We’ve talked about the ‘flipped classroom’ model a couple times on this blog. Here’s a nice example of a teacher doing this in real life. It looks like it’s working for him and his students!
Here’s a link to the full article, though the video does a nice job summing up the article.
I’ve been motivated by George Hart and Zachary Abel to make my own mathematical sculpture with found objects :-). A few former students dropped by to visit me this afternoon and I put them to work making this (they had no where to be, right!?):
A cardboard stellated icosahedron
It’s a stellated icosahedron, made out of these little triangular pyramids. I did not make the pyramids, they came to me this way. Can you guess what their original purpose was?
Pop quiz: What do you think this is??
My wife and I redid our kitchen a few years ago, and I saved twenty of these from (did you guess it yet?) the packaging our cabinets came in. For each cabinet, there are 8 of these keeping the corners safe. The construction process was pretty straight forward, but here are some photos documenting the event.
Construction begins
Every vertex looks like this on the inside.
Almost done!
The last piece goes on.
Here are some more views of the icosahedron. The icosahedron has a symmetry group of size 60.
There are 15 pairs of opposite edges, each with 2-fold symmetry (for a total of 15 orientations, not counting the identity)
There are 10 pairs of opposite faces, each with 3-fold symmetry (for a total of 20 orientations, not counting the identity)
There are 6 pairs of opposite vertices, each with 5-fold symmetry (for a total of 24 orientations, not counting the identity)
So (1 identity) + (15 edge symmetries) + (20 face symmetries) + (24 vertex symmetries) = 60 total orientations.
Now I just need to find a large enough Christmas tree upon which to put this giant star!
I’ve mentioned Khan Academy lots of times before, and other resources that allow teachers (math teachers in particular) to “flip the classroom.” Here’s a nice graphic that summarizes the model and provides a bit of research in favor of it. I haven’t been bold enough to try it, but I’d like to experiment in the next few years. It seems like you wouldn’t have to buy into the model 100%; you could use the flipped classroom model sometimes, and the traditional model other times.
Also, it occurs to me that this discussion is most relevant and most revolutionary in the math classroom. English and History classes have always used this flipped classroom model, to some extent–you read outside of class, then come to class to discuss the material. Historically, it’s math teaching that has been lecture-based. So maybe we’re just catching on to something that English and History teachers have known all along: the real thinking and learning happens when the student is involved–talking, speaking, doing, practicing, experimenting.
I have a small problem with the following language in our Algebra 2 textbook. Do you see my problem?
Horizontal Line Test
If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.
Here’s the issue: The horizontal line test guarantees that a function is one-to-one. But it does not guarantee that the function is onto. Both are required for a function to be invertible (that is, the function must be bijective).
Example. Consider 
Wrong. The mapping given is not invertible, since there are elements of the codomain that are not in the range of 
This might seem like splitting hairs, but I think it’s appropriate to have these conversations with high school students. It’s a matter of precise language, and correct mathematical thinking. I’ve harped on this before, and I’ll harp on it again.
Certain Uncertainty: Schrödinger Equation
From Wired.com…
9 Equations True Geeks Should (at Least Pretend to) Know
By Brandon Keim
Even for those of us who finished high school algebra on a wing and a prayer, there’s something compelling about equations. The world’s complexities and uncertainties are distilled and set in orderly figures, with a handful of characters sufficing to capture the universe itself.
For your enjoyment, the Wired Science team has gathered nine of our favorite equations. Some represent the universe; others, the nature of life. One represents the limit of equations.
We do advise, however, against getting any of these equations tattooed on your body, much less branded. An equation t-shirt would do just fine.
The Beautiful Equation: Euler’s Identity
Also called Euler’s relation, or the Euler equation of complex analysis, this bit of mathematics enjoys accolades across geeky disciplines.
Swiss mathematician Leonhard Euler first wrote the equality, which links together geometry, algebra, and five of the most essential symbols in math — 0, 1, i, pi and e — that are essential tools in scientific work.
Theoretical physicist Richard Feynman was a huge fan and called it a “jewel” and a “remarkable” formula. Fans today refer to it as “the most beautiful equation.”
I was glad to see that the first “must have” equation was Euler’s Identity (note that “Euler’s Identity” is the accepted name for this, not to be confused with Euler’s Formula or Euler’s Polyhedron Formula or any of the other amazing facts named for Euler). I think there’s large consensus in the math community that this is, indeed, a breathtaking equation. It may not be the most fundamentally important, but it definitely showcases why mathematicians delight in math.
I’m ashamed to say it, but I hardly knew any of the other equations. I knew Boltzman’s equation; Maxwell’s equations and Schrödinger’s equation have come up in some of my graduate coursework, but the others I hadn’t ever seen. One might argue that the other equations are not so important. (If you like arguing about such things, join those commenting on the article). You should still look through the list yourself; how many of these equations do you know?
Granted, this was a general article that encompased all “true geeks” not just math geeks. But still, don’t we all want to be a true geek?
(Oh, and happy birthday to Johan (III) Bernoulli, who had no notable equations named for him :-))
I’ve posted on infinite series a few times before. But I was inspired to touch on the topic again because I saw this post, yesterday, over at the Math Less Traveled. Actually, the post isn’t really about infinite series as much as it is about p-adic numbers and zero divisors. I’m excited to read more from Brent on this subject. But I digress.
The point I want to make with this post is that students struggle with wrapping their minds around convergent infinite series, and yet they live with them all the time. Students have inconsistently held beliefs about infinite sums.
The simplest convergent series is a geometric series 
for 
But this proof isn’t very satisfying for the student encountering infinite series for the first time ever. Evaluating the limit feels like ‘magic.’ The idea of adding up an infinite amount of things and getting a finite value is unsettling. I admit, it sounds like quite a lot to swallow. That being said, however, students have no problem declaring the infinite series
to be 
Once they finally shake their denial, they can more easily accept the convergence of other infinite series like 
So there it is. An inconsistently held belief about infinite sums. To the student: You cannot have it both ways. Either you must agree with, or deny, both of the following equations:
But to believe one equation is true and the other is only ‘kind of’ true is inconsistent. I rest my case. 🙂
Wow, if the title of this post didn’t grab you, I don’t know what will. Pretty riveting, right? Has anyone ever thought of teaching math with applications? </end sarcasm>
This is the basic thesis of a recent article from the NY Times, “How to Fix Our Math Education” by Sol Garfunkel and David Mumford:
THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.
All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.
Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a “reform” version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.
For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a “group of transformations” or a “complex number”? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.
from toothpastefordinner.com
To be fair, the real thesis–if you read further in the article–is that we should primarily teach applications and math can swoop in and rescue us if and when it’s needed:
Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.
If you haven’t gathered already, I don’t agree at all with this thesis. It’s my opinion that math should be taught as math, respected as its own field of study, and a valuable part of a high school liberal arts curriculum. Students should value math for its inherent, abstract beauty. Applications are of course a must in any course. But I find that in the text resources I’ve used, the applications are often contrived. Extremely contrived. Doing math should feel like playing a game, like working on a puzzle, or like arguing.
In fact, when high school students ask why are we learning this?, I FIRST respond with the things I just said: It’s part of a liberal education; it will make you a well-rounded, intelligent person who can hold conversations with other smart people in other fields; and it’s fun. I mention SECOND what applications exist for the math we’re learning. For high school students, if we’re honest, most of them will never need any of the math we’re teaching. Seriously. If you’re not working in a math or science field, when was the last time you had to factor a polynomial?
The authors go on to say “Science and math were originally discovered together, and they are best learned together now.” But this is not universally true. In many cases, the ‘useless’ math was developed first (think of number theory for instance) and then only later were applications discovered (think of the RSA or El Gamal public key encryption schemes).
So why learn math? There was a nice post about this yesterday on one of my new favorite blogs, dy/dan, titled “Cornered By The Real World.” He highlights this great article by Samuel Otten in this August’s Math Teacher magazine. I highly recommend reading the whole article. As a taste, I’ll include the same snippet that Dan shared:
I believe that thinking and acting as if the justification for teaching and learning mathematics is found solely in everyday applications can be dangerous. Mathematics does not exist only to serve other professions, nor is it merely a collection of algorithms and procedures for dealing with real-world situations. Such a mind-set essentially paints our discipline into a weak and lonely corner and leaves undefended many of its greatest aspects.
I could say more about this, but I’ll spare you. I’m passionate about making math a subject worth learning all on its own. If I say more, I’ll start to sound like Paul Lockhart.
This summer I finally finished reading the Manga Guide to Calculus by Hiroyuki Kojima and Shin Togami. Here are my two cents:
The Manga Guide to Calculus is chocked full of great mathematics and lots of quality comic art (the author went to great lengths to ensure it was authentic manga, with illustrations by popular artist Shin Togami).
That being said, I don’t think anyone could ever learn Calculus using this book. In fact, I think Kojima must know that. He never claims this can be used as a textbook replacement. The math isn’t presented in a very systematic way, and there are very few real exercises for the reader. Right from the beginning he puts heavy emphasis on linear approximation. He takes a very different approach to presenting Calculus than a math book would. It is a story most of all. Kojima, in his preface, says its a great book for those who already have Calculus knowledge–both for those who love Calculus and for those who have been “hurt by it.” I tend to agree.
As for the story, well, it’s a bit contrived. But what story that tries to smuggle in some math doesn’t seem a little contrived? Sometimes it’s a bit of a stretch and the story suffers. You should still give it a chance, though.
So to those looking for a Calculus textbook, you need to look elsewhere. For instance, I was looking for things I might be able to use in the Calculus class I teach, but couldn’t find much usable content. But for those interested in math and are looking for a fun read, I would recommend picking it up.
I just wanted to let everyone know that I’m back, and excited for the new school year. I took a little summer break from blogging, just like I did from teaching :-).
Today was our first day of school at Richard Montgomery High School, where I teach. (And we actually did have school today, even though we had both a 5.8 earthquake and an ‘historic’ hurricane in the last week.) So welcome back to all my current and former students!
I love this discussion at squareCircleZ. All my readers should check it out. Which is the graph of arccot(x)?
from squarecircleZ
from squarecircleZ
I especially like this controversy because some big players have weighed in on each side. Mathcad and Maple prefer the first interpretation, Mathematica and Matlab prefer the second.
For a more thorough treatment, check out the original post here. Three cheers for great math blogging! 🙂
Random Walks 