# Category Archives: Geometry

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 and it did about a dozen graceful, perfect three-point hypocycloids before falling off the table.

Filed under Geometry, Math, Problem Solving

## 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 still need to work on my skills before I teach with it. Also, I need to figure out exactly what I am going to teach. I’m going to be using Euclid’s Elements as the textbook for the course, eschewing commercial textbooks for both monetary and educational reasons. But I’m not totally sure what I’m going to have students do, exactly. So I’ll be reading through the Elements and possibly thinking out loud here on the blog about how to incorporate a 2000-year old mathematical work with modern open-source dynamic geometry software in an engaged classroom. I’m calling it “ancient-future geometry”, whatever it turns out being.

Second, I’ll be working on our dual-degree Engineering program to try and make it a little easier to schedule and complete. This is hard-core administrative stuff, interesting to nobody but a select few geeks like me.

Third, I’ll be working to further my programming skills with MATLAB and Python. I picked up a lot of MATLAB programming to get ready for the course this past semester, but that seemed only to highlight how much more I needed to learn. And I watched enough of this MIT computing course over Christmas break that I want to do the whole thing now that I have some time.

Fourth, I’ll be attending the American Society for Engineering Education conference in Louisville next month. Part of that experience is a day-long minicourse titled “Getting Started in Engineering Education Research”. I’ll be taking my participation in that minicourse as the kickoff to a concerted effort to get into the scholarship of teaching and learning. Along with the minicourse I’ll be reading through some seminal SoTL articles this summer, and probably blogging what I’m thinking.

Fifth, and finally, I’ll be mapping out some incursions of the inverted classroom model in my Calculus course this fall. More on that later as well.

For now, my family and I are heading out to Tennessee on vacation to visit family and hang out. I’ll be off the grid for a week or so. Enjoy yourselves and stay tuned!

## 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 course. Sadly, the extras have outlived my knowledge of differential geometry. (Sorry, Dr. Ratcliffe.)
• In the second semester of my graduate school intro abstract algebra class, my prof gave us an assignment to write a computer program to calculate information about certain kinds of rings. This was a small assignment in a class full of big ideas, but I had to go back and re-learn my Pascal in order to write the program, and the idea of writing computer programs to do algebra was a great “extra” that again has stuck with me.

Today I really like to build in an “extra”, usually having something to do with technology, into every course I teach. In calculus, my students learn Winplot, Excel, and Wolfram|Alpha as part of the course. In linear algebra this year I am introducing just enough MATLAB to be dangerous. I use Geometers Sketchpad in my upper-level geometry class, and one former student became so enamored with the software that he started using it for everything, and is now considered the go-to technology person in the school where he teaches. In an independent study I am doing with one of my students on finite fields, I’m having him learn SAGE and do some programming with it. These “extras” often provide an element of fun and applicability to the material, which might be considered dry or monotonous if it’s the only thing you do in the class.

What kinds of “extras” were standouts for you in your coursework? If you’re a teacher, what kinds of “extras” are you using, or would you like to use, in your classes?

## 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 induction. The induction step is trivial; focus on the induction step. Here we assume that $1 + 2 + 2^2 + \cdots + 2^k = 2^{k+1} - 1$ for all natural numbers less than or equal to $k$ and then prove:

$1 + 2 + 2^2 + \cdots + 2^{k+1} = 2^{k+2} - 1$

Here we have to prove two expressions are equal. Here’s what the typical “backwards” proof would look like (click to enlarge):

A student may well come up with this as his/her proof. It’s not a bad initial draft of a proof. Everything we need to make a totally correct proof is here. But the backwards-ness of it — all stemming from the first line, where we have assumed what we are trying to prove — needs fixing. Here’s how.

First, note that all the important and correct mathematical steps are taking place on the left-hand sides of the equations, and the right-hand sides are the problem here. So delete all the right-hand sides of the equals signs and the final equals sign.

Next, since the problem with the original proof was that we started with an “equation” that was not known to be true,  eliminate any step that involved doing something to both sides. That would be line 4 in this proof. This might involve some re-working of the steps, in this case the trivial task of re-introducing a -1 in the final steps:

You could reverse these first two steps — eliminate all “both sides” actions and then get rid of the left-hand sides.

Then, we need to make it look nice. So for n = 1 to the end, move the $(n+1)^\mathrm{st}$ left-hand side and justification to the $n^\mathrm{th}$ right-hand side:

Now we have a correct proof that does not start by assuming the conclusion. It’s shorter, too. Really the main thing wrong with the “backwards” proof is the repeated — and, notice, unnecessary — assertion that everything is equal to the final expression. Remove that assertion and the correct “forwards” proof is basically right there looking at you.

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Filed under Abstract algebra, Geometry, Math, Problem Solving

## 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 contact with the key ideas.

Example: A rigorous course in geometry goes beyond just memorization of formulas, applications to simple geometric exercises, and “hand-waving” attempts at proof. Instead, such a course treats details as important, the ability to explain on a deep level the truth of formulas and results as a key goal for students, and sets a high bar for the exactitude of mathematical arguments. Euclid’s “Elements” for example is the prototype of the rigorous treatment of geometry. It’s not a difficult work to understand, necessarily; in fact one of the enduring qualities of the Elements is the clarity and precision of not only each individual proposition but also in how the overall collection of propositions fits together. By contrast, many modern books on geometry are highly non-rigorous, omitting details, putting theorems out of order, and defining a proof as a “reasonable explanation” only.

Is rigor good? It depends on the audience and the goals of the class. When I teach a geometry course for junior and senior Math Education majors, rigor is of the utmost importance because I want those pre-service teachers to go into their classrooms with tough, precise minds for the sake of their students. If I were to teach a geometry class for fifth-graders, on the other hand, I think rigor would obscure the subject, and I would depend a lot more on intuitive explanations and perhaps constructivist techniques for discovering key ideas in geometry and save rigorous proofs for another day. Similarly, when I teach calculus at my college, the audience is about 50% business majors, and so we designed the course not to cover much theory. This is not a rigorous treatment of calculus, but it is more effective for the students than if we included the epsilon-delta proofs and what not.

The quest for more rigor is most important in the post-calculus courses I teach (geometry, abstract algebra, and introduction to proof). These are subject areas where precision and detail-orientation are essential for a complete understanding of the material. Students are not allowed to give examples when a proof is called for, and I nitpick every little thing in their proofs up to and including the choice of punctuation and prepositions. [If any of the five who took this course from me this past semester are reading this, feel free to chime in with an “Amen.” – RT.] At the calculus level and below, I lay off on the theory but the rigor in the course comes from getting details of mechanical calculations right. And this is a big issue, because students in high school are generally taught only to produce a correct answer, not a clear and detailed solution. I am on a mission to make sure students can not only get right answers but also communicate their methods in a clear and audience-appropriate way, and that’s what “rigor” looks like there.

[After-the-fact note: To clarify, in calculus I insist on details in mechanical calculations but also on the details of processes and in paying attention to nuances in solving application-style problems. For example, students know that if you just set $f''(x) = 0$ and solve for $x$, that this doesn’t give you an inflection point; and in an optimization problem you can’t just find the critical number of the model function, you must also test it with the First or Second Derivative Test to see if it really yields a maximum. Or at least, they don’t complain when they forget to do it and I take off points!]

I have two kids, ages 3 and 5. (There’s a third one on the way in three weeks, but that’s another story!) I’m pretty rigorous with them, too — when the 5-year old says “Mimi comed to our house this weekend” I correct her grammar, and she gets it right the next time. You have to do it in a gentle way, but getting details right now will help them get the more complicated things right later. If I were to project myself out of higher ed and into the K-12 sphere I could see my teaching being “rigorous” in that kind of way — insisting that kids get the details right and not gloss over things, but doing it in a lovingly persistent way. I wish more K-12 teachers would do this, though, because it’s obvious from my freshmen in the last 4-5 years that this isn’t happening (or at least it’s not sticking).

[Final note: That last sentence isn’t a slam on either my freshmen, who were really quite excellent this year in calculus, or their teachers. It’s an observation, and I stand by it. I can show you their work at the beginning of the semester if you don’t believe me. Why this kind of “rigor” is not sticking with them is something I can’t fully explain because I don’t know what was going on with them in high school. Is it them? Is it their teachers? Is it the system? Is it the preponderance of standardized testing, which makes rigor more or less irrelevant? Comment!]

Filed under Calculus, Education, Geometry, High school, Teaching

## 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.

Filed under Education, Geometry, Grading, Math, Problem Solving, Teaching

## 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 completely agree. It seems to me that the reason Geometry gets sandwiched between Algebra I and Algebra II is that people want to use algebra concepts in geometry. But I think that doesn’t necessarily have to be the case. If you look at the source — Euclid’s Elements — you will not find a drop of algebra in it. All the concepts that we, today, would label as being algebra or number theory or what-have-you are just latter-day retrofittings of Euclid’s ideas. Euclid himself phrased everything in terms of geometry, with the algebra and number theory done in terms of commensurable lengths and other geometric terminology. I wouldn’t go so far as to say Euclid knew nothing of algebra or number theory, but if you follow Euclid you don’t need algebra, as we know it, at all in your geometry.

That would leave a geometry course that is mainly about logical reasoning, cogent organization of facts, objective deductions from data, and clear exposition of an argument. One might add to this list the art/craft of forming conjectures from experimentation and then writing an argument in favor of your conjectures, which is astoundingly simple these days thanks to Geometers Sketchpad and other fun, low-cost dynamic geometry software packages. (My students who use Sketchpad in their student teaching report, to a person, that students really turn on when they use Sketchpad and do some very good mathematics, for 8th-9th graders.) This sounds like precisely the kind of foundation, and buffer zone, that students need to acquire before tackling algebra with a view towards understanding how it works rather than just memorizing facts. (Indeed, memorizing facts in algebra is quite hard unless you understand why the facts work.)

Of course, if you ask ten people whether they liked their geometry class in school, eight will probably say “no” and seven of those eight will say it was because of “proofs”. But I wonder what that really means. Perhaps, having gotten a taste of equation solving in algebra and therefore acquiring the “there’s only one right answer and I have 30 seconds to find it” mentality about mathematics, they are spoiled for ever encountering mathematics as it really is (which is something that geometry is a lot closer to than algebra I). Perhaps they had a geometry teacher who was not really good at, trained in, or interested in math at all — or someone who was like so many teachers out there who “just love kids” but who choose not to translate that love into teaching their kids how to think well.

But I think if you put a geometry class like what I described above into the hands of a competent, mathematically astute teacher with a mind to help his/her students become excellent thinkers, a year of that could very well change a generation of kids.

Filed under Geometers Sketchpad, Geometry, High school, Math, Teaching

## 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 that would be of use to an educated adult learner with little math background but a lot of genuine interest. Leave your suggestions in the comments.

Filed under Education, Geometry, Math, Problem Solving

## 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 roll these days, what with this new edition of Greenberg and with Rogawski’s excellent new calculus text. (Disclosure: I was a reviewer for Rogawski.)  I don’t advocate for textbook use often, but if you have to use one, use a good one!

Filed under Education, Geometry, Higher ed, Math, Textbooks

## Calculus is older than we thought

It turns out that according to a recent discovery in an ancient manuscript, calculus might first have been discovered not by Newton or Leibniz in the 1700s but by Archimedes a millenium earlier:

For seventy years, a prayer book moldered in the closet of a family in France, passed down from one generation to the next. Its mildewed parchment pages were stiff and contorted, tarnished by burn marks and waxy smudges. Behind the text of the prayers, faint Greek letters marched in lines up the page, with an occasional diagram disappearing into the spine.

The owners wondered if the strange book might have some value, so they took it to Christie’s Auction House of London. And in 1998, Christie’s auctioned it off—for two million dollars.

For this was not just a prayer book. The faint Greek inscriptions and accompanying diagrams were, in fact, the only surviving copies of several works by the great Greek mathematician Archimedes.

The kind of mathematics that Archimedes was doing look a lot like standard problems on integration that Calculus II students mutter about today — finding the areas of curved figures, finding the volumes of solids via cross-sectional area sums, and so on.

The article goes into depth about what makes Archimede’s work such a breakthrough, namely the willingness to work with “actual infinity” instead of “potential infinity”. Contemporaries like Aristotle believed that actual infinity didn’t exist; instead, the world is full of potential infinities. For example, lines that are infinitely long do not exist, only lines which are finite but could be extended, hypothetically, to infinite lengths.  The article points out that today, we don’t use actual infinity but rather potential infinity, which is the basic underpinning of the concept of the limit which in turn is what all of calculus is based on.

This is a major discovery — the kind that makes you think about the what-if questions of how the world might have turned out differently if calculus had taken off 700 years prior to Newton. And it’s somehow fitting that high-tech microscopic methods, developed no doubt using the very calculus that Archimedes must have envisioned, were used to extract Archimedes’ handwriting from the copied-over manuscript.