Thursday, January 30, 2014

Identifying Effective Representations

At a recent PD, the Pre-Calc/Calc teachers at my site worked on the following question while we were on the topic of Identifying Effective Representations.

The verbal instruction given was: Solve the problem in as many ways as you can. The written instructions are as follows:

Solve the following problem, using alternative representations where appropriate.

A woman was 3/8 of the way across a bridge when she heard the Orient Express approaching the bridge behind her at 60 mph. Because she was a mathematics student, she quickly calculated that she could save herself by running to either end of the bridge at top speed. How fast could she run?

Go ahead. Try it out.

In the limited time given, we were able to find two representations that led to the solution. For me, it felt more like the same representation solved two different ways. The traditional way, if you could call it that, required more work as we solved a system of equations to get the solution. The other method we found required a little more thinking about the situation, but the path to the solution was much simpler and shorter. Can you find other effective representations?

The idea is that as we move to Common Core, it is not sufficient to just get an answer. Students are expected to solve a problem using multiple approaches and this was an example of that. How realistic is it that students will have the kind of insight that allows them to solve this kind of problem using multiple approaches and, more importantly, how do you go about teaching it in the limited time we have?

I would love to see your approach and your diagrams as you work on this problem. Link or comment below. I'll update this post with my work in a week or so.

Wednesday, July 17, 2013

Fun Little Area of Quadrilateral Question

Fun little problem about the area of a quadrilateral tweeted by @daveinstpaul. Try it out.

If you're stuck, try the following applet and click on the checkboxes labeled "Show Hint ##"

mrhodotnet, 17 July 2013, Created with GeoGebra

Some questions that popped in my mind as I worked on this question:
  1. Is the shaded quadrilateral unique? Is this the only quadrilateral that fits the criteria (given sides and angles)?
  2. Is it possible to draw another diagram with the same given sides and angles but with different shape?
  3. Isn't it the case that given only 4 sides of a quadrilateral, there isn't enough information to determine area? If so there may not be enough information to determine area, but if it were the case the question would be pointless.
  4. Is it possible that the area of the shaded quadrilateral is constant regardless of the shape?
  5. If the area is constant, let's choose the diagram that is the most calculation friendly and find the area.
  6. I have the area now. How do I know it's constant? Can I show this? What principles are at work? What can I learn about the special cases (the diagrams that I drew) to show it for the general case?
  7. Is it possible to show this using trigonometry? How should I break down the quadrilateral? A trapezoid and a triangle? Two triangles? Should I tackle the unshaded area first?
  8. Let's label the unknown sides and use Pythagorean Theorem to find the relationship between the missing sides. Do the equations simplify?
It was some time later, while doing chores, when it hit me. There is a kind of joy and excitement when an idea strikes and it's immediately clear that it'll work. A simple auxiliary line segment is all it was, probably immediately obvious to Geometry teachers. It's interesting how this question was one auxiliary line segment away from being boring. Glad it was omitted from the original.

This is my first time embedding a tweet in a post and embedding a GeoGebra html5 applet instead of a java applet. People really make these things easy to do nowadays. Thanks Twitter and GeoGebra!

Update: Removed tweet embedding until I figure out how to embed without including media.

Thursday, June 27, 2013

Monty Hall Problem through Philosophical Chairs

This is an activity based on Philosophical Chairs that I first learned from AVID teachers at my school. I don't have my handout readily available but found one from San Diego COE on Philosophical Chairs. From their handout:
In theory, learning happens when students use critical thinking to resolve subsequent conflicts, which arise when presented with alternative perspectives, ideas or contradictions to what they have previously learned or believed. "Philosophical Chairs" is a technique to allow students to critically think, verbally ponder and logically write their beliefs.

Essentially, Philosophical Chairs is a way to organize discussion on debatable topics and then have the students reflect and justify their position in writing. These were yes/no questions and students had to choose between these two options and undecided. The desks are typically arranged in a horseshoe seating arrangement with yes and no on the sides, and undecided in the middle. 3 Signs are posted so that students can move to the corresponding area once they decide on a position with respect to the question/topic being discussed. They are called upon to justify their position in the "hot seat" in the middle of the room. Only one speaker is allowed at a time. Other students are invited or "volunteered" to join in with questions, comments, or present their arguments, however, they must summarize the previous speaker before they're allowed to do so. They will be given opportunity to move as their minds change.

The activity seemed tailored for ELA and Social Studies. If you do a quick search online you'll also find plenty of examples for those classes. Perhaps I haven't looked hard enough, but I haven't seen one for math yet. I didn't want discussion on whether reliance on calculators for computation has become a crutch for math students or whether all students should be required to take up to algebra 2 or any other math education topics. I wanted math topics. I was about to throw this activity into the cool-if-I-taught-other-subjects folder until I heard a presenter use the word paradoxical. I don't remember the context, but it gave me ideas.

The Monty Hall problem (wikipedia) isn't really a paradox, but it's infamous for being so counter-intuitive that even PhDs get it wrong. Here's a wording of the problem popularized by Marilyn vos Savant in her column "Ask Marilyn":

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
If this is your first encounter with this problem, stop reading now. Take some time and think about the problem.

The correct answer is you switch to door No. 2 for a better chance to win. It's not really paradoxical, it's really just counter-intuitive that when left with only 2 choices (door No. 1 and door No. 2), where the chance of winning seems to be 50%, it is more advantageous to switch. This is one of those problems where intuition works against you. Often, even after being presented with simulations and calculations, it's hard for people to wrap their minds around it.

I chose this activity so students would have an opportunity to get an understanding of the problem and be able to put into words their reasoning and also evaluate the reasoning of others. Instead of asking just one question, I would ask series of questions that led them to the answer. They also get an intuitive feel for why the initial conditions make a difference and opens the door for discussion about assumptions and about what probability really is. Plus, is there a probability or statistics class that doesn't cover or at least mention this problem?

I made the following modifications/additions:
  • Instead of presenting just one question, I would present several different scenarios to help students clarify their thinking
  • I told students that by the end of the day all students would get the correct answer so the goal is not to justify their position with a personal opinion, but to help each other understand the solution through discussion, questioning, and mathematical arguments. For this to happen, they must be open-minded.
  • Instead of yes/no/undecided, for this problem the students chose between yes/no/doesn't matter
  • To better maintain order with multiple questions and with the movement in the classroom, I organized each question into a round that consisted of: a) silent reading of question b) students move to the appropriate areas after deciding on their position c) call students to justify their position to start discussion d) ask for volunteers if necessary e) students ask questions f) students can switch positions before next round g) periodically I ask why students changed their minds
    I had the following questions on PowerPoint and also on a handout with the wording of the problem.
    1. (Original Problem) In the case above, Monty Hall (the host) asks you - do you want to stay with Door 1 or switch to Door 2? If you want a better chance of winning, should you switch from your original choice? Does it make a difference?
    2. Imagine there are 1000 doors and only 1 prize. You pick a door. Now he eliminates 998 other doors that don't contain the prize so that 2 doors are left, one with the prize and one with no prize. Should you switch from your original choice? Does it matter?
    3. Suppose there are 1000 doors. 500 (half) of which have the prize. You select a door, and all doors are eliminated so that 2 doors remain. The options are eliminated such that one has the prize and one doesn't. Should you switch from your original choice? Does it matter?
    4. Suppose there are 1000 doors. 999 of which have the prize. You select a door, all doors are eliminated so that 2 doors remain. The options are eliminated such that one has the prize and one doesn't. Should you switch from your original choice? Does it matter?
    5. Suppose there are 1000 doors. 1 of which has the prize. You select a door, the host opens a door without a prize and says you have the choice of 1 door or the 998 other doors. Should you switch from your original choice? Does it matter?
    6. Suppose there are 3 doors and 2 prizes. You select a door and the host eliminates an option so that one has the prize and one doesn't. Should you switch from your original choice? Does it matter?
    7. Suppose there are 3 doors and there’s only 1 prize. You select 1, and the host asks you if you want to keep the door you selected or the 2 that you didn’t select. Should you switch? Does it matter?
    8. Suppose there are 3 doors and there’s 2 prizes. You select 1, and the host asks you if you want to keep the door you selected or take the other 2 that you didn’t select. Should you switch? Does it matter?
    9. (Original Problem) Should you switch from your original choice? Does it matter?
    I tried to emphasize mathematical justification. Some students will offer reasons like "I didn't switch because I trust my gut instincts" or "I don't like switching because I'm not a quitter." There will be times where students don't really have a position and can't provide an argument, just move on and return to the student later and ask him/her to summarize views presented by others. It takes a few rounds before students would automatically rephrase the previous speaker's argument before presenting their own. An interesting thing is most students will answer question 4 correctly but not 2. There might be some kind of bias at work here. Usually, by question 5 most students will choose correctly. At some point, someone realizes initial conditions make a difference in this problem and there's a discussion to be had about assumptions. To conclude this activity after discussion, I call on students for the correct answer to each question. Once the students are done with discussion, they write their reflection and justification in the back along with answering these activity related questions:
    1. What strategy would give you an advantage: staying with the original choice or switching from the original choice? Explain.
    2. What is the difference between the scenarios presented in questions 2, 3, and 4? Explain the strategy you would employ for each (switch/don't switch/doesn't matter).
    3. What was the most frustrating part of today's discussion?
    4. What was the most successful part?
    5. What statements led you to change your seat or to stay sitting in your position today?
    6. What is probability to you?
    7. What would you change about your participation in today’s activity? Do you wish you had said something that you did not? Did you think about changing seats but didn’t? Explain.
    After we conclude the Philosophical Chairs part, we would continue with simulations of the problem and perform an analysis using tree diagrams to calculate probability. If there's more time I would play the MythBusters clip on the Monty Hall problem.

    This is not the standard Philosophical Chairs where you have a debatable topic and students can hold valid differing point of view based on their beliefs and values. But if you focus on the critical thinking, the formulation of problem, persuasive argumentation aspect of the activity, I think it can provide a framework for having good discussion about counter-intuitive topics in math. The main difference is that there is a correct answer in the end. The journey to get there is what's valuable.