Author: samjshah

Mathematics in Context

I have a bit of teacher-musing, and then I want to share a funny story that happened in calculus class yesterday.

MUSING

Everyone is always saying “make math relevant for your students!” Well, great. Sure. Okay. Will do. I think that’s a bunch of hooey. It’s not about making math relevant, but making math into problem solving skills [1]. I don’t care if students will ever have to use what they learn in my class in the “real world” (what world are they living in now?). But I do care that they can see a problem and break it apart into its component pieces. That they can go down dead ends without getting frustrated. That they can see that there are often multiple ways to solve a problem. Sure, sometimes one solution is more elegant than another, but that can be okay… because sometimes problems don’t have elegant solutions at all!

All this being said, I’m terrible at doing this. At least, I’m terrible at doing this in my non-accelerated classes (which are my Algebra II and Calculus classes). I stick close to the basic skills and concepts, we don’t do too much investigation; I’m focused on making sure they can do basic skills. That they can verbally explain the concepts. Which is unfortunate, because after students learn basic skills, they should be given the opportunity to draw connections, hit dead ends, and all that good stuff I just listed above.

As a teacher, it’s so easy to make excuses (we have a fast-paced curriculum, the class period is only 50 minutes, my students are working on so many different ability levels, there isn’t enough time in the day to design these classes) of why not to do it. I also think that investigative work doesn’t go well in the non-accelerated classes.

But that’s probably a function of me not knowing how to do it right — how to design and implement these sorts of lessons without spending too much time at home or in class working on them.

FUNNY STORY

So back to what I was saying… the mantra “make math relevant” actually took a funny turn after my calculus class yesterday. To set this up, I have to remind you that I teach at an independent school. Tuition is high and students tend to come from wealthier backgrounds. It’s a different world [see my post about that here]. Anyway, after school, one of my calculus students said “Mr. Shah, I have a math question for you. It’s not related to what we’ve been learning in class.”

Turns out, he found out that he and one of the other seniors both resort at the exact same place in the summer. And after thinking about what a strange coincidence it was, he wanted to know “what are the chances that two seniors at this school both resort at the same place?”

So that’s math in context for my students, apparently. 

(These are the sorts of moments that I realize that this world is so different than the world I grew up in. But I really like these kids.)

If you care, my answer was that we can definitely figure it out together. I then told him about the birthday problem (how many people do you need in a room before the chances that two of them have the same birthday reaches 50%?), and how we could use that as an analogy to solving our problem.

In the birthday problem, you have n people in a room, and each of these people have a birthday from the calendar year (each person could have one of 365 birthdays). In our problem, we have 80 seniors, of whom “only “n resort. Each person who resorts goes to one of a certain number of places (we’d have to do a back of the envelope calculation/Fermi problem to find an approximate number of resorts that people from my school could go to).

Then the analysis would be the same as the analysis for the birthday puzzle.

If you’re curious, you need about 23 people in a room to have a 50% chance that two people in the room have the same birthday. Of course, you’re going to need 366 people in a room to guarantee that two people in the room have the same birthday. (Do you see why?)

[1] That’s not to say that I don’t think the the content is unimportant. The content is always my primary focus. And I guess when I say “relevant,” I mean specifically “real world applications.”

Blog Review of “Wild About Math”

Sol Lederman, at the blog Wild About Math, had an idea of a blog-review-exchange (he writes reviews of math-related blogs, in return for math blogs reviewing his) [see his post here]. I like this idea, if for no other reason that I get introduced to new math blogs through reading his reviews. So here I go.

Wild About Math has been in existence since October 2007. His first post clearly articulated the goal for starting his blog: “This blog is the expression of a life-long passion that I’ve had for all things mathematical. My sincere desire is to share articles, reviews, and links to products, services, and web-sites that inspire people of all ages to enjoy Math.” Sol is uncharacteristic in that he was able to continually maintain the original intent of his blog for months (to the present!), without giving into laziness or devolving into a forum for personal screeds (although I do have a soft spot in my heart for the blogs with the latter).

My favorite aspect of Sol’s blog is the Monday Math Madness competition, which is alternates between Wild About Math and Blinkdagger. [The current MMM problem is here.] In recent months, besides the mini-blog-reviews, most of the blog posts on Wild About Math are related to Monday Math Madness. The problems are easy to state, range from easy to difficult, but always are engaging. The best part is that one (usually) doesn’t need more than precalculus to solve the problems, and usually less. Some of the best questions take forever to think through, even though they don’t require higher level mathematics. Those are the best kinds! I have even gotten one of my high school students hooked — and he is sending in solutions to the competitions when he has time!

Although the original intent of the blog has been maintained, the nature of the content has shifted. As I noted, in recent months, Sol has focused on nurturing his (wonderful) Monday Math Madness contests. However, before this contest started, Sol had frequently posted about everything and anything math, like neat websites about fractals or speed multiplication. To see the nature of these posts, some of Sol’s favorites in fact,  you can check them out here. And I don’t know if this is still happening, because I am not a subscriber and didn’t know about this until looking through the back-blog-posts, but Sol offers (or used to offer) a supplemental periodic email called “Math Bites”.

Some of my favorite posts:

  1. Experience With Math Camps? (which inspired me to write my own recollections! [1])
  2. Uncountably Many Errors in In Texas Math Books (the type of information that I’d miss if I didn’t read this blog)
  3. Review: Numbers Juggling (a review of a math website which goes in depth – we need more reviews like these)
  4. A Hard But Fascinating Puzzle (a problem which got me thinking deeply!)
  5. TI-Nspire Inspires Math Students (a review of the TI-Nspire)

Overall, this is definitely a blog worth keeping in your reader (or putting into your reader if it’s not there yet) for the Monday Math Madness contests. I do wish, however, that at least once in a while Sol would continue to post same types of posts — the great reviews of books, math resources, and calculators, the “for fun” math puzzles, for the math news — that defined the blog before Monday Math Madness came into the picture.

[1] You can read it here.

WebAssign: A Quick Question

I was reading Teaching College Math and came across a glowing review of WebAssign. (Her great powerpoint here.) Wow! Math help websites have come a long, long way! I was wondering if anyone out there uses it, and if so, what the pros and cons of it are? Are your students satisfied with it, or do they complain about how it works?

Why I’m acute to this: In grad school, I took a few undergrad French and German classes, and we had web assignments too. Unfortunately, although a great idea in theory, those assignments were a nightmare in practice. Missing letters, a different way to say things, a forgotten umlaut or accent, an extra space, etc., would render an entire question wrong. Or you could be completely correct, and you’d still be marked wrong. Everyone hated it. I want to make sure that WebAssign doesn’t have those sorts of annoying bugs. I want to make sure if you write y=\frac{x-1}{2x-1}, it would be marked the same as if you had written y=\frac{1-x}{1-2x}.

Movies about Mathematics

I asked my department head if we had a budget for DVDs, so that we could start creating a small DVD library for us to use (in class, in mathclub). She said yes, and put me in charge of finding DVDs. I’ve ordered a bunch, but tonight, I came across 0ne more that I want so dearly that I wrote an email pleading to case get the $35 to purchase it!

*****


Julia Robinson and Hilbert’s Tenth Problem

*****

Other DVDs that we’ve ordered and that I’m excited about include:

*****


Hard Problems (two youtube clips from the movie: clip 1, clip 2)

*****


Chaos (a series of  24 lectures, 30 minutes each, from the Teaching Company)

*****


N is a Number: A Portrait of Paul Erdos (on youtube: part 1, part 2, part 3, part 4, part 5, part 6)

*****


The Elegant Universe (on String Theory)

*****

I also got the old classics: Stand and Deliver, Good Will Hunting, and A Beautiful Mind.

And I will also soon be downloading this movie on various Dimensions.

I wish that NOVA’s The Proof (about Andrew Wiles solving Fermat’s Last Theorem) was out on DVD, but alas, no such luck. It is on VHS and on youtube (part 1, part 2, part 3, part 4, part 5).

I haven’t watched Dangerous Knowledge — a documentary on Cantor, Boltzmann, Godel, and Turing — yet (I don’t like the trope of mathematician as crazed genius), but it’s on youtube here: part 1, part 2, part 3, part 4, part 5, part 6, part 7, part 8, and part 9 and on Google Video here.

And finally, I wanted to show my Algebra II class something about fractals tomorrow, since we introduced complex numbers today. I didn’t do much searching, but I did find Arthur C. Clark’s movie on Fractals, which — sans the annoyingly trippy music — doesn’t seem too bad: part 1, part 2, part 3, part 4. It isn’t very up-to-date, but it does have a lot of famous people talking about fractals.

Any more recommendations? Throw ’em in the comments below.

Is it bad that…

I’m already counting the days until Spring Break? In the last few days before winter break ended, I’ve been inspired to start on some teaching projects… but I didn’t because I know I won’t have the time to finish them. I should have started them earlier in break. This always happens to me, where I initially want to be a complete schlump for break and then regret it in a fit of flurry at the end. Here’s what I didn’t do, that I hoped to do.

1. Create a project and rubric where calculus students video an object changing position over time (and velocity over time), and then analyze it using LoggerPro.

2. Create a topological project for my multivariable calculus students.

3. Come up with a revised, scaled back video project for my Algebra II students, like last year’s project.

4. Think about a string hanging from the ceiling of a classroom, creating a pendulum like Foucault’s Pendulum. I would give it a start and watch it trace out some crazy spirographic curves. I wonder how these curves connect up with polar coordinates/polar equations. I don’t want to look it up, but instead explore and think about it myself. (I don’t want to use it to study the motion of the earth.) Maybe expand my study to harmonographs and lissajous curves. Could be good for mathclub.

5. Think about creating a second semester class blog for my Alg II class, like this one here. I really like the idea of students having to communicate what they learned for others to refer to. It’s valuable (if done well) on so many levels.

6. Compile all the class data from this linear regression sheet to make one giant data set for us to look at together. I really dropped the ball on this — I wanted to do it but never had the time/motivation to enter all the data in an Excel sheet. I should have had students all enter it in a Google Spreadsheet. Sigh.

7. Start working on making an interesting 3 day calculus midterm review.

Yeah, it’s pretty bad that I didn’t do any of that. Sigh.

Venn Diagrams and Formulas

At the math office today, two math teachers were discussing probability. Two things were surprising about it. One, it was a Saturday afternoon part of our winter break, so no teachers should have been on campus. (We’re a dedicated lot, us math teachers.) Two, the topic they were discussing was so simple, and yet, it reveals the real mind-bending character that probability has on us and our students.

Question 1: What is the probability that you draw a heart or a queen from a deck of cards?
Question 2: What is the probability that you roll a die and get a number less than 4 or an odd number?

Both questions are simple enough. The first one is 16/52 (because there are 16 cards which are hearts or queens in a deck). The second one is 4/6 (because you can roll a 1, 2, 3, or 5).

The problems are seemingly the same. Let’s now look at this problem from the perspective of a venn diagram.

picture-1If we want to know the probability that event A or event B occurs, we clearly can see that we have:

P(A\text{ or }B)=P(A)+P(B)-P(A \text{ and } B)

(We have to subtract that last term, because we added that overlapping section twice when we took P(A)+P(B).)

Let’s apply that to our two questions:

Question 1: We have a probability of: P(\text{Heart or Queen})=P(\text{Heart})+P(\text{Queen})-P(\text{Heart and Queen}). Clearly P(\text{Heart})=13/52 and P(\text{Queen})=4/52. What is P(\text{Heart and Queen})? Intuitively — or using our venn diagram — we know it is 1/52. And the answer works out correctly to 16/52.

Question 2: We have a probability of:  P(\text{less than 4 or odd number})=P(\text{less than 4})+P(\text{odd number})-P(\text{less than 4 and odd number}). Clearly P(\text{less than 4})=3/6 and P(\text{odd number})=3/6. What is P(\text{less than 4 and odd number})? Intuitively — or using our venn diagrams — we know it is 2/6. And the answer works out correctly to 4/6.

However, let’s say we wanted to calculate P(\text{Heart and Queen}) and P(\text{less than 4 and odd number}) (the overlapping regions) mathematically? It turns out that there is something fundamental that makes these two problems different. In question 1, the two events (drawing a heart / drawing a queen) are independent. In question 2, the two events (rolling a number less than 4 / rolling an odd number) are dependent. For the first question, you can say that P(\text{Heart and Queen})=P(\text{Heart})P(\text{Queen}) while in the second problem you cannot do that.

Recall that the definition of independence of two events A and B is if P(A|B)=P(A).

Checking the first question for independence, we see that the probability of drawing a heart given that you already have a queen is 1/4, and that is the same as the probability of drawing a heart (1/4). (Similarly, the probability of drawing a queen given that you already have a heart is 1/13, and that is the same as the probability of drawing a queen (1/13).) So the two events are independent.

Checking the second question for independence, we see that the probability of rolling and odd number given that you have rolled a number less than 4 is 2/3, while the probability of rolling an odd number is 1/2. (Similarly, the probability of rolling a number less than 4 given that you’ve rolled an odd number is 2/3, while the probability of rolling a number less than 4 is 1/2.) So the two events are dependent.

The teacher who brought up this problem was grading exams, and one student had calculated P(\text{Heart and Queen})=P(\text{Heart})P(\text{Queen}). And seeing the two problems were almost identical, calculated P(\text{less than 4 and odd number})=P(\text{less than 4})P(\text{odd number}) — which, as we know, isn’t right for dependent events.

What we were discussing is how we could explain to the student that the two situations are different, even though on the surface the questions seem like they are of the same form. In other words, is there a conceptual — non mathematical — way to explain that the first question involves independent events while the second question involves dependent events? It certainly isn’t intuitive, at least not to me.

Analyzing Parametric Equations

I saw a tweet that sumidiot posted on the parametric equations x(t)=t+c*\sin(2t) and y(t)=\cos(t) and spent a good 30 minutes thinking about the value of c which makes the graph intersect itself exactly once. I was going to post about my solution, but I was beaten to the punch. And thank goodness, because there are graphs and everything on sumidiot’s solution, which you should read here.

I still haven’t fully evaluated twitter, which I will post about later once I come to some solid conclusions. But I’m leaning towards liking it. I’ve got lots of good links, anyway. If you want to be my twitter friend, I’m here!