Friday, February 28, 2014

Linkfest for Friday, February 28

Calculus: Optimizing the Goodness

Optimization may be the best application topic taught in standard math courses.

Why do we use math in the real world? Because we want to make something better. I don't care for the problems that are about maximizing profit, as if other considerations aren't relevant. But when we make something, we are trying to satisfy a need, and we have particular goals - maximizing how well we achieve those goals can sometimes be modeled mathematically.

Making a box that maximizes volume  with a given amount of material is a simple problem that makes sense to all of us, and that represents well the class of problems we can solve with optimization. So that's how I begin our study of optimization (probably thanks to another blogger, though I don't remember now who it might be). Yesterday I brought in origami paper and showed the class how I would fold a simple box. Their task was to make the box with the biggest volume.

I've done this lesson before, but I added a few details this time that improved it dramatically. I want to write up what happened so I'll remember it next semester.

This is a class with over forty students. I set them in groups of four, had one person come up to get four sheets of the origami paper, and then showed them my steps. I fold in half twice in each direction, reverse two folds so the sides can come up, and make a triangle at each corner that can then fold out over the edge to lock the corner in place.

That method gives a particular shape of box, so then I show them how to vary the height:
  • Fold wherever you want for your height. 
  • Fold in half so you can copy this fold to the opposite edge.
  • Fold one corner onto this fold, creating a 45-45-90 isosceles triangle.
  • This give the position for the other sides, so that they'll be the same size.
  • Now fold your four sides up, and make the triangular corner bits as before, folding over the edge to lock.
  • When the box is made, measure to determine length and width (which are equal), and also the height.
  • Determine volume.
As they worked, I ran off to get the rulers I'd forgotten. I think that had unintended benefits. Before I left, I asked who had done origami before. I asked the others to look at the hands up, and turn to them for help. Then I ran off, and left them to it. I think the learned helplessness that schooling engenders in so many students was minimized by my absence.

After I passed out the rulers to each group that didn't have their own (love those students who carry rulers around with them), I asked whether we wanted to measure using inches or centimeters. I heard lots of them say centimeters and went with that. Good thing, because that worked better with decimals, and gave us an even measurement on the paper, which was exactly 15cm by 15cm. Then I asked who had their box measured, and began to make a table on the board.

After I wrote the height Edwin had given me, I asked how I would find volume. The student I asked didn't know (which surprised me). I asked them to visualize little cubic centimeters filling the box. I could see that wasn't enough, so I had them all visualize a big box on my desk, 3 feet by 4 feet by 2 feet high. Then I used my hands to describe a cubic box, 1 foot in each direction. How many of these fit along the back of the big box? Yes, four. And how many rows of four? Three. So how many are in this bottom layer? Twelve. And how many layers? Two. So the total is ... twenty four. And now we can see that the volume of any rectangular box must be...  I think they had all memorized V = LWH before, but few of them had seen why it must be so. This visualizing helps them see where the formula comes from, and is a start for many on the journey toward being able to visualize things like volumes of rotation.

I put the height Edwin had given me onto my diagram of a paper with the fold lines, and talked my way through finding the proper length and width. I asked Edwin to find the volume. I repeated with seven more volunteers. For each one, I'd figure the height as I did the first time, saying I didn't want to go by their measurements, in case they measured high and got an unfair advantage. That allowed me to prime them to notice that we were always subtracting twice their height from the width of the paper.

As I added entries to the table, I also added them to a graph of height versus volume. I noticed that it looked like it could be a parabola, though we didn't yet know its shape. I talked about the extreme points - how a height of 0 or 7.5cm would give a volume of 0 - and added those to the graph.

I knew the lesson was going well when one of the students wondered how we could figure out the best volume. We had not yet written a function, but we were getting close - as a group! I got to talk about modeling and mathematizing. I also mentioned that experimenting with particular cases was one of my favorite ways to get more grounded in what might be going on in a particular problem.

I got to point to my graph, with a curve running through the points (Can I do that on Desmos?), and asked what calculus would say about this problem. They were able to tell me that the highest point was where the derivative was 0. I drew the tangent onto the graph and wrote y'=0. 

So that student's question got us to introduce a variable. Height was where we had started, so that became x. Then I got to ask them to think in their groups about how to find the length and width. Many of them were ready to tell me that was 15-2x. (This is one of the harder steps to my students, who are unfortunately not used to thinking for themselves in math.)

I had them work in groups on simplifying the equation we had written, and finding the x values that would make the derivative equal zero. I pointed out that it was a cubic, with a double root, which they should know how to graph. I sketched in the full graph past x=0 on the left and past x=7.5 on the right, and got to mention the domain of the math function (all reals) versus the domain based on this problem, which is 0 < x < 7.5.

The quadratic we got from the derivative isn't easy to factor, so we used the quadratic formula (giving me a chance to be silly and sing it). We saw that one of the solutions, x=7.5, would give a volume of 0, which is a minimum. I talked about how we would know that the other point at x=2.5 is a max, even without a graph, since the second derivative, V" = 24x-120, is negative there.

As I summarized what we had discovered together, I said that this problem was unusual in that it didn't seem to have two equations, where most optimizing problems will. That had been my impression in the past. But as I described where we began, wanting to maximize V = L*L*H, I saw that it was like most optimizing problems - a function with two variables. And our constraint, which I now labeled, was L = 15-2H.

I moved on to a quick walk through the problem of fencing a pen attached to a barn. I said I was imagining I had a barn and a big enough yard to have a goat. I want the pen as big as possible given 200 feet of fencing.  Students were able to give me all of the steps for this problem.

I think the best thing I did in this lesson was to draw the graph of height versus volume, so they could see that the derivative of the volume function gives us the maximum.

Teaching a big group energizes me. I feel like this is the best I've done yet teaching Calc I. Whether they are really getting it is unclear to me - the downside of such a big group.

On Monday I'll have them work in groups on two or three problems.

Any suggestions for your favorites?

Tuesday, February 25, 2014

One Good Thing: A Student Who Learned to Think

There's a blog called One Good Thing, where teachers tell about something good from their day with students, that has often provided me with sweet bits of inspiration. Maybe I'll ask to be added to their set of authors. For now, I'll share here...

After my calculus class, as I headed through the halls back to my office, a student from last semester said hi, and we stopped to talk. He told me that I had taught him to think, and that he was acing his next math course (Finite Math), which felt like a breeze to him after pre-calc. He said he had learned to question everything. He had been struggling in my course, and suddenly it all clicked during the final, and he earned a B on it.

He said at first he thought I wasn't such a good teacher, because he had been confused by the way I left things open. It took a while for him to understand that I was trying to get them to think about why things work.

He said that he's doing better in all his classes, that I changed his life!

Wow! I needed to write this down so I can read it on days that I don't remember how much I can do for students. I told him he had made my day.

Saturday, February 22, 2014

Papert on Girls and Computers

In Edward Fiske's interview of Seymour Papert in the New York Times, almost thirty years ago:

Q: Boys tend to be more interested in computers than girls. Is that something that troubles you? 

A: It does trouble me, and it's a reflection of the general phenomenon. It's not the computer as such that's more attractive to the boys than to girls. It's the fact that the computer comes out of a male technological, technocratic, white-dominated culture. The computer as we know it was made by engineers who like to think in a very systematic, organized, top-down, highly planned way. 

Not everybody likes to think like that, but science and mathematics instruction in our schools is powerfully biased against people with a more artist-like style of thinking. They react against a culture that has no room for intuition, no empathy, no communication about what you're doing. They react against a culture where the emphasis is on linear thinking, on individual work and on making a product that works rather than a product that you can talk about with other people. 

The computer, though, allows you to approach technical subjects, and mathematical ones too, more like the artist who creates by a negotiation of the object you're trying to create. There's no incompatibility between that intuitive kind of thinking and being able to do mathematics in a very creative way. We're making pockets of computer culture where learning is very personalized, where you can build up from the bottom and still structure it from the top. You can make something and change it. You can let it grow the way a painting on the canvas grows in a kind of negotiation between you and the product. 

Has our perception of computers changed enough in these thirty years to make programming more welcoming for girls? 

I found this when I searched on "Seymour Papert girls". I was looking for a passage that I think is in his book Mindstorms, about how showing the kids projects that involved designing rooms got the girls much more involved. Something like that. I couldn't find the passage. Can anyone help me?

Friday, February 21, 2014

Revisiting a Lesson: Derivatives of Sine and Cosine

A year ago, I posted the handouts I had made for this lesson. They provide detailed explanations of:
  • how we know sin(x+h) = sin x cos h + cos x sin h
  • the squeeze theorem 
  • how we know (sin x) / x approaches 1 as x approaches 0
  • how we know (cos x - 1) / x approaches 0 as x approaches 0

Thursday Morning Lecture
I love lecturing! I agree that it's not usually an effective way to teach. I keep that in mind when I lecture, and do everything I can to draw my students in. I started out by telling them that as they go on in math it becomes more and more about proof. I warned them it would take a lot of effort to work their way through the reasoning I presented, and asked them to please tell me whenever something wasn't making sense.

I used index cards to call on people, and asked them to provide some of the simpler steps. I asked for the whole class at once to call out the very simplest parts: "sin2 x + cos2 x = ... "  I used different colors of marker to point out various triangles we were considering. I had the students do a few algebra steps individually at their desks before I wrote them on the board. Except for redrawing the diagram and doing those algebra steps, I told them there was no need to take notes, since I would be giving them a handout at the end.

I asked afterwards how many were able to stay with it. I think over half of them raised their hands.

As we worked our way through this diagram, the students were getting a much-needed (for most of them) review of trig.

I need to modify this so that it doesn't look like there's a straight line formed by the terminal sides of α + β  and -β. Darker lines and bigger angle and point identifications would be nice too.

We used this to prove our trig identity, and used that to get as far as...

Next Week
We are through two of the four pages of my handout. For homework, I told them to use their calculators to fill in a table of values, evaluating the two expressions in the limits for  h = .1, .01, .001, etc.  On Monday, we'll look at the squeeze theorem to help us find these two limits, which will finish off our proof. (Maybe I'll give the second handout at the beginning of class, and ask them at the end of class which worked better for them - getting the handout afterward like we did on Thursday, or getting it before.) 

This proof is a good review of trig, a good way to see the power of the squeeze theorem, and a good way to think about limits from some new perspectives. (We have not done a unit on limits. I skip that chapter, and come back to it during our last unit, before considering integration.)

I was so excited after my 80-minute performance, I wanted to make a video of it. I don't know if it would be useful to anyone else, but a number of my students would have liked to watch it this weekend. A lot of what I do right in my lectures would be hard to reproduce on a video - I need the audience to get me pumped up, and I need to see a confused face to realize that I should say more. But one of my students agreed to be my audience, and another offered to join her. So I might find a way to do my video lesson with just a bit of student participation. If it works out well, maybe I can do a bunch. We'll see...

Just in case, I set up a youtube channel for Math Mama. (Thank goodness that name wasn't taken!)

An Alternate  Proof
When I got home from work, I re-read my blog post from last year. One of the commenters had posted a very different proof for the derivative of sine and cosine. It's very short and very visual. I love it. One of my reactions was to worry that dragging my students through the longer proof was unnecessary.  But I think our work will help the students better understand the sorts of proof that use lots of algebra, and will give them a great context for the squeeze theorem. I also think this alternate proof isn't quite complete. (At some angles, adding a bit to the angle increases the cosine instead of decreasing it. What then?) But it is so cool, I have to show it to my students. I think I'll wait a day or two, so they'll have done all they want with our conventional proof first.

What do you think? Would you use this alternate proof?

Full-time Math Teaching Position at Contra Costa College

The college I work at is hiring. Time got away from me, and I forgot to post this when it first came up. The deadline for applying is next Friday. (Application process is completely online.) If you're interested, check it out. You can email me if you have questions: mathanthologyeditor on gmail.

Wednesday, February 19, 2014

Linkfest for Wednesday, February 19

Tuesday, February 18, 2014

Re-post: My Top Ten Issues in Math Education

Alexandre Borovik invited me to join the writers at The DeMorgan Forum. My first post over there, at his request, is a slightly revised version of My Top Ten Issues in Math Education, originally posted in 2010.

Monday, February 17, 2014

Linkfest for Monday, February 17

Sunday, February 16, 2014

Math Teachers at Play #71 (with 71 links)

Back in 2009, the first time I hosted a Math Teachers at Play Blog Carnival post, we were at #11. Seems like those smaller numbers almost always had something interesting going on. For 71, it's a stretch...

What's Special About This Number has:
  • 71 divides the sum of all the primes before it (i.e., 2 + 3 + 5 + 7 + 11 + ... + 67 is divisible by 71)
  • 71 = (4! + 4.4)/.4 (representation of numbers using only four 4's)
  • 71 = 362 - 352 = 36 + 35
  • 71 - 1 = 1 x 2 x 5 x 7 and 71 + 1 = 3 x 4 x 6 products of partitions of consecutive numbers)
  • 712 = 7! + 1
  • 712 = 27 + 173 (sum of prime powers of two prime numbers)
  • 713 = 357911 (consecutive odd numbers) 

 Number Gossip has:
  • 71 is the only two-digit number n such that (nn-n!)/n is prime.
  • 71 is the 2nd Google number. The nth Google number is the first n-digit prime found in the decimal expansion of e. They are named Google numbers because of the unusual hiring ad that Google put up.
2, 71, 271, 4523, 74713, ...

71, Mark Gonyea
71, Brent Yorgey

And here are a few images of the number 71 itself. Mark Gonyea is a designer. Brent Yorgey and Richard Schwartz are mathematicians. Posters are available for the numbers 1 to 100 from each of these artists.

Richard Schwartz

Math teachers at play know that math is best learned when the student is thoroughly engaged, through their body, their imagination (story-telling), or the world of games. I've started out this month's post with those three categories. (Most of the submissions this month described hands-on, or feet-on, activities. It's as if there had been a theme agreed upon without anyone mentioning it.) Some of the following posts are from submissions, and others are posts that I wanted to share from my internet wanderings. This post has 71 links. (You might need to digest it in smaller bites.) Enjoy!

Learning with Our Bodies

Julie, at Highhill Education, shares her family's mandala art, and the geometry they learned while doing it. (Julie is based in Germany. In the U.S., I've found some of the best inexpensive books come from Dover - here are some of their mandala coloring books.)

Jennifer Bardsley, at Teaching My Baby to Read, works (plays) with her son, exploring rotational symmetry using cookie cutters and flour.

Ticia, at Adventures in Mommydom, shares her hands-on fraction lessons.

Margo Gentile, at Margo's Math and More, was inspired by all those snow days, and created some wonderful mazes in the snow for her dog and kids to navigate. (Bummer! I don't have any snow here in California to try this in.)

Lilac, at Learners in Bloom, wrote Combinatorics in Kindergarten, her story of making clothes for the bears, so her daughters could count how many outfits Little Bear could wear. She also made a Feed the Clown game to help her daughters have fun practicing basic addition facts.

Maria Droujkova, at Moebius Noodles, describes how a student combined ideas from the Moebius Noodles book, making mirror books to create fractal stars.

Nicora, at Bridging the Gap, suggests that we should let students break things to help them learn fractions. "Give them lots and lots of experiences where they have to break up things evenly and share things fairly." Discuss, put back together, discuss some more.

Steven Strogatz describes the math found in our cowlicks and fingerprints.


Denise, at Let's Play Math!, shares the power of stories: "The mere hint of fantasy adventure can change graphing equations from boring to cool." She used  Dan Wekselgreene's inequalities lesson based on the adventures of Ohio Jones. (Denise has written extensively about Fibonacci and Alexandria Jones. I think they must all be related somehow.)

Advertisers tell stories to convince us to buy what they're selling. Often their stories are deceptive. Mr. Honner describes how Prudential's 'oldest person you know' ad subtly points in the wrong direction.

There is a story mathematicians like to tell, of how the young Carl Gauss was asked, along with his classmates, to add up the numbers from 1 to 100, perhaps to give the schoolmaster a bit of time to relax. As the story goes, Carl saw a nice trick, and wrote just the answer down, turning it in almost immediately. Brian Hayes was curious about the historical accuracy of this story, and researched it. His article (in American Scientist) is quite intriguing.

Alexandre Borovik has written a delightful story about Anthony the Ant, and his discoveries of his world (a piece of paper, which gets folded into a cube).

Games & Puzzles

John Golden, at Math Hombre, has a games page that looks marvelous!

Did you know that some coins cost more to make than their face value?! Dan, at Math for Love, used this to make a math lesson, and one of his students came up with a great question. The puzzle she posed is whether or not you can come up with coins that are worth one dollar and cost one dollar to make. And another coins puzzle, from Nathan Kraft, at Out Rockin' Constantly: Which is worth more, a pound of quarters, or a pound of dimes?

I don't know if this counts as a puzzle - more like a problem-solving challenge. Nat Banting, at Musing Mathematically, asks what pattern the wet part of a tire makes as the tire rolls along.

How fast can you decide how many dots you saw? (The creators of this simple game have not left their names. They write: "Subitizing is the ability to immediately recognize the quantity of a small number of objects without counting. Research has shown subitizing to be foundational to basic arithmetic and other math skills. Many children who struggle with basic math also have trouble subitizing.")

Mike, at Spiked Math, has created a very visual puzzle.

Greg Ross, at Futility Closet, has given us the lovely puzzle you see below. You might word it differently, depending on your student's stamina and mathematical sophistication. (Instead of asking for proof, maybe just ask them to find all the midpoints first. Then ask if they can pick points so none of the midpoints will occur at an intersection. Finally, see if they can figure out why five points will always produce at least one line whose midpoint is on an intersection.)

Math Education

Crystal Wagner, at Triumphant Learning, knows that problem solving is at the heart of mathematics, and gives some guidelines and resources for keeping problem-solving at the heart of your math lessons. (I'd like to add two more resources to her list: The Art of Problem Posing, by Stephen Brown, who advocates for students to pose problems, like this. And for advanced math students, The Art and Craft of Problem Solving, by Paul Zeitz.)

Jenny, at Elementary, My Dear, Or Far From It, describes the benefits of confusion.  She also linked to this piece by Jeffrey McClurken, and this piece by Annie Murphy Paul. (It's hard for students in the U.S. to understand how useful confusion is - as a stage in learning anything new. I want to share all their ideas with my college students. Maybe I'll write a post consolidating it all...)

Dan Finkel, at Math for Love, gives an inspiring description (using his goats for comparison) between the fearful learner, who leaves others in charge, and the adventurous learner who takes charge both of their own learning and of the math problem at hand.

Mama Squirrel, at Dewey's Treehouse, wrote How I became a Math Teacher? to describe her journey into math teaching and her thoughts about the matter.

Cathy, at Math Babe, interviews the lead writer of the common core math standards.

Megan Hayes-Golding describes why ranking tasks are especially valuable as learning tools.

Alexandre Borovik, at the De Morgan Forum points to a paper [pdf] by Herbert Wilf, who argues that there is no useful math education research out there. The abstract states:

We examine a number of papers and a book, all of which have been cited, by people who are knowledgeable in the field, as being good examples of “research in mathematics education.” We find specific serious flaws, indeed fatal flaws, in all of them, so that no conclusions of any interest follow as a result of any of the “research” that is reported in these works. We have found no evidence that the research paradigm, involving test and control groups, randomized trials, etc., which is invaluable in the life sciences, is of any use whatever in studying mathematics education and we urge that it be abandoned, in favor of human-to-human discourse about how we can improve curricula and teaching.
Also at the De Morgan Forum are the results of a study that found that practice at "guesstimating" can speed up math ability.  (Hmm, isn't this research on math education? Maybe Wilf has different sorts of research in mind. I'd enjoy discussing his paper with anyone interested.) And another: Can you imagine a whole post on 3 - 1 = 2? Alexandre Borovik has translated a paper originally written in Russian by Igor Arnold, which gives 20 different problems that all boil down to finding 3 - 1.

Bruno Reddy, at Mr. Reddy's Math Blog, posted some interesting videos from a workshop he attended. If you teach students with limited English language proficiency, you may find this valuable.

Geoff Krall, at Emergent Math, is thinking about how to make his classroom a safe place for taking risks. That's a common theme at Cheesemonkey's blog.

Visual Math

Dan Walsh, at Dan's Geometrical Curiosities, saw this, and just had to figure it out. The mathematical description of what's happening is called curves of pursuit.

Here's your chance to make a bit of mathematical art: at Ahh...

If you were going to try to figure out how far it is to the horizon, what sort of picture would you draw? Bryan Meyer, at Doing Mathematics, thinks we can learn a lot about students' thinking by discussing the pictures they draw to solve problems with.

The two mathematicians described at the beginning of this post, Brent Yorgey and Richard Schwartz, have made their images to help people visualize factors, prime numbers, and composite numbers. Jeffrey Ventrella has the same goal with his composite number tree and his book, Divisor Drips and Square Root Waves (link is to a fascinating online version). There is also this intriguing Prime Number Patterns applet, by Jason Davies.


A simulation of glacier movement can be run backwards to predict where things were in the past. When the remains of some hikers who were lost almost 90 years ago were recently found, the simulation was used to figure out where they most likely were when they died.

Rachel Thomas, at +plus magazine, writes about the math of bubbles, which inspired the architecture of the National Aquatic Centre in Beijing, built for the Olympics.

The Rubik's cube has over 43 quintillion (4.3x1019) positions. It has recently been shown that there is a way to move it to each different position in sequence, without ever repeating a position. (This is called a Hamiltonian circuit.) Thanks to Robert Talbert for pointing this out on his Casting out Nines blog.

Caroline Chen has written a very readable account of the strange proof of the ABC Conjecture.

The Journal of Humanistic Mathematics has some interesting articles in its current issue, including one on Gallileo and Aristotles' Wheel [pdf] describing a paradox and how mathematicians think about it.

Joselle Kehoe, at Mathematics Rising, writes about brain research showing that the part of the brain that deals with counting deals also (and earlier in our evolution) with representing the fingers.

(from Pat Bellew)

Valentine's Day Math

Mr. Honner wishes us all a Happy Permutation Day.

Laura, at Math for Grownups, shared this quickie video (she calls if a gif, my son calls these vines) valentine.

A Few More Tidbits

Video Helpers for Algorithms and Problem-Solving (from Prairie Creek Community School)

Recursion (links to other collections of math links)

Brie Finegold, at Blog on Math Blogs, has ideas about how to get your friend to like math.

Math Munch comes out weekly. Anna Weltman, Justin Lanier, and Paul Salomon say: "We write Math Munch to help more kids find something mathematical that they love." Here's a post I liked on art and math, one of their favorite topics, I think. And this post is full of puzzles.
Don’t miss the 107th Carnival of Mathematics (our sister blog carnival).

That rounds up this edition of the Math Teachers at Play carnival. I hope you enjoyed the ride.
The next installment of our carnival will open sometime during the second week of March. If you would like to contribute, please use this handy submission form. Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Past posts and future hosts can be found on our blog carnival information page. We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the Math Teachers at Play blog carnival, please speak up!
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