# Category Archives: Geometry

January 3, 2012, 1:59 pm

# Three things I learned about teaching by taking a short course

One of the main reasons I’m at the AMS/MAA Joint Meetings this week is to take an MAA short course on discrete and computational geometry. That course is wrapping up this afternoon, and it’s been a good experience. I came into the course with zero knowledge of computational geometry, a within-$$\epsilon$$-of-zero knowledge of algorithms, and an extremely rusty skill set in topology. But I’m coming out with an appreciation for this subject and, hopefully, a basis for pushing farther into the field and eventually contributing something new.

Teachers ought to take courses more often. Apart from being intellectually satisfying, it’s useful to be on the receiving end of academic teaching in one’s own discipline every now and then because it helps you remember what it’s like to be in the shoes of your own students. Here are some things I’ve re-learned about being a student in a math…

May 26, 2011, 8:01 pm

As part of preparing for our impending move from Indy to Grand Rapids, my family and I have made a couple of visits to the area. These by necessity combine business with pleasure, since our three kids (ages 2, 5, and 7) don’t handle extended amounts of business well. On the last visit, we spent some time at the Grand Rapids Childrens Museum, the second floor of which is full of stuff that could occupy children — and mathematicians — for hours. This “exhibit” was, for me, one of the most evocative. Have a look:

I asked this on Twitter a few days ago, but I’ll repost it here: In the spirit of Dan Meyer’s Any Questions? meme, what questions come to mind as you watch this? Particularly math, physics, etc. questions.

One other thing — just after I wrapped up the video on this, someone put one of the little discs rolling on the turntable…

May 28, 2010, 11:39 am

# Summer plans

I’m still in recovery mode from this past semester, which seemed somehow to be brutal for pretty much everyone I know in this business. But something that always helps me in this phase is thinking about what I get to do with the much lighter schedule that summertime affords. Here’s a rundown.

Mostly this summer I will be spending time with my family. On Mondays and Fridays, I’ll be home with my two daughters. On Wednesdays I’ll have them plus my 16-month old son, plus my wife will have that day off. On Tuesdays it’ll be just the boy and me. So I plan lots of trips to the zoo, the various parks around here, and so on.

I still have plenty of time to work, and I have a few projects for the summer.

First, I need to get ready for my Geometry class this fall. I am making the move from Geometer’s Sketchpad to Geogebra this fall, and although I took a minicourse at the ICTCM on Geogebra, I…

January 20, 2010, 10:31 am

# Courses and "something extra"

Some of the most valuable courses I took while I was in school were so because, in addition to learning a specific body of content (and having it taught well), I picked up something extra along the way that turned out to be just as cool or valuable as the course material itself. Examples:

• I was a psychology major at the beginning of my undergraduate years and made it into the senior-level experiment design course as a sophomore. In that course I learned how to use SPSS (on an Apple IIe!). That was an “extra” that I really enjoyed, perhaps moreso than the experiment I designed. (I wish I still knew how to use it.)
• In my graduate school differential geometry class (I think that was in 1995), we used Mathematica to plot torus knots and study their curvature and torsion. Learning Mathematica and how to use it for mathematical investigations were the “something extra” that I took from the …

December 31, 2009, 5:30 am

# How to convert a "backwards" proof into a "forwards" proof

Dave Richeson at Division By Zero wrote recently about a “proof technique” for proving equalities or inequalities that is far too common: Starting with the equality to be proven and working backwards to end at a true statement. This is a technique that is almost a valid way to prove things, but it contains — and engenders — serious flaws in logic and the concept of proof that can really get students into trouble later on.

I left a comment there that spells out my  feelings about why this technique is bad. What I wanted to focus on here is something I also mentioned in the comments, which was that it’s so easy to take a “backwards” proof and turn it into a “forwards” one that there’s no reason not to do it.

Take the following problem: Prove that, for all natural numbers $$n$$,

$$1 + 2 + 2^2 + \cdots + 2^n = 2^{n+1} – 1$$

This is a standard exercise using mathematical…

December 19, 2008, 9:54 am

# What does academic rigor look like?

I got an email from a fellow edu-blogger a couple of days ago asking for my input on the subject of academic rigor. Specifically this person asked:

Is the quest for more rigor an issue for you? Is it good, bad, meaningless? What does rigorous teaching look like in your classroom?

I hope she doesn’t mind my sharing the answer, because after writing it I thought it’d make a good blog post. I said:

For me, “rigor” in the context of intellectual work refers to thoroughness, carefulness, and right understanding of the material being learned. Rigor is to academic work what careful practice and nuanced performance is to musical performance, and what intense and committed play is to athletic performance. When we talk about a “rigorous course” in something, it’s a course that examines details, insists on diligent and scrupulous study and performance, and doesn’t settle for a mild or  informal…

August 13, 2008, 5:15 pm

# What are some fatal errors in proofs?

The video post from the other day about handling ungraded homework assignments went so well that I thought I’d let you all have another crack and designing my courses for me! This time, I have a question about really bad mistakes that can be made in a proof.

One correction to the video — the rubric I am developing for proof grading gives scores of 0, 2, 4, 6, 8, or 10. A “0″ is a proof that simply isn’t handed in at all. And any proof that shows serious effort and a modicum of correctness will get at least a 4. I am reserving the grade of “2″ for proofs that commit any of the “fatal errors” I describe (and solicit) in the video.

July 25, 2008, 2:36 pm

# Geometry first

Jackie at Continuities is wondering whether the usual path through high school mathematics — Algebra I, then Geometry, then Algebra II, etc. — is out of order, and whether geometry ought to come first:

As far as I can tell the only difference between Alg II and Pre-Calc is that trig is taught during Pre-Calc and Pre-Calc introduces the concept of the limit. Functions are developed a bit more rigorously too.

The first semester of Algebra II is mostly a repeat of Algebra I as they’ve forgotten it with the year “off” during Geometry.

Why not then teach Geometry first? I’m talking about plane and solid geometry with an emphasis on reasoning, and right angle trig. Obviously there would need to be some supplementing needed (work with radicals, solving equations). Most students have “seen” the solving of equations in 8th grade (Have they mastered it? No, of course not).

I…

March 10, 2008, 8:16 pm

# Suggestions for an adult math convert?

I got an email this afternoon from a reader who is interested in learning mathematics — as an adult, post-college. The reader has an advanced degree in a humanities discipline and never studied mathematics, but recently he’s become interested in learning and is looking for a place to start.

I recommended The Mathematical Experience by Davis and Hirsch, How to Solve It by Polya, and any good college-level textbook in geometry (like Greenberg, or for a humanities person perhaps Henderson). I felt like these three books give an ample and accessible start at — respectively — the big picture and history of the discipline, the methodology of mathematicians, and a first step into actual mathematical content.

But what I thought this was an interesting question, and I wonder if the other readers out there would have similar suggestions for books, articles, movies or documentaries… anything…

November 20, 2007, 12:25 pm

# Greenberg geometry text updated

I got a nice surprise in the mail this morning — a review copy of the fourth edition of Marvin Greenberg’s classic text Euclidean and Non-Euclidean Geometries. It seems like this book has been in the third edition since time immemorial. I used the third edition in my first year of teaching after graduate school, 10 years ago, and loved the depth and clarity of the writing. That much seems not to have changed. There are some significant rearrangements and updates to the material, and overall the book just looks a lot nicer (And the color scheme matches my blog, to boot!) There don’t seem to be a lot of good intro-level geometry texts out there — and there are a lot of bad ones — so a new Greenberg is a nice early Christmas present. It’s the kind of book that makes you want to sit down and work through it just so you can learn geometry from back to front.

Freeman textbooks are on a…

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