Inferential Statistics: A Different Approach

For the longest time I've given thought to providing instruction on inferential statistics in a unique fashion.  If you're an AP Stat teacher, it means a departure from the One-Proportion Z-test, Two-Proportion Z-test, One Sample t-test, Two-Sample t-test, Matched Pair t-test, Chi-Squared Test(s), t-tests for slopes of regression lines.

So here's how I'd start...all data collection.  Spend a couple of days collecting data for each situation.  One of the essential questions I'd like my students to explore throughout the year is "Which model is the most appropriate for data you have collected?"  Here's where we go into depth about why certain models are more appropriate than others...

Data to Collect

1. Number of victories in 100(or so) games of Rock-Paper-Scissors
2. Toss a thumbtack and record proportion of "up" 
3. Drop a piece of buttered toast 50(or so) times and measure how many times it lands "buttered-side down"
4. Give a dummy homework assignment and measure the proportion in each class that complete the assignment
5. Compare batting averages of two baseball players
6. Time how long it will take kids to walk to the pool and back
7. Prices of items at clothing stores (found through browsing catalogs online)
8. Number of each type of animal cracker per box
9. How long it will take you to sort beans on to bulls-eyes with a dominant/non-dominant hand
10. Give the ol' Memory Experiment(groups rate sentence on how hard they are to pronounce/how easily they can form a vivid mental image) and compare number correct for each group
11. Count the number of each color of M&M you receive in a sample of M&M's
12. Change drop-height/rotor length of paper helicopters and record the time it takes to fall

After you spend about a week or so doing data collection, ask students to reflect on how data was collected. Notice also that some activities are done the same way (measuring proportions/means).  I'm fairly certain this has to be done to guide reflections, make kids confused, and ultimately learn something about making generalizations (mathematical modeling at its finest).

Ideas for Reflection:

1. What was measured in each data collection?  How does it compare with other types of data?
2. Which activities were useful for making comparisons?
3. If we're not making a comparison, what can we do with the data we collect?
4. Does it matter than some samples are smaller than others?  
5. Create a display for each activity with the raw data.  Which models tend to be the most appropriate?

I can see this being two weeks of AP Stat where kids think about collecting data and fitting similar models to similar methods of data collection.  Once they start fitting models to each situation for comparison, then you bring about some hypothesis testing procedures.

If you're a Stat teacher or not, provide your suggestions and ideas for data to collect.  It'd be great to get a new type of data to collect from somebody outside of the Stat realm.

AP Stat Lesson: Confidence Intervals (Graduation Party)

The Excel file: Graduation Party simulation.

What this Excel file does is simulates a student sending out 1500 invitations to a graduation party.  There is a true proportion of people that will attend, but it is unknown (see the "Population" tab of the Excel spreadsheet is completely blacked and password protected).  If you'd like the unlocked version, feel free to get in touch with me and I can send it along.  Students will conduct samples of 20, 50, and 100 to estimate the true proportion, and once they've generated a sufficient number of each sample size, they'll take a guess as to what the true proportion is.  Discussion follows as to which sample was most helpful to make the guess from.  Most guesses are that the true proportion is between 0.2 and 0.3.

They choose one of their sample proportions for sample size 100 and create a confidence interval for 4 different confidence levels: 68%, 90%, 95%, 99.7% (not randomly thought up by any means).  I chose these confidence levels because in the past I've seen students not associate confidence intervals with a middle percentage.

Collect students intervals using this form: One-Proportion Z-intervals Data Collection Form 
Display their responses here: One-Proportion Z-Intervals Raw Data (pay attention to both tabs, one has intervals and one has whether or not the interval captures the true proportion)

Once they've done some thinking, they will open this Excel file(One-Proportion Z-intervals Displays of Each Interval), providing a visual of each confidence interval.

We follow with having students lead their own discussion.  They'll begin by posting comments to a specific page of the class wiki, in order to get them to jot down an initial reaction to see if what they thought still holds up, or if their thinking needs revision.  A whole class discussion follows, and I challenge them to not allow me to speak for 10 minutes.  This can be difficult for me, but it is extremely difficult for them.

AP Stat Lesson: Unstructured Investigation

Give kids this: Hank Aaron - Home Runs by Pitcher

Let all hell break loose.

I've become quite a fan of "unstructured time" as of late, and I think this is perfect for an AP Statistics class.  I can see my baseball fans in class leaping at this opportunity to explore some baseball stats.  Whatever they wish to investigate, they are free to do so.  For the non-baseball fans, it may be just an opportunity to learn something about baseball.  I owe them some data on what they're interested in.  I look forward to hearing what a student that knows very little about baseball has to say.

Possible investigations:
1.  Comparison of Barry Bonds(or any other great home run hitter) to Hank Aaron. 
2.  Are pitchers today better (as a whole) than the ones Hank Aaron faced? (this comes from listening to Colin Cowherd say that Babe Ruth hit his home runs against guys who drove a milk truck in the off-season)
3.  Just how biased is this website?

Let them come up with how they're going to do any of the above. 

You Say It Best When You Say Nothing At All (about Hypothesis Testing)

I really like sharing things that work really well.  I don't know if it's a good idea on it's own, but this worked great considering we've done a lot of remediation (for students that needed it) and a lot of projects that delve deeper into statistics.  They are working with basics of statistics all the time, so tying it together into making statistical inferences is fairly easy when they have a good foundation.  I'd also like to think it has something to do with the way we've interacted with hypothesis tests in class.

Thumbtacks - Introduction
It begins with this form (Inference for Proportions) and handing out one thumbtack.  In their own brain, students decide what they think it is and what it would take to convince them it was wrong.  Then they toss the tack to be able to compare their observations to a model they've developed.  Sounds a lot like your entire hypothesis testing/inferential statistics unit.

The NCAA Basketball Tournament - A Basic Example
Kids then made predictions for the NCAA tournament and we tested just how good they were at doing so by comparing their proportion of correct first round picks to randomly guessing (p = 0.5).  A big question that came up was "Are we just doing this for the first round?" and in true teacher fashion I said, "Yes.  Now how come we're only doing it for the first round?"  Cue a killer discussion about large enough sample sizes and the Success/Failure condition.

Back to Thumbtacks - Put it into practice
Fire up the laptops and open up what the rest of your classmates thought (What They Thought).  Immediately they began to think "Why did this kid think they were incorrect when they got a lower proportion that what they thought they needed to be incorrect?"...a not so formalized way to think about a standard of proof and a low enough p-value to reject the null hypothesis.  This was one of those points in class where I said nothing and let their brains piece together what they were looking at.  I clarified what we were looking at, asked them to pick case that they thought was theirs and test the original hypothesis.

Conclusions
On the board, write your p-value and whether or not you rejected your original hypothesis.  As a class we'll have a look at everyone's p-values and decisions, then decide who has correctly rejected/not rejected.  They all argue about what p-value is considered "low enough" that you have to reject.  One of those moments where again, I say nothing and they develop an understanding of alpha-levels.  Not so formal...yet.

Projects/Practice

Pick another one of those contexts from the Inference for Proportions form and investigate it.  I think I'm going to add some more situations/contexts.  I'm also not sure that they ever need to fill out that form more than once...

What's Left to Do?
Sit back, relax, and let the 5's on the AP exam roll in.  Dress it up.  Put all the formal AP Exam terms/vocab/stuff to what they've already understood.  Then....
1.  So what really is the true proportion (Confidence Intervals)
2.  Is what we got really that different? (two proportions)
3.  Repeat procedure for sample means instead of proportions

Feel free to go to our class wiki for any supplemental exercises/materials.

Challenges of Project Based Learning (PBL) in AP Stat

Doing project based learning (PBL) in AP Stat this year has been a challenge. Ultimately, the best part about it is the learning experiences and opportunities it provides students. Every time a project is completed I think of 300 things that could be changed to make it, excuse me for using the phrase "100% better" (kind of an inside joke).  Below I've listed my major concerns about these projects, and the solutions I'm considering.  Your input on any of these is greatly appreciated.

Concerns and Solutions

1. The project was not rigorous enough. It covered too many skills too broadly, or too few skills in unnecessary depth.

     I want to make sure I create a sample project for each project to see just how in depth the projects go. I'm guilty of doing a bare-bones project example (okay, sometimes not even one at all).  There, I said it, I don't always do the project I assign.  The reason for this is to learn alongside of my students.

2. Some kids put a lot of work into a project that just doesn't really address much content, does so incorrectly, or doesn't really get into depth.

     Do it over. It's worth the learning experience of starting from scratch and completing the project again. 

3. What are they actually learning and can they replicate it

     Most of the time a project will involve them learning a new piece of technology as well as learning an AP Stat concept in greater detail.  I'm not sure these projects translate very well to getting an answer correct on the AP exam. Honestly, I want my projects to be far removed from getting right answers on an AP exam.

The Stat Project Process

1. Skills Organization - lay out the content related skills you will be addressing in your project

     Example: conditions for using the binomial/geometric probability distributions, calculating probability for each distribution, determining expected value and standard deviation for a probability distribution

2. Place context on each skill - group brainstorming to see what context fits each skill the best

     Example: highlight the difference between the two probability distributions by filming students walking downthehallway until we observe one of them wearing earbuds (geometric). Compare with a binomial distribution, showing 10 kids walking down the hall, 5 of which are wearing earbuds. 

 

3. Storyboard/Product: what multimedia can we put together, how does it flow, how does everything fit together?  

     Here's where students choose a tool that meets their project's needs.  

4. Edits - does anything need to be rethought or redone as something better?  

     High school students seem to miss this step in almost everything they do, once the "be done" mentality takes over. Sometimes the "do it over" option is the best learning experience.  I've found I've spent more time suggesting they edit and critique their own work and each others' work, and it's made a world of difference in overall quality of product and understanding of statistics.  

Since I don't believe in giving deadlines for learning, when a student asks when their projects are due, I tell them that they may turn them in whenever this process is completed.  With most projects I honestly don't think this process is ever completed.

Today's AP Stat Lesson: EXCEL HEAVY - VLOOKUP(RANDBETWEEN(NERD, GEEK), STATGEEKS,2)

Today's(tomorrow's) plan for AP Statistics is a little Excel heavy,  something that I hope carries over for my students into college and beyond. Is there a standardized test that measures a student's increased proficiency at Microsoft Excel, or other computer apps for that matter?  Most commands involve looking up a value at random between zero and one hundred.

Each student has been keeping track of the number of sheets of paper received in each class, each day, over the span of about 2 months. The focus is on teacher created paper, so if a student uses a piece of their own notebook paper it doesn't count.  Incidentally, om pretty sure that AP Stat is dead last for every one of my students major subjects (AP Stat instructor pats himself on back).

Using this info, they are going to create a probability model for the number of sheets of paper received for a single class.

For AP Stat...

SHEETS OF PAPER    0     1      2 

PROBABILITY         0.90 0.05 0.05  

Now that the probability model is created the Excel fun can begin.  Number all cells in one column from 1-100 to represent a possible outcome. Place each possible outcome in the second column according to it's the probability you observed. Example: in the chart above, the probability of receiving 0 sheets of paper was 0.90, so spaces 1-90 would be 0. The probability of receiving 1 was 0.05, so 91-95 would be 1, and 96-100 would be 2.  Repeat for all other possible outcomes. Suggest to students that they choose a class that is at least manageable as far as different numbers of sheets of paper.  Yes, there is a way to make Excel do this, but that is a little too awesome for an AP Stat class.  

Create a new sheet for simulating new days of each class. For the first day, we are going to "lookup" random values from the previous sheet to see how many sheets of paper we will receive. The command for doing this... VLOOKUP(RANDINT(1,100),in the previous sheet, give the value in the second column in the row that the random number is in). 

Example:  =VLOOKUP(RANDBETWEEN(1,100),Sheet2!A$1:B$100,2)

Keep the dollar signs so that when you drag to autofill the formula they continue to look within the same array of cells. Autofill about 200 or 300 of these cells...or 500 :)

Next we're going to calculate the average number of sheets of paper we've received each day. AVERAGE(cell to the left and all cells above). 

Example: =AVERAGE(B2:B6)

Now we can look at the average over the long run (1000 days, or even more) and begin to build our definition of Expected Value.  Then, later, we'll do expected value the easy way in Microsoft Excel using the formula.  I wanted them to wrap their brains around the definition of Expected Value before they began using the formula: E(X) = SUM[X*p(X)].  It's pretty cool to see just how long of a run you need to make the simulation average approach the expected value.

The lesson is a bit like a cooking show, but it's the first time we are in Excel. The kids use their own data, so that's enjoyable/unique for most of them. The bigger idea of simulating outcomes is very powerful in learning statistics. I love nothing more than a student centered approach, but I would hate to say "discover how to use vlookup and randint functions in Excel. If anyone has a way to take a constructivist approach to learning Microsoft Excel, I'd love to hear it.

The Excel spreadsheet that students create

My Test with No Right Answers

At the slightly more than halfway point in the year, I decided to give my students an assessment on skills we would have learned previously this year.  In an AP Stat class, this type of exercise is meant to take the place of the 2 weeks of "review" that most educators wind up with at the end of the year before the AP Exam.

Here's the test: AP Statistics Mid-Year Assessment .  Using this link (and excusing the poor formatting that comes with inserting images into Google docs) they are to create a Google Doc and share it with me that is essentially their stream of consciousness  statistical reasoning.  They were permitted to use the class wiki, their Stat textbook (BVD - Stats: Modeling the World), and any other online resources available to them.  Big thanks to David Wees (@davidwees) for the data, graph, and the article!

They were not permitted to use each other as a resource, as this was an assessment to see what they know as individuals.  This makes me uncomfortable, since it doesn't necessary follow the cooperative model I've we've built the classroom on.  Next time I give an assessment, I'll allow them to use each other as a resource as well, but the task needs to be uber-authentic.  Suggestions?
I'll share a conversation I had with a student about this, since it made my day.

"Mr. C, can we use each other as a resource?" - Student
"I'd prefer not, since I want to see what you know." - Me
"But in 'real life' we'd be allowed to use each other." - Student
"Arrggghhh, why must you make that argument?" - Me
"Because you taught me to." - Student

First thing a Stat teacher, or anyone familiar with Statistics will notice is that for most of the questions there's not really one "right" answer.  The purpose behind this is to allow kids to think and write what they know, in regard to the skills being assessed.  I'm loving the thought that has gone into each response.  It feels like I've really touched on something here, something that does a little bit better than the strategy of "write everything you know about the topic" for the poor test taker/assessment do-er.

So what resource did kids use the most?  At quick glance, my class wiki comes in first, with the Stat textbook coming in second.  I'm hoping this spurs more content creation on the class wiki, so that it can develop into the ultimate online stat textbook.  Yes, I am still living that dream.

Reassessment Fridays (SBG): A Love Story

“Every student did their own thing in class today.”
How often is that the summary of a mathematics class?  How about an art class? 


Today was the standard reassessment day for my AP Stat classes.  I wasn’t sure how I was going to let students reassess when I started Standards Based Grading.  I’ve come up with using every Friday as a reassessment day.



I let students pick 2 skills to reassess on.  They pick them Wednesday, I create reassessments Thursday, they take them on Friday.  The reassessors are given a mail merged document that is their assignment for that day’s reassessment.  Should they choose not to formally reassess in class, they always have the option to do a project on their own to show their mastery of any of the skills they have learned.  This has been a system that works for the students that want to reassess, but it works even better for those not reassessing.


Today about a third of each class was reassessing.  So what do the other students in class do?  They play.  They learn.  They collaborate:

1. One student began conducting simulations to simulate random clicks in Minesweeper.  Two others quickly became interested and a Minesweeper collaborative was born.
2. About 10 different students were editing 10 different pages of the class wiki to make it an online textbook.  
3. They were gathering data for different projects they hope to complete later in the year
1. a project where they wish to determine the proportion of teachers at school that have tattoos
2. simulating hands for various different card games
3. the true proportion of dives by a diver that are “ripped”

I look forward to my “Reassessment Fridays” because of the classroom vibe.  I’ll provide a few ideas for what can be worked on, then they get started doing things that they want to do.  Today’s ideas: Conduct a simulation, finish an old project, start a new project, edit the class wiki, create models of data collected this week.  I’m immediately working on making this the classroom activity for every day, not just Friday.  


My favorite part about it is that I am providing them time to be as smart as they want to be in class.  I’m not tricking them into learning with some gimmick.  I’m not having them finish a worksheet for points.  I’m not asking 26 of them to answer the same question.