# Bringing the Problem to Physical Reality: Trigonometry

This post is part of a series of lessons about immersing students in an environment so that they can ask questions of their “surroundings” instead of their teacher.  See my other posts and an introduction.

The night before I wanted to teach trigonometry to my students I gave them the following diagram.

This diagram took up an entire side of a piece of paper.  On the other side of the sheet I gave a huge, 4-column, 37-row chart.  The first four rows are below.

 Angle x-coordinate y-coordinate ratio of y/x 0 degrees 10 degrees 20 degrees

and so on, down to 360 degrees.

I included the instructions below.

“This is a diagram showing a circle with a radius of one.  There is a 40-degree angle drawn on top of it.  Please check now to see that the angle ends at (approximately) the point (0.76, 0.64).  We can approximate to the hundredths place from this diagram.

“On the opposite side of this sheet you will see a table with rows for every angle between 0 and 360 degrees, in increments of ten degrees.  In the row for 40 degrees, please enter 0.76 under the x-coordinate, because 0.76 is the x-coordinate of the end of the 40-degree angle on the diagram.  Please enter 0.64 under the y-coordinate, because 0.64 is the y-coordinate of the end of the 40-degree angle.  Finally, please enter 0.84 under “ratio of y/x,” because 0.64/0.76 is approximately 0.84.

“Your homework is to fill out this chart completely, for all 35 of the other angles listed.  The coordinates you get will vary as you choose other angles.  You will need a protractor to draw angles – please actually draw the angles you need to measure, and do not attempt to estimate angles without a protractor.  If you find a logical shortcut, you may use it.”

Of course, the “x-coordinate” column is what we call $cos(\theta)$, the “y-coordinate” column is $sin(\theta)$, and the “ratio” column is $tan(\theta)$, but the students don’t need those names yet, so I didn’t introduce them.  I wanted to meet the students at their own fluency level, and they are perfectly comfortable with x- and y-coordinates, protractors, angles, and ratios (this is a precalculus class).

Why would I start with “there’s a function called sin(x) that gives you the y-coordinate of a point at the end of an angle inside a unit circle?”  We focused on the calculation of cosine and sine before we had the names, and calculated it many times.  In our discussion of our values, the students were getting frustrated with saying “the x-coordinate of the point at the end of the angle where it intersects the circle,” and so I innocently mentioned that this is what mathematicians call “cos(x).”  Pretty shorter, huh?  They dug it.  Some of said, “oh, this is how you calculate cosines?”

The work they did at home varied.  The clever idea here is that you can actually fill out the entire chart with only nine measurements by using various symmetries. I would say that most students made 18 measurements – the x- and y-coordinates in the first quadrants – and then filled out the rest of the chart.  Some students came in with fully half of the measurements made but their chart incomplete, giving variations of “I can’t be bothered to do this kind of grunt work” (and I don’t grade homework).  Some students made all 70 measurements.  All of my students made some attempt at the homework – I think that somehow it’s kind of a fun activity!

This simple circle, with its highly-structured instructions and almost no student initiative, has the benefits of an immersive physical environment.  The students measure values directly off of the circle, and when they see the symmetries involved they can check them for themselves.  During class, when we’re talking about various properties of cosine and sine, they can check values directly.  When I eventually ask about 45 degrees and 15 degrees, they are fluent in this calculation and can easily adjust what they are doing.  When I ask about circles with different radii, they have intuitive guesses (some right and some wrong: great!) about what will happen to the x-coordinates and y-coordinates.  They are so comfortable with the measurements at this point that they can see the logic in using $2\cdot cos(x)$ instead of making a new cosine function for a circle of radius two.

Learning happens when you’re comfortable enough with a situation that you can experiment.  You have to be able to change the initial conditions a little bit, see what happens, and use that to find a pattern, to form a generalization in your mind.  This circle gives students something they can experiment with in a way that triangles don’t.  While they could certainly construct right triangles, it would be too hard for them, and take too long – they would be able to quickly try different angles, or to see all of the angles at once like they can with the circle.  The symmetry is hidden with the triangles, but it’s glaring with the circle.  This circle, even though it’s just a circle on a piece of paper, is a whole environment which the students can explore.  And, importantly, they are already good at the skills they need to explore it.  They can find new information with old skills.  That’s what this series is about!

Update 2/1/2011: Changed evaluation of sin(40) from 0.72 to 0.64.  I’m that powerful.

# Monty Hall, right?

For the last three weeks I’ve been studying probability with my Algebra 2 students, thinking: “too bad there’s no good way to teach this.”  Probability only works on large scales, and then it really only approaches working.  The kids all get the basic stuff, like coin flips having 50/50 odds, they all don’t get the more complicated stuff like standard deviation.  They lack the tools for problems with continuous distribution, and as far as I can tell that leaves us with carnival games and card tricks.  Which only work on average.  By being my interested, lively self during class I managed to interest half of them for most of the time (what are the odds that you’re interested in this question, Johnny?).  But it was a hard unit for me.

Today, last day of class, the final is over, and I think to myself, “ok, let’s talk about the Monty Hall problem.”  (Knowing what the problem is is necessary to understanding this post.)  I bring the requisite bowls and candy prizes.

“Who wants to play?” I ask, and J raises her hand.  They don’t know the rules yet.  I just say, “One of these three doors (bowls) has a tootsie-roll pop in it.  If you guess which one, you get to keep the candy!”

So she guesses, and gets it right, and I give her the lolly pop, arcing it to her desk.

“Who else wants to play?”  More hands go up, and we play several more times with this simple rule.  “What’s the probability of winning?” I ask, and they easily respond with some form of one third.

On the fifth round, L is playing, and she makes her choice.  I say, “L, your guess was as good as any other.  But I want to give you another chance.  I will show you that this bowl over here <lifting bowl> is empty!”  She, and the rest of the class, stare at me blankly.  “Do you want to keep your original guess, or switch!?”  I say this channeling Regis Philbin on the $500,000 question. She stays. “Is that your final answer?” She says that it is, and with a flourish I show her that she is right; she’s won the candy! Congratulations! Congratulations all around! We play again. I stop after showing an empty bowl to ask about the odds of the last two unknown bowls, and the entire class is quite confident that there is a 50% chance that the candy is under bowl 1, and a 50% chance that the candy is under bowl 2. This is intuitively obvious and not true. Excellent. They still don’t see anything fishy when we play another time, and again I ask about the chances of the bowls. I say, “So, when you chose at first, you had a 1/3 chance of winning. Now you say that the same bowl you picked has a 1/2 chance of winning. How could flipping over an empty bowl over here improve the chances that this bowl is a winner?” The students sort of stammer – they aren’t fluent in this language and don’t have the vocabulary to convince me that it’s true. A student in the back starts to give an example. He says, “Imagine if you had fifty bowls,” and I immediately whirl around and draw 50 bowls on the board. It takes a long time, and I make a spectacle out of it. 50 bowls. One of them has candy in it. “Ok,” he continues, a little incredulous that I actually just drew fifty marks on the board. “And the player chooses one, so -” “Which one?” I interrupt. “Uhh…” “Please come up and circle one!” Does this bowl have the candy? I say, “What are the odds that you just chose the right bowl?” He says, “1/50,” and the rest of the class agrees. I say, “Ok. So now, as the host, I’m going to open all the doors but the one you chose and one other. If you chose right at the beginning, the extra door will be empty. If you chose wrong, it will have the candy in it. Here we go!” And I make a big, slow deal of erasing all but one other mark. The original choice and one other bowl remain. What are the odds that the candy is in the other bowl? “Which bowl do you think has the candy?” And they all think that the lower left bowl probably has the candy. Several students are laughing at this point. “What are the odds?” I ask, and they falter! “Aren’t they 50/50?” I ask. “There are two things, and one of them has the candy, and we don’t know which one, right? So what’s the big deal?” H protests, “but in this example you chose the right one to leave! Obviously it has the candy!” “Isn’t that what I did before?” The rest of the lesson goes on like you’d expect. We make a tree diagram. I extend the example: imagine you had to pick the correct blade of grass out on the lawn, and then I went and mowed all the other blades of grass except one. Do you think you’d pick correctly first, or second? Etc. We work out numerical probabilities. But this lesson is different from the others in probability in that the kids are engaged. This problem should have been first. I wish I based the whole subject of probability around it: • The students can describe the problem well. They think they are fluent at first but later find they are not, and are intrigued to find better ways to describe what is going on. No other probability scenario I found had this quality. • We don’t have to try a million times before the probability gives actual results. Because of the role of the host, knowing probability actually helps kids win the game in an interesting way (much better than “you should bet on 7 because it comes up 1/6 of the time which is more than other numbers”). • This game is actually fun to play. Who knows why. The switching thing is great. All of my quarter and die games were flops. For the last five years. I thought I would like teaching probability, but I don’t, because it claims to be so practical and is actually so impractical. Do you have any ways of teaching it that you like? # In Which Riley Admits To and Eventually Embraces Being Sappy Our most important job is as a role model for children. Our students spend a huge percentage of their lives with us and our colleagues – they’re not only learning math, right? They’re learning how to speak to people, how to treat people, how to hold others responsible and be responsible to others, manners… they are learning everything. And yet the professional development I go to focuses only on the math, and the tests my kids take focus only on the math, and the grades I give are only on the math, and it is so easy to forget that there is so much more going on here and that it is so much more important than the math. Other people are doing better work than me in their treatment of SBG, math curriculum, interactive, thought-provoking lessons, etc, and I’m so glad of that. My skills in these areas have soared enormously this year from reading your great blogs (see list on the sidebar). Thank you! What I don’t see as much of is the explicit focus on how to help our students become better people. Even on my blog, where every other post seems to be about how to empower students constructively, I often steer away from the topic, and you (on statistical average) don’t read as much when I do write it! Here’s why: it’s intractable, it’s easy to be wrong, we have other stuff to do, and we’re not explicitly held accountable for our social curriculum. But it’s the point. Teaching our students about community and society is the whole point! I was prodded in to seeing this imbalance in emphasis just last week, actually, when I was thinking about how to make my blog more popular (#okIadmitit). I have to tell you that when I noticed nobody (small exaggeration) reads my homey-values posts and that everyone (big exaggeration) reads my technical posts I was disheartened. I thought no one was interested in the really important part of school! Then I realized all the reasons that these core values are so much harder to handle than concrete techniques. And then! @samjshah, @jybuell, @monk51295, and @mctownsley wrote and said that they thought these are important ideas! I am surprised by how affected I was by these tweets – a mixture of “They really do like me!” and “oh, of course people care about the really important part of school! We’re all so bogged down that we just don’t have a lot of time to focus on it!” After all, we all focus on community, responsibility, and respect in some way. None of us lets students swear and insult each other, or flip their desks upside down, or paint on the walls. We all teach kids how to be members of society automatically, by being good members of society ourselves. When we write a good lesson plan, more often than not it is good because it empowers students and helps them interact in a good way. I just want to focus on social skills and norms specifically. I want to improve in this dimension the same way I’ve been improving in assessment and feedback, lesson plans, and classroom management! Do I just miss these sessions at NCTM, or is there a whole strand about “Personal Responsibility and Other Social Skills in the Classroom?” With this new realization I feel more free to spend a large part of my writing time on the social values of teaching. I’ve organized a page about it: http://larkolicio.us/blog/?page_id=412. Don’t get me wrong: I’m still a huge dork and will be posting geogebra applets, and I’m still passionate about math curriculum and will be posting lesson plans. As we move in to summer I will be writing more sparsely, and more specifically about camp (as education). But I will no longer feel that no one wants to hear about my hippier (certainly not hipper) side. Thanks for reading, you guys. This blog is living up to its name for me. PS: You know what profession has great professional development opportunities? The camp director profession. I run a summer camp (still accepting campers 9-13!) and so I get to go to conferences that focus on these important skills instead of the random skills like factoring that we deem vital for our future mathematicians (or whatever). Like a quarter of the sessions are fun games you can teach to kids to build skill xyz. You should go to an ACA conference even if you have never been to summer camp. # Bringing the Problem to Physical Reality: Modeling a Pendulum’s Period This post is part of a series of lessons about immersing students in an environment so that they can ask questions of their “surroundings” instead of their teacher. See my other posts and an introduction. I wanted to teach data modeling. The students already knew the basic shapes of lines, hyperbolas, parabolas, cubic functions, square roots, etc, and how to graph them, but we hadn’t talked about what to do with real data. The tempting approach is to give out 25 points and say “find an equation based on the functions you know that goes close to these points.” In fact, that’s what I did in my first year. This is terrible. In my second year, still burning from the bored stares of the first year, I convinced our facilities manager to climb something like nine meters up a terrifying ladder to hang a couple of ropes from the roof of our gym. Later, I brought my class to the gym with bunch of weights between 2 and 10 pounds. I described a couple properties of pendulums, did the whole omg-its-going-to-smash-his-nose-oh-whew-he’s-fine trick, and finally defined “period,” complete with a tutorial of how to measure it (give all the kids stop watches, have everyone time it at once, and then give an official measurement so they can see if they were too slow or too fast or what). I made sure NOT to say that the period of a simple pendulum is determined by its length alone, and that weights have no effect. They do not know at this point that period vs. length is a square-rooty function. Then, I said, “please make your pendulum swing so that its period is within a tenth of a second of 4.5 seconds.” This is great. Here’s why: the students are conversationally fluent in this environment. They know that pendulums swing back and forth, they know how to tie knots, lift weights, push things, measure time and distance, and they consider themselves experts in this area. In fact, they are experts – they have been practicing physical interactions every day for the last 16 or 17 years! But they lack the technical precision to describe it well enough to predict specific measurements. When they realize they actually don’t know how to achieve a specific period, they will be intrigued. Some will think they do know how, but then be unable to do it, and they will be even more intrigued. This is why you spend hours to create a real environment. They are good at reality, and feel comfortable in it. They have vocabulary to talk to each other about it. Compare this with discussing square root regression in class – they’ve learned about root functions and practiced for, what, 2 hours total (against 75,000+ hours of practice with physical motion and perception). They don’t feel comfortable talking about what they’ve learned yet, and they certainly don’t have the vocab to experiment with what they’ve learned by tweaking it in small ways. In the real world, they can easily say “let’s try pushing harder” or “add more weight!” and see results instantly. Anyway, the students spend the next 40 minutes trying things, different weights, different initial velocities, different angles, and different pendulum lengths. Only the lengths make a difference. They naturally figure out which direction makes the period longer and which makes it shorter, and I jump on that opportunity to point out that they’re already fitting a pattern to what they’re seeing (aka translating reality into math). They try a bunch of different lengths, but never get that close to 4.5 seconds. “Why not?” I ask innocently. Some say they don’t know, others say they can’t reach high enough. I ask, “Well, how high will you have to go?,” again innocently enough. “I could bring a ladder to the next class I suppose, but I don’t want to go to the trouble of bringing the wrong length. Those things are really awkward and weigh a ton.” Guys, they are hooked right now. My question was so natural, so authentic that they want to find me an answer just as a matter of course. I obviously need that information to help them out, not just to be an obnoxiously unhelpful math teacher. As homework they look at their data, figure out how they have to organize it to get me an answer more precise than “it needs to be higher,” and bring that answer to me at the next class period. We go to the facilities office, get the appropriate ladder, and try out their answer, which turns out to be astonishingly close to 4.5 seconds (easily within our measuring precision). A cheer goes up! Now, I’m happy, but not that happy, because every group used a line to model the data. It turns out that a line is a pretty good model for pendulums as long as you stay in a pretty small range. So I asked the kids, “how high would you have to be to get the period to be one second?” Their linear models told them something like 16 meters, and it was obvious (after converting to feet) that the ceiling was nowhere near 16 meters tall. We spent maybe 20 minutes at this problem, talking about domains of models, practicing using the models to predict various values, etc. A ton of normal curriculum at the board fell in to this discussion, and they got a lot of personal practice time (please always mix practice in to your lessons). I suggest that we use a smaller pendulum to get more data. This required changing our models to be functions of the length of the rope instead of the height off the ground (if all of my kids graduated knowing only how to change the parameters of a function I would consider myself a success). I bring out a few pendulums of the 1-, 2-, and 3- meter variety (I brought some twine and a pair of scissors to class) and we get some more data points. CLEARLY not linear now. We get MORE practice fitting the same data with a different type of graph. Now we can compare models, learn about different kinds of errors, answer fun questions like “so how high do you think the gym ceiling is?” etc. I hope I’m making the benefits of an immersive environment clear. When you are working with new mathematical concepts, students cannot play with them or experiment with them because they don’t understand them well enough. If you give them a framework with which they are comfortable, they can feel confident enough to experiment, think, discuss, and ultimately learn. Additionally, when students are asking questions of the environment instead of you, they are responsible for their own questions and data. You won’t accidentally give away the lesson by providing only and all of the pertinent data. If you do this pendulum thing, I recommend you measure while you can reach the whole rope. I forgot until just after the facilities manager had climbed down from the ceiling, and I wasn’t about to ask him to go back up there. It needs to be a thin, light, sturdy rope that doesn’t stretch too much, and it can’t rub too much at the top when it swings. Please emphasize, organize, and enforce safety procedures. You can hurt someone with 15 pounds on an 8-meter pendulum (attached with knots by kids with Velcro shoes). # Two and a Half Ways to Make Your Next Test More Readable Math questions are hard to read. It’s easy to mix up numbers, mis-attribute modifiers, confuse powers with multipliers, etc. Unless you’re testing for reading skill, it’s important to put work into making questions as readable as possible. Here are two (and a half) ideas you can use on your next test. ### One: Repeat the Question in an Answer Box My test questions include space for work and a very specific location for answers: the Answer Box. The answer box is the dark box you see in the lower-right corner of the question area. I pre-print that box on every test question I ever give so that, regardless of the sloppiness of a student’s work, I can see what they thought the final answer should be. I indicate the type of answer I expect right in the box. My hope is that even a student with low English-reading skills will be guided by that indicator to provide at least the right type of answer. ### Two: Provide a graphical representation with your text, where possible. In this problem I wanted to test students’ ability to transform parabolas. I accompany the text with a picture of what I mean. Without the picture, “compressed by a factor of 2” is ambiguous. When a student answers $y=\frac 1 2 (x+2)^2 +1$ I don’t want to have to guess if he doesn’t know what to do or if he just got right and left mixed up. In the previous example, the MS clipart of a die reminds ELL students what a die is (not obvious from the word itself!). ### Two and a Half: Easy Stuff Actually, more like five tenths of a big idea: 1. Use large fonts (obviously easier to read) 2. Include grids on your graphs (the grids above are more obvious in print) 3. Use the same format every time (students know what to expect even when they don’t command the language) 4. Number and name questions with the standard they are testing (make double-sure ELL students know what you’re asking) 5. Leave room between things (not only room for work. Spaces help us separate ideas) If you don’t make your questions crystal clear, you can’t be sure you’re testing what you think you’re testing. Reading is hard, everyone messes it up occasionally, and students who have only been learning English for a couple years need more help than you think! Of course, I’m just a math teacher, not a design expert or language coach. What have you used to make your print materials more readable? # Averaging Sucks. Where’s the Software? Which of these pieces of feedback is better? Fig 1. You're good at tree diagrams and bad at area diagrams. Let's work on area diagrams. Fig. 2: You suck. Try studying harder. And yet, this is what my grading software looks like to students: Fig. 3: As a student, you are a 93%. I assume you can tell why. In an era of easy software development, why do we put up with this? Do you know of anything better? I’m quitting my job to make something better. I’m terrified. # How to Teach Curiosity There are ways that I have found to boost and encourage curiosity in my classroom. Some of my techniques you can try tomorrow; others rely on a basic classroom culture that takes a long time and careful planning to build. I’ll touch on the easy ones first. ## Easy Ways to Teach Curiosity ### Labels When you see it, name it. The label is more important than praise. Here’s some psychology for you that I learned from an actual psychologist: people adopt the labels that they hear applied to themselves. As a person of authority, if you say to someone, “nice question! More evidence that you’re a curious person!” (or whatever version of this feels natural to you), he will actually start to actively think of himself as curious. If two persons of authority give him that label he will almost certainly assimilate the label into his self image. It was hard for me to believe in the power of this technique at first. How could such a small thing change the basic nature of another human? An alternative example helped me believe: what if you called a student “stupid” or “bad at math” once a week or once a month? Such a student would start to think of himself as stupid and bad at math. For some reason that’s easier to believe in than positive labels, but why should it be? I’ve been intentionally using specific, positive labels for about a year (and I mean I throw them out all the time) and the results are tangible. It’s mind control and it’s manipulative, but… whatever. I say go for it. Please only do it if it’s true. Uncle Ben, right? Please do not tell a student who asked a boring question that he is curious – you’ll water down your power and give the kid delusions about himself. And specificity reigns here: please say “this shows that you’re curious because you asked about xyz even though it’s not obviously related,” not “that question shows curiosity” and certainly not “that was a curious question!” 😉 Try not to say “great question” without saying why it was a great question. Evidence that makes your claim believable is vital to the success of this technique, but it doesn’t have to be a lot of evidence. ### Games and Secrets Make your review into a treasure hunt that takes you around the building or the room. Tell a story without telling the ending. Make a show of explaining a theorem or puzzle quietly to one student or a group of students and then audibly say “don’t tell anyone else.” This is more incitement of curiosity than teaching of curiosity, but it can complement your other techniques well. ## Harder Ways to Teach Curiosity ### Modeling This is also psychology: people do what they see other people doing. If you can spare a few minutes of your lesson to be interested in a tangential question in a way that includes the students, you can show that curiosity is a norm in your class. This is hard because not every class period offers up an interesting side-note, and even if it does, you have to be careful not to interfere with your plan so much that learning is affected too much. Before and after class are a good time to try to be interested in what your students are up to and what they’re learning that interests them. Even your most bored student has something that they’re learning at the moment (bike tricks, knitting, texting, a videogame). Hopefully it’s not something totally passive. ### Finding Interesting Problems I’m sorry for even including this. It’s really hard. Check out Dan Meyer’s WCYDWT, Shawn Cornally’s “How I Teach Calculus: a Comedy,” Kate Nowak’s and Sam Shah’s whole blogs, etc. ### Make Students Responsible for Their Own Learning Shawn Cornally writes, “I then have them construct a grant out of their group’s best/most interest­ing question.” Students who are responsible for choosing an idea based on how interesting it is must evaluate how interesting ideas are. This is the beginning of curiosity. Students who are responsible for writing about an interesting idea in an interesting way must become interested in the idea and find ways to study it. This is full-fledged curiosity in a way that “Complete items 1-4” will never teach. I achieve this effect in my classes by involving students in their own assessment, which is a little bit easier than involving them so deeply in the curriculum itself (but also less effective, I’m sure). However you do it, increasing responsibility will also teach independence and curiosity. I mean, what is curiosity besides a feeling of confident independence? # Tweeting Your Own Horn I grew up a privileged kid. Not private-boarding-school privileged, but certainly computer-in-my-room, internet-access privileged. I learned most of what I know about programming and the internet in my spare time. I helped create and run the very-successful Uniball, a huge learning experience for me: I dealt with hundreds of players in my leagues and on the chat service, dealt with technical issues, and even supplied and hosted a game server, all for free (1500 hours and 18,000 lines of code). I made free game frameworks, free time-card applications, ray-tracers. I contributed to an open-source physics simulation engine. My interests shifted towards education and I started contributing for free to the blogosphere, writing this blog, leaving comments on the blogs of others, and now tweeting (follow @rileylark). I’ve done all of this for no money; the experience and the learning has been its own reward (and I got a job at Microsoft for my Uniball work). In this community of free contributions, doing anything to make money is frowned upon, actually. There’s a pressure not to be making money and not to be looking for the most followers, etc. Your work should speak for itself, and your learning and fun should be the only reward! Well, I have an announcement to make. Next year I will be drastically reducing my hours at school in order to write software to better support teachers in the switch to and maintenance and improvement of standards-based grading. My steady yearly income will drop to something like$10,000, which is, as they say, not enough.  I’m betting a year of my life on the software I plan to create.

Developing great software is hard, but I can do it – I have tons of practice, both professionally, academically, and as a hobby.  What I don’t have practice with is intentional networking.  About 180 people follow this blog (thanks!  I’m glad you like it!), but I’m going to need more exposure than that if I’m going to support myself on my own handiwork – I know what your budgets look like.

Now, finally, the point: the last 15 years of internet social experience are telling me not to publicize.  Don’t add “Tweet this post” buttons to your posts, and don’t put ads on your pages!  Looking forward to a profession based on the internet, though, shows me that I have a lot to learn about marketing and revenue.  So, I’m adding these things to my blog as experiments.  If you think they’re trashy, please let me know.  If you don’t notice them, please let me know.  If you like tweeting about stuff when you’re given a little box, please let me know!

I’d love to hear from you in the comments.  What place does marketing have in our blogging community?  What about money?  How do you feel about tweeting your own horn?

# A Probability Discussion

There’s a major problem with probability: it doesn’t work very well.  For example, common knowledge tells me that in a fair coin toss I have 50% odds of flipping heads and 50% odds of flipping tails.  But flip a coin once and you’ll either get 100% tails or 100% heads.

Now, I get it.  I mean, don’t worry.  After many flips, blah blah blah.  We can study a fair coin toss for two weeks, counting total possibilities, various games, expected values, special orders with surprising probabilities, etc, but none of this helps me guess the next flip at all.  Even in the next six flips there’s a depressingly low chance of getting three heads and three tails.  We certainly can’t rely on it happening in our math class while we try to convince people of what “50% likely” means!

I believe in probability, but it is indemonstrable.  Once you do something, it 100% happened despite its 50% (or 5%) odds.  The challenge for us, then, is to reconcile these seemingly-disparate pieces of evidence.

I chose to let the students have a 30-minute discussion about the issue and I brought several props.  My goal was to get the students to seriously consider their beliefs about probability – to be wary before they were confident.  By the end of this conversation, I want some students to doubt that heads and tails are equally likely.

I started the conversation by showing them this quarter, through which I have inserted a piece of paper:

After the requisite jokes about how hard it was to get that piece of paper through the quarter, I asked, “What are the odds of flipping tails?”

“50%,” one student immediately asserted.  The other students all agreed (I was surprised that they all agreed).  Experimentally, we showed that the odds are not even – there was a strong correlation between the side that starts up on your thumb and the side that lands up on the table, apparently because the paper drastically reduced the coin’s tendency to flip at all.  It sort of parachuted down instead of flipping and rolling.  One of the students tried cutting the paper (which started as a clean square) to see if she could affect the odds.  The students’ assumptions were incomplete and wrong, which is one of my favorite ways of getting them interested.

Then I asked another set of questions, this time with a plain coin and no paper:

“When I hold this coin way above the table with tails facing up and drop it, what are the odds that tails will land up?”

50%, Riley, [obviously | I think | assuming it bounces].

“Ok… when I hold this coin one inch above the table with tails facing up, what are the odds that tails will land up?”

100%, Riley, it won’t even flip.

But at two inches, the coin bounced a bit, and at three inches it was hard to tell what was going to happen.  “At what height does it become 50%, class?”

They got my point.  If we can tell what is going to happen after an inch, why can’t we tell what is going to happen in a foot?  Philosophical questions abound here, right?  And they started to doubt themselves about everything.  They started to see that there’s no way to test the probability with perfect accuracy.  They started to see that you might be able to rig a coin flip to turn out the same every time.  They started to see the effects of human bias in selection games (I had a deck of cards too).

By the end of this discussion, my students were using phrases like “assuming that there is enough complexity” and “assuming that no one is cheating” and “assuming that the sides are equally heavy.”  They saw the assumptions that go into a statement like “heads and tails are equally likely,” and saw that we can’t rely on that information in all cases.  I sacrificed 30 minutes of class during which they could have practiced permutations and combinations, but I hope that their increased savvy about statistics will pay off.