Why Your Iowa Caucus Math is (Probably) Wrong

Why Your Iowa Caucus Math is (Probably) Wrong

by Devon Cantwell

February 2, 2016

 

The Iowa Caucuses are now the most recent example of why we need to revamp our emphasis on some statistics education in the US.

 

One of the most popular articles floating around today, in the aftermath of a brutal and tight Monday night at the Caucuses, is poor math analysis about how improbable it was for Clinton to win 6/6 coin tosses last night.

 

The most popular theory claims  that the probability for this event is the following:

 

P(HHHHHH)= (½*½*½*½*½*½)= 1/64 or 1.56% probability of this event occurring.

 

This is not accurate.

 

To get a true understanding of the math behind whether this event was probable or not, we need to understand two major statistical tools: the Law of Large Numbers (LLN) and simulation modeling.

Law of Large Numbers

This central theory, one that I taught my AP Statistics students many moons ago, asserts that as you do more trials of an independent event (like a coin toss), your results should begin to norm and approach a limit– that of the expected value of the event. In the case of a coin, we would expect numbers to even out, starting at 30 samples, at about .50 for the probability of flipping heads in 30 or more samples. 30 is the minimum value for LLN to apply and for a sample or study to be considered “normal.”

In a study from 2004 by Bruce E. Trumbo in the  Department of Statistics at California State University, Hayward, we are given a simulation of ten coin tosses. Here, i represents each coin toss occurrence. Toss represents the result of the toss, i. Ri represents the actual value of our desired event (in this case, flipping Heads on a coin)

SOURCE: http://www.sci.csueastbay.edu/~btrumbo/Stat3401/Hand3401/LLNCoinTossB.pdf

As you can see here, the first five flips in this simulation result in H (Heads), with the last flip landing on tails. Through the simulation of ten coin tosses, we begin to approach our expected value of .5, but still land at .70, or in other terms, the coin landing on Heads 7 out of 10 of the times.

Simulations and Sample Size

The next piece to consider is: how common is it that the event of P(HHHHHH) will occur?

A November 2005 publication from Cornell helps us understand this below:

SOURCE: http://www.cs.cornell.edu/~ginsparg/physics/info295/mh.pdf

 

In the case of the simulations and example above, if we were to have flipped coins across 50 caucus sites 100 times in a row (just to be fair, you know), we would actually expect at least 27 of those sites to report 6 consecutive Head flips for their sample of coin flips.

If we had kept flipping this coin, it would have been far less likely that it would have continued to be Heads– that probability substantially drops after 8 consecutive heads, going from 17% to 8%. The probability begins to decrease almost exponentially (becoming about half each time).

Conclusion

The clear conclusions we can build from these mathematical understandings is that in the case of the Iowa Caucuses last night, our sample size was too small to rely on the expected value of .5 for the six coin tosses, and that all in all, the probability of this event across a large number of samples in this event would happen about 55% of the time. Perhaps the Sanders camp should switch to calling Heads for the rest of the the campaign trail.

Algebra: Necessary. But, is it evil?

Opening up my computer this sunny, Sunday morning, I ran across the headline “Is Algebra Necessary?,” and nearly spit out my coffee. For those that don’t know, in a few short weeks, I will starting my first year of teaching 7th graders at Davis Hills Middle School in Huntsville, AL– in that dreaded, hated subject of math.

I’ve never been naturally gifted in math or science myself. Going through school, my natural talents were always in the social sciences and humanities. However, this preference for liberal arts education never stopped me from taking 5 math classes during high school. As much as math can be grueling, frustrating, or boring, the reality is that math isn’t optional. Math is everything. We need math every day of our lives- from things as simple as measuring ingredients for a cake, building a fence, balancing a check book or calculating a bill to things as complex as fixing an engine or developing new GPS software.

The concerns raised by Dr. Hacker, in my opinion, speak more to issues with our educational system as a whole, issues that I would like to quick address in this post.

1. “Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.” 

In short- we teach students facts and formulas to memorize, not HOW to consider, manipulate, and think about math.  This failure to teach critical thinking on a large scale in this country is exactly what has prompted a national movement toward the Common Core State Standards Initiative, an initiative that my state and district have mandated participation in, starting this year. The Common Core, as it is called for short, approaches necessary skills of math through a lens of problem solving and critical thinking- not just a mere memorization of formulas. I agree with Dr. Hacker in this respect. When we merely teach our students tricks and formulas, they have no reason to remember, care, or be invested in our subject, and the course becomes essentially useless to their education and development as individuals. However, when we let students discover, create, and ultimately own their math education, these processes of critical thinking can only enhance their ability to flourish in fields outside of STEM.

2. “A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.”

Yes, because students are consistently struggling in a subject, we should just stop teaching it. No. No, no, no. This logic is a mirror reflection of this generation’s ridiculous grade inflation and reluctance to do anything outside of our comfort zone in terms of success. I know that my personal struggle to succeed with math and science was lined with a series of failures initially- but my perseverance through the subject was vital to my character development.

If anything, Dr. Hacker’s rationale on this point only enforces why, more than ever, it is crucial for us to push higher math education in the United States. The high drop out rates, alarming percentages of students who need to take remedial math courses in college, or need to retake math courses in their entirety does send a loud and clear message to teachers like myself though- we need to dramatically change the way we teach and apply math principles. As a whole, the nation’s approach to teaching math seems to fullfil Einstein’s famous quote that the definition of insanity is “doing the same thing over and over again and expecting different results.” Dr. Hacker provides an alternative of teaching “quantitative reasoning” at ages as young as kindergarden- I completely agree. The more students understand solid connections to the real world, the more that they will internalize the knowledge we are teaching them.

3. “Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.”

Dr. Hacker, with all due respect to you as a scholar and writer, for someone who relies on statistical analysis as a crucial part of your field, you seem to have no idea what goes into statistics. Please tell me, how in the world can I expect my students to analyze their data regarding personal habits and academic performance, or to interpret a scatterplot with information, without having a strong foundation in Algebraic concepts such as expression, the number system, and proportions and probability? You can’t piecemeal mathematical concepts. They all play an important relational role to one another, and students must have the whole picture before we ask them to pick apart the pieces.

The Bottom Line: Mathematics is not the intrinsic problem. The way that we teach it is. We need to invest our students, our society, in the need and application of this vital subject. Math, just like English, social sciences, theatre, music, and science (as well as many other subjects), contribute to well-rounded and active citizenship. Without the ability to communicate across disciplines and lines of difference, our society will continue to stagnate in development.

Ten Things Harry Potter Taught Me About College

10. You will have to make some of the most important decisions of your life/college early in school. 

Remember when Draco comes up to Harry in before his sorting and gives him the ultimatum between success and glory with his crew of thugs and being friends with Ron? And Harry begs the hat to put him in anywhere but Slytherin as a result? Think of how those books would have been rocked had Harry befriended Draco instead. College forces you to make crucial social and academic decisions sometimes as soon as weeks before you start. Which dorm will you live in? Potluck or best friends for roommates? Do you join a sorority or a fraternity? Do you chose your schedule based off of the hot pepper and easiness ratings on ratemyprofessor? Some of these decisions you’ll look at fondly four years later. Some you’ll regret, but learn from nonetheless.

9. You can’t always get what you want, but if you try, sometimes you get what you need

Okay, so this one is borrowed a bit from Mick Jagger as well, but exemplified throughout the whole HP series. Whether it’s Harry getting an “E” on the Potions OWL exam that he needed an “O” in, or whether it’s a rejection letter from a major or program you are applying for- opportunities will present themselves to you. College is full of opportunities to get back on the hippogriff if you get buck off early. It will take a little extra elbow grease, but it will work out in the end.

8. Constant Vigilance 

Mad-Eye Moody would make an excellent freshman mentor. This mantra doesn’t solely apply to your physical safety (though you should be sure to take care of yourself). It also applies to trying to maintain the delicate balance of your life for four years. Take a couple mental health days. Get that second helping of butter-beer. But also make sure you using your Skiving Snackboxes wisely- the last thing you want is to get booted out of the castle because of your grades.

7. Just take a chance and ask him/her out 

Yes, you might end up looking like a fool. Yes, the girl you end up snogging to make your best friend jealous might end up being bat shit crazy. But, you’ll figure out what characteristics you do/don’t want in a mate. Maybe you’ll figure out that Cho Chang just isn’t your type. Or maybe Victor Krum will just fade out of your life after he goes back to Bulgaria, and you’ll just become distant pen pals. This doesn’t mean you should date everyone you see, but giving some people the benefit of the doubt with a first date can lead to some great things.

6. Your friendships are only as deep as you invest in them 

That Patronus Charm might be useful for fighting off Dementors, but it won’t get you anywhere if you keep that shield up through all of your friendships. If you are 100% honest with your close friends about your dreams, problems, successes, and failures, you will get so much more out of those relationships than if you pretend that you’ve had your past erased by a Memory Charm. Harry, Ron, and Hermione had their rough patches, but in the end, because Ron and Hermione were eventually honest about their feelings for each other, because Harry confided in them about the horcruxes, and because they all put the highest value on their friendship investment, everything worked out. Not much is constant in college- it can be a rough ride. But the ride is so much more fun if you have friends by your side

5. Every school has a Dumbledore and McGonagall- find them and learn from them

At the age of 18 or 19, you feel pretty damn invincible. The reality is, you probably have about as much common sense, and still have about as much to learn, as the 11 year-old First Years at Hogwarts. That’s where your Dumbledores and McGonagalls come in. Seek out mentors who are accomplished, eager to work with students, a little cynical about the rules and administrative structure, and learn from them. These professors want to see their students flourish, and will provide you with valuable insight to accomplish your goals. Be warned though- this process will likely not be easy. These professors generally push students to perform beyond their comfort zone and thus may be the most challenging (yet rewarding) classes you will encounter in your time in college.

4. Take some fluffy, interesting courses

You can take the Time-Turner route of course scheduling for eight straight semesters. But you’ll be more thankful to yourself if you throw in that occasional, fluffy, Divination course. There is a lot to be said for maintaining your mental health during college.

3. Some of your most important lessons won’t be in the classroom

I’m pretty sure we can all agree that History of Magic and Astronomy were pretty useless courses when it came down to the bare bones. A few valuable critical thinking skills were enhanced, but the most important moments of learning happen through kinesthetic and experiential learning. Harry wouldn’t have been as nearly prepared to destroy horcruxes had he not participated in the Triwizard Tournament. As much as Dolores Umbridge will insist that you only need to know the theory- practicing what you learn in an environment outside of the classroom should, and generally is, the outcome we strive for. This is one of the reasons that maintaining your social and professional networks are important. For most, a 4.0 shows diligence and a certain level of intellect, but a 3.7 as a result taking a couple nights to grab some mead at the Leaky Cauldron might serve you better in the long run.

2. Money doesn’t equal happiness 

The Weasleys had a few tough instances in terms of school financing, but overall, the cash strapped family was happy and healthy. Contrasted with the disfunction of the Malfoy family, it’s clear that although money allows you the opportunity to buy your way into the Slytherin Quidditch team, it’s not a causal factor in generating your success and happiness. You (almost certainly) will make stupid financial decisions in college, accrue debt, and eat many a package of Ramen. Cherish these days- it builds strong character and humbles you. Something about being consistently broke creates this universal bond with your peers too, and it’s nice to be in on that in some ways.

1. Establish your values early, and stick to them

At the end of the day, a person can only be judged on their values and actions. At the end of your time in college, you should be able to proudly on the decisions that you have made- regardless of the specific outcome. Staying true to yourself throughout college is vital to your growth as a student and person. Stick by your true friends, be true to them. Try to do right whenever possible. Your dreams and goals may require some growing pains, but rarely will they require you to compromise your values and to do harm. If they do, you need to reevaluate your goals and pathway.

How to Graduate (Properly) from the University of Kansas

“June 12, 2008
Devon Kyla Cantwell
Dear Devon:
Congratulations! You have been admitted to the University of Kansas.”

Four years ago, I was less than thrilled to have this email in my inbox. Unlike many of my peers, KU was not a first choice. Or a second. Or a twelfth. In fact, I didn’t even apply to KU until the end of May. The *fantastic* economy in 2008 caused me to have to reneg on my orignal plan to trot off to Boston and come to KU.

Now, a little older, (slightly) less stubborn, and certainly more humbled, I’ve joined the ranks of a proud and loud Jayhawk Nation, and I’m certainly counting my blessings on that decision. After a little reflection on my graduation from this fine institution two weeks ago, I’ve decided that there are a few essential elements to a Jayhawk graduation (elements that I almost overlooked as well).

1. Do all the cheesy graduation hoopla. All of it- the parties, the graduation pictures all around campus, decorating your cap, inviting your family/friends/dog/cat/anyone who will listen. Even 10 days after my ceremony, I’m glad that I can go back and nostalgically look at pictures as I’m packing to move twelve hours away from the place I’ve called home for the past four years.

2. Actually sit through the ceremony. To be honest, I legitimately cannot remember the name of our keynote speaker, but it’s certainly a bonding experience to collectively make cracks with your peers about the cliche pointless lines in every speech. Also, you’ll have a pretty sweet “First Picture as an Alum” picture in the University of Kansas photo albums.

3. SUNSCREEN. Worst mistake of my graduation day. I am still partially rocking a hot one shoulder strap sunburn

4. They will take pictures. Like, professional pictures. And they will email you no less than 273 times to beg you to order prints before they give up. Do yourself a favor, get up a little earlier the morning of graduation, feel good about how you look, and rock that hideous cap and gown.

5. Plan to walk with someone and/or a group. Share the your day with someone. Graduation is going to be kind of a roller-coaster of emotions. You’ll have plenty of time to reflect on your own before and after.

6. That’s adorable that you thought you would find parking close to the stadium 30 minutes before your ceremony. Naw dawg, park and walk. You’ll be glad you did later.

7. Look out for your mentors. Hug them. Thank them. Take a picture with them in their wizarding/Leonardo da Vinci robes if you can snag one (seriously- those things make priceless pictures- second biggest regret of my graduation ceremonies was not snagging that picture with my advisor).

8. Treat everyone like it’s the last time you will see them. For the majority of people- this is probably going to be true.

9. Eat before the ceremony and down some coffee. You’ll thank yourself later. Especially when your best friends parents send you the rest of the way down the hill with a bottle of champagne.

10. Take some pictures at/during the ceremony. But also make sure you come back and take pictures later that day or after when you are a little more rested, not dripping in sweat, and the campus  is less crowded.

11. Hunt for the mascots to snag photos with. I still have to find Centennial Jay. Lazy bum.

12. Have fun, be proud. College is hard, the job market stinks, and it’s pretty much all downhill from here. Let people brag about you and rejoice in it.