Archive for the ‘Algebra’ Category

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

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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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Playing to learn math?

This presentation, built yesterday, is my philosophical argument for why we should be actively pursuing games as a way to teach algebra. In fact, you’ll find that many of the definitions of games and game design principles sound like they are describing algebra. Of course, the presentation misses something without my accompanying talk, but it has enough to get you started thinking about where we could be going with math education.

It took me 12 hours just to build the presentation you see below … and collaboration with my assistant and an illustrator.  I’ve been obsessively reading and thinking about this topic for about two years.

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Prime Number Manipulatives

tcm_blog_button2For those of you who are curious what we were actually doing in class in yesterday’s post (Record with a Document Camera and a Flip), we were “fingerprinting” the factors of composite numbers with their prime compositions. The wooden “prime number tiles” are created using the backs of Scrabble tiles and the white tiles you see on the board are the student version of the prime number tiles (each of them cut out a sheet of their own prime number tile manipulatives at the beginning of class).  Just for the record, I would like to confess to destroying more than 5 Scrabble games in the last 6 months.

I tell students that the prime numbers are like the building blocks of the whole numbers, in the same way that the nucleotide bases (adenine, guanine, cytosine, and thymine) are the building blocks of DNA. It was ironic, then, that at the end of our prime factorization fingerprinting, the board looked (to me) the way an agarose gel electrophoresis looks.


You can also use the prime number tiles to line up the prime factorizations of two numbers (leaving spaces when the prime is not in both numbers) and read the GCF as the intersection of the two lines of tiles and the LCM as the union of the two lines of tiles.



I’ve tried explaining these concepts in writing before (circling common factors and using colors or highlighting to show the steps), but the ability to easily rearrange the prime numbers after they’ve been found makes all the difference.  Students find the prime factorization using a factor tree or short division, then place out all the tiles for each number.  At this point, they can rearrange each row into least-to-greatest order.  Then line up the primes, bumping the whole row when there is no overlap in primes between the composite numbers.

We also were able to look at more interesting problems now that students could “see” the inner workings of the factorizations better.  For example, find a pair of numbers with a GCF of 42 and an LCM of 4620 (you may not use the “trivial” answer of 42 and 4620).  By constructing a “board” of prime number tiles, starting with the factorization of the GCF and the LCM, students begin to see which primes have to be in the two numbers and which can only be in one of the two numbers.  Note that there will be more than one answer here … which is another interesting discussion to have!

I’ve never seen “prime number tiles” in any math manipulative kit … they were just something I created last semester.  I think this method of using Prime number tiles would also be helpful with explaining how to find the GCF in the factoring unit of algebra, and with explaining how to find the LCD in fractions.

Feel free to print the prime number tiles and use them in your own classes if you’d like.

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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:

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:

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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Algebra Balance Scales

There are lots of “games” out there about solving equations, but I haven’t found a single one that is more than algebra homework dressed up with pretty packaging.  The “games” are all of the same format.  We’ll give you problems, you give us answers and we’ll reward you (or your character) if you get them right.  These are not teaching games, these are just more of the same kind of practice that you would find in an algebra text.

There is one applet that is worthy of mention, though.  The Algebra Balance Scales from the National Library of Virtual Manipulatives is quite good.  It isn’t billed as a game, but when you’re using it, you feel like you are playing a game because you’re interacting with the algebra on the screen.


I recorded an example to show my students how it works.

An interesting assignment for an online or hybrid class would be to have THEM record an example explaining the process (you could, for example, use Jing like I did) and turn in the link.

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Intriguing Inverses

Finally, I’ve made a sequel to Funky Function Notation. Here is Intriguing Inverses. Feel free to embed in courses or use in the classroom or presentations. Enjoy.


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Modern War follows a Linear Regression

Just watched an amazing 7-minute TED talk on The Mathematics of War where an interdisciplinary team of researchers (physics, mathematics, economist, intelligence, computers) figured out how to mine data from public streams of information to collect and analyze modern warfare.

It turns out that when they began plotting the number killed in an attack with the frequency of those attacks, they found the data was linear. Not only was this relationship linear, but the same linear relationship then appeared in every modern war they looked at (with slopes that varied slightly).


So, next they modeled the probability of an event where x people are killed.


Finally, they went back to each conflict to try to understand the meaning the slope of the line.  It turns out that the alpha value (which hovers around 2.5) has to do with the organizational structure of the resistance.  If the resistance becomes more fragmented, it is pushed closer to 3.  If the resistance becomes more organized, it is pushed closer to 2.

Anyways, it’s only seven minutes.  You should definitely watch it!

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Assess Math Study Skills

Last week I was in Denver for a 1-day math conference and one of the speakers was Paul Nolting (who has written several books about math study skills).

One of the resources that he passed along to us was an online Math Study Skills Evaluation.  Paul suggested that rather than discussing a bad test during office hours, you have the students take the survey and bring the printout with them for discussion during office hours.

Although the survey printout refers to specific pages in Paul’s book, Winning at Math, it also tells the students a bit about why this particular behavior might be causing problems.  Here is an example of the results:

Winning At Math Survey Results

Especially for those of us that teach developmental math courses (although good for any student that is struggling), this survey would be a great activity to do right after the first exam.  Our students often focus on not being “smart enough” to do math, and this should bring the focus to the student not having the appropriate study skills.

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Funky function notation

At the UMD faculty workshop, one of the participants had an idea for using an Animoto video.  She suggested it might be a good way to break up a long lecture time.  This got me thinking about short lessons (like the CommonCraft videos).  Just because the video is short, it doesn’t mean it’s not effective.

I thought I would try out a short video of my own using Animoto.  This one is called “What is function notation?”  If the video doesn’t load for you, go directly to the site here or see the YouTube rendition here.  Either video can be embedded if you’d like to use them in a course shell.


You might be interested in the process I used to build this.  For Animoto, you need a file folder with image files.  First, I created a deck of 75 PowerPoint slides (those being relatively easy to edit).  Then I printed from PowerPoint to SnagIt (because of a special SnagIt save option). Then I saved the SnagIt file as jpg files, where each slide is saved as an individual image file. This gave me a folder of all the slides, but with each slide saved as an image.

I then uploaded the 75 images into Animoto and made sure they were in the proper order (for some reason the last slide fell first and had to be moved back to the last position). You choose the slides you want to “focus” on – places where the reader may need an extra second to think or read. Choose some music (preferably without words), and finally, choose the speed. I tried it at regular speed first (no way), but settled on 1/2 speed as a good speed to show the slides.

I don’t have any student guinea pigs at the moment, so someone play it for your students and let me know what they think! I was toying with the idea of explaining a theorem next.

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Explaining Finance

CommonCraft has three new videos about math (well, technically financial math). I wonder if they’ll tackle the Gaussian Copula Function next?

Saving Money in Plain English

Investing Money in Plain English

Borrowing Money in Plain English

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