Learning at Scale Slides from ICTCM

Mar 11, 2017 by

Learning at Scale: Using Research To Improve Learning Practices and Technology for Teaching Math

In the last 5 years, there has been a rise in what we might call “large-scale digital learning experiments.”  These take the form of centralized courses, vendor-created courseware, online homework systems, MOOCs, and free-range learning platforms. If we mine the research, successes, and failures coming out of these experiments, what can we discover about designing better digital learning experiences and technology for the learning of mathematics?


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Abandoning Ship on Wolfram Alpha?

Dec 13, 2011 by

I am really getting fed up tired of having to explain Wolfram Alpha graphs to students.  For some reason, the default in Wolfram Alpha is to graph everything with imaginary numbers.  This results in bizarre-looking graphs and makes it near-impossible to use Wolfram Alpha as a teaching tool for undergraduate mathematics, a real shame.  Now that Google has entered the online graphing fray, I have a wary hope that the programmers at Wolfram Alpha might finally (after two years of waiting) fix the problem.

Here are a few examples.  I’ll show you the graph in Wolfram Alpha, on a TI-84 Plus emulator (TI-SmartView), from Google Search, and from Desmos Graphing Calculator.  These are all the “default” looks.  Wolfram Alpha consistently shows this confusing imaginary view as the default whenever working with graphs involving variables in radicals.

Example 1: f(x)=\sqrt[3]{x}(x+4)

Example 2: f(x)=\log{x}

Example 3: f(x)=\sqrt{x^2-9}

I was hoping to really teach my College Algebra students to use Wolfram Alpha next semester.  But, between the Logarithm Issues and this graphing issue, I’m afraid I’m going to have to abandon ship on using Wolfram Alpha as a teaching tool for students. Students simply don’t have enough mathematical sophistication to look at the graphs and realize that they aren’t seeing what they are supposed to be seeing and I’m seeing far too much confusion on assessments that are caused by the oddities in graphs and logarithms on Wolfram Alpha.  What a shame that we can’t work this out, huh?

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Collection of Math Games

Nov 14, 2011 by

To view the collections of Math Games, hover over the Games Menu, and go to one of the dropdown categories.




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NYT Opinionator Series about Math

Apr 13, 2010 by

For a few months now, the NYT Opinionator Blog has been hosting a series of pieces that do a phenomenally good job of explaining mathematics in layman’s terms.

The latest article is about Calculus (with a promise of more to come): Change We Can Believe In is written by Steven Strogatz, an Applied Mathematician at Cornell University.

There are several other articles in this series, and if you haven’t been reading them, you really should go check them out.  Assign them.  Discuss them in your classes.

Given the discussions we’ve been having about teaching Series and Series approximations lately on Facebook, Twitter, and LinkedIn, I wonder if he’d consider writing an article explaining “Why Series?” to students.

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Logarithm Graphs in Wolfram Alpha

Mar 16, 2010 by

At the Wolfram Alpha Workshop at ICTCM, there was universal disappointment about the fact that you cannot get a graph of a logarithm that is only over the real numbers.  We tried everything we could think of to remove the complex part of the graph.

Personally, I have tried and tried and tried and tried to explain the problem with this in the feedback window for Wolfram Alpha, but been universally unsuccessful.   Every time I suggest a change, I am told that the “After review, our internal development group believes the plots for input “log(x)” are correct.” … yes, I know that … that doesn’t mean it’s the answer that most people will be looking for.

I find it ironic that “inverse of e^x” produces the graph we’d like to see, and even gives log(x) as an equivalent.


But then ask for a graph of  log(x) or ln(x) and the graph will always include the solution over the complex numbers.


What’s worse is that W|A inconsistently decides when to use reals only and when to use both complex and real numbers.  For example, the output for y=ln(x), y=x includes the complex numbered plot, while the output for y=ln(x), y=2x-3 includes only the Reals.  What!?!  Actually, I have some idea why this is … it seems that in some cases, if the extra graph intersects the real part of the log graph, then you get reals only.  If the graphs do not intersect, then you get real+complex.  For example compare the output for y=ln(x), y=2x-3 to the output for y=ln(x), y=2x+5.  On the other hand, when I tried to show a graph transformation, like y=ln(x) with y=ln(x)+4 (including the extra graph y=4x-3), I was back to getting the graphs with complex numbers again. Maddening.


We spend a LOT of time in the algebra and precalculus levels working with  transformations of graphs, understanding inverse functions, and specifics like the domain of a graph.  We can’t use Wolfram Alpha for any of these topics with regards to logarithms because of the way the graphs look.  I can live with the fact that W|A uses log(x) instead of ln(x) … it’s not great, and is confusing to students, but I can explain it and live with it.  But as long as the Wolfram Alpha graph includes the complex number system with no way to see the graphs on only the reals, we’ll have to pull out that old-fashioned graphing calculator to teach this section, and that’s a shame.

I’ve also heard the argument that we should just include the domain we want to see in the W|A input.  For example, y=ln(x), x>0.  But how is a student, learning logs for the first time, supposed to recognize that this is required?  After all, the graph they see when they first try W|A with y=ln(x) leads them to believe that y=ln(x) has a domain that includes all real numbers but zero.  This argument also means that to show graph transformations, we need to use much more complicated graphing commands, restricting each domain separately (to tell you the truth, I have not yet figured out a way to do it … although I suspect it’s possible).

It seems to me that there are two obvious solutions to this math teaching nightmare, and I can’t imagine why either one wouldn’t serve all parties using Wolfram Alpha (both high-level mathematicians, and the rest of us):

Solution #1: Use a toggle-able option to see the graph with only reals or both complex and reals  (I would prefer a default to the Real numbers graph, since my guess would be that the majority of the world’s population would be looking for that one).


Solution #2: Display TWO graphs.  Show a graph of the logarithm that is only on the real number system.  Then, below it, show a graph that includes both the complex and real number systems.


That’s all – end of rant.  This is the one thing I absolutely hate about Wolfram Alpha.  And I’m guessing that I’m not alone here.  Please can’t we just find a solution without hearing “After review, our internal development group believes the plots for input “log(x)” are correct.” again?

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Math Videos at the Sputnik Observatory

Oct 26, 2009 by


The Sputnik Observatory, is dedicated to providing a venue for viewing and sharing ideas and philosophies of contemporary culture.  Jonathan Harris, who worked on the mindblowing sociological website We Feel Fine, is the site director and blog creator for Sputnik Observatory.  Sputnik also has a host of codirectors with diverse backgrounds in journalism, architecture, and ballet.  Members of Sputnik have spent the last ten years interviewing scientists, philosophers, academics, and the like.  They have over 200 videos of conversations on themes such as coherence, interspecies communication, and urban metabolism.

Sputnik Observatory is a New York not-for-profit educational organization dedicated to the study of contemporary culture. We fulfill this mission by documenting, archiving, and disseminating ideas that are shaping modern thought by interviewing leading thinkers in the arts, sciences and technology from around the world. Our philosophy is that ideas are NOT selfish, ideas are NOT viruses. Ideas survive because they fit in with the rest of life. Our position is that ideas are energy, and should interconnect and re-connect continuously because by linking ideas together we learn, and new ideas emerge.”

Here are some of the short interviews that involve mathematics (and all really COOL mathematics).  All of these can be embedded into course shells.

Will Wright – Possibility Space

Ian Stewart – Alien Mathematics

Ian Stewart – Pattern-Seeking Minds

Lord Martin Rees – Simple Recipe

Trevor Paglen – Geologic Agents

Jacques Vallee – Information Universe

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