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.

Free Responses and Projects

Going into this school year, my intention was to resume blogging regularly. I wrote my first post back in September, but now it’s been a month and a half. My hope had been to write biweekly. Anyway, I won’t bore any readers with the reasons.

If you read my last post, my plan was to change my assessments significantly. And, so far, I have followed through on that. I have given three tests all have been made up of only Free Response Questions. The students have completed one project and will do one more before Thanksgiving. My thought right now is to give a multiple choice assessment before the end of the semester. But that will be the only one.

I just finished grading the last test and it appears that the students are getting better at answering the Free Response Questions.

The next project will be on data gathering techniques. The students, working in pairs, will have to choose a topic, create a questionnaire, devise a sampling strategy, and discuss bias. Their population will have to be found within our school so that they can select the sample and distribute their questionnaires. The questionnaires will be made online using Google Forms. The data they collect can be used for future projects or assessments.

The sixth Free Response Question on the AP Statistics exam is always a longer one called an Investigative Task. My plan is to try to set the project up in some ways like an extended investigative task. I will try to make it so that in the process of doing the project, they will have to address the same sorts of issues that might arise on an investigative task.

Anyway, that’s what’s going on for now.

A New Beginning

Last year, I started this blog as a way to journal about my experience as a first-year AP Statistics teacher. I wanted to share my experiences to help me reflect on my practices and to hopefully initiate some discussions with whomever read my blog and dared to comment. I knew that there would be other teachers out there in a similar position as I and that my posts might help them think through their own situations. I also hope experienced statistics teachers might stumble onto my blog and provide insights. In the end, the blog mostly served the first purpose, and that was fine. But, then I lost my drive to blog and I posted less regularly and then stopped altogether.

Now I begin again. I want to continue my professional development and I think blogging will be a terrific means to do that. Maybe some discussion will ensue.

Last year, I was anxious about getting through all of the material in time for the exam. To my amazement, I succeeded. Not only did my students cover every topic on the AP Statistics syllabus, but we still had two weeks to review before the exam.

However, in reflection, I do not feel my students were served best by zooming through the material. A few of them did tell me that they liked that we covered all the material (and finished the entire book). But depth of understanding was sacrificed for breadth and I hated that.

The students' results on the exam also suggested that some changes needed to be made. The students took a mock AP exam as their spring final exam two weeks before the actual AP exam. Based on their performance on that exam and based on their work all year long, I had expected them to score a bit higher. But, when I think about how hard they worked, and how some of them did not work too hard, I wasn't altogether surprised. Those who did the best had the best work ethic. Those who succumbed to senioritis largely did not pass.

This year, one change I am planning is to not worry about covering every single topic on the syllabus. I teach at an independent school (polite speak for private school) and we have only around 140 teaching days, I think, before the exam. My plan is cover 80-90% of the material, but to go into greater depth. The instructions on the AP exam itself state that they do not expect students to know everything. My class looked at scoring and were surprised to learn that you could get 50% of the multiple choice items wrong and still have a very good chance of scoring a 4 or 5.

Another change is that I am not going to give them very many multiple choice assessments. Last year, I gave the students 10 multiple choice items on every test and they had about 11 tests. And then they had 40 more multiple choice items on the mock exam. But I don't think one needs to practice multiple choice tests. Certainly not to the extent that I did last year. And I detest multiple choice items. So, I plan to give them the 40 problems on the mock exam and then maybe two other multiple choice assessments. That's it.

A third change is that I am going to incorporate more project work this year. When I have taught non-AP Statistics, I have always taught it purely project-based as I feel this is the best way to learn the material and the best way for me to evaluate their understanding. It feels more authentic and it removes test anxiety and the time constraints of tests. I will likely write an entire post on projects later.

But now, I must rest.

Thanks for reading.

Blackjack or Bust

Over three weeks in to the second semester and, unsurprisingly, the seniors in my AP Statistics course are finding it harder and harder to get motivated to do school. That means it’s time for some blackjack.

At Tuesday’s evening class, we dealt some cards, made some bets, and recorded lots of data. I have done a blackjack project several times now and it always grabs the students’ attention and interest. Some students learn the game for the first time while others learn the finer points such as splitting and doubling down. But all enjoy watching their piles of chips go up and down.

While they play, the students record such things as card values, number of hits, wins, losses, pushes, and busts. In groups, the students will then use the data to create their own examples that are meant to illustrate the various probability concepts we have studied thus far. These topics include conditional probability and independence, random variables, the binomial and geometric distributions, and sampling distributions. The groups are tasked with teaching their topics to their classmates. As a part of their presentation, a group will also provide a sample problem for everyone to try to solve.

I have come to strongly value students taking over class and teaching their peers. Explaining the intricacies of challenging mathematical ideas is not easy. And many students have anxiety about speaking in front of the class. But it is so worth it. From my experiences with these student presentations, I have seen my students gain deep understanding of the topics and I think the students appreciate having class led by someone other than their teacher. After reading Jo Boaler’s book, Mathematical Mindsets, I now know that research has shown that explaining work to others creates deep learning.

In addition to the presentation, students have the opportunity to collaborate with others. Group work is another prominent part of my classes. I think my favorite classes are when each group is humming along, discussing ideas, solving problems, checking each other’s answers, and helping one another increase their mathematical knowledge. It provides opportunities for students to explain ideas in a small group setting that might be easier than in front of the whole class. Once again Dr. Boaler notes in her book that the research backs the practice.

The project will give the students the opportunity to review many different ideas. I like the idea of doing review now, even though a little more than three months still remain before the AP exam. I want my students to engage with the material as many times as they can before May. I think it’s a bit silly to think that a student can really learn and understand a concept the first time they see it. The more times they try to reason through a binomial distribution problem, the more familiar they will become with the idea. I have taught random variables ten times now, and I am better this year with them than I was last year.

Finally, projects are such a refreshing change from the in-class timed tests. I like that my students get a chance to show what they know in a much different setting. They are not sitting, working alone, trying to remember when to use which formula. Rather, they are up, working with others, and they can look up information in their books, in their notes, and even on the internet. They can meet with me to talk about their project. The students can choose to spend many hours in preparation so that they can produce exactly the quality of project they want. I am going to try to see if I can test less next year and still prepare my students well for the AP exam. I believe it can be done. Do you? Anybody out there teach project-based AP Statistics course? I would love to hear about your experiences.

Halfway there... a self-evaluation

My greatest concern going into this school year was to effectively teach AP Statistics. Though I have been teaching statistics for ten years, I had never taught AP before and I was worried about pacing and the quality of the lessons and assignments. I was worried about sacrificing pedagogical beliefs for the sake of getting through the syllabus in time and preparing students for the format of the test.

The course is going better than I had hoped. I greatly value projects in lieu of tests. Whereas a test is completed in a set amount of time, contains closed questions, and is done individually, a project is done over a number of days where students can choose how much time to spend on it, is more open-ended, and is done in a group of 2-4 students. Tests have very little value in helping students improve on their knowledge and are limited in their ability how much information they provide regarding what a student knows. Projects can help a student to better understand material and are provide richer information about what a student can or cannot do. My non-AP Statistics course was always completely project-based. I wanted to carry over as much as I could from the non-AP course into my AP course, but I feared the pace would make this difficult. I was optimistic that I could incorporate two projects this year. Happily, the students were able to complete two projects in the first semester alone. I now expect them to complete two more in the second semester at least.

The pace has been fine. While I am going faster than I ever did in my non-AP course, the pace has not been so fast that students are being left behind. The faster pace requires students to be more self-motivated, but, maybe because it carries the AP designation, students have been up to the challenge. Also, I have been able to adjust the pace at times based on feedback from my students. I had concerns that there would not be enough flexibility to adjust the pace mid-semester. But there proved to be more flexibility than I had anticipated.

Back in August, I started this blog regarding my teaching of AP Statistics. I am happy with how blogging has gone. This makes post number fourteen. Though my audience is quite limited, people have been at least clicking on the blog links I share on Twitter. One Twitter user even added one of my blog posts to the Math Education Daily and that particular post had a lot more page clicks. Blogging has allowed me frequent opportunities to reflect on my teaching and that alone has helped me to design a better course. It has provided little so far in professional discourse. I hope to get back to blogging weekly and I hope that my regularity can increase the chances of future discussions about my posts. I have engaged in a couple Twitter discussions with other math teachers. I would like to do more of that going forward.

[Regarding Twitter, I very much want to participate in Twitter Stats Chat. This is definitely a goal of mine. I have yet to figure out how to do that. I know the hashtag (#statschat) and I know that it starts at 6 pm (Pacific Time) on Tuesdays. I have TweetDeck set up for #statschat. But I never see a bunch of posts starting at the specified time in my feed. Grumble.]

In looking over the various assessments I have designed for my AP Statistics students, I am very pleased with the progress they are making. The projects and tests show that they are learning the material at an acceptable rate. Predictably, the multiple choice items have been the least helpful. I do not like such test questions. I have felt compelled to include ten with each chapter test so as to better prepare the students for the AP exam which will contain forty multiple choice questions. However, I am questioning how much practice students really need with such types of questions. This re-evaluation has partly been fueled by Jo Boaler’s book, Mathematical Mindsets, which I am almost done reading. In the book, Dr. Boaler presents research that affirms my views that students do not need practice for standardized tests. Rather, students can actually perform better on standardized tests if they have learned to be confident mathematical problem solvers. For the next chapter test, I have included ten multiple choice questions. But, I have made one adjustment that was suggested by a Twitter user: students can designate a first-choice answer and a second-choice answer. A correct first-choice answer earns two points while a second-choice answer earns one point. However, I think I will include more short answer questions and fewer multiple choice questions on future tests.

The Weekly Menu

This semester, I have started serving something new in AP Statistics: the Weekly Menu. (Click here to see the Weekly Menu for this week.)

After each class meeting, the Chefs du Jour are expected to add one note taken from class. That note can be anything the student thinks is useful or important to remember. Other students in the class can see the notes posted by their classmates and add to them or even edit them. The Topics du Jour and the SMART Board notes for the day are also posted.

One goal of the Weekly Menu is to get students more invested and active in their learning. I am currently reading Jo Boaler’s book, Mathematical Mindsets, and it inspired me to be more intentional in trying to develop student self-awareness and responsibility. Dr. Boaler outlines Assessment for Learning (page 149 of this book) which “teaches students responsibility for their own learning.” I hope that by assigning some responsibility to students to fill in the document, they will take greater ownership of the course. There is also a level of peer accountability since all students will see what the Chefs have or have not added to the document.

Another goal of the Weekly Menu is to provide students the opportunity to reflect more often on what has been covered in class and what they should be learning.

Additionally, I am hopeful the Weekly Menu will serve as a good review tool. They can use it to review for tests and even for the final exam and AP exam. We will also use it to review what we did the previous day and use it to transition to the next topic.

Students can also use the Weekly Menu as another way to see what is coming up that week. And, if they miss class, the Weekly Menu can get an idea of what happened in class that day.

So far, the feedback I have gotten from students has been somewhat positive. We are only on the second week, so it is still in its infancy. Colleagues of mine like the idea, but ultimately it is meant to be a tool to help students, so it is their opinion that I will value most.