Some probability problems for your consideration

As I was preparing this tweet on AMS prizes given at the 2016 Joint Mathematics Meetings…


…I checked the AMS Annual Survey of the Mathematical Sciences to determine that in 2015 roughly 30% of full-time faculty in the mathematical sciences were female. Then, I wrote and did the following math problem:

Question #1: Suppose that 20 people are being chosen for various awards. Assume that for each award, there is a committee that narrows the field of candidates down to a set of 10 finalists who are deserving of the award, and 3 of them are female. Assume that each committee independently chooses the winner from this set of finalists in such a way that each of the finalists is equally likely to win the award. What is the probability that none of the 20 people are women?

Answer: (0.7)^20 ≈ 0.0007979

That seems really unlikely.

Some people are going to argue that because women have been excluded from the mathematical sciences to a greater degree in our recent history, the population of senior people who have made big contributions to their fields is going to skew even more male than the general population of full-time faculty.  (BTW, not all of the awards above are for senior people who’ve made big contributions.) OK, so let’s consider another problem.

Question #2: Keep everything from Question #1 the same but this time let’s assume that x (in %) of the finalists for each award are men. What would x have to be so that the probability of not having any women chosen is something more likely, like 25%?

Answer: x^20=0.25 means x ≈ .933 ( or 93.3%)

Is it the case that there are so few women deserving of awards in mathematics?

I assert that we don’t suffer from a lack of talented women and people of color in the mathematical sciences. The problem is that we don’t nurture talented women and people of color, don’t recognize their talent, and don’t take enough steps to make sure our award processes are unbiased, fair, and equitable. (There is plenty of evidence that women are just as implicitly biased against women as men are.)

Look at the dramatic change in the representation of people of color in this year’s Oscar nominations, compared to a mere two years ago when #OscarsSoWhite was trending. Did the field suddenly produce a bunch of new Black artists and producers and directors in those two years? No.  So why can’t those of us in the mathematical sciences also try harder to make sure that we make sure award winners are more broadly representative?

Every time a women or person of color looks at a list of winners like this one, they receive messages that they aren’t welcome in the field.

Important caveats:

  • These two problems are just food for thought. I’m not implying that the actual awarding of the AMS prizes is anything like what is described above.
  • I’m not suggesting that any of the selection committee members were deliberately being racist or sexist.
  • I’m not suggesting that any of the winners of the awards were not deserving and should return their prizes.
  • Also, my tweet and this post is about the underrepresentation of women, and I also recognize that there is an underrepresentation of African-American, Hispanic/Latinx, and Native American people among the award winners.

It would be interesting to go back in history to look at the demographics of all AMS prize winners.

Recommended reading:

Addressing the underrepresentation of women in mathematics conferences” by Greg Martin

The MAA’s guidelines for award selection committees on how to avoid implicit bias.



Institutional and Personal Roadblocks to Implementing Active Learning

As I posted last October, the evidence in support of the benefits of active learning for student learning is very robust. The CBMS put out a strong statement advocating for “institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into post-secondary mathematics classrooms.”

Active learning has been around a while. Why hasn’t it been more widely adopted? Debbie Gochenaur (Shippensburg University) and Larissa Schroeder (University of Hartford) organized a panel discussion at the 2017 Joint Mathematics Meetings on this subject. Larissa, Kimberly Presser (Shippensburg University), and I spoke.  Here are some of my thoughts on the institutional and personal roadblocks that prevent faculty from implementing active learning in mathematics courses.

Here’s a link to Debbie’s notes on active learning based on her presentations at 2017 JMM.

Roadblock #1: Department/institution culture is often not set up to encourage teaching innovation

Three instances of this roadblock:

  • In our discipline, lecture is still dominant and it is uncomfortable to go against the dominant culture.
  • If departments don’t seem something as needing to be fixed, then they won’t invest the time to fix it. In many institutions, high DFW rates are not seen as a problem. Sometimes, they are even seen as a good thing because of a commitment to upholding “high standards” and seeing Calculus courses (or intro math) as gatekeeper courses.
  • Reappointment/promotion practices and departmental culture often discourages teaching innovation. Many people will say that teaching matters, but when the rubber meets the road, what do senior people in department and on reappointment committees really think about spending time on teaching instead of research? At many schools, especially research-intensive schools, spending time on teaching is a hindrance. Unless we get our institutions to expect faculty to spend on teaching, folks won’t do it.

Some possible strategies to overcome these roadblocks:

  1. Find allies. Are there folks in your department who use active learning? Talk to them. Look for people at your institution in other disciplines. If your institution has a teaching and learning center, find folks there who can support you.
  2. Don’t be naive about your departmental/institutional culture. Be politically savvy. Especially if you’re a junior or pre-tenure person, know who you can trust to tell you the truth about the prevailing attitudes and practices are at your institution. Get clarification from them about what the tenure and promotion expectations are. How is teaching innovation viewed? Is it ok to not get stellar teaching ratings one semester when you try something new as long as you can describe what you did, show evidence of self-reflection and improvement? Most administrators are reasonable and understand that tenure processes are not supposed stifle teaching innovation. If your department is not supportive, seek administrative support.
  3. Have evidence about active learning at your fingertips. There is a ton of research about active learning. Read and use the CBMS statement about active learning. If there are other folks at your institution who have had success with active learning, talk to them and find out how they measured success. Be aware of common myths about active learning and how to rebut them. (Common myths: You can’t “cover” as much material if you use active learning; or you can’t do active learning if you have too many students in a class.)
  4. Sometimes the active learning folks can behave like they’re members of a cult, so be careful about how you talk about active learning with your colleagues who aren’t into it. In your zeal for a particular active learning technique, don’t forget that any particular teaching strategy is not going to be the magic bullet that will solve everyone else’s problems. We should try not to alienate people but to welcome them into dialogue with us about teaching.

Roadblock #2: It’s difficult to find adequate training and support to implement active learning.

The truth is that you can go to a bunch of sessions about about inquiry-based learning (IBL) and still not know how to implement it. In fact, we know that is unlikely that people actually change their teaching practices as a result of attending a one-off workshop or session about a particular teaching strategy.

Learning a new teaching approach requires a lot of direct interaction with other people who are thinking about and working on similar problems. This is why the previous suggestions about finding allies is doubly important. You want to find people who can observe your classes and give feedback. You should also try to visit others classrooms. If you have the opportunity to co-teach with someone that you trust, that is often a great way to really work on your teaching.

Look for more in depth training. There are multi-day institutes during the summer, MAA minicourses, etc. Again, seek out folks in the teaching and learning center to support your efforts.

Roadblock #3: Many students don’t expect to be active in college classes. Change is uncomfortable.

If you spring active learning on your students, sometimes that can back fire.

Be transparent about why you’re using active learning. Don’t surprise them by it. Talk about why you’re using active learning on the first day of class. Tell them how to prepare for class to engage. Introverted, English language learners, and students with learning differences need heads up and time to prepare.

Roadblock #4: We are our own best and worst critic.

Don’t worry about getting it 100% right the first time. Improving teaching is a lifelong process anyway. I will never be perfect at my teaching. When trying something new, the most important thing is quickly adjust based on feedback. I strongly recommend exit tickets at the end of your class to get feedback. Then in the next class, refer back to specific comments that you received (good and bad) and show students that you’re taking their feedback seriously.

Don’t feel like you have to dive into the deep end. For example, don’t feel like you have to flip all of your classes. That’s one way to do it, but I having done it, I don’t recommend that unless you’re willing to put in the time and you have support. You can gradually develop more active learning strategies in your teaching repertoire over time.

There are lots of different forms of active learning. If one form of active learning doesn’t work for you, there is always something else to try. Not all of them are as time intensive as group work, or project-based learning. You can use clickers in class. You can set aside time for a quick think/pair/share. The key about active learning is that you are not doing all of the intellectual heavy lifting for students–you just have to get them ready to do it. Find ways to get students to prepare for class by doing the reading (or watching videos) so that they are able to take on some of those tasks on their own in class. That way you still “cover” the same material but you involve them actively in some of their own intellectual development.

If you’re pre-tenure or worried about employment security, take time to document your process as you teach. Jot down your thoughts at the end of class before you forget them. Use those in your tenure packet later. Use your exit tickets and teaching evaluations as evidence of self-reflection and improvement.

We often hold up others as models for ourselves. That is great to be inspired by others, but you have to be yourself in the classroom. And in fact, if you aren’t authentically yourself, students will know. So take the time to be introspective and find out what works for you and don’t apologize for being yourself.

What roadblocks have you encountered or observed to implementing active learning? How did you overcome those roadblocks?