What a cool idea: a free software package that simulates a planetarium on your PC!
Preloaded with 600,000 stars, all you have to do is set your coordinates and start gazing!
You can search by name for various heavenly bodies, map the constellations, and even get close-ups of observable planets!
I’m not sure if this technically qualifies as a harmonograph, but the images produced by this cool apparatus definitely remind me of Lissajous curves.
The video of this device in action, from Robert Howsare, is definitely worth a look.
An interesting challenge would be to come up with the parametric equations of these curves based on the angular velocity of the two turntables, much like what can be done with actual harmonographs.
This is an interesting article by Jonah Lehrer of the Wall Street Journal about the limits of standardized testing.
https://www.wsj.com/articles/SB10001424052748704471904576230931647955902
Lehrer discusses the results of a study from the 1980s in which psychologist Paul Sackett attempted to measure the speed of supermarket cashiers. A short “check-out test” was developed which involved scanning a small number of items. The test was administered, and resulted in a list of the fastest cashiers.
What is interesting is that when Sackett compared the results of the test with long-term data collected by the electronic scanning systems, there was a surprisingly weak correlation between the results of the speed test with the data from regular usage. That is to say, there was no real connection between being fast on the test and being a fast on a day-to-day basis.
Sackett’s misconception, and perhaps one held by many, is that there is a natural correlation between maximum performance (that on a short test) and typical performance (that is, under normal, day-to-day circumstances). Tests like the SAT, the GRE, and other high stakes tests, are tests of maximum performance. Our educational system relies on these more and more, but are we sure they measure what we assume they measure?
Lehrer points out that individual success is determined more by character traits like perseverance and self-control, but of course, it’s hard to capture that in a timed, multiple choice exam.
Finding 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.
This is a nice collection of links to profiles and biographies of women in mathematics.
http://darkwing.uoregon.edu/~wmnmath/People/index.html
In addition to resources about historical figures, this list contains references to many living mathematicians.
And here is another nice collection of biographies of female mathematicians from the University of St. Andrews.