Teaching the Big Ideas

This summer, as part of my professional development, I am taking an online Stanford course entitled "Mathematical Mindsets". This is a course developed by Jo Boaler, Cathy Williams, and others at YouCubed.org, a part of the Graduate School of Education at Stanford.

As part of this course, the question just posed to me was, "Do you think you could teach to big ideas, instead of all the isolated content standards? Why or why not?" For some reason, I have been working on this course through the lens of my 10th grade integrated math course. But, this question resonated with me in regards to AP Statistics.

One of my great difficulties in my first two years of teaching the course was getting through all of the content in time for the AP exam in May. I have been relying a lot on our textbook. So, my goal has largely been to cover as much of the text as possible. The book is broken up into 12 chapters and each chapter has two or three sections.

Each chapter has its own main idea. But, each section within a chapter has its own ideas. So, I feel like I have been teaching AP Statistics by focusing a lot on the isolated content standards.

I know there are big ideas in this course. Without looking it up, I think there are four. So, I wonder now if I could teach AP Statistics better focusing less on the nitty gritty and more on the big ideas and then study problems and ideas in each. I may not get to every individual smaller idea, which might cause some problems on the multiple choice problems. But, I think this approach might help students do a lot better on the free response questions.

Teaching this way certainly seems possible. I wonder if anybody reading this understands what I am writing and has any experiences that they would be willing to share. Have you tried teaching AP Statistics by focusing on the big ideas? Was it successful? Did you try this multiple times? Has anyone tried this approach but thrown it out because they felt it didn't work?

Improving student learning

It is the middle of summer. Temperatures are in the 90s and 100s. And here I am writing about my AP Statistics course.

This fall, I will embark on my third year teaching AP Statistics. I have not been generally pleased with the learning of my students in the first two years. A very low percentage of my students have passed the AP exam. I do not blame myself too much for these scores. My students thus far have not been terribly dedicated to this course. Senioritis interferes significantly.

But, there were some students that I think worked hard enough to get a three on the exam and fell short. I do think I provided ample opportunity for these students to succeed. I believe they had meaningful learning experiences that could have prepared them well for the exam.

I want better for them. I want to improve my course. Not just to improve AP scores. But to up the interest in the material and the course. I believe if students find the course more interesting, their learning will improve. I also want to make the material easier to access.

My first move is to start the year off with data gathering unit that will culminate in a questionnaire project. Working in pairs, students will design a questionnaire to gather data on an issue of their choice (such as abortion, immigration controls, the legalization of marijuana, etc.). The questionnaires will be distributed to the students and teachers at our school and the data we collect will be used throughout the year.

My second move is to increase the number of projects. Last year, there were four projects. The other main assessment I used was free response questions. So, this year, I am hoping to do six projects and reduce the number of free response questions.

I have been teaching a non-AP statistics course for many years now. That course has always been project-based. So, I have plenty of experience from which to draw. I am hopeful that project work will lead to deeper understanding.

Of course, today, one of our administrators wrote in an email to me that she hopes AP classes will be gone from our school in one or two years. That was a bit demotivating, but I owe it to my students to make the course better. Hopefully, that will be motivation enough.

Homework? To what end?

Is homework necessary to deliver the AP Statistics curriculum?

This week, my class will meet four times and the students have four homework assignments (see below) that they should work on prior to each class. I expect that they should spend 30 to 45 minutes on each assignment. So, I am asking my students to spend around six hours this week on AP Statistics. And this is a fairly typical week. We have about 31 weeks total from August until May for the students to prepare for the exam (vacation weeks excluded). So, that works out to about 186 hours of preparation for a 3 hour exam.

The AP courses are supposed to be college courses. Yet, we have far more class meetings than any college course I ever took. And the course is spread out over a year rather than a semester. My college courses met for 50 minutes, three times a week, over about fourteen weeks (though it may have been fewer than that). If we assume that a one-hour assignment accompanies each class meeting, then that would work out to around 77 hours of study in a semester. We should probably round that up as that does not account for projects or papers. So, let’s say 90 hours.

How is it that a college course expects a student to learn the same material in roughly half the time as a high school student?

Is homework necessary?

I would like to perform an experiment to investigate this question further. That just does not seem practical, or maybe ethical is a better choice of words.

If I were to eliminate homework altogether, my students would still have about 103 hours of preparation time which should exceed that of any college student. Add in another 20 hours maybe for project work and get a total of 123 hours which would be about 33% more than a college student. That seems like a more appropriate balance. Especially when you figure that the course would be delivered over twice the number of weeks allowing for the work to be presented in smaller chunks, and thus more easily digested.

In my mind, this should be feasible. If the AP Statistics course truly is a college course, that is. My class consists of eleven twelfth graders and two juniors. The twelfth graders are six months away from taking college courses. Do we really expect to be preparing them for college if the transition is so sharp?

Using an activity to teach about unbiased estimators

Today in AP Statistics, we had one of our better classes, so I wanted to share it with you. We did an activity to better understand the differences between parameters and statistics and what makes for an unbiased estimator. I based it off of an activity in our textbook and it worked beautifully.

Each of my 12 students wrote their heights in centimeters on a slip of paper and put that paper in a box. Each group then took turns randomly selecting a sample of size three with replacement. The groups recorded the data and calculated the sample mean and the sample range. Each group then wrote up their data and their statistics on the board. I then had each group take another SRS of size three and report their findings. Meanwhile, two students were tasked with finding the population mean and population range, but they were asked to not share these numbers with the class.

So, now we had eight samples, eight sample means, and eight sample ranges. I briefly talked about how statistics were used to estimate parameters.

I then asked the students which sample mean was the best estimate of the population mean and to explain their thinking. Four students were willing to offer their opinions and they gave reasons such as it seemed like the number in the middle. Interestingly, two students selected 172 cm and two others selected 168 cm.

I then asked the same question regarding the sample ranges. This discussion went differently. One student immediately realized that the largest value, 30 cm, was the closest to the population range. The same student continued and argued effectively that the range was at least 33 cm and that no sample would ever have a range greater than the population range.

We observed and discussed how the various sample statistics deviated from each other. We discussed the centers of the sample distributions. And then I asked the class which of the two statistics they thought was unbiased. One student bravely spoke up and said the sample ranges were unbiased and offered up his reasoning. The others listened and then, respectfully, argued against his claim and presented their own reasoning.

The population parameters were then shared. We then drew dotplots of the sampling distributions and added in the population values to the dotplots so the students could visualize how the sample ranges were all less than the population range and how the sample means fell on either side of the population mean. We also discussed how the sample ranges deviated on average more from the population range than the sample means deviated from the population mean.

The dotplots were very effective. We talked about how the population parameter is the target and the sample statistics are our attempts to hit that target. Using this lens, it was easy to see how the sample mean was a better estimator than the sample range.

This was only a 35-minute class, so we did not get into how we could improve the accuracy of our estimates (so how to affect variability), but that would be an easy extension. Again, a dotplot could be used to show various levels of accuracy dependent on sample size.