While most mathematicians would agree that Gauss was correct in pointing out that concepts, not symbol manipulation, are at the heart of mathematics, his words do have to be properly interpreted. While a notation does not matter, a representation can make a huge difference. The distinction is that developing or selecting a representation for a particular mathematical concept (or notion) involves deciding which features of the concept to capture.

For example, the form of the ten digits 0, 1, … , 9 does not matter (as long as they are readily distinguishable), but the usefulness of the Hindu-Arabic number system is that it embodies base- 10 place-value representation of whole numbers. Moreover, it does so in a way that makes both learning and using Hindu-Arabic arithmetic efficient.

Likewise, the choice of 10 as the base is optimal for a species that has highly manipulable hands with ten digits. Although the base-10 arithmetic eventually became the standard, other systems were used in different societies, but they too evolved from the use of the hands and sometimes the feet for counting: base-12 (where finger-counting used the three segments of each of the four fingers) and base-20 where both fingers and toes were used. Base-12 arithmetic and base-20 arithmetic both remained in regular use in the monetary system in the UK when I was a child growing up there, with 12 pennies giving one shilling and 20 shillings one pound. And several languages continue to carry reminders of earlier use of both bases — English uses phrases such as “three score and ten” to mean 70 (= 3x20 + 10) and French articulates 85 as “quatre-vingt cinq (4x20 + 5).

Another number system we continue to use today is base-60, used in measuring time (seconds and minutes) and in circular measurement (degrees in a circle). Presumably the use of 60 as a base came from combining the finger and toes bases 10, 12, and 20, allowing for all three to be used as most convenient.

These different base-number representation systems all capture features that make them useful to humans. Analogously, digital computers are designed to use binary arithmetic (base 2), because that aligns naturally with the two states of an electronic gate (open or closed, on or off).

In contrast, the shapes of the Hindu-Arabic numerals is an example of a superfluous feature of
the representation. The fact that it is possible to draw the numerals in a fashion whereby each
digit has the corresponding number of angles, like this

On the other hand, the huge difference a representation system can make in mathematics is indicated by the revolutionary change in human life that was brought about by the switch from Roman numerals and abacus-board calculation to Hindu-Arabic arithmetic in Thirteenth Century Europe, as I described in my 2011 book The Man of Numbers.

Of course, there is a sense in which representations do not matter to mathematics. There is a legitimate way to understand Gauss’s remark as a complete dismissal of how we represent mathematics on a page. The notations we use provide mental gateways to the abstract notions of mathematics that live in our minds. The notions themselves transcend any notations we use to denote them. That may, in fact, have been how Gauss intended his reply to be taken, given the circumstances.

But when we shift our attention from mathematics as a body of eternal, abstract structure occupying a Platonic realm, to an activity carried out by people, then it is clear that notations (i.e., a representation system) are important. In the early days of Category Theory, some mathematicians dismissed it as “abstract nonsense” or “mere diagram chasing”, but as most of us discovered when we made a serious attempt to get into the subject, “tracing the arrows” in a commutative diagram can be a powerful way to approach and understand a complex structure. [Google “the snake lemma”. Even better, watch actress Jill Clayburgh explain it to a graduate math class in an early scene from the 1980s movie It’s My Turn.]

A well-developed mathematical diagram can also be particularly powerful in trying to understand complex real-world phenomena. In fact, I would argue that the use of mathematical representations as a tool for highlighting hidden abstract structure to help us understand and operate in our world is one of mathematics most significant roles in society, a use that tends to get overlooked, given our present day focus on mathematics as a tool for “getting answers.” Getting an answer is frequently the end of a process of thought; gaining new insight and understanding is the start of a new mental journey.

A particularly well known example of such use are the Feynmann Diagrams, simple visualizations to help physicists understand the complex behavior of subatomic particles, introduced by the American physicist Richard Feynmann in 1948.

A more recent example that has proved useful in linguistics, philosophy, and the social sciences is the “completion diagram” developed by the American mathematician Jon Barwise in collaboration with his philosopher collaborator John Perry in the early 1980s, initially to understand information flow.

A discussion of one use of this diagram can be found in a survey article I wrote in the volume Handbook of the History of Logic, Volume 7, edited by Dov Gabbay and John Woods (Elsevier, 2008, pp.601-664), a manuscript version of which can be found on my Stanford homepage. That particular application is essentially the original one for which the diagram was introduced, but the diagram itself turned out be to be applicable in many domains, including improving workplace productivity, intelligence analysis, battlefield command, and mathematics education. (I worked on some of those applications myself; some links to publications are on my homepage.)

To be particularly effective, a representation needs to be simple and easy to master. In the case of a representational diagram, like the Commutative Diagrams of Category Theory, the Feynmann Diagram in physics, and the Completion Diagram in social science and information systems development, the representation itself is frequently so simple that it is easy for domain experts to dismiss them as little more than decoration. (For instance, the main critics of Category Theory in its early days were world famous algebraists.) But the mental clarity such diagrams can bring to a complex domain can be highly significant, both for the expert and the learner.

In the case of the Completion Diagram, I was a member of the team at Stanford that led the efforts to develop an understanding of information that could be fruitful in the development of information technologies. We had many long discussions about the most effective way to view the domain. That simple looking diagram emerged from a number of attempts (over a great many months) as being the most effective.

Given that personal involvement, you would have thought I would be careful not to dismiss a novel representation I thought was too simple and obvious to be important. But no. When you understand something deeply, and have done so for many years, you easily forget how hard it can be for a beginning learner. That’s why, when the MAA’s own James Tanton told me about his “Exploding Dots” idea some months ago, my initial reaction was “That sounds cute," but I did not stop and reflect on what it might mean for early (and not so early) mathematics education.

To me, and I assume to any professional mathematician, it sounds like the method simply adds a visual element on paper (or a board) to the mental image of abstract number concepts we already have in our minds. In fact, that is exactly what it does. But that’s the point! “Exploding Dots” does nothing for the expert. But for the learner, it can be huge. It does nothing for the expert because it represents on a page what the expert has in their mind. But that is why it can be so effective in assisting a learner arrive at that level of understanding! All it took to convince me was to watch Tanton’s lecture video on Vimeo. Like Tanton, and I suspect almost all other mathematicians, it took me many years of struggle to go beyond the formal symbol manipulation of the classical algorithms of arithmetic (developed to enable people to carry our calculations efficiently and accurately in the days before we had machines to do it for us) until I had created the mental representation that the exploding dots process capture so brilliantly. Many learners subjected to the classical teaching approach never reach that level of understanding; for them, basic arithmetic remains forever a collection of incomprehensible symbolic incantations.

Yes, I was right in my original assumption that there is nothing new in exploding dots. But I was also wrong in concluding that there was nothing new. There is no contradiction here. Mathematically, there is nothing new; it’s stuff that goes back to the first centuries of the First Millennium—the underlying idea for place-value arithmetic. Educationally, however, it’s a big deal. A very big deal. Educationally explosive, in fact. Check it out!

Original author: Mathematical Association of America

]]>The most common (I believe) use of clickers is to provide students with frequent quiz questions to check that they are retaining important facts. (The early MOOCs, including my own, used simple, machine-graded quizzes embedded in the video lectures to achieve the same result.) And a lot of that research I just alluded to showed that the clickers achieve that goal.

So too does the latest study. All of which is fine and dandy if the main goal of the course is retention of facts. Where things get messy is when it comes to conceptual understanding of the material—a goal that almost all mathematicians agree is crucial.

In the new study, the researchers looked at two versions of a course (physics, not mathematics), one fact-focused, the other more conceptual and problem solving. In each course, they gave one group fact-based clicker questions and a second group clicker questions that concentrated on conceptual understanding in addition to retention of basic facts.

As the researchers expected, both kinds of questions resulted in improved performance in fact- based questions on a test administered at the end.

Neither kind of question led to improved performance in a problem-based test questions that required conceptual understanding.

The researchers expressed surprise that the students who were given the conceptual clicker questions did not show improvement in conceptual questions performance. But that was not the big surprise. That was, wait for it: students who were given only fact-based clicker questions actually performed worse on conceptual, problem solving questions.

To those of us who are by nature heavy on the conceptual understanding, not showing improvement as a result of enforced fact-retention comes as no big surprise. But a negative effect! That’s news.

By way of explanation, the researchers suggest that the fact-based clicker questions focus the student’s attention on retention of what are, of course, surface features, and do so to the detriment of acquiring the deeper understanding required to solve problems.

If this conclusion is correct—and is certainly seems eminently reasonable—the message is clear. Use clickers, but do so with questions that focus on conceptual understanding, not retention of basic facts.

The authors also recommend class discussions of the concepts being tested by the clicker questions, again something that comes natural to we concepts matter folks.

I would expect the new finding to have implications for game-based math learning, which regular readers will know is something I have been working on for some years now. The games I have been developing are entirely problem-solving challenges that require deep understanding, and university studies have shown they achieve the goal of better problem-solving skills. (See the December 4, 2015 Devlin’s Angle post.) The majority of math learning games, in contrast, focus on retention of basic facts. Based on the new clickers study, I would hypothesize that, even if a game were built on math concepts (many are not), unless the gameplay involves active, problem-solving engagement with those concepts, the result could be, not just no conceptual learning, but a drop in performance on a problem solving test.

Both clickers and video games set up a feedback cycle that can quickly become addictive. With both technologies, regular positive feedback leads to improvement in what the clicker- questions or game-challenges ask for. Potentially more pernicious, however, that positive feedback will result in the students thinking they are doing just fine overall—and hence have no need to wrestle more deeply with the material. And that sets them up for failure once they have to go beneath the surface fact they have retained. Thinking you are winning all the time seduces you to ease off, and as a result is the path to eventual failure. If you want success, the best diet is a series of challenges— that is to say, challenges in coming to grips with the essence of the material to be learned—where you experience some successes, some failures from which you can recover, and the occasional crash-and- burn to prevent over-confidence.

That’s not just the secret to learning math. It’s the secret to success in almost any walk of life.

Original author: Mathematical Association of America

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*ELC17 Embraces Cutting Edge Theme at August Event*

**Maricopa, AZ June 7 ^{th}, 2017- **Approximately 45% of all jobs will be replaced by artificial intelligence, machine learning or cognitive systems. The World Economic Forums identifies 5 technology trends that are transforming global commerce: internet of things, artificial intelligence, advanced robotics, wearables and 3D printing. If you are not embracing these emerging technologies, your organization will be left behind.

*“Digital is the wires, but digital intelligence, or artificial intelligence as some people call it, is about much more than that. This next decade is about how you combine those and become a cognitive business. It’s the dawn of a new era.”-* VIRGINIA M. ROMETTY, CHAIRMAN, PRESIDENT AND CHIEF EXECUTIVE OFFICER, IBM

**Where does a leader begin this journey?**

The Enterprise Learning! Conference is singularly focused on delivering the answers you need. ELC17 features 6 keynoters who are embracing these technological forces and re-inventing their organizations’ products, services and learning ecosystems. Invest 2-days at the Enterprise Learning! Conference and create your action plan to thrive in this age of disruption.

**Step1: Keynote: Thriving in the Age of Disruption**

In this keynote, Sundar Nagarathnam, SVP of Salesforce, and Kathy Breis, GM of Learning@Cisco reveal how to thrive in the age of disruption. They will share their strategies for innovation, collaboration and the rapid transformation spurred by the digitalization of the enterprise.

Salesforce, named Most Innovative Company by Forbes 7 years in a row, delivers the world’s #1 CRM solution. Salesforce is rapidly evolving, offering Lightning for mobile apps, visual data analytics (Wave) and AI (Einstein). Enterprises can leverage these sophisticated cognitive systems without internal software engineering teams.

Cisco has spent the past three decades helping companies seize the opportunities of tomorrow through the transformation of how people connect, communicate and collaborate. Three of the most critical business issues facing Cisco and many organizations today are reskilling the workforce for continual transformation, improving employee engagement to drive productivity and agile responsiveness; and sharing institutional knowledge and best practices across the organization.

**Step 2: Keynote: The Future Workplace Experience: 10 Rules for Mastering Disruption**

In a business landscape rocked by constant change and turmoil, companies like Airbnb, Cisco, GE Digital, Google, IBM and Microsoft are reinventing the future of work. What is it that makes these companies so different? They’re strategic, they’re agile, and they’re customer focused. Most importantly, they’re game changers. And, their workplace practices reflect this.

In this keynote, Kevin Mulcahy, partner, Future Workplace, will present an actionable framework for meetings today’s toughest business disruptions head-on. He will guide you step-by-step through the process of recruiting top employees and building an engaged culture- one that will drive your company to long-term success. He will provide you 10 rules for rethinking, reimagining and reinventing your organization.

**Step 3: Keynote: The Future Learning Ecosystem**

Globalization, social media, ever-increasing computing power, and the proliferation of low-cost advanced technologies have created a level of worldwide complexity and rapid change never seen. To remain competitive in this environment, today’s workers, military members, and civil servants require an expanded set of competencies, higher levels of nuanced skills such as critical thinking and emotional intelligence, and more efficient and agile pathways to expertise. Achieving these outcomes depends, at least in part, on enhancing our learning ecosystem. This includes, for instance, identifying new ways to empower our personnel to learn anytime/anywhere, enhancing the quality of learning delivered, and better tailoring learning experiences to individuals’ needs.

In this keynote, Dr. Jennifer Vogel-Walcutt, Innovation Director of ADL, will outline a vision for the future of learning, painting a picture of the “art of the possible” and proposing a roadmap that outlines five enabling conditions needed to achieve this future vision.

**Step 4: View the complete ELC17 conference program at**: http://2elearning.com/images/ELC2017/ELC17_ConferenceGuide.pdf

**Step 5: Register today**at: http://www.elceshow.com. Register by July 1st and save up to $500.

**About the Enterprise Learning! Conference**

The Enterprise Learning! Conference 2017 hosts global thought leaders and executives from corporate enterprise, government agencies, higher education and non-profit organizations. This conference reveals how leaders are building high-performance organizations in the age of digital disruption. ELC17 serves the robust $243 billion enterprise learning market expanding at 17% CAGR.

ELC17 convenes over 125 award-winning learning professionals to share the best practices of high performance organizations, lessons learned, and future strategies. Invest 48 hours at ELC17, and discover how to engage teams, build a productive learning culture, measure impact and embrace the future digital enterprise.

**Who Should Attend**

Executives charged with driving enterprise performance via learning and workplace technologies, including HR, Talent, Development, Training, E-learning, Project Management, Education, Sales & Service should attend ELC17. Government, non-profit agencies and educational institution leaders are also in attendance to collaborate on the now and the next in learning. Attending this conference is an amazing opportunity to meet colleagues from across the globe. Registration is now open at: http://www.elceshow.com.Register by July 1^{st} and save up to $500.

*9th Annual Enterprise Learning! Conference Announces 6 Keynotes and 2 Awards Events at August 29th-30th Conference in San Diego, CA*

Elearning! Media Group, the leader in learning and workplace technology media, announced the Enterprise Learning! Conference 2017 (ELC17) keynotes and event agenda. Registration is also now open. The event takes place August 29-30, 2017 in San Diego, CA. The theme is “Building the High-Performance Organization in the Age of Disruption.”

The Enterprise Learning! Conference 2017 hosts global thought leaders and executives from corporate enterprise, government agencies, higher education and non-profit organizations. This conference reveals how leaders are building high-performance organizations in the age of digital disruption. ELC17 serves the robust $243 billion enterprise learning market expanding at 17% CAGR.

ELC17 convenes over 125 award-winning learning professionals to share the best practices of high performance organizations, lessons learned, and future strategies. Invest 48 hours at ELC17, and discover how to engage teams, build a productive learning culture, measure impact and embrace the future digital enterprise.

“There is no better location to share what’s now and next than California,” said Catherine Upton, ELC17 conference chair. The rate of technological innovation is disruptive to our organizations. At ELC, attendees will meet leaders from Salesforce, NASA’s Jet Propulsion Labs,Zappos, T-Mobile and Cisco; all are embracing innovation to re-invent learning within their organizations.”

ELC17 Keynotes Announced

ELC17 theme of Building the High-Performance Organization in the Age of Digital Disruption. The digital evolution is just beginning; AI, Machine Learning and Immersive learning is progressing rapidly and will change the workplace, our jobs and roles. Discover how to harness the age of disruption by attending these keynotes at ELC17.

Keynote: Thriving in the Age of Disruption

Speakers: Sundar Nagaranthnam, SVP, Salesforce University, Salesforce

& Kathy Bries, GM, Learning@Cisco, Cisco

Keynote: Breaking the Rules: Creating the Contemporary Learning Organization

Speaker: Anthony Gagliardo, Head of HR & Training, NASA JPL

Keynote: The Future Work Experience: 10 Rules for Mastering Disruption

Speaker: Kevin J. Mulcahy, Partner, Future Workplace

Keynote: Learning Ecosystems for Tomorrow’s Workplace

Speakers: Dr Jennifer Vogel-Walcutt, Director of Innovation, ADL, Dept. of Defense, & Tina

Marron-Partridge, VP, Global Talent Director, IBM Watson (invited)

Keynote: Building the Culture of WOW at Zappos.com

Speaker: Erica Javellana, Speaker of the House, Zappos.com

Keynote: Helping Employees Thrive in the Age of Disruption

Speaker: Joe Burton, CEO, Whil Concepts, Inc.

Celebrating Excellence

ELC17 provides executives an engaged environment to network, share and learn from leaders across the globe. Coupled with cutting edge research, expert learning technologists and two prestigious industry award programs- Learning! 100 and Learning! Champions- this is the “Must Attend” forum for learning and performance executives. Registration is now open at: http://www.ELCEShow.com Register by July 1st and save $500.

Who Should Attend

Executives charged with driving enterprise performance via learning and workplace technologies, including HR, Talent, Development, Training, E-learning, Project Management, Education, Sales & Service should attend ELC17. Government, non-profit agencies and educational institution leaders are also in attendance to collaborate on the now and the next in learning. Attending this conference is an amazing opportunity to meet colleagues from across the globe. Registration is now open at:http://www.elceshow.com. Register by July 1st and save up to $500.

About Elearning! Media Group

Elearning! Media Group is owned by B2B Media Group LLC. Elearning! Media Group consists of eleven media products including: Elearning! Magazine, Government Elearning! E-Magazine, e-mail newsletters, Alerts, Websites, Web seminars, the Enterprise Learning! Summit and Enterprise Learning! Conference. Elearning! Media Group serves the $243 billion learning & workplace technology market. Suppliers and practitioners can follow us: online at www.2elearning.com; on Twitter: @2elearning or #ELCE; via Facebook: Elearning! -Magazine or LinkedIn: Elearning! Magazine Network or Enterprise Learning! Conference.

Enterprise Learning! Events

Since 2008, Enterprise Learning! Events bring onsite and online audiences together to learn, network and share. Mark your calendar for Enterprise Learning! Conference on August 29-30, 2017 in San Diego, CA. Enterprise Learning! Conference hosts the Learning! 100 and Learning! Champion Awards. The Enterprise Learning! Conference Online is an on-demand event available to all ELC17 conference attendees, and online only attendees after the live event. For more information about the Enterprise Learning! Conference visit http://www.elceshow.com.

--*Those with higher education see better health, graduation rates & technology access.*

By Catherine Upton, *Elearning!* Magazine

National Center for Education Statistics released the “Condition of Education 2017” report this week; a congressionally mandated annual report summarizing the latest data on education in the United States. The report is designed to help policymakers and the public monitor educational progress. This year’s report includes 50 indicators on topics ranging from prekindergarten through postsecondary education, as well as labor force outcomes and international comparisons.

Highlights:

-Graduation Rates Higher for 4-year Post Secondary Schools

Among first-time college students, the percentage of students who were still enrolled or had graduated after 3 years was higher for students who began at 4-year institutions (80 percent) than for those who began at 2-year institutions (57 percent).

-Access to Internet at Home Increases with Parent’s Education Level

The percentage of students who use the Internet at home varied by parental education level in 2015, ranging from 42 percent for children whose parents had not completed high school to 71 percent for those whose parents had completed a bachelor’s or higher degree.

-Disability rates are inversely-related to education level

Sixteen percent of 25- to 64-year-olds who had not completed high school had one or more disabilities in 2015, compared to 4 percent of those who had completed a bachelor’s degree and 3 percent of those who had completed a master’s or higher degree.

Download full report at: https://nces.ed.gov/pubs2017/2017144.pdf

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As the business world becomes ever more complex (and less and less forgiving), the ability to show data in a meaningful way—without overwhelming people or leading them to the wrong conclusions—has become a foundational skill for *all *leaders. The good news is there are lots of tips and tricks you can use to create easy-to-understand, high-impact dashboards. Here, the authors of *The Big Book of Dashboards* offer 9 Dos and Don'ts to keep in mind:

**DO experiment, iterate, and most importantly, *** get feedback.* You'll be amazed what happens when you show your work to somebody who hasn't lived with the dashboard as long as you have.

"Do you think all the dashboards that made it into the book 'got it right' the first time?" asks Wexler. "So many of the dashboards had to endure an 'extreme vetting process' where we would ask, 'Why are those bars red? Why is that text over there? What happens if you move that important component from the bottom right—where I didn't even see it—to the upper left?'"

"Getting feedback early also turns passive participants into project owners who will want to see the project succeed," adds Cotgreave.

**DON'T emulate snazzy infographics from ****Time**** and ****USA Today**** to garner user engagement. **Your organizational dashboard probably isn't appearing in a magazine or public website where it must compete with pictures of kittens and puppies. If you really want to engage your audience, you should personalize the dashboard for each viewer (people like things that are about themselves even more than kittens and puppies). It's easier to do than you might think.

**DO make it personal. **You want to create things that are meaningful to the people viewing them, and one of the best ways to do that is to include the viewer *in* the visualization. You can do this by creating dashboards that make it easy for people to see how their department, region, store, etc., is doing with respect to others (see below).

**DO mix and match the best elements from different dashboards and from different industries. **Yes, each industry has its own challenges, and you want to be inspired by solutions you can relate to, but you'll be missing great stuff if you don't look beyond your silo. One of the best examples is a mobile-ready dashboard that professional soccer players use to see how well they performed across different measures.

"The technique to show the most recent match compared with previous matches is **jaw-droppingly effective** and will work in any situation where you need to compare current with past performance," says Shaffer.

**DON'T use red and green. **At least don't use the hues of red and green that you usually see in things like balanced scorecards as a startlingly large percentage of the population cannot discern any differences between those two colors! If you know a little bit about the general population and color-vision deficiency (a.k.a., "color blindness"), you'll make things that more people can use and that will likely be considerably more attractive than a traffic light.

**DON'T try to answer every question in one dashboard. **"We have yet to hear anyone say, 'This dashboard would be better if you overstuffed it with even more information,'" says Wexler. "By all means, answer the most important questions (at least the ones you know about) but expect your work to generate as many new questions as it answers! It's the first step in building an open, data-driven company as it encourages individuals to explore data on their own, discover new insights, and then publish their own dashboards."

**DO design with mobile devices in mind. **You want to make it as easy as possible for people to explore the information they need, when they need it, and that means creating dashboards that are mobile-friendly.

**DO use BANs (big-ass numbers) on your dashboards. **These are large, key performance indicators that start as anchor points to the dashboard. They can serve as conversation starters and finishers, can provide context to adjacent charts, and can serve as a universal color legend. They are also wonderful anchor points for people who may be unfamiliar with a dashboard as they practically scream, "Start here!"

**DON'T use the same old chart for your data. **You've got a dataset that has time in it. Don't assume a line chart will be the best choice. There are many ways of visualizing time. Some will reveal insights the standard timeline cannot show. If you're looking at seasonality, or changes in rank, or part-to-whole changes over time, the timeline is sometimes the worst choice. Consider the question being asked of the data, then iterate through the different chart types to find the most appropriate.

**About the Authors:**

**Steve Wexler, Jeffrey A. Shaffer**, and **Andy Cotgreave** are coauthors of *The Big Book of Dashboards: Visualizing Your Data Using Real-World Business Scenarios*.

**Steve Wexler** has worked with ADP, Gallup, Deloitte, Convergys, Consumer Reports, *The Economist*, ConEd, D&B, Marist, Tradeweb, Tiffany, McKinsey & Company, and many other organizations to help them understand and visualize their data. Steve is a Tableau Zen master, Iron Viz champion, and training partner. To learn more, visit DataRevelations.com.

**Jeffrey A. Shaffer** is vice president of information technology and analytics at Recovery Decision Science and Unifund. He is also adjunct professor at the University of Cincinnati, where he teaches data visualization, and was named the 2016 Outstanding Adjunct Professor of the Year. To learn more, visit DataPlusScience.com.

**Andy Cotgreave** is technical evangelist at Tableau Software. He has over 10 years' experience in data visualization and business intelligence, first honing his skills as an analyst at the University of Oxford. Since joining Tableau in 2011, he has helped and inspired thousands of people with technical advice and ideas on how to build a data-driven culture in a business. To learn more, visit GravyAnecdote.com.

**About the Book:**

*The Big Book of Dashboards: Visualizing Your Data Using Real-World Business Scenarios *(Wiley, April 2017, ISBN: 978-1-119-28271-6, $49.95) is available at bookstores nationwide. For more information, please visit thebook's page on www.wiley.com.

Be that as it may, when I woke up this morning and went online, two fascinating stories were waiting for me. What’s more, they are connected – at least, that’s how I saw them.

First, my Stanford colleague Professor Jo Boaler sent out a group email pointing to a New York Times article that quoted her, and which, she noted, she helped the author to write. Titled "No Such Thing as a Math Person," it summarizes the consensus among informed math educators that mathematical ability is a spectrum. Just like any other human ability. What is more, the basic math of the K-8 system is well within the capacity of the vast majority of people. Not easy to master, to be sure; but definitely within most people’s ability. It may be defensible to apply terms such as “gifted and talented” to higher mathematics (though I will come back to that momentarily), but basic math is almost entirely a matter of wanting to master it and being willing to put in the effort. People who say otherwise are either (1) education suppliers trying to sell products, (2) children who for whatever reason simply do not want to learn and find it reassuring to convince themselves they just don’t have the gift, or (3) mums and dads who want to use the term as a parental boast or an excuse.

Unfortunately, the belief that mathematical ability is a “gift” (that you either have or you don’t) is so well established it is hard to get rid of. Part of the problem is the way it is often taught, as a collection of rules and procedures, rather than a way of thinking (and a very simplistic one at that). Today, this is compounded by the rapid changes in society over the past few decades, that have revolutionized the way mathematics needs to be taught to prepare the new generation for life in today’s – and tomorrow’s – world. (See my January 1 article in The Huffington Post, "All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime," and its follow up article (same date), "Number Sense: the most important mathematical concept in 21st Century K-12 education.")

With many parents, and not a few teachers, having convinced themselves of the “Math Gift Myth,” attempts over the past several decades to change that mindset have met with considerable resistance. If you have such a mindset, it is easy to see what happens in the educational world around you as confirming it. For instance, one teacher commented on The New York Times article:

“Excuse me? I'm a teacher and I refute your assertion. I have seen countless individuals who have problems with math – and some never get it. The same goes for English. But, unless you've spent years in the classroom, it takes years to fully accept that observation. The article's writer is a doctor, not a teacher; accomplishment in one field does not necessarily translate readily to another.”

Others were quick to push back against that comment, with one pointing out that her final remark surely argues in favor of everyone in the education world keeping up with the latest scientific research in learning. We are all liable to seek confirmation of our initial biases. And both teachers and parents are in powerful positions to pass on those biases to a new generation of math learners.

In her most recent book, Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching, Prof Boaler lays out some of the considerable evidence against the Math Gift Myth, and provides pointers to how to overcome it in the classroom. The sellout audiences Boaler draws for her talks at teachers conferences around the world indicates the hunger there is to provide math learning that does not produce the math-averse, and even math-phobic, citizens we have grown accustomed to.

And so to that second story I came across. Hemant Mehta is a former National Board Certified high school math teacher in the suburbs of Chicago, where he taught for seven years, who is arguably best known for his blog The Friendly Atheist. His post on May 22 was titled "Years Later, the Mother Who 'Audited' an Evolution Exhibit Reflects on the Viral Response." Knowing Mehta’s work (for the record, I have also been interviewed by him on his education-related podcast), that title hooked me at first glance. I could not resist diving in.

As with The New York Times article I led off with, Mehta’s post is brief and to the point, so I won’t attempt to summarize it here. Like Mehta, as an experienced educator I know that it requires real effort, and courage, to take apart ones beliefs and assumptions, when faced with contrary evidence, and then to reason oneself to a new understanding. So I side with him in not in any way trying to diminish the individual who made the two videos he comments on. What we can do, is use her videos to observe how difficult it can be to make that leap from interpreting seemingly nonsensical and mutually contradictory evidence from within our (current!) belief system, to seeing it from a new viewpoint from which it all makes perfect sense – to rise above the trees to view the forest, if you will. The video lady cannot do that, and assumes no one else can either.

Finally, what about my claim that post K-12 mathematics may be beyond the reach of many individuals’ innate capacity for progression along that spectrum I referred to? Of course, it depends on what you mean by “many”. Leaving that aside, however, if someone, for whatever reason, develops a passionate interest in mathematics, how far can they go? I don’t know. Based on a sample size of one, me, we can go further than we think. I look at the achievement of mathematicians such as Andrew Wiles or Terrence Tao and experience the same degree of their being from a different species as the keen-amateur- cyclist-me feels when I see the likes of Tour de France winner Chris Froome or World Champion Peter Sagan climb mountains at twice the speed I can sustain.

Yet, on a number of occasions where I failed to solve a mathematics problem I had been working on for months and sometimes years, when someone else did solve it, my first reaction was, “Oh no, I was so close. If only I had tried just a tiny bit harder!” Not always, to be sure. Not infrequently, I was convinced I would never have found the solution. But I got within a hairsbreadth on enough occasions to realize that with more effort I could have done better than I did. (I have the same experience with cycling, but there I do not have a particular desire to aim for the top.)

In other words, all my experience in mathematics tells me I do not have an absolute ability limit. Nor, I am sure, do you. Mathematical proficiency is indeed a spectrum. We can all do better – if we want to. That, surely is the message we educators should be telling our students, be they in the K-8 classroom or the postgraduate seminar room.

Gifted and talented? Time to recognize that as an educational equivalent of the Flat Earth Belief. Sure, we are surrounded by seemingly overwhelming daily experience that the world is flat. But it isn’t. And once you accept that, guess what? From a new perspective, you start to see supporting evidence for the Earth being spherical.

Original author: Mathematical Association of America

]]>What makes one CEO better than another? McKinsey Group studied 600 CEOs tenures across the Fortune 500 from 2004 to 2014. How was success defined? Performance. These CEOs lead organizations that exceeded their industry averages. Some took laggard companies and reinvented them, while others exercised operational disciplines or strategically levers to reinvent their companies. Exceptional CEOs are defined as those that achieve 500% or greater growth in stakeholders returns during their tenure.

There were three distinct strategies exceptional CEOs embrace are:

Strategy 1: Hire External Candidates

The high performing CEO is twice as likely to be an external hire and 1.5 times more likely than the top 25% of high performing CEOs. The external CEO brings ‘fresh blood’ into the company and is more likely to question the status quo and tap strategic levers. Today, 55%of CEOs are internal hires.

Strategy 2: Take Strategic Actions

Sixty percent of exceptional CEOs conducted strategic review of the organization within 24 months of taking the reins regardless of the performance of the company. These leaders are 19% more likely to use cost reductions, the exceptional CEOs were significantly more likely to launch initiatives than the average CEO, thereby building strategic momentum. At the same time, exceptional CEOs are 48% less likely to reorganize the company, 40% less likely to launch new products, and 23% less likely to shuffle management teams.

Strategy 3: Achieve Organizational Balance

According to the study, ‘exceptional CEOs are less likely than the average CEO to undertake organizational redesign or management-team reshuffles in the first two years in office. This could be a function of the strategic game they were playing… since there are only so many initiatives and changes that organizations and people can absorb in a short space of time. Indeed, since the exceptional group contained an above-average proportion of outsider CEOs launching fundamental strategic rethinks, the data may reflect a sequencing of initiatives, with structural change following strategic shifts.’

Aspiring CEOs can learn much from this study. Think like an outside. Use strategic review process. Stage change over time, not all at once.

]]>By Ryan Rose

In fantasy sports, participants put together a “dream team” based on predictive analytics, such as how this quarterback has performed statistically in the past three games or how that pitcher has interacted with left-handed batters historically. Similarly, horseracing fans take predictive analytics very seriously. Bets can be determined on such granular details as how that colt performs on mud versus turf.

For those not initiated with this admittedly geeky pastime, here’s a basic breakdown of how fantasy sports work: Essentially, it’s an online game in which participants assemble imaginary or visual teams comprised of real players of real teams. These teams compete based on the statistical performance of those players in real games. This performance is then converted into points that are compiled and totaled according to a roster selected by the fantasy team’s manager.

I’ve never liked football, but I’m crazy about fantasy football. And that’s because I love statistics and especially, analytics.

Essentially, when it comes to fantasy sports, you’re looking to create the best team you can create. It used to be done all on graph paper, but these days, it's digital. In a way, it’s like dungeons and dragons but for jocks.

Computers can help you put together these all-start fantasy teams. If we’re talking football, then you pick a wide receiver, a quarterback, etc. – someone to represent all of the positions you’d have on an IRL (in real life) team. The same goes for any other fantasy sport.

It all comes down to looking at the predictive analytics of each player to determine how they would perform in a given setting.

So how does this translate to the workplace?

It might sound strange, but when it comes to putting together strong teams of engaged employees, a similar strategy can be used.

In the same way that you would look for the best running back or shortstop, now you’re looking at the stats around each expert in each subject matter. Basically, you’re drafting/building a team of knowledge experts.

Traditionally, many learning professionals in corporate settings have looked to intuition or hearsay to identify individuals as experts. In fact, many organizations struggle when trying to accurately identify skills strengths and gaps in their teams because these decisions are often gut-driven rather than based in analytics. What’s more, many

organizations don’t know how or where to pull the data needed to create the most successful teams possible.

Analytics are the opposite of the gut reaction. There is data everywhere to pull insights from – it’s just a matter of harnessing it. For most organizations, this doesn’t require adding additional tools or software; typically, the data is there but it’s either not being used or not used effectively.

Analytics can be pulled from formal training records, such as learning management systems, formal classes the employees’ have taken, certifications and degrees they might hold, etc. There’s also analytics to be pulled from individuals’ social learning activities – which encompass activities such as blogs they read and interact with, social media activity, and online collaboration tools. With a system in place, data from all of these can be tracked, collected and analyzed in a much more effective manner than say, giving people written assessments to fill out.

Peer assessments, be they formal or informal, also are a key asset. The trick is to ensure there is a unified place where all of this information can be pulled together into once place, where that data can then be weighed against the actual work being done.

Once this information is evaluated, learning leaders can create teams of varied skill sets and levels of expertise to optimize performance. It’s also important to keep teams balanced as things change – an employee leaves the company, gets promoted, for instance. When you have all of the data available in one easy-to-view spot, that makes it easy to ensure you’re keeping the balance even when the roster changes.

**Want to Learn More?**

Learn more about how to gamify learning, by subscribing to Elearning! Magazine at www.2elearning.com

]]>For some reason, once a number has been given names like “Golden Ratio” and “Divine Ratio”, millions of otherwise sane, rational human beings seem willing to accept claims based on no evidence whatsoever, and cling to those beliefs in the face of a steady barrage of contrary evidence going back to 1992, when the University of Maine mathematician George Markovsky published a seventeen- page paper titled "Misconceptions about the Golden Ratio" in the MAA’s College Mathematics Journal, Vol. 23, No. 1 (Jan. 1992), pp. 2-19.

In 2003, mathematician, astronomer, and bestselling author Mario Livio weighed in with still more evidence in his excellent book The Golden Ratio: The Story ofPHI, the World's Most Astonishing Number.

I first entered the fray with a Devlin’s Angle post in June 2004 titled "Good Stories Pity They’re Not True" [the MAA archive is not currently accessible], and then again in May 2007 with "The Myth That Will Not Go Away" [ditto].

Those two posts gave rise to a number of articles in which I was quoted, one of the most recent being "The Golden Ration: Design’s Biggest Myth," by John Brownlee, which appeared in Fast Company Design on April 13, 2015.

In 2011, the Museum of Mathematics in New York City invited me to give a public lecture titled "Fibonacci and the Golden Ratio Exposed: Common Myths andFascinating Truths," the recording of which was at the time (and I think still is) the most commented-on MoMath lecture video on YouTube, largely due to the many Internet trolls the post attracted—an observation that I find very telling as to the kinds of people who hitch their belief system to one particular ratio that does not quite work out to be 1.6 (or any other rational number for that matter), and for which the majority of instances of those beliefs are supported by not one shred of evidence. (File along with UFOs, Flat Earth, Moon Landing Hoax, Climate Change Denial, and all the rest.)

Needless to say, having been at the golden ratio debunking game for many years now, I have learned to expect I’ll have to field questions about it. Even in a media interview about a book that, not only flatly refutes all the fanciful stuff, but lays out the history showing that the medieval mathematician known today as Fibonacci left no evidence he had the slightest interest in the sequence now named after him, nor had any idea it had several cute properties. Rather, he simply included among the hundreds of arithmetic problems in his seminal book Liber abbaci, published in 1202, an ancient one about a fictitious rabbit population, the solution of which is that sequence.

What I have always found intriguing is the question, how did this urban legend begin? It turns out to be a relatively recent phenomenon. The culprit is a German psychologist and author called Adolf Zeising. In 1855, he published a book titled: A New Theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole nature and art, accompanied by a complete summary of the prevailing systems.

This book, which today would likely be classified as “New Age,” is where the claim first appears that the proportions of the human body are based on the Golden Ratio. For example, taking the height from a person's naval to their toes and dividing it by the person's total height yields the Golden Ratio. So, he claims, does dividing height of the face by its width.

From here Zeising leaped to make a connection between these human-centered proportions and ancient and Renaissance architecture. Not such an unreasonable jump, perhaps, but it was, and is pure speculation. After Zeising, the Golden Ratio Thing just took off.

Enough! I can’t bring myself to continue. I need a stiff drink.

For more on Zeising and the whole wretched story he initiated, see the article by writer Julia Calderone in business Insider, October 5, 2015, "The one formula that's supposed to 'prove beauty' is fundamentally wrong."

See also the blogpost on Zeising on the blog misfits’ architecture, which presents an array of some of the battiest claims about the Golden Ratio.

That’s it. I’m done.

Original author: Mathematical Association of America

]]>Devlin makes a pilgrimage to Pisa to see the statue of Leonardo Fibonacci in 2002. |

Before long, a major publisher contracted me to publish a collection of my MicroMaths articles, which I did, and following that Penguin asked me to write a more substantial book on mathematics for a general audience. That book, Mathematics: The NewGolden Age, was first published in 1987, the year I moved to America.

In addition to writing for a general audience, I began to give lectures to lay audiences, and started to make occasional appearances on radio and television. From 1991 to 1997, I edited MAA FOCUS, the monthly magazine of the Mathematical Association of America, and since January 1996 I have written this monthly Devlin’s Angle column. In 1994, I also became the NPR Math Guy, as I describe in my latest article in the Huffington Post.

Each new step I took into the world of “science outreach” brought me further pleasure, as more and more people came up to me after a talk or wrote or emailed me after reading an article I had written or hearing me on the radio. They would tell me they found my words inspiring, challenging, thought-provoking, or enjoyable. Parents, teachers, housewives, business people, and retired people would thank me for awakening in them an interest and a new appreciation of a subject they had long ago given up as being either dull and boring or else beyond their understanding. I came to realize that I was touching people’s lives, opening their eyes to the marvelous world of mathematics.

None of this was planned. I had become a “mathematics expositor” by accident. Only after I realized I had been born with a talent that others appreciated—and which by all appearances is fairly rare—did I start to work on developing and improving my “gift.”

In taking mathematical ideas developed by others and explaining them in a way that the layperson can understand, I was following in the footsteps of others who had also made efforts to organize and communicate mathematical ideas to people outside the discipline. Among that very tiny subgroup of mathematics communicators, the two who I regarded as the greatest and most influential mathematical expositors of all time are Euclid and Leonardo Fibonacci. Each wrote a mammoth book that influenced the way mathematics developed, and with it society as a whole.

Euclid’s classic work Elements presented ancient Greek geometry and number theory in such a well-organized and understandable way that even today some instructors use it as a textbook. It is not known if any of the results or proofs Euclid describes in the book are his, although it is reasonable to assume that some are, maybe even many. What makes Elements such a great and hugely influential work, however, is the way Euclid organized and presented the material. He made such a good job of it that his text has formed the basis of school geometry teaching ever since. Present day high school geometry texts still follow Elements fairly closely, and translations of the original remain in print.

With geometry being an obligatory part of the school mathematics curriculum until a few years ago, most people have been exposed to Euclid’s teaching during their childhood, and many recognize his name and that of his great book. In contrast, Leonardo of Pisa (aka Fibonnaci) and his book Liber abbaci are much less well known. Yet their impact on present-day life is far greater. Liber abbaci was the first comprehensive book on modern practical arithmetic in the western world. While few of us ever use geometry, people all over the world make daily use of the methods of arithmetic that Leonardo described in Liber abbaci.

In contrast to the widespread availability of the original Euclid’s Elements, the only version of Leonardo’s Liber abbaci we can read today is a second edition he completed in 1228, not his original 1202 text. Moreover, there is just one translation from the original Latin, in English, published as recently as 2002.

But for all its rarity, Liber abbaci is an impressive work. Although its great fame rests on its treatment of Hindu-Arabic arithmetic, it is a mathematically solid book that covers not just arithmetic, but the beginnings of algebra and some applied mathematics, all firmly based on the theoretical foundations of Euclid’s mathematics.

After completing the first edition of Liber abbaci, Leonardo wrote several other mathematics books, his writing making him something of a celebrity throughout Italy—on one occasion he was summonsed to an audience with the Emperor Frederick II. Yet very little was written about his life.

In 2001, I decided to embark on a quest to try to collect together what little was known about him and bring his story to a wider audience. My motivation? I saw in Leonardo someone who, like me, devoted a lot of time and effort trying to make the mathematics of the day accessible to the world at large. (Known today as “mathematical outreach,” very few mathematicians engage in that activity.) He was the giant whose footsteps I had been following.

I was not at all sure I could succeed. Over the years, I had built up a good reputation as an expositor of mathematics, but a book on Leonardo would be something new. I would have to become something of an archival scholar, trying to make sense of Thirteenth Century Latin manuscripts. I was definitely stepping outside my comfort zone.

The dearth of hard information about Leonardo in the historical record meant that a traditional biography was impossible—which is probably why no medieval historian had written one. To tell my story, I would have to rely heavily on the mathematical thread that connects today’s world to that of Leonardo—an approach unique to mathematics, made possible by the timeless nature of the discipline. Even so, it would be a stretch.

In the end, I got lucky. Very lucky. And not just once, but several times. As a result of all that good fortune, when my historical account The Man of Numbers: Fibonacci’s Arithmetic Revolution was published in 2011, I was able to compensate for the unavoidable paucity of information about Leonardo’s life with the first-ever account of the seminal discovery showing that my medieval role-model expositor had indeed played the pivotal role in creating the modern world that most historians had hypothesized.

With my Leonardo project such a new and unfamiliar genre, I decided from the start to keep a diary of my progress. Not just my findings, but also my experiences, the project's highs and lows, the false starts and disappointments, the tragedies and unexpected turns, the immense thrill of holding in my hands seminal manuscripts written in the thirteenth and fourteenth centuries, and one or two truly hilarious episodes. I also encountered, and made diary entries capturing my interactions with, a number of remarkable individuals who, each for their own reasons, had become fascinated by Fibonacci—the Yale professor who traced modern finance back to Fibonacci, the Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci's astonishing story, and the remarkable widow of the man who died shortly after completing the world’s first, and only, modern language translation of Liber abbaci, who went to heroic lengths to rescue his manuscript and see it safely into print.

After I had finished the Man of Numbers, I decided that one day I would take my diary and turn it into a book, telling the story of that small group of people (myself included) who had turned an interest in Leonardo into a passion, and worked long and hard to ensure that Leonardo Fibonacci of Pisa will forever be regarded as among the very greatest people to have ever lived. Just as The Man of Numbers was an account of the writing of Liber abbaci, so too Finding Fibonacci is an account of the writing of The Man of Numbers. [So it is a book about a book about a book. As Andrew Wiles once famously said, “I’ll stop there.”]

This post is adapted from the introduction of Keith Devlin’s new book Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, published this month by Princeton University Press.

Original author: Mathematical Association of America

]]>CLOSE More Options

If you don't honor and treasure what's closest to you, it's really hard to ask, believe and receive something that's even broader and larger.

Original author: Jeanne Meister

]]>For many of us, the linked video of Hans Rosling's 2006 TED Talk was our first encounter with the Swedish academic and his powerful data visualization tools Trendalyzer and Gapminder. I don't know when he first gave that talk, but over the ensuing years he gave essentially the same presentation (audience-interactive performance would be a better description) many times, albeit always with up-to-date data. He was right to do so. At the heart of his presentations was one message. Data matters. Provided you collect them with care, analyze them properly, and present them in honest ways that the human mind can readily grasp, numbers are one of Humankind's most powerful tools.

For all his engaging presentation skills, the numbers were at the heart of Rosling's talks. It was not his oratory that convinced us, in an instant, that our preconceptions of our world were wrong -- often violently so. It was the data -- the numbers displayed on the screen in front of us. For that reason, I decided that I would let Hans and his graphs occupy all of this month’s post. There was neither need nor place for my words.

As it happens, Rosling's death comes at a moment in time when people in highly powerful positions are waging an assault on scientific facts, on numerical data, and indeed on truth in general. I did not want to detract from using my MAA blog to pay respect to the passing of a great numerist (I had to make up a word to adequately describe him) by inserting this observation into what I wanted to be one final platform for him to spread his message. Like Hans, I wanted his numbers to do the talking.

But here, in the comments section, I feel free to speak as an individual mathematician. An attack on truth is an attack on Society in general. Those of us whose lives revolve around discovering and communicating numerical and mathematical truth have a duty to speak up forcefully, in opposition. If our Society loses the respect for, and dependency on, truth, the loss of mathematics will be the least of our worries.

Original author: Mathematical Association of America

]]>
*By Ryan Rose, head of customer experience & product design, Cisco Collaborative Knowledge, Cisco *

We live in a world that seems almost predisposed to tell us where we are weak. When you ask most people to point out their weaknesses, the answer comes rather easily. Ask them about their strengths, and that question tends to be harder for them. Most individuals, in fact, aren't likely to recognize their own strengths unless someone else points them out.

This sort of “evolve or die” mentality is pervasive, and it can often inhibit learning, especially at the organizational level.

What if we changed our approach to view our talents as our greatest opportunities for success? That’s the idea put forth in the 2003 research paper“Investing in Strengths,” written by Donald O. Clifton and James K. Harder of the Gallup Foundation. It wasn’t a new idea at the time, and it certainly isn’t radical now. However, when you really look at the structure of most learning programs and organizations, it becomes pretty obvious that time and investment are addressing weaknesses rather than strengths.

During our formative years, we focus on development and overcoming weaknesses. It’s important for students who struggle with math or reading, for example, to have extra assistance and encouragement in these areas. Unfortunately, this mentality seems to continue in most corporate learning environments, where the focus could – and should – be on investing in developing employees’ strengths.

A few years ago, Pew Research conducted a study examining this situation. Researchers decided to look at two groups of people: 100 slow-to-medium readers and 100 high-performance readers. Both groups were sent to a speed reading course.

Not surprisingly, the lower group saw an average of 150 percent improvement. Less expected was that the high performers saw an average of 700 percent improvement. In other words, their strengths were multiplied by seven!

According to Cliftonand Harder, “When people become aware of their talents, through measurement and feedback, they have a strong position from which to view their potential. They can then begin to integrate their awareness of their talents with knowledge and skills to develop strengths.”

This finding provides organizations and individuals with huge opportunity. People who use their strengths every day are six times more likely to be engaged on the job, according to Gallup’s "State of the American Workplace"report.

Effective managers will ensure that teams have a mix of strengths and weaknesses. In a collaborative environment, everyone is able to build together and ultimately achieve more. A team with a mix of talents harnesses the power of collaboration to learn to build on strengths, not just shore up resources. The Gallup report found that when managers focused on their employees’ strengths, active disengagement – when employees are not only unhappy but might be undermining the organization as well – fell dramatically, to about 1 percent. Imagine if this shift happened in all workplaces – productivity would soar.

Organizations must work to help individuals identify their strengths and provide access to the people, communities, courses, and other tools neededto build on these strengths. Rather than looking at the situation as “evolve or die,” the rallying cry should be “collaborate and live.”

**About Ryan Rose**

Ryan Rose is Head of Customer Experience & Product Design for the new Cisco Collaborative Knowledge social learning platform developed by Cisco.

Rose leads the Digital Strategy Team within Learning@Cisco that supports the development, build-out, and delivery of Cisco Collaborative Knowledge. Rose is responsible for the overall User Experience design, Customer Experience and Engagement for Cisco Collaborative Knowledge, and supports the development of the solution’s asynchronous collaboration tools.

In addition to his role developing Cisco Collaborative Knowledge, Rose co-manages the Cisco.com Web Team that updates and maintains the web properties that make up the widely-acclaimed Training and Events section; supported the Learning@Cisco Content Development Team in the Quality Assurance Services group; and acts as a user experience, documentation, and best practices consultant across Cisco supporting a number of software development initiatives.

]]>1 + 2 + 3 + 4 + 5 + . . . = –1/12

This identity had been lying around in the mathematical literature since the famous Indian mathematician Srinivasa Ramanujan included it in one of his books in the early Twentieth Century, a curiosity to be tossed out to undergraduate mathematics students in their first course on complex analysis (which was my first exposure to it), and apparently a result that physicists made actual (and reliable) use of.

The sudden explosion of interest was the result of a video posted online by Australian video journalist Brady Haran on his excellent Numberphile YouTube channel. In it, British mathematician and mathematical outreach activist James Grime moderates as his physicist countrymen Tony Padilla and Ed Copeland of the University of Nottingham explain their “physicists’ proof” of the identity.

In the video, Padilla and Copeland manipulate infinite series with the gay abandon physicists are wont to do (their intuitions about physics tends to keep them out of trouble), eventually coming up with the sum of the natural numbers on the left of the equality sign and –1/12 on the right.

Euler was good at doing that kind of thing too, so mathematicians are hesitant to trash it, rather noting that it “lacks rigor” and warning that it would be dangerous in the hands of a lesser mortal than Euler.

In any event, when it went live on January 9, 2014, the video and the result (which to most people was new) exploded into the mathematically-curious public consciousness, rapidly garnering hundreds of thousands of hits. (It is currently approaching 5 million in total.) By February 3, interest was high enough for The New York Times to run a substantial story about the “result”, taking advantage of the presence in town of Berkeley mathematician Ed Frenkel, who was there to promote his new book Love and Math, to fill in the details.

Before long, mathematicians whose careers depended on the powerful mathematical technique known as analytic continuation were weighing in, castigating the two Nottingham academics for misleading the public with their symbolic sleight-of- hand, and trying to set the record straight. One of the best of those corrective attempts was another Numberphile video, published on March 18, 2014, in which Frenkel give a superb summary of what is really going on.

A year after the initial flair-up, on January 11, 2015, Haran published a blogpost summarizing the entire episode, with hyperlinks to the main posts. It was quite a story.

[[ASIDE: The next few paragraphs may become a bit too much for casual readers, but my discussion culminates with a link to a really cool video, so keep going. Of course, you could just jump straight to the video, now you know it’s coming, but without some preparation, you will soon get lost in that as well! The video is my reason for writing this essay.]]

For readers unfamiliar with the mathematical background to what does, on the face of it, seem like a completely nonsensical result, which is the MAA audience I am aiming this essay at (principally, undergraduate readers and those not steeped in university-level math), it should be said that, as expressed, Ramanujan’s identity is nonsense. But not because of the -1/12 on the right of the equals sign. Rather, the issue lies in those three dots on the left. Not even a mathematician can add up infinitely many numbers.

What you can do is, under certain circumstances, assign a meaning to an expression such as

X1 + X2 + X3 + X4 + …

where the XN are numbers and the dots indicate that the pattern continues for ever. Such expressions are called infinite series.

For instance, undergraduate mathematics students (and many high school students) learn that, provided X is a real number whose absolute value is less than 1, the infinite series

1 + X + X2 + X3 + X4 + …

can be assigned the value 1/(1 – X). Yes, I meant to write “can be assigned”. Since the rules of real arithmetic do not extend to the vague notion of an “infinite sum”, this has to be defined. Since we are into the realm of definition here, in a sense you can define it to be whatever you want. But if you want the result to be meaningful and useful (useful in, say, engineering or physics, to say nothing of the rest of mathematics), you had better define it in a way that is consistent with that “rest of mathematics.” In this case, you have only one option for your definition. A simple mathematical argument (but not the one you can find all over the web that involves multiplying the terms in the series by X, shifting along, and subtracting—the rigorous argument is a bit more complicated than that, and a whole lot deeper conceptually) shows that the value has to be 1/(1 – X).

So now we have the identity

(*) 1 + X +X2 + X3 + X4 + … = 1/(1 – X)

which is valid (by definition) whenever X has absolute value less than 1. (That absolute value requirement comes in because of that “bit more complicated” aspect of the rigorous argument to derive the identity that I just mentioned.)

“What happens if you put in a value of X that does not have absolute value less than 1?” you might ask. Clearly, you cannot put X = 1, since then the right-hand side becomes 1/0, which is totally and absolutely forbidden (except when it isn’t, which happens a lot in physics). But apart from that one case, it is a fair question. For instance, if you put X = 2, the identity (*) becomes

1 + 2 + 4 + 8 + 16 + … = 1/(1 – 2) = 1/(–1) = –1

So you could, if you wanted, make the identity (*) the definition for what the infinite sum

1 + X + X2 + X3 + X4 + …

means for any X other than X = 1. Your definition would be consistent with the value you get whenever you use the rigorous argument to compute the value of the infinite series for any X with absolute value less than 1, but would have the “benefit” of being defined for all values of X apart from one, let us call it a “pole”, at X = 1.

This is the idea of analytic continuation, the concept that lies behind Ramanujan’s identity. But to get that concept, you need to go from the real numbers to the complex numbers.

In particular, there is a fundamental theorem about differentiable functions (the accurate term in this context is analytic functions) of a single complex variable that says that if any such function has value zero everywhere on a nonempty disk in the complex plane, no matter how small the diameter of that disk, then the function is zero everywhere. In other words, there can be no smooth “hills” sitting in the middle of flat plains, or even one small flat clearing in the middle of a “hilly” landscape—the quotes are because we are beyond simple visualization here.

An immediate consequence of this theorem is that if you pull the same continuation stunt as I just did for the series of integer powers, where I extended the valid formula (*) for the sum when X is in the open unit interval to the entire real line apart from one pole at 1, but this time do it for analytic functions of a complex variable, then if you get an answer at all (i.e., a formula), it will be unique. (Well, no, the formula you get need not be unique, rather the function it describes will be.)

In other words, if you can find a formula that describes how to compute the values of a certain expression for a disk of complex numbers (the equivalent of an interval of the real line), and if you can find another formula that works for all complex numbers and agrees with your original formula on that disk, then your new formula tells you the right way to calculate your function for any complex number. All this subject to the requirement that the functions have to be analytic. Hence the term “analytic continuation.'

For a bit more detail on this, check out the Wikipedia explanation or the one on Wolfram Mathworld. If you find those explanations are beyond you right now, just remember that this is not magic and it is not a mystery. It is mathematics. The thing you need to bear in mind is that the complex numbers are very, very regular. Their two-dimensional structure ties everything down as far as analytic functions are concerned. This is why results about the integers such as Fermat’s Last Theorem are frequently solved using methods of Analytic Number Theory, which views the integers as just special kinds of complex numbers, and makes use of the techniques of complex analysis.

Now we are coming to that video. When I was a student, way, way back in the 1960s, my knowledge of analytic continuation followed the general path I just outlined. I was able to follow all the technical steps, and I convinced myself the results were true. But I never was able to visualize, in any remotely useful sense, what was going on.

In particular, when our class came to study the (famous) Riemann zeta function, which begins with the following definition for real numbers S bigger than 1:

(**) Zeta(S) = 1 + 1/2S + 1/3S + 1/4S + 1/5S + …

I had no reliable mental image to help me understand what was going on. For integers S greater than 1, I knew what the series meant, I knew that it summed (converged) to a finite answer, and I could follow the computation of some answers, such as Euler’s

Zeta(2) = π2/6

(You get another expression involving π for S = 4, namely π4/90.)

It turns out that the above definition (**) will give you an analytic function if you plug in any complex number for S for which the real part is bigger than 1. That means you have an analytic function that is rigorously defined everywhere on the complex plane to the right of the line x = 1.

By some deft manipulation of formulas, it’s possible to come up with an analytic continuation of the function defined above to one defined for all complex numbers except for a pole at S = 1. By that basic fact I mentioned above, that continuation is unique. Any value it gives you can be taken as the right answer.

In particular, if you plug in S = –1, you get

Zeta(–1) = –1/12

That equation is totally rigorous, meaningful, and accurate.

Now comes the tempting, but wrong, part that is not rigorous. If you plug in S = –1 in the original infinite series, you get

1 + 1/2-1 + 1/3-1 + 1/4-1 + 1/5-1 + …

which is just

1 + 2 + 3 + 4 + 5 + …

and it seems you have shown that

1 + 2 + 3 + 4 + 5 + . . . = –1/12

The point is, though, you can’t plug S = –1 into that infinite series formula (**). That formula is not valid (i.e., it has no meaning) unless S > 1.

So the only way to interpret Ramanujan’s identity is to say that there is a unique analytic function, Zeta(S), defined on the complex plane (apart from at the real number 1), which for all real numbers S greater than 1 has the same values as the infinite series (**), which for S = –1 gives the value Zeta(–1) = –1/12.

Or, to put it another way, more fanciful but less accurate, if the sum of all the natural numbers were to suddenly find it had a finite answer, that answer could only be –1/12.

As I said, when I learned all this stuff, I had no good mental images. But now, thanks to modern technology, and the creative talent of a young (recent) Stanford mathematics graduate called Grant Sanderson, I can finally see what for most of my career has been opaque. On December 9, he uploaded this video onto YouTube.

It is one of the most remarkable mathematics videos I have ever seen. Had it been available in the 1960s, my undergraduate experience in my complex analysis class would have been so much richer for it. Not easier, of that I am certain. But things that seemed so mysterious to me would have been far clearer. Not least, I would not have been so frustrated at being unable to understand how Riemann, based on hardly any numerical data, was able to formulate his famous hypothesis, finding a proof of which is agreed by most professional mathematicians to be the most important unsolved problem in the field.

When you see (in the video) what looks awfully like a gravitational field, pulling the zeros of the Zeta function towards the line x = 1/2, and you know that it is the only such gravitational field there is, and recognize its symmetry, you have to conclude that the universe could not tolerate anything other than all the zeros being on that line.

Having said that, it would, however, be really interesting if that turned out not to be the case. Nothing is certain in mathematics until we have a rigorous proof.

Meanwhile, do check out some of Grant’s other videos. There are some real gems!

Original author: Mathematical Association of America

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Finding out what you love to do is a great feat in and of itself.

Original author: Jeanne Meister

]]>It’s been two years since I last wrote a post to this blog. Originally, that hiatus came about because other issues took most of my time, and besides, my Coursera MOOC Introduction to Mathematical Thinking had reached a steady state, and the only input required of me was to show up on the class discussion forum once a day for the ten weeks the course ran, and interact with the class—or at least, the minority of registered students who were themselves active on the forums.

My absence became considerably longer because, faced with a need to generate revenue in order to survive as a company, Coursera rebuilt their course platform, tailoring it to a more production-line, largely instruction-based approach to education than the “Let’s take a typical, highly interactive Stanford course structure and make it available on the Web” that inspired many of the original MOOC pioneers, of which I was one. (Strictly, the Stanford group were x-MOOC pioneers, in deference to the Canadians who, a few years earlier, had created the very first MOOCs, subsequently renamed c-MOOCs, with the “c” standing for “connectivist”, not “Canada”. See here for the tortuous history.)

Coursera‘s shift—I would not go as far as to call it a pivot, though I think that Silicon Valley beloved term does apply to pioneering xMOOC-provider Udacity‘s move to the corporate training market—was totally understandable, given their need to survive as a for-profit company. Unfortunately, many of the features they eliminated from their original platform left my course completely un-runnable.

It has taken almost a year, working with a Coursera engineer, for me to rebuild my original MOOC into something that will run on the new platform. That new edition of MathThink, as we insiders all called the course, launches on January 9. On the whole, I am pretty happy with it. I think many people will find it useful. But it is no longer the course I originally created (or rather, ported from the real classroom to a virtual one), and after the first run in early 2017, I will no longer play an active role. (That great, 1972 Douglas Trumbull sic-fi movie Silent Running comes to mind, with me corresponding to the Bruce Dern character. I know, I view education in a romantic way. I think most educators do. Why else would we do it?)

At some point, I may find the energy and the enthusiasm to re-create something like the original MathThink on the Open EdX platform, which, as a nonprofit, Open Source academic project, is not subject to the commercial pressures on a for-profit company. (Though in a subsequent post I will indicate why I think it will differ in significant ways from my original MathThink.)

While taking a successful course and removing features I found valuable was a frustrating task, the reason I approached it in a sanguine fashion was that I had long been aware of the numerical realities of my course. Of the 40,000 or so who would typically register, around 5,000 would complete the Basic Course, lasting eight weeks, and of them around 1,000 would continue to complete the two-week capstone experience I called Test Flight, where they were given an opportunity to experience the role of a professional mathematician. Of those, at most maybe 100 (that is one hundred) would have actually had the fully interactive experience I had been trying to take from my physical classroom onto the Web, involving regular interactions with me and my small army of volunteer Teaching Assistants.

For reasons I gave in an earlier post on this blog, on January 2, 2014, I was more than happy to put in the effort to reach that one hundred or so students around the world. For me, the “massive” part—being able to “reach” tens of thousands of students—was simply a way to find that one hundred or so I would react with on a daily basis for ten weeks. That was more than ten times the number of students I would really interact with in a physical classroom. I was sorry to have to give up that twice-yearly fix of global mathematical outreach, where I was given a real opportunity to change a small number of lives dramatically for the better (and, like all MOOC instructors, I was able to do just that). But there was no way I could feel bitter about it. No company can survive when its core market is one hundred, and moreover, many of that one hundred were unable to pay anything for the experience.

So, the old MathThink is dead. Long live the new MathThink.

Meanwhile, get ready to meet MOLEs: Massive Online Learning Experiences. That concept is the (potential) jewel I was able to identify when I was raking through the ashes of my original MOOC. As I continually tell my students (including my MOOC students), “failure” is something to be looked upon, not as an ending, but as a learning opportunity that can lead to starting something new. (Google “Edison + lightbulb + failure”.)

Original author: Keith Devlin

]]>It’s been two years since I last wrote a post to this blog. Originally, that hiatus came about because other issues took most of my time, and besides, my Coursera MOOC Introduction to Mathematical Thinking had reached a steady state, and the only input required of me was to show up on the class discussion forum once a day for the ten weeks the course ran, and interact with the class—or at least, the minority of registered students who were themselves active on the forums.

My absence became considerably longer because, faced with a need to generate revenue in order to survive as a company, Coursera rebuilt their course platform, tailoring it to a more production-line, largely instruction-based approach to education than the “Let’s take a typical, highly interactive Stanford course structure and make it available on the Web” that inspired many of the original MOOC pioneers, of which I was one. (Strictly, the Stanford group were x-MOOC pioneers, in deference to the Canadians who, a few years earlier, had created the very first MOOCs, subsequently renamed c-MOOCs, with the “c” standing for “connectivist”, not “Canada”. See here for the tortuous history.)

Coursera‘s shift—I would not go as far as to call it a pivot, though I think that Silicon Valley beloved term does apply to pioneering xMOOC-provider Udacity‘s move to the corporate training market—was totally understandable, given their need to survive as a for-profit company. Unfortunately, many of the features they eliminated from their original platform left my course completely un-runnable.

It has taken almost a year, working with a Coursera engineer, for me to rebuild my original MOOC into something that will run on the new platform. That new edition of MathThink, as we insiders all called the course, launches on January 9. On the whole, I am pretty happy with it. I think many people will find it useful. But it is no longer the course I originally created (or rather, ported from the real classroom to a virtual one), and after the first run in early 2017, I will no longer play an active role. (That great, 1972 Douglas Trumbull sic-fi movie Silent Running comes to mind, with me corresponding to the Bruce Dern character. I know, I view education in a romantic way. I think most educators do. Why else would we do it?)

At some point, I may find the energy and the enthusiasm to re-create something like the original MathThink on the Open EdX platform, which, as a nonprofit, Open Source academic project, is not subject to the commercial pressures on a for-profit company. (Though in a subsequent post I will indicate why I think it will differ in significant ways from my original MathThink.)

While taking a successful course and removing features I found valuable was a frustrating task, the reason I approached it in a sanguine fashion was that I had long been aware of the numerical realities of my course. Of the 40,000 or so who would typically register, around 5,000 would complete the Basic Course, lasting eight weeks, and of them around 1,000 would continue to complete the two-week capstone experience I called Test Flight, where they were given an opportunity to experience the role of a professional mathematician. Of those, at most maybe 100 (that is one hundred) would have actually had the fully interactive experience I had been trying to take from my physical classroom onto the Web, involving regular interactions with me and my small army of volunteer Teaching Assistants.

For reasons I gave in an earlier post on this blog, on January 2, 2014, I was more than happy to put in the effort to reach that one hundred or so students around the world. For me, the “massive” part—being able to “reach” tens of thousands of students—was simply a way to find that one hundred or so I would react with on a daily basis for ten weeks. That was more than ten times the number of students I would really interact with in a physical classroom. I was sorry to have to give up that twice-yearly fix of global mathematical outreach, where I was given a real opportunity to change a small number of lives dramatically for the better (and, like all MOOC instructors, I was able to do just that). But there was no way I could feel bitter about it. No company can survive when its core market is one hundred, and moreover, many of that one hundred were unable to pay anything for the experience.

So, the old MathThink is dead. Long live the new MathThink.

Meanwhile, get ready to meet MOLEs: Massive Online Learning Experiences. That concept is the (potential) jewel I was able to identify when I was raking through the ashes of my original MOOC. As I continually tell my students (including my MOOC students), “failure” is something to be looked upon, not as an ending, but as a learning opportunity that can lead to starting something new. (Google “Edison + lightbulb + failure”.)

Original author: Keith Devlin

]]>The author climbing the locally-notorious Country View Road just south of San Jose, CA |

As regular readers may know, one of my consuming passions in life besides mathematics is cycling. Living in California, where serious winters were wisely banned many years ago, on any weekend throughout the year you are likely to find me out on a road- or a mountain bike.

Being also a lover of well-designed technology, I long ago switched to using tubeless tires on my road bike. Actually, it’s bikes, in the plural—my road bikes number four, all with different riding conditions in mind, but all having in common the same kind of ultra- narrow saddle that non-cyclists think must be excruciatingly painful, but is in fact engineered to be the only thing comfortable enough to sit on for many hours at a stretch. [Keep going; I am working my way to making a mathematical point. In fact, I am heading towards THE most significant mathematical point of all: What is the secret to doing math?]

Road tubeless tires have several advantages over the more common type of tire, which requires an airtight innertube. One advantage is that you need inflate them only to 80 pounds per square inch, as opposed to the 110 psi or more for a tubed tire, which provides even more comfort over those many hours in the saddle.

You need tire pressures 3 or more times that of a car tire because of the extremely low volume in a road-bike tire, which sits on a 700 cm diameter wheel with a rim whose width is between 21 mm and 25 mm. It is that high pressure that made the manufacture of tubeless wheels and tires for bicycles such a significant challenge. How can you ensure an almost totally airtight fit when the tire is inflated, and it still be possible for an average person to remove and mount a deflated tire with their bare hands. (Tire levers can easily damage tubeless wheels and tires.) We are almost to the secret to doing math. Hang in there.

Clever design: Tubeless rims and tires on a road bike wheel. |

The airtight fit is possible precisely because of that relatively high pressure inside the tire—80 psi is over five times the air pressure outside the tire. (An automobile tire is inflated to roughly twice atmospheric pressure, much lower.) The cross-sectional photo on the left shows how a tubeless tire has a squared-off ridge that fits into a matching notch in the rim. The more air pressure there is in the tire, the tighter that ridge binds to the rim, increasing the air seal.

The problem is, as I mentioned, getting the tire on and off the rim. The tire ridge that fits into the rim-notch has a steel wire running through it, and its squared-off shape is designed to make it difficult for the tire to separate from the rim—that, after all, is the point. To solve the mounting/removal problem, the wheel has a channel in the middle, as shown more clearly in the photo on the right.

To mount the tire, you push the two tire-rims into that channel, one after the other. By the formula for the circumference of a circle, when a tire rim is in that center channel, you have just over 3 times the depth of the channel of superfluous tire length to play with, roughly 12mm of tire looseness. The idea is to use that “looseness” to work your way around the wheel, pushing (actually rolling) first one tire edge over the wheel rim and into the channel, then the other. Once the tire is seated on the rim, inflating it with a hand pump forces the tire rims out of the channel into the notches. To remove the tire after it is deflated, you push the two tire rims into the channel and reverse the process.

That, at least, is the theory. Putting theory into practice turns out to be quite a challenge. When I first started to use road tubeless tires, several years ago, I read several online manuals and watched a number of YouTube videos demonstrating how to do it, and could never do it. I usually ended up taking the wheel and tire to my local bike shop, where the mechanic would do it for me with seeming ease before my eyes. “Fifteen dollars, please.”

But what would happen if I had a flat on one of the remote rides I regularly do in the mountains that surround Silicon Valley, where I live? One major advantage of tubeless tires is that, even if they puncture, usually the air leaks out only very slowly, and can generally be stopped by inflating the tire from a small pressure-can of air and liquid latex you carry in your back pocket, which seals the hole. Which is how I was always able to get to a bike shop where someone else could solve the problem for me. But a major puncture in the remote, with no cell phone access, could leave me dangerously stranded. Clearly, I had to learn how to do it myself.

From now on, when I say “change a tubeless tire”, you can interpret it as “do mathematics”. The secret is coming up. Moreover, it is coming with a moral that those of us in mathematics education ignore at our students’ peril.

What I find cool is that, for me I somehow stumbled on the secret to doing math fairly early in life, before math had become such a problem that I felt I could never do it. But taking up cycling later in life, when I had a fully developed set of metacognitive skills, I approached the problem of changing a tubeless tire in much the same way as many people—including, I suspect, the mechanics in my local bicycle shop—see math. Namely, people like me (and that smart kid sitting in the front row in the school math class) make doing math look effortless, but many people feel they could never master it in a million years.

Nothing, surely, can look less requiring of skill or expertise than putting a tire on a bicycle wheel. (This is why I think this is such a great example.) Surely, you just need to read an instruction manual, or perhaps have someone demonstrate to you. But no matter how many times I read the instructions, no matter how many times I viewed—and re-viewed—those how-to YouTube videos, and no matter how many times I stood alongside the bike shop mechanic and watched as he quickly and effortlessly put the tire onto the wheel, I could never do it.

Just think about that for a moment. For some tasks, instruction (on its own) just does not work. Not even for the seemingly simple task of changing a bicycle tire. And yet we think that forcing kids to sit in the math class while we force-feed instruction will result in their being able to do math! Dream on.

What does work, in fact what is absolutely necessary, both for changing tubeless tires and doing math, is that the learner has to learn to see things the way the expert does. And, since instruction does not work, that key step has to be made by the learner. All that a good teacher can do, then, is find a way to help the learner make that key leap. [That short initial word “all” belies the human expertise required to do this.]

Clearly, when I was, yet again, standing in the bicycle repair shop, watching the mechanic change my tire, what he was doing—more precisely, what he was experiencing—was very different from what I was doing and experiencing when I tried and failed. What was I not getting?

My big breakthrough finally came the one time when the mechanic, holding the wheel horizontally pressed to his stomach, while manipulating the tire with both hands, told me what he was really doing. “You have to think of the tire as alive,” he said. “It wants to be sitting firmly on the rim” [that, after all, is what it was—expensively—designed for], “but it is not very disciplined. It’s like a small child. It moves around and resists your attempts to force it. You have to understand it, and be aware, through your hands, of what it is doing. Work with it—be constantly aware of what it is trying to do—so you both get what you want: the tire gets onto the wheel, where it belongs, and you can inflate it and get back on your bike (where you want to be).”

Fanciful? Maybe. But it worked. And it continues to work. As a result, not only can I now change my tubeless tires, it has for me become “mindless and automatic,” as effortless (to me) as Picasso drawing a simple doodle on a restaurant napkin to pay the bill for his meal was to him. (I thought that if you got this far, you deserved a second example with greater cultural overtones.)

It took many years for Picasso to learn to draw the way he did (and for the marketplace to assign high value to his work), but that does not mean his work was not skillful; rather, he simply routinized part of it. When I watch a film of him at work, I see superficially how he created, and it looks routine and effortless, but I do not see his canvas as he did, and I could not draw as he did.

Likewise, my skill in fitting a tubeless tire, now effortless and automatic, is a result of my now seeing and understanding what earlier had been opaque.

I admit that it is far easier to learn to mount a tubeless tire on a road bike wheel than to draw like Picasso. But I am less sure the difference is so great between changing a tubeless tire and being able to solve any one particular kind of math problem. Still, no matter how great the difference in the degree of skill required, it is possible to learn from the analogy.

Given what I have said here, will reading this essay mean you can go out and immediately be able to change a tubeless tire? Have I just made a case for instruction working after all? It’s possible—for changing bicycle tires, but surely not for painting like Picasso. Instruction can and does work, and it is an important part of learning. But my guess is you would find my words are not enough. I think that the reason that one piece of bike-shop instruction was so instantly transformative for me was that I had spent an aggregate of many hours struggling to change my tire and failing. I had reached a stage where the effective key was to get inside the mind of an expert. But the ground had to be prepared for that simple revelation to work.

In education, as in so many parts of life, there are no silver bullets. But given enough of the right preparation—enough experience acquired through repeated trying and failing—an ordinary lead bullet will do the job.

----

This month’s column is loosely adapted from a passage of my forthcoming book Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, due out in March.

Original author: Mathematical Association of America

]]>Keith Devlin mails his completed election ballot. What does math have to say about his act? |

A classic example is how we count votes in an election, the topic of an earlier Devlin’sAngle post, in November, 2000. In that essay, I looked at how different ways to tally votes could affect the imminent Bush v. Gore election, at the time blissfully unaware of how chaotic would be the process of counting votes and declaring a winner on that particular occasion. The message there was, particularly in the kinds of tight race we typically see today, the different ways that votes can be tallied can lead to very different results.

Everything I said back then remains just as valid and pertinent today (mathematics is like that), so this time I’m going to look at another perplexing aspect of election math: why do we make the effort to vote? After all, while elections are sometimes decided by a small number of votes, it is unlikely in the extreme that an election on the scale of a presidential election will hang on the decision of a single voter. Even if it did, that would be well within the range of procedural error, so it makes no difference if any one individual votes or not.

To be sure, if a large number of people decide to opt out, that can affect the outcome. But there is no logical argument that takes you from that observation to it being important for a single individual to vote. This state of affairs is known as the Paradox of Voting, or sometimes Downs Paradox. It is so named after Anthony Downs, a political economist whose 1957 book An Economic Theory of Democracy examined the conditions under which (mathematical) economic theory could be applied to political decision-making.

On the face of it, Downs’ analysis does lead to a paradox. Economic theory tells us that rational beings make decisions based on expected benefit (a notion that can be made numerically precise). That approach works well for analyzing, say, why people buy insurance year after year, even though they may never submit a claim. The theory tells you that the expected benefit is greater than the cost; so it is rational to buy insurance. But when you adopt the same approach to an election, you find that, because the chance of exercising the pivotal vote in an election is minute compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the cost. So you should opt out. [The same observation had in fact been made much earlier, in 1793, by Nicolas de Condorcet, but without the theoretical backing that Downs brought to the issue.]

Yet, many otherwise sane, rational citizens do not opt out. Indeed, society as a whole tends to look down on those who do not vote, saying they are not "doing their part." (In fact, many countries make participation in a national election obligatory, but that is a separate, albeit related, issue.)

So why do we (or at least many of us) bother to vote? I can make the question even more stark, and personal. Suppose you have intended to "do your part" and vote. You wake up on election morning with a sore throat, and notice that it is raining heavily. Being numerically able (as all Devlin’s Angle readers must be), you say to yourself, "It cannot possibly affect the result if I just stay at home and nurse my throat. I was intending to vote, after all. Changing my mind about voting at the last minute cannot possibly influence anyone else. Especially if I don’t tell anyone." The math and the logic, surely, are rock solid. Yet, professional mathematician as I am, I would struggle out and cast my vote. And I am sure many Devlin’s Angle readers would too – most of them, I would suspect.

So what is going on? We can do the math. We are good logical thinkers. Why don’t we act according to that reasoning? Are we conceding that mathematics actually isn’t that useful? [SPOILER: Math is useful; but only when applied with a specific purpose in mind, and chosen/designed in a way that makes it appropriate for that purpose.]

Which brings me to my main point. To make it, let me switch for a moment from elections to the Golden Ratio. In April 2015, the magazine Fast Company Design published an article titled "The Golden Ratio: Design’s Biggest Myth," in which I was quoted at length. (The author also drew heavily on a Devlin’s Angle post of mine from May 2007.)

With a readership much wider than Devlin’s Angle, over the years the Fast Company Design piece has generated a fair amount of correspondence from people beyond mathematics academia, often designers who have not been able to overcome drinking Golden Ratio Kool-Aid during their design education. One recent email came, not from a designer but a high school math teacher, who objected to a statement the article quoted me (accurately) as saying, “Strictly speaking, it's impossible for anything in the real-world to fall into the golden ratio, because it’s an irrational number.” The teacher had, it was at once clear to me, drunk not just Golden Ratio Kool-Aid, but Math Kool-Aid as well.

In the interest of full disclosure, let me admit that, in the early part of my career as a mathematics expositor, I was as guilty as anyone of distributing both Golden Ratio Kool-Aid and Math Kool-Aid, to whoever would drink it. But, as a committed scientist, when presented with evidence to the contrary, I re-examined my thinking, admitted I had been wrong, and started to push better, more honest products, which I will call Golden Ratio Milk and Mathematical Milk. I described Golden Ratio Milk in my 2007 MAA post and peddled it more in that Fast Company Design interview. Here I want to talk about Mathematical Milk.

The reason why the Golden Ratio’s irrationality prevents its use in, say architecture, is that the issue at hand involves measurement. Measurement requires fixing a unit of measure – a scale. It doesn’t matter whether it is meters or feet or whatever, but once you have fixed it, that is what you use. When you measure things, you do so to an agreed degree of accuracy. Perhaps one or two decimal places. Almost never to more than maybe twenty decimal places, and that only in a few instances in subatomic physics. So it terms of actual, physical measurement, or manufacturing, or building, you never encounter objects to which a numerical measurement has more than a few decimal places. You simply do not need a number system that has fractions with denominator much greater than, say, 1,000,000, and generally much less than that.

Even if you go beyond physical measurement, to the theoretical realm where you imagine having an unlimited number decimal places available, you will still be in the domain of the rational numbers. Which means the Golden Ratio does not arise. Irrational numbers arise to meet mathematical needs, not the requirements of measurement. They live not in the physical world but in the human imagination. (Hence my Fast Company Design quote.) It is important to keep that distinction clear in our minds.

The point is, when we abstract from our experiences of the world around us, to create mathematical models, two important things happen. A huge amount of information is lost; and there is a significant gain in precision. The two are not independent. If we want to increase the precision, we lose more information, which means that our model has less in common with the real world it is intended to represent. Moreover, when we construct a mathematical model, we do so with a particular question, or set of questions in mind.

In astronomy and physics, and related domains such as engineering, all of this turns out to be not too problematic. For example, the simplistic model of the Solar System as a collection of point-masses orbiting around another, much heavier, point-mass, is extremely useful. We can formulate and solve equations in that model, and they turn out to be very useful. At least they turn out to be useful in terms of the goal questions, initially in this case predicting where the planets will be at different times of the year. The model is not very helpful in telling us what the color of each planet’s surface is, or even if it has a surface, both of which are certainly precise, scientific questions.

When we adopt a similar approach to model money supply or other economic phenomena, we can obtain results that are, mathematically, just as precise and accurate, but their connection to the real world is far more tenuous and unreliable – as has been demonstrated several times in recent years when those mathematical results have resulted in financial crises, and occasionally disasters.

So what of the paradox of voting? The paradox arises when you start by assuming that people vote to choose, say, a president. Yes, we all say that is what we do. But that’s just because we have drunk Election Kool-Aid. We don’t actually behave in accordance with that statement. If we did, then as rational beings we would indeed stay at home on election day.

Time to throw out the Kool-Aid and buy a gallon jug of far more beneficial Election Milk: (Presidential) elections are about a society choosing a president. Where that purpose impacts the individual voter is not who we vote for, but in providing social pressure to be an active member of that society.

That this is what is actually going on is illustrated by the fact that U.S. society created, and millions of people wear, "I have voted" badges on election day. The focus, and the personal reward, is not "Who I voted for" but "I participated in the process." [For an interesting perspective on this, see the recent article in the Smithsonian Magazine, "WhyWomen Bring Their “I Voted” Stickers to Susan B. Anthony’s Grave."]

To be sure, you can develop mathematical models of group activities, like elections, and they will tend to lead to fewer problems (and "paradoxes") than a single-individual model will, but they too will have limitations. All mathematical models do. Mathematics is not reality; it is just a model of reality (or rather, it is a whole, and constantly growing, collection of models).

When we develop and/or apply a mathematical model, we need to be clear what questions it is designed to help us answer. If we try to apply it to a different question, we may get lucky and get something useful, but we may also end up with nonsense, perhaps in the form of a "paradox."

With both measurement and the election, as is so often the case, one benefit we get from trying to apply mathematics to our world and to our lives is we gain insight into what is really going on.

Attempting to use the real numbers to model the acts of measuring physical objects leads us to recognize the dependency on the physical activity of measurement.

Likewise, grappling with Downs Paradox leads us to acknowledge what elections are really about – and to recognize that choosing a leader is a societal activity. In a democracy, who each one of us votes for is inconsequential; that we vote is crucial. That’s why I did not just spend a couple of hours yesterday making my choices and filling in my ballot and leaving it at that. I also went out earlier today – in light rain as it happens (and without a sore throat) – and put my ballot in the mailbox. Yesterday I acted as an individual, motivated by my felt societal obligation to participate in the election process. Today I acted as a member of society.

As a professional set theorist, I am familiar with the relationship between, and the distinction between, a set and its members. When we view a set in terms of its individual members, we say we are treating it extensionally. When we consider a set in terms of its properties as a single entity, we say we are treating in intensionally. In an election, we are acting intensionally (and intentionally) – at the set level, not as an element of a set.

* A shorter version of this article was published simultaneously in The Huffington Post.

Original author: Mathematical Association of America

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