I always preach mathematical fearlessness to my kids, my colleagues, and just about anyone who will listen. I go on and on about the importance of being able to sit with a new problem for more than 5 minutes, or even 5 days, and mulling it over, poking around, trying whatever you can to solve it without giving up or losing interest or Googling the answer. I always want my students to try

*something*in lieu of just staring, and I am usually stingy with hints.One thing - quite possibly the only thing - I miss about math grad school is the feeling of sitting down and figuring out a really, really difficult problem. At the time it felt a bit like "intellectual masturbation" because that's all I would do and I felt like I was contributing absolutely nothing to the world, but it definitely helped me develop my mathematical fearlessness. Since I started teaching it is rare that I actually sit down with a truly challenging problem and force myself to think it through from start to finish. So much of what I do is in a rush - I don't have time to

*really*think about a problem because I am trying to plan two lessons for the next day, so instead I think about it for 5 minutes and then look at how someone*else*did it. Shameful, I know. The question presented itself to me: Can I actually practice what I preach? Am I still mathematically fearless?My office-mate presented me with this problem about a week ago:

**You have a square dartboard. What is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge?**I sat down to solve it and was absolutely stumped. I had no idea where to start besides drawing a little picture. Said office-mate told me that he had banged his head against the wall and couldn't figure it out, which made me a little disheartened because I consider him to be a much more clever problem solver than I am. [Lesson one about how my kids feel: It's difficult for them to actually believe in their abilities when they look around and see classmates who they consider to be "smarter" who are also struggling with the problem.]

I took the problem home and presented it to my boyfriend, who really is the smartest math guy I know. He struggled with it for a little bit, went inside the bedroom, and came out about 30 minutes later announcing that he had solved it

*and*that he would not tell me how he did it. Fine. Be that way. At that point I was still really struggling; I didn't even feel like I had a solid starting point. I went into the bedroom and I have to admit that I glanced over at his clipboard where he had solved the problem and saw some nasty math that I didn't like. My heart sunk even more. [Lesson 2 about how my kids feel: When they see another student's solution and don't immediately understand it - and how could you really immediately understand someone else's solution to a problem? - they tend to give up because they think that they could*never*think of that solution. The thought doesn't even cross their mind that maybe they can come up with another way to solve the problem.]So I gave up for a few days, thinking about the problem a little bit here and there but never hard enough so that I'd feel like a failure if I didn't figure it out. [Lesson 3: Not trying hard is how kids avoid feeling like failures.] This went on until yesterday afternoon. My boyfriend and I spent the afternoon at my favorite neighborhood coffee shop, basking in the San Diego sunshine. I was working on my end-of-year comments when I suddenly remembered the dartboard problem. I asked him to tell me how he had solved it. He looked at me somewhat disappointedly. "Really? But then you'll never keep thinking about it your way." At that point I still didn't have a "way" but a small fire lit inside of me - How could I not have a "way"? Some idea, some line of reasoning? What would I say to a kid who asked me for a help with a problem and didn't have anything of his own to show? So I told him to wait a sec, and I took out a piece of paper and started working. I came up with what seemed like a great solution with a simple answer. Boyfriend checked the work and agreed, but then asked me to look over his solution because he had gotten a completely different answer and had been sure he was correct. As he was explaining it, he immediately found an error in his reasoning which made his method quite complex to carry out. However, we then went back to

*my*method and found an error, so we worked it through again together. I now have a solution that I'm pretty happy with and that is*completely*different from his solution, and it feels so good that it is mine. He was right - if I had looked at his solution first, I would have never had the guts or desire to come up with my own. [Lesson 4: The lengthy process really is worth it! Any human being - teacher, student, adult - feels amazing after coming up with a clever answer to a problem that once seemed insurmountable.]I would be happy to post my solution if anyone is interested; I would also like feedback because I wouldn't say I'm 100% confident. More importantly, though, I feel like this was an exercise in my own mathematical fearlessness. I gained a new respect for my students. As I know they do, I felt anxious, inadequate, and angry at various points in this problem-solving process. My new question is: How can I better support them so that they actually want to stick it out until the end?

I think telling your story at some early point in the semester would go a long way toward supporting your students in sticking it out. Struggle is vital to science, and seeing someone else's example always seems to help.

ReplyDelete"In light of knowledge attained, the happy achievement seems almost a matter of course, and any intelligent student can grasp it without too much trouble. But the years of anxious searching in the dark, with their intense longing, their alterations of confidence and exhaustion and the final emergence into the light -- only those who have experienced it can understand it." - Albert Einstein on Relativity

“Problems worthy of attack prove their worth by hitting back.” Piet Hein

Oh, and thanks for the blog. Keep up the good work!

p.s. I came up with my answer for the square dartboard. I'm curious about yours.

It sounds like you have pretty good insight into this process. I loved hearing about your students collaborating to solve problems. I am working through the Project Euler problems and I love how after you solve a problem you get access to the problem's forum. Seeing how others solved it helps me for next time.

ReplyDeleteThe first step is having a good problem. These unfortunately are scattered across the net and amongst various people's heads. I am usually inspired by blogs like Dan Meyer, puzzles from Gardner, sites like Cut the Knot and mathchallenge.

The next step is creating a culture and community of learning and interest. This is why I grade so little of our math. I have seen motivation and interest die off when there is a grade attached, plus competition and frustration arise.

I await for a future Infinigons post where you let me know the third and fourth step!

Thank you so much for posting this problem. I had a ton of fun figuring it out. I've taught geometry, calculus, and some probability things, and I have to say I had to use all 3 of those areas of my brain. I THINK I'm right .... so I'd also like a comparison with you .... and I want to see what other ways people used to solve it.

ReplyDeleteUpdate - I posted my attempted solution here: http://tinyurl.com/3bwd5bj ... I'm still a little doubtful because I can't seem to simplify my answer for the life of me, so any thoughts would be most appreciated! It's possible I'm not seeing something in the setup, or that there's some trick for simplifying at the end that I'm missing...

ReplyDeleteI plugged your answer into my graphing calculator, and it's the same answer as I got. Cool! I approached it differently, and I have to go out now, but I'll post later.

ReplyDeleteEssence: I also divided the square into basically your "1/8th" of the region. Then I considered parabola knowledge and basically broke the "area" I had to find into 2 almost like you did. Then I multiplied that area by 8 (for total dart success) and divided it by 16 (probability is the area of successful dart placement divided by whole area of square where my square was a 4x4 square).

Okay, that probably makes no sense in words ... more later, but again, thanks for posting the fun challenge.

I completely agree with Pat's comment. One of the best ways to help your students is to share your experiences with them. Knowing that you still struggle at times will help them connect to since as much as they want you to be perfect, they also want to see that you are vulnerable too.

ReplyDeleteShare this with all students and all teachers. When we open up to our kids we allow the relationship to grow deeper and the connection becomes stronger, and in my opinion, the learning too. Thank you for sharing this and, of course, your incredible thoughtfulness.

@Ms. Cookie, I'm curious about this "parabola knowledge"

ReplyDelete@Everyone - Eureka! sqrt(3-2sqrt(2)) = sqrt(2)-1!!! That simplifies the answer significantly. Revised solution here: http://tinyurl.com/3bhxz58

@Jenny - Thank you for reading my blog and for supporting me always. You are the BEST :)

ReplyDeleteYou might want to check these out as well if you haven't yet. They might serve you well in your new position:

ReplyDeletehttp://www.mathcircles.org/

http://wildaboutmath.com/2011/06/18/review-mathematics-education-for-a-new-era-video-games-as-a-medium-for-learning/

Okay, here is my solution to the target problem: http://www.box.net/shared/jpnqyy8dcc5opux9gqbm

ReplyDeleteIt's not as neatly written as yours, but this is basically what I did. And I couldn't quickly find a way to combine my 5 jpegs into one other than a word document .... and it's 2meg!

The 'intellectual masturbation' comment about grad school is exactly how I felt. I started a PhD program at UCSD, but quit it as quick as I could.

ReplyDeleteI love your description of the frustration, and avoiding something you think you might fail at anyway. The hardest part about math might just be keeping enough faith in yourself to sludge through the muck until you get some insight.

Ms. Cookie - how cool! I would've never thought to use parabolas like that.

ReplyDeleteThanks for this problem. I started with solving it if the dartboard were a circle, and that got me wondering... what if the dartboard were an equilateral triangle.

ReplyDeleteHere's my solution. It's a lot like yours, I think...

http://mathforum.org/blogs/max/problem-solving-journal/

Also, I'm really curious what your method was vs. your boyfriend's method. What was different about them? Did your two different insights help you figure out the glitches in each others' problem solving?

ReplyDeleteThanks for sharing this problem. I had a lot of fun/frustration while working this out, mostly fun. I had quite a few false starts and made arithmetic and algebraic errors; it took a bit of patience. Makes me think that maybe my students need to see me struggle with a question more often and how much time and persistence it takes to solve problems.

ReplyDeleteHere's a post where I pose an extension to the problem about n-sided regular polygons:

http://mrhodotnet.blogspot.com/2011/06/square-dartboard-probability.html

I hope I didn't make mistakes. Here I attempt a solution and verify your and Ms. Cookie's answer for n=4 (square dartboard).

http://mrhodotnet.blogspot.com/2011/06/ngon-dartboard-probability.html

Have you ever started reading a book, and then slowly realize that maybe you already read it, like maybe 5-10 years ago? That just happened to me for the first time with a math problem! Weird feeling! I <3 problems like this that use all the parts of your math brains. Thank you!

ReplyDelete@Kate - So do I! I also love problems that can be stated in one simple sentence that anyone can understand. You could pose this problem to a middle schooler and they'd understand the question, and maybe even come up with some clever approach to solving it that doesn't use integrals!

ReplyDeleteThere is a doubly interesting relationship between this problem, and the one I posted yesterday:

ReplyDeletehttp://blog.matthen.com/post/6738340459/this-is-a-good-example-of-a-curve-that-can-be-made

(check out the comments), it all boils down to this picture:

http://www.matthen.com/misc/twosquares.gif

(4*(2^.5)-5)/5 ... take the difference of two integrals (one of which is the area of a triangle) and divide it by the area of another triangle

ReplyDeleteAm I missing something ... if the dartboard is circular then the the dart is closer to the center than the edge if it lands within half the radius of the circle. Can't we just compare the ratio of the areas then? If the circular dartboard has a square frame around it then the denominator changes, but the numerator stays the same.

ReplyDeleteHere is a the full process I went through to solve this problem:

ReplyDeleteStep 1: See a reference to the problem on Dan Meyer's blog and thought it sounded interested. Find this page

Step 2: Grumble to myself about pet peeve with probability problems using dart boards since most people actually aim for the middle, so have strong aversion to using a dark board as a model for randomness. Consider adding line "I am so bad at darts that I have an equal chance of hitting any spot on the target" to turn problem into joke instead.

Step 3: Spend 1 minute running through and throwing out trivial solutions.

Step 4: Realize that it is some kind of curve, and that problem can be simplified by looking at smaller triangles (gotta love symmetry).

Step 5: Consider a generalized algebraic solution - then realize the problem has a high chance of containing messy powers and roots. Lose motivation, go with simple numerics. Setting square area to 1 means just finding area under curves with no need for division at end.

Step 6: Go out on my inlines and forget about problem. Nothing on paper at this point.

Step 7: Unable to forget about problem while skating. Run through obvious curve types while sakting: parabolic, hyperbolic, trig, etc. Almost run into a biker while distracted. Realize I need to just sit down and solve the simple equation.

Step 8: Get home and draw a square on an XY axis around the origin, about 1 minute of simple algebra give the equation of the parabola.

Step 9: Smack myself on the head and say "D'oh" for not remembering that this is actually one of the definitions of a parabola - all the points equidistant from a point and a line.

Step 10: Spend a few minutes drawing a nice picture in GeoGebra mainly because GeoGebra is really cool and really fun to use.

Step 11: Pick on of the many options for integrating to find the area under the parabolas.

Step 12: Roll my eyes at ending up with messy powers and roots. Do them anyway.

Step 13: Type my integral into Wolfram Alpha to check the answer - but mostly as an excuse to use that site since it is even cooler than GeoGebra.

Step 14: Check my answer against Allison's and feel satisfied that it is the same.

Step 15: Want to continue with this since it was fun. Think about Mr. H's extension to N-sided polygons - see the basic idea but realize the calculations are likely to be a bit messy. Attention span wavers.

Step 16: Lament privately that I can't give this to my students since I am doubtful there is a way to do this without integals (I would be curious if someone found one) and I don't currently teach that level of math.

Step 17: Realize that I could give the first step of the problem to advanced students - have them just find the equation of the parabolas. Possibly have the really good ones describe the area and put some sort of clever upper and lower bound on the area.

This is the (very positive) direction in which we are taking Math instruction over the next several years. By giving students challenging tasks and allowing them to struggle, collaborate, and revise ideas based on feedback, we are creating mathematical *thinkers* that are far more adaptable than "procedure followers." The Common Core standards will require educators at all levels to embrace this methodology, and the Shell Center is putting together a set of tasks like these that are aligned to the Common Core standards.

ReplyDeleteI know it's the right way to go because, when I attended a PD on this idea 2 weeks ago, they gave us sample math tasks that *I* enjoyed solving (like this one!). I got the same answer as you and used basically the same methodology except that I used the "upper" triangle so I didn't need to split the integral.

- Julien

Damn it, I got "one third of (13 - 8*sqrt(2))". Must have done something wrong in my three pages somewhere. Had fun, though, and will try to correct it later.

ReplyDeleteMy dartboard went from (0,0) to (2,2) so that the center was at (1,1) and I used an upright parabola, which I think is easier to integrate. Mind you, I did have to deal with a *lot* of surds, so maybe it's not easier.

Hopefully I can extract some parts of the problem for the delectation of students. Thanks for the stimulus!

If you figure out how to teach this skill of risking failure and trying really hard, please come teach me? I'm working on writing my first paper and I'm still not very good at it. I occasionally try to teach it to my students, and it almost always goes poorly!

ReplyDeleteI'm glad to see that you seem to be happy and doing well!

Anna

Sweet. Ms. Cookie - my solution was almost identical to yours, except my calculus skills are a bit rusty so I only integrated y=-x^2/4 + 1 from 0 to -2+2sqrt(2), and then subtracted that integral by the triangular area A=0.5(-2+2sqrt(2))^2 which does not lie inside the 1/8 slice. Then I divided by the area of the 1/8 slice to get the same probability.

ReplyDeleteI like this problem.

Allison, I'm making the answer key for the puzzles in the book (

ReplyDeletePlaying With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers), so I *finally* did this problem, just now. I loved it. I did the portion above the diagonal, so the integration would be easier. My only mistake was in simplifying the square root messes. Wolfram wouldn't do that for me. (Very interesting to bump up against its limitations.)I'll send you my write-up.