Signed Numbers: Colored Counters in a “Sea of Zeros”

Nov 10, 2011 by

The “colored counter” method is an old tried-and-true method for teaching the concept of adding signed numbers.  However, to show subtraction with the colored counter method has always seemed painful to me … that is, until I altered the method slightly.

Now all problems are demonstrated within a “Sea of Zeros” and when you need to take away counters, you can simply borrow from the infinite sea.  Voila!  Here’s a short video to demonstrate addition and subtraction of integers using the “Sea of Zeros” method.  You can print some Colored Counter Paper here.

Video: Colored Counters in a Sea of Zeros

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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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Algebra is Weightlifting for the Brain

Mar 29, 2010 by

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.


I’ve loaded the uncut, unedited video that I took of the presentation to YouTube.  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 “Algebra is Weightlifting for the Brain” here.

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Prime Number Manipulatives

Feb 19, 2010 by

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 for Inquiry Based Learning in Calculus

Jan 22, 2010 by


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:^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.

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Teaching Math with Technology (Discussion Panel)

Dec 3, 2009 by

While I was at Wolfram Alpha Homework Day, I participated in a Panel Discussion about the Myths about Teaching with Technology. The panel ran 30 minutes and was mediated by Elizabeth Corcoran. There were three of us (all women, weirdly enough), Debra Woods, a mathematics professor at the University of Illinois at Urbana-Champaign; Abby Brown, a math teacher at Torrey Pines High School; and myself.

I no longer remembered anything that I said in this panel, so it was fun to watch the discussion from an outside point-of-view. I am glad to see that I talked about the value of play during the discussion, because I am finding more and more that introducing play (and exploration) back into learning makes a big difference in engagement and in retention of the subject.

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