tag:theconversation.com,2011:/columns/kevin-knudson-161348Orders of magnitude – The Conversation2016-10-05T14:56:02Ztag:theconversation.com,2011:article/665322016-10-05T14:56:02Z2016-10-05T14:56:02ZThe Nobel Prize for Physics goes to topology – and mathematicians applaud<figure><img src="https://cdn.theconversation.com/files/140538/section/width496/6493y623-1475674497.jpg" /><figcaption><span class="caption">Math doesn't get its own Nobel, but is the foundation for much Prize-winning research.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/tereneta/88098709">Tim Ereneta</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span></figcaption></figure><p><a href="https://sharepoint.washington.edu/phys/people/Pages/view-person.aspx?pid=85">David Thouless</a>, <a href="http://physics.princeton.edu/%7Ehaldane/">Duncan Haldane</a> and <a href="https://vivo.brown.edu/display/jkosterl">Michael Kosterlitz</a> received the <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/">2016 Nobel Prize for Physics</a> for their work on exotic states of matter. They were inspired by the observation that some materials have unusual electrical properties – and their investigations led them to topology. That’s the branch of mathematics concerned with the properties of geometric objects that don’t change when bent or stretched (though torn would be a different story). As there is no Nobel Prize for mathematics, the topology community is understandably excited by this recognition of the utility of our discipline.</p>
<p>The old saw is that a topologist is a mathematician who cannot tell the difference between a doughnut and a coffee cup. (This joke is getting tiresome, but we stick with it anyway.) Both objects have just one hole and it’s easy to see how to deform one to the other.</p>
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<figcaption><span class="caption">Mmm…doughnut. (Video by Jim Fowler)</span></figcaption>
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<p>Topology aims to classify these spaces via indirect means. Since it’s often rather difficult to demonstrate how to deform a particular space to make it look like another, topologists develop mathematical machinery that takes spaces as an input and produces an algebraic object. This output might just be a number or it could be more complicated, but the machine should take spaces that are “the same” and spit out the same result. This allows us to distinguish spaces – two inputs are different if the corresponding outputs are different.</p>
<p>For example, it may seem obvious that a doughnut and a sphere are distinct objects, but just because you cannot see how to deform one to the other it doesn’t follow that it’s impossible. Topology comes to the rescue, however. One of many ways to show that a sphere and a doughnut aren’t the same is to compute their <a href="https://en.wikipedia.org/wiki/Fundamental_group">fundamental groups</a>. This is an algebraic object built from considering loops in the space.</p>
<p>A useful way to visualize loops is to imagine a rubber band lying on the surface of an object. First consider the sphere. Any loop on the sphere contains a disc inside it, and now you can imagine shrinking that loop down to a single point by pulling it through the disc. So there aren’t any interesting loops on the sphere – they are all deformable to a single point.</p>
<p>That’s not true for the doughnut, however. In fact there are lots of interesting loops on its surface (we are dealing with a hollow doughnut; there’s nothing but air inside). One such loop is obtained by drawing a circle around a vertical cross section (the blue loop in the figure below). Another arises from a horizontal cross-section (the red loop). It’s impossible to contract these loops down to a the same single point, so the fundamental groups of the sphere and doughnut aren’t the same and thus, they are different objects.</p>
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<span class="caption">Loops on a doughnut.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:ToricCodeTorus.png">Woottonjames</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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<h2>The topology of materials</h2>
<p>Topology works in all dimensions, but physics is mostly concerned with our three-dimensional universe (well, that’s not always true – just ask <a href="https://en.wikipedia.org/wiki/String_theory#Extra_dimensions">string theorists</a>). When studying electrical properties of materials, we are definitely dealing with three dimensions. Even a thin wire has length, width and height. For a fixed electrical conductor, say a copper wire, it’s usually possible to determine the relationship between the voltage placed on the wire and the current that flows. Sometimes, however, materials experience an electrical phase transition (<a href="https://en.wikipedia.org/wiki/Superconductivity#Superconducting_phase_transition">superconductivity</a>, for example, which is obtained by lowering the temperature of the material) and the usual equations governing voltage and current break down. </p>
<p>Thouless, Haldane and Kosterlitz discovered that mathematically these <a href="https://en.wikipedia.org/wiki/Kosterlitz%E2%80%93Thouless_transition">transitions</a> correspond to an abrupt change in the topological type of the material. Certain thin films can be considered as being two-dimensional – imagine a surface that’s only one atom thick – and electrical current often flows in channels on the surface with low resistance. It turns out that there are points where the electrons flow around in a circular motion, sometimes clockwise and sometimes counterclockwise, and the number of such points can change as the material undergoes a phase transition.</p>
<p>Mathematicians immediately recognize this type of space from a first course in algebraic topology – it’s a plane with a few points removed and its fundamental group is very easy to compute. It turns out that the number of these types of points completely determines the topological type of the space.</p>
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<span class="caption">How can you scientifically describe a black hole without math?</span>
<span class="attribution"><a class="source" href="http://www.jpl.nasa.gov/spaceimages/details.php?id=PIA16695">NASA/JPL-Caltech</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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<h2>Topology elsewhere in physics</h2>
<p>Einstein’s <a href="https://en.wikipedia.org/wiki/General_relativity">general theory of relativity</a> posits that space-time is curved by gravity. The equations also imply the existence of black holes, which in mathematical terms correspond to <a href="https://en.wikipedia.org/wiki/Singularity_theory">singularities</a>, points in a space where all hell breaks loose (so to speak). A typical example familiar to calculus students is a point on the graph of a function where the derivative fails to exist. Much more complicated examples are possible and the space around such points can have interesting topology. Around ordinary points, space looks like a three-dimensional ball, but around singularities space can be knotted in unusual ways. Of course, we can’t experience this ourselves, but we can model it mathematically.</p>
<p>Topology has provided a framework in physics in other ways, such as the development of <a href="https://en.wikipedia.org/wiki/Topological_quantum_field_theory">topological quantum field theories</a>. <a href="https://en.wikipedia.org/wiki/String_theory">String theory</a> is a generalization of this idea in which particles are modeled by one-dimensional objects called strings. These theories, unlike Einstein’s four-dimensional spacetime, require extra dimensions to be consistent – either 10, 11 or 26 depending on which model you prefer. Why don’t we observe these dimensions? The prevailing interpretation is that they are “small” and curl up on themselves so that we don’t notice. These extra dimensions form a type of space familiar to algebraic geometers called a <a href="https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold">Calabi-Yau manifold</a>. </p>
<p>So it seems that a great deal of theoretical physics is based in sophisticated mathematics. Using ideas from topology, algebraic geometry and abstract algebra, not to mention differential equations and probability, physicists attempt to make sense of our universe. While math may not have its own Nobel Prize, many of the significant advances in other disciplines would not be possible without the development of sophisticated mathematics to provide the proper language for stating the results (Heisenberg’s <a href="https://en.wikipedia.org/wiki/Uncertainty_principle">uncertainty principle</a>, for example).</p>
<p>This is all heady stuff. In the end, though, the discoveries made by Thouless, Haldane and Kosterlitz have led to practical devices currently in use in industry (for example, efficient hard drives in computers) and may lead to advances in <a href="https://en.wikipedia.org/wiki/Quantum_computing">quantum computing</a>. Understanding how electrons move in materials is crucial to building better computers and instruments, and it’s exciting for us mathematicians to know that topology can help get us there.</p><img src="https://counter.theconversation.com/content/66532/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/647872016-09-15T01:59:02Z2016-09-15T01:59:02ZWhat exactly does 'instantaneous' mean?<figure><img src="https://cdn.theconversation.com/files/136342/section/width496/tcktw3fg-1473692698.jpg" /><figcaption><span class="caption">Lightning moves pretty quickly; would you call it instantaneous?</span> <span class="attribution"><a class="source" href="http://www.noaanews.noaa.gov/stories2013/images/lightning_safety_300.jpg">Steven Vanderburg, NOAA</a></span></figcaption></figure><p>How short is an “instant”? Is it a second? A tenth of a second? A microsecond? You might think all of these qualify. What about 100 years? That certainly doesn’t seem like an instant, and to a human being, it isn’t, since we’d be lucky to have a lifespan that long. But to a giant sequoia, say, 100 years is no big deal. And in geological terms it’s practically nothing.</p>
<p>How should we make sense of the idea of an instant? Does it cloud our judgment when we make decisions, both as individuals and as a society? Are we moving too slowly on solving big problems because we don’t see them happening “instantly”?</p>
<h2>What does math say?</h2>
<p>When Newton and Leibniz developed the calculus, they were forced to confront the infinitely small. The goal was to understand the idea of the “instantaneous velocity” of an object – that’s the speed at which something is moving at a particular instant in time (think of your car’s speedometer reading). They took the following approach.</p>
<p>We know how to compute average speed over some time interval: Simply take the total distance traveled and divide by the total time. For example, if the object travels 1 meter in 1 second then the average speed is 1 m/s. But what if you have a better measuring device? Say instead you can discover that the object really traveled 20 cm in the first 10th of a second. Then the average speed over that interval is 2 m/s and you’d probably agree that is a better approximation to what we mean by the instantaneous velocity of the object at that point. </p>
<p>But it’s still just an approximation. To get the real value, you would need to take smaller and smaller time intervals and have increasingly accurate measuring equipment. In the 17th century, the way mathematicians got around this was to talk about <a href="https://en.wikipedia.org/wiki/Infinitesimal">infinitesimals</a>, quantities that were not zero yet were smaller than any positive number you can think of, including really tiny fractions like 1/1,000,000,000,000,000,000,000,000,000.</p>
<p>Some scientists of the day, as well as various institutions (the Jesuits, for example), rejected this idea as nonsense. Indeed, the idea that one could divide things forever flew counter to the Platonic ideal of indivisibles (also called atoms) and therefore did not sit well with the Renaissance embrace of ancient Greek philosophy. There’s a great book about this called <a href="http://us.macmillan.com/infinitesimalhowadangerousmathematicaltheoryshapedthemodernworld/amiralexander">“Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World”</a>; I recommend it heartily. Still, this is how calculus was done until <a href="https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy">Cauchy</a> introduced the formalism of <a href="https://en.wikipedia.org/wiki/Limit_of_a_function">limits</a>, thereby pushing infinitesimals <a href="http://plato.stanford.edu/entries/continuity/">out of the picture</a>. Roughly speaking, a function <em>f</em> has limit <em>L</em> as <em>x</em> approaches <em>a</em> if the values of <em>f</em>(<em>x</em>) can be made arbitrarily close to <em>L</em> by taking <em>x</em> sufficiently close to <em>a</em>. The precise mathematical definition of this idea obviates the need for the old-fashioned use of infinitesimals.</p>
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<span class="caption">When you’ve got all this, what’s one more?</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=88719397">Cash image via www.shutterstock.com.</a></span>
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<p>Still, it’s a shame that infinitesimals fell out of favor, because they’re really useful for thinking about relative scale. An example I always give my students when talking about the reverse problem of dealing with the infinitely large is to talk about money. If you are a billionaire, meaning you have roughly 10⁹ dollars, you sure don’t care about 100 (or 10²) dollars. That’s a difference of seven orders of magnitude, and from your billionaire point of view it’s pointless to get upset over 100 dollars (indeed, you have 10 million hundred-dollar bills at your disposal).</p>
<p>In a similar way, infinitesimals help us deal with the infinitely small – a microsecond (1 millionth of a second) is a short amount of time, but it’s huge relative to a picosecond (10⁻¹² of a second). In mathematical terms, if <em>dx</em> denotes a small amount (like a microsecond) then its square (<em>dx</em>)² (a picosecond) is negligible. So when you’re working on time scales in the seconds you don’t really care about microseconds, and when you’re working on microsecond scales you don’t really care about picoseconds. </p>
<p>(By the way, our words for time are <a href="https://en.wikipedia.org/wiki/Minute">based on these relative notions of smallness</a>. A minute is so named because it was considered small relative to an hour. Seconds were once called “second minutes” to indicate their relative insignificance.) </p>
<h2>What’s your point of view?</h2>
<p>I bring this up because a pair of articles I read recently made me wonder if our human-influenced idea of “instantaneous” is leading us to unfortunate decisions.</p>
<p>Question: <a href="https://www.1843magazine.com/features/the-human-layer">Has the planet entered a new geological epoch</a>, the so-called “<a href="https://theconversation.com/an-official-welcome-to-the-anthropocene-epoch-but-who-gets-to-decide-its-here-57113">Anthropocene</a>”? <em>Homo sapiens</em> has undoubtedly influenced the Earth’s environment, and some geologists are arguing for a change to the <a href="http://www.stratigraphy.org/ICSchart/ChronostratChart2016-04.jpg">International Chronostratigraphic Chart</a>, the official timeline of periods, eons and other geological timescales. (We currently live in the Holocene epoch, already distinguished by the appearance of human beings on the scene.)</p>
<p>I’m not a geologist, so I cannot comment on whether or not this is something we should do, but the obvious first problem to be solved would be settling on a start date for this proposed epoch. Should it be the beginning of the <a href="https://en.wikipedia.org/wiki/Industrial_Revolution">Industrial Revolution</a> in the late 18th century? What about the beginning of <a href="https://en.wikipedia.org/wiki/Mining">mining</a> in ancient Egypt around 2500 BC? Or how about the mid 20th century, as <a href="http://doi.org/10.1177/2053019614564785">others have argued</a>? </p>
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<span class="caption">Compared to Earth’s existence, yours doesn’t even look like a blip in time.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Geologic_Clock_with_events_and_periods.svg">Woudloper</a></span>
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<p>The Earth is roughly <a href="https://en.wikipedia.org/wiki/Age_of_the_Earth">4.5 billion years old</a>. Even if we decided this new epoch began 3,000 years ago, that is still effectively now in geological terms. There have been a million and a half 3,000-year periods in the planet’s life. When things move on such timescales, perhaps we’re just splitting hairs when thinking about when to declare something like this has begun.</p>
<p>Climate change presents another example. <a href="http://www.nytimes.com/2016/09/04/science/flooding-of-coast-caused-by-global-warming-has-already-begun.html?_r=0">Sea levels are rising</a>, but the change is not immediately noticeable. Still, by the end of the 21st century, even the most <a href="https://www.ipcc.ch/pdf/unfccc/cop19/3_gregory13sbsta.pdf">conservative estimates</a> suggest a three- or four-foot rise, with some scientists predicting it will be double that amount.</p>
<p>Why all the <a href="http://wndbooks.wnd.com/the-greatest-hoax/">denialism</a> and resistance to action, then? Aside from the obvious political disagreements, there is a more basic cause for the inertia: We don’t see it happening in real time. Sure, we notice there’s not as much snow in the winter as there was when we were kids or that the streets flood in Miami Beach on sunny days at high tide nowadays, but that could just be a fluke, right? Don’t we need more data?</p>
<p>In human terms, these changes are not instantaneous, but in the Earth’s climate cycle they effectively are. We are waiting for some catastrophic event to clearly tell us the climate has officially changed, but it simply takes longer than that. We’re looking for a sign on our human timescale, which is just infinitesimal from a geological viewpoint. But once a few more billion years have passed, some future entity will be able to spot the turning point – though not down to the year or century (a geological instant).</p>
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<a href="https://cdn.theconversation.com/files/137435/area14mp/image-20160912-19222-o0ppnr.jpg"><img alt="" src="https://cdn.theconversation.com/files/137435/width754/image-20160912-19222-o0ppnr.jpg"></a>
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<span class="caption">Six meters of sea level rise would cover the coastal areas marked in red.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:6m_Sea_Level_Rise.jpg">NASA</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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<h2>Fast or slow, it comes down to scale</h2>
<p>In the absence of catastrophic planetary events, such as a large meteor collision, significant change to the Earth takes time. But it’s important to keep in mind that our relatively short lifespans distort our perception of “instantaneous” events.</p>
<p>As far as the planet is concerned, with its phases measured in the tens or hundreds of millions of years, things are moving pretty quickly. A 1℃ increase in global temperature in 100 years is very fast. If we use this to approximate the future, we quickly see that the planet would be virtually uninhabitable within a few hundred years. The real dynamics are complicated, of course, but perhaps we should keep this simple calculus in mind as we attempt to craft sustainable solutions. Scale is everything and our idea of small doesn’t necessarily align with reality.</p><img src="https://counter.theconversation.com/content/64787/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/557402016-03-04T19:18:30Z2016-03-04T19:18:30Z'The Math Myth' fuels the algebra wars, but what's the fight really about?<figure><img src="https://cdn.theconversation.com/files/113910/width496/image-20160304-17753-1dd1vyc.jpg" /><figcaption><span class="caption">A confused student might not be leaving a math classroom....</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=295860566">Student image via www.shutterstock.com.</a></span></figcaption></figure><p>I discovered recently that my calculus students do not know the meaning of the word “quorum.” Since a course in American government is a high school graduation requirement in most states (including here in Florida), I was taken aback.</p>
<p>How should I react? Should I take to the editorial pages of <em>The New York Times</em>, bemoaning the state of civics education? Should I call out political scientists and high school history teachers for their failures?</p>
<p>Surely you’d admonish me to calm down a bit and perhaps not venture into disciplines where I’m not an expert.</p>
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<span class="attribution"><a class="source" href="http://thenewpress.com/books/math-myth">The New Press</a></span>
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<p>Yet Andrew Hacker, professor emeritus of political science at the City University of New York, recently <a href="http://www.nytimes.com/2016/02/28/opinion/sunday/the-wrong-way-to-teach-math.html">took this exact approach</a> to attack the teaching of algebra in American schools. He also <a href="http://thenewpress.com/books/math-myth">wrote a book</a>. And he’s <a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html">done it before</a>.</p>
<p>Nor is he alone. Novelist Nicholson Baker <a href="https://harpers.org/archive/2013/09/wrong-answer/">wrote a piece</a> for <em>Harper’s</em> in 2013 that got the math community talking. The real target of Baker’s piece was the accountability movement and the associated standardized testing, but he chose mathematics as his straw man because it (a) is easy, and (b) will sell magazines. He manages to boil the modern course in Algebra II down to this:</p>
<blockquote>
<p>It’s a highly efficient engine for the creation of math rage: a dead scrap heap of repellent terminology, a collection of spiky, decontextualized, multistep mathematical black-box techniques that you must practice over and over and get by heart in order to be ready to do something interesting later on, when the time comes.</p>
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<p>At least Baker is an entertaining writer.</p>
<p>Hacker makes many of the same points in his <em>Times</em> articles, decrying algebra as a high school graduation requirement that holds back far too many students from having a productive life. He argues instead for “numeracy” and suggests what such a course should contain. It’s mostly statistics and financial mathematics, and lessons in visualizing and analyzing data.</p>
<p>To fight off the counterassertion that it’s possible to learn this material in a high school advanced placement statistics course, Hacker comes up with lists of obscure terminology: “The A.P. [Statistics] syllabus is practically a research seminar for dissertation candidates. Some typical assignments: binomial random variables, least-square regression lines, pooled sample standard errors.”</p>
<h2>It’s not just happening in math</h2>
<p>Every subject in school has been broken down into a string of often unrelated facts or tasks, not just mathematics.</p>
<p>I recall an episode from my own son’s experience in ninth grade while taking “Honors Pre-AP English I” (yes, that’s the real name of the course, not some Orwellian nightmare). His teacher led the class through the “CD/CM method” of essay writing, which goes like this. Fill out a worksheet with the “funnel” (4-7 sentence introduction), the thesis statement, and then for each of three paragraphs create 11 (!) sentences – the topic sentence (fine) and then CD#1, CM#1, CD#2,CM#2,…,CD#5,CM#5. What is a CD, you ask? Concrete Detail. A CM? Comment, of course.</p>
<p>Now, this is really just a superextended outline for an essay, but my son was extremely frustrated by this, eventually exclaiming, “I just want to write the damn paper!”</p>
<p>Is this example from the humanities really any different from what Hacker and Baker complain about?</p>
<p>Hacker is not completely wrong, however. School mathematics <em>has</em> largely been drained of context and beauty. University mathematicians complain about this, too.</p>
<p>For example, my son has also brought home worksheets full of dozens of polynomials with the simple instruction: Factor. But why?</p>
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<span class="caption">Light rays striking a parabolic mirror reflect to a common point called the focus (point F above).</span>
<span class="attribution"><span class="source">created in Geogebra by the author</span></span>
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<p>There is no context given for why we care about polynomial equations, no discussion of why parabolas (graphs of quadratic equations) are useful things. Maybe we should explain that without parabolas, we wouldn’t have good headlights on our cars or all those pretty pictures of deep space from the Hubble telescope. But just as mathematicians would not argue for the elimination of English or civics from the high school curriculum, Hacker shouldn’t be arguing for the elimination of algebra.</p>
<p>Let’s be honest. Mostly because of the accountability movement and high-stakes testing, K-12 education suffers from these types of problems in every subject. Picking on math alone because it’s particularly vexing for some people is unsporting.</p>
<h2>Credibility gap</h2>
<p>Of course, Hacker and Baker have proposals for how to fix this mess. The problem is that the major prerequisite for much of what Hacker proposes is, ironically, algebra. Not so much the grinding, symbol-driven form of algebra taught in school today, but algebra nonetheless. Reading bar graphs in the newspaper is a skill that we should expect high school graduates to be able to do, but nontrivial calculations with data require at least some facility with algebra. Hacker surely knows this, but it would undermine his argument to admit it.</p>
<p>He’s certainly not wrong that some students fall by the wayside, and the way we teach algebra and geometry in the middle grades is largely to blame. Stanford mathematician Keith Devlin wrote a <a href="http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html">wonderful response</a> to Hacker’s recent piece, pointing out how his ideas may actually be correct but misguided:</p>
<blockquote>
<p>Not only did that suggestion [the elimination of algebra from the high school curriculum] alienate accomplished scientists and engineers and a great many teachers – groups you’d want on your side if your goal is to change math education – it distracted attention from what was a very powerful argument for introducing the teaching of algebra into our schools, something I and many other mathematicians would enthusiastically support.</p>
</blockquote>
<p>Unfortunately, Hacker undermines his credibility by stating falsehoods. For example, he claims “Coding is not based on mathematics … Most people who do coding, programming, software design, don’t do any mathematics at all.” It may be true that these individuals are not crunching numbers all day (that’s what software is for, of course), but the algorithmic processes underlying coding are the very essence of mathematics. To say otherwise is just delusional.</p>
<p>Hacker also asks, “Would you go to a mathematician to tell us what to do in Syria? It just defies comprehension.” Actually, it shouldn’t. The Central Intelligence Agency and other national security groups <a href="https://www.cia.gov/careers/opportunities/analytical">employ thousands of mathematicians to analyze data</a> associated with foreign affairs, looking for patterns amid the chaos. So, Hacker is just plain wrong about some things, even if his overall idea has merit. </p>
<h2>We’re all on the same team</h2>
<p>You see, college math professors <em>know</em> there is a problem with K-12 mathematics. We see the results in our classrooms on campus. As much as Hacker would like to believe his <em>ad hominem</em> assertions about math faculties at high schools and colleges, we really just want our students well-prepared for the beautiful, fascinating and, yes, useful material we have to offer.</p>
<p>Algebra is a beautiful baby; it would be a shame to throw it out with some dirty bathwater.</p><img src="https://counter.theconversation.com/content/55740/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/547172016-02-19T11:08:56Z2016-02-19T11:08:56ZExtreme numbers: the unimaginably large and small pop up in recent experiments<figure><img src="https://cdn.theconversation.com/files/111848/width496/image-20160217-19245-q5aano.jpg" /><figcaption><span class="caption">It's a lot of grains of sand, but numbers can get a whole lot bigger....</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/hisgett/2289454268">Tony Hisgett</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>The physics world erupted in celebration this month with the confirmed <a href="http://www.nytimes.com/2016/02/12/science/ligo-gravitational-waves-black-holes-einstein.html?_r=0">discovery</a> of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (<a href="http://www.ligo.org/">LIGO</a>) group. Predicted by Einstein a century ago, the discovery verifies his description of the universe in which space and time can warp and bend.</p>
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<img alt="" src="https://cdn.theconversation.com/files/111566/width754/image-20160215-22560-d43qif.jpg">
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<span class="caption">Orbiting black holes generate gravitational waves.</span>
<span class="attribution"><span class="source">NASA</span></span>
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<p>And what is the evidence gathered by LIGO? A billion years ago, a pair of black holes of masses about 30 times that of the sun collided, releasing about three solar masses’ worth of energy in the form of gravitational waves. Those waves traveled through space and reached the LIGO antennas, one in Louisiana and one in Washington, seven milliseconds apart, vibrating the mirrors at the end of each antenna’s 2.5-mile-long vacuum tube by a mere four thousandths the diameter of a proton.</p>
<p>I’m no physicist, but the LIGO numbers intrigue me. In fact, I’ve noticed quite a few huge (and tiny) numbers in recently announced scientific advances, which got me to thinking about how real physical situations force us to deal with numbers so extreme they’re inconceivable.</p>
<p>Let’s unpack these numbers. The gravitational waves travel at the speed of light, so the black holes that generated them were roughly one billion light-years away from Earth. That’s more than 6 billion trillion miles, or 6 x 10²¹ miles. The energy that created the waves is roughly equivalent to the light output of a billion trillion suns. And the end result was nudging a pair of mirrors by 4 x 10⁻¹⁸ meters, an unfathomably small distance.</p>
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<figcaption><span class="caption"><em>Powers of Ten,</em> by Charles and Ray Eames.</span></figcaption>
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<p>A great visualization of these scales can be seen in the classic film <em>Powers of Ten</em>, created by Charles and Ray Eames in 1977. It doesn’t go out as far as the black holes that created the gravitational waves, nor does it go as small as the movement of the LIGO mirrors, but it does give a great sense of the scales involved in the recent announcement.</p>
<h2>Even larger numbers</h2>
<p>Here’s a question: say you have 128 tennis balls. How many different ways can you arrange them so that each ball touches at least one other? You can stack them, lay them out in various grids, stack the layers and so on. There are probably a lot of configurations, right?</p>
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<a href="https://cdn.theconversation.com/files/111567/area14mp/image-20160215-22560-fex3rr.jpg"><img alt="" src="https://cdn.theconversation.com/files/111567/width754/image-20160215-22560-fex3rr.jpg"></a>
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<span class="caption">How many ways can tennis balls be arranged?</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/atomictaco/5390499643">Atomic Taco</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>This question was <a href="http://www.cam.ac.uk/research/news/how-many-ways-can-you-arrange-128-tennis-balls-researchers-solve-an-apparently-impossible-problem">answered recently</a> by a team of researchers at Cambridge University. The number of possible arrangements is on the order of 10²⁵⁰; that’s a 1 with 250 zeroes after it. To give a sense of how large this number is, note that there are only about 10⁸⁰ atoms in the universe. In fact, if we packed the known universe with protons, there would be only about 10¹²⁶ of them. So if we could somehow encode each configuration of the tennis balls on an atom (or even a subatomic particle), we would be able to get through only about the cube root of the total number of possibilities. </p>
<p>Since it’s impossible to actually count all the arrangements of the balls, the team used an indirect approach. They took a sample of all the possible configurations and computed the probability of each of them occurring. Extrapolating from there, the team was able to deduce the number of ways the entire system could be arranged, and how one ordering was related to the next. The latter is the so-called <a href="https://en.wikipedia.org/wiki/Configuration_entropy">configurational entropy</a> of the system, a measure of how disordered the particles in a system are.</p>
<p>This may seem like an odd calculation to make, but it is an important question in granular physics. This is the study of the behavior of materials that are granular in nature, such as sand or snow. If we wish to understand how sand dunes form and evolve over time, or how avalanches happen, we must first be able to enumerate the possible initial configurations of the particles. Clearly, 128 particles is nowhere near a large enough number for us to begin to understand a sand dune, but it’s a start. And the methods employed for this study may yield insights that will help attack bigger systems.</p>
<h2>Still bigger numbers</h2>
<p>A number such as 10²⁵⁰ is enormous, but relative to numbers “close” to infinity it is effectively zero. At scales like this, I find it comforting to turn to literature and philosophy. In “<a href="http://eduscapes.com/history/contemporary/babel.pdf">The Library of Babel</a>,” the fascinating short story by Jorge Luis Borges, we learn about a certain library in which each book has 410 pages, and each page has 40 lines of 80 characters. The alphabet in use has 22 letters and three punctuation marks, making a total of 25 orthographic characters. We are told that every possible book is somewhere in this imagined library. So, how many books are there? First note that there are 410 x 40 x 80 = 1,312,000 characters in each book and since we have 25 choices for each character, there are 25¹³¹²⁰⁰⁰ possible books. As a power of 10, that’s roughly 10¹⁸³⁴⁰⁹⁷.</p>
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<a href="https://cdn.theconversation.com/files/111995/area14mp/image-20160218-1274-b5a2mf.jpg"><img alt="" src="https://cdn.theconversation.com/files/111995/width754/image-20160218-1274-b5a2mf.jpg"></a>
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<span class="caption">A brick and mortar library, however grand, holds a beyond-minuscule amount of books compared to the imagined Library of Babel.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:George-peabody-library.jpg">Matthew Petroff</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>If we can’t wrap our heads around 10²⁵⁰, how are we to manage a number like this? Borges’ fictional library tells us how. While we can’t possibly enumerate a catalog of all the books, we can imagine any book we like. There is a completely blank book. There is a book with a single comma in the middle of page 204 and nothing else. There are actually 1,312,000 books with a single comma and nothing else (just in each of the possible locations). There is a book with only the letter y in every spot. This article you’re reading right now appears exactly as it is written (by spelling out the numbers and ignoring extraneous punctuation) in an enormous number of books in the library (10 to a very large power, certainly more than 1.7 million). It appears in every language on the planet (suitably translated into the alphabet). </p>
<p>If you want to play around with this idea, there is an <a href="https://libraryofbabel.info/">online Library of Babel</a> that catalogs every possible page of 3200 characters. This amounts to only about 10⁴⁶⁷⁷ books, a tiny fraction of the total library, but it’s great fun to search for strings of characters. Jonathan Basile, the site’s creator, has devised a scheme for cataloging the books based on Borges’ description of the library as a collection of hexagonal cells with a certain number of books on each shelf (only four of each cell’s six walls contain shelves). For example, the phrase “when in the course of human events” occurs by itself at the top of page 186 of volume 21 on shelf 1 of wall 3 of a hexagon labeled with a 3254-digit identifier in base 36. Whew. </p>
<p>And yet, despite the enormity of the Library of Babel, the number of books is less than the <a href="http://www.mersenne.org/primes/?press=M74207281">largest known prime number</a>, discovered in January 2016. The Mersenne number M74207281 = 2⁷⁴²⁰⁷²⁸¹ - 1 has more than 22 million digits, way more than the puny number of books in the library (only about 1.8 million digits). And there are surely larger primes out there (Euclid <a href="https://en.wikipedia.org/wiki/Euclid's_theorem">told us so</a>), with billions, trillions, or 10²⁵⁰ digits.</p>
<h2>Should we care?</h2>
<p>So, are these unimaginable numbers actually good for anything? In a practical sense, no. They are simply too large to be useful in everyday scientific computation (we need big primes for encryption algorithms, but not <em>that</em> big). And once you’ve counted every subatomic particle in the universe, there’s probably not much need for a bigger number. They do provide fertile ground for thought experiments, though, and illustrate the human capacity to ponder the unreasonably large (and small, too).</p><img src="https://counter.theconversation.com/content/54717/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/524912015-12-21T12:47:59Z2015-12-21T12:47:59ZA purported new mathematics proof is impenetrable – now what?<figure><img src="https://cdn.theconversation.com/files/106736/width496/image-20151219-27875-yjavw6.jpg" /><figcaption><span class="caption">Wait, what was that? You lost me.</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=306402011&src=id">Notations image via www.shutterstock.com.</a></span></figcaption></figure><p>What happens when someone claims to have proved a famous conjecture? Well, it depends. When a paper is submitted, the journal editor will pass it off to a respected expert for examination. That referee will then scan the paper looking for a significant new idea. If there isn’t one, then the whole argument is unlikely to get much more scrutiny. </p>
<p>But if there is a kernel of a new approach, it will be checked carefully. Additional experts may be consulted. Eventually the mathematics community may reach consensus that the argument is correct and the conjecture becomes a theorem. This can happen outside the formal refereeing process thanks to preprint servers such as the <a href="http://arxiv.org">arXiv</a>, but in the end, enough expert referees have to give the work their imprimatur before the paper is finally published in a journal.</p>
<p>In my mathematical career, there have been a few such big announcements, the most well-known being Andrew Wiles’ <a href="https://en.wikipedia.org/wiki/Wiles'_proof_of_Fermat's_Last_Theorem">solution</a> of Fermat’s Last Theorem in 1994. Grigori Perelman’s <a href="https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture">proof</a> in 2003 of the Poincaré Conjecture comes to mind as well. Now a reclusive yet respected Japanese mathematician has put forth a solution to another notorious problem.</p>
<p>In those earlier examples, the stature of the mathematicians involved made other experts interested in verifying their results. But what if the proposed solution is impenetrable? What if it reads, as University of Wisconsin Math Professor Jordan Ellenberg <a href="https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/">put it on his blog</a>, like mathematics from the future, full of new concepts and definitions that are disconnected from current language and techniques? If the author is relatively unknown it may be dismissed, or even ignored. But if the mathematician has a reputation for being careful and producing solid results, what then?</p>
<h2>The ABC conjecture</h2>
<p>Shinichi Mochizuki of the <a href="http://www.kurims.kyoto-u.ac.jp/en/index.html">Research Institute for Mathematical Sciences</a> at Kyoto University is such a mathematician. In August 2012, he posted a series of four papers on his personal web page claiming to prove <a href="https://en.wikipedia.org/wiki/Abc_conjecture">the ABC conjecture</a>, an important outstanding problem in number theory. A proof would have Fermat’s Last Theorem as a consequence (at least for large enough exponents), and given the difficulty of Wiles’ proof of Fermat’s Last Theorem, we should expect a proof of the ABC conjecture to be similarly opaque.</p>
<p>The conjecture is fairly easy to state. Suppose we have three positive integers <em>a,b,c</em> satisfying <em>a+b=c</em> and having no prime factors in common. Let <em>d</em> denote the product of the distinct prime factors of the product <em>abc</em>. Then the conjecture asserts roughly there are only finitely many such triples with <em>c > d</em>. Or, put another way, if <em>a</em> and <em>b</em> are built up from small prime factors then <em>c</em> is usually divisible only by large primes.</p>
<p>Here’s a simple example. Take <em>a=16</em>, <em>b=21</em>, and <em>c=37</em>. In this case, <em>d = 2x3x7x37 = 1554</em>, which is greater than <em>c</em>. The ABC conjecture says that this happens almost all the time. There is plenty of numerical evidence to support the conjecture, and most experts in the field believe it to be true. But it hasn’t been mathematically proven – yet.</p>
<p>Enter Mochizuki. His papers develop a subject he calls <a href="https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory">Inter-Universal Teichmüller Theory</a>, and in this setting he proves a vast collection of results that culminate in a putative proof of the ABC conjecture. Full of definitions and new terminology invented by Mochizuki (there’s something called a Frobenioid, for example), almost everyone who has attempted to read and understand it has given up in despair. Add to that Mochizuki’s odd refusal to speak to the press or to travel to discuss his work and you would think the mathematical community would have given up on the papers by now, dismissing them as unlikely to be correct. And yet, his previous work is so careful and clever that the experts aren’t quite ready to give up.</p>
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<h2>A meeting at Oxford</h2>
<p>The <a href="http://www.claymath.org/">Clay Mathematics Institute</a> and the <a href="https://www.maths.ox.ac.uk/">Mathematical Institute</a> at Oxford recently sponsored a <a href="https://www.maths.nottingham.ac.uk/personal/ibf/files/iut-sch1.html">meeting</a> about Mochizuki’s work. He was not in attendance, but many of the world’s leading number theorists and arithmetic geometers were. The goal was not to verify the proof of the ABC conjecture, but rather to equip experts in the field with enough background and information to at least begin to read through the papers carefully. There are many summaries of the meeting online (Stanford Math Professor <a href="http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/">Brian Conrad’s</a> is particularly detailed and illuminating), and some attendees <a href="https://twitter.com/search?q=mochizuki%20shinichi&src=tyah">tweeted</a> about it.</p>
<p>The general feeling was one of frustration, especially during the last two days when audience members repeatedly asked for illustrative examples, were promised they were coming, but then they never materialized. Mathematicians have little patience for being led down a rabbit hole, but the potential payoff in this case may persuade some to at least go in a little deeper.</p>
<h2>Prognosis</h2>
<p>It’s not clear what the future holds for Mochizuki’s proof. A small handful of mathematicians claim to have read, understood and verified the argument; a much larger group remains completely baffled. The December workshop reinforced the community’s desperate need for a translator, someone who can explain Mochizuki’s strange new universe of ideas and provide concrete examples to illustrate the concepts. Until that happens, the status of the ABC conjecture will remain unclear.</p>
<p>There’s a general sense among nonmathematicians that the subject is either right or wrong, and the truth is easily discovered. While our discipline does insist on rigorous, logical proof of correctness, we often argue over the details. This is good for mathematics since it generally leads to better exposition and streamlined proofs.</p>
<p>These arguments have happened before. Wiles’ proof of Fermat’s Last Theorem was scrutinized thoroughly, and an error was found which had to be corrected. Perelman’s work on the Poincaré Conjecture was only a detailed sketch of a proof which required hard work on the part of others to be made rigorous. Mochizuki’s work may eventually pass the test, but it could take many years before we get to a clean version that can be more widely understood.</p><img src="https://counter.theconversation.com/content/52491/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/497462015-11-20T11:12:17Z2015-11-20T11:12:17ZThe rush to calculus is bad for students and their futures in STEM<figure><img src="https://cdn.theconversation.com/files/101493/width496/image-20151110-5460-1v3e63d.JPG" /><figcaption><span class="caption">The author, teaching at the very front of his calculus class.</span> <span class="attribution"><span class="source">Kevin Knudson</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>Two years ago I taught a section of Calculus I to approximately 650 undergrad students in a large auditorium. Perhaps “taught” isn’t the right word. “Performed,” maybe? Unsurprisingly, my student evaluation scores were not as high as they usually are in my more typical classes of 35 students, but I do remember one comment in particular: “This class destroyed my confidence.” According to a <a href="http://www.nctm.org/Publications/Mathematics-Teacher/2015/Vol109/Issue3/Insights-from-the-MAA-National-Study-of-College-Calculus/">new report</a> from the Mathematical Association of America (MAA), this outcome is common, even among students who successfully completed a calculus course in high school. So what is going on? </p>
<p>Former MAA president <a href="http://www.macalester.edu/%7Ebressoud/">David Bressoud</a> led this five-year comprehensive study funded by the National Science Foundation. He’s been thinking about this problem for many years and has synthesized a huge amount of data measuring high school and college calculus enrollments. I heard Bressoud <a href="http://www.macalester.edu/%7Ebressoud/talks/2010/UFL-transition.pdf">speak</a> about some preliminary results of the study a few years ago, and one piece of data stuck in my head: in the mid-1980s, when I was in high school, approximately 5% of high school students took an AP exam in calculus.</p>
<p>That aligns with my personal experience in which there were about 150 students in my entire North Carolina county taking calculus in any given year (out of roughly 3,000 high school seniors). Nationally, about 60,000 students took an AP calculus exam my senior year (1987). Today? That number has risen to nearly 350,000 students taking an AP exam in calculus in 2011 (roughly 15% of high school students). As one of my colleagues remarked after Bressoud’s talk, it’s not as if the talent pool has gotten that much deeper in the last 30 years. This tripling of the proportion of students taking these exams feels wrong somehow. </p>
<h2>Why the dramatic increase?</h2>
<p>There appear to be at least two driving forces behind the rush to calculus. </p>
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<a href="https://cdn.theconversation.com/files/102074/area14mp/image-20151116-4947-11gmow6.png"><img alt="" src="https://cdn.theconversation.com/files/102074/width237/image-20151116-4947-11gmow6.png"></a>
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<span class="caption">Breakdown of all Advanced Placement exams taken in 2013.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:AP_Exams_Taken_in_2013.svg">Ali Zifan</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>One is college admissions. Students and their parents seek an advantage in the increasingly competitive admissions tournament, and the number of AP courses taken is a metric that is easy for students to boost. The increase in the number of AP exams taken is not unique to calculus; indeed, the total population of <a href="http://apreport.collegeboard.org/">students taking exams</a> doubled between 2003 and 2013, with the number of exams administered increasing by 150% over that period. As the name “Advanced Placement” suggests, these exams often yield college credit for students; this appeals to parents, as well, since it ostensibly lowers tuition costs later.</p>
<p>Another factor that must be considered is the <a href="http://www.nytimes.com/roomfordebate/2014/06/03/are-new-york-citys-gifted-classrooms-useful-or-harmful/americas-future-depends-on-gifted-students">overall decline in support</a> for enhanced education for gifted students. In an era of shrinking education budgets, school administrators find it tempting to conflate advancement with enrichment. Pushing gifted students ahead at a faster rate via AP courses is seen as a solution for meeting the needs of advanced students.</p>
<p>This approach may be dangerous in any discipline, but it is especially problematic in mathematics, where a strong foundation is key to success in upper division courses. The general strategy in high school is one of uniform advancement – taking advanced coursework in all disciplines under the assumption that gifted students are exceptional in every subject. In the drive to make it to calculus by the senior year, students often rush through algebra and geometry in lockstep with their gifted peers whether they are ready for it or not.</p>
<p>The end result is a group of students who have “succeeded” in high school calculus without really having the proper foundations, a tower built on sand. It is quite possible for students to learn the mechanics of many categories of calculus problems and to answer questions correctly on exams without really understanding the concepts. To quote the MAA’s report:</p>
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<p>In some sense, the worst preparation a student heading toward a career in science or engineering could receive is one that rushes toward accumulation of problem-solving abilities in calculus while short-changing the broader preparation needed for success beyond calculus.</p>
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<span class="caption">Calc students’ favorite friend: the graphing calculator.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/cdnphoto/4537872477">Gene Wilburn</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
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<h2>College versus high school calculus</h2>
<p>There are two flavors of AP calculus, AB and BC. The former is equivalent to a typical first-semester college course, while the latter covers the first two semesters. Exams are scored from 1 to 5; most universities grant credit for a score of 3 and up.</p>
<p>Many students take Calculus I again at their universities, even if they have a passing score on the AP exam. There are many reasons for this: some colleges insist (engineering programs in particular) and many medical schools <a href="http://www.cse.emory.edu/sciencenet/additional_math_reqs.pdf">demand</a> that applicants take the course at a university. Or students may not feel particularly confident about their abilities. In my own experience, the number of students retaking the calculus course is very high – in a typical section of engineering calculus, up to 90% of my students have taken it in high school. While there are some positive aspects to retaking the course, there are downsides, the most notable of which is overconfidence and a student’s misplaced certainty that he or she already knows the material.</p>
<p>A typical first-semester calculus course consists of 45 lectures delivered three times per week over a 15-week term. The pace is quick. Contrast that with a typical high school Calculus AB course, which meets five days per week for 180 class meetings. The college course covers the same material in a quarter of the time; students must therefore have solid skills in algebra and geometry along with good study and work habits to succeed.</p>
<p>So this is the crux of the problem: students lacking the requisite foundational abilities may not succeed because the college faculty member expects them to be at ease with these more basic ideas, freeing them to absorb and understand the new, more conceptual material. The rush to AP Calculus has instructed students in the techniques for solving large classes of standard calculus problems rather than prepare them for success in higher mathematics.</p>
<p>It’s precisely this disconnect that causes students to lose their confidence if they don’t do well in university calculus. All through high school, the evidence suggested that they were “good at math” because they succeeded in parroting what they saw demonstrated in class. Parroting is not learning, however, and may hide a student’s true abilities.</p>
<h2>What to do?</h2>
<p>The authors of the MAA report sum it up best:</p>
<blockquote>
<p>Students are better prepared for post-secondary mathematics when they have developed an understanding of the undergirding principles which, when accompanied by fluent and flexible application of the concepts and procedures of precalculus mathematics, enable them to understand calculus as a coherent and broadly applicable body of knowledge.</p>
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<p>Like so many issues in K-12 education, the reasons that we have gotten to the current state are manifold, and reversing trends is difficult. But if we want to advance STEM education and continue to produce a high-quality technical workforce we must confront this issue. We need to stop the rush to calculus and focus instead on a thorough grounding in algebra, geometry and functions.</p>
<p>Calculus is one of the great intellectual achievements of the last 400 years; shortchanging it by reducing its beauty and utility to a list of problems to be checked off a rubric does a disservice to everyone.</p><img src="https://counter.theconversation.com/content/49746/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/465852015-09-09T10:25:07Z2015-09-09T10:25:07ZThe Common Core is today's New Math – which is actually a good thing<figure><img src="https://cdn.theconversation.com/files/94197/width496/image-20150908-4358-zdmhft.jpg" /><figcaption><span class="caption">Change can be a good thing – really.</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-182868605/stock-photo-frustrated-father-throws-up-his-hands-in-despair-frustrated-elementary-age-boy-lays-his-head-on.html">Homework image via www.shutterstock.com.</a></span></figcaption></figure><p>Math can’t catch a break. These days, people on both ends of the political spectrum are lining up to deride the <a href="http://www.corestandards.org/">Common Core standards</a>, a set of guidelines for K-12 education in reading and mathematics. The Common Core standards outline what a student should know and be able to do at the end of each grade. States don’t have to adopt the standards, although many did in an effort to receive funds from President Obama’s <a href="http://www2.ed.gov/programs/racetothetop/index.html">Race to the Top</a> initiative.</p>
<p><a href="http://www.usnews.com/news/special-reports/a-guide-to-common-core/articles/2014/02/27/who-is-fighting-against-common-core">Conservatives</a> oppose the guidelines because they generally dislike any suggestion that the federal government might have a role to play in public education at the state and local level; these standards, then, are perceived as a threat to local control.</p>
<p><a href="https://www.laprogressive.com/fighting-common-core/">Liberals</a>, mostly via teachers’ unions, decry the use of the standards and the associated assessments to evaluate classroom instructors.</p>
<p>And parents of all persuasions are panicked by their sudden inability to help their children with their homework. Even <a href="http://www.newyorker.com/news/daily-comment/louis-c-k-against-the-common-core">comedian Louis CK got in on the discussion</a> (via Twitter; he has since deactivated his account). </p>
<blockquote>
<p>My kids used to love math. Now it makes them cry. Thanks standardized testing and common core!
— Louis CK (@louisck) April 28 2014</p>
</blockquote>
<p>In the middle are millions of American schoolchildren who are often confused and frustrated by these “new” ways of teaching mathematics.</p>
<p>Thing is, we’ve been down this path before.</p>
<h2>The old New Math</h2>
<p>When the Soviets launched Sputnik in 1957, the United States went into panic mode. Our schools needed to emphasize math and science so that we wouldn’t fall behind the Soviet Union and its allegedly superior scientists. In 1958, President Eisenhower signed the <a href="http://www.britannica.com/topic/National-Defense-Education-Act">National Defense Education Act</a>, which poured money into the American education system at all levels. </p>
<p>One result of this was the so-called New Math, which <a href="https://en.wikipedia.org/wiki/Secondary_School_Mathematics_Curriculum_Improvement_Study#Curriculum">focused more on conceptual understanding of mathematics</a> over rote memorization of arithmetic. Set theory took a central role, forcing students to think of numbers as sets of objects rather than abstract symbols to be manipulated. This is actually how numbers are constructed logically in an advanced undergraduate mathematics course on real analysis, but it may not necessarily be the best way to communicate ideas like addition to schoolchildren. Arithmetic using number bases other than 10 also entered the scene. This was famously spoofed by <a href="https://en.wikipedia.org/wiki/Tom_Lehrer">Tom Lehrer</a> in his song “New Math.”</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/UIKGV2cTgqA?wmode=transparent&start=0" frameborder="0" allowfullscreen></iframe>
<figcaption><span class="caption">This 60’s song about New Math gives us a glimpse of what the ‘old math’ was like.</span></figcaption>
</figure>
<p>I attended elementary school in the 1970s, so I missed New Math’s implementation, and it was largely gone by the time I got started. But the way Lehrer tries to explain how subtraction “used to be done” made no sense to me at first (I did figure it out after a minute). In fact, the New Math method he ridicules is how children of my generation – and many of the Common Core-protesting parents of today – learned to do it, even if some of us don’t really understand what the whole borrowing thing is conceptually. Clearly some of the New Math ideas took root, and math education is better for it. For example, given the ubiquity of computers in modern life, it’s useful for today’s students to learn to do binary arithmetic – adding and subtracting numbers in base 2 just as a computer does. </p>
<p>The New Math fell into disfavor mostly because of complaints from parents and teachers. Parents were unhappy because they couldn’t understand their children’s homework. Teachers objected because they were often unprepared to instruct their students in the new methods. In short, it was the <em>implementation</em> of these new concepts that led to the failure, more than the curriculum itself.</p>
<figure class="align-center zoomable">
<a href="https://cdn.theconversation.com/files/94201/area14mp/image-20150908-14047-1q4g6nd.JPG"><img alt="" src="https://cdn.theconversation.com/files/94201/width754/image-20150908-14047-1q4g6nd.JPG"></a>
<figcaption>
<span class="caption">Give us our New Math!</span>
</figcaption>
</figure>
<h2>Those who ignore history…</h2>
<p>In 1983, President Reagan’s National Commission on Excellence in Education released its report, <a href="http://www2.ed.gov/pubs/NatAtRisk/index.html">A Nation at Risk</a>, which asserted that American schools were “failing” and suggested various measures to right the ship. Since then, American schoolchildren and their teachers have been bombarded with various reform initiatives, privatization efforts have been launched and charter schools established.</p>
<p>Whether or not the nation’s public schools are actually failing is a matter of serious debate; indeed, many of the claims made in A Nation at Risk were <a href="http://eric.ed.gov/?id=EJ482502">debunked</a> by statisticians at Sandia National Laboratories a few years after the report’s release. But the general notion that our public schools are “bad” persists, especially among politicians and business groups. </p>
<p>Enter Common Core. Launched in 2009 by a consortium of states, the idea sounds reasonable enough – public school learning objectives should be more uniform nationally. That is, what students learn in math or reading at each grade level should not vary state by state. That way, colleges and employers will know what high school graduates have been taught, and it will be easier to compare students from across the country. </p>
<p>The guidelines are just that. There is no set curriculum attached to them; they are merely a list of concepts that students should be expected to master at each grade level. For example, here are the <a href="http://www.corestandards.org/Math/Content/3/NBT/">standards</a> in Grade 3 for Number and Operations in Base Ten:</p>
<ul>
<li><p>Use place value understanding and properties of operations to perform multi-digit arithmetic.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/1/">CCSS.Math.Content.3.NBT.A.1</a>
Use place value understanding to round whole numbers to the nearest 10 or 100.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/2/">CCSS.Math.Content.3.NBT.A.2</a>
Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/3/">CCSS.Math.Content.3.NBT.A.3</a>
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (eg, 9 × 80, 5 × 60) using strategies based on place value and properties of operations.</p></li>
</ul>
<p>There is a footnote that “a range of algorithms may be used” to help students complete these tasks. In other words, teachers can explain various methods to actually accomplish the mathematical task at hand. There is nothing controversial about these topics, and indeed it’s not controversial that they’re things that students should be able to do at that age.</p>
<p>However, some of the new methods being taught for doing arithmetic have caused confusion for parents, causing them to take to social media in frustration. Take the 32 - 12 problem, for example:</p>
<figure class="align-center zoomable">
<a href="https://cdn.theconversation.com/files/93956/area14mp/image-20150904-14609-18h7ni1.jpg"><img alt="" src="https://cdn.theconversation.com/files/93956/width754/image-20150904-14609-18h7ni1.jpg"></a>
<figcaption>
<span class="caption">Just because you didn’t learn it that way doesn’t make it inscrutable or wrong.</span>
</figcaption>
</figure>
<p>Once again, it’s the <em>implementation</em> that’s causing the problem. Most parents (people age 30-45, mostly), remembering the math books of our youth filled with pages of exercises like this, immediately jump to the “Old Fashion” (sic) algorithm shown. The stuff at the bottom looks like gibberish, and given many adults’ <a href="https://theconversation.com/when-parents-with-high-math-anxiety-help-with-homework-children-learn-less-46841">tendency toward math phobia/anxiety</a>, they immediately throw up their hands and claim this is nonsense.</p>
<p>Except that it isn’t. In fact, we all do arithmetic like this in our heads all the time. Say you are buying a scone at a bakery for breakfast and the total price is US$2.60. You hand the cashier a $10 bill. How much change do you get? Now, you do <em>not</em> perform the standard algorithm in your head. You first note that you’d need another 40 cents to get to the next dollar, making $3, and then you’d need $7 to get up to $10, so your change is $7.40. That’s all that’s going on at the bottom of the page in the picture above. Your children can’t explain this to you because they don’t know that you weren’t taught this explicitly, and your child’s teacher can’t send home a primer for you either.</p>
<figure class="align-center zoomable">
<a href="https://cdn.theconversation.com/files/94200/area14mp/image-20150908-15659-ep1zt2.jpg"><img alt="" src="https://cdn.theconversation.com/files/94200/width754/image-20150908-15659-ep1zt2.jpg"></a>
<figcaption>
<span class="caption">New ways to learn can be better for students – if rolled out appropriately.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/departmentofed/9610695698">US Department of Education</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>Better intuition about math, better problem-solving</h2>
<p>As an instructor of college-level mathematics, I view this focus on conceptual understanding and multiple strategies for solving problems as a welcome change. Doing things this way can help build intuition about the size of answers and help with estimation. College students can compute answers to homework problems to 10 decimal places, but ask them to ballpark something without a calculator and I get blank stares. Ditto for conceptual understanding – for instance, students can evaluate <a href="https://en.wikipedia.org/wiki/Integral">integrals</a> with relative ease, but building one as a limit of <a href="https://en.wikipedia.org/wiki/Riemann_sum">Riemann sums</a> to solve an actual problem is often beyond their reach.</p>
<p>This is frustrating because I know that my colleagues and I focus on these notions when we introduce these topics, but they fade quickly from students’ knowledge base as they shift their attention to solving problems for exams. And, to be fair, since the K-12 math curriculum is chopped up into discrete chunks of individual topics for ease of standardized testing assessment, it’s often difficult for students to develop the problem-solving abilities they need for success in higher-level math, science and engineering work. Emphasizing more conceptual understanding at an early age will hopefully lead to better problem-solving skills later. At least that’s the rationale behind the standards.</p>
<p>Alas, Common Core is in danger of being abandoned. Some states have already <a href="http://academicbenchmarks.com/common-core-state-adoption-map/">dropped the standards</a> (Indiana and South Carolina, for example), looking to replace them with something else. But these actions are largely a result of mistaken conflations: that the standards represent a federal imposition of curriculum on local schools, that the <a href="http://www.parcconline.org/about">standardized tests</a> used to evaluate students <em>are</em> the Common Core rather than a separate initiative.</p>
<p>As the 2016 presidential campaign heats up, support for the Common Core has become a political liability, possibly killing it before it really has a chance. That would be a shame. The standards themselves are fine, and before we throw the baby out with the bathwater, perhaps we should consider efforts to implement them properly. To give the Common Core a fair shot, we need appropriate professional development for teachers and a more phased introduction of new standardized testing attached to the standards.</p>
<p>But, if we do ultimately give in to panic and misinformation, let’s hope any replacement provides proper coherence and rigor. Above all, our children should develop solid mathematical skills that will help them see the beauty and utility of this wonderful subject.</p><img src="https://counter.theconversation.com/content/46585/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/449632015-08-03T10:19:08Z2015-08-03T10:19:08ZCan math solve the congressional districting problem?<figure><img src="https://cdn.theconversation.com/files/90486/width496/image-20150731-17164-uo2zqf.png" /><figcaption><span class="caption">The original 1812 gerrymander district designed to favor Massachusetts governor Elbridge Gerry.</span> <span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/File:The_Gerry-Mander_Edit.png">Elkanah Tisdale</a></span></figcaption></figure><p>Lost amidst the frenzy of coverage of the Supreme Court’s rulings about the Affordable Care Act and same-sex marriage was a <a href="http://www.usatoday.com/story/news/nation/2015/06/29/supreme-court-arizona-congress-maps/27400015/">case</a> involving the constitutionality of an <a href="http://www.scotusblog.com/2015/06/opinion-analysis-a-cure-for-partisan-gerrymandering/">independent commission to draw congressional districts</a> in Arizona.</p>
<p>Through a ballot measure in 2000, the state amended its constitution to create a nonpartisan group to draw up new districts; the ultimate goal is to reduce gerrymandering. Named for the salamander-shaped district drawn by Massachusetts governor Elbridge Gerry in 1812, <a href="http://www.washingtonpost.com/news/wonkblog/wp/2015/03/01/this-is-the-best-explanation-of-gerrymandering-you-will-ever-see/">gerrymandering</a> occurs when a state legislature draws voting district lines in a manner that benefits the ruling party at the expense of the opposition.</p>
<p>The goal is to consolidate power for the party in control, making it effectively impossible for the opposition to gain seats. Many state legislatures have engaged in this process recently, prompting grassroots movements advocating independent commissions to draw districts. The Supreme Court ruled 5–4 that Arizona’s commission is constitutional.</p>
<p>This begs the question: is there a truly unbiased method for drawing fair districts that yield more competitive elections?</p>
<p>As it turns out, there are mathematical methods that could fit the bill.</p>
<h2>Requirements of congressional districts</h2>
<p>There are three primary <a href="http://redistricting.lls.edu/where.php">requirements</a> in federal law when drawing congressional districts: they must distribute population evenly, be connected and be “compact.” The last term has never been rigorously defined. The Voting Rights Act of 1965 also insists on some guarantees of representation for minority voters.</p>
<p>Over the years state legislatures have employed various strategies to meet all these criteria – which has led to some interesting districts.</p>
<figure class="align-center ">
<img alt="" src="https://cdn.theconversation.com/files/89828/width754/image-20150727-7637-1mkm1jq.png">
<figcaption>
<span class="caption">Florida’s 5th Congressional District.</span>
<span class="attribution"><span class="source">US Dept. of the Interior</span></span>
</figcaption>
</figure>
<p>For instance, Florida’s 5th Congressional District is one of the nation’s most gerrymandered. It is connected geographically (barely), but it’s probably not what most reasonable people would call compact since it stretches 140 miles from parts of Jacksonville in the north to Orlando in the south. A portion of its border runs along West 13th Street in Gainesville, dividing the college town in half.</p>
<figure class="align-center ">
<img alt="" src="https://cdn.theconversation.com/files/89827/width754/image-20150727-7659-1yid8mh.png">
<figcaption>
<span class="caption">Florida’s 3rd Congressional District.</span>
<span class="attribution"><span class="source">US Dept. of the Interior</span></span>
</figcaption>
</figure>
<p>My own district, Florida’s 3rd, shares that border along 13th Street in Gainesville. Because the lines are drawn this way, the western half of the city, which generally votes for Democratic candidates in local elections, is included in a large rural district represented by Ted Yoho, one of the most conservative Republican members of the House. Geographically, however, the 3rd district looks completely reasonable.</p>
<p>Little wonder the Florida Supreme Court <a href="http://www.tampabay.com/news/politics/stateroundup/florida-supreme-court-orders-new-congressional-map-with-eight-districts-to/2236734">ruled</a> this summer that the 5th district, as well as several others in the state, must be redrawn without political bias.</p>
<p>Of course, gerrymandering is not restricted to any particular political party. Legislatures controlled by the Democratic Party have abused their power to draw districts (for example, <a href="https://en.wikipedia.org/wiki/Illinois%27s_4th_congressional_district">Illinois’ 4th Congressional District</a>).</p>
<p>Most people agree that gerrymandering is bad, but it’s not obvious what to do about it.</p>
<h2>Splitline districting</h2>
<p>One might approach voting reform by either changing the way we tabulate votes (for instance, via <a href="http://rangevoting.org/RangeVoting.html">score voting</a>, or <a href="https://theconversation.com/how-to-make-the-house-of-representatives-representative-32921">fair majority voting</a>) or by drawing the districts differently.</p>
<p>One unbiased way to draw districts is via the <a href="http://rangevoting.org/SplitLR.html">shortest splitline algorithm</a>. It works like this. Suppose a state is to be divided into <em>N</em> districts. Let <em>A</em> be the largest integer less than or equal to <em>N/2</em> and let <em>B</em> be the smallest integer greater than or equal to <em>N/2</em>. Then <em>N = A + B</em> (for example, 9 = 4 + 5).</p>
<p>Now find the shortest straight line that divides the population of the state into the ratio <em>A:B</em>. Ties are broken by choosing the line that is closest to north–south (other choices are possible). You then have two “substates” that need to be divided into <em>A</em> and <em>B</em> districts, respectively. Repeat the algorithm until the state is divided completely. Below, compare the actual (top) and splitline (bottom) districts for Florida as they were in 2009.</p>
<p><figure class="align-center ">
<img alt="" src="https://cdn.theconversation.com/files/89831/width754/image-20150727-7646-1aeluhn.gif">
<figcaption>
<span class="caption">Florida’s congressional districts, 2009.</span>
<span class="attribution"><span class="source">Florida Office of Economic & Demographic Research</span></span>
</figcaption>
</figure> </p>
<p><figure class="align-center zoomable">
<a href="https://cdn.theconversation.com/files/89832/area14mp/image-20150727-7662-19xufl1.png"><img alt="" src="https://cdn.theconversation.com/files/89832/width754/image-20150727-7662-19xufl1.png"></a>
<figcaption>
<span class="caption">Florida shortest splitline districts, 2009.</span>
<span class="attribution"><a class="source" href="http://rangevoting.org/SplitLR.html">Center for Range Voting, algorithm by Warren D Smith, software by Ivan Ryan</a></span>
</figcaption>
</figure> </p>
<p>It’s not so easy to see, but, particularly in South Florida, there are some rather bizarre boundaries to the existing districts. The splitline algorithm eliminates these.</p>
<p>One obvious downside to this approach is that it ignores natural and political boundaries. There may be good reasons to put an entire city into one district, for example, but the algorithm might not make that happen.</p>
<p>One obvious advantage, however, is that the algorithm has no political loyalties or biases; it simply divides the population evenly into polygonal chunks on a map. </p>
<h2>Drawing districts randomly</h2>
<p>In a 2014 <a href="http://arxiv.org/abs/1410.8796">paper</a>, mathematicians Jonathan Mattingly and Christy Vaughn introduced a probabilistic method for drawing districts. They were motivated by the fact that in North Carolina’s 2012 election, a majority of voters selected Democratic candidates, yet only four of the state’s 13 districts had a Democratic winner. </p>
<p>Their method considers the set of all possible divisions of the state into 13 districts with roughly equal population such that each district is connected and “compact.” They also toss out those districts that are not “simply connected” in the sense that they entirely enclose another district – imagine a circular district containing another circular district.</p>
<p>They then define a class of <a href="https://en.wikipedia.org/wiki/Probability_measure">probability measures</a> on the set. This is a function that essentially gives the likelihood of a particular element of the set being chosen at random. Think of rolling a die – the probability measure assigns the value one-sixth to each of the six outcomes. The number of such divisions of the state is unimaginably large (on the order of 10²⁷⁸⁵), so it’s effectively impossible to compute the probability distribution exactly. But, there are <a href="https://en.wikipedia.org/wiki/Monte_Carlo_method">methods to estimate</a> the function and therefore obtain useful results.</p>
<p>With these estimates in place, Mattingly and Vaughn ran simulations using the actual votes cast in 2012 to determine the outcome of the election using various randomly chosen redistrictings. Of 100 such maps, more than half had either seven or eight Democratic representatives, and all of them had between six and nine.</p>
<p>They estimate the probability of only four Democrats being elected in a particular districting – remember, that’s the actual election outcome in real life – to be very small, raising the question of whether the current congressional district map of North Carolina results in representation that reflects the “will of the people.”</p>
<h2>Should we bother?</h2>
<p>One approach is to do nothing and leave the system as it is, accepting the current situation as part of the natural ebb and flow of the political process. But when one political party receives a majority of votes nationally yet does not have control of the House of Representatives – as occurred in the 2012 election – one begins to wonder if the system needs some tweaks.</p>
<p>The advantage of using mathematics is that it’s built on cold logic rather than political heat. But, there is no perfect algorithm (and there are <a href="http://www.ams.org/samplings/feature-column/fc-2014-08">others</a> not mentioned here), so the optimal solution will likely require a mixture of science and art.</p><img src="https://counter.theconversation.com/content/44963/count.gif" alt="The Conversation" width="1" height="1" />
Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.