Category: Math for Teachers

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Why National Polls Don’t Matter: Electoral college math

This post makes me scared. Not because the math is challenging or because I’m worried about the election. I’m afraid of looking partisan or being accused of ideology. (It’s happened before!) But I can’t avoid election math any longer, so I’ve decided to take the plunge — today and Monday — into these shark-infested waters, trusting that my readers (and new guests) will put away their partisan differences if only for a few hours. Do for the sake of the math.

There’s no denying the math that goes on in elections. There are polls, ad buys, the number of minutes each candidate has spoken during debates — and yes, the electoral college. Whatever you may think of our dear map, it is how elections are decided in this country — for the most part.

There’s no reason to expect a repeat of Election Day (and the weeks following) 2000 this year. So I thought it would be a good idea to review the electoral map — from a mathematical perspective — so that we can better understand its power. First some history.

During the Constitutional Convention in 1787, the founding fathers quickly rejected a number of ways to select the country’s president: having Congress choose the president, having state legislatures choose and direct popular vote. The first two ideas were tossed based on fears of an imbalance of power — giving Congress or the states too much control. They also worried that a direct popular vote would be negatively influenced by the lack of consistent communication. In other words, without information about out-of-state candidates, voters would simply choose the candidate from their own states. And then there was the very real fear that a candidate without a sufficient majority would not be able to govern the entire nation.

So, these fine men drew up a fourth option: a College of Electors. The first design, which is outlined in Article II of the U.S. Constitution, was pitched after four Presidential elections, after political parties emerged. Much of the original system remained, but the 12th Amendment to the Constitution instituted a few changes to reflect the country’s new party system. Here what the electoral college looks like today:

  • The Electoral College consists of 538 electors.
  • Each state is allotted the same number of electors as it has Congress members (Senators and Representatives)
  • Therefore, representation in the Electoral College is dependent on each state’s population. More populous states have more electoral votes; less populous states have fewer electoral votes.
  • The 23rd Amendment to the Constitution gives the District of Columbia 3 electoral votes, event though it is not a state.
  • Each state has its own laws governing how electors are selected. Generally, electors are selected by the political parties themselves.
  • Most states have a “winner takes all” system, which means that the candidate with the majority of the direct popular votes in the state gets all of the electoral votes.
  • However, Maine and Nebraska have a proportional system, which means the electoral votes can be divided between candidates.

Whew!

Some basic calculations allow the media and election officials and the candidates themselves to make really good predictions on election night in most situations. But the electors don’t officially cast their votes until the first Monday after the second Wednesday in December. Then, on January 6 of the following calendar year in a joint session of Congress, the electoral votes are counted, and the President and Vice-President are declared. (Got all that?) Almost always, though, the losing candidate concedes the election on election night or the next day, making the electoral vote and counting a mere formality.

The thing that makes this complex is that each state has a different number of electoral votes. In order to win the presidential election, a candidate must secure at least 270 electoral votes. And that’s why you’re probably seeing a red and blue (and purple?) map in your newspaper, on television and online.

In my state, there is no question which candidate will take all of the electoral votes. Maryland has been staunchly Democratic for several decades. And there’s no mystery about Texas, which is about as red as a state can get. But if it were a contest between Maryland’s and Texas’ electoral votes, Governor Romney would win. That’s because Texas has 38 electoral votes, while Maryland has 10.

Right now, there are lots and lots of predictions out there concerning how the electoral college will vote. (Personally, I think Nate Silver0 of the New York Times is the most reliable source. Dude has a killer math brain, correctly predicting the electoral college outcomes in 49 of the 50 states in the 2008 election. In that same election, he correctly predicted all of the 35 Senate races.) But there’s little doubt about many of the states. A few swing states will certainly claim this election: Colorado, Florida, Iowa, New Hampshire, Ohio, Virginia and Wisconsin. Mathematically speaking, we’re talking about 89 votes:

  • Colorado: 9
  • Florida: 29
  • Iowa: 6
  • New Hampshire: 4
  • Ohio: 18
  • Virginia: 13
  • Wisconsin: 10

Now out of those, which states would you guess the candidates really want to win? Yep, the ones with the highest number of electoral votes. So to them, the most important states in these last days of the campaign are Florida, Ohio and Virginia. (Where do you draw the line? I chose more than 10 electoral votes.)

If you live in one of these three states, you are acutely aware of this fact. Unless you don’t have a television set or listen to the radio or have a (really) unlisted phone number.

So what does this mean? Right now, it means that President Obama is likely to win the election. There are scenarios that show the opposite outcome — and there are even a few that produce a tie. However, most political analysis says that it’s Obama’s to lose at this point. This is despite the fact that most polls show the popular vote at a statistical dead heat (in other words, any lead by either candidate is well within the margins of error).

Because our founding fathers made a decision that we wouldn’t elect our presidents with a direct popular vote. What matters in these last days are the popular votes in the swing states — most importantly Florida, Ohio and Virginia — though there are scenarios that give Mitt Romney the edge without winning all of the swing states.

If you are a complete geek about election numbers, do visit Silver’s FiveThirtyEight blog at the New York Times. His math is good, regardless of what some conservative pundits have claimed in recent weeks.

EDITOR’S UPDATE: Sam Wang of the Princeton Election Consortium also has great analysis. Hurricane Sandy has messed with his servers, so the site looks pretty rudimentary, but he is updating his site regularly. It’s pretty cool to compare Silver’s and Wang’s conclusions — especially on a day-by-day basis.

I also highly recommend a really slick interactive tool put out by the New York Times. It graphically illustrates ways in which the electoral votes could swing the election in either way, based solely on the math. Unlike Silver’s blog, this section does not offer a prediction of who will will win, but describes the various scenarios for each candidate.

Whatever you think of the candidates and the issues, vote. No matter what, vote. Our votes — even outside swing states — matter. It’s our responsibility as U.S. citizens to declare our preferences. And in my mind, if you don’t vote, you can’t complain.

Coming on Monday… a look at the polls themselves. What makes a good poll? How should we average folks interpret polls? Can they really tell us what’s going on?

What are your thoughts on the math of the electoral college? (I get it. These discussions can get heated. Please be respectful in your comments. I will not approve or will delete any comments that I deem outside the bounds of civility. Thank you for playing nice.)

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Saving Face: Avoiding performance math

If there’s one thing most folks assume about me, it’s this: That I am some sort of mathmagician, able to solve math problems in a single bound — quickly, in public and with a permanent marker.

Nothing could be farther from the truth.

I don’t like what I call performance math. When I’m asked to divvy up the dinner tab (especially after a glass of wine), my hands immediately start sweating. When friends joke that I can find 37% of any number in my head, I feel like a fraud. I’m not your go-to person for solving even the easiest math problem quickly and with little effort.

Truth is I really cannot handle any level of embarrassment. And I’m very easily embarrassed. I’m the kind of person who likes to be overly prepared for any situation. This morning, before contacting the gutter company about getting our deposit back because they hadn’t shown up, I had to re-read the contract and literally develop a script in my head. What if I misunderstood something and was — gasp! — wrong about the timeline or terms of our contract?

Oh yeah, and I hate being wrong. About anything.

In short, I’m not much of a risk taker. Unlike many of my friends and some family members, I can’t stand the thought of failing publicly. Imagine writing a math book with this hang up! Thank goodness for two amazing editors, who checked up behind me.

I’m also not a detailed person. Not one bit. I’m your classic, careless-mistake maker — from grade school into grownuphood. I’m much more interested in the big picture, and I am easily lured by the overreaching concepts, ignoring the details that can make an answer right or wrong.

For years and years, I worried about this to no end. How could I be an effective teacher, parent, writer, if I didn’t really care about the details or I had this terrible fear of doing math problems in public? What I learned very quickly in the classroom was this: Kids needed a math teacher like me, to show them that failing publicly is okay from time to time and that math is not a game of speed or even absolute accuracy. (It’s never a game of speed. And it’s frequently not necessary to have an exact answer.)

Two weeks ago, as I sat down with my turkey sandwich at lunch, the phone rang. It was a desperate writer friend who was having some trouble calculating the percentage increase/decrease of a company’s revenue over a year. (Or something like that. I forget the details. Go figure.) She really, really wanted me to work out the problem on the phone with her, and I froze. I felt embarrassed that I couldn’t give her a quick answer. And I worried that I would lose all credibility if I didn’t offer some sage insight PDQ.

But since I have learned that math is not a magic trick or a game of speed, I took a deep breath, gathered my thoughts and asked for some time. Better yet, I asked if I could respond via email, since I’m much better able to look at details in writing than on the phone. I asked her to send me the information about the problem and give me 30 minutes to get back with her.

Within 10 minutes, I had worked out a system of equations and solved for both variables. She had her answer, and I could solve the problem without the glare of a spotlight (even if it was only a small spotlight).

My point is this: Math isn’t about performing. If you like to solve problems in your head or rattle off facts quickly or demonstrate your arithmetic prowess at cocktail parties, go for it. That’s a talent and inclination that I sometimes wish I had. But if you need to retreat to a quiet space, where you can hear yourself think and try out several methods, you should take that opportunity.

Anyone who criticizes a person’s math skills based on their ability to perform on cue is being a giant meanie. And that includes anyone who has that personal expectation of himself. There’s no good reason for math performance — well, except for Mathletes, and those folks have pretty darned special brains.

Do yourself a favor and skip math performance if you need to. I give you permission.

Do you suffer from math performance anxiety? Where have you noticed this is a problem? And how have you dealt with it?

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Engineering tops highest-earning degrees — again

With the economy still struggling along and a price of a college degree outpacing ordinary inflation, more and more personal finance experts are suggesting that students choose a major based on its earning potential. And true to form, this year’s American Community Survey data shows that STEM (science, technology, engineering and mathematics) degrees continue to promise much higher incomes than even business degrees. And so today, instead of interviewing someone about how they use math in their job, I thought I’d take a look at this data.

In 2011, 59 million Americans (25 years and older) held bachelor’s degrees. The most popular degree is business (20%), with education coming in second (12%). In fact, those with business degrees were the most likely to be employed. But here’s where the rubber hits the road: those with engineering degrees continue to out-earn business majors by about $25,000 a year (based on median salaries).

Yes, you read that right.

And the hits keep coming (again, based on median salaries): those with mathematics, computer science or statistics degrees earn $13,000 more each year, as do those with physical science degrees. Even if a STEM degree holder was not working in that humanities degree holders were (naturally) at the low end of the earning potential, along with education,

But money isn’t everything. Those in STEM careers are more likely be employed in full-time, year-round jobs. (Curiously, teachers aren’t considered year-round employees, which I think skews the data somewhat.) The mathy/sciencey types are also less likely to be unemployed.

I am not one to suggest that someone get a degree merely for the earning potential. If you don’t want to be an engineer, don’t major in that field. It sounds a little woo-woo, but I firmly believe in the general idea that we should all be following our bliss (and being smart about what that means financially).

Where I think this data matters — big time — is much farther down the educational ladder. Students who learn to love (or at least appreciate) STEM subjects are much more likely to consider these as a field of study. On the other hand, many of you can personally attest to the fact that it’s hard to fall in love with these subjects — and play catch up with the concepts and foundation needed to excel in them — when you’ve learned to hate them or have zero confidence in your abilities.

In other words, the work starts in elementary and middle school. For students reach their real earning potential and for employers to find qualified experts for the jobs that they do have, we really must make STEM a priority in these grades. That doesn’t mean more testing or introducing concepts at a younger age. (In my opinion, those strategies are counterproductive.) It means finding truly gifted STEM teachers who are able to motivate their students and overcome our epidemic of mathematics anxiety and general apathy towards the subject.  It means approaching STEM subjects with excitement and a sense of discovery. It means encouraging, not discouraging, exploration in these subjects.

So I ask you: What are you doing to help with this?

Interested in how things broke down numerically? Here are a few median salaries from the American Community Survey:

  • Engineering, $91,611
  • Computers, mathematics, statistics, $80,180
  • Physical and related sciences, $80,037
  • Business, $66,605
  • Literature and languages, $58,616
  • Education, $50,902
  • Visual and performing arts, $50,484

What do you think? Should college students choose a degree based on earning potential? Or should they “follow their bliss”? How can schools help students develop an interest in the fields that offer a higher earning potential? Share your comments!

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Math at Work Monday: Joe the Platform Consultant

In the IT field, there are many machines and programs that are really confusing and difficult to understand. Not only do we have to trust and depend on these machines, but also the people who service them. Joe Thompson is one of the good guys. He provides assistance to the users and companies when they need it most. From consulting to maintenance, Joe and his colleagues are there for us when our technology isn’t working quite right. (Joe is also one of my former geometry students. It’s been great to reconnect with him and see how accomplished he is now!)

Can you explain what you do for a living?

Red Hat’s consultants help customers get our products working when they have specific needs that go beyond the usual tech support.  We are essentially advanced computer system administrators on whatever our customers need us to be to get Red Hat’s products to work for them.  Common consulting gigs are setting up Red Hat Satellite to manage the customer’s servers, or doing performance tuning to make things run faster or a “health check” to verify things are running as efficiently as possible.

We just put out a marketing video about our consulting for public-sector clients, actually:

https://www.youtube.com/watch?v=eMzANG3Yhlk(We do more than just public sector and cloud, of course.)

When do you use basic math in your job?

The most common is when tuning a system to perform well, or configuring various things.  Unit conversions and base conversions are especially important.

IT has a long-running math issue actually: does “kilo” mean “1000” (a round number in base 10), or “1024” (a round number, 10000000000, in base 2)?  There are various ways people try to indicate which is intended, like using a capital K vs. a lowercase k, or using KiB vs. KB.  This matters in a lot of cases because when you get up into large data sizes, the difference between round numbers in base 10 and base 2 gets pretty big.  A 1-TB hard drive (a typical size today, maybe even a little small) is a trillion bytes — 1000 to the fourth power, not 1024 to the fourth power.  The difference is about 10% of the actual size of the drive, so knowing which base you’re dealing with is important.

Then there are units that have to be converted.  A common adjustment for better performance is tweaking how much data is held in memory at a time to be transmitted over the network, which is done by measuring the delay between two systems that have to communicate.  Then you multiply the delay (so many milliseconds) by the transmission speed (so many megabits or gigabits per second) and that gives the buffer size, which you have to set in bytes (1 byte = 8 bits) or sometimes other specified units.Sometimes software writers like to make you do math so they can write their code easier.  If a program has options that can either be on or off, sometimes a programmer will use a “bitfield” — a string of binary digits that represent all the options in a single number, which is often set in base 10.  So if you have a six-digit bitfield and want to turn off everything but options 1 and 6, you would use the number 33: 33 = 100001 in binary.

Do you use any technology (like calculators or computers) to help with this math? Why or why not?

I’ve always done a lot of arithmetic in my head and I can at least estimate a lot of the conversions without resorting to a calculator.  I’ll break out the calculator if the math is long and tedious though, like averaging a long column of numbers, or if I need a precise answer quickly on something like how many bytes are in 1.25 base-10 gigabits — I can do the billion divided by 8 and come out with 125 million bytes per base-10 gigabit, and then multiplying by 1.25 I know I’m going to be in the neighborhood of 150 million bytes, but I need the calculator to quickly get the exact answer of 156250000 bytes.  If I’m on a conference call about that kind of thing I’ll use the calculator more than otherwise.Google introduced a new feature a couple of years ago that will do basic math and unit conversions for you, so if I’m deep into things or just feeling lazy I can also just pull up a web browser and type “1.25 gigabits in bytes” in the search bar, and Google does it all for me.  But recently I noticed I was reaching for the calculator more, and arithmetic in my head was getting harder, so I’ve been making a conscious effort to do more head-math lately.

How do you think math helps you do your job better?

Without math, I couldn’t do my job at all 🙂 Even so little a thing as figuring out how long a file will take to transfer takes a good head for numbers.  As soon as you dig under the surface of the operating system, it’s math everywhere.

How comfortable with math do you feel? Does this math feel different to you ?

I’m pretty comfortable with math.  A lot of my off-time hobbies touch on computers too so it’s a lot of the same math as work even when I’m not working.

What kind of math did you take in high school? Did you like it/feel like you were good at it?

I took the standard track for an Advanced Studies diploma from grades 8-11 (Algebra I, Geometry, Algebra II, Advanced Math), plus AP Calculus my senior year, and always did well. I didn’t expect to like Geometry going in because it’s not one-right-answer like a lot of math, but I ended up enjoying the logical rigor of proofs.  (Though I do recall giving my Geometry teacher fits on occasion when my proofs took a non-standard tack…)

Did you have to learn new skills in order to do the math you use in your job? Or was it something that you could pickup using the skills you learned in school?

Most of it was learned in school, although base conversion isn’t something we spent a lot of time on.  I got good at it through long, frequent practice as you might guess…

Do you have a question for Joe? Send me your question and I will forward it to him.

Photo Credit: Dan Hamp via Compfight cc

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Pre-Algebra on Facebook: How Mark Zuckerberg helped a frustrated parent

Wednesday on Facebook, I had the most amazing experience. Suffering from an all-day migraine, I had spent the afternoon bored out of my mind, obsessively checking Facebook while the television droned in the background. At one point, this status update from my friend Alyson appeared in my feed:

ALGEBRAAAAAAAAAAAAAAAAAAA!!! (Shaking fist angrily in air at math gods)

I was Batman and here was the bat signal. How could I help?

The first response was from someone I didn’t know and very typical: “Outside of college, you don’t really need it, right?” I rolled my eyes inwardly and thought about why Alyson might need to solve an algebra problem. Then I remembered her incredibly bright son, who is completely enamored with computers. I mean in love with the machines. I’d bet my last dollar that the boy will find himself programming or engineering or something in STEM as an adult. In other words, he would need algebra.

I posted a few questions to see how I could help, and eventually Alyson posted the original equation to solve:

algebra fractions equation
fraction algebra
fraction algebra

Whew! It is a doozy, right? Alyson had one very specific question: how to handle the last term of the equation: . I told her the simple answer — that it was the same thing as . Still a teacher at heart, I wanted to see what she could do with that information. Was it enough to help her solve the problem?

Meanwhile lots of other people were chiming in, and Alyson was expressing lots of feelings:

And just so everyone knows, I suck at fractions. Always have, always will. When I took SAT and ACT and whatever else, I literally turned all fractions into decimals because I can never remember how to add, subtract, divide, multiply, etc. fractions.

I’m close to crying…I still don’t understand what you’re saying. He worked the whole thing out at got what my online algebra check thing says is a wrong answer, and I’m trying to work it out so I can figure out how to get the RIGHT answer and I really do think I’m going to cry…

Frustration cry. Because I didn’t think I’d ever use math. And I was wrong. For the record. Sorry, Mrs. Blankenship.

This is a super smart lady. She edits college-level courses of all kinds, and she’s had a successful freelance writing career for many years. And I can completely identify with her frustration. I’d been struggling with Venn diagrams and conditional statements all day. No wonder I had a migrane.

But then something really amazing happened. Really amazing. A mutual facebook friend and writer, Jody (owner of Charlotte on the Cheap) tagged us both in her status update:

Do I have it right? Do I?

She had attached this photo:

At 6:15 on a Wednesday evening, she had not only worked out a challenging pre-algebra problem but also taken the time to scan it and post on Facebook. She was so excited. And, yes, she had gotten the correct answer.

She had also done it differently than I did. But that’s not even the best part. Alyson saw Jody’s process and looked carefully — very carefully. She posted this:

I worked through it on my own twice using your strategy, which ended up making a lot of sense to me once I talked it out a few times. So now I can explain it to [my son] and actually have a clue what I’m talking about. THANK YOU.

Within an hour, another of Alyson’s friends had posted one more way to do the problem. It was a smorgasbord of solutions!

But here’s the very best part: with all of these threads, there were very few people chiming in to say that they were too dumb to help or “who cares?” In fact, I saw many more people posting things like this:

This I can do. Proof reading for grammar errors…….not so much!

I will be glad to do some algebra when the time comes.

 I love math, call me, text me pictures!!!! I will PM you my number.

Why WHY WHY are you having an algebra party without ME?! I love me some equations!

It wasn’t a complete love-fest, but it was worlds different than I’m used to seeing. The tenor of the discussion was supportive and positive, rather than defeated. Sure, there’s was lots of frustration. And I’m betting that there were lots of people reading the threads and thinking, “Good god, I’m going to be in BIG trouble when my kid takes algebra.” But what played out in the end was a good experience — not just getting the right answer but learning different ways to approach the problem.

I originally became a math teacher because I was convinced of two things: math is important and anyone can do math. For years, I’ve felt pretty alone in those two estimations — especially after leaving the classroom. Yet, here was a community of people who were working from the same premise, encouraging Alyson and excitedly trying out the problem themselves.

I can’t think of a better way to end Back-to-School month at Math for Grownups. If you parents can express this enthusiasm — along with your frustration, if you have any — you’ll be doing your kids a big favor. It’s the pushing through and looking for ways to understand things differently that makes a difference. Imagine how much more empowered and confident our kids will feel if they get the message that math is important and that they can do it.

What positive messages about math have you seen lately? Have you found ways to be more encouraging about math with your own kids? Share your thoughts in the comments section.

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Forget the Ping Pong — Let’s Do Some Math

Today’s guest post is from Laura Overdeck, the ingenious creator of Bedtime Math. Don’t know what that is? Keep reading. And then go sign up for their daily email. You won’t regret it!

As kids go back to school, it’s natural for parents to look at their bright-eyed offspring and wonder what they’ll go on to do in life. Chances are they’re hoping their kids will major in something substantial in college to lay the foundation for a great career. Unfortunately, that isn’t what we see happening: According to the National Center for Education Statistics, today more U.S. undergraduates are majoring in “leisure and recreation studies” than in all physical sciences combined (chemistry, physics, astronomy…you get the idea). And there are more than twice as many majoring in leisure studies as in math.

Why does this happen? Why is our next generation running scared from the subjects that involve math? Can we make math the fun leisure activity that kids flock to do?

The problem is that our culture doesn’t view math the way we view reading. Everyone knows to read to children from a young age, and most educated parents do. By the time kids get to kindergarten, even if they can’t read yet, they probably have a warm, cozy feeling about books. To them, reading is a leisure activity, an activity you do for fun when you have free time. Math, by contrast, doesn’t get the same warm, fuzzy introduction as the bedtime story. While there are plenty of magnetic number sets and 1-10 counting puzzles, for most families math at home stops as soon as the children reach toddlerdom. As a result, for a lot of kids their first real experience with math happens in school, with all the associations of homework, drilling and tests. That’s not going to make it feel like leisure. It feels like a chore.

It’s no wonder, then, that kids start off on the wrong foot with their relationship with math.

Kids should view math as fondly as they think about storybooks. While your child plugs away at the usually dry math material from school, it’s good to counter that with fun math at home as an antidote. By the way, that doesn’t mean forced, contrived set-ups where your child can see right through to your intentions. Luckily, there are plenty of fun activities ripe with “stealth” math that may be part of your day already:

  • Baking: Doubling recipes requires multiplying; cutting in half requires dividing; measuring 1/4-cups or 1/4-tsps uses fractions. Any time you bake, you’re quantifying ratios to make magic in the oven. Toddlers can participate by counting out chocolate chips, and of course eating them as a bonus.
  • Building: Anything that involves measuring gets kids counting, adding, and multiplying. Lego and other building toys revolve around numbers, too. And we all see how kids can entertain themselves with a bunch of cardboard boxes, especially if told they’re off limits. Cutting up pieces to assemble a fort all revolves around measurement.
  • Planning: For example, setting up party favors. They’re all sold in different quantities: 10 in one pack, 24 in another, 18 in a third. If there are 16 kids coming over, how many packs of each do you need, and what’s left over? Even putting out breakfast or dinner takes some planning and counting.
  • Sports and exercise: Kids love stopwatches, and watching the seconds tick off gives kids great exposure to counting. Distances and heights require measurement, and even counting jumping jacks, baseball swings, or the kicks to get a soccer ball to the end of a field, can involve numbers.

All of these activities contain a ton of math as a natural part of the process, and this list is just a start. When kids get absorbed in a favorite activity, they don’t even notice they’re learning, just like when they read a bedtime story.

To that point, math can become a part of bedtime as well. Six months ago I founded Bedtime Math, a free website that offers a fun nightly math problem every night – about electric eels, chocolate chips, zip lines into the neighbor’s pool.  Again, the idea is to take kids’ absolute favorite topics and sneak some math in there. After just six months we have over 20,000 people following us through the daily email, the website or Facebook, and parents have written that their previously math-resistant children now ask for Bedtime Math at night, thanks to this new spin on numbers.

Again, it’s all about catching kids while they still think numbers are fun, and building on that mindset. By making math a fun part of favorite natural routines, kids will think of numbers as recreation instead of compulsory drudgery. When they enter school with that new world view, they will have an entirely different, incredibly positive experience with math at school. And with that foundation, maybe they won’t have to sink to majoring in leisure studies when they grow up.  They’ll major in math instead – for fun.

Laura, you’re singing my song! The beauty of Bedtime Math is that it offers three levels of difficulty. Parents don’t have to try to figure out what they should expect their toddler to do or how difficult the math should be for their 2nd grader. And I guarantee — GUARANTEE! — that if you give this a try, you’ll find yourself injecting math all over the darned place.

So what do you think about these sneaky math ideas? Are you ready to throw away the worksheets and flash cards? Have you figured out some easy ways to encourage your kids to do math — without their even knowing it? Share your ideas in the comments section.

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More Sneaky Math for Kids: Guest post from Erin Flynn Jay

When you’re balancing work and home and all sorts of other responsibilities, it can be downright overwhelming to consider doing everyday math with them, so they can perform well in school. That’s why I invited author Erin Flynn Jay to guest post on this very topic. Her recently published book,  Mastering the Mommy Track, tackles many of the questions all of us working parents have, and today she addresses math.

How do your kids do with their math homework? Is it a struggle to get them to concentrate, or do they have a good handle on calculations?

I grew up watching my mother tutor grammar school kids in math at our home. She was also a substitute math teacher at our local schools when I was in grammar school and high school. Because I am not a math whiz, I asked for her viewpoint on this blog post (she read this and offered a critique).

Kids need examples, which will allow them to understand numbers better.

— Beginning when they are toddlers, help them count their snack food like Cheerios or Goldfish from one to ten. It’s wise to get them counting at the earliest age possible.

— When you take your kids grocery shopping, explain to them what your purchases cost. If they are learning how to add numbers in school, ask them for a total. You could ask them, “Okay we have this corn which is $2, chicken for $8 and lemonade for $2. How much money do I need? What is 2 plus 8 plus 2?”

— Give your kids a weekly or monthly allowance depending on your budget. Take them to the pizza place or movie theater and ask them to pay for their purchase themselves. This way, they can understand the value of a dollar or 50 cents more easily. They can get change back and will get a better grasp of what their favorite items actually cost.

— Finally, teach them how to measure their TV shows in 30 minute intervals. For example, you could say, “Alright, you can watch your show for 15 minutes before bedtime.” When the time is up, let them know 15 minutes has passed and it’s time to pack it in.

One final suggestion is to check with your local librarian or bookstore–get recommendations for age appropriate math books. Read them one math book per night.

When your kids sit down to do their math homework, they will perform better if you have introduced basic math concepts at the preschool age. Repetition will reap results.

Erin Flynn Jay is a writer and publicity expert, with articles appearing in a diverse list of publications, including careerbuilder.com, MSN Careers and Wealth Managers. Order her book Mastering the Mommy Track at Amazon.com or barnesandnoble.com. 

Here at Math for Grownups, you’ve gotten a lot of ideas on how to sneak math into your kids’ everyday lives — from reading time to when you’re on the road. What suggestions have you tried? How have they worked out? I’d love to hear about your successes (and yes, failures)! 

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The Problems with PEMDAS (and a solution)

If you’re on Facebook, you’ve probably seen one of a variety of graphics like the one above.

The idea is to solve the problem and then post your answer. From what I’ve observed, about half of the respondents get the answer correct, while the other half comes to the wrong answer. The root of this problem? The order of operations.

Unlike reading English, arithmetic is not performed from left to right. There is a particular order in which the addition, subtraction, multiplication, and division (not to mention parentheses and exponents) must be done. And for most of us old-timers, that order is represented by the acronym PEMDAS (or its variations).

P – parentheses
E – exponents
M – multiplication
D – division
A – addition
S – subtraction

I learned the mnemonic “Please Excuse My Dear Aunt Sally” to help me remember the order of operations.

The idea is simple: to solve an arithmetic problem (or simplify an algebraic expression), you address any operations inside parentheses (or brackets) first. Then exponents, then multiplication and/or division and finally addition and/or subtraction.


But there really are a lot of problems with this process. First off, because multiplication and division are inverses (they undo one another), it’s perfectly legal to divide before you multiply. The same thing goes for addition and subtraction. That means that PEMDAS, PEDMSA, and PEMDSA are also acceptable acronyms. (Not so black and white anymore, eh?)

Second, there are times when parentheses are implied. Take a look:

If you’re taking PEMDAS literally, you might be tempted to divide 6 by 3 and then 2 by 1 before adding.

Problem is, there are parentheses implied, simply because the problem includes the addition in the numerator (top) and denominator (bottom) of the fraction. The correct way to solve this problem is this:

So in the end, PEMDAS may cause more confusion. Of course, as long-time Math for Grownups readers should know, there is more than one way to skin a math problem. Okay, okay. That doesn’t mean there is more than one order of operations. BUT really smart math educators have come up with a new way of teaching the order of operations. It’s called the Boss Triangle or the hierarchy-of-operations triangle. (Boss triangle is so much more catchy!)

The idea is simple: exponents (powers) are the boss of multiplication, division, addition, and subtraction. Multiplication and division are the bosses of addition and subtraction. The boss always goes first. But since multiplication and division are grouped (as are addition and subtraction), those operations have equal power. So either of the pair can go first.

So what about parentheses (or brackets)? Take a close look at what is represented in the triangle. If you noticed that it’s only operations, give yourself a gold star. Parentheses are not operations, but they are containers for operations. Take a look at the following:

Do you really have to do what’s in the parentheses first? Or will you get the same answer if you find 3 x 2 first? The parentheses aren’t really about the order. They’re about grouping. You don’t want to find 4 + 3, in this case, because 4 is part of the grouping (7 – 1 x 4).  (Don’t believe me? Try doing the operations in this problem in a different order. Because of where the parentheses are placed, you’re bound to get the correct answer more than once.)

And there you have it — the Boss Triangle and a new way to think of the order of operations. There are many different reasons this new process may be easier for some children. Here are just a few:

1. Visually inclined students have a tool that suits their learning style.

2. Students begin to associate what I call the “couple operations” and what real math teachers call “inverse operations”: multiplication and division and addition and subtraction. This helps considerably when students begin adding and subtracting integers (positive and negative numbers) later on.

3. Pointing out that couple operations (x and ÷, + and -) have equal power allows students much more flexibility in computing complex calculations and simplifying algebraic expressions.

Even better, knowing about the Boss Triangle can help parents better understand their own child’s math assignments — especially if they’re not depending on PEMDAS.

So what do you think? Does the Boss Triangle make sense to you? Or do you prefer PEMDAS? What to learn to solve these and other problems, buy the book that will help grown-ups like you with these and other math problems here.

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3 Ways to Turn Reading Time Into Math Time (Psst… You may be doing these already!)

Yesterday, Karma Wilson stopped by to describe how she uses math in her work as a children’s book author. She’s the author of such wonderful best sellers as Bear Snores On and Bear Says Thanks. And she got me thinking about the days when I used to read her (and other’s) books to my daughter (who is almost a teenager now!).

I guess because I like math and learned to incorporate numbers and math concepts into my everyday life pretty quickly, I naturally looked for the math hidden within each book I read to my daughter. But after having lots of conversations with other parents, I’m realizing that many don’t see the connection between literacy and numeracy. (Remember, numeracy is to math as literacy is to reading.) So I thought I’d share my ideas today in this special post.

It’s not actually very difficult to bring math into almost any picture book. But perhaps parents think they should shoot for the stars — asking kids to add, subtract, multiply or divide. Unless that’s part of the plot, I’m guessing such ideas come off a bit forced. Or they may just suck all of the fun out of sitting down and reading with your child. So try these suggestions on for size.

Count

In Bear Snores On (illustrated by Jane Chapman), a hibernating bear is visited by a variety of forest animals who host a mid-winter party in his cave. The easy thing to do here is to count the animals. How many are there by the middle of the book? How many are there by the end of the book? When you count, point at each animal and say the number out loud. As your child grows, ask him to point and count.

This process is a really easy way to reinforce something called one-to-one correspondence (or as we math geeks write, 1:1 correspondence). It’s so stupidly easy, that it’s hard to imagine kids aren’t just born knowing this. But the idea is that each animal in the book corresponds to a specific number. If you count the animals in the order that they appear in the book, the mouse will always be 1 and the raven will always be 7.

Addition

In Karma’s A Frog in the Bog (illustrated by Joan Rankin), the protagonist (yes, a frog in a bog) makes a very large meal of a variety of different critters in a variety of different numbers.  There is one tick, two fleas, etc. This is the perfect time to prompt some addition. After the frog eats the tick and both fleas, how many critters are there in all? Count them in the picture, again by pointing to each one and saying the numbers out loud.

Tiny ones won’t necessarily learn to add simply by reading this book, but asking these kinds of questions can help lay the groundwork for an idea called counting up. In this process, when kids are adding two numbers, they start with one and count up using the second number. So 2 + 3 becomes 2, 3, 4, 5 — visually counting from 2 to 5 using fingers or something else concrete.

Proportions

Whopper Cake (illustrated by Will Hillenbrand) features a grandpa who is making a mighty big cake for Grandma’s birthday. From the title, we adults can predict that this cake is going to be huge, but your little one may not have those literary skills down pat. Guide him to that place by pointing out the sizes of things. The recipe calls for two cups of sugar, but Grandpa puts in 23 pounds of the sweet stuff. “Is that more or less sugar?” you can ask. Same with the eggs, which are increased from four to 84. Again, this may seem really basic, but young children (preschool and kindergarteners) are just learning about size comparisons. Until we teach them about bigger and smaller, they don’t have a clue.

Bonus: Have your child play with measurements and follow an algorithm (or a set of steps) by making a Whopper Cake from the recipe at the end of the book.

So, there you have it — a few ways to turn reading time into math time. These ideas don’t look so hard, do they? They should seem pretty darned simple. And they pay off, big time. Not only will your child be learning some basics about the foundation of numeracy and mathematics — he’ll likely pick up the message that math is everywhere and even a little bit of fun. Not even Bear should sleep through that!

What are ways you’ve incorporated math into your kids’ nightly reading? Share your ideas in the comments section.

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Math Homework: 5 important ways to help your kid

By 8:30 on Tuesday night, I was ready to go home and curl up with a good book. But there I was, crammed into a windowless computer lab with 25 other exhausted parents, listening to the new math teacher describe how math instruction would work this year.

He described how the Common Core standards will change math education and showed off the fancy online curriculum that our school is lucky to have. Then he asked for questions — and the parents pounced. Poor guy.

See, this fellow is exactly what students need. He’s tough; he’s smart; and he thoroughly understands a critical element of mathematics education: Kids have got to take risks that might not lead to a solution. Just like Sir Isaac Newton and Albert Einstein and Ada Byron Lovelace (yes, she’s Lord Byron’s daughter and the founder of scientific computing) went down long and winding roads to their discoveries, our kids must do the same.

But the parents were having none of that. The homework that Mr. T is sending home each night is really challenging. Really challenging. My daughter was complaining and crying and slamming doors because of it. And I know we weren’t alone in our little nightly soap opera.

As the parents got more frustrated and asked more and more questions about grading and building confidence and avoiding stress, I realized that they were missing the whole point. As parents, it’s not our job to shelter our kids from struggle and frustration. I was having a really hard time resisting the urge to step up to the front of the room and do some damage control.

So I figured I should take this opportunity to share my ideas here. Fact is, Common Core may mean that your child is more frustrated. But there are ways to cope.

1. Get proactive

What do the Common Core objectives say? Well, they’re no big secret. Check out this grade-by-grade list. I want you to notice something really, really important: the list of concepts your child is expected to grasp by the end of the year is pretty darned short. At the same time, these ideas are pretty robust. The objectives cover less material and fewer facts, but they do so more deeply.

Armed with some information about Common Core, you will be better able to set the parameters around what your child is learning at home. If solving for x is not on that list, don’t expect your child to do it. But if ratios are, it could be helpful for you to brush up on those concepts. (See Wednesday’s post for help on this.) But not so you can walk your child through a process. (Keep reading for more info on that.)

2. Meet the teacher

And at this meeting, don’t get hung up on grades and tests. Ask her what her teaching philosophy is. Ask what she wants you to do to help support your child’s learning. It is very possible that you’re making assumptions about your role. Depending on your child’s age, you might need to offer a great deal of help. Or you might need to back off. Your child’s teacher can tell you for sure.

If your child has math anxiety, this is a great time to share that with the teacher. Sometimes even the best teachers inadvertently send messages to their kids that unnecessarily ups the anxiety. (Some struggle is good; too much can shut down the pathways of critical thinking.) Offering the teacher a little background in your kids’ previous math experiences can be really useful.

3. Trust

This is probably the hardest step, but unless you have really good reason not to, you must trust your child’s teacher. Seriously. In my observation, many parents think they understand everything about teaching, simply because they were once students.* That approach undermines teachers’ authority and ignores their education and expertise. It’s actually pretty insulting in some ways. Just because you can flush a toilet doesn’t mean you are a plumber. The same goes for teaching.

Teachers are not just experts in their field of study (math, Spanish, English, science); they’re experts in pedagogy, which is the practice of teaching. And pedagogy is much more mysterious than trigonometry or set theory.  It’s where the science and art of teaching collide. The way in which topics are introduced and explored in the classroom is a careful dance. Sadly, some of this can be undone at home, during the homework wars.

Unless you believe your child’s teacher is downright incompetent, you’ve got to trust that she knows what she’s doing. Chances are, there’s very good reason she sent home those challenging problems.

*This goes for homeschooling parents, too. Anyone who has been successful with homeschooling will tell you that there’s a lot to learn about pedagogy — from the developmentally appropriate times to introduce certain concepts to proven ways to encourage exploration and discovery.

4. Stop spoonfeeding

Especially when kids enter middle school, we parents need to back off — big time. Yes, we want them to succeed. But what may be even more important is this lesson: failure is a part of learning.

I don’t mean that you should be okay with a failing grade or ignore his bellowing, “I DON’T UNDERSTAND!!’ But at some point (very soon!), you must stop checking his assignments or walking him through each and every problem. You also need to endure his frustration. When children make mistakes, they can learn from them. When they struggle, they learn they can overcome adversity.When you swoop in to rescue your child from struggle and frustration, you are actually depriving him of these important lessons. (If you want your kid to live in your basement, rent-free, after graduation, ignore the above.)

Check with your child’s teacher about the grading process for homework. Will he be expected to get the answers right? Or is the teacher merely expecting an honest effort? If effort is the main theme (and I hope it is!), quit trying to explain to your child how to do the work. Instead, offer support and encouragement. If you believe your child can succeed, he’ll believe it too.

5. Get curious

One of the best ways to get involved with your child’s education is to ask questions. Kids are rarely given an opportunity to verbalize what they understand about math. Curiosity is a is a huge gift you can give your kids. But in case you’re stuck, try keeping these questions in your back pocket for stressful times:

— What do you know about the problem? (Encourages your child to think critically about the information included in the problem.)

— What are you being asked to do? (Prompts the child to identify the question in the problem.)

— If your math teacher were here, what would she say? (Demonstrates an alliance between yourself and the teacher, and gives you information about her expectations.)

— What ideas do you have for solving the problem? (Helps students identify problem-solving techniques, like making a list, guess-and-check, drawing a picture, etc.)

Read through the above questions again. What do you notice about them? Yep, not a one of them has anything to do with getting the answer. Not a single one. And that’s because it’s not your job to find the answer. Your job is to help your child move towards an answer, not solve the problem for him.And with questions like these, you’re helping your child see math as a process, not merely a solution.

This is hard work. Even with my background in math education, I’m struggling with homework histrionics. It is no fun to come home from a long day at work, only to be pulled into an emotional tangle over math. But I will guarantee this: If you’re working with a good teacher and you practice the steps above, your child will learn to feel very confident in his math skills. And he’ll be a much better grownup for it.

What do you think about this advice? Which of these steps are you already practicing? Which do you think are challenging to implement? Are there any that you think are downright wrong? Share your feedback in the comments below. And if you have further questions, ask them!

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Zombie Math!

It’s a special Thursday edition of Math for Grownups, and today we have a guest post from Bon Crowder of Math Four, a great blog that was featured in the Wall Street Journal last month. Here’s her creative take on math and zombies.

Math isn’t only in real life, it’s in our fantasy and fear worlds too!

I’m a huge fan of The Walking Dead – the popular zombie show on AMC that returns October 14.

While watching season two, I started pondering if our heroes even had a chance against the hoards of zombies.

I realized that regardless of how science explains the start of a zombie epidemic, the way it continues and the way to stop it is explained with math.

Zombies make other zombies.

If you’re bitten or otherwise infected by a zombie, you’ll turn into a zombie yourself. Since zombies never sleep, and are constantly on the lookout for human victims, they have the ability to create many more zombies very quickly.

Killing zombies is a chore.

According to The Walking Dead the only way to kill a zombie is by ceasing brain function. In other words: removing or impaling the brain.

That sounds simple enough. But just watch one episode and you’ll see how challenging it can be!

Do we have a chance against a zombie epidemic?

There are around 30 humans in The Walking Dead. Let’s assume that represents reality: there are only 30 humans on the entire planet, and the rest of them have been turned to zombies (or will be soon).

We can do a little math to figure out how long it would take for our 30 heroes to rid the world of this epidemic.

There are a little over 7 billion people in the world. That’s 7,000,000,000. (A whole bunch of zeros, I know.)

Each hero will be responsible for killing about 230 million zombies. That’s 7,000,000,000 zombies ÷ 30 heroes.

(Notice I’m rounding like crazy – a fun thing to do when estimating anything. Including zombie deaths.)

Suppose now that each of our heroes could be expected to live 60 more years.

60 years • 365 days = about 22,000 days of life left.

We can calculate how many zombies each hero must kill per day:

230,000,000 zombies ÷ 22,000 days = over 10,000 zombies each day!

Um… we only have 24 hours in a day. That’s 1,440 minutes or 86,400 seconds. So each hero has to kill one zombie every 8 seconds.

Yipes!

Check out Bon’s blog at MathFour.com. 

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5 Math Resources for Confused Parents

So your kid needs some help with her math homework. Do you understand what she’s doing? Chances are, it’s not so cut and dry these days — and not because you don’t remember your middle school math lessons. Two things are going on in math ed: 1) concepts and processes are being taught differently, and 2) kids are getting more complex lessons earlier on.

All of this may leave you feeling completely helpless.

Luckily, there are some great resources out there that are there just to help you. Here are my top five.

Your child’s teacher

This is a really obvious idea, but not everyone thinks of it right away. Or maybe, like a lot of parents, you feel intimidated by the teacher or you don’t know how to ask for help. There are exceptions to the rule, but most teachers are eager to speak with parents, not only about their kids’ progress but about the best ways to help their child succeed. Find out how he or she prefers to communicate — email, phone or in person. Then use that resource as much as you possibly can.

Online textbook resources

Do you know what curriculum your child is using in math class? If not, find out, because today publishers are putting a wide-range of resources online — just for parents. This is especially true for discovery-based math programs, like Everyday Math and Investigations. The publishers of these programs know that they’re challenging for parents to grasp (since we learned very different ways of doing the math), so they’ve included very strong parent components.

math.com

This really simple website offers quick reviews of basic math ideas. Forgotten what a GCF is? You can find out here. Don’t remember how to solve for x in a proportion? This is a great place to start. Math.com also includes lists of formulas and some basic online tools, like a scientific calculator.

Purplemath

Focused entirely on algebra, purplemath is where you can find help with solving quadratic equationsor graphing linear equalities. Each concept includes a detailed lesson that walks you through the process and examples. Believe me, it’s been an invaluable tool for my addled brain!

The Math Forum

Ask Dr. Math has been around since 1992, so the site has amassed a wealth of questions from math students and answers from real-live math professors. Because it is generally focused on pedagogy (the concepts behind teaching mathematics) and higher-level math, it may seem a bit overwhelming. But if you search the archive, it is likely someone has asked the very question you have. You can also submit your own questions. But don’t expect an immediate response. This site is not designed for quick, individual feedback.

So there you have it, my top five resources for parents with math questions. Got any others to share? If so please include them in the comments section. Sometimes we need all the help we can get!