Introductory experience – practice – quiz – review
Each of the chapters in this unit was introduced with a real-life example. After this, the example was connected into an example of a problem that they would encounter and I modeled how to solve it. For the next day or two, students had work time in class to do practice problems, during which I would circulate the room and help students with questions. Finally, students took a quiz on the material before we moved to the next topic. Quizzes were graded overnight and returned the next day. I went over common mistakes with the group and took individual questions for a short time before recollecting the quizzes. The sections below outline the introductory experiences for four of the topics.
Binomial theorem
I created a game where students formed groups of six and used the “random” function on their calculators. When it was the group’s turn, everyone pushed the enter key. They counted the number of “winners” on their team and reported it to me, where I marked it on a histogram on the board. After playing a number of rounds, we analyzed the distribution and discussed why it was not always the expected number (2) of winners per game, yet followed a predictable pattern.
Confidence intervals
I probed the class about the statements made by gallop.com about what Americans think about a variety of topics: how could they say what Americans think when I’m an American and I wasn’t asked? How sure were they about their statements? How did they get so sure? How many people did they ask? The class discussion that followed put out a number of preconceptions about the degree of accuracy of published data (ranging from “they’re sure” to “they just guessed”). This bridged well into a discussion of margin of error and confidence levels.
Hypothesis testing
We spent the days leading up to this introduction playing Monty Hall’s 3 door game from “Let’s Make a Deal”. The students guessed one of 3 doors on the board, I removed a door, and students decided to keep their first choice or switch. Because of the fixed probability of the first choice being 1/3 and the student having only 2 choices, there is twice as great of a chance of winning by switching. However, most students didn’t notice this nuance of the game and used each strategy half of the time. At the end of the game, we compared the actual data to the guess of 50/50 odds using the new concept of a hypothesis test. Students learned how to use a set of data to show that a hypothesis is very unlikely to occur.
During this unit, I also introduced the idea of a null and alternative hypothesis using the example of a court of law. The null, or default hypothesis is that the defendant is innocent. If the evidence is sufficiently convincing where it is very unlikely that the defendant is innocent, we convict and say the person is guilty. This analogy also proved useful in the discussion of error in the next chapter.
Power
Using the court of law example again, I introduced Type I and II errors as wrongly convicting someone or letting a guilty person go free, respectively. Choosing the appropriate alpha level for the situation naturally followed in this discussion.
Lesson plans: see the linked files at the bottom of the page for a sample of lesson plans during this part of the unit (including reflection on what succeeded or needed work).
Minute to Win It games and data collection
Each student self-selected into a team of 3-4 students. I asked students to choose a game from the TV show “Minute To Win It”. These games are fun challenges made out of household items where players have one minute to win the game. Examples included stacking 4 chapstick containers, attaching pedometers to each limb and trying to hit a threshold number of “steps”, or trying to blow all but the bottom card off of a deck of cards resting on the mouth of a glass soda bottle. Students chose a game, acquired the needed supplies, and decided on a number of categorical and quantitative variables to measure.
On the two game days, each group was paired with another group to play each other’s games until everyone was able to play every game. The groups collected times and scores and recorded them in a shared Google spreadsheet. Many groups chose to do a comparison study between left and right hand performance, first and second trials, or two different versions of the game.
After the games were played, each group had a spreadsheet with 24 students’ data. For one part of the project, they looked at data from all students to study the population. They generated a histogram to visualize the distribution of one of their variables, and then went through a painfully long process to generate 24 random samples of 8. They plotted these next to the original distribution to demonstrate how the central limit theorem works.
Minute to Win It team paper
Then they took a random sampling of 8 of their 24 students to use as the “sample” that they would perform inference on. This was the basis for the paper they would write. To guide students through the paper, I provided a very detailed rubric and a sample paper that I wrote. Since the papers were written using Google Docs, both the team and I had access to the document at all times. Thus, each night between classes during the week of writing, I left students many comments to guide them during the next day’s work. The paper included an explanation of their game and the experimental conditions, a two-way table of counts, a binomial distribution, a significance test, a confidence interval, a discussion of error. Each paper also had an abstract, introduction, and conclusion to put it in the style of professional technical papers.
After writing the paper and finding confidence intervals and performing tests of significance, they checked their inferential work against the population data they also collected. They wrote a short section in a separate document on whether their tests performed as desired or if any error occurred due to chance. 2 of 8 groups had an incorrect confidence interval, which is not statistically unlikely when working with 90% confidence intervals.
Unit Review
Before taking the unit test, I returned all of the quizzes back to the students. Students had a chance to make sure they still understood old topics and had a second chance to ask questions about topics they struggled with the first time through. For students who needed extra practice, I put problems online on a new class wiki. Students found this practice to be a very helpful check on their understanding.