Archive for the ‘Calculus’ Category

Collection of Math Games


The page of digital and non-digital games has grown too long and unwieldy, so I’ve finally taken the time to reorganize the content by topic area. I’ve also added all the new “Block” games on various topics in Trigonometry, Rational Exponents, and Logarithms.

If you’ve bookmarked the old Games page, you’ll see that it now just tells you how to find the new sub-pages.

Direct links to the new game pages are below:

I’ve also decided to collect your suggestions for other digital and/or paper games, puzzles, and manipulatives  using a Google Form, but before you submit a game for me to review, PLEASE check it against my criteria for Lame Games.

Submit your suggestions here.

 

 

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New Math Game: Antiderivative Block


Here’s a game I created last week called “Antiderivative Block” to encourage students to (1) learn their derivative rules well (2) begin thinking about derivatives backwards, and (3) to learn to be careful not to mix up derivatives and antiderivatives.

Two students playing the Antiderivative Block game.

Here’s the game board of a well-played game:

Game board (6x6 grid) that is almost completely used up.

The rules are very simple (they are described on the game pdf), but the game play is complex enough that you really have to be on your toes to play.  Here are a couple of students demonstrating how to play:

I have to say that watching students play this game was the most fun I have ever had in a math class.  They quickly got very competitive and I heard several students in both classes say something like “I really need to learn these derivatives” – even when you think you have won the game, it can be lost by missing a negative on an answer.  Within 10-minutes, students from different pairs were challenging each other to matches (winners played winners).  Some won on mathematical skill alone (being better at the derivatives than their opponent), some won by playing the game well (and knowing their math).  Their attempts to psych each other out and cross-group banter had me laughing so hard in one class that I was crying.

Another interesting side effect of this game was that one of my ESL students suddenly got much better at correctly saying the math because his opponents wouldn’t let him claim spaces if he said “sine x squared” instead of “sine squared x” … I think his understanding of how to SAY the math had improved ten-fold by the end of the hour.

Don’t let the calculus nature of this game fool you.  You could build the exact same game for learning trig values of special angles, for learning to simplify exponential expressions, for exponential and log functions.  As a matter of fact, on the very same day I built this game, I instantly modified it for learning vocabulary in my MathET class (lucky for me, every student already had a set of small vocabulary cards that were the same size as the gameboard spaces).  Here they are playing Geometry Vocabulary Block:

Four students playing Geometry Vocabulary Block on two game boards.

We also had one group of three players (we used red chips for the third player) and everyone who tried the 3-player game said that the gameplay was very different than the 2-player game.  So a simple alteration to the game is just to change the number of players.  The students also suggested that they wanted more cards to move into the game board so that the problems were always fresh.

Three students playing the 3-player version of the game.

P.S. Sorry about the strange RSS problem this week. It was not intentional.  Just a misguided WordPress plugin that I tried.  Needless to say, it has been disabled.

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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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Calculus Tweetwars: The End


The Calculus Tweetwars: Act 3

We hope you’ve enjoyed the production of “The Calculus Tweetwars” and thanks to all that participated by interacting with the characters.
Also, check out our mention in the Chronicle of Higher Education Newton and Leibniz Duke it out on Twitter

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Calculus Tweetwars: 1676-1698


In case it’s difficult to follow the events in “real time” …

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The Calculus Tweetwars (1661-1675)


Installment #1

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The Calculus Tweetwars


I wanted to wait until I was SURE that this was going to happen before I mentioned it here.  My Honors Calculus II students have decided to “tweet” The Calculus Wars for modern times.

Their assignment was to read “The Calculus Wars” by Jason Socrates Bardi, and then come up with a project (individually or collectively) that requires them to further explore something from the book.  A few years ago, I had one student in this course and he build the Leibniz Calculating Machine the animation software Blender (you can see it here).

Anyways, this year, there are three students.  During our discussion of the book, we observed that the scientists involved were like the bloggers and tweeters of their time, sending and publishing an incredible amount of correspondence (some anonymous) via really old-fashioned mail (i.e. SLOW).  Then we wandered into what it would look like if the Calculus Wars happened today and all the characters were in Facebook (friending, unfriending, fan pages, wall posting, etc.).  Ultimately, the students decided to work together to create a modern-day recreation of The Calculus Wars.  Facebook turned out to be too difficult (each follower would have to “friend” each character in order to see the storyline play out).

The students have written a rather lengthy script that includes a rather large cast of characters.  In order to get the twitter accounts, they had to first get email addresses for each character.  Let’s just say we now know how many email or twitter accounts you can set up on one IP address before you get blocked for the day.

We originally tried to use Google Wave to build the script (since it allows for simultaneous collaboration), but it proved to be too glitchy and clunky to get the job done.  About two weeks ago we began transferring the entire script to a Google Doc instead (which, surprise! Also allows simultaneous multi-user collaboration now).  The script is now built as a table so that we could map out the years (1661-1726) against the dates of tweeting, tweets, and who is responsible for putting up the tweets.  There are just a few tweets per year in these early years, but when the Calculus Wars heat up, it will be a lot of work to get all the tweets up properly.

The Calculus Tweetwars started yesterday, and you don’t need a twitter account to follow it.  Just visit the CalcWars Twitter List several times a day to see what’s happened in the lives of Newton, Leibniz, and others.  If you DO have a twitter list, you can just follow the list, and you’ll see all the characters show up in your tweetstream.  Please feel free to interact with the characters as if they were members of your own PLN (personal learning network).

This might seem like a strange academic project to you, but the purpose was to increase awareness of what the Calculus Wars were, and help students see math as something that has not always been so static.  Given that they already have 67 followers after 24 hours, I’d say that the students will be successful with their mission to educate others.

Again, you can follow the project (for the next two weeks) here: http://twitter.com/#list/busynessgirl/calcwars

Enjoy!

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NYT Opinionator Series about Math


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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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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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.”

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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:

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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.

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  1. Start by exploring numbers under 50 (42, 37, 15, 29).
  2. Now ask students to figure out where the pattern changes (hint: it’s between 50 and 100).
  3. Explore numbers in the next tier and see if they can figure out at what number the next place digit gets added.
  4. 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.

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

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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.

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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.

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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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