Fibonacci Flushers

While travelling in Europe, I became fascinated with the variety of toilet-flushing mechanisms I encountered.  The typical toilet had a low-flow / high-flow option (which I imagine saves a great deal of water in the long run) , and a lot of creativity emerged in the way this two-flush system was implemented.

While documenting the many ways to flush, I found this rectangular model oddly familiar and appealing.

And then it hit me:  this looks like the golden rectangle!

The golden ratio has long been used by artists and architects to create aesthetically pleasing work.  It is, after all, the divine proportion.  Could it be that these toilet-makers took their cues from the masters of art and math?  I had to find out.

I dropped my image into Geogebra and took some measurements.

The total length of the rectangle divided by its height is around 1.71.  So, it’s not quite the golden ratio, but it’s pretty close.  This flush-design is about 90% divine, I suppose.

Maybe their next design will be closer to the perfect proportion.

Differentiating Consumer Quantities

One thing I’d like to see more of in math education is an appreciation for practical consumer math.  For example, suppose your favorite cereal brand unceremoniously starts arriving in a smaller box.

Students should learn to be on the lookout for things like this.  They should develop a quantitative curiosity, exploring the various ways that quantity can be disguised.

And ultimately be able to put together a real quantitative analysis that helps them make good decisions.

This is math that everyone can use!

The Write Angle for Teaching Math: Why Write in Math Class?

Math WritingFinding ways to get students to write about mathematics has played a pivotal role in my development and growth as a math teacher.  Mathematical writing challenges students to express their ideas clearly and efficiently; it forces students to stop thinking of mathematics as merely equations and answers; and it opens up a new and unexpected dialogue between math teacher and student.

I have always found great value and pleasure in writing.  It is a valuable skill, a necessary tool of scholarship, and a powerful creative outlet.  And now I see its value as a math teacher.  The more my students write, the more useful and interesting we all find it.

In this post, I’ll first address the question “Why Write in Math Class?”.

Why Write in Math Class?

There are infinitely many good reasons to write.  I’ll offer three that have been on my mind lately.

1)  Writing is a fundamental mathematical skill

Many people might not be aware of it, and many might not admit it, but good writing is a fundamental mathematical skill.  A proof isn’t a proof unless others understand it, and that can’t happen if it isn’t written clearly and concisely.  Also, it’s great when we find the right answer in a math problem, but as most teachers know, it’s usually more important to understand the problem-solving process than to get the right answer.  Good writing skills help narrate and record that process, and make that process available for teachers and peers to understand.

2)  Writing is an indispensible professional skill

I’ve had many different jobs in my life, and being a good writer made me more effective at all of them.  Whether designing technology systems, meeting with clients or consultants, talking through project specifications, or working on a team, being able to document and communicate effectively about the process gave me an edge.   Writing about mathematical ideas and procedures can be hard, but it’s great training for thinking and communicating about the kinds of open-ended problems students will face in the real world.

3)  Writing helps me understand my students better

By regularly interacting with my students through writing, I get to know them in a significantly different way than through their work on exams and homework.  Through various writing activities, I can develop a better sense of what kinds of math problems they like, what kinds of problem-solving techniques they are most comfortable with, and of course, what kinds of ideas are difficult for them to consume.  Getting a different look at how my students think mathematically is incredibly valuable as a teacher, and it can be extremely fun, too!  Giving students the chance to think and write creatively about math almost always produces something unexpectedly wonderful!

For more resources, see my Writing in Math Class page.

Statistical Baseball Predictions

As the 2012 Major League Baseball season gets under way, it’s a good time to check in on the predictions of Bruce Bukiet, mathematics professor at the New Jersey Institute of Technology.

http://m.njit.edu/~bukiet/baseball/baseball.html

Using the performance data from all of the expected players, Bukiet applies a mathematical model to predict the final win-loss standings for every team in the league.  Last year, Bukiet’s model correctly identified six of the eight playoff teams.

Unlike most predictors (my statistics friends would probably prefer I use the term projectors), Bukiet does not seem shy about comparing his past predictions to the actual results.

An interesting mathematical question would be “How can we measure how accurate these predictions really are?”

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