End of Year Project Topics

This is an incredibly interesting list of project ideas that my students have generated.  I thought I'd share what they're capable of when they have zero restrictions...

10-11 End of Year Project Ideas 

Some highlights if you don't feel like reading through the whole thing...

1. Mythbusters: Mac vs PC
2. Value of a Power Hitter or Contact Hitter in Fantasy Baseball
3. Developing software that is usable exclusively in an AP Stat class
4. Using a test vs. a qualitative measure to assess learning
5. Profiles of countries in a state of unrest to predict revolt (the wiki for this project)
6. Price differences between Ebay and craigslist
7. The chance of finding a 15 to 64 year old male in Luxembourg that is a noble compared to a similar-aged male in Denmark being a noble.  (easily the most unique project ever done)
8. Using Wikipedia, how long it takes for random words using the formula of always clicking the first blue, unbolded and unitalicized word, lead to the end word of Philosophy. On the theory that every word in Wikipedia will eventually lead to Philosphy. Words will be chosen via randomized dictionary words. (based on the fact that clicking the first blue, unbolded, unitalicized word of every page eventually leads to Philosophy)

With a little bit of freedom (okay, a lot of freedom) they come up with some awesome topics and they really enjoy their work in class during these couple of weeks.

How will I grade them?  By writing a list of what the student did well and what the student did not do well. 

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.

Who decides what kids should learn?

Stop me if you've heard these before...
"Kids these days can't do simple math without a calculator!"
"Kids these days can't write well at all!"
"Kids these days are lazy!"

So what?  What gives you the right to tell students today what they should learn?  You've never met my students, so in my view you have zero authority to tell them what they should be learning.  On the flip side, if you've never met my students, why is it okay to tell them what they don't need? (as you begin making those budget cuts to eliminate foreign languages and the arts)

Ultimately our kids should have the freedom to learn whatever they want.  There should be no reason that every student should have to take Algebra II before they graduate high school.  If they're interested in it or would like to try mathematics, then go for it.  If they have an interest in art, why would we then tell them that they can only take one art course this semester since they have to take 6 other subjects they don't care about?

Imagine if we required every student to take a painting and drawing class every year from 7th grade to 12th grade.  Why does that sound so blasphemous, yet we can easily require them to take a math class (or two) every year from 7th grade to 12th grade?

Don't get me wrong, I see the great benefits in students taking any mathematics courses.  I'm a math teacher.  I want kids to discover their own interest in learning mathematics on their own schedule, not on some mandated timetable. 

AP Stat Lesson: Type I and Type II Errors

THE EXCEL SPREADSHEET
Type I and Type II Errors (housed on Box.net...is there a better way to do this?)
Directions for the activity contained in the spreadsheet.

SKILLS ADDRESSED
Statistical significance, Confidence Intervals related to hypothesis tests, Type I Error, Type II Error, Power, Alpha, Beta

THE CONTEXT
A factory is producing pharmaceutical grade glass vials.  Quality control engineers are employed to see if the factory is producing items at or below the industry standard of 5% defects.  They conduct a sample of size 100 (Mistake #1: I know this violates np>10, but it turns out that you wind up failing to reject a lot and it leads to a good understanding of Type II error) and determine the proportion of their sample that is defective. (Mistake #2: sampling 100 items and getting a proportion defective of 0.063 defective is impossible.  They will need to round to a whole number of successes when using the graphing calculator.)  Based on the results of this hypothesis test, they will decide if the factory must undergo a quality control review or continue with business as usual.


THE ACTIVITY
1ST PART - CONDUCT THE TEST, DECIDE WHETHER TO REJECT OR FAIL TO REJECT
It's dynamic.  Each kid will receive a randomly generated proportion.  They may do this up to 50 times.
First part of the activity: conduct the one proportion z-test using your graphing calculator.

2ND PART - DECIDE IF YOU MADE THE RIGHT DECISION
Unlock the spreadsheet (password: apstat5).  Have them change the fill color of the "True Proportion" column to reveal the true proportion of items that are actually defective.  They then evaluate their decision as to whether it was correct or incorrect.  Cue a whole class discussion on the 4 different scenarios of errors, then slap the AP Stat vocabulary on.

REFLECTION
Two huge mistakes that led to an amazing understanding of errors.  An overly planned lesson would have avoided these mistakes.  It also would not have generated a discussion on appropriate assumptions and conditions for inference.  An overly planned lesson would have also not brought up the question of "How come I'm failing to reject so much when it's false?"  An awesome comment: "I would not have learned this that well if I didn't have to think about those things."

This was 3 days worth of 46-minute classes.  Let's see how they do on the assessment of these skills.

THE SLIDESHOW

10-11 Errors

View more presentations from christjt.

Coming soon to this post...
The Google Form Assessment
A better version of the spreadsheet that allows any null hypothesis and any sample size.
100 comments on how to make this even better (hopefully)

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.

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

"AP Kids" are not the only ones who need quality learning experiences

After hitting the Google Reader hard while at Subway the other day (probably should've been eating lunch and relaxing instead of working) I was inspired by @InnovativeEdu's post on asking kids to design their own learning. Having just switched to the project-based learning format for my classroom, I love to share what my students are doing at these professional development sessions we conduct.

The question that always comes up is "Yeah, but what level do you teach?"  When my response is AP Statistics, it's immediately dismissed since "they are AP kids."  So what if I said, "I had this great lesson where I stood and lectured for 46 minutes with zero audience participation!"  I can almost guarantee to have the same response, "Well, your lecture worked so well because they're AP kids, no way that would work with my 4.0's!"

As an educator in a professional development workshop, why not spend that time to think of ways to reach the kids that are not "AP kids".  It seems like that would be a better use of time than to confirm your suspicions that there just isn't anything that works to educate those that are not taking Advanced Placement courses.

The students in 4.0 classes are the ones that have been most vocal about being not interested in what you have to say.  They are students that are completely unwilling/unmotivated to work unless it interests them.  Know what is especially uninteresting...x's, y's, and slopes of lines.  But these lower level courses cover basic equation solving and "find the slope" the exact same way, over and over again from the time the student is in 9th grade until 12th grade.

Great lessons, quality education, and interesting projects shouldn't be reserved for the best and brightest students.  The 4.0 students don't need more lectures and more basic junk that they don't care about.  They need to be interested, first and foremost.  They don't need more discipline or a rigid classroom structure.  They've told you 100 times that they hate that environment, so stop imposing it on them.

When I see my "AP kids" work on a project that they're excited about (sampling teachers in the school to see if they have tattoos, experiment on whether or not people can walk and text, see how often radio stations repeat certain songs) they aren't excited about it because they're "AP kids".  They aren't excited about it because I threatened them with detention if they showed a lack of enthusiasm.  They're excited about it because they had the choice in what they wanted to do.  They're students, there is no way they are so drastically different than their peers that just so happened to not do well in one math class so they were forced to slide down the ladder and be stuck in "4.0 world".

Oh, and they learn way more from me doing their own projects than they ever could from answering some multiple choice and some free response questions for me.

Reading through state standards(PA), going "mental"

I am spending my snow day planning for a presentation on technology integrated K-6 math instruction.  The focus of my session is on generating student inquiry and keeping technology completely transparent.  I am keeping this presentation aligned to state standards and vision, and doing so makes me very uncomfortable.  These standards/essential questions/competencies seem to all center around being able to generate a correct answer for a test.  This elementary math curriculum framework can be found at the Pennsylvania Standards Aligned System Website.  Here's a few phrases I don't particularly care for:

Taken from the 2nd Grade Mathematics list of Big Ideas and Essential Questions:
1.  "How do we know when it is appropriate to estimate or when it is appropriate to use mental math for an exact answer?"
The person that included the phrase "mental math" is clueless in the area of mathematics.  Estimation and approximation are (in my view) much more "mental" than development of an "exact" answer, yet they are projected here as completely non-cerebral tasks.

2.  "Develop extended understanding of multiple models, and properties of addition and subtraction, leading to fluency with efficient, accurate and generalizable methods to add and subtract multi-digit whole numbers and develop quick recall of addition and related subtraction facts. Select and apply appropriate methods to estimate sums and differences or to calculate them mentally."
Again, here they reference calculation as being done "mentally" and estimation as something that's done as an alternative to thinking.  I also don't like the use of the words "efficient" and "quick recall" as they imply that the student that adds two numbers in 10 seconds is somehow better than a student that adds two numbers in 10 minutes.

As we move towards common core standards and the like, is this the language that is to be used?  If so, mathematics instruction will never be more than an instruction of process.  A focus on efficiency over a focus on an understanding of mathematics keeps us at this procedural level.  Maintaining that estimation is done non-mentally, now we're completely missing the boat.  Right answers are not the most important part of learning mathematics.  Isn't it time we start asking our students to experiment and create in their math classes, instead of simply generate the same right answer that 25 other classmates generated?