Tag Archives: collaborative learning

Gender and Mathematics

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This morning’s New York Times had a headline reading: “Girls Lead in Science Exam, but Not in the United States.” The article started with a rather fascinating graph showing country-by-country performance on the OECD test with a display of the percentage gap between male and female students. In the United States, the average scores were 509 for males and 495 for females; thus the males outperformed females by 14 points, or around 2.7%. Compare this with Japan’s data: Average scores of 534 for males and 545 for females gave the girls about a 2% lead.

Both the graph and accompany article interested me enough that a printed copy can now be found on my office door, along with my own editorial remark at the top. (See photographic proof.)

I found the Times article through my Twitter feed. Other interesting articles that hit my feed were a blog post by Hariett Hall (“Gender Differences and Why They Don’t Matter So Much“) and a 2005 article from Time magazine on “The Iceland Exception: A Land Where Girls Rule in Math.” [Michael Shermer linked to Hall’s article, and I shared the link about to the Iceland article.]

After I posted the Iceland article, John Wilson (@jwilson1812) asked for my opinions “about what this report from Iceland might suggest, what’s generalizable, what isn’t, and so on.” In this post I’m hoping to capture a longer response than what 140-characters would allow.

1. The United States has a gender discrepancy problem in mathematics.
To me, this point seems somewhat obvious. But given the headline from Hall’s article, and other comments, conversations, and feedback I’ve received over the last decade or two, it also seems clear that it isn’t obvious to everyone. I mean “problem” in the above statement as in, “Something we ought to be concerned with pondering and understanding, and (if possible) fixing.”

2. A partial fix could be fixing the educational and employment climate.
As the Times article points out,

Researchers say cultural forces keeping girls away from scientific careers are strong in the United States, Britain and Canada.

Hall’s article points out that men and women are different, and that their skills, interests, and aptitudes are shaped both by biology and by culture. Talking about how biological differences may (or may not) influence mathematical aptitude gets murky very quickly, and I am certainly not qualified to say anything one way or the other. On the other hand, talking about how cultural differences influence mathematical aptitude is a conversation we ought to have frequently.

3. How can we fix the problem?
The real answer to this question is, “I don’t know.” But I have a lot of hunches.

Hunch #1: We need more collaborative classrooms.
Somewhere a long time ago I read about a study done on middle school aged children playing soccer during recess or physical education classes. The students were separated by gender. In each group, researchers looked at what happened if a soccer player were injured during the game. With the boys’ game, an injury momentarily paused play; a spectator was swapped for the missing team member; the game quickly resumed. With the girls’ game, an injury stopped play. The girls (on both teams) decided they’d rather not play without their injured friend on the field, and so they took up to doing another activity altogether.

I think this parable fits with how I picture what happens in math classrooms. While I’ve taken lots and lots of math classes, I was never able to take a class that would fit any description other than “traditional, chalk-talk, lecture-style, definition-theorem-proof.” The math classes I saw as a student were like the boys’ soccer game: If one student fell behind, or got confused, or failed at mastering a concept, the class would pause, remove the “injured” participant, and continue moving forward. The aim of the class was the soccer game itself and not who was playing and who wasn’t. In my experiences, math classrooms are places where students practice an individual sport (like tennis) concurrently. They are not places of collaboration or conversation or team work. The coach is interested in keeping the game moving forward (even if dropping players is necessary).

I think this is bad for a few reasons. But the top reason is that I think it gives everyone (both women and men) the false impression that mathematics is an individual sport where the performance of the athlete is a solo endeavor. But real mathematics is nothing like this. As mathematicians, collaboration is essential. We publish papers together. We give weekly colloquium addresses to teach each other new ideas and to solicit help on tough problems. We travel to conferences to have conversations with others and work through problems as a team. Why do our classrooms give the opposite impression of how mathematics is done?

Showing the world (and girls especially) that mathematics is not done in isolation is crucial. I believe that marketing mathematics as a collaborative, socially-based adventure would attract more girls to become mathematicians and scientists of all types.

Hunch #2: Attract, hire, and retain more female math professors.
I did my undergraduate work at U.C. San Diego where I was a “Pure Mathematics” major. At the time I was there (early 2000s), the department had about 55 full-time tenured math faculty members. Of those, 5 were female. [See their department directory today for comparison.] One of the women professors mentioned that, at the time, among the “Top 25” math departments, U.C.S.D. had the highest percentage of tenured female math professors. What percent is 5/55? About 9%. This statistic was quoted with pride: “We are so great to have so many women! Among the math professors, only 90% of them are male here! Fantastic job!”

I think our cultural conception of what “Mathematics Professor” looks like needs to change. Yes, there are plenty of math professors I know who fit the stereotype exactly. But then there are those who look like me. The way we shift the stereotype is to disprove it. We need more minority math professors, we need more female math professors, we need more math professors who aren’t 60-year-old white males with chalk dust on their pants.

On keeping women in science: One thing obviously in need of repair in academics is promoting careers that allow for a work-life balance. Right now, I am expecting my second child. When I complained recently to colleagues about the “Leave Policy for New Faculty Parents,” one responded, “Well, when each of my five children were born, I was back at work the next week.”

I wish I could say this were not the norm. But it reminded me of a conversation I had 10+ years ago, when one of the women faculty at U.C.S.D. told me about giving birth on Thursday and being back teaching classes the following Monday.

I love my job, I love my co-workers, I love my students, I love being in the classroom. But my employer’s Leave Policy, combined with the remarkable and surprising lack of empathy from colleagues about said Leave Policy, has certainly made me consider jumping ship. Academia needs to wake up and offer a family-friendly, parent-friendly work environment where people are valued for being people first (and professors second).

Hunch #3: We need to teach teachers differently.
As an educator, it’s difficult to structure one’s classroom in a way dramatically different from the one you were in as a student. You think back, “How was I taught this idea?” and that’s the easiest answer to, “How will I teach this idea to my own students?” You can see this all over the math community as the traditional, blackboard-based, definition-theorem-proof machine chugs chugs chugs along. Thankfully, there’s been a giant movement in recent years toward changing the idea of what a classroom should look like. (See my earlier ideas about collaboration.)

Given that we are all inclined to teach the way we were taught, and given that for a very long time it was accepted dogma that boys always outperform girls in mathematics, it’s easy to see how this idea could still linger. Not that I think any particular person goes into their calculus classroom and says, “Sorry ladies, everyone knows you don’t have the skills to be really good at this.” But I do think (and I have seen ways) that this underlying stereotype has affected the way people teach.

My Conclusions
1. The gender imbalance in mathematics has some cultural factors.
2. We ought to be concerned with what those factors are, and how to change them.
3. Changing them is a process that will definitely take a lot of time and probably take a lot of money.
4. My best strategy at overcoming this problem is this: Become a female math prof who posts blog articles about the gender imbalance in mathematics. Unfortunately, this strategy is probably not widely implementable. It definitely takes a lot of time. An easier thing to do is to support and encourage those who are doing this or things similar to it.
5. My next best strategy for overcoming the problem is: Seek out like-minded people and work together to figure out how we can change the math culture. 

As I said at the beginning of this, I know there is a problem and I don’t know it’s solution. But I’d be happy to hear what you think it might be.

Rational Power Functions

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One of the topics our PreCalculus syllabus (Math 111) covers is “Rational Power Functions.” Since functions like f(x)=x^n are called power functions, the rational power functions would be those of the form f(x)= x^{p/q} (where p/q \in \mathbb{Q} ). Unfortunately, this topic isn’t covered in our textbook (Zill’s “Essentials of Precalculus with Calculus Previews“).

Tom Kunkle, on his Math111 Homepage, provides his students a nice summary of how these functions behave: See ratpowfunc.pdf. When thinking about this week’s Lab Assignment, my goal was to give students practice on drawing graphs of functions like

y = -(x-5)^{4/3}+1
Lab Description
I created nine cards, which I’m calling “Function Cards.” Each card is a half-page (printed on heavy-duty card stock). On one side, the card has a Graph of some translated rational power function. On the other side, the card has an Equation of a translated rational power function. However, the equation and the graph are not the same function!

Here are the directions I provided to students:

  1. When you get your Function Card, flip it to the Equation side. Make a note of the card number. Write down the equation.
  2. As a group, work together to sketch a graph of the equation. You may use the Rational Power Function handout (if you like). You may NOT use a calculator. Work together! Your sketch should clearly display & label all intercepts, any cusp points, any vertical asymptotes, any locations where the tangent line is vertical, and the parent function.
  3. Once your group has agreed upon what you think the graph looks like, draw your sketch on the Answer Sheet, along with the information listed in Step 2. Then go around to the other groups. Find the Function Card with the graph that matches your sketch. Ask nicely, then take it from that group.
  4. Return to Step 1.

The students will have the entire class period to create graphs of each of the nine Functions. Since they will track down the graph most closely matching theirs, they will have a chance to check their answers. Even once they see the answer (i.e., the graph), they are still asked to do some thinking since they have to provide information about the graph’s important features.

Hopefully this process will work. The only thing that concerns me is how long it will take them. I am really terrible at gauging how long it takes students to think about things. My best guess was that it would take about 4 minutes to sketch a quick graph, and then another 3 minutes to write down its important features. So, (7 minutes)x(9 Function Cards) should be a little over an hour.

I picked the nine functions from Tom’s “Homework Handout” on this Section, available here. If the weather complies, maybe we can do this activity outside. The only thing better than math class is math class in the sunshine.

First Day Activity

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I really enjoyed today’s “First Day” activity in Precalculus. I found the idea on Becky Lyon’s blog; you can also find her on Twitter: @rhlyon.

I had the students find someone to work with and told them one member of the pair would be the Explainer and the other would be the Grapher. The Explainer was supposed to sit facing the projector screen, while the Grapher was supposed to sit facing the door (i.e., away from the projector where they could not see it at all).

The idea of the activity is this:

  1. Display a picture or graph on the projector screen for about one minute.
  2. The Explainer has to describe the graph only using words — no hand gestures allowed!
  3. The Grapher tries to re-create the picture or graph from the description.

The pictures I used started out easy (a giant smiley face) and got progressively more difficult. To give you an idea, I uploaded the exact graphs I used to my public Dropbox space: It’s http://dl.dropbox.com/u/59433434/111-Day1.pdf. After we were done, we went through the graphs together and talked about what descriptions had been given and what people could have said to make it easier.

This gave us a great opportunity to review vocabulary like “degree” and “vertex” and “parabola” and “quadratic” and “intercept” and “slope” and “local maximum [versus global maximum]”. It also gave me feedback as to what the “groupthink” occurred and at what level my students are starting. (For instance, some of them volunteered the idea that an even-degree root of a polynomial behaves differently on a graph than an odd-degree root!)

They seemed to enjoy the activity. It helped cement for them that I will expect them to do things in class, not just be. And, happily, it gave them the opportunity to practice my Friendship Policy.

My three favorite comments from today include:

  • “This will be the most FUN class!” –a student said to her friend, at the end of class
  • An e-mail I received after class said, “What a great first day of class! Super exciting and thank you for your approach!“
  • A fantastic Tweet (admittedly from a calculus, not precalculus, student):

     

Friendship Policy

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I have my first course meetings this morning. Right now I’m enjoying a one-hour break between classes in what will become my Office Hours once students figure out what Office Hours are for. I thought I’d take the time to write about an important topic I covered during today’s PreCalculus class.

A Very Important Course Policy:

One of the notable policies I have on my syllabus is called my Friendship Policy: Students in my courses are required to make two friends from class. For those of you who, like me, haven’t been a college student in a number of years, this policy may seem very silly and totally unnecessary! However, the policy has an important function at fixing a “problem” I noticed a few semesters ago.

Before class, I would find students sitting on benches in the hallway for several minutes waiting for the previous class to end. There would be, say, ten or twelve students all from the same course, standing in the same hallway, and it was library silent. No person was talking to any other person! Instead, every single one of them was texting someone on their phone, checking Facebook on their iPad, playing a game on their laptop, etc. Eventually they would all enter the same classroom and continue their technologically dependent anti-social activities.

When I pointed this out to my students, they had never noticed this phenomenon and they didn’t understand why I thought it was weird!

“Back in my day,” says the professor…

There were no cell phones. In order to fill the awkward silence, students in my classes would talk to each other, real-time, face-to-face. Sure, we would talk about course-related things like homework or exam studying, but we would also talk about social activities or sporting events or movies or whatever. This is how we made new friends.

I realize that students in my class have lots of friends. (Otherwise, who would they be constantly texting?) But I still have not figured out how they make new friends. Hence the birth of my Friendship Policy:

Friendship Policy:

You are required to make friends with students in this class. If you are absent from class, your friends will be very happy to lend you their notes to copy! In fact, I think cooperative learning is so important I am going to leave blank space on this syllabus for you to write down the names of two of your class friends and their contact information.

After explaining all this to the students, they usually look at me with confused faces until I say something along the lines of, “Friendship Time: Commence!” and then stare at my wristwatch expectantly. Within seconds, the room explodes in conversation. Occasionally, I have to nudge some of the shy students in the right direction.

Results and Analysis

After several classes over several semesters, this policy seems to make a big difference. First, no one sits before class in techno-quiet. They talk to each other, get to know each other, and occasionally I have caught them teaching each other how to do math problems. Second, I no longer get e-mails asking, “What did you cover in class yesterday?” Third, I learn a lot from my students by participating in before class conversations. For example, in this morning’s class, one student is here on a golf scholarship from Sweden! (How awesome is that!)

I still have two more classes this morning. We’ll see how those groups take to forced friendship-making time.

Project Based Learning

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Our classes for the Fall 2012 semester start today. Thankfully, my teaching schedule doesn’t include Tuesdays, so I don’t start until tomorrow! I’m hoping to use Tuesdays this semester to work on several other projects, including adding more blog postings. Wish me luck.

This semester I’ll be teaching two sections (Section 05 and 17) of our Pre-Calculus class (Math 111) and one section (Section 05) of our Calculus I class (Math 120). Each class meets for 50-minutes per day on Mondays, Wednesdays, and Fridays, and an additional 75-minutes on Thursdays. The longer meetings on Thursdays will be useful in my current quest to incorporate Project Based Learning (“PBL”) into my classes.

I’ve begun the task of designing “Lab assignments” for students to work on, in small groups, during our Thursday meetings. Ideally they would be assignments that require no pre-lecture and ask the students to draw from their course content knowledge to form connections between ideas. By working together in a group, the students could collaborate (hopefully allowing for some peer instruction), ask questions, have a discussion, and digest what we’ve talked about during our other class meetings. According to my calendar, the students will have ten lab assignments over the course of the semester.

Yesterday I began working on the third lab assignment for Calculus. The topics covered earlier that week will be limits at infinity; asymptotic behavior; and continuity. I found an activity called “Carousel Game” from the NCTM‘s Illuminations series and modified it for my class. Here’s a brief overview of this lab:

  • Topic: Graphing rational functions
  • Goal: To correctly determine the equation that corresponds to the problem situation or graph
  • Technology Required: None allowed!
  • Warm-up: Vocabulary assessment, including: asymptote,  rational function, exponential function, end behavior, domain, range
  • Activity: Students will use a description or a graph to find the equation for twelve functions
  • Assessment: After finding the functions, students will find domain, range, vertical asymptotes, horizontal asymptotes, and all intercepts. This will be turned in and graded.

I also uploaded a copy of the lab instructions to my public Dropbox. If you are interested in seeing the entire lab, check it out here: http://dl.dropbox.com/u/59433434/120-Lab2.pdf. (Notice that it’s 120-Lab2, even though I mentioned before it is really our third lab — I start numbering things with zero.)

I’m hoping to reuse this activity in Pre-Calculus later in the semester, once we cover material about rational functions.

Statistics Group Projects: In Progress

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The Elementary Statistics students in my course are wandering the streets of downtown Charleston gathering data for their group projects. There are five groups and here are the questions they are asking (along with their best guess as to what they will find):

  • Do you own a bike? Best guess: At least 30% “Yes” response rate
  • Do you have a passport? At most 40%
  • Are you on vacation? About 30%
  • Do you consume alcohol? At least 80%
  • Have you ever had a fake ID? Around 50%

I look forward to seeing their data!

Statistics Group Project

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Project Motivation
There are two class meetings left in my “Elementary Statistics” summer course. This class time will be devoted to students working together on a group project. Last semester when I taught this course for the first time I really wanted to implement some type of end-of-term project. I wanted the project to be collaborative in nature since both my own experiences and recent research in education have shown that students explaining concepts to each other is as important to their learning process as hearing their instructor’s explanations. I also wanted the project to be somewhat self-designed by the groups themselves. It was my hope that giving them some freedom in their projects would increase their interest level in what they were doing.

The topics we finished covering at the end of the course were about creating confidence intervals and performing hypothesis tests (sometimes called tests of significance). Because we discussed this material so recently, it seemed appropriate to have this be the jumping-off point for the projects.

Project Introduction
I wanted the students to have experience going out into the “real world” to gather data, so the project asks them to conduct interviews with people they find around campus. Since it’s only a week-long project (instead of over an entire semester), to make things easier each group has to agree on a single”Yes” or “No” question to ask their random sample. There are three rules for the question.

  • First, each member of the group must agree with the group’s decision on the question. They have to discuss different ideas, vote on them, and eventually reach consensus.
  • Second, the question must be “interesting.” This is hard to define, but basically I want them to avoid boring questions like “Are you a human being?” or “Have you ever been to Mars?” that will result in boring data.
  • Third, the question must be “appropriate” — it has to be something each group member would feel comfortable asking a perfect stranger or their grandmother or their kid brother. (Hopefully they would know to avoid offensive or disrespectful or inappropriately personal questions, but who knows?)

Once they have chosen their question, each individual is asked to guess (to the nearest 10%) what proportion of interviewees will answer “Yes” to the question. After reaching an individual conclusion, the groups discuss what they expect as a group. I wrote a handout describing the “What” and “How” of prior probability distributions and each group works on creating [a very basic] one before they are allowed to leave to gather data.

Project Report
The groups have the rest of the class time to gather data together. I tried to avoid giving them much direction on who they should interview, or where they should find the people, or what types of people to ask. (For instance, do they want to focus on College of Charleston undergrads, or are tourists okay too?) I suggested to them that they need to keep in mind a lot of the ideas we discussed in the class, like:

  • What’s an appropriate sample size?
  • What sampling method should we use? (Convenience, cluster, stratified, systematic, etc.)
  • Should we expect bias in our data? If so, what types? (Sampling bias, response bias, nonresponse bias, etc.)
  • Can we do anything to eliminate bias?

Eventually the groups must produce a typed project report, outlining their process from how they decided on a question and constructed their prior to where they conducted their interviews. They must use the methods of inferential statistics that we learned in our class to create a confidence interval for the proportion of subjects who said “Yes” and give a correct interpretation of the confidence interval. They also have to perform a one-proportion hypothesis test. They are expected to use their prior probability distribution to formulate a claim to test. They are graded on both their data analysis and interpretation of results.

Project Grading
I created a grading rubric for the project. It’s available as a public Dropbox file: 104-project-rubric.pdf A colleague looked over it in the copy room and commented, “You sure are overly detailed with that thing!” This is probably a fair criticism, but mostly I was trying to avoid hearing lots of student questions that boiled down to, “What is the least I have to do in order to get an A?”

Something that comes up a lot in discussions about graded collaborative assignments is the “slacker problem”, i.e., How do you keep students from getting by doing zero work? I don’t have a good answer for this. I know that when I was a student, I was annoyed by free-loaders, so I have empathy for students who feel the same way. One of the categories on my grading rubric is “Teamwork Assessment.” Each student must individually send me a confidential e-mail discussing how their group functioned as a team and how they contributed to the overall project. They are asked to give their team a grade of how well they worked together. It was my hope that telling them on the project rubric that (a) they are responsible for their group functioning as a team and (b) they are also responsible for ensuring they contribute to the group that it would create a cultural pressure toward equal collaboration. I can’t say for sure how successful this was last semester, however I was happy that out of nearly 100 students, I only had one or two complain about slackers in their groups.

Looking Forward
This summer’s class has a strong group mentality, I think partly because we have been spending ten hours a week together in class. I hope that this will contribute to great collaborative effort toward these projects. I am also excited to see what questions they will ask and what their data will show. I’ll end this post with a few questions I remember from the class projects last spring:

  • Do you have a fake ID?
  • Do you own an iPhone?
  • Will you vote to re-elect Barack Obama?
  • Did you drink alcohol last weekend?
  • Do you have blue eyes?
  • Do you have a car on campus?