical research. There are consulting opportunities in business and industry. My PhD is in statistics, so finding these opportunities might be easier for me than for people in more abstract fields, but there are dozens of underpublicized success stories of people who make contributions in this more mundane way.<br /><br />I feel fortunate to have an enjoyable, rewarding and I hope useful career in mathematics and statistics. My nonmathematical friends see me as a math nerd who sees the world differently than they do, and some of my mathematical colleagues probably see me as a slacker who doesn’t really ‘do’ math. That’s a balance I like.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com2tag:blogger.com,1999:blog-7775513617406888811.post-58565286643262405222009-04-01T13:20:00.005-04:002009-04-03T09:57:34.768-04:00Graphical Dysfunction<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_Kom-mudcX5I/SZH-DML-zbI/AAAAAAAAAXE/psceXzRPFDw/s1600-h/onmeasurement.gif"></a><p><strong><span style="font-size:130%;">Frank Swenton - Middlebury College</span></strong><br /></p><p>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.<br /><br />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.<br /><br />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. <em>X</em> 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.”<br /><br />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.<br /></p><img id="BLOGGER_PHOTO_ID_5319775163108133794" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 350px; CURSOR: hand; HEIGHT: 249px; TEXT-ALIGN: center" alt="" src="http://2.bp.blogspot.com/_Kom-mudcX5I/SdOjU1IBi6I/AAAAAAAAAbs/71gQul3oOYg/s400/am2.jpg" border="0" /><br />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.<br /><br />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 <em>X</em> to <em>Y</em> is as a particular subset of the Cartesian product <em>X <span class="Apple-style-span" style="font-style: normal;">x </span>Y</em>. However, consider the logical definition of the implication “if <em>P</em>, then <em>Q</em>”: do we properly conceptualize it as “<em>Q</em> or not <em>P</em>,” or is this merely the logical statement that defines its formal meaning? A function, while encoded as a subset of <em>X <span class="Apple-style-span" style="font-style: normal;">x </span>Y</em>, is conceptually no more a subset of <em>X <span class="Apple-style-span" style="font-style: normal;">x</span> Y</em> 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.<br /><br />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.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com11