## Archive for the ‘Wolfram Tech’ Category

## Wolfram Alpha for Inquiry Based Learning in Calculus

Now that all my Calculus II students know about Wolfram Alpha (*I showed them*), I have to make sure that the assignments I ask them to turn in can’t just be “walphaed” with no thought. In Calc II, our topics list includes a lot of “techniques-oriented” topics (integration by partial fractions, integration by parts, etc.) and because of the need to keep this course transferable to 4-year schools, I can’t really get around this. So now I’m in the position of having to reconcile the use of technology that easily evaluates the integrals with making sure that students actually understand the techniques of integration. There are two ways I’m tackling this:

**1. CCC (Concept Compare Contrast) Problems:** I’m writing problems that focus on understanding the mathematical process and the compare/contrast nature of math problems. While Wolfram Alpha can evaluate the integrals for them, the questions I’ve asked require (I hope) a deeper level of understanding about what happens when the techniques are used. Here’s an example from my recent problem set:

There are two pairs of problems below that are exactly the same. You won’t see why until you do the integration, showing all the steps. Find the pairs and then explain how the matched integrals are fundamentally the same.

**2. Inquiry Based Learning:** One appropriate use for any CAS (computer algebra system) is to use it as a way for students to explore problem types that they have not learned about yet. Here’s a definition of IBL, in case you’re not familiar with the terminology:

Designing and using activities where students learn new concepts by actively doing and reflecting on what they have done. The guiding principle is that instructors try not to talk in depth about a concept until students have had an opportunity to think about it first (Hastings, 2006).

It is relatively easy to use IBL in the really low levels of math (K-6) where there is not as much abstraction of concepts. However, with the introduction of variables, rules, theorems, and definitions that come later in math, the use of IBL requires either that the instructor act as the inquiry tool or the use of CAS.

Back to the point (how to use Wolfram Alpha to do this): I could have just taught the integral techniques straight up … here’s the technique, now apply it … repeat. But learning the technique is not anywhere near as important (at least, in my mind) as learning to decide **when** to use a technique, i.e. what makes one integral different from another?

This semester, I’m doing it backwards. In the problem set **before** we look at specific techniques of integration, the students will use Wolfram Alpha to evaluate twenty integrals. Then they will look for patterns in the answers and the problems, and try, on their own, to make sense of what kinds of problems solicit different answers. After they understand what characteristics make one integral fundamentally different (in technique of integration) from another, **then** we’ll look at how each technique works. Below, you see a few examples of the integrals the students will explore. You can view the whole assignment here.

For years, we’ve had CAS tools like Derive, Mathematica, Sage, Maple, etc. However, the use of these programs traditionally required so much coding minutia that the IBL often got lost in the coding. How do I know? Because this was *my* experience as a student. I had instructors that tried to teach me this way. All I remember is how painful the coding was. I followed the directions in the labs, I typed what I was supposed to type, and I answered the questions that were put forth to me. But in the end, I never sat down at a computer and generated my own inquiries. The details of using the programs were so painful that I just didn’t have any desire.

Here’s the sum total of the directions that were necessary for me to teach students how to evaluate integrals in Wolfram Alpha:

For example, here’s how to do the first one: http://www.wolframalpha.com/input/?i=integrate+10/(x^2-16)

I think Wolfram Alpha is a game-changing CAS (*and no, I’m not being paid by someone to say this*). For better or for worse, my students are now using W|A on their own, without any prompting from me. Their evidence of usage is showing up in emails, in discussion board questions, and in questions they ask in the classroom. Maybe my class is unusual because I’ve given them the first push… but it’s just a matter of time before W|A is discovered by your students too.

NOTE: If you’ve missed all the other posts I’ve made about Wolfram|Alpha, you can see all of them here.

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## My Interview at Wolfram Alpha HomeworkDay

Believe it or not, it was scarier to watch the video than to do the interview! I think I will tuck my hair behind my ears next time.

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## Wolfram|Alpha: Recalculating Teaching & Learning

My talk today at the 2009 International Mathematica User Conference:

For at least a decade, we have had the ability to let CAS software perform computational mathematics, yet computational skills are still a large portion of the mathematics curriculum. Enter Wolfram|Alpha. Unlike traditional CAS systems, Wolfram|Alpha has trialability: Anyone with Internet access can try it and there is no cost. It has high observability: Share anything you find with your peers using a hyperlink. It has low complexity: You can use natural language input and, in general, the less you ask for in the search, the more information Wolfram|Alpha tends to give you. Diffusion of innovation theories predict that these features of Wolfram|Alpha make it likely that there will be wide-spread adoption by students. What does this mean for math instructors?

This could be the time for us to reach out and embrace a tool that might allow us to jettison some of the computational knowledge from the curriculum, and give math instructors greater flexibility in supplemental topics in the classroom. Wolfram|Alpha could help our students to make connections between a variety of mathematical concepts. The curated data sets can be easily incorporated into classroom examples to bring in real-world data. On the other hand, instructors have valid concerns about appropriate use of Wolfram|Alpha. Higher-level mathematics is laid on a foundation of symbology, logic, and algebraic manipulation. How much of this “foundation” is necessary to retain quantitative savvy at the higher levels? Answering this question will require us to recalculate how we teach and learn mathematics.

There are two videos embedded in the slideshow. You should be able to click on the slide to open the videos in a anew web browser. However, if you’d **just** like to watch the video demos, here are direct links:

- Complexity and Relative Advantage: Comparing the use of Mathematica to Wolfram|Alpha (for educational purposes, 2 minutes)
- Trialability and Observability: Demonstration of how the use of W|A is likely to spread via direct online contact (sounds like a virus, doesn’t it? 5 minutes)

Note that I’ve turned ON commenting for these two video demonstrations and I will try to load them into YouTube later this weekend.

There are several other posts about Wolfram|Alpha that you may want to check out:

- What we’re doing with Wolfram|Alpha
- Don’t get Wolfram|Alpha? Try these examples
- Shaking Up Math Education
- Impact of Wolfram|Alpha on Math Ed
- Chemistry Takes a Hit Too
- Implications for Math Instructors

If you were at the live version of this talk, and you would like to rate the presentation, you can do so here at SpeakerRate.

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## What we’re doing with Wolfram|Alpha

Originally, I started this post with the title “What I’m doing with Wolfram|Alpha” and then I revised it, because it’s not just **me** using Wolfram|Alpha. My students are using it too. Here are some of the things we’re doing:

**Discussion Boards: Wolfram|Alpha + Jing = Awesome**

Before Wolfram|Alpha, it could take several steps to get a graph or the solution to solving an equation to the discussion board in an online class. You had to use some program to generate the graph or the equations, then make a screenshot of the work, then get that hyperlink, image, or embed code to the discussion board.

With Wolfram|Alpha, sometimes a simple link suffices. Suppose, for example, I needed to explain the last step in a calculus problem where the students have to find where there is a horizontal tangent line. After finding the derivative, they have to set it equal to zero and solve the equation (and calculus students notoriously struggle with their algebra skills). Rather than writing out all the steps to help a student on the discussion board, I could just provide the link to the solution and tell them to click on “Show Steps.”

Sometimes, a bit more explanation may be required, and in these circumstances, Jing + Wolfram|Alpha really comes in handy. For instance, I needed to show how to reflect a function over the line *y*=1.

Here’s what the reflection over y=1 looks like. If you graph y=sqrt(x) and y=-sqrt(x)+1 you will see that they are not reflected over y=1.

Here’s another example of Wolfram|Alpha + Jing:

**Classroom Demonstrations**

We’re also finding that Wolfram|Alpha can be a good program to use for exploratory learning. One of the subjects we cover in *Math for Elementary Teachers* (MathET) is ancient numeration systems. Rather than just tell students how the Babylonian number system worked, students can use Wolfram|Alpha to explore the number systems until they’ve worked out the pattern.

- Start by exploring numbers under 50 (42, 37, 15, 29).
- Now ask students to figure out where the pattern changes (hint: it’s between 50 and 100).
- Explore numbers in the next tier and see if they can figure out at what number the next place digit gets added.
- Discuss how a zero is written (and why this is problematic).

**Supplement to Online Course Shell**

Another topic in Math for Elementary Teachers is learning to perform operations in alternate-base systems (like Base 5 and Base 12). You can easily supplement your online course shell by providing additional practice problems and then linking to the answers with Wolfram|Alpha.

- Find the sum of 234 and 313 in base 5. (answer)
- Subtract 234 from 412 in base 5. (answer)
- Multiply 234 by 3 in base 5. (answer)

**Student Projects**

Wolfram|Alpha has also started making its way into student projects because of the ease of just linking to the mathematics instead of writing out or drawing the math. Here are a few examples.

For one of the calculus learning projects, the group built a mindmap that demonstrates the graphs and translations of exponential and logarithmic functions.

Another group recorded some help tutorials on using Wolfram|Alpha for evaluating limits. Here are two of their videos (one with sound and one without).

Several of the MathET students have used Wolfram|Alpha and Wolfram Demonstration links as they mapped out the concepts in our units.

**Checking Solutions and Writing Tests
**

Personally, I’m finding that I use Wolfram|Alpha from a simple calculator to a CAS for checking answers as I write a test. I’ve also been snagging images of graphs from Wolfram|Alpha to use on tests (use Jing for simple screenshots). Here’s a short 1-minute tutorial on how to change the plot windows to get the image you desire.

**Homework Day**

Oh, I almost forgot to tell you. I’ll be down in Champaign, IL for the rest of the week at Wolfram Research. Tomorrow I’ll be one of the “experts” participating in Wolfram|Alpha Homework Day (a live, interactive web event). The events begin at noon (CST) and end around 2am. I’ll be interviewed somewhere around 3 pm and participate in a panel discussion about technology and math education at 8pm.

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## Igniting a new Math War?

Check out the story published in The Chronicle of Higher Education A Calculating Web Site Could Ignite a New Campus ‘Math War’.

There’s one small error I think I should point out in the story. “In other words, it can instantly do all the homework and test questions found in many calculus textbooks.” Replace the word “all” with the word “some.”

**Possibly Related Posts:**

- TED Talk on Wolfram Alpha
- Wolfram Alpha for Inquiry Based Learning in Calculus
- My Interview at Wolfram Alpha HomeworkDay
- Wolfram|Alpha: Recalculating Teaching & Learning
- What we’re doing with Wolfram|Alpha

## Don’t get W|A Implications? Try these examples.

Wolfram|Alpha is a “computational search engine” built by Wolfram Research (the developers of Mathematica). W|A (pronounce this as “walpha” if you’d like) is similar in appearance to the search engines that we are used to and easy to use. It’s not that W|A will replace other search engines, because it won’t. It’s more of a missing piece in the search engine puzzle. W|A provides a collection of data, formulas, computations, and interpretations that are different from other search engines.

Although the media has stressed data-driven examples (for example, type your first name to see a graph of the frequency of that name over time), the ability of W|A to function as a combination of CAS and natural language computational system is stunning. Let me illustrate with a couple of examples for you to try yourself. Simply follow the links below to see how W|A handles these search requests:

• 126 (make sure to click on “other historical numerals”)

• convert 125 m^3 to gallons

• sphere r=7 cm

• Line (2,7) and (3,1/2)

• Solve x^2-6x=16 (make sure to click on “show steps”)

• 4 – x^2

• Triangle 7,8,9

• x^2-y^2=9

• limit x->3 (x-3)/(x^2-9) (again, make sure to click on “show steps”)

• integral (x^2)sin(x^3) (“show steps”)

• sum 1/n^2

• New York City, Chicago

• convert 78 to base 5

There are several differences between W|A and traditional CAS systems. The first, which you should have noticed after those examples, is that the less you ask for, the more you get. W|A just assumes you want all relevant computations and information that it can generate: graphs, solutions, alternate forms, derivatives, integrals, area under the curve (if bounded), and steps (if they are available). W|A provides quick and painless access to all sorts of data that has been organized so that it can be cross-referenced. In this sense, W|A could be a valuable tool for us in helping students to see the connections between concepts within mathematics and in relating mathematics to the real world.

On the other hand, you’re probably also seeing that there could be implications with academic dishonesty, especially in online and hybrid courses. We will all have to individually decide whether W|A is off limits, and if so, how we can possibly enforce it. Ready or not, W|A is now available on any computer with Internet access and on most SmartPhones.

It is up to us to think about (with as much advanced notice as possible) how we want to embrace, adopt, accommodate, or regulate the use of W|A in our courses. This is a conversation we should have in every department at every level of mathematics, including both full-time and part-time instructors. It is a conversation that we should have with our colleagues in the partner disciplines and with our colleagues at our transfer institutions.

**Note: **There are additional resources you may wish to view. A longer analysis regarding the rate of adoption and the impact of large-scale change in mathematics on the higher education system can be found here. A wiki (WalphaWiki) documenting some of the capabilities of W|A for math courses and the implications for teaching has also been started by Derek Bruff.

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## Impact of Wolfram Alpha on Math Ed

I’ve had almost two weeks to think about the impact of Wolfram|Alpha (abbreviated as W|A, and now pronounced by me as “Walpha”), and I’m ready to share some of my thoughts with you.

After spending hundreds of hours* *reading more than 200 papers on innovation in math instructional practices, change in higher education, and diffusion of innovation theory, it is strange to suddenly find myself observing the possibility of a sudden shift in math education caused by a new innovation. I liken it to being a vulcanologist who has, up until this point, been observing a dormant volcano and then quite unexpectedly, it begins rumbling.

Please keep in mind that these are **my own** predictions and thoughts, for better or for worse.

1. The adoption rate of W|A amongst **students** in higher education will be extremely fast.

I’ve examined the attributes and variables that affect the diffusion of innovations, and found that every single one points to a fast adoption amongst students. Because W|A is free and similar to other technologies they know how to use (designed like a search engine), it has *relative advantage* over other CAS technologies. With prior CAS technologies, you had to know exactly what series of steps or commands to write in order to extract the outcome you desired, but with W|A, the less you ask for, the more you get out. W|A just assumes you want all relevant information it can generate. W|A is easily *trialable* – anyone with Internet access can try it. Not only that, but *observability* is also high – simply use a hyperlink to share what you’re doing in W|A with others. Compound this ease of observability with the incredible connectedness of the student population in the U.S. (Facebook, MySpace, etc.), and you can see why I don’t think it will take long for W|A to spread to the undergraduate population of math students.

Most students take their math classes for one reason: they are required to for their degree. W|A will provide solutions to problems, relevant mathematical information, and in many cases, steps for how the solution was obtained. Thus, for the reason that it appears to be a means to an end (getting through that math course with the least pain possible), using W|A to help complete assignments for math courses will be extremely *compatible* with the belief systems of these students.

2. There will be a sizable group of math instructors that **immediately** shifts to using Wolfram Alpha in instruction, and thus, begins to shift the curriculum in those classes away from computational mathematics.

I’ve already outlined many reasons why students will be fast adopters, and for the most part, these are the same reasons that instructors will be fast adopters (high *relative advantage*, low *complexity*, good *trialability* and *observability*). The main difference between the student and instructor populations will be the *compatibility* between their beliefs systems and the innovation. This is the only attribute where the adoption rate of W|A might be slowed. For example, Computer Algebra System (CAS) technology (TI-89 calculators, Maple, Mathematica, etc.) has been around for at least 10 years, and yet CAS is **not** widely adopted in math courses (see the latest CBMS Statistical Report).

That’s not to say that math instructor beliefs aren’t compatible with the use of CAS Technologies. I suspect that many, like myself, simply found that *implementation* of CAS in the classroom was too difficult. In my case, I questioned how could I ask my students, who already had a non CAS-calculator in-hand from high school, to pay for extra software or another calculator to adopt the curriculum to CAS-inclusion. To teach using software (before students all began buying laptops), we would require computer labs and site licensing, and this was not in the budget for many of us. Whether it was calculators or software, either decision would require students to spend more money, and thus, these were decisions that would likely have required department buy-in.

What does this mean for the adoption of W|A today? Instructors who already teach with CAS technologies will easily make the shift to using W|A. Instructors who liked the *idea* of teaching with CAS, but were unable to implement for logistical reasons, will quickly also quickly make the shift to using W|A (you may think I’m full of it here, but I already know of several who have **already** changed their courses). The real beauty of W|A being free is that individual instructors, under the umbrella of academic freedom, do not have to ask their departments or colleagues for buy-in. Shoud they? **Yes. ** And if they are under some kind of contract to teach in a prescribed manner given by the department, then they should **definitely** ask. But for the majority of us, if we just decided to change our courses tomorrow, very little could be done to stop it.

For my complete analysis of the rate of diffusion of W|A, you can download the 2-page analysis or view the slides that compare CAS and W|A.

3. There will be a sizable group of math instructors that attempt to either ignore W|A or put up an active resistance to it.

While some instructors will actively ignore the existence of W|A (look at the theory of cognitive dissonance), some will just passively miss it for a while (you know, by ignoring that email that they get sent that warns them to take a look at what W|A does).

However, given that there are still pockets of instructors and departments in the U.S. where graphing calculators are still not allowed, some instructors will likely react with resistance (i.e. we still don’t change anything) or possibly even with the charge that using W|A is cheating. For these instructors, compatibility of beliefs is not there.

4. We **can** change if we do so by focusing on **areas of agreement** instead of disagreement.

Mathematicians in higher education have been divided over reform teaching for 20 years now. Much like some of the political hot potatoes of our time (which shall go unnamed here for fear of blog spammers), it is unlikely that the two camps of traditionalists and reformists will ever sway followers from the opposite side. However, we can hope to agree on a middle ground. I think we would all agree that we want to make sure that math instruction focuses on learning* concepts*. I think we would all agree that *some* understanding of algebraic manipulation is important to lay the foundational structure upon which the rest of mathematical understanding is laid. I think we would all agree that there is *some* set of fundamental skills that must be learned extremely well in order to progress to higher levels of mathematics. Finally, I think we would all unanimously agree that we wish we had more time in our classes to be flexible in what we teach – to bring in interesting mathematical examples from the world around us even if the math doesn’t directly relate to the topic of the day’s lesson.

Perhaps this is the time for us to reach out and embrace a tool that might allow us to jettison some of the computational knowledge from the curriculum, and give math instructors greater flexibility in supplemental topics in the classroom. Just like each English instructor has their own favorite books to teach with, math instructors have their own favorite topics they wish they could share with students: fractals, trend analysis, network theory, number theory, modern algebra … maybe these finally get a turn at the table.

One more thing. You (my readers) have to understand how scary this whole thing might be for some math instructors. I still think that anyone who is not a *little *scared by the changes that W|A brings hasn’t thought about it enough yet. I’ve always been an instructor that lived for change, and I’ve been uneasy since W|A launched on May 15. I have no doubt that I **can** change my courses to adapt to the new environment, and I know that in the end, the changes will be good ones – but the thought of changing so much across the board in all my courses is a daunting one.

We math folks were attracted to mathematics for its beauty, its power, and its logic. In the classroom, we have always been the beneficiaries of its non-changing nature. Algebra is algebra and calculus is calculus. In all the languages of the world, algebra and calculus have been fundamentally the same for hundreds of years. You can walk into any colleague’s class and cover for them as long as they tell you which topic to launch into. This has been a fairly easy world for us to inhabit and teach in up to now. So now, things change. And probably, they change quickly.

5. We all need to keep the system in mind.

None of us teaches in a vacuum. You cannot make major changes to your course without at least *considering* the impact that it will have when those students move to the next course, the next instructor, or the next college. Make sure that your course changes still provide sufficient “math backbone” to span students successfully to the next level of mathematics. For more on this, view the slides starting on slide #13.

Personally, I do plan to change my courses to incorporate W|A in the fall, and let me tell you that I’m grateful to have another couple of months to think about exactly how to do it. To those of you who are already using W|A this summer – you are the pioneers! Please blog, write, comment, or email about how it is going and advice for making it work.

Note: Derek has also put up a WalphaWiki where we can all begin to document how W|A handles traditional math topics and the impacts this will have on our courses.

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## Chemistry takes a hit too

So, perhaps we’re struggling a bit with the possibility of our course curriculum shifting a bit in math with the introduction of Wolfram|Alpha (or walpha, as I’ve started calling it – they should have picked a shorter name). If you’re not struggling a bit, perhaps you haven’t played with it enough yet to really grasp the implications here.

Just so you know, whether they know it or not, the chemistry educators are going to have to think about this too. Here are some examples:

Walpha will convert units.

It will give you all the chemistry of an element, a compound, or something more complicated (make sure to click on “more”).

It will compare the chemistry of several elements or compounds.

It will solve a chemical formula for a specific variable (make sure to click on “show steps”).

Walpha will do chemical conversions for you or do formula calculations.

If there’s a chemist in your circle of friends, perhaps it’s time they got to join in the fun!

**Possibly Related Posts:**

- TED Talk on Wolfram Alpha
- Wolfram Alpha for Inquiry Based Learning in Calculus
- My Interview at Wolfram Alpha HomeworkDay
- Wolfram|Alpha: Recalculating Teaching & Learning
- What we’re doing with Wolfram|Alpha

## Implications for Math Instructors

They say a picture is worth 1000 words. Then here are about 15,000.

I’ve taken screenshots of several examples of the algebra through calculus that WolframAlpha will do.

You can see the album of screenshots here: **http://tinyurl.com/q49xy8**

I’ll let you see the implications for yourself.

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