Archive for the ‘MathStats’ Category

Using Math to Understand the Future

Futurist Peter Bishop was one of the keynote presenters at MichMATYC 2010 this year.  He spoke to us about what a futurist does, and shifted our paradigms about how to look at data trends to one that is more mindful of the cone of plausibility.  Don’t know what that is? Well, watch the talk!  If you don’t have a lot of time, then watch the last 20 minutes.  You can also get the slides here.

If you’re interested in the other sessions at MichMATYC 2010, many of the slide decks are posted in the Resources Tab of the MichMATYC 2010 Website.

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Interactive Simulations from PhET

PhET (Physics Education Technology) consists of a group of scientists, software engineers, and science educators from the University of Colorado at Boulder, who are striving to create effective, interactive learning tools.  Their work spans the fields of physics, biology, chemistry, earth science, and mathematics.  Much care has been taken in the design of each simulation.  The developers use a research-based  strategy to implement the most effective visual cues for learning.  User-interviews are held routinely.   Members of PhET have learned that animated responses are effective, as well as,  the use of a “click and drag interface”.  There’s more:  each simulation comes with lesson plans that have been submitted by instructors.  The simulations are free and require flash and java to run.

Here is an applet on estimation:


Here is Calculus Grapher, which shows the relationship between a function and its derivative.


And a Plinko Probability applet that shows a histogram that approaches the Binomial Distribution.


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Mindmaps for Learning

I’ve been using a web-app called Mindomo for about two years now. With it I am able to map out ideas and create interactive sets of resources in a non-linear fashion. You may have seen some of my resources or been in a presentation where I used one of these maps:


I think that using these interactive maps gives three main advantages:

  1. If you present with a map, you are no longer forced into a linear presentation and can easily respond and adapt to audience questions.
  2. The audience can play along during the presentation, wandering off to explore the areas of the map that interest them most.  This is the same idea behind Edward Tufte’s “supergraphic” – a data-rich resource that the audience becomes engaged with, each person in their own context.
  3. The process of creating a mindmap helps to organize resources and ideas, think of applications to ideas, fosters thinking about comparisons and contrasts, and helps you to see the holes where information or resources are missing, all in a very visual manner.

It is this third item that has me particularly intrigued.  When I begin building a new presentation, I now find it helpful to organize a mindmap as one of the first activities I do.  The process of building the map teaches me more than I would ever learn on my own.

This year I’m planning to put this idea to the student test and have each student in my MET class (Math for Elementary Teachers) create a Mindomo mindmap for one of the units as one of their four Learning Projects.  The Mindomo accounts are free (for up to 6 maps) as long as you are willing to live with a 1-inch wide strip of advertising on the right-hand side.

I had been stressing over the need to create a tutorial video, but one of our workshop participants (Rose Jenkins of Teching Up) has created a fabulous video on getting started with Mindomo (click here for her tutorial).  I’m planning on just sending my students right to Rose’s video for their introductory tutorial on using Mindomo.mindomo2

Rose has also got an interesting idea for pushing out a partially-created mindmap to her statistics students, and then asking them to add the appropriate resources and annotations to the map (Read her post, Mapping Out Math).  It was a little tricky to figure out HOW to create a map and then share it to students in a way that makes each copy their own, but Rose made a tutorial about THAT too! (click here for the tutorial about sharing maps)

Kudos to Rose for taking charge of a set of tutorials that really needed to be made!

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Mathematics of Coercion


Bruce Bueno de Mesquita, a consultant to the CIA and DOD, uses mathematical analysis to predict the outcome of “messy” human events in this 2009 TED Talk: Three predictions on the future of Iran, and the math to back it up.  He claims that we can use mathematics to predict the outcomes of complex negotiations or situations involving coercion (everything that has to do with politics and business).

His modeling is based in Game Theory, which (he says) is based on three assumptions that (1) people are rationally self-interested, (2) that people have values and beliefs, and (3) people face limitations.  The CIA verifies the predictive ability of the model, claiming it is correct 90% of the time even when the experts are wrong.

To build a model of the outcomes, he says he need to know (1) Who has a stake in the decision? (2) What do they say they want? (3) How focused are they on one issue compared to other issues? (4) How much persuasive influence could they exert?  Using this, we can predict behavior by assuming that everybody cares about two things: the outcome (effect on their career) and the credit (ego).  In the model, you must be able to estimate people’s choices, chances they are willing to take, values, and beliefs about other people.  Believe it or not, history is not necessary for the model.

Other than the mention of mathematics and a really general look at game theory, there was not a lot of mathematics in this talk.  There was one concrete mathematical example that you might be able to utilize in one of your classes (especially if you teach a little combinatorics as part of Probability and Statistics or Liberal Arts Mathematics):

To build a model that predicts the outcome of complicated social events, we need to look at the interactions between all of the people who have input in the decision-making (the influencers).  The number of interactions between n influencers is n!  If we double the number of influencers in the interaction, does that double the number of interactions?  (to use this example, play from 4:24 to about 7:15)

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A Random Walk (so to speak)

Here is a video that can be understood by all levels of mathematics students called the “Law of Large Numbers.” This one shows, in several situations, how the center of gravity of randomly-moving particles becomes more stable as the number of particles increases.

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TED features Mathemagics

This little 15-minute TedTalk, Arthur Benjamin races calculators from the audience doing the calculations in his head. He is very entertaining and I think students would enjoy the presentation, since they are often under the impression that nobody has every done calculations without a calculator. He squares a random 4-digit number from the audience in his head just as fast as the calculators.

For those of you that teach probablity, he does a neat trick guessing the missing digits of 7-digit numbers (this is somewhere around the 7-minute mark).

For elementary ed, there is a nice “number” on guessing the day of the week when someone was born based on the birth date (8-9 minute mark).

For you algebra teachers, he does the square of a 5-digit number, going through the thought process out loud (starting around the 11-minute mark). Cleverly, he squares the number by breaking it into a binomial:
(57683)2 = (57000+683)2 = 570002 + 57000*683*2 + 6832

I’ll let you work on that one in your head…

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Math Games at Interactivate

I stumbled across a nice website last week called Interactivate with 140 (or so) interactive Java-based activities for algebra, geometry, probability, statistics, etc. Many of the activities are modifications of other activities, but still there are at least 30 unique game-based activities here to help your students learn.

I particularly liked “Algebra Four” (a play on the game Connect Four). I am teaching my algebra students all about solving equations right now and this would give them some good practice. The student can choose the level of difficulty (one-step, two-step, distributive, etc.). So conceivably, a student could first play at the one-step level, then the two-step level, then add the distributive property, and work their way up. This is a two-player game, which is really the only drawback, as a student at home would have to play against themself (or convince someone else to play an algebra game with them… hmm… unlikely). I do like the timer, which would encourage the student to get faster at solving equations. And if a student doesn’t want to play against the timer, it could just be set for a high time.

Another nice game here is the “function machine” like we’ve seen in textbooks, only this one is really a machine where you (or the student) inputs values, and it (the machine) processes the values and outputs them. The “game” here is to guess the function. I could see using this one in a classroom when we talk about function notation, and writing out the function notation process of each “guess” and “answer” to the side of the projected machine. A small improvement on this game would be to show the function notation to the side and then run a version that would let the all-knowing instructor input “x” at the end of the game, to show that if you input “x” or “a” into the function notation, the function notation shows you exactly what happens during the processing.

One last gem from this site is Area Explorer, which shows the student a graph of connected, shaded squares, and asks the student to find the area and perimeter. What I like is that it emphasizes the underlying principle behind area and perimeter (area is counting unit squares in the interior of the figure and perimeter is counting unit lines on the perimeter of the figure). Our students today seem to often miss this concept alltogether and just want to boil everything down to a formula, so I think this would be a great activity to have students try on their own.

All in all… kudos to this site for creating some really nice interactive materials!

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