Wednesday, September 19, 2012

On the Block All Things Are Possible

That was almost definitely the best class of the week.

Lucas walked in just before the bell wondering how long it would take to walk the length of the Great Wall of China.

Somehow, some way, my heart has not yet closed to hearing such questions.

"I wonder indeed, child! The floor is yours! Here, have this class of 12 rambunctious geniuses! Here, have some Internet! Have a projector! Let's rock this bitch out!"

You don't fiddle around with a class a little, you don't know what's possible.


Everybody looked a scosh alarmed. I perched up on a desk in the back and willed myself to not say much of anything. And good thing because dear God, these shoes.

They clarified the question on their own. They were riveted, every one. They were jumping up to open google maps. I suggested Google Earth and showed them how to measure distances. And the children, they were glowing. They were addicted. d=rt had its best day in a while. It's not explicitly part of this course, but let's not kid ourselves that all these cherubs have it on lockdown.

It took them all of 20 minutes to figure out, with satisfying certainty, how long it would take to walk the great wall of China. On periods, that would have been half the period, and there goes today. On the block, there were 70 minutes remaining, which if Miss Nowak can't adjust on the fly her treatment of parallel lines cut by a transversal to fit in 70 minutes, one should feel free to question her qualifications.

Progress on the Wanted Board

No one has brought me a post-it dodecahedron yet. (Boo! I was hoping to decorate with them.) But two other problems have been solved.

The first one, with the pentagram, took over a week. But lots of kids tried it. I know because I caught them doodling it in class.

The second one only took three days. It was this old chestnut:

Mr. Wolverton had a party and invited Mr. Zimmerman, Mr. Horst, and Miss Nowak. When they arrived, and because they are math teachers and they enjoy this sort of thing, Mr. Wolverton said, “Here’s a riddle. I have three daughters. The product of their ages is 72. The sum of their ages is the same as my house number. How old are my daughters?” 
The guests went outside to look at the house number. When they came back in, they said, “This problem can not be solved!” 
Mr. Wolverton said, “Oh! I forgot the most important clue. My youngest daughter prefers strawberry ice cream.”

And now I have this one up there, inspired by Paul Lockhart's new book.

Draw any quadrilateral.

Any old quadrilateral will do. Make it as ridiculous as you want. Make it concave or convex. Make it as wacky as you like. Make it disgustingly irregular.

Find the midpoint of each side.

Connect these four midpoints with four non-intersecting line segments, to make a new quadrilateral.

Something beautiful will happen.

What is it?

Now, explain why.

I have found that this proof is at the edge of what kids in regular Geometry can do, and it's beyond the edge for many of them. But I'm hoping some older kids might have fun revisiting Geometry. I'm hoping they don't realize they're doing a proof until it's too late. I left that word out intentionally, since it has teeth and would scare many of them off.


Tuesday, September 18, 2012

Fourth and Final Week of New Bloggers!

Here is our fourth installment of new bloggers! If you needed any evidence that our teachers are a national treasure, look no further. Passion, thoughtfulness, smarts - our weird and awesome little corner of the Internet is rocking it right now.

Sorry to pick favorites, but this is my personal favorite from this batch: Rachel Tabak @ray_emily has a blog named Writing to Learn to Teach. The fourth post for the Blogging Initiation is titled Error Analysis, Decimal Operations, and Being Less Lame and the author sums it up as follows: "Kids who would likely have been utterly lost received mini-lessons from peers. (Also, I could spot those kiddos and work with them fairly easily, given that I was NOT in front of the class blah-blah-blahing. Oh – in case it wasn’t clear: There was no blah-blah-blahing. I just let kids dive in.)" A memorable quotation from the post is: " Now that I’ve moved from moping to actually determining what in particular was not working—a necessary step that I’d pathetically avoided, previously—it’s time to get to work."

Amy Zimmer has a blog named Ms. Z Teaches in Mathland. The fourth post for the Blogging Initiation is titled Pictures! and the author sums it up as follows: "Looking at ways to engage students" A memorable quotation from the post is: "I love when math students get to shine using all their intelligences."

Sarah Miller has a blog named Proof in the City. The fourth post for the Blogging Initiation is titled Small change with a big payoff and the author sums it up as follows: "One thing I've realized this year is that I need to spend time at the beginning of my lessons reminding kids what we are working on and where we left off last class period. It helps get them focused, and makes it possible for me to move forward with the topic at hand." A memorable quotation from the post is: "They don't hang on my every word, they don't look forward to what new math knowledge they can get today, they don't take a moment to reflect on where we are and what we are learning before class starts."

vanvleettv @vanvleettv has a blog named Everything's Rational. The fourth post for the Blogging Initiation is titled New Blogger Initiation Week 4: Solving Equations Shout Out and the author sums it up as follows: "The post is about a post I came across by a fellow new blogger. It is a resource and insight as to how she teaches solving equations." A memorable quotation from the post is: "As I was cruising through the Mathblogotwittosphere, I came across a post by MathyMissC called reteaching solving equations and wanted to give some props to her."

Alex Freuman @freuman has a blog named Math Teachering. The fourth post for the Blogging Initiation is titled One way to get students to ask questions... and the author sums it up as follows: "For me, this is an effective way to get students thinking and asking questions. It targets quieter and more timid students." A memorable quotation from the post is: "It is very natural for me to pause during a lesson and say, "Any questions?" My feeling is that many students interpret this as, "If you're a little too slow to keep up with my pace, confess now.""

helen oehrlein has a blog named Bowditch's Apprentice. The fourth post for the Blogging Initiation is titled Wonderful Course for NYC/Long Island Teachers and the author sums it up as follows: "In my last post, I mentioned a great course I had taken that really moved me along s a math teacher. The coordinator of the program saw my post, and asked me to let teacher know that there are openings in the latest course, which starts on Oct. 3. It was the most valuable PD I have ever done." A memorable quotation from the post is: "Each class a different experienced teacher shared their best ideas and lessons. I learned so much, and wished that this had been part of my teacher training."

Thursday, September 13, 2012

Online Population Projection

For some reason, this came to my attention. Because, math was wrong? What?


Ben Foster anticipated Facebook's billionth user would login last month, in August 2012. He wasn't alone. Here was NBC News' technology blog, reporting on a prediction by iCrossing:


And Bloomberg Businessweek would only call it for "later this year:"

Which all seem to presume linear growth from here on out. Indefinitely? I don't know. Which just makes me wonder, to what extent can we apply usual population growth-type logic to online populations? If Facebook were growing within an environment with biological limiting factors, we would have expected what we've already seen, for example, exponential growth at first. For quite a while, Facebook was growing at a lovely exponential clip of approximately 10% a month. This shows their growth for December 2004 through December 2009, with an exponential regression to fit:



However, maintaining that growth rate was clearly unsustainable. If it were, Facebook's population would have reached seven billion In June of 2012. Obviously, that didn't happen.



In biological populations with finite space and resources, we expect growth that looks exponential at first, but due to limiting factors, levels off eventually. And, indeed, Facebook's growth did not continue exponentially.



And I suppose it has looked rather linear for a while, but I'm not sure that's the best model. The rate of increase has slightly decreased the past year or so (shout-out to the second derivative!) 

If we apply a logistic model to the data so far, we get:



Which has Facebook reaching a billion users in April of 2013, and predicts its eventual population will top out at less than 1.1 billion.

But this all raises more questions than it resolves. Facebook may be approaching its maximum realistic number of users within the United States. However, as far as I understand, it has lots of room to grow in other huge markets. So this logistic growth model is flawed as well. I'll cop to not understanding how graphing calculators come up with logistic regression equations, like, at all. At least not with nearly the depth that I understand how they calculate linear regression equations. I simply know how to apply it as a blunt instrument to a table of values. On the other hand, linear growth, as the news organizations have used, has not panned out - as we're past August 2012, and have not reached a billion users yet. Have worldwide, internet ecosystem limiting factors unavoidably kicked in already? Should we expect another period of exponential growth in the future, if it catches on in India? Are there reasons to think this linear-looking growth will continue for a long time? And for how long? I don't know if any of these are answerable! But I do love the questions.

Here is a Geogebra file, if you want to play.

Wednesday, September 12, 2012

We Got Tricks

So, I don't love tricks?  You know, the "don't worry about the why, kid, just remember this little song" variety. But we're in the weeds of finding areas and perimeters of composite shapes. Of the quarter-circle stuck on a rectangle stuck to a triangle variety. Yes I want them to have a intuitive grasp that the distance around a circle is a little more than three diameters, and yes I want them to see a circle deforming into a rectangle whose width is r and length is half a circumference. And yes we are mostly solving problems with tracks and whatnot.

Buuuut....I am not confident about how much of that they are going to be able to retrieve when they are sitting for their SAT's a year and a half from now. So, we are shamelessly using a trick, that a student heard somewhere else, for keeping circumference and area of a circle straight. "Chocolate pi is delicious, and Apple pis are, too." I'm storing it in the same dark place as "All Students Take Calculus," as in, it makes me feel a little dirty, but it seems to be a necessary evil.

Sunday, September 9, 2012

Week 3 of the New Bloggers

Katie Cook @kjgolickcook is my Real! Life! Colleague! She is very funny and very awesome and finally has a blog (yaaayaayaaay! (Kermit arms)) named Math Teacher by Day. Her post for this event is Why Do We Have to Learn This? and includes three very solid responses to that question. I'm looking forward to good things from Katie's blog - she's a long ball hitter, that one.

Chris Rime @chrisrime has a blog named Partially Derivative. The third post for the Blogging Initiation is titled NBI #3 — In which a student learns the truth about math class and the author sums it up as follows: "All about how I respond to that infamous, "When are we ever going to use this?" question. With super-bonus material related to the apparent difference between illiteracy and innumeracy." A memorable quotation from the post is: "This is as real as the world gets."

Sarah Miller has a blog named Proof in the City. The third post for the Blogging Initiation is titled Math Quotes and the author sums it up as follows: "I wrote about one math quote that I like, and how it has affected my teaching." A memorable quotation from the post is: "I even found a quote to explain why I love quotes."

Carl Edgren and Hannah Schuchhardt @hschuchhardt, @carledgren have a blog named Teaching Systematically. The third post for the Blogging Initiation is titled Get to the point. and the author sums it up as follows: "Students often just want to know the answer to problems - mathematics is the process of completing a problem for the correct answer. We need to show them that learning mathematics is about analyzing the world around us!" A memorable quotation from the post is: "The students simply want us to get to the point already."

helen oehrlein has a blog named Bowditch's Apprentice. The third post for the Blogging Initiation is titled A Hodgepodge of Ideas? and the author sums it up as follows: "I have come to believe that Precalculus is not a hodgepodge of ideas, but a study of the deeply related Mandala of the Functions (as Berlinski sees it). Furthermore, although these functions have much in common, their differences shine a light on why we really do need so many different functions." A memorable quotation from the post is: "Inverse functions are so useful that we truncate the trigonometric functions in order to create them, and we actually have a totally artificial, man-made inverse of the exponential function, namely the logarithmic function."

Mark Davis @graphpapershirt has a blog named Graph Paper Shirt. The third post for the Blogging Initiation is titled Where's All the Stuff Come From? and the author sums it up as follows: "This post is about how students come to class with misconceptions about photosynthesis and where plant material comes from. It's such an important concept to understand because it encompasses the fundamental idea of how the sun ultimately powers (almost) all life on earth." A memorable quotation from the post is: "To make a long post short, the students learn that even some of the brightest students in our country don’t know something as fundamental as where plant material comes from, and ultimately their food, and themselves!"

Jasmine has a blog named Jazmath. The third post for the Blogging Initiation is titled I Was Never Good at Math Either and the author sums it up as follows: "Parents night is an important time to show families how class really runs and the attitudes that we hope to instill in students. I try to run the ten minutes that I get with each set of parents as if it's a real class period." A memorable quotation from the post is: "I want to keep the focus on how we deal with discomfort and what we want to model for our children."

Kelly Berg @kmbergie has a blog named The M Stands for Math. The third post for the Blogging Initiation is titled Sharper questions and the author sums it up as follows: "I took a risk to teach material with meaning instead of with boring-ness. My risk paid off to the student's benefit. Students were involved and engaged. BAM!" A memorable quotation from the post is: "And right there I went from concrete to abstract AND they had better understanding of the concept because there was meaning attached to it."