Category: Math for Teachers

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For the Love of Math

Last Friday, my family adopted a sweet, little poodle puppy, named Zipper. The foster mother, Sally, had brought him from a Mexico shelter to her own home in Silver Springs, Md. During the home visit on Friday, we talked about our careers, and I mentioned that I write about math. That’s when she told me about her neighbor, the mathematician and novelist.

“You two should meet!” she said. Apparently, we have some of the same ideas about math.

Well, I did “meet” Manil Suri today, via the pages of the New York Times op-ed section. His excellent piece, “How to Fall in Love with Math” points out some ideas I’ve been extolling for years — along with a couple that I might have said were hogwash a couple of weeks ago.

As a mathematician, I can attest that my field is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall.

Yes, yes, and again I say, yes! Mathematics is not exclusively about numbers. Hell, arithmetic is only a teeny-tiny fraction of what mathematics really is. Mathematics is the language of science. It’s a set of systems that allow us to categorize things, so that we can better understand the world around us.

Math is a philosophy, which I guess is what makes us math geeks really different from the folks who are merely satisfied with knowing how to reconcile their accounting systems or calculate the mileage they’re getting in their car. We mathy folks are truly interested in the ideas behind math — not just the numbers.

Last week, I attended a marketing intensive, a workshop during which I outlined my current career and explored how I want to take things to the next level. I’m ready to think bigger, and I need a plan to get me there.

The other entrepreneurs there thought there was real value in my creating a coaching service for entrepreneurs. My services would center around the numbers that these folks need to make their businesses survive and thrive. Marketing numbers, sales numbers, accounting numbers. They resisted the word “math” and advised me to really underscore the numbers.

From a purely marketing standpoint, I completely get it. I don’t have so much of a math wedgie that I can’t understand that the word “numbers” may be less threatening than “math.” So why not just go for it?

But the entire process left me thinking about what it is that draws me to mathematics. And ultimately what will drive me in a career, what moves me to get up in the morning and say, “Let’s go!” If you’ve been around here long, you know that it ain’t the numbers, sisters and brothers.

At the same time, I can’t say that I love math. But maybe that’s semantics, too. For the last two years, I’ve said that I’m attracted to how people process mathematics. But isn’t that just philosophy? So, isn’t that just math? This is what Suri had to say:

Despite what most people suppose, many profound mathematical ideas don’t require advanced skills to appreciate. One can develop a fairly good understanding of the power and elegance of calculus, say, without actually being able to use it to solve scientific or engineering problems.

Think of it this way: you can appreciate art without acquiring the ability to paint, or enjoy a symphony without being able to read music. Math also deserves to be enjoyed for its own sake, without being constantly subjected to the question, “When will I use this?”

At first, I disagreed with Suri’s thesis that math is worth loving — for math’s sake alone. But his analogy here is right on target. I couldn’t paint my way out of a paper bag, but each and every time I see “Starry, Starry Night” at MOMA, I catch my breath.

We come back to a failure to educate, as Suri so wonderfully elucidates in his piece. When we allow people who hate — or don’t appreciate — math to teach the subject, well, does anyone think that’s a good plan?

At any rate, I hope you’ll take a look at Suri’s piece. Meantime, I’m going to reach out to him to share my appreciation of math. Maybe there is a way — beyond teaching — for me to make a living as a math evangelist.

What do you think? Do you notice a difference between mathematics and numbers? Have you changed your mind about math in recent years or month? Please share!

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Dear Math Teacher

I’ve been talking with grownups about math for more than three years now. Parents, 20-somethings, writers, DIYers, seniors… they all have something in common: a piss-poor relationship with math.

This bad attitude is probably your fault.

The stories I hear — over and over and over again — all point to a major breakdown in the educational system. Sure, we can blame standardized testing or the state standards themselves. Middle school teachers can blame elementary school teachers. High school teachers can blame middle school teachers. College professors can blame high school teachers. And by all means, let’s not leave out the parents.

But you, dear math teacher, have control over only one thing: yourself. So what are you going to do about it? Here are some ideas.

Be Nice

If I hear one more math teacher opining about how dumb his students are, I think I might scream. Why do people teach, if they don’t like their students enough to be nice to them? Your students aren’t dumb. They’re uneducated. And guess whose job it is to educate them? If they come to your class unprepared, tough noogies. You get the kids you get. You were hired to overcome those obstacles. That’s the job, and if you can’t deal with it, perhaps this isn’t the right profession for you.

Don’ take your frustrations out on your students. Quit talking down to them. Quit berating them in public. Quit rolling your eyes or slamming doors. Be a grownup. They’re kids, and they respond to kindness and respect. Give it to them, and you’ll likely see motivation.

Inspire

You don’t have to be Martin Luther King, Jr. or Oprah. But lose the this-is-good-for-you-so-do-what-I-say attitude. It doesn’t work.

Look, you teach math for some important, personal reason. What is it that motivates you? Dig deep, find that thing, and share it with your students. Go for that spark every single day. It’ll make you feel better and get your students motivated.

It’ll also make your job MUCH easier. An inspired kid will work, will stop playing around when you ask her to, will make deeper connections. An inspired kid will meet you halfway. This gives you more energy to devote to that kid who is still messing around in the back of the room or who is ready for the next unit before the rest of the class. Inspiration means autonomous learning.

Teach Students, Not Math

Wait, did you actually think you were going to teach math? Sorry, but that’s not the job. Math teachers don’t get to immerse themselves in math all day long. Nope, your job is to teach kids. Whiny, pain-in-the-butt kids who are more interested in last night’s episode of Pretty Little Liars than their upcoming geometry test. BECAUSE THEY’RE KIDS!

Whether you like it or not, most of your students don’t give a flying flip what x is. Most adults don’t care either. You want your students to learn math? Recognize each and every student as a person, not a container to be filled with math facts. Let them experience the subject for themselves. Let them teach you.

The most effective teachers have students who will follow them to the ends of the earth. And that’s no accident. Students of all ages can spot a bullshitter in two seconds flat. They yearn for genuine relationships with adults. You give them that when you recognize that math isn’t the be-all-end-all of their day. You give them that when you see them as a whole person, not just a math student.

Be Real, But Not Too Real

Having a bad day? Own it. Frustrated with how things are going? Take responsibility. All classrooms — even the most traditional — are two-way streets. When you are real with your students, they’ll be real with you.

But expect to get some pretty raw stuff in return. Kids can’t act like adults, because — guess what? — they aren’t gown up yet. You’re modeling for them every, single day what it’s like to be a grownup. When you react to their realness with childish behavior, well, that’s a pretty strong message.

And for goodness sakes, draw some lines. Sure, you hate standardized testing. (What teacher doesn’t?) But really, do you need to share that ad nauseam with your students? Heck, you might have tremendous disdain for how administrators are running the place, but keep your trap shut on that subject. These kids aren’t your friends. And again — they’re not grownups.

Notice something? There’s not a single number, mathematical concept or teaching strategy in the above advice. I really believe this from the bottom of my heart: It’s not about the math. It’s about how you relate to your students. Every.single.time. You have way, way more power than you can even fathom. Your students carry the messages you send to them — throughout their lives. Try it. Ask five friends about their math education. I guarantee that four of them will have detailed, sad stories about why they hate math.

You have a chance to turn this around for thousands of students. And honestly, if you’re not up to the task, get out of the freaking way. Let someone else do it. Because you can do a hell of a lot more damage to one student than a kid could ever do to you.

Sincerely,

Laura Laing (informal therapist to math-haters of all ages)

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Getting Aware of Common Core Standards

Not all of us are parents or teachers, but I’ve long asserted that education is a “public good,” something that each and every one of us should be very, very concerned with. When kids don’t graduate or graduate with poor critical thinking skills, a lack of curiosity of the world around them or a dearth of basic math, reading and writing abilities, everyone suffers. And in a world where STEM-based employers are recruiting and paying more, we owe it to the next generation to do better.

(This is not to say that our educational system doesn’t have some absolutely enormous issues in other areas. Perhaps the biggest problems our schools face are not academic at all. I believe that if our country took a good, hard look at poverty, violence and teacher care, we’d make huge strides in the right direction. But this post is about academics.)

Enter the Common Core Standards. For decades, each state has developed and cultivated its own standards – or objectives required by each basic course, from history to language arts to biology. But over the last 20 years, a movement has grown to standardize these objectives across the country. With this umbrella of standards, what little Johnny is learning in Arkansas will be similar to what little Patrice is learning in Maine.

Right now, the Common Core Standards only cover English (language arts) and math. They’ve been adopted by 45 states. (Alaska, Nebraska, Texas and Virginia haven’t adopted them at all, and Minnesota adopted only the English language arts standards.) Standards for other subjects are in the works, including science and social studies.

For the last six months, I’ve been writing and editing curricula designed to meet the Common Core Standards for mathematics. I’ve gotten a pretty good feel for what they are, and I have to say that I like them for the most part. Here are some general thoughts I have:

Students will learn certain concepts earlier. I haven’t spent much time with the elementary level standards, but at least in middle and high school, various mathematical topics will be introduced earlier in the standards. For example, exponential functions (an equation with x as an exponent, like with exponential decay or compound interest) is covered in Algebra I, rather than Algebra II. 

The result is two-fold. As the standards are rolled out, some students will be left behind. In other words, kids who started school without Common Core may have a hard time catching up or bridging the gap. Second, students will have the opportunity to learn more mathematics throughout their high school career. The idea is to better prepare them for STEM in college and careers.

The emphasis is on critical thinking. This part, I love, love, love. For example: geometry proofs are back! And rather than compartmentalizing the various branches of mathematics, students will make connections between them. I just wrote a lesson that looks at how the graphs, equations and tables for various functions – linear, quadratic and exponential – are alike and dissimilar. Previously, students may never have seen these functions together in the same unit, much less the same lesson.

This means that assessments will change. Students will be asked to explain their answers or verbalize the concepts. Expect to see much more writing and discussion in math class.

Applications, applications  applications. Math is no longer done for math’s sake. And this couldn’t be better news. As I’ve said here many times before, math is pointless until it’s applied. Students should get this first-hand with Common Core, which outlines very specific applications for various concepts.

The idea here is to demonstrate that the math they’re learning is useful. The result? Hopefully more students will choose to enter STEM careers or major in these fields in college.

Students learn in different ways. Modeling plays a big role in the new standards, which means that students can approach the math in a variety of ways – from visualizing the concepts to using manipulatives like algebra tiles to working out equations in more traditional ways to graphing. This way, students can enter the material from a variety of different doors. And that can translate to greater success.

Sure, there is a lot to be concerned about (most especially the gap that we expect to see in student performance), but from my perspective the Common Core Math Standards are a step in the right direction. It’s important to know that these do not form a federal curriculum; the states are still responsible for choosing curricula that meet these standards, and education resource companies are scrambling to meet these meets. (That means I’m very, very busy these days!) It’s also important to know that chucking old ideas and implementing new ones puts a huge burden on already over-taxed schools and school systems. Finally, there is no doubt that this initiative was driven by the textbook companies, which means we’re still beholden to politics and capitalism.

But in looking at the standards alone, I think Common Core is excellent. If we can implement the standards well and keep them in place for a while, I think our kids will benefit.

What do you think of Common Core? Share your thoughts in the comment section.

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Daily Digits: My math day

Most folks readily tell me that they don’t do any math in a day. Not a stitch. So maybe they don’t sit down and solve for x or graph a quadratic equation or use the Pythagorean Theorem. But we all do math every day. And I decided to prove it.

It was last Tuesday — a pretty regular day.

April 2, 2013

6:00 a.m.: Review to-do list, estimating the time that each item would take. Count up the number of hours estimated to be sure not to exceed eight hours, while leaving time for lunch and exercise.

7:00 a.m.: Track all Weight Watchers points that I expect to use for the day, by planning what I’ll have to eat for breakfast, lunch, dinner and snacks. Allow the online program to add everything up, but pay close attention that my breakfast and lunch are around 6 points each and that I’m using less than 8 points from my weekly extra points.

10:00 a.m.: Review invoicing for first quarter. Within bookkeeping program, look at the data in a variety of ways: bar graphs, showing income for each month, and tables showing the income for each client. Compare income to goals and adjust expectations where necessary.

11:00 a.m.: Set budget for new book postcard, using designer’s estimates. Compare costs of a fewer number of cards to the costs of a much larger run. Table the decision to think about things.

12:00 a.m.: Attend weekly Weight Watchers meeting, and learn that I lost 0.4 pounds last week. Spend meeting mentally calculating how that could have happened, given the fact that I didn’t stay within my allotted daily points for a few days. Remember that balancing the equation of caloric intake and output, with variables like water retention, is way too complex for mental math. Decide to just feel fortunate and proud.

1:00 – 3:30 p.m.: Outline online lesson about linear, quadratic and exponential functions. (Yes, this is where I and the rest of the world differs! But I wanted you to know that this curriculum doesn’t appear out of thin air.)

4:00 p.m.: Meet with potential photographer for our wedding. Count backwards from the start of the wedding to estimate the time necessary and the cost of a second photographer. Mentally calculate how much over our budget we’d go if we hired this photographer. (Everything goes over budget, I’ve found.)

6:30 p.m.: Meet a friend for drinks at a local restaurant. Scan menu for lowish-calorie drink, decide that since a cosmo is the same points as a glass of wine, why not have the pink drink in the fancy glass?

7:30 p.m.: Get the check. Find the tip by taking 10% of the bill and doubling it. Then split the check evenly since we got the same drink and shared an appetizer.

11:30 p.m.: Daughter can’t sleep. Mentally add up the number of hours of sleep we can each expect to get if she would just fall asleep right now. Finally she dozes off.

And there you have it — my math day. As you can see, the math was tucked into various nooks and crannies. If I hadn’t been paying attention, I wouldn’t have even noticed it. And most of it had nothing to do with the way I learned to do math at school.

So what about you? Here’s my challenge: Just for today, jot down when you’ve used math. Then share what you learned about yourself in the comments section. Did you find that you used math more than you thought? Did you discover that you’re using a kind of math that you never, ever expected? I want to know!

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Boston Marathon: How FBI profilers use math

We can all agree that the horrific events at Monday’s Boston Marathon sent a chill down our country’s collective spine. The two bombs that exploded have made us afraid and sad and hopeless. One message that seemed to ease many’s pain and fear was from Mr. Rogers, who once said:

When I was a boy and I would see scary things in the news, my mother would say to me, ‘Look for the helpers. You will always find people who are helping.’ – Fred Rogers

This is an amazing idea in the midst of the mayhem and terror that followed the explosions. There were dozens and dozens and dozens of people who ran toward the bomb sites, because that’s what they do – help those in need.

In the days that have followed, the FBI and others have been investigating the explosions, gathering information that will likely lead to an arrest and hopefully a conviction. Our natural question in these situations is, “Why?” Catching the person or people who did this will help us find that answer.

It shouldn’t surprise you to know that these investigators will depend on mathematics to help them solve this crime. From measuring the trajectory of the shrapnel to piecing together a timeline of events, math is a critical component in investigation.

A while back, I had the pleasure of interviewing Mary Ellen O’Toole, a former FBI profiler and author of Dangerous Instincts: How Gut Feelings Betray Us. She answered my questions about how she used math as a profiler. And I’m betting that this holds true for the investigation in Boston, as well.

Math at Work Monday: Mary Ellen the FBI profiler

Can you explain what you do for a living?

For half of my career, I worked in Quantico, at the FBI’s Behavioral Analysis Unit, the very unit that is the focus of the television show Criminal Minds. While there I tracked down, studied, and interviewed some of the world’s most infamous criminals, and I analyzed their crime scenes, too. These criminals included Gary Ridgeway (the Green River Killer), Ted Kaczynski (the Unabomber) and Derrick Todd Lee (the serial killer of Baton Rouge.) I worked everything from white-collar crime to work place and school violence to kidnappings to serial murder.

Since my retirement in 2009, I’ve worked as a consultant to law enforcement, corporate security, administrators, and many other professionals. I also teach at the Smithsonian, FBI Academy and many other locations.

When do you use basic math in your job?

As I and other profilers worked to solve a crime, we used every type of math from basic addition to geometry and pattern analysis to statistics and probability to reasoning and logic.

Read the rest of the interview.

If you’d like to share your wishes for the victims of the Boston Marathon bombing, please feel free to do so in the comments section. 

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Sharing Awareness with Kids: Bedtime Math

One of the questions I get most often from parents is this: How can I help my kids from being anxious about math like I am? And for a math nerd like me, the answer is pretty simple. I’m unnaturally aware of the math around me. Because of my background and experience — and maybe even the way my brain is wired to see patterns in damned near everything — I can weave math into just about any situation I come across.

(Go ahead, try me. Post a situation in the comments section, and I’ll bring the math. It’s a game I used to play with my daughter, until she got really tired of losing.)

But for most parents, this level of math awareness is just not as simple to access. This is where Laura Bilodeau Overdeck comes in. With degrees in astrophysics and public policy, Overdeck is probably a little like me — finding math in everything and pointing it out to her kids at every turn. But she didn’t just keep this to herself. Nope, she launched Bedtime Math, a really simple idea designed to help parents inject a little math in their kids’ everyday lives.

Each day, she and her crackerjack team send out an email to subscribers (it’s free!) that offers three math questions — one for Wee Ones, one for Little Kids and one for Big Kids — that are centered on a little story or current event. Yesterday, the theme was tongue twisters. On Wednesday, it was hopping.

During Math Awareness Month, Overdeck and her team have introduced a series of mini, math videos. And these things are funny. The first is about ninja training — what kid (or parent) wouldn’t want to find out what happens? Check it out below:

I can’t tell you how much I love Bedtime Math. If you have little kids, give it a shot. You’ll probably learn something too — and you might even raise your awareness of the math around yourself.

Are you a Bedtime Math subscriber already? How do you use it with your kids? What do your kids think about it? Share in the comments section.

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Math Awareness Month: What’s Your Story?

Lots of people make one of two incorrect assumptions about me. I’m a writer, so they initially assume that I don’t have a good relationship with math. And when they find out that I have a degree in math, they assume that I love to sit down and solve trigonometry problems all day long.

Sure, I like math. I’ve said it a hundred times – math is a useful tool. I feel comfortable using math to figure out problems that I have, like how much fabric I need to order to recover my couch or the number of calories in a 3/4 serving of granola. (Yes, I actually do that second thing on a regular basis.) But I’ve never been head-over-heels in love with math.

What do I really love? A good story. And so for the third year, I’m inviting you to share your math story. Telling others how you came to love, like or hate math is an interesting process. I’ve learned that education – and particularly teachers – make a huge difference in how people feel about math. Get a great teacher, and you have a much better chance of at least coming out of the class appreciating math. But a burned out, cynical or, worse, mean teacher can destroy any positive feelings a student might be cultivating about math.

Why share your story now? Well, April is Math Awareness Month, which is not about appreciating math. Nope. The goal of this month is to simply encourage people to notice the math around them. (Which is also my personal goal with this blog and my book.) The first step can be telling the story that helped form your impressions of math. Is there something blocking your understanding or appreciation? Could be.

Before you share your story, you may want to read some others’. Start with mine, and then check out how math almost ruined Lisa Tabachnick Hotta‘s life and how Siobhan Green learned to use math, despite an early bad experience.

Then tell your story in the comments section. Do you like math, hate it, don’t care one way or the other? Does math make your hands sweat? Why do you think you have these feelings about math? Do you have a sad story — giving up and giving in? Or did you triumph? Whatever your personal experience, I want to hear from you. And if you’d like more space, feel free to contact me about a special guest post.

So what are you waiting for? Share your story today.

P.S. The official theme for this year’s Mathematics Awareness Month is sustainability. That’s a wonderful topic, but I think for many of us, it’s a little esoteric. So I’m going to pull back and focus on some more mundane topics this month. However, math educators should check out the Math Awareness Month website for ideas on how to relate this theme to the classroom. There are some really cool resources there.

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Math Summer Camps: Guest post by Lynn Salvo of MathTree

So last summer, I wrote about my disdain for math-geared summer camps. And I was summarily schooled by my friend Lynn Salvo, founder of MathTree, which offers summer camps in Virginia, Washington D.C., Maryland and Delaware. She was right, of course, and I invited her to share why a summer camp centered on math can be a rewarding experience for parents and students. Mind changed. (Thank you, Lynn.)

While summer is a great time to kick back and recharge, the down side is that kids forget a lot of math over the summer.  Studies show that during the lazy months of summer, all kids suffer from “brain drain” or the loss of learning. In fact, students lose (on average) 2.6 months of mathematical competency in June, July and August.  Only the most math-minded and determined parents can find the math in everyday life to keep math going over the summer.  A couple of weeks of a math camp anywhere in the summer can bridge the long gap.

I am president of  MathTree, which I founded in 1999 to address this very issue.  We have been providing math camps for children ages five to15 throughout the DC, Virginia, Maryland and Delaware area ever since, mostly in the summer but also during long school breaks.  Children love our camps and return summer after summer.  Some have even grown up to be instructors for us themselves.

If you’re reading this blog, you probably know math is not the most popular subject.  MathTree would have gone out of business long ago if we had not found a formula that works to provide a great summer math experience for kids.  In a typical school setting, younger children are taught math by amazing elementary school teachers who are generalists, not specialists. And unfortunately, many of them don’t really like math all that well.  Summer is a great time to give your child an opportunity to work with folks who love math, love kids, and love teaching kids math!

So what should you be looking for and how can you evaluate the math (summer) camp possibilities you are considering for your child?  Here are some questions:

  • Does the camp provide different ways of learning there is not time for in a packed school curriculum?
  • Does the camp promise an adventure?  Will the camp creatively lead my child on a mathematical exploration?  At MathTree we have our own mathical characters, including Princess KrisTen; Grouper, the Regrouper; and Numero, the Number Wizard, which we use to happily engage our campers in fun math exploration.
  • Is my child going to be set up to make mathematical discoveries?
  • Will my child play fun math-rich games?  For instance, we play games such as Ten Mingle to learn numbers that add up to 10 or Product Parfait to master multiplication facts.

Other important questions should include:

  • Will my child actively engage with people who can sense subtlety and nuance in my child’s understanding or will s/he be babysat by an electronic device?
  • Does the camp provide a sustained and focused learning experience? Does it develop momentum and go deep into math?
  • Does the camp focus on why, not just how?  Will my child learn concepts, not just processes?  For example, will my child learn what division is, not just how to do it?
  • Does the camp provide an enriching head start on the big ideas coming in math in the next school year?
  • Will my child be placed with mathematical peers or lumped with others of the same age or grade regardless of where they are mathematically? It’s critical that your child is neither frustrated (too hard) nor bored (too easy).
  • How will my child be assessed?  Will my child’s understanding be monitored in multiple ways, even in simple conversations?
  • What is the staff to camper ratio?  Your child may have suffered already in a large class.

I firmly believe that parents should always look for classes with less than 15 campers where there is a teacher and an assistant.  Here is what the teacher can do in that setting:

  • Actively engage your child in the learning
  • Tune in to your child — your child is not a number!
  • Embrace your child’s unique personality
  • Notice if your child looks confused
  • Jump in quickly and “unconfuse” your child.

Here’s what the assistant can do in that setting:

  • Handle routine tasks so the teacher can be fresh and creative with the class
  • Check children’s work quickly
  • Prepare rich math materials for children to use and store them after use so there is more quality class time
  • Provide a challenge if your child gets ahead of the group.

Whichever camp you choose, take the selection of your children’s summer math camp seriously.  You want your child to come away loving math (more) and you want to feel you got a high return on your investment.

MathTree has been growing our children’s love of math since 1999. For more information about MathTree and registration, please go to www.MathTree.com.  MathTree provides summer camps for kids in 25 locations in DC, DE, MD, and VA.  Use our camp locator to find a MathTree camp near you.

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Math for Grownups Math for Teachers Math for Writers Statistics

That’s So Random: Getting sampling right

On Wednesday, we talked about sample bias, or ways to really screw up the results of a survey or study. So how can researchers avoid this problem? By being random.

There are several kinds of samples from simple random samples to convenience samples, and the type that is chosen determines the reliability of the data. The more random the selection of samples, the more reliable the results. Here’s a run down of several different types:

Simple Random Sample: The most reliable option, the simple random sample works well because each member of the population has the same chance of being selected. There are several different ways to select the sample — from a lottery to a number table to computer-generated values. The values can be replaced for a second possible selection or each selection can be held out, so that there are no duplicate selections.

Stratified Sample: In some cases it makes sense to divide the population into subgroups and then conduct a random sample of each subgroup. This method helps researchers highlight a particular subgroup in a sample, which can be useful when observing the relationship between two or more subgroups. The number of members selected from each subgroup must match that subgroup’s representation in the larger population.

What the heck does that mean? Let’s say a researcher is studying glaucoma progression and eye color. If 25% of the population has blue eyes, 25% of the sample must also. If 40% of the population has brown eyes, so must 40% of the sample. Otherwise, the conclusions may be unreliable, because the samples do not reflect the entire population.

Then there are the samples that don’t provide such reliable results:

Quota Sample: In this scenario, the researcher deliberately sets a quota for a certain strata. When done honestly, this allows for representation of minority groups of the population.  But it does mean that the sample is no longer random. For example, if you wanted to know how elementary-school teachers feel about a new dress code developed by the school district, a random sample may not include any male teachers, because there are so few of them. However, requiring that a certain number of male teachers be included in the sample insures that male teachers are represented — even though the sample is no longer random.

Purposeful Sample: When it’s difficult to identify members of a population, researchers may include any member who is available. And when those already selected for the sample recommend other members, this is called a Snowball Sample. While this type is not random, it is a way to look at more invisible issues, including sexual assault and illness.

Convenience Sample: When you’re looking for quick and dirty, a convenience sample is it. Remember when survey companies stalked folks at the mall? That’s a convenience or accidental sample. These depend on someone being at the right (wrong?) place at the right (wrong?) time. When people volunteer for a sample, that’s also a convenience sample.

So whenever you’re looking at data, consider how the sample was formed. If the results look funny, it could be because the sample was off.

On Monday, I’ll tackle sample size (something that I had hoped to include today, but didn’t get to). Meantime, if you have questions about how sampling is done, ask away!

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One in a Million: How sample bias affects data

Continuing with our review of basic math skills, let’s take a little look-see at statistics. This field is not only vast (and confusing for many folks) but also hugely important in our daily lives. Just about every single thing we do has some sort of relationship to statistics — from watching television to buying a car to supporting a political candidate to making medical decisions. Like it or not, stats rule our world. Unfortunately, trusting bad data can lead to big problems. 

First some definitions. A population is the entire group that the researchers are interested in. So, if a school system wants to know parents’ attitudes about school starting times, the population would be all parents and caregivers with children who attend school in that district.

A sample is a subset of the population. It would be nice to track the viewing habits of every single television viewer, but that’s just not a realistic endeavor. So A.C. Nielsen Co. puts its set-top boxes in a sample of homes. The trick is to be sure that this sample is big enough (more on that Friday) and that its representative.  When samples don’t represent the larger population, the results aren’t worth a darn. Here’s an example:

Ever hear of President Landon? There’s good reason for that. But on Halloween 1936, a Literary Digestpoll predicted that Gov. Alfred Landon of Kansas would defeat President Franklin Delano Roosevelt come November.

And why not? The organization had come to this conclusion based on an enormous sample, mailing out 10 million sample ballots, asking recipients how they planned to vote. In fact, about 1 in 4 Americans had been asked to participate, with stunning results: the magazine predicted that Landon would win 57.1% of the popular vote and an electoral college margin of 370 to 161. The problem? This list was created using registers of telephone numbers, club membership rosters and magazine subscription lists.

Remember, this was 1936, the height of the Great Depression and also long before telephones  and magazine subscriptions became common fixtures in most families. Literary Digest had sampled largely middle- and upper-class voters, which is not at all representative of the larger population.  At the same time, only 2.4 million people actually responded to the survey, just under 25 percent of the original sample size.

On Election day, the American public delivered a scorching defeat to Gov. Landon, who won electoral college votes in Vermont and Maine only. This was also the death knell for Literary Digest, which folded a few years later.

This example neatly describes two forms of sample bias: selection bias and nonresponse bias. Selection bias occurs when there is a flaw in the sample selection process. In order for a statistic to be trustworthy, the sample must be representative of the entire population. For example, conducting a survey of homeowners in one neighborhood cannot represent all homeowners in a city.

Self-selection can also play a role in selection bias. If a poll, survey or study depends solely on participants volunteering on their own, the sample will not necessarily be representative of the entire population. There’s a certain amount of self-selection in any survey, poll or study. But there are ways to minimize the effects of this problem.

Nonresponse bias is related to self-selection. It occurs when people choose not to respond, often because doing so is too difficult. For this reason, mailed surveys are not the best option.  In-person polling has the least risk of nonresponse bias, while telephone carries a slightly higher risk.

If you’re familiar with information technology, you know the old adage: Garbage in, garbage out. This definitely holds true for statistics. And this is precisely why Mark Twain’s characterization of number crunching — “Lies, damned lies and statistics” — is so apropos. When the sample is bad, the results will be too, but that doesn’t stop some from unintentionally or intentionally misleading the public with bad stats. If you plan to make good decisions at any point in your everyday life, well, you’d better be able to cull the lies from the good samples.

If you have questions about sample bias, please ask in the comments section. Meantime, here are the answers to last Wednesday’s practice with percentage change problems: –2%, 7%, –6%, –35%. Friday, we’ll talk about sample size, which (to me) is a magical idea. Really!

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Math for Grownups Math for Teachers

Math Warriors: The comedic side of math

As I continue to crawl from under a mountain of work, I thought I’d share a really cool webseries that I discovered late last year. It’s not clear if the series will continue this spring or not, but you can take a look at the first two seasons on YouTube or the Math Warriors‘ website. 

Not since Tina Fey and Lindsay Lohan have math geeks been so cool. Math Warriors satirical look at a fictitious rivalry between Harvard’s and Yale’s math teams. And tucked away in each episode is a little bit of math. See if you can find it.

Season 1, Episode 1

Season 1, Episode 2

Season 1, Episode 3

What do you think? Stay tuned for more episodes. Or, if you can’t wait, check out the Math Warriorswebsite.

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Basic Math Review Math for Grownups Math for Parents Math for Teachers

Multiplying and Dividing — Integer Style

Continuing on in our review of basic math, I welcome you to Day 2. The answers to Day 1 questions are at the bottom of the post — along with new questions. But first, let’s learn how to multiply and divide integers.

Let’s say you have a bank account with a service fee of $15 per month. If that amount was deducted every single month, how can you represent the yearly amount for these fees? Well, you would multiply -$15 (the fee is negative because it’s taken out of the account) by 12 (the number of months in the year). But how the heck do you multiply negative and positive numbers? Let’s find out.

Remember integers — those negative and positive numbers that aren’t fractions, decimals, square roots, etc.? I like to think of them as positive and negative whole numbers (though most real mathematicians would argue against that classification). On Wednesday, you learned how to add and subtract these little buggers. (Check out the post here, if you missed it.)  Today, we multiply and divide.

Her’s the really good news: it is way, way easier to multiply and divide integers than to add and subtract them. First, though, it’s a good idea to understand how the rules work. When you first started multiplying numbers, you did things like this:

2 x 3 = 2 + 2 + 2 = 6

In other words “2 x 3” is the same thing as adding up three 2s. Get it? And because you started working with positive numbers when smacking a girl upside the head meant you “like-liked” her, you know without a shadow of a doubt that the answer is positive.

Let’s see what happens when you multiply a negative number by a positive number:

-2 x 3 = -2 + -2 + -2 = -6

Now to understand this, you need to either pull up your mental number line and count or remember the addition rules from Wednesday’s post. When you add two numbers with the same sign, add the numerals and then take the sign. So -3 + -3 is -6.

But what about multiplying two negative numbers? Admittedly, this is a little trickier to explain. It helps to look for a pattern using a number line. Let’s try it with -2 x -3.

-2 x 2 = -4
-2 x 1 = -2
-2 x 0 = 0
-2 x -1 = ?
-2 x -2 = ?

Based on the pattern shown on the number line, what is -2 x -1? What is -2 x -2? If you said 2 and 4, you are right on the money.

And now we can summarize the above with some rules. Believe me, this is one math concept that is much, much easier to remember with the rules. Still, if knowing why helps anyone get it, I’m all for pulling back the curtain.

When multiplying integers:
If the signs are the same, the answer is positive;
If the signs are different, the answer is negative.

Bonus: The same rules work for division. That’s because division is the inverse (or opposite) of multiplication.

When dividing integers:
If the signs are the same, the answer is positive;
If the signs are different, the answer is negative.

The only tricky part is this: Sometimes it seems that if you are multiplying or dividing two negative numbers, the answer should be negative. It’s a trap! (Not really, but you could think of it that way, if it helps.) The key in multiplying and dividing integers is noticing whether the signs are the same or different.

In fact, if you are doing a whole set of these kinds of problems, you can simply run through the problems and assign the signs to the answers — before even multiplying or dividing. (I tell students to do this all the time, because I think it helps them to remember the rules.)

4 x -3 → signs are different → answer is negative
-4 x -3 → signs are the same → answer is positive
-4 x 3 → signs are different → answer is negative
4 x 3 → signs are the same → answer is positive

Then all you’d need to do is the multiplication itself:

4 x -3 = -12
-4 x -3 = 12
-4 x 3 = -12
4 x 3 = 12

And like I said, division works the same way:

-24 ÷ -2 = +? = 12
24 ÷ -2 = -? = -12
24 ÷ 2 = +? = 12
-24 ÷ 2 = -? = -12

Got it? Try these examples on your own.

1. 5 x -6 = ?

2. -18 ÷ 9 = ?

3. -20 ÷ -4 = ?

4. 8 x 4 = ?

5. -2 x 7 = ?

Questions? Ask them in the comments section. Up Monday are fractions. If you can’t remember how to add, subtract, multiply or divide fractions or mixed numbers, tune in. 

Answers to Wednesday’s “homework.” (It’s not really homework, I promise.) -10, -4, 2, -15, -2. How did you do?