tag:theconversation.com,2011:/global/topics/calculus-22686/articlesCalculus – The Conversation2018-01-31T11:42:20Ztag:theconversation.com,2011:article/906792018-01-31T11:42:20Z2018-01-31T11:42:20ZWhy colleges must change how they teach calculus<figure><img src="https://images.theconversation.com/files/203461/original/file-20180125-102758-1aqnvq9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Many college students who take calculus fail to earn a C or better. Could 'active learning' help turn things around?</span> <span class="attribution"><a class="source" href="https://pixabay.com/en/differential-calculus-board-school-2820657/">pixabay</a></span></figcaption></figure><p>Math departments <a href="https://www.asee.org/public/conferences/8/papers/3737/download">fail too many calculus students</a> and <a href="http://www.siam.org/reports/gaimme-full_color_for_online_viewing.pdf">do not adequately prepare</a> those they pass.</p>
<p>That is the <a href="https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/crafty/summer-reports/workshop-engineering">message</a> heard from engineering colleges across the country. Calculus has often been viewed as a tool for screening who should be allowed into engineering programs. But it appears to be failing in that regard, too. That is, it is preventing students who should be proceeding from going on, and it is letting students through who do not have the mathematical preparation that they need.</p>
<p>Each fall, according to forthcoming data expected to be published <a href="http://www.ams.org/profession/data/cbms-survey/cbms2015">here</a> this spring, more than <a href="http://www.ams.org/profession/data/cbms-survey/cbms-survey">300,000</a> students enroll in first-semester calculus in colleges and universities throughout the U.S. Most of those students are aiming for a degree in engineering, physics, chemistry, computer science or the biological sciences. About a quarter of them will <a href="http://www.ams.org/notices/201502/rnoti-p144.pdf">fail to earn the C or higher</a> needed to continue. Many more are so discouraged by their experience that they <a href="https://nces.ed.gov/pubs2018/2018434.pdf">abandon their career plans</a>. </p>
<p>While some blame may rest with inadequate preparation, most colleges and universities try to <a href="https://www.maa.org/sites/default/files/PtC%20Technical%20Report_Final.pdf">place students</a> so that only those who are prepared are allowed into calculus. In fact, most of these students have <a href="http://www.nctm.org/Publications/Mathematics-Teacher/2015/Vol109/Issue3/Insights-from-the-MAA-National-Study-of-College-Calculus/">already passed calculus while in high school</a>. Comparing data from student <a href="https://heri.ucla.edu/publications-tfs/">intended majors</a> when they enter college to the <a href="https://nces.ed.gov/programs/digest/">number of bachelor’s degrees</a> awarded each year, we lose 45 percent of the students who want to be engineers. For instance, 184,000 entered in 2011 with the intention of majoring in engineering, but only 98,000 graduated with this major in 2015. Similarly, we lose almost 35 percent of those heading into the biological sciences (167,000 entered in 2011, but only 110,000 graduated in 2015), and 30 percent of those pursuing physics or chemistry (43,000 entered in 2011, but just 30,000 graduated in 2015).</p>
<p>At the same time that a new National Science Board report indicates that China is <a href="https://www.nsf.gov/statistics/2018/nsb20181/report/sections/overview/r-d-expenditures-and-r-d-intensity">overtaking</a> the United States in research and development, we are bleeding large numbers of potential researchers. Calculus is one of the chief barriers to their progress. We don’t have measures of the fraction, but both the <a href="https://energy.gov/sites/prod/files/Engage%20to%20Excel%20Producing%20One%20Million%20Additional%20College%20Graduates%20With%20Degrees%20in%20STEM%20Feburary%202012.pdf">“Engage to Excel”</a> report from the President’s Council of Advisors for Science and Technology and the work of <a href="http://www.jngi.org/wordpress/wp-content/uploads/2015/06/TAL-1-and-TAL-R-briefing-paper-_3_.pdf">Elaine Seymour and Nancy Hewitt</a> criticize the nature of mathematics instruction as a primary reason for leaving STEM fields.</p>
<h2>An ‘active learning’ approach</h2>
<p>We now know that much of the problem rests with an outdated mode of instruction, a lecture format in which students are reduced to scribes. This may have worked in an earlier age when calculus was for a small elite group that excelled in mathematics. Today, professions that require calculus make up <a href="https://www.nsf.gov/statistics/2018/nsb20181/digest/sections/u-s-s-e-workforce-trends-and-composition">5 percent</a> of the workforce, a proportion that is growing at a rate that is <a href="https://www.nsf.gov/statistics/2018/nsb20181/report/sections/science-and-engineering-labor-force/u-s-s-e-workforce-definition-size-and-growth#growth-of-the-s-e-workforce">50 percent higher</a> than overall job growth. We can no longer afford to ignore what we know about how to improve the student experience, both inside and outside the classroom. </p>
<p>There are a variety of approaches that are known to promote active engagement with mathematics, helping students to understand and be able to use their mathematical knowledge outside of their math class. They generally go under the name “active learning.” One such technique is to challenge students with a question that uses the knowledge they have, but in an unfamiliar way. One classic example is the following, which can be used to spur discussion. The question is not novel, but using questions like this to force students to use their knowledge within a classroom setting is one example of active learning.</p>
<p>Imagine that there is a rope around the equator of the Earth. Add a 20-meter segment of rope to it. The new rope is held in a circular shape centered about the Earth. Then the following can walk beneath the rope without touching it:</p>
<p><code>a) an amoeba</code></p>
<p><code>b) an ant</code></p>
<p><code>c) I (the student)</code></p>
<p><code>d) all of the above</code></p>
<p>(The answer is (d), according to <a href="http://www.math.cornell.edu/%7EGoodQuestions/GQbysubject_pdfversion.pdf">this paper</a> from the GoodQuestions Project at Cornell University.) </p>
<p>Another example is to require students to read the relevant section of their textbook or watch a video before class, answer questions about the material to ensure they have done the assignment, and then describe their own questions and uncertainties. These can be great launch points for classroom interaction.</p>
<h2>Efforts to change</h2>
<p>Active learning does not mean ban all lectures. A lecture is still the most effective means for conveying a great deal of information in a short amount of time. But the <a href="https://www.nap.edu/download/13362">most useful lectures</a> come in short bursts when students are primed with a need and desire to know the information. A lecture is a poor substitute for giving students the time they need to discover the answers themselves.</p>
<p>The presidents of the professional societies in the mathematical sciences have <a href="https://www.cbmsweb.org/2016/07/active-learning-in-post-secondary-mathematics-education/">endorsed</a> these methods of teaching. Yet despite clear <a href="http://www.pnas.org/content/111/23/8410.full">evidence</a> that they greatly improve student learning, science and mathematics faculty have been <a href="https://journals.aps.org/prper/abstract/10.1103/PhysRevPhysEducRes.12.010103">slow</a> to adopt them. Part of this is reluctance on the part of faculty to diverge from what worked for them. But a greater obstacle is that faculty <a href="http://homepages.wmich.edu/%7Echenders/Publications/2012HendersonPRST-PER_RBISFactors.pdf">need</a> both departmental encouragement and a supportive network if they are to make the transition to more effective teaching.</p>
<p>Several organizations are working to build these networks. One of the most recent examples is a $3 million five-year project funded by the National Science Foundation. It is called <a href="http://www.aplu.org/projects-and-initiatives/stem-education/seminal/about-seminal/index.html">SEMINAL</a>, an acronym for Student Engagement in Mathematics Through an Institutional Network for Active Learning. Through this initiative, 12 public universities, led by San Diego State University, the University of Colorado-Boulder and the University of Nebraska-Lincoln, will work together to show how active learning can be implemented and supported in mathematics classes from precalculus through higher forms of calculus.</p>
<p>The days when we could afford to teach mathematics as an elite subject, that has often disadvantaged women and students from underrepresented minority groups, are long gone. According to the federal <a href="https://nces.ed.gov/programs/digest/">Digest of Education Statistics</a>, women earn 57 percent of all bachelor’s degrees but only 20 percent of engineering degrees, 38 percent of degrees in the physical sciences, and 43 percent in the mathematical science. African-Americans earn over 10 percent of all bachelor’s degrees, but only 4 percent of engineering degrees and 5 percent of degrees in the physical or mathematical sciences.</p>
<p>While active learning approaches help all students, they <a href="https://link.springer.com/article/10.1007/s10755-013-9269-9">have been shown</a> to be most effective for the students who are at the greatest risk of failing to earn a satisfactory grade or dropping out of the sequence of courses needed for their intended career. The demographics of those entering the workforce are changing, and we can no longer afford to ignore traditionally underrepresented groups of students. White students, which were 73 percent of all high school graduates in 1995, will account for less than 50 percent by 2025. Black students will comprise 14 percent and Hispanic students 27 percent <a href="http://launchings.blogspot.com/2017/08/changing-demographics.html">of these graduates</a>. If the United States is to maintain its preeminence in science and technology, it will require a skilled workforce whose racial and ethnic makeup reflects the diversity of this country. This workforce needs sophisticated mathematical skills, increasingly including a working knowledge of calculus.</p><img src="https://counter.theconversation.com/content/90679/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>David Bressoud receives funding from the National Science Foundation to study calculus instruction in the United States and serves as an advisor to the APLU SEMINAL Project. He is Director of the Conference Board of the Mathematical Sciences that works to coordinate the activities of the professional societies in mathematics including the Mathematical Association of America and the American Mathematical Association of Two-Year Colleges.</span></em></p>Each year large numbers of college students drop plans to become engineers or scientists because of poor performance in calculus. A new 'active learning' approach could help turn things around.David Bressoud, DeWitt Wallace Professor of Mathematics and Director of the Conference Board of the Mathematical Sciences, Macalester CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/843322017-09-21T11:02:15Z2017-09-21T11:02:15ZFive ways ancient India changed the world – with maths<figure><img src="https://images.theconversation.com/files/186896/original/file-20170920-16437-hxdak9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bakhshali manuscript.</span> <span class="attribution"><span class="source">Bodleian Libraries, University of Oxford</span></span></figcaption></figure><p>It should come as no surprise that the first recorded use of the number zero, <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> to be made as early as the 3rd or 4th century, happened in India. Mathematics on the Indian subcontinent has a rich history <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">going back over 3,000 years</a> and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.</p>
<p>As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">trigonometry, algebra, arithmetic and negative numbers among other areas</a>. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.</p>
<h2>The number system</h2>
<p>As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as <a href="https://www.ancient.eu/The_Vedas/">the Vedas</a>. In these texts, numbers were commonly expressed as <a href="http://www.thehindu.com/sci-tech/science/understanding-ancient-indian-mathematics/article2747006.ece">combinations of powers of ten</a>. For example, 365 might be expressed as three hundreds (3x10²), six tens (6x10¹) and five units (5x10⁰), though each power of ten was represented with a name rather than a set of symbols. It is <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">reasonable to believe</a> that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=126&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=126&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=126&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=158&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=158&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=158&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Brahmi numerals.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Brahmi_numerals#/media/File:Indian_numerals_100AD.svg">Wikimedia</a></span>
</figcaption>
</figure>
<p>From the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">third century BC</a>, we also have written evidence of the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html">Brahmi numerals</a>, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics. </p>
<h2>The concept of zero</h2>
<p>Zero itself has a much longer history. The <a href="http://www.bodleian.ox.ac.uk/news/2017/sep-14">recently dated first recorded zeros</a>, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the <a href="https://www.scientificamerican.com/article/history-of-zero/">Babylonian and Mayan cultures in the early centuries AD</a> and arguably in <a href="https://www.scientificamerican.com/article/history-of-zero/">Sumerian mathematics as early as 3000-2000 BC</a>.</p>
<p>But only in India did the placeholder symbol for nothing progress to become a <a href="https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232">number in its own right</a>. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the <a href="https://blog.sciencemuseum.org.uk/illuminating-india-starring-oldest-recorded-origins-zero-bakhshali-manuscript/">democratisation of mathematics</a>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">No abacus needed.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though <a href="http://www.springer.com/gb/book/9780387407371">Fibonnacci’s book liber abaci</a>. </p>
<h2>Solutions of quadratic equations</h2>
<p>In the seventh century, the first written evidence of the rules for working with zero were formalised in the <a href="https://archive.org/details/Brahmasphutasiddhanta_Vol_1">Brahmasputha Siddhanta</a>. In his seminal text, the astronomer <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">Brahmagupta</a> introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.</p>
<h2>Rules for negative numbers</h2>
<p>Brahmagupta also demonstrated rules for working with negative numbers. He referred to <a href="https://nrich.maths.org/5961">positive numbers as fortunes and negative numbers as debts</a>. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.</p>
<p>This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Negative cows.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that <a href="https://books.google.co.uk/books?id=STKX4qadFTkC&pg=PA56&redir_esc=y#v=onepage&q&f=false">negative numbers were absurd</a>. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.</p>
<p>For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total. </p>
<h2>Basis for calculus</h2>
<p>This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his <a href="https://books.google.co.uk/books?id=CXG6CgAAQBAJ&pg=PA165&lpg=PA165&dq=Leibniz+zero+negatives+calculus&source=bl&ots=NsKOzdZL7Y&sig=dE2KJvCXPFovF4uyFdgHMJOAQr8&hl=en&sa=X&ved=0ahUKEwjdxKv8_LPWAhXhAcAKHR0XBcUQ6AEIMjAC#v=onepage&q=Leibniz%20zero%20negatives%20calculus&f=false">development of calculus</a> in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=759&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=759&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=759&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=954&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=954&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=954&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Leibniz: Beaten to it by 500 years.</span>
</figcaption>
</figure>
<p>But <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bhaskara_II.html">Indian mathematician Bhāskara</a> had already discovered many of Leibniz’s ideas <a href="https://ijrier.com/published-papers/volume-1/issue-8/origin-of-concept-of-calculus-in-india.pdf">over 500 years earlier</a>. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Doiphantine” equations, that <a href="https://www.amazon.co.uk/Mathematical-Achievements-Pre-modern-Mathematicians-Elsevier/dp/0123979137#reader_0123979137">would not be rediscovered in Europe for centuries</a>.</p>
<p><a href="https://link.springer.com/referenceworkentry/10.1007%2F978-1-4020-4425-0_8683">The Kerala school of astronomy and mathematics</a>, founded by <a href="https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama">Madhava of Sangamagrama</a> in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would <a href="http://www.jstor.org/stable/1558972?origin=crossref&seq=1#page_scan_tab_contents">later be repeated in Europe</a> including Taylor series expansions, infinitessimals and differentiation. </p>
<p>The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation <a href="http://www.cbc.ca/news/technology/calculus-created-in-india-250-years-before-newton-study-1.632433">suffers from the Eurocentric bias</a>, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by <a href="http://m.ranker.com/list/famous-mathematicians-from-india/reference?page=1">providing key players at the forefront of every branch of mathematics</a>.</p><img src="https://counter.theconversation.com/content/84332/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christian Yates does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>High school students can blame ancient India for quadratic equations and calculus.Christian Yates, Senior Lecturer in Mathematical Biology, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/842322017-09-20T14:14:28Z2017-09-20T14:14:28ZNothing matters: how the invention of zero helped create modern mathematics<figure><img src="https://images.theconversation.com/files/186837/original/file-20170920-16403-yazsqf.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>A small dot on an old piece of birch bark marks one of the biggest events in the history of mathematics. The bark is actually part of an ancient Indian mathematical document known as the Bakhshali manuscript. And the dot is the first known recorded use of the number zero. What’s more, researchers from the University of Oxford <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> the document is 500 years older than was previously estimated, dating to the third or fourth century – a breakthrough discovery.</p>
<p>Today, it’s difficult to imagine how you could have mathematics without zero. In a positional number system, such as the decimal system we use now, the location of a digit is really important. Indeed, the real difference between 100 and 1,000,000 is where the digit 1 is located, with the symbol 0 serving as a punctuation mark.</p>
<p>Yet for thousands of years we did without it. The Sumerians of 5,000BC employed a <a href="http://www.storyofmathematics.com/sumerian.html">positional system</a> but without a 0. In some <a href="https://www.scientificamerican.com/article/what-is-the-origin-of-zer/">rudimentary form</a>, a symbol or a space was used to distinguish between, for example, 204 and 20000004. But that symbol was never used at the end of a number, so the difference between 5 and 500 had to be determined by context.</p>
<p>What’s more, 0 at the end of a number makes multiplying and dividing by 10 easy, as it does with adding numbers like 9 and 1 together. The invention of zero immensely simplified computations, freeing mathematicians to develop vital mathematical disciplines such as algebra and calculus, and eventually the basis for computers. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/pV_gXGTuWxY?wmode=transparent&start=0" frameborder="0" allowfullscreen></iframe>
</figure>
<p>Zero’s late arrival was partly a reflection of the negative views some cultures held for the concept of nothing. Western philosophy is plagued with grave misconceptions about nothingness and the mystical powers of language. The fifth century BC <a href="https://plato.stanford.edu/entries/parmenides/">Greek thinker Parmenides</a> proclaimed that nothing cannot exist, since to speak of something is to speak of something that exists. This Parmenidean approach kept prominent <a href="http://www.nothing.com/Heath.html">historical figures</a> busy for a long while. </p>
<p>After the advent of Christianity, religious leaders in Europe argued that since God is in everything that exists, anything that represents nothing must be satanic. In an attempt to save humanity from the devil, they <a href="http://yaleglobal.yale.edu/history-zero">promptly banished</a> zero from existence, though merchants continued secretly to use it.</p>
<p>By contrast, in Buddhism the concept of nothingness is not only devoid of any demonic possessions but is actually a central idea worthy of much study en route to nirvana. <a href="http://www.huffingtonpost.com/lewis-richmond/emptiness-most-misunderstood-word-in-buddhism_b_2769189.html">With such a mindset</a>, having a mathematical representation for nothing was, well, nothing to fret over. In fact, the English word “zero” is <a href="http://www.etymonline.com/index.php?term=zero">originally derived</a> from the Hindi “sunyata”, which means nothingness and is a central concept in Buddhism.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=299&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=299&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=299&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=376&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=376&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=376&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Bakhshali manuscript.</span>
<span class="attribution"><span class="source">Bodleian Libraries</span></span>
</figcaption>
</figure>
<p>So after zero finally emerged in ancient India, it took almost 1,000 years to set root in Europe, much longer than in China or the Middle East. In 1200 AD, the Italian mathematician Fibonacci, who brought the decimal system to Europe, <a href="http://www.springer.com/gb/book/9780387407371">wrote that</a>: </p>
<blockquote>
<p>The method of the Indians surpasses any known method to compute. It’s a marvellous method. They do their computations using nine figures and the symbol zero.</p>
</blockquote>
<p>This superior method of computation, clearly reminiscent of our modern one, freed mathematicians from tediously simple calculations, and enabled them to tackle more complicated problems and study the general properties of numbers. For example, it led to the work of the seventh century Indian <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">mathematician and astronomer Brahmagupta</a>, considered to be the beginning of modern algebra.</p>
<h2>Algorithms and calculus</h2>
<p>The Indian method is so powerful because it means you can draw up simple rules for doing calculations. Just imagine trying to explain long addition without a symbol for zero. There would be too many exceptions to any rule. The ninth century Persian mathematician Al-Khwarizmi was the first to meticulously note and exploit these arithmetic instructions, which <a href="https://books.google.co.uk/books?id=zTQrDwAAQBAJ&pg=PA47&lpg=PA47&dq=al+khwarizmi+abacus&source=bl&ots=PakFxbCVwk&sig=FWjwHlnppHAU9i_zgAficOcw4ug&hl=en&sa=X&ved=0ahUKEwii-46257PWAhUhBcAKHaWtCRcQ6AEIajAP#v=onepage&q=al%20khwarizmi%20abacus&f=false">would eventually</a> make the abacus obsolete.</p>
<p>Such mechanical sets of instructions illustrated that portions of mathematics could be automated. And this would eventually lead to the development of modern computers. In fact, the word “algorithm” to describe a set of simple instructions <a href="https://en.oxforddictionaries.com/definition/algorithm">is derived</a> from the name “Al-Khwarizmi”.</p>
<p>The invention of zero also created a new, more accurate way to describe fractions. Adding zeros at the end of a number increases its magnitude, with the help of a decimal point, adding zeros at the beginning decreases its magnitude. Placing infinitely many digits to the right of the decimal point corresponds to <a href="https://www.youtube.com/watch?v=JmyLeESQWGw&list=PLYoCqdGsxmn9HOU84Ln2PhPKpxRfaEO9h&index=17">infinite precision</a>. That kind of precision was exactly what 17th century thinkers Isaac Newton and Gottfried Leibniz needed to develop calculus, the study of continuous change.</p>
<p>And so algebra, algorithms, and calculus, three pillars of modern mathematics, are all the result of a notation for nothing. Mathematics is a science of invisible entities that we can only understand by writing them down. India, by adding zero to the positional number system, unleashed the true power of numbers, advancing mathematics from infancy to adolescence, and from rudimentary toward its current sophistication.</p><img src="https://counter.theconversation.com/content/84232/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ittay Weiss does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Turning zero from a punctuation mark into a number paved the way for everything from algebra to algorithms.Ittay Weiss, Teaching Fellow, Department of Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/780172017-06-14T02:23:21Z2017-06-14T02:23:21ZThe rise – and possible fall – of the graphing calculator<figure><img src="https://images.theconversation.com/files/172964/original/file-20170608-32318-zz67zc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Graphing calculators – like the ones used in this seventh grade Dallas classroom – have become ubiquitous in U.S. education.</span> <span class="attribution"><a class="source" href="http://www.apimages.com/metadata/Index/AP-A-TX-USA-F-MATH-INITIATIVE/77d11887816444b08f34dd2e0e50289a/9/0">AP Photo/Tony Gutierrez</a></span></figcaption></figure><p>The first handheld graphing calculator, <a href="http://americanhistory.si.edu/collections/object-groups/handheld-electronic-calculators">the Casio fx-7000G</a>, appeared in 1985. </p>
<p>Since then, graphing calculators have become a common – and controversial – tool for learning mathematics. These devices can do all of the calculations of a scientific calculator, plus graph equations, make function tables and solve equations. Many have the ability to do statistical analysis and even some calculus.</p>
<p>Advocates claim that the calculators provide students access to more powerful mathematics. Critics worry that they might hurt students’ fluency in basic math and standard algorithms. </p>
<p>Today, some teachers are replacing expensive graphing calculators with free apps that can do more. But even after decades of use, graphing technology of any sort in the classroom <a href="https://www.theatlantic.com/education/archive/2016/12/the-conundrum-of-calculators-in-the-classroom/493961/">still sparks debate</a>.</p>
<p>As mathematics educators, we think the graphing calculator transformed American classrooms for the better. Whether teachers continue to use these tools or ditch them in favor of new ones, graphing technology will likely always have a place in secondary math education. </p>
<h1>Understanding math</h1>
<p>Math educators often talk about <a href="http://www.davidtall.com/skemp/pdfs/instrumental-relational.pdf">two kinds of understanding</a>.</p>
<p>“Instrumental understanding” comes from mastering a procedure or memorizing a fact, without really understanding the mathematics behind it. It’s knowing how, but not why. The saying “It is not ours to wonder why, just invert and multiply!” – which is sometimes used to teach division of fractions – captures this sort of understanding.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/173422/original/file-20170612-2090-9483pa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/173422/original/file-20170612-2090-9483pa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=791&fit=crop&dpr=1 600w, https://images.theconversation.com/files/173422/original/file-20170612-2090-9483pa.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=791&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/173422/original/file-20170612-2090-9483pa.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=791&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/173422/original/file-20170612-2090-9483pa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=994&fit=crop&dpr=1 754w, https://images.theconversation.com/files/173422/original/file-20170612-2090-9483pa.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=994&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/173422/original/file-20170612-2090-9483pa.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=994&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The first scientific graphing calculator.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/51764518@N02/15810483905/in/photolist-q67S36">51764518@N02/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>In contrast, “relational understanding” is a kind of connected, conceptual understanding. People with a relational understanding don’t just know how to invert and multiply, they know why such a procedure results in the quotient of two fractions.</p>
<p>Advocates for graphing calculators in school saw promise in the tool’s ability to <a href="http://www.nctm.org/Standards-and-Positions/More-NCTM-Standards/An-Agenda-for-Action-(1980s)/">help students develop relational understanding</a>. While the calculator takes care of the “how,” students can focus on “why.” </p>
<p>The impact is quite clear in the Advanced Placement (AP) calculus program, which started to require graphing calculators in their courses and on their exams <a href="http://www.andover.edu/gpgconference/documents/four-decades.pdf">in 1995</a>. Prior to 1995, AP calculus exam questions probed almost solely for the students’ ability to use rules to find derivatives and integrals of functions. After 1995, there was a marked shift away from this instrumental understanding and toward questions that probed for relational understanding.</p>
<p>As exams evolved, so too did teaching <a href="https://www.macalester.edu/%7Ebressoud/pub/launchings/launchings_06_10.html">philosophies</a>. The AP program required teachers to use graphing calculators in their courses. This was not just so students would learn how to use the calculator. Rather, the focus of instruction shifted so that students could learn mathematics through the graphing calculator. </p>
<p>For example, by using the graphing and zoom features of the graphing calculator, students could compare and contrast the local and global behavior of functions such as <em>y</em> = <em>x</em>² and <em>y</em> = <em>x</em>² + 2. By zooming in, students can see that in any local area, the graphs are clearly different. By zooming out, students can see that globally, the graphs are basically identical. Through exploration such as this, students gained relational understanding of infinite limits. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/173424/original/file-20170612-10795-15jqaok.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/173424/original/file-20170612-10795-15jqaok.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=201&fit=crop&dpr=1 600w, https://images.theconversation.com/files/173424/original/file-20170612-10795-15jqaok.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=201&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/173424/original/file-20170612-10795-15jqaok.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=201&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/173424/original/file-20170612-10795-15jqaok.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=252&fit=crop&dpr=1 754w, https://images.theconversation.com/files/173424/original/file-20170612-10795-15jqaok.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=252&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/173424/original/file-20170612-10795-15jqaok.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=252&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">In the local area (-6 < x < 6) shown at left, the graphs of y = x² + 5 and y = x² are clearly different. But zooming out (at right), the graphs of the two functions are basically identical.</span>
<span class="attribution"><span class="source">Frederick Peck</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h1>Does the tech help or hurt?</h1>
<p>Still, <a href="https://doi.org/10.1111/j.1949-8594.2007.tb17776.x">some teachers wondered</a> whether <a href="https://www.jstor.org/stable/27968919">this shift in instruction</a> in AP calculus and across the K-12 curriculum would have a negative effect on students’ instrumental understanding. After all, if a machine is doing the calculating for you, why bother to learn it?</p>
<p>After more than three decades of research, the findings are clear. Graphing calculators have a positive effect on <a href="https://doi.org/10.1111/j.1949-8594.2006.tb18067.x">students’ relational understanding</a> and a slight positive effect <a href="https://doi.org/10.1007/978-0-387-73315-9_60">on their instrumental understanding</a>.</p>
<p><a href="http://www.ti-researchlibrary.com/Lists/TI%20Education%20Technology%20%20Research%20Library/Attachments/122/GC%20in%20secondary%20math%20-%20research%20findings%20and%20implications%20-%20Burrill%202002%20-%20(yellow%20book)%20CL2872,%20review.pdf">Another review</a>, conducted by respected researchers in math education (but funded by a calculator company), came to a similar conclusion.</p>
<p>In other words, students who use graphing calculators in school know at least as many basic facts and are at least as good at doing standard algorithms as students who do not use graphing calculators. However, students who use graphing calculators have a deeper understanding of the “whys” behind those algorithms. </p>
<p>Of course, there are many individual studies that show negligible or even negative effects of graphing calculators. But overall, when the technology is paired with appropriate instructional techniques, the result is more and better mathematics learning.</p>
<h1>The rise of graph apps</h1>
<p>Today, online and app-based technology, such as <a href="https://www.desmos.com/calculator">Desmos</a> and <a href="https://www.geogebra.org/graphing">GeoGebra</a>, <a href="https://www.bloomberg.com/news/articles/2017-05-12/startup-targets-the-ti-calculators-your-kid-lugs-to-class">aim to replace</a> the role of stand-alone graphing calculators in school.</p>
<p>As with graphing calculators, for these new technologies to have a positive effect on student learning, teachers have to <a href="http://blog.desmos.com/post/150453765267/the-desmos-guide-to-building-great-digital-math">adapt their instruction</a>, changing <a href="https://theconversation.com/how-math-education-can-catch-up-to-the-21st-century-77129">what they teach and how they teach it</a>. </p>
<p>For example, technology can help students connect graphical representations with algebraic equations. We recently observed seventh grade students in Missoula, Montana doing that using Desmos. Students graphed three different linear equations, each with a different coefficient for the <em>x</em>-term. Based on this exploration, students made conjectures about the role of the coefficient, and used the app to test their conjectures – for example, <a href="https://www.desmos.com/calculator/ylc2sdigoi">by using a “slider” to dynamically vary the coefficient</a>. The flexibility of the technology also encouraged students to pose and explore their own questions.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/172758/original/file-20170607-29588-mbwaim.PNG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">On the Desmos app, students can test their questions by dynamically varying the coefficient using a slider.</span>
<span class="attribution"><span class="source">David Erickson</span></span>
</figcaption>
</figure>
<p>Even as these cloud- and app-based tools provide powerful technology for free on smartphones and other personal devices, the expensive graphing calculator (which typically costs between US$80 and $150) <a href="https://www.washingtonpost.com/news/innovations/wp/2014/09/02/the-unstoppable-ti-84-plus-how-an-outdated-calculator-still-holds-a-monopoly-on-classrooms/">continues to be a stalwart in K-12 math classes</a>, with year-over-year units sales <a href="http://www.cbs58.com/news/digital-calculators-replacing-ti-84-graphing-calculator">increasing in 2015 and 2016</a>.</p>
<p>There has recently been <a href="https://mic.com/articles/125829/your-old-texas-instruments-graphing-calculator-still-costs-a-fortune-heres-why">much</a> <a href="http://money.cnn.com/2017/05/12/technology/ti-84-graphing-calculator/index.html">online</a> <a href="http://www.popularmechanics.com/technology/apps/a26520/free-online-tool-graphic-calculator/">hand-wringing</a> over this state of affairs, with commentators aghast at the continued dominance of graphing calculators when compared to inexpensive or free apps. The debate even provoked <a href="https://education.ti.com/en/success-story/why-graphing-calculators-add-up">a response</a> from the president of Texas Instruments, the dominant graphing calculator company in the U.S. </p>
<p>We agree with many of the points in favor of the new apps. But there is no doubt that, when properly used, graphing technology – whether on a calculator, computer screen, tablet or smartphone – is a powerful tool for helping students learn mathematics.</p><img src="https://counter.theconversation.com/content/78017/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Las personas firmantes no son asalariadas, ni consultoras, ni poseen acciones, ni reciben financiación de ninguna compañía u organización que pueda obtener beneficio de este artículo, y han declarado carecer de vínculos relevantes más allá del cargo académico citado anteriormente.</span></em></p>Teachers are starting to ditch graphing calculators for math apps. Was the expensive tech ever worth it – or is it just holding students back?Frederick Peck, Assistant Professor of Mathematical Sciences, The University of MontanaDavid Erickson, Professor, Teaching and Learning, The University of MontanaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/771292017-06-02T02:52:29Z2017-06-02T02:52:29ZHow math education can catch up to the 21st century<figure><img src="https://images.theconversation.com/files/171036/original/file-20170525-23232-hokg03.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A student in Cape Coast solves a math problem.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/worldbank/5321246556/in/photolist-97dMdh-8rMuqU-8rMtNd-7tub8y-97aGaD-8rJogv-fAuDrE-fAfmgK-eokjMs-8rJoSr-fAuC9W-fAfkp2-8rJo7R-2HGirM-fAuBG5-fAuDPW-8rJnn8-fAfjkT-8rJp4i-fAfkv4-fAfkAp-fAfmUr-fAfmPB-fAuDJY-8rJp7V-5ua7bz-fAuCwJ-fAuBM1-5dsFUh-fAuDfj-8rMufd-8cwJHq-fAfjZi-5domkt-8rJoNM-8q1J3r-fAfmzM-5dom5D-5cWBvd-5domaD-8rJoZi-8rMu6s-fAfm4t-8rMuc3-aYTnYM-aRAYUk-8rMuGq-fAfkMr-fAuCZ7-7tq9jP">World Bank/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>In 1939, the fictional professor J. Abner Pediwell published a curious book called “<a href="https://www.amazon.com/Saber-Tooth-Curriculum-Classic-Abner-Peddiwell-ebook/dp/B00G6DSY7Q">The Saber-Tooth Curriculum</a>.”</p>
<p>Through a series of satirical lectures, Pediwell (or the actual author, education professor <a href="http://education.stateuniversity.com/pages/1783/Benjamin-H-R-W-1893-1969.html">Harold R. W. Benjamin</a>) describes a Paleolithic curriculum that includes lessons in grabbing fish with your bare hands and scaring saber-toothed tigers with fire. Even after the invention of fishnets proves to be a far superior method of catching fish, teachers continued teaching the bare-hands method, claiming that it helps students develop “generalized agility.” </p>
<p>Pedwill showed how curricula can become entrenched and ritualistic, failing to respond to changes in the world around it. In math education, the problem is not quite so dire – but it’s time to start breaking a few of our own traditions. There’s a growing interest in emphasizing problem-solving and understanding concepts over skills and procedures. While memorized skills and procedures are useful, knowing the underlying meanings and understanding how they work builds problem-solving skills so that students may go beyond solving the standard book chapter problem. </p>
<p>As education researchers, we see two different ways that educators can build alternative mathematics courses. These updated courses work better for all students by changing what they teach and how they teach it.</p>
<h1>New paths in math</h1>
<p>In math, the usual curricular pathway – or sequence of courses – starts with algebra in eighth or ninth grade. This is followed by geometry, second-year algebra and trigonometry, all the way up to calculus and differential equations in college. </p>
<p>This pathway still serves science, technology, engineering and mathematics (STEM) majors reasonably well. However, some educators are now concerned about students who may have other career goals or interests. These students are stuck on largely the same path, but many end up terminating their mathematics studies at an earlier point along the way. </p>
<p>In fact, students who struggle early with the traditional singular STEM pathway are more likely to fall out of the higher education pipeline entirely. Many institutions have identified <a href="http://www.npr.org/sections/ed/2014/10/09/354645977/who-needs-algebra">college algebra courses</a> as a key roadblock leading to students dropping out of college altogether. </p>
<p>Another issue is that there is a <a href="http://www.cnbc.com/2015/06/15/math-science-skills-add-up-to-more-job-opportunities-survey.html">growing need</a> for new quantitative skills and reasoning in a wide variety of careers – not just STEM careers. In the 21st century, workers across many fields need to know how to deal effectively with data (statistical reasoning), detect trends and patterns in huge amounts of information (“big data”), use computers to solve problems (computational thinking) and make predictions about the relationships between different components of a system (mathematical modeling). </p>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">New technology offers unprecedented mathematical capabilities.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/kosheahan/4009828196/in/photolist-77kqdU-4HZ3fV-4J4cuJ-4TC8zV-21EH7M-fh2dMQ-8Ahkos-5LJbpH-QMsaX-aPr5mn-6MH67-CpmDp-aPqYNZ-aPr3AP-CiKXa-aPr1sV-Cpmvg-zQ3Q3-6HVf6B-4v6ue-QMbRz-CpmoE-4J4dCo-4PENW-CiKQr-5CmU4d-bRwBMZ-QLpSA-QLpAj-zQ3fK-bsKvA-4sobRc-zQ37h-4ssaJm-FvWeu-h5PJV-4so3kv-h5PbY-4ssfLQ-4ss6eb-h5Pjw-4so6QX-4ss4xE-4so5Ec-4sseyU-njnKGG-4sseHA-4so3sa-4ss4tJ-4ssae7">kosheahan/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>What’s more, sophisticated computational tools provide us with mathematical capabilities far beyond arithmetic calculations. For example, large numerical data sets can be visually examined for patterns using computer graphing software. Other tools can derive predictive equations that would be impractical for anyone to compute with paper and pencil. What’s really needed are people who can make use of those tools productively, by posing the right questions and then interpreting the results sensibly.</p>
<p>The quest to improve student retention has led schools to consider other pathways that would provide students with the quantitative skills they need. For example, <a href="http://www.educationworld.com/a_curr/mathchat/mathchat025.shtml">courses that use spreadsheets</a> extensively for mathematical modeling and powerful statistical software packages have been developed as part of <a href="http://epublications.bond.edu.au/cgi/viewcontent.cgi?article=1178&context=ejsie">an alternative pathway</a> designed for students with interests in business and economics. </p>
<p>The Carnegie Foundation for the Advancement of Teaching has created alternative math curricula called <a href="https://www.carnegiefoundation.org/in-action/carnegie-math-pathways/">Quantway and Statway</a> as examples of alternative pathways – used primarily in community colleges – that focus on quantitative reasoning and statistics/data analysis, respectively. </p>
<h1>Lectures aren’t enough</h1>
<p>These alternative pathways involve activities that go beyond students writing examples down in their notebooks. Students might use software, build mathematical models or exercise other skills – all of which require flexible instruction.</p>
<p>Both new and old pathways can benefit from new and more flexible methods. In 2012, the President’s Council of Advisors on Science and Technology <a href="http://files.eric.ed.gov/fulltext/ED541511.pdf">called for a 34 percent increase</a> in the number of STEM graduates by 2020. Their report suggested current STEM teaching practices could improve through evidence-based approaches like active learning.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Calculating the best way to learn math.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/schoolgirl-glasses-solving-math-problem-on-167441195?src=msBh2Y81MF_nzOMV89qUTw-1-39">ESB Professional/Shutterstock</a></span>
</figcaption>
</figure>
<p>In a traditional classroom, students act as passive observers, watching an expert correctly work out problems. This approach doesn’t foster an environment where mistakes can be made and answers can be questioned. Without mistakes, students lack the opportunity to more deeply explore how and why things don’t work. They then tend to view mathematics as a series of isolated problems for which the solution is merely a prescribed formula. </p>
<p>Mathematician <a href="http://launchings.blogspot.com/2011/07/the-worst-way-to-teach.html">David Bressoud</a> summarized this well:</p>
<blockquote>
<p>“No matter how engaging the lecturer may be, sitting still, listening to someone talk, and attempting to transcribe what they have said into a notebook is a very poor substitute for actively engaging with the material at hand, for doing mathematics.”</p>
</blockquote>
<p>Conversely, classrooms that incorporate active learning allow students to ask questions and explore. Active learning is not a specifically defined teaching technique. Rather, it’s a spectrum of instructional approaches, all of which involve students actively participating in lessons. For example, teachers could pose questions during class time for students to answer with an electronic clicker. Or, the class could skip the lecture entirely, leaving students to work on problems in groups. </p>
<p>While the idea of active learning has existed for decades, there has been a greater push for widespread adoption in recent years, as more scientific research has emerged. <a href="http://www.pnas.org/content/111/23/8410.full">A 2014 analysis</a> looked at 225 studies comparing active learning with traditional lecture in STEM courses. Their findings unequivocally support using active learning and question whether or not lecture should even continue in STEM classrooms. If this were a medical study in which active learning was the experimental drug, the authors write, trials would be “stopped for benefit” – because active learning is so clearly beneficial for students. </p>
<p>The studies in this analysis varied greatly in the level of active learning that took place. In other words, active learning, no matter how minimal, leads to greater student achievement than a traditional lecture classroom. </p>
<p>Regardless of pathway, all students can benefit from active engagement in the classroom. As mathematician <a href="http://www.jstor.org/stable/2319737?seq=1#page_scan_tab_contents">Paul Halmos</a> put it: “The best way to learn is to do; the worst way to teach is to talk.”</p><img src="https://counter.theconversation.com/content/77129/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mary E. Pilgrim receives funding from National Science Foundation. </span></em></p><p class="fine-print"><em><span>Thomas Dick receives funding from National Science Foundation.</span></em></p>By embracing a style beyond the typical classroom lecture, math education can serve all of our students better.Mary E. Pilgrim, Assistant Professor of Mathematics Education, Colorado State UniversityThomas Dick, Professor of Mathematics, Oregon State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/754512017-05-11T01:04:29Z2017-05-11T01:04:29ZArguments why God (very probably) exists<figure><img src="https://images.theconversation.com/files/168651/original/file-20170509-11018-iusi2b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Does God exist?</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/archangel_raphael/2752361370/in/photolist-5cdz9o-5MPMT7-dYqzeW-8CnEND-5fwBBf-4RCzdy-oAuPVx-8CqHw3-jYB59K-dGkGY9-ebb3z9-bmUXpi-4Ryqd4-dLCYTb-4Rywh2-sbYb8p-xAoUt-dzwwnv-3PXafs-56eQ59-9239RA-KW6Wt-95umWj-4QJfRg-HBvebJ-8WdEEe-fvURzu-DaZN4P-cCGNP-egguiY-CrdPGK-5PjMfs-4sgKsE-cS8iCu-8bdnPN-72mX75-sj6AoA-641YFp-eirqWh-2cNR4L-91vXMw-91vXPL-91Myv-8QedDi-2WhJzn-hoCmR-6w29Dw-7dd9kJ-oprMU-5Qo3Vj">Michael Peligro</a>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span></figcaption></figure><p><em>Note from Editor of The Conversation US: This is a revised version of the original piece. We have done so to make explicit the author’s expertise with regard to the subject of this article. We have also incorporated important context that was missing in the original version.</em></p>
<p>The question of whether a god exists is heating up in the 21st century. According to a <a href="http://www.pewforum.org/religious-landscape-study">Pew survey</a>, the percent of Americans having no religious affiliation reached 23 percent in 2014. Among such “nones,” <a href="http://www.pewresearch.org/fact-tank/2015/11/04/americans-faith-in-god-may-be-eroding/">33 percent said</a> that they do not believe in God – an 11 percent increase since only 2007. </p>
<p>Such trends have ironically been taking place even as, I would argue, the probability for the existence of a supernatural god have been rising. In my 2015 book, <a href="http://www.jstor.org/stable/j.ctt1cg4m4s">“God? Very Probably: Five Rational Ways to Think about the Question of a God,”</a> I look at physics, the philosophy of human consciousness, evolutionary biology, mathematics, the history of religion and theology to explore whether such a god exists. I should say that I am trained originally as an economist, but have been working at the intersection of economics, environmentalism and theology since the 1990s.</p>
<h2>Laws of math</h2>
<p>In 1960 the Princeton physicist – and subsequent Nobel Prize winner – <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/1963/wigner-bio.html">Eugene Wigner</a> raised a <a href="https://www.dartmouth.edu/%7Ematc/MathDrama/reading/Wigner.html">fundamental question</a>: Why did the natural world always – so far as we know – obey laws of mathematics?</p>
<p>As argued by scholars such as <a href="https://www.brown.edu/academics/applied-mathematics/philip-j-davis">Philip Davis</a> and <a href="http://www.math.unm.edu/%7Erhersh/">Reuben Hersh</a>, <a href="https://books.google.com/books?id=lMdz84dWNnAC">mathematics exists</a> independent of physical reality. It is the job of mathematicians to discover the realities of this separate world of mathematical laws and concepts. Physicists then put the mathematics to use according to the rules of prediction and confirmed observation of the scientific method.</p>
<p>But modern mathematics generally is formulated before any natural observations are made, and many mathematical laws today have no known existing physical analogues. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/168655/original/file-20170509-11023-1xtngmq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/168655/original/file-20170509-11023-1xtngmq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=399&fit=crop&dpr=1 600w, https://images.theconversation.com/files/168655/original/file-20170509-11023-1xtngmq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=399&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/168655/original/file-20170509-11023-1xtngmq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=399&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/168655/original/file-20170509-11023-1xtngmq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=502&fit=crop&dpr=1 754w, https://images.theconversation.com/files/168655/original/file-20170509-11023-1xtngmq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=502&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/168655/original/file-20170509-11023-1xtngmq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=502&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Einstein Memorial, National Academy of Sciences, Washington, D.C.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/wallyg/136352792/in/photolist-d3QWd-bMjP2P-a9Wixm-88biBa-arc7Xp-gmg5g-9Axhbh-9ED6od-7H6zwA-kspmmF-mwM7A8-rcBYua-5jNiBM-dwrSTk-bFnrEt-Bmt7Gf-fKCrGc-3ikCMe-mkxs3-pnx7k2-9EAbAT-5jSyTC-9ED6yq-9EAbPD-36oWNQ-6vAiEJ-6vAitA-5jSz51-6vAioA-6vw6Bc-9ED6rb-9ED6sW-6vAi9J-9EAbVR-pHw41u-4Xpc3u-6vAieo-9LWE8f-ed5Hqe-fmWi5A-NiFw7-5V3mPF-6vAiiG-4zSaJC-5hMACB-5tkt3x-dzXjD-odeh3T-7jxVer-CNWNZ">Wally Gobetz</a>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Einstein’s 1915 general theory of relativity, for example, was based on theoretical mathematics developed 50 years earlier by the great German mathematician <a href="http://www.storyofmathematics.com/19th_riemann.html">Bernhard Riemann</a> that <a href="http://www.penguinrandomhouse.com/">did not have any known practical applications</a> at the time of its intellectual creation. </p>
<p>In some cases the physicist also discovers the mathematics. Isaac Newton was considered among the greatest mathematicians as well as physicists of the 17th century. Other physicists sought his help in finding a mathematics that would predict <a href="https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA459&lpg=PA459&dq=%22An+Ocean+of+Truth,%22&source=bl&ots=q6FyxbOuQh&sig=0kW1di2-2C3MhwYBClurJPdE234&hl=en&sa=X&ved=0ahUKEwjr3r_NqePTAhWC54MKHSNhCeYQ6AEITzAM#v=onepage&q=%22An%20Ocean%20of%20Truth%2C%22&f=false">the workings of the solar system</a>. He found it in the mathematical law of gravity, based in part on his discovery of calculus. </p>
<p>At the time, however, many people initially resisted Newton’s conclusions because <a href="https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA459&lpg=PA459&dq=%22An+Ocean+of+Truth,%22&source=bl&ots=q6FyxbOuQh&sig=0kW1di2-2C3MhwYBClurJPdE234&hl=en&sa=X&ved=0ahUKEwjr3r_NqePTAhWC54MKHSNhCeYQ6AEITzAM#v=onepage&q=%22An%20Ocean%20of%20Truth%2C%22&f=false">they seemed to be “occult.”</a> How could two distant objects in the solar system be drawn toward one another, acting according to a precise mathematical law? Indeed, Newton made strenuous efforts over his lifetime to find a natural explanation, but in the end he could say only that <a href="https://books.google.com/books/about/The_Religion_of_Isaac_Newton.html?id=jipvnQAACAAJ">it is the will of God</a>. </p>
<p>Despite the many other enormous advances of modern physics, little has changed in this regard. As <a href="https://www.dartmouth.edu/%7Ematc/MathDrama/reading/Wigner.html">Wigner wrote</a>, “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.” </p>
<p>In other words, as I argue in my book, it takes the existence of some kind of a god to make the mathematical underpinnings of the universe comprehensible.</p>
<h2>Math and other worlds</h2>
<p>In 2004 the great British physicist <a href="https://www.maths.ox.ac.uk/people/roger.penrose">Roger Penrose</a> put forward a vision of a universe composed of <a href="http://chaosbook.org/library/Penr04.pdf">three independently existing worlds</a> – mathematics, the material world and human consciousness. As Penrose acknowledged, it was a complete puzzle to him how the three interacted with one another outside the ability of any scientific or other conventionally rational model. </p>
<p>How can physical atoms and molecules, for example, create something that exists in a separate domain that has no physical existence: human consciousness?</p>
<p>It is a mystery that lies beyond science.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/168656/original/file-20170509-7918-1llyulm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/168656/original/file-20170509-7918-1llyulm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/168656/original/file-20170509-7918-1llyulm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/168656/original/file-20170509-7918-1llyulm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/168656/original/file-20170509-7918-1llyulm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/168656/original/file-20170509-7918-1llyulm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/168656/original/file-20170509-7918-1llyulm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Plato.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/elizabethe/8650802082/in/photolist-ebrC53-22n4Ao-6XiYSp-mDJXUH-7krE6e-7ghijy-8LbhtW-63AHwi-gBbuuu-9wE4y1-gFZkdg-bXTHV-9hJkks-gBkNF3-deKLMZ-gBimB1-4Kh7ix-aPBXUV-gBkAK6-qLjMXy-9wE4C5-8U3S7g-5bNfij-gBm3tJ-gBdkBV-6PjxqB-hYAeXG-btMFXK-dqmEB2-gBkn65-dW2Pz7-7LFfiy-wc2hAE-eL82wz-2qq9aG-qEBFER-9wB5E4-2NqSdJ-2g5pn-59Asp6-GuBJ-qEBFKv-uLex4-inFRzM-bLCKM6-5PPUjM-Nk1suL-5ZMBfG-aUZYb8-8PXnEP">Elizabethe</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>This mystery is the same one that existed in the Greek worldview of Plato, who believed that abstract ideas (above all mathematical) first existed outside any physical reality. The material world that we experience as part of our human existence is an imperfect reflection of these prior formal ideals. As the scholar of ancient Greek philosophy, <a href="https://lucian.uchicago.edu/blogs/news/2010/08/20/ian-mueller-scholar-of-ancient-greek-philosophy-and-math-1938-2010/">Ian Mueller</a>, writes in <a href="https://www.elsevier.com/books/mathematics-and-the-divine/koetsier/978-0-444-50328-2">“Mathematics And The Divine,”</a> the realm of such ideals is that of God. </p>
<p>Indeed, in 2014 the MIT physicist <a href="https://www.reddit.com/r/IAmA/comments/2e3t77/i_am_max_tegmark_an_mit_physics_professor/">Max Tegmark</a> argues in <a href="https://books.google.com/books/about/Our_Mathematical_Universe.html?id=BiKDvgAACAAJ">“Our Mathematical Universe”</a> that mathematics is the fundamental world reality that drives the universe. As I would say, mathematics is operating in a god-like fashion.</p>
<h2>The mystery of human consciousness</h2>
<p>The workings of human consciousness are similarly miraculous. Like the laws of mathematics, consciousness has no physical presence in the world; the images and thoughts in our consciousness have no measurable dimensions. </p>
<p>Yet, our nonphysical thoughts somehow mysteriously guide the actions of our physical human bodies. This is no more scientifically explicable than the mysterious ability of nonphysical mathematical constructions to determine the workings of a separate physical world.</p>
<p>Until recently, the scientifically unfathomable quality of human consciousness inhibited the very scholarly discussion of the subject. Since the 1970s, however, it has become a leading area of <a href="https://global.oup.com/academic/product/consciousness-9780199739097?cc=us&lang=en&">inquiry among philosophers</a>.</p>
<p>Recognizing that he could not reconcile his own scientific materialism with the existence of a nonphysical world of human consciousness, a leading atheist, <a href="http://ase.tufts.edu/cogstud/dennett/">Daniel Dennett</a>, in 1991 took the radical step of <a href="https://books.google.com/books/about/Consciousness_Explained.html?id=d2P_QS6AwgoC">denying that consciousness even exists</a>. </p>
<p>Finding this altogether implausible, as most people do, another leading philosopher, <a href="http://philosophy.fas.nyu.edu/object/thomasnagel">Thomas Nagel</a>, <a href="https://global.oup.com/academic/product/mind-and-cosmos-9780199919758?cc=us&lang=en&">wrote in 2012</a> that, given the scientifically inexplicable – the “intractable” – character of human consciousness, “we will have to leave [scientific] materialism behind” as a complete basis for understanding the world of human existence. </p>
<p>As an atheist, Nagel does not offer religious belief as an alternative, but I would argue that the supernatural character of the workings of human consciousness adds grounds for raising the probability of the existence of a supernatural god.</p>
<h2>Evolution and faith</h2>
<p>Evolution is a contentious subject in American public life. <a href="http://www.pewresearch.org/fact-tank/2017/02/10/darwin-day/">According to Pew,</a> 98 percent of scientists connected to the American Association for the Advancement of Science “believe humans evolved over time” while only a minority of Americans “fully accept evolution through natural selection.” </p>
<p>As I say in my book, I should emphasize that I am not questioning the reality of natural biological evolution. What is interesting to me, however, are the fierce arguments that have taken place between professional evolutionary biologists. A number of developments in evolutionary theory have <a href="https://books.google.com/books?id=fvmv2kU6PrYC">challenged</a> <a href="http://www.penguinrandomhouse.com/books/300564/why-evolution-is-true-by-jerry-a-coyne/9780143116646/">traditional Darwinist</a> – and later neo-Darwinist – views that emphasize random genetic mutations and gradual evolutionary selection by the process of survival of the fittest. </p>
<p>From the 1970s onwards, the Harvard evolutionary biologist <a href="http://www.nytimes.com/2002/05/21/us/stephen-jay-gould-60-is-dead-enlivened-evolutionary-theory.html">Stephen Jay Gould</a> created controversy by positing a different view, <a href="http://www.hup.harvard.edu/catalog.php?isbn=9780674024441">“punctuated equilibrium,”</a> to the slow and gradual evolution of species as theorized by Darwin. </p>
<p>In 2011, the University of Chicago evolutionary biologist <a href="http://www.thethirdwayofevolution.com/people/view/james-a-shapiro">James Shapiro</a> argued that, remarkably enough, many micro-evolutionary processes worked as though guided by a purposeful “sentience” of the evolving plant and animal organisms themselves. “The capacity of living organisms to alter their own heredity is undeniable,” <a href="http://www.thethirdwayofevolution.com/people/view/james-a-shapiro">he wrote</a>. “Our current ideas about evolution have to incorporate this basic fact of life.” </p>
<p>A number of scientists, such as <a href="https://www.nih.gov/about-nih/who-we-are/nih-director/biographical-sketch-francis-s-collins-md-phd">Francis Collins</a>, director of the U.S. National Institutes of Health, “see no conflict between believing in God and accepting the contemporary theory of evolution,” as the American Association for the Advancement of Science <a href="https://www.aaas.org/sites/default/files/QA_Evolution_0.pdf">points out.</a> </p>
<p>For my part, the most recent developments in evolutionary biology have increased the probability of a god. </p>
<h2>Miraculous ideas at the same time?</h2>
<p>For the past 10,000 years at a minimum, the most important changes in human existence have been driven by cultural developments occurring in the realm of human ideas. </p>
<p>In the Axial Age (commonly dated from 800 to 200 B.C.), world-transforming ideas such as Buddhism, Confucianism, the philosophies of Plato and Aristotle, and the Hebrew Old Testament almost <a href="http://www.hup.harvard.edu/catalog.php?isbn=9780674061439&content=reviews">miraculously appeared</a> at about the same time in India, China, ancient Greece and among the Jews in the Middle East, <a href="http://www.hup.harvard.edu/catalog.php?isbn=9780674061439&content=reviews">groups having little interaction</a> with one another. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/168658/original/file-20170509-7904-18g5e6p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/168658/original/file-20170509-7904-18g5e6p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/168658/original/file-20170509-7904-18g5e6p.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/168658/original/file-20170509-7904-18g5e6p.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/168658/original/file-20170509-7904-18g5e6p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/168658/original/file-20170509-7904-18g5e6p.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/168658/original/file-20170509-7904-18g5e6p.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Many world-transforming ideas, such as Buddhism, appeared in the world around the same time.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/toofarnorth/2823588421/in/photolist-62hFvB-fibjQb-4cVFiJ-5H8HUd-C2ndV-7GG2Fz-CMFevp-MZ7ub-4uem3q-7G15Wb-5H4rdP-5ivCsZ-abNL6Z-S745UE-buJSFq-DASDo8-7FW9mX-7GG2va-afm5zm-7G15SG-7G16fS-7GG2S8-nEfEnS-dvB1wA-5izUKE-5ivCy2-i568PS-rzW68-8nZ4HH-6QkHJQ-RHCDvx-5qxVPm-naTuwk-5BTpt3-i56a2U-aoGX3g-5CAT8N-3qo1At-2sm31A-5yn3-ooAKm-miHypu-F2LFi-mwGDT9-5dHrPL-6HonDA-rLmSz-5qUmsd-6hyasC-4Tzfz">Karyn Christner</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>The development of the scientific method in the 17th century in Europe and its modern further advances have had at least as great <a href="http://www.cambridge.org/us/academic/subjects/history/global-history/european-miracle-environments-economies-and-geopolitics-history-europe-and-asia-3rd-edition?format=PB&isbn=9780521527835#M6WWJdxjxfZQZiIi.97">a set of world-transforming consequences</a>. There have been <a href="http://press.uchicago.edu/ucp/books/book/chicago/B/bo9419313.html">many historical theories</a>, but none capable, I would argue, of explaining as fundamentally transformational a set of events as the rise of the modern world. It was a revolution in human thought, operating outside any explanations grounded in scientific materialism, that drove the process. </p>
<p>That all these astonishing things happened within the conscious workings of human minds, functioning outside physical reality, offers further rational evidence, in my view, for the conclusion that human beings may well be made “in the image of [a] God.”</p>
<h2>Different forms of worship</h2>
<p>In his commencement address to Kenyon College in 2005, the American novelist and essayist David Foster Wallace said that: <a href="http://www.newyorker.com/books/page-turner/this-is-water">“Everybody worships</a>. The only choice we get is what to worship.” </p>
<p>Even though Karl Marx, for example, condemned the illusion of religion, his followers, <a href="https://www.upress.pitt.edu/BookDetails.aspx?bookId=35157">ironically, worshiped Marxism</a>. The American philosopher <a href="http://philosophy.nd.edu/people/alasdair-macintyre/">Alasdair MacIntyre</a> thus wrote that for much of the 20th century, Marxism was the <a href="http://undpress.nd.edu/books/P00260">“historical successor of Christianity,”</a> claiming to show the faithful the one correct path to a new heaven on Earth. </p>
<p>In several of my <a href="https://books.google.com/books/about/Reaching_for_Heaven_on_Earth.html?id=oKm2AAAAIAAJ">books</a>, I have <a href="http://www.psupress.org/books/titles/0-271-02095-4.html">explored</a> how Marxism and other such “economic religions” were characteristic of much of the modern age. So Christianity, I would argue, did not disappear as much as it reappeared in many such disguised forms of <a href="http://www.marketsandmorality.com/index.php/mandm/article/view/1095">“secular religion.”</a> </p>
<p>That the Christian essence, as arose out of Judaism, showed such great staying power amidst the extraordinary political, economic, intellectual and other radical changes of the modern age is another reason I offer for thinking that <a href="http://www.jstor.org/stable/j.ctt1cg4m4s">the existence of a god is very probable</a>.</p><img src="https://counter.theconversation.com/content/75451/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Robert H. Nelson does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>There remain many mysteries that are beyond science. Does that mean that a God truly exists? A scholar gives reasons for this possibility.Robert H. Nelson, Professor of Public Policy, University of MarylandLicensed 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://images.theconversation.com/files/136342/original/image-20160901-1012-shlkpz.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C1479%2C1058&q=45&auto=format&w=496&fit=clip" /><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>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137664/original/image-20160913-4989-10ksej.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<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>
</figcaption>
</figure>
<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>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=575&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=575&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=575&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=723&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=723&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137431/original/image-20160912-19258-17danh0.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=723&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<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>
</figcaption>
</figure>
<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>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=300&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=300&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=300&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=377&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=377&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137435/original/image-20160912-19222-o0ppnr.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=377&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<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>
</figcaption>
</figure>
<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" />
An instant likely feels different to a person, or a redwood, or a gnat. What's infinitely small for one might be a whole lifetime for another – and that scale influences the choices we make.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/556882016-03-14T10:06:19Z2016-03-14T10:06:19ZPi pops up where you don't expect it<figure><img src="https://images.theconversation.com/files/114522/original/image-20160309-13712-g9bqus.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Pi is at the center of all circles.</span> <span class="attribution"><a class="source" href="https://upload.wikimedia.org/wikipedia/commons/8/8c/Matheon2.jpg">Holger Motzkau</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Happy Pi Day, where we celebrate the world’s most famous number. The exact value of π=3.14159… has fascinated people since ancient times, and mathematicians have computed <em>trillions</em> of digits. But why do we care? Would it actually matter if somebody got the 11,137,423,895,285th digit wrong?</p>
<p>Probably not. The world would keep on turning (with a circumference of 2πr). What matters about π isn’t so much the actual value as the <em>idea</em>, and the fact that π seems to crop up in lots of unexpected places.</p>
<p>Let’s start with the expected places. If a circle has radius r, then the circumference is 2πr. So if a circle has radius of one foot, and you walk around the circle in one-foot steps, then it will take you 2π = 6.28319… steps to go all the way around. Six steps isn’t nearly enough, and after seven you will have overshot. And since the value of π is irrational, no multiple of the circumference will be an even number of steps. No matter how many times you take a one-foot step, you’ll never come back exactly to your starting point.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=650&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=650&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=650&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=817&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=817&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114518/original/image-20160309-13712-1om653l.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=817&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Calculating the area of a circle with wedges.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:CircleArea.svg">Jim.belk</a></span>
</figcaption>
</figure>
<p>From the circumference of a circle we get the area. Cut a pizza into an even number of slices, alternately colored yellow and blue. Lay all the blue slices pointing up, and all the yellow slices pointing down. Since each color accounts for half the circumference of the circle, the result is approximately a strip of height r and width πr, or area πr<sup>2</sup>. The more slices we have, the better the approximation is, so the exact area must be <em>exactly</em> πr<sup>2</sup>. </p>
<h2>Pi in other places</h2>
<p>You don’t just get π in circular motion. You get π in <em>any</em> oscillation. When a mass bobs on a spring, or a pendulum swings back and forth, the position behaves just like one coordinate of a particle going around a circle. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/9r0HexjGRE4?wmode=transparent&start=0" frameborder="0" allowfullscreen></iframe>
<figcaption><span class="caption">Simple harmonic motion is another view of circular motion.</span></figcaption>
</figure>
<p>If your maximum displacement is one meter and your maximum speed is one meter/second, it’s just like going around a circle of radius one meter at one meter/second, and your period of oscillation will be exactly 2π seconds.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=480&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=480&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=480&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=603&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=603&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114722/original/image-20160310-26261-66iv16.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=603&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The area of the space under the normal-distribution curve is the square root of pi.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:E%5E(-x%5E2).svg">Autopilot</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Pi also crops up in probability. The function
f(x)=e<sup>-x²</sup>, where e=2.71828… is Euler’s number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of π. </p>
<p>How did π get into it?! The two-dimensional function f(x)f(y) <a href="https://www.google.com/#q=z+%3D+exp(-(x%5E2%2By%5E2">stays the same if you rotate the coordinate axes</a>. Round things relate to circles, and circles involve π. </p>
<p>Another place we see π is in the calendar. A normal 365-day year is just over 10,000,000π seconds. Does that have something to do with the Earth going around the sun in a nearly circular orbit? Actually, no. It’s just coincidence, thanks to our arbitrarily dividing each day into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.</p>
<p>What’s <em>not</em> coincidence is how the length of the day varies with the seasons. If you plot the hours of daylight as a function of the date, starting at next week’s equinox, you get the same sine curve that describes the position of a pendulum or one coordinate of circular motion.</p>
<h2>Advanced appearances of π</h2>
<p>More examples of π come up in calculus, especially in
infinite series like <br>
1 - (<sup>1</sup>⁄<sub>3</sub>) + (<sup>1</sup>⁄<sub>5</sub>) - (<sup>1</sup>⁄<sub>7</sub>) + (<sup>1</sup>⁄<sub>9</sub>) + ⋯ = π/4<br>
and <br>
1<sup>2</sup> + (<sup>1</sup>⁄<sub>2</sub>)<sup>2</sup> + (<sup>1</sup>⁄<sub>3</sub>)<sup>2</sup> + (<sup>1</sup>⁄<sub>4</sub>)<sup>2</sup> + (<sup>1</sup>⁄<sub>5</sub>)<sup>2</sup> + ⋯ = π<sup>2</sup>/6<br>
(The first comes from the <a href="http://mathworld.wolfram.com/TaylorSeries.html">Taylor series</a> of the arctangent of 1, and the second from the <a href="http://mathworld.wolfram.com/FourierSeries.html">Fourier series</a> of a sawtooth function.) </p>
<p>Also from calculus comes Euler’s <a href="https://www.math.toronto.edu/mathnet/questionCorner/epii.html">mysterious equation</a>
<br>
e<sup>iπ</sup> + 1 = 0
<br>
relating the five most important numbers in mathematics: 0, 1, i, π, and e, where i is the (imaginary!) square root of -1.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114520/original/image-20160309-13712-hc11ya.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A graph of the exponential function y=e^x.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Exp.svg">Peter John Acklam</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>At first this looks like nonsense. How can you possibly take a number like e to an imaginary power?! Stay with me. The rate of change of the exponential function f(x)=e<sup>x</sup> is equal to the value of the function itself. To the left of the figure, where the function is small, it’s barely changing. To the right, where the function is big, it’s changing rapidly. Likewise, the rate of change of any function of the form f(x)=e<sup>ax</sup> is proportional to e<sup>ax</sup>.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=582&fit=crop&dpr=1 600w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=582&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=582&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=732&fit=crop&dpr=1 754w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=732&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/114521/original/image-20160309-13693-16ccfxu.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=732&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The relationship between an angle, its sine, cosine and a circle.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Sin-cos-defn-I.png">345Kai</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>We can then <em>define</em> f(x)= e<sup>ix</sup> to be a complex function whose rate of change is i times the function itself, and whose value at 0 is 1. This turns out to be a combination of the trigonometric functions that describe
circular motion, namely cos(x) + i sin(x). Since going a distance π takes you halfway around the unit circle, cos(π)=-1 and sin(π)=0, so e<sup>iπ</sup>=-1. </p>
<p>Finally, some people prefer to work with τ=2π=6.28… instead of π. Since going a distance 2π takes you all the way around the circle, they would write that e<sup>iτ</sup> = +1. If you find that confusing, take a few months to think about it. Then you can celebrate June 28 by baking <em>two</em> pies.</p><img src="https://counter.theconversation.com/content/55688/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Lorenzo Sadun has received funding from the National Science Foundation. </span></em></p>We know pi appears when we talk about circles. But it appears in many other places, too. Why, pi, why?Lorenzo Sadun, Professor of Mathematics, University of Texas at AustinLicensed 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://images.theconversation.com/files/101493/original/image-20151110-5460-1v3e63d.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><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>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=464&fit=crop&dpr=1 600w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=464&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=464&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=583&fit=crop&dpr=1 754w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=583&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=583&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<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>
</figcaption>
</figure>
<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>
<blockquote>
<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>
</blockquote>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=399&fit=crop&dpr=1 600w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=399&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=399&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=501&fit=crop&dpr=1 754w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=501&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=501&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<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>
</figcaption>
</figure>
<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>
</blockquote>
<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" />
More students are taking Advanced Placement calculus in high school. They may be learning techniques for solving certain problems at the expense of the mathematical foundations they need to advance.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.