Monthly Archives: July 2013

Exponent rules and Witchcraft

I just received this email from a student:


3^0 = 3^(1 + -1) = (3^1)*(3^-1) = 3 * (1/3)

Talk about an accomplished summer.

This group in Algebra 2 took a lot of convincing. I went through about four or five different approaches to proving this. They objected to using laws of exponents since 30 is one of the rules of exponents. They didn't like writing out factors and dividing them out. They didn't like following patterns. While they did accept that they could use the exponent rule as fact, they didn't like doing this. I really liked that they pushed me so far on this, and I don't entirely believe that their disbelief was simply a method of delaying the lesson of the day.

Whatever it was that led this particular student to have such a revelation, it makes me incredibly proud that this student chose to follow that lead, especially given that it is the middle of summer vacation. Despite labeling the content of the course 'witchcraft', I'm marking this down in the 'win' column.

Half Full Activity - Results and Debrief

Screen Shot 2013-07-10 at 7.07.48 AM

If you haven't yet participated, visit and see what it's all about. If I've ever written a post that has a spoiler, it's this one.

First, the background.

"A great application of fractions is in cooking."

At a presentation I gave a few months ago, I polled the group for applications of fractions. As I expected, cooking came up. I had coyly included this on the next slide because I knew it would be mentioned, and because I wanted the opportunity to call BS.

While it is true that cooking is probably the most common activity where people see fractions, the operations people learn in school are never really used in that context. In a math textbook, using fractions looks like this:

Screen Shot 2013-07-10 at 7.15.13 AM

In the kitchen, it looks more like this:

A recipe calls for half of a cup of flour, but you only have a 1 cup measure, and to be annoying, let's say a 1/4 cup as well. Is it likely that a person will actually fill up two 1/4 cups with flour to measure it out exactly? It's certainly possible. I would bet that in an effort to save time (and avoid the stress that is common to having to recall math from grade school) most people would just fill up the measuring cup halfway. This is a triumph of one's intuition to the benefits associated with using a more mathematical methods. In all likelihood, the recipe will turn out just fine.

As I argued in a previous post, this is why most people say they haven't needed the math they learned in school in the real world. Intuition and experience serve much better (in their eyes) than the tools they learned to use.

My counterargument is that while relying on human intuition might be easy, intuition can also be wrong. The mathematical tools help provide answers in situations where that intuition might be off and allows the error of intuition to be quantified. The first step is showing how close one's intuition is to the correct answer, and how a large group of people might share that incorrect intuition.

Thus, the idea for half full was born.

The results after 791 submissions: (Links to the graphs on my new fave are at the bottom of the post.)


Screen Shot 2013-07-10 at 7.42.14 AM
Mean = 50.07, Standard Deviation = 8.049


Screen Shot 2013-07-10 at 7.47.10 AM
Mean = 42.30, Standard Deviation = 9.967


Screen Shot 2013-07-10 at 7.50.55 AM
Mean = 48.48, Standard Deviation = 14.90


Screen Shot 2013-07-10 at 7.55.34 AM
Mean = 51.16, Standard Deviation = 16.93

First impressions:

  • With the exception of the trapezoid, the mean is right on the money. Seems to be a good example of wisdom of the crowd in action.
  • As expected, people were pretty good at estimating the middle of a rectangle. The consistency (standard deviation) was about the same between the rectangle and the trapezoid, though most people pegged the half-way mark lower than it actually was on the trapezoid. This variation increased with the parabola.
  • Some people clicked through all four without changing anything, thus the group of white lines close to the left end in each set of results. Slackers.
  • Some people clearly went to the pages with the percentage shown, found the correct location, and then resubmitted their answers. I know this both because I have seen the raw data and know the answers, and because there is a peak in the trapezoid results where a calculation error incorrectly read '50%'.

    I find this simultaneously hilarious, adorable, and enlightening as to the engagement level of the activity.

Second Impressions

  • As expected, people are pretty good at estimating percentage when the cross section is uniform. This changes quickly when the cross section is not uniform, and even more quickly when a curve is involved. Let's look at that measuring cup again:

    In a cooking context, being off doesn't matter that much with an experienced cook, who is able to get everything to balance out in the end. My grandmother rarely used any measuring tools, much to the dismay of anyone trying to learn a recipe from her purely from observing her in the kitchen. The variation inherent in doing this might be what it means to cook with love.

  • My dad mentioned the idea of providing a score and a scoreboard for each person participating. I like the idea, and thought about it before making this public, but decided not to do so for two reasons. One, I was excited about this and wanted to get it out. Two, I knew there would probably be some gaming the system based on resubmitting answers. This could have been prevented through programming, but again, it wasn't my priority.
  • Jared (@jaredcosulich) suggested showing the percentage before submitting and moving on to the next shape. This would be cool, and might be something I can change in a later revision. I wanted to get all four numbers submitted for each user before showing how close that user was in each case.
  • Anyone who wants to do further analysis can check out the raw data in the link below. Something to think about : The first 550 entries or so were from my announcement on Twitter. At that point, I also let the cat out of the bag on Facebook. It would be interesting to see if there are any data differences between what is likely a math teacher community (Twitter) and a more general population.

This activity (along with the Do You Know Blue) along with the amazing work that Dave Major has done, suggests a three act structure that builds on Dan Meyer's original three act sequence. It starts with the same basic premise of Act 1 - a simple, engaging, and non-threatening activity that gets students to make a guess. The new part (1B?) is a phase that allows the student to play with that guess and get feedback on how it relates to the system/situation/problem. The student can get some intuition on the problem or situation by playing with it (a la color swatches in Do You Know Blue or the second part of Half Full). This act is also inherently social in that students easily share and see the work of other students real time.

The final part of this Act 1 is the posing of a problem that now twists things around. For Half Full, it was this:

Screen Shot 2013-07-10 at 8.37.30 AM

Now that the students are invested (if the task is sufficiently engaging) and have some intuition (without the formalism and abstraction baggage that comes with mathematical tools in school), this problem has a bit more meaning. It's like a second Act 1 but contained within the original problem. It allows for a drier or more abstract original problem with the intuition and experience acting as a scaffold to help the student along.

This deserves a separate post to really figure out how this might work. It's clear that this is a strength of the digital medium that cannot be efficiently done without technology.

I also realize that I haven't talked at all about that final page in my activity and the data - that will come later.

A big thank you to Dan Meyer for his notes in helping improve the UI and UX for the whole activity, and to Dave Major for his experience and advice in translating Dan's suggestions into code.



The histograms were all made using If you haven't played around with this yet, you need to do so right away.





Raw Data for the results presented can be found at this Google Spreadsheet.

Technical Details

  • Server side stuff done using the Bottle Framework.
  • Client side done using Javascript, jQuery, jQueryUI, Raphael for graphics, and JSONP.
  • I learned a lot of the mechanics of getting data through JSONP from Chapter 6 of Head First HTML5 Programming. If you want to learn how to make this type of tool for yourself, I really like the style of the Head First series.
  • Hosting for the app is through WebFaction.
  • Code for the activity can be found here at Github.

Visiting the Museum of Math in NYC

I was reminded by John Burk that the Museum of Math had been opened since I was last in New York. A good friend was in town on a weekend getaway from summer coursework in preparation for starting teaching math this fall. It was the perfect motivation to make time for visiting the museum sooner rather than later.




I was really impressed with the museum when I first walked in. The message that mathematics can be a language of play and exploration is emphasized from the experiential exhibits on the top floor. From pulling a cart that rolled across various solids to working to tesselate a hyperbolic surface with polygons, there is lots to touch, pull, and do to play with mathematical concepts. Without a doubt, these activities "stimulate inquiry, spark curiosity, and reveal the wonders of mathematics" as the mission statement aspires to do. The general organizing idea of the activities on the top floor is to provide a really interesting, perplexing object or concept to play with, and then dip into the mathematics surrounding that play for those that are interested. One could miss the kiosks that explain the underlying concepts and still feel satisfied with the overall experience.


The museum had a good mix of activities from different fields of mathematics. Most exhibits were built around a visually defined task that required little explanation in order to start playing with it and understanding its rules through that play. For me, the activities with the largest initial investment/perplexity ratio were the line of lasers that made it possible to see the cross section of a shape and the wall that generates a fractal tree from images of you and a neighbor. Building objects that roll along a particular path was also incredibly engaging for me.

The challenge for a museum like this is to be intriguing without being tricky or elitist. Given that many people experience anxiety about mathematics for all sorts of reasons, I am absolutely sure that the museum's exhibit designers worked extremely hard to make the bar for entry for these activities as low as possible with a high ceiling. To this end, I think the museum has done a fabulous job. Most exhibits make clear what they are all about and give visible feedback on a visitor's progress toward reaching the goal. The only area for growth that jumped out at me was an expansion of the role of the staff circulating among the exhibits. They were extremely knowledgeable of the content of the exhibits and were really excited to share what they knew about them. Even as someone that enjoys mathematics, however, there were some exhibits that left me wondering whether I was playing with it correctly. I saw the Shape Ranger exhibit as a puzzle to figure out, for example. I could see others leaving it when the activity didn't clearly define visually what the score of the activity represented. I envision an expanded role of staff members helping nudge visitors to understand the premises of the more abstract exhibits through careful questioning and good examples of how to play.

This was not a deal breaker for me, however, and didn't seem to be for the crowd of people in attendance. The number of smiling kids and adults enjoying themselves is clearly the best indication that the museum is doing a great job of fulfilling its mission. This museum has the same sort of open-ended atmosphere that the Exploratorium in San Francisco creates for its visitors, and that puts it in very respectable company.