Saturday, October 29, 2011

Webinar Today: The GoodQuestions Project, Using Qustions that Deepen Understanding

I ended yesterday's post with this teaser:
If you're interested in thinking about good questions for this vote-and-discuss type of interaction, join us at the free webinar I'll be hosting tomorrow. I'll be interviewing Maria Terrell, founder of the Good Questions Project, as part of the MathFuture webinar series. It starts at 11am Pacific time / 2pm Eastern time. 
 But I neglected to tell you how to join us. Here are Maria Droujkova's instructions:

How to join

  • Follow this link at the time of the event:
  • Saturday, October 29th 2011 we will meet online at 11am Pacific, 2pm Eastern time. WorldClock for your time zone.
  • Click "OK" and "Accept" several times as your browser installs the software. When you see Session Log-In, enter your name and click the "Login" button
  • If this is your first time, come a few minutes earlier to check out the technology. The room opens half an hour before the event.


The recording will be at

About Good Questions

The GoodQuestions project seeks to improve calculus instruction by adapting two methods developed in physics instruction — ConcepTests and Just-in-Time-Teaching. GoodQuestions is a pedagogical strategy that aims to raise the visibility of the key concepts and to promote a more active learning environment. The essence of the approach is to develop questions that
  • stimulate students’ interest and curiosity in mathematics;
  • help students monitor their understanding;
  • offer students frequent opportunities to make conjectures and argue about their validity;
  • reflect the role of student prior knowledge and misconceptions in building conceptual understanding;
  • provide instructors with frequent formative assessments of what their student are learning;
  • support instructors efforts to foster an active learning environment.

Can you raise the visibility of key calculus concepts, promote a more active learning environment, support young instructors in their professional development in their early formative teaching experiences, and improve student learning? We think the answer is yes, if you ask students Good Questions and encourage them to refine their thinking with their peers. What makes a question good? Imagine a classroom where the instructor pauses every fifteen minutes or so to ask a highly conceptual multiple choice or True-False question. For example True or False: You were once exactly π feet tall. Students think about the question independently and register their vote. As the instructor uses that feedback to start to assess the state of the class’s understanding, students are encouraged to discuss their answers with someone sitting near by, preferably a student who is thinking about the problem differently. As the room erupts with inquires of “what did you think? ” and “why did you think that?”, and with replies of “well I’m not sure, but I think...”, the instructor listens in on conversations. Students share their reasoning, argue its validity, and work together as they think more deeply about what the question means.

Event Host

MariaSheaTerrell.jpgMaria Shea Terrell writes:
My recent interests in geometry have included tensegrities and the history of geometrical optics and linear perspective. I am collaborating with a group of faculty and graduate students in an effort to improve undergraduate mathematics instruction through a project we call GoodQuestions. The project is developing materials to help instructors engage students in meaningful discussions about key concepts in calculus. At a recent MER (Mathematicians in Education Reform) workshop I presented a paper about my recent experience in the project.

Friday, October 28, 2011

Peer Instruction, Good Questions, and a Good Day in Class

When I see the effort many other teachers (especially bloggers) put into their daily lessons, I often feel embarrassed. On Wednesday I had a vague idea of what we'd cover, a handout from my Murder Mystery, and no notes. And yet it went great. (It does help that I've taught the course many times.)

I'm doing a mini-quiz each day right as students come in, to get them to come on time. It's also helping me see what needs more work. I make up the problems pretty much on the spot. I gave the same type of problems today that I gave yesterday - 4 basic log problems:
Quiz #10
  1. log3243
  2. log264
  3. log24   (Oops, way too easy, I meant this to be  log42!)
  4. log5(1/5)

Then we started in on the murder mystery. [Read my previous post for details.] We had discussed it a bit before, and they were supposed to have completed assignment 1:
We want to think about how a hot cup of coffee cools off.

1. What would be a reasonable starting temperature?

2. After about how long would it be cold?

3. About what temperature is it when it’s cold? (Why?)

4. Now let time be the x-axis (t-axis) and temperature (T) be the y-axis, and (on graph paper) graph temperature versus time for a cop of coffee, using what you know from common sense. Does a straight line graph make sense for this?
Many of them had answers that were way off, so we discussed en masse. We talked about the physics involved (though we have not yet mentioned Newton's Law of Cooling), and I got estimates from them for the first 3 questions and built the beginnings of a graph, with (0,160) and (60,80) plotted, and a dotted line at y=80. [We were saying the coffee started out at 160 degrees, and after 60 minutes it had cooled to the air temperature, 80 degrees.]

Then I had them get in their groups (front two people push their desks sideways, and all four push the desks closer, it's very quick), and draw a quick graph using that framework. Lots of hesitation, I had to prod them to just guess. I told them I had seen two types of graphs as I'd walked around the room:

As we talked about them, it turned out we needed two more graphs:

I labeled them A, B, C, and D, and asked the groups to discuss and each group would vote on the one they thought best represented the cooling coffee. (A is two straight line segments, a common hypothesis from students. B is exponential decay. C is a straight line segment, then a curve, then another straight line.  D looks kind of logistic.) As I waited, I realized this was much like the Peer Instruction championed for physics courses by Eric Mazur. Interestingly, D got the most votes. (It was 0, 2, 2, and 5.) We talked some more about the physics of it, and decided to measure actual coffee the next day. (They had gotten their 30th donut point the day before for catching my 30th mistake, so we decided to have donuts and coffee on Thursday.)

I left this question open at the end of class.  In the past, students have often looked up rate of cooling in the textbook or online, and have mentioned Newton's Law of Cooling. The initiative they take is great, but I'm sorry to see them just following a formula after that. This class came in Thursday not having done that. We had hot coffee and a thermometer from the chem lab. We got our data, and will look at it on Monday. This class has engaged more with the project than any other class in my memory. (Granted, I do have a bad memory.)

If you're interested in thinking about good questions for this vote-and-discuss type of interaction, join us at the free webinar I'll be hosting tomorrow. I'll be interviewing Maria Terrell, founder of the Good Questions Project, as part of the MathFuture webinar series. It starts at 11am Pacific time / 2pm Eastern time.

Wednesday, October 19, 2011


I've dreamed of offering a one-unit course on fractions for our students. (I teach at a community college.) So many of them really struggle with fractions, and don't really understand why fractions work the ways they do. I'd like a course that's just pass-fail, with no grades otherwise. I'd like no textbooks (unless there's something really good for adults that I haven't seen yet). I'd like it to be about understanding, but to also include practice to cement that understanding.

Uri Treisman showed years ago that an excellent strategy for getting students up to speed was to have them work in groups on extra challenging problems.

MBP, at Rational Expressions, just offered up a good problem that is challenging enough to make me work hard, and approachable enough for the students in a fractions class to work on as their 'Research Into Fractions'. MBP and I have different requests stemming from this problem. MBP wants to know what makes a problem 'hooky'. (If you can't answer that, maybe you can offer MBP an example of a problem that really hooked you.) I want to know what other problems would be good for my imagined students in this imagined class. Problems that involve fractions, and make the students work hard with fractions, that start out approachable, and have enough hook to get the students working persistently.

Here's my version of the problem MBP offered:

Any fraction of the form 1/n is called a unit fraction. 1/2 can be written as the sum of two other unit fractions (1/3+1/6).
  1. Can this be done for all unit fractions? 
  2. Find a rule for the number of ways to do this.

Got any challenging fraction problems that a newbie might enjoy chewing on?

Thursday, October 13, 2011

Richmond Math Salon - This Saturday, 2 to 5pm

Lots of fun. See the video here. Call me for more info: 510-236-80 four four.

Saturday, October 1, 2011

Interesting Number Puzzle, but I'm Stuck

Head on over to Think Again, for Jan's latest puzzle. No one's been commenting, and I'm curious what other people think of this.

Julia Robinson Mathematics Festival

I had a great time today, volunteering at the Julia Robinson Mathematics Festival. There were 16 tables. Each table had two volunteers helping kids with a page full of interesting problems, that generally started pretty easy, and ended up quite challenging.

Ours was the Multiplication Table - each participant's first job was to create a multiplication table, and look at the pattern of even and odd numbers. They were asked to describe the pattern, and explain why it turns out the way it does. The next few questions, about which numbers show up the fewest and most times, stretched kids a bit more. Which numbers show up an odd number of times? Some saw it immediately, and some weren't ready for that question. If we made the table bigger, what would be the first number with 10 factors? One girl I worked with saw it pretty quickly. What about 11 factors? I didn't even get that far...

The 3 hours were up before I knew it, and then we had a presentation by Karl Schaeffer, of Math Dance. It was fabulous! He had us get in groups of 3, and try to swap places with neighbors to make each permutation* just once. Then we tried it in groups of 4. Then we did windmills with our arms. You'd be surprised how many ways there are to do that (clockwise or counter, both arms the same or different, in phase or out). There was even more.

Check out their video:

I had just ordered his book last week. I can't wait to play with it.

*Permutation means arrangement. You can arrange the three letters A, B, and C in a bunch of ways: ABC, ACB, BAC, BCA, CAB, and CBA. How many ways could you arrange 4 letters?
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