A First Day Statistics Activity

I have the honor of again teaching our undergraduate statistics course in the School of Education, better known here as EDUC 4716 Basic Statistical Methods. Perhaps the most interesting thing about the course is that it's not required for any education programs, minors, or certificates. Instead, the course attracts students largely from the Department of Speech, Language, and Hearing Sciences (who don't need it to graduate, but do need it to apply to grad school or, more recently, to get certified) and sociology majors. So how does this course end up in the School of Ed? Probably due to the legacy we have in quantitative methods, thanks to people like Robert Linn, Gene Glass, Lorrie Shepard, and now faculty like Derek Briggs and Greg Camilli. Somehow all of their hard work and success filters down and gives a relative stats-hack like me a chance to teach undergrads.

Many of my students are upperclassmen and have spent much of their college experience avoiding math courses. In fact, on last year's FCQ (Faculty Course Questionnaire) my students' average rating for the item "Personal interest in this subject prior to enrollment" was a 1.8 out of 6 -- a response the university tells me is at the 0th percentile across campus. I like to think of this as a great opportunity in a "nowhere to go but up" kind of way, a chance for me to change the way students think of mathematics and see themselves as mathematical beings. Then again, it's hard to make big changes in only 15 class meetings of 2.5 hours each. If I'm going to make a difference, class has to get off to a solid start.

My opening activity this year started with the preparation of four simple index cards with different distribution shapes:
Four common distributions, clockwise from top left: normal, left skewed, right skewed, and normal.

I have the benefit of a small class of 14 students. So I cut my graphs into a total of 14 pieces:

14 pieces for 14 students. Note on the bottom I've provided the hints A, B, C, and D.

When class started, I mixed up the graph pieces and handed one to each student. Then I told the class to find the other people in class who had the graph pieces that aligned with theirs. Once they had a completed graph, form a group at one of the tables and discuss which of the following they thought their group's graph might describe:
  • People born each month of the year
  • Student GPAs at this university
  • Student heights at this university
  • Starting salaries of new graduates from this university
It took my class less than 3-4 minutes to find their groups and then I gave them another 3-4 minutes to discuss what their graph shape might describe. As a class, I had each group share their ideas and then we discussed them. Not everybody agreed initially about which shape matched which description, which led into important comments about how we might think about unbiased sampling of students and imagining different scales and labels along the horizontal axes.

So in less than 15 minutes I combined group-making, statistics, and active, student-centered problem solving into one activity. This activity also gets students thinking about distribution shapes, which I sometimes worry we ignore in the rush to calculate centers and spreads. If you're wondering how to adapt this for your classroom, I offer these suggestions:
  • If you have a few more students, cut more slices.
  • If you have twice as many students, consider making two of each distribution shape and scaling the x-axis to match one of 8 potential descriptions. (i.e., a normal distribution scaled for heights in inches could be distinguished from one scaled for SAT scores.)
  • If you want to use this for Algebra 1, you can make graphs that describe things like, "Toni walked to the bus stop at 2 mph, rode the bus at 30 mph to the bike shop, then rode a bike back home at 12 mph." Such an activity begins CPM's Algebra Connections and was the inspiration for my activity.
  • If you want to use this for Algebra 2 or higher, you can use graphs of functions that students will become familiar with (parabolas, cubics, hyperbolas, etc.). I don't think it's worth fretting over vocabulary at this point -- just give students an opportunity to think about how the functions behave and what phenomena they could possibly model.

Nielsen's Reinventing Discovery (2011) in the Context of Education Research

As a Ph.D. student I've taken my share of methods courses, giving me skills in everything from ethnography to ANOVA. But as important as those things are, I've sensed that there are new research methods emerging thanks to technological advancements and online communities. Our lives are too data-rich and our means of communication are too plentiful to limit ourselves to the same methods for research -- and learning -- that we used just 10 years ago.

Even though I feel like I live in the thick of this revolution, engaging with teachers and researchers on Google+ and Twiter, I wanted a broader perspective on how researchers use networks to make new discoveries. For this I turned to Michael Nielsen's book Reinventing Discovery: The New Era of Networked Science. Although Nielsen is a pioneer in quantum computing, I hoped to find some ideas that I could apply to a social science like education research.

Nielsen uses a variety of examples and concepts to describe what works and what doesn't (or hasn't) in networked science. Instead of listing them here, watch this TEDx talk by Nielsen:

If that talk wasn't long enough for you, Neilsen held a longer talk at Google that is worth checking out.

As much as I like Neilsen's example of Tim Gowers's Polymath Project, I can't imagine a direct translation to education research. One of the beautiful aspects of mathematics is that it usually doesn't require conducting an experiment, interviewing subjects, sampling a population, or agreeing on a conceptual framework -- the kinds of things that make social science untidy and difficult. Frankly, if solving problems in education were structured like proving mathematical theorems, I think we'd be solving more problems and finding better solutions than we are currently.

Neilsen's story about Qwiki hits home for me. For some time now I've imagined creating and maintaining a wiki that essentially translates the contents of the NCTM's Second Handbook of Research on Mathematics Teaching and Learning into knowledge that teachers could access and use. Just like Qwiki, it's easy to get math teachers and educators to agree that this would be a great resource to have. Unfortunately, I'm not sure how a math education wiki like the one I've imagined would avoid Qwiki's fate. Without incentives for experts to contribute and maintain the site, I'd probably spend more time fighting spam than helping teachers.

Neither the Polymath Project or Qwiki offer a blueprint for a new kind of mathematics education research. Thankfully, Nielsen describes some general characteristics for successful networked science. First, in his chapter titled "Restructuring Expert Attention," Nielsen suggests networked science has these attributes:
  • Harnessing Latent Microexpertise -- The project must allow even the narrowest of expertise. A 3rd-year algebra teacher might not have the broad expertise of an experienced math education researcher, but that 3rd year teacher might have small elements of expertise that exceed that of the recognized experts.
  • Designed Serendipity -- The project needs to be easy to follow and encourage participation from a variety of experts. You want problems to be seen by many in the hopes that just a few will think they have a solution they wish to contribute.
  • Conversation Critical Mass -- One person's ideas need to be seen by others so they create more ideas, and the conversation around all the contributions keeps the project going.
  • Amplifying Collective Intelligence -- The project should showcase the fact that collectively we are smarter than any one individual.
Those are all great characteristics of any project. But what makes this any different than any traditional, offline project? Nielsen offers several suggestions. Unlike a large group project with clear divisions of labor, technology allows us to divide labor dynamically. Wikipedia certainly would not have grown the way it did if labor had been divided statically between a set of contributors. Also, networked science uses market forces to direct the most attention to the problems of greatest interest. Lastly, contributing to an online project rarely feels like committee work, and participants can more easily ignore poor contributions or disruptive members.

Projects like Wikipedia and Linux exhibit the above attributes, but Nielsen explains that such projects needed something extra in order to scale to thousands of participants. Nielsen describes these in a chapter called "Patterns of Online Collaboration," and they are: (1) being modular, (2) encouraging small contributions, (3) easy reuse of earlier work, and (4) signaling to what needs attention. When I look at this list and think of Wikipedia, I can see how well a wiki or open source software project fosters these patterns. But how do we build such a project in education? Given Nielsen's framework above, a project that would interest me needs three key aspects:
  • The content of the project has to be something that both teachers and researchers can contribute, such as a collection of math tasks, curriculum plans, or perhaps pedagogical techniques.
  • Teachers need to be able to easily use and modify each other's content.
  • (This one's the crux!) When teachers use content, there needs to be a way to collect and submit feedback about the use of that content, and that feedback becomes data that researchers can use not only to improve the content of the site, but to produce new and traditional reports of research.
It's that last bullet that's the hardest but most intriguing. There are so many places to get lesson ideas on the internet, but I don't know of any that collect data about the effectiveness of the lesson in a format suitable for research. Khan Academy claims to do this this kind of data collection internally, but KA is a closed project that lacks nearly all of the attributes Nielsen has described in his book. The project I want needs to be an open one, with all of its moving parts exposed and no more owned or identified with a single participant as Jimmy Wales is identified with Wikipedia. If you have ideas for what such a project could/should look like, leave them in the comments!