Archive for the ‘Wolfram Alpha’ Category

Navigating WolframAlpha Pro Features


Last week I had to do a workshop about WolframAlpha, and I noticed that there are three different feature sets: not logged in, logged in, logged in to Pro.

I needed to know which login settings provided which features (especially for giving workshops and working with students), so I decided to be thorough about it.  You can download the PDF of this document, Guide to Wolfram Alpha Features, as well.

Hope this makes the decision-making a little easier for you!

 

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Abandoning ship on using Wolfram Alpha with Students


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:

Example 2:

Example 3:

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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Timeline of the Rise of Data


When Wolfram Research set out to build Wolfram Alpha, they set out to make all knowledge computable.  Last week they published a Timeline of Systematic Data and the Development of Computable Knowledge.

You can interact with the timeline online, but far cooler (I think) will be hanging the 5-foot poster of the timeline ($7.25 + shipping) that links data and computable knowledge with history, science, and culture on the walls of our Math ELITEs.

The blog post about the timeline is pretty interesting too, discussing which civilizations have tracked the most data.

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Wolfram Alpha in a Nutshell


This little video is reminiscent of the “in Plain English” videos that we all love.  If you’re trying to explain Wolfram Alpha to a non-math person, try Wolfram Alpha in a Nutshell.

Thanks to the Tech the Plunge blog for pointing this one out.

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Giving up Calculation by Hand


This is scary stuff for math professors, but with the arrival of amazing programs like Wolfram Alpha, we’re going to have to start paying attention to the signs of change.  I talked to Conrad Wolfram (at Wolfram Alpha Homework Day) when he was still formulating what he wanted to say at this TED Talk.  I think it’s worth 18 minutes of your time to watch Teaching kids real math with computers.

Here’s an outline of the Conrad Wolfram’s argument (which I am paraphrasing/quoting here):

What’s the point of teaching people math?

  1. Technical jobs (critical to the development of our economies)
  2. Everyday living (e.g. figuring out mortgage, being skeptical of government statistics)
  3. Logical mind training / logical thinking (math is a great way to learn logic)

What IS math?

  1. Posing the right questions.
  2. Convert from real world to mathematical formulation
  3. Computation
  4. Convert from mathematical formulation BACK to real world

The problem? In math education, we’re spending about 80% of the time teaching students to do step 3 by hand.

Math is not equal to calculating, math is a much broader subject than calculating.  In fact, math has been liberated from calculating.

Should we have to “Get the basics first”?  Are the “basics” of driving a car learning how to service or design the car?  Are the “basics” of writing learning how to sharpen a quill?

People confuse the order of the invention of the tools with the order in which they should use them in teaching. Just because paper was invented before computers, it doesn’t necessarily mean you get more to the basics of the subject by using paper instead of a computer to teach mathematics.

What about this idea that “Computer dumb math down” … that somehow, if you use a computer, it’s all mindless button-pushing.  But if you do it by hand it’s all intellectual.  This one kind of annoys me, I must say.  Do we really believe that the math that most people are actually doing in school practically today is more than applying procedures to problems they don’t really understand for reasons they don’t get? … What’s worse … what they’re learning there isn’t even practically useful anymore.  It might have been 50 years ago, but it isn’t anymore.  When they’re out of education, they do it on a computer.

Understanding procedures and processes IS important. But there’s a fantastic way to do that in the modern world … it’s called programming.

We have a unique opportunity to make math both more practical and more conceptual simultaneously.

Personally, I’m all for it.  But how?  That’s the question.  How to shift and incredibly complex and interconnected system of education? How to train tens of thousands of teachers and faculty to teach a new curriculum that they themselves never learned?  Hmmm … it seems that we might need some help, maybe a new paradigm for education itself.  It’s coming.

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Shifting Assessment in a World with WolframAlpha


I let my students use Wolfram Alpha when they are in class and when they are doing their homework (um, how would I stop them?).  Because of this, I’ve had to shift how I assess on more formal assignments.  For the record, it’s the same adjustment you might make if you were using ANY kind of Computer Algebra System (CAS).

The simplest shift is to stop asking for the answers to problems, and just give the students the answers.  After all, they live in a world where they CAN easily get the answers, so why pretend that it’s the answers that are important?  It’s the mathematical thinking that’s important, right?  Giving the students the answers turns problems into “proofs” where the evaluation (grade) is based on the thought-process to get from start to finish. It also eliminates the debate about whether to award points for a correct answer with no correct process.

Here are two examples of problems from a recent Calculus exam (old and new wording).

I wish I had thought to do this years ago, because students who insist on just doing the “shortcut” (and not learning what limits are all about) now have nothing to show for themselves (the answer, after all, is right THERE).

Again, a student that knows the derivative rules might get the right answer, but the right answer is now worth zero points.  The assessment is now clearly focused on the mathematical thinking using limits.

Another reason that I really like this is that it allows students to find mistakes that they are capable of finding “in the real world” where they can quickly use technology to get an answer.  They are now graded solely on their ability to explain, mathematically, the insides of a mathematical process.

Wolfram Alpha also allows me to pull real-world data into my tests much faster.  Here’s a question about curve shape (the graph is just a copy/paste from W|A):

If you haven’t begun to think about how assessment should change in a world with ubiquitous and free CAS, you should.  You don’t have to change all your problems, but I think some of them should change.  Otherwise, we’re just testing students on the same thing that a computer does, and that doesn’t sit well with me.  If you can be replaced by a computer, you’re likely to be replaced by a computer.  Let’s make sure we’re teaching students how to think mathematically, not how to compute mathematically.

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Math Technology to Engage, Delight, and Excite


Back in May 2010 I presented a keynote at the MAA-Michigan meeting in Ypsilanti.  Even though it sounds like it’s about math, it’s really more about a philosophy of using technology to engage students.  Yes, the examples are in the context of math, but if you’re involved with educational technology in any way, I think much of the talk is applicable to all subjects.

We’re in a recession and so is your department budget.  Luckily for you, there are lots of great programs and web resources that you can use to teach math, and most of these are free.  Use the resources in this presentation to tackle the technology problems that haunt you and capture the attention of your math classes with interactive demonstrations and relevant web content.

Here is the video, audio, and slides from my keynote talk “Math Technology to Engage, Delight, and Excite” from the MAA-Michigan meeting in May 2010.  There is also an iPad/iPod-friendly version here.

In case you’re wondering, the PIP video was recorded from a Flip Video camera that was affixed to one of the seats in the auditorium with masking tape.  It’s not elegant, but it works.

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TED Talk on Wolfram Alpha


The talk is titled “Computing a Theory of Everything

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Algebra is Weightlifting for the Brain


This was my presentation on Friday in Austin, Texas at the Developmental Education TeamUp Conference.

The process of learning algebra should ideally teach students good logic skills, the ability to compare and contrast circumstances, and to recognize patterns and make predictions. In a world with free CAS at our fingertips, the focus on these underlying skills is even more important than it used to be. Learn how to focus on thinking skills and incorporate more active learning in algebra classes, without losing ground on topic coverage.

Algebra Is Weightlifting For The Brain

View more presentations from wyandersen.
UPDATE: By popular demand (this presentation has been viewed 2,000+ times in 3 days), I’ve loaded the uncut, unedited video that I took of the presentation to my Screencast account.  I’m not going to claim the video recording is great (recorded with a Flip Video Camera sitting on a table), but you’ll get to hear the audio and more of the details.  View it here.

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Logarithm Graphs in Wolfram Alpha


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.

inverse-of-ex

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

problem-with-logs3

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.

log-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).

log-toggle-option

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.

log-two-graph-solution

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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