AP Statistics “Best Practices” 2014

Last week, I arrived home after 8 days in Kansas City, where I participated in the AP Statstics Exam reading. It’s hard work, filled with long days of grading papers. But all the readers seem to take some sadistic delight in this work, and the professional connections made through the week are outstanding.

One of the highlights of the week is Best Practices Night, organized by my friend Adam Shrager. This year, 20 or so different folks presented 5-minute looks into their classrooms.  Below are summaries of some of my personal favorites. You can check out all of the presentations on Jason Molesky’s StatsMonkey site

GUMMI BEARS – KEVIN DiVIZIA

You’ll find that AP Stats teachers enjoy candy….too much so at times my doctor tells me. Last year, Kevin shared his data collection activity with stomp rockets.  This year, Kevin upped the ante, with an activity where students launch Gummy Bears, Gummy Worms and other candies using catapults.  Which type of candy flies farthest? What can we say about the consistancy of the launches? I’m looking to incorporate this into my 9th grade class as an introduction to variability and estimation.

Gummis

Kevin’s presentation on the StatsMonkey site is Keynote. I have converted it here to Powerpoint for us non-Keynote users.

MORBID MATH – BRIANNA KURTZ

Stats teachers have many data collection activities in their arsenal, but this idea from Brianna wins the prize for most off-beat concept. In this activity, students are asked to estimate life expectancy in a population. To collect data, the class uses something readily avilable every day: the obituaries. This presentation was one of the clear highlights of the evening, with many in attendance wondering what a class taught by the hysterically entertaining Brianna would be like!  Visit StatsMonkey for her activity worksheet, and use the dead as data!

zpuzzles Z-PUZZLES – CHRISTINE WOZNIAK

Jigsaw puzzles make for great reviews in just about any math class.  Here, Christine shares puzzles she uses to review the Normal Distribution. Cut out the pieces, find the probabilities and solve the puzzle!  Template included.

SAMPLING USING BEADS – PAUL RODRIGUEZ

Paul is part of the AP Stats Test Development Committee, and always has great ideas for the Stats Classroom. At the reading, Paul shared his sampling activity, using Air Gun ammo of different colors (and slightly different sizes) to draw small samples from a large population. Using a paddle made from pegboard, random samples can be drawn, leading to a first discussion on inference. Paul promises to share the plans for building your own sampling paddle, so check back on StatsMonkey often!

UPDATE: Paul’s presentation has been uploaded to the StatsMonkey Site, along with plans for making your own sampling paddles.

STARBUSTS AND R-SQUARED – DOUG TYSON

I appreciate presentations where speakers attempt to de-tangle a tricky concept in math class. Having students move beyond a “canned” understanding of the coefficient of determination and towards a real understanding of predictive improvement based on an explanatory variable is a worthwhile lesson. In his activity, Doug Tyson challenges students to grab as many Starburst candies (see…I told you Stats folks like cnady) as possible in their hand, then examines the predictive value of using hand size to estimate the number of grabbed candies.  How much are our predictions improved by thinking about hand size, as opposed to thinking about the mean?

There’s so much more sharing goodness on the StatsMonkey site, including:

  • A review of Geddit, for formative assessment
  • A QR code scavenger hunt
  • Hershey Kisses and Confident Intervals, which I used in my class this year

Soon, I will post more resources shared by Chris Franklin, who gave a brief history of stats education during her Professional Night presentation.

Estimation and Anchoring

A recent post by my Stats-teacher friend Anthony, “Wisdom of the Crowd“, reminded me of an estimation activity I have used many times in my 9th grade Stats class.  The activity is based on a chapter from John Allen Paulos’ book A Mathematician Reads the Newspaper.

You’ll need two groups of students; 2 different classes will do.  Each student uses an index card or a scrap of paper to write responses to 2 survey questions. I warn the students beforehand that the questions may seem strange: just do your best to answer as best you can.

  • Question 1: Do you believe the population of Argentina is MORE or LESS than 10 million people?
  • Question 2: Estimate the population of Argentina.

Allow a few moments between the questions for the inevitable blank stares and mumbling.  Then collect the responses.

For the second group, you will ask the same two questions, except that the first question will replace 10 million with 50 million.  After you have data from both groups, write it on the board or print it and hand it out. It’s time to analyze and compare. Challenge students to communicate thoughts about center and spread. Also, which group’s data do they feel does a better job of estimating question 2?  It’s a neat activity, and while you will receive some strange responses as estimates, and students will generally guess higher on question 2 if they have been anchored to the 50 million number.  Some guidelines for this activity are avilable.  Have fun!

According to Google, the actual population of Argentina is around 41 million.

 

 

When Student Choice is a Struggle

Like most of the East Coast, schools here still have quite a ways to go before enjoying summer. I see my students for one more full week before final exam review begins and finals are given; a time which becomes more crazy as I travel to Kansas City for the AP Stats reading (or…Stats Christmas in June!)

It’s a starange time of year for AP Stats.  The College Board exam was given on May 9, and students took a final exam in my class before then, so we have been done with new material for some time now.  With a full 3 weeks (or more) between the exam and the end of the school year, it’s a time to take my foot off the gas from day-to-day material, but I still need to see my kids engaged in statistics.  Our culminating event, Stats Fair, provides a chance to highlight our program and keep the statistical ball rolling.  There’s really only one requirement for Stat Fair: design a project of your choosing which serves as evidence of your statistical learning. At the Fair, students show off their work to invited guests and fellow students (you can see pictures from previous fairs on my school website).  Teams must also provide printed documentation of their project to me.  It’s a great opportunity to be creative, study something you are passionate about, and explore something new.  There’s just one little problem…

Most student project ideas suck

Yep.  After a year of learning about experimental design, the role of randomness, and all sorts of nifty confidence intervals, many of my 17 year-old students will revert back to their 6th grade dopplegangers; proposing scientific studies of their peers’ favorite colors or chocolate chip cookie preference or how much honors’ kids backpacks weigh. Sigh….

Maybe I’m just jaded.  I warn the students early-on that it is likely I will reject their first 5 stats fair ideas.  It’s not that I am intentionally trying to be mean, rather I want my students to pick something memorable, something they could speak passionately about in front of others.  Working with students to develop their concepts could be the most frustrating part of my academic year.  Why is it so difficult for students to develop a “good” concept?

  • Despite a year full of examples and articles, it’s still a tough leap to the “real world” of teenagers.
  • Developing a good concept takes deep thought, revision, patience and reflection; not always teenage qualities.
  • The best concepts often contain a high dose of creativity – not something we are always accustomed to in math class.
  • It’s the end of the year, and the beach awaits

But all is not lost!  Today’s class started with a rousing success: a student, who had earlier proposed a study of NBA player ages (which was going nowhere), finally moved towards one of his passions – music. Using an app on his iphone, he tested the ability of peers to detect high and low pitches in mHz.  This led him today to some independent study online of the human ear, and reflection on the data he had gathered.

Another group is using their passion for fashion to see just how “skinny” jeans are these days, comparing waist sizes from different stores.  Some interesting data coming from this.  Another group is testing the “locally grown produce” claim of supermarkets…neat stuff!  And I’m looking forward to the random study of our school’s wireless device access – just how slow is it?  It’s the interesting projects which keep me coming back, and make this class memorable – like the team a few years back who entered and won the American Stats Association poster competition with their Bacterial Soap review.

Stats Fair is next Friday.  Look forward to sharing pictures and reflections!

Your Official Guide to Math Classroom Decorations

Pi Digits

The most recent challenge by the MTBOS (Math-Twitter-Blog-oSphere) is to share what’s on your classroom walls.  (Follow the action on twitter, #MTBoS30)

This post will go beyond my own classroom, and take you on a tour of many classrooms of my colleagues.  Here I present to you the Official Guide to Math Classroom Decoration.

To rank these items, I will be using the “Justin Scale”, an internationally-accepted scale of math beauty.  It is based on the works of Justin Aion, who is an expert on classroom decoration.  Seriously, you should be following Justin’s Blog for his daily classroom obsessions.

Here’s how the “Justin Scale” works

  • 1 Justin = an insult to scotch tape
  • 2 Justins = better than having a blank wall; marginally stimulating mathematically
  • 3 Justins = setting the tone for an engaging math experience
  • 4 Justins = cool beans!

You can see it’s pretty scientific.  Now, on to the decor!


PROCEDURE POSTERS

ImaginaryExponents

In the history of math posters, has any student ever looked at one of these and thought “hey, so THAT’S how you add fractions”…seriously?  Sure, these posters are well-intentioned, but they are boring as heck and suck any imagination out of math class.  Also, I have to cover them up anytime the SAT comes around.

VERDICT:

1.5 JUSTINS


MATH T-SHIRTS

Shirts

I like to have items around my room which tell a story. Maybe they are stories of past students or experiences; other times they remind me of math nuggets I pull out once a semester. These shirts are from a number of Muhlenberg College Math Contests from the past few years, each with a neat math concept from the year of the contest.  On the left, the 28th year celebrated 28, a perfect number. 27 is a cubic formula, and the 31st features the Towers of Hanoi.   Full disclosure, I designed the 16th shirt as an undergrad.

VERDICT:

3 JUSTINS


TI CALCULATOR POSTER

TI Calc

Go to any math conference and you’ll find gaggles of math teachers walking around the vendor area with swag bags, free stuff the many companies have for you. TI posters are one of the most popular items, and you’ll find many math classrooms sporting these artifacts of math boredom.  “It was free, therefore I must place it on my wall”

These posters fill lots of space and give your room the right dose of geekiness.  And a reminder of the vast machine TI is.  Have any english teachers ever placed a large photo of a typewriter on their wall?  Nope.

VERDICT:

1 JUSTIN


INFOGRAPHICS

Stats Info

Data Scientists

So many cool infographics to choose from, so little toner. Love posting these guys all over my room; love it even more when I find kids checking them out just before the bell.  But they are a pain to print, and they age badly.

 

 

 

 

 

 

 

 

VERDICT:

2.5 JUSTINS


ASSORTED MATH HUMOR / INSPIRATIONAL POSTERS

Posters

Usually purchased by rookie teachers, you will find these posters at your local teacher supply store.  Hunting season for these posters is short, running from early August to mid-September, so get yours while they last.  “Is that a cat hanging from a tree”….why yes, yes it is….

VERDICT:

1 JUSTIN


PICTURES OF INTERESTING THINGS

Things

You don’t need to try hard to find neat stuff for your classroom.  A colleague of mine, who often teaches geometry, has pictures of neat things above his board.  Here’s your challenge: find your favorite items from 101qs.com, print them, and post them all over the place. The conversations start themselves.

VERDICT:

3 JUSTINS


 STUFF KIDS MADE AND DID

Kid-CreatedAnytime you can post, share and provide inspiration through student work, it’s bonus time.  Here, an oragami construction a student made for me a few years ago watches over class, and posters sharing pictures from Stats Fair in years past take over my bulletin board.  Also, I have a John McClane action figure on this board….and you can’t blaspeheme Nakatomi Plaza….never forget!

 

VERDICT:

4 justins

 

What Betty Crocker Can Teach Us About the Common Core

Despite my attempts to maintain a somewhat healthy diet, I still succomb often to sweets. If there are cookies or cupcakes in the teacher planning room, I’m there…and often regretting the indulgence later.

I especially enjoy chocolate cake.  My fingers tremble in anticipation just as I type those wonderful words – chocolate cake.

It’s a great day for baking, so I did an online search for a real kick-butt chocoloate cake recipe.  There were many, many great candidates, but I stumbled upon a recipe touting itself as Heavenly Chocolate Cake

Heavenly, you say?…tell me more…

Need to make sure I have all of the ingredients around, or else it is off to the grocery store:

  • Eggs
  • Milk
  • Sugar
  • Flour

Check, check, check, check….we are good so far.

  • Bicarbonate of Soda – uhhhh…what?
  • Instant espresso – is this really necessary?
  • Powdered gelatine

{{Sigh}}….let’s look ahead. Maybe I can skip some of this stuff?  Perhaps the instructions will give me an out here:

Let chocolate mixture cool to room temperature. Whip the double cream to soft peaks and fold into the chocolate mixture.

What are soft peaks? And is folding just a fancy way of stirring, or is that whisking?

Be careful not to over cook and curdle the mixture. Pour egg-milk mixture through a strainer into the melted chocolate. Melt the gelatine and water

Do I own a strainer? And how do I know when I have reached the event horizon for curdling?

OK, I surrender. I’m probably a little over my head here.  Fortunately, there is an option for the cooking-challenged like me:

Box

StepsThanks to Mrs. Crocker and her boxed wonderfulness, I can make a tasty cake in just 30 minutes!  Eggs, oil and water.   And just 3 simple steps: heat – stir – bake. These are some steps I can get behind!

And just that quickly, I am enjoying cake!  It’s the way I have always made cake, and the cake has always been quite tasty. My mom made cakes using this recipe; don’t go telling me that your cake is any better!

But in my heart, I know it’s no match for the Heavenly Chocolate Cake, which I salivate for.  I once had a cake like that which a neighbor made: such a memorable cake – I want more of that cake! So many sophisitcated flavors.  I can admire its beuaty, subtlety, its intricacies and I am aspire to be just half of the kitchen pro my neighbor is.


It occurs to me that Betty Crocker’s cake products share a lot with the ongoing debate over the necessity of Common Core math methods: the cakes you bake are simple and satisying, but in no way are they a suitable replacement for the genuine cooking experience in both the path taken, and the finished product.

NOTE: I understand that the Common Core does not suggest a method for mathematical operations.  Many of the methods confused as Common Core methods have been around for quite a long time, and are commonplace in math programs. Ideally, it would be wonderful to discuss these methods separate from the Common Core debate.  My intent here is provide a justification for seemingly more convoluted methods, through the lens of the Standards for Mathematical Practice.

A worksheet full of correct answers doesn’t mean you are good at math, in the same way that successfully baking a Betty Crocker cake isn’t cooking.  There’s a real disconnect over what it means to do math.  And the disconnect is not just between what educators expect from students and what parents hope to see from schools.  There are also wide differences from teacher to teacher, and school to school.  Yesterday, I ran across a post concerning an “insane how-to-add guide” which represents the worst of both worlds: a frustrated parent wondering why so many steps are needed to add, and a weak addition “guide” which is overly helpful.  Math, like cooking, cannot be diluted down to simple steps without a loss in complexity and reflection. In my recent post on Common Core subtraction, I suggested that reflection and adaptation are far more important to me as a math teacher than filled worksheets.

Betty Crocker = Core Standards. Gourmet cooking = Standards for Mathematical Practice.  I’m still here enjoying my cake, and I’ll likely make Betty Crocker cakes again.  Maybe next time I’ll toss in some extra chocolate chips, but the cake won’t be much different from the mandated recipe.  A true chef can experiment with flavor profiles, adjust and develop new ideas for cakes.  They can “Make Sense of Cakes and Persevere is Baking Them” (even if a few attempts don’t taste so heavenly) and “Construct their own Recipes and Critique the Recipes of Others”.  The end result – the cake – is still the star of the show.

Basic skills matter.  Being “right” matters.  But true chefs, and mature math students, can demonstrate understanding through explanation, exploration, and tackling rich problems.

The recipe with the least number of steps ultimately leads to a less-satisfying product. I recognize that my cake is good, but not great.  I’d really like to experience the Heavenly cake, but understand that it will require time and effort.  Worksheets allow for lists of correct answers, but this is not the most-satisfying mathematics. Effective math teachers cause thinking. We can add fractions, but what next? How do we use this skill? How can we extend it to other ideas? Can we explain how to add fractions to other?  There may be some brain sweat, and many eggs to crack, before we reach our goal.  When we start building new flavor profiles from fractions, exponents, graphs and equations….that’s when we are doing math.

And now, off for some jogging to burn off the cake….

 

 

 

 

 

Don’t F*$& ing Curse in Math Class

For the first time in many years, I find myself teaching a unit on polynomials to 9th graders.  Time to back up one of my pet peeves, and put my money where my mouth is. Some of my recent tweets may provide some clues to one of my least-favorite math acronyms….

My students seem amused by my swear cup…

I shared my thoughts on binomial multiplication, and gave a little plug to Nix The Tricks in the recent ATMOPAV (Association of Math Teachers of Philadelphia and Vicinity) newsletter.  The article is reproduced below, and I hope you enjoy it.  I serve as second vice-president of this organization, and invite you to visit our website and enjoy our spring newsletter.


A CROWD-SOURCED MATH PUBLICATION:NIX THE TRICKS 

There are many words which have “curse” status in my classroom. Some of these words are universally agreed to be “bad” – words which will result in a fast trip out of my class, and probably a phone call home. But other words are on a second tier of curses – words which make me cringe, and which require a donation to the math swear jar.

Like Foil.

Yes, that FOIL.  Our old “First – Outside – Inside – Last” friend. It’s banned from my classroom.

It’s not that FOIL is bad…heck, it’s quite a universal term in the math world.  The problem is that FOIL, while well-intentioned, is a trick.  It’s a trick for a specific situation: multiplying two binomials.  What happens when we multiply a monomial by a binomial, or even a binomial and a trinomial? I suggest FOSSIL here, to account for the Stuff inSide.

The problem with FOIL is that it removes the most important math property involved in the multiplication from the conversation: the distributive property.  And we replace this key property with a cute acronym which is only useful to one specific scenario.

Last year on my blog (mathcoachblog.com) I proposed a list of terms often overheard in math class which require some re-evaluation.  Terms which confound the deeper mathematics happening, and which distract from genuine understanding.  Besides FOIL, I also proposed the “Same-Change-Change” method for subtracting integers, and “cancelling like terms”.  Many teachers I follow on Twitter shared similar thoughts about not only terms, but also short-cuts often presented in math class.  Tina Cardone, a teacher from Massachusetts, started a Google Doc where teachers could contribute not only tricks, but proposed replacements for classroom shortcuts.  The response from the Twitter-world was robust, with not only tricks and terms proposed, but also conversations regarding best practices for concept attainment.

The response was so overwhelming that Tina compiled the online discussions into a free, downloadable resource for teachers: Nix The Tricks.  The document can be found at www.nixthetricks.com, and a printed version is now available on Amazon.

Nix The Tricks currently contains over 25 “tricks” used in math classes, categorized by concept. Along with a description of the trick, suggested fixes to help students develop deeper understanding of the underlying mathematics are presented.

The “Butterfly Method” for adding fractions is an example of the math tricks found in the document.  Do a quick Google search for “butterfly method adding fractions” and you’ll find many well-intentioned teachers offering this method as a means to master fraction addition.  But is student understanding of fraction operations enhanced by this method?  What are the consequences later in algebra when the same student, who mastered butterflies, now must add rational, algebraic expressions?  How should this topic be approached in elementary school in order to develop ongoing understanding?  Download the document and find commentary on this, and many other math tricks.

I am proud to have been part of this project, and continue to seek out new “tricks” to add to the mix.  The document is a tribute to the power of Twitter, where many conversations developed while debating the validity and helpfulness of tricks.  The group continues to seek new ideas to make Nix The Tricks grow.  To participate, follow me (@bobloch) or Tina Cardone (@crstn85) on Twitter, or contribute your ideas on the website: www.nixthetricks.com

 

Hitting the Home Stretch: Exponents, GCF’s and LCM’s

This is a busy stretch in my school year.  My 2 Prob/Stat classes are nearing the end of new material with PA Keystone Exams in Algebra 1 looming. For my College Prep class, about half have not yet taken the Keystone while the rest took it last year as 8th graders. In Academic, all students will take the Keystone in May.  Combine this with my AP Stats class taking their final exam this week, with the AP Exam next week, and my track and field meet responsibilities building as the season reaches its peak; it’s a hectic time of year.

In both of my Prob/Stat classes, we are beginning unit on polynomials.  The Prob/Stat class is a course we offer between Algebra 1 and 2. While the course contains much Prob/Stat material, we also clean up some concepts from algebra.  Unlike other algebraic units like systems of equations where there are many rich examples and opportunities to differentiate, the start of a polynomials unit often feels static. Here are two activities I have used this week for Laws of Exponents and GCFs/LCM’s.

LAWS OF EXPONENTS – TRUE/FALSE GROUPS

This activity worked equally well in my college-prep group (for whom this was review material), and my academic group (where this was mostly new).  The file below contains 16 cards with numeric statements.  Break your class into teams of 2, 3 or 4.  The job of the group is to identify the true statements and the false statements.  For this activity I banned all calculators.

The letters on the cards are not anything the kids need to worry about, but allow me to easily check progress. The cards with vowels are all the false statements.

I was surprised at how much trouble my college-prep group had with building the correct false pile.  To reach some consensus on the false pile, I asked every group to hold up one card they knew to be absolutely, positively false.  Many groups mistakenly agreed that any number raised to the zero power was worth zero, which led to a class argument on who was right.

Tomorrow, we will look more closely at the falses.  In the file above, note that the cards are arranged in groups of 4. In the first group, we will review the addition rules for exponents; then the subtraction rule in the next group of 4; then the multiplication rule for the next group.  In the end, this felt much more satisfying, with increased engagement and peer discussion than simply listing rules on the board.

GCFs and LCMs SPEED DATING

The speed dating concept is one many math teachers have stolen from the great Kate Nowack, and it worked perfectly in my Academic class to work through greatest common factors and least common multiples.  After doing just one example on the board, desks were arranged  into a pairs facing each other, down one long row.

Speed Dating

All students were given a card with a monomial.  They then worked with their partner facing them, and found the GCF of the two monomials.  The first time around, my co-teacher and I provided help to just about all groups.  After teams found their GCF’s, all students on the right-hand side stood and moved down one seat and worked with their new partner.  There were so many plusses to this activity:

  • all students were repsonsible for their own monomial
  • all students were engaged: no hiding behind a worksheet
  • students worked together, and with different partners each time

Some of the cards I handed out are shown here.  I tried to have a variety of cards which clearly shared factors, with different powers of x and y.

cards

I was very impressed with how my class performed on this activity, and we moved onto a second round where LCM’s were found. This time I had students trade cards, and the left-hand side shifted down each time.

Let your kids work together, discuss and find patterns – the notes then write themselves.