mental math


Today’s interview is with Mina Greenfield.  She has been a speech-language pathologist for sixteen years.  I enjoyed hearing not only about the math involved in her job but also about her work with children on the autism spectrum.  People like Mina are becoming needed more and more as autism is on the rise. I’m so thankful that she has dedicated herself to this important job.

Can you explain what you do for a living?

I am a clinician in a private school for students on the autism spectrum. I work on interdisciplinary teams that include classroom teachers, teaching assistants, occupational therapists, and social workers. When most people think of a “speech therapist”, they think of kids that can’t say their R’s or S’s. However, my work takes a broader look at communication. Can they understand what they hear or read? Can they express their ideas? And can they use language to communicate effectively with others?

When do you use basic math in your job?

I use basic math in my job to calculate my billable hours (each 15 minute segment counts as a unit) and to compare my “scheduled vs. actual” therapy time for the week (i.e. I was scheduled to do 23.5 hours of therapy time, but a kid was absent so my actual time was 22.5). I also use math when scoring standardized tests and interpreting test scores on incoming reports. When looking at standardized tests, usually the mean =100 and the standard deviation (SD) is 15. Therefore scores between 85 and 115 are considered to be within the average range. If I read a report on a new kiddo and I see language scores that are in the 60’s or 70’s (or lower), I will be keeping a close clinical eye on him. Percentile ranks also make frequent appearances in assessments.

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

I use a widget calculator on my desktop for daily and weekly billable hours. I’ve always been good at mental math so it makes that process much quicker. When scoring standardized tests, there’s a lot of basic addition to determine a raw score, but then you use the manual to look up corresponding scores which does not require math.

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

My ability to do mental math makes my job much quicker which I suppose makes me do my job more efficiently (better). I’ve been in the field long enough that I don’t have to “think” about standardized scores and what they mean. If I see a certain number, I know it indicates a certain strength or deficit.

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

For my purposes, I feel comfortable with math all of the time. Again, I’m very thankful I’m good at mental math.

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

I took them all…Geometry, Trigonometry, Calculus, and AP Calculus. I also took statistics in college.

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 pick up using the skills you learned in school?

I picked up the math at my current job pretty quickly. I think compared to other professions, it’s “basic” math. (maybe?)

Questions for Mina?  Let me know, and I’ll pass them on.

Photo Credit: fotoroto via Compfight cc

Last week, we had some fun with the order of operations at the Math for Grownups facebook page.* Turns out remembering the order that you should multiply, add, etc. in a math problem is a tough thing for adults to remember. Imagine how kids feel! But this is a really simply thing that you can apply to your everyday life — all the while, reminding your kid how it goes.

First off, here’s the problem that we considered on facebook last week:2 • 3 + 2 • 5 – 2 = ?The answer choices were 38 and 14.I would say that the responses split pretty evenly. Lots of folks chose the incorrect answer first and then realized their mistakes.

So what’s the correct answer? 14. Why? Because of the order of operations. A lot of us learned the order of operations — or the set of rules that establishes the order we add, subtract, multiply, divide, etc. — with a simple mnemonic:Please Excuse My Dear Aunt SallyORParentheses, Exponents, Multiplication, Division, Addition, SubtractionORPEMDAS

(Before going further, I must acknowledge that there are some problems with this approach. First off, it doesn’t really matter if you add before your subtract or multiply before you divide. Those operations can be done in either order with no problem. Second, many teachers are approaching this differently, a topic that I’ll explore in September.)

If you do the operations in the wrong order — add before you multiply, for example — you’ll get the wrong answer. And that’s how people got 38, instead of 14. They simply did the math from left to right, without regard to the operations.CORRECT2 • 3 + 2 • 5 – 2 = ?6 + 2 • 5 – 2 = ?6 + 10 – 2 = ?16 – 2 = 14INCORRECT2 • 3 + 2 • 5 – 2 = ?6 + 2 • 5 – 2 = ?8 • 5 – 2 = ?40 – 2 = 38

All of this is well and good, but what does it have to do with the real world? How often are you faced with finding an answer to a problem like the one above? And that’s exactly what one reader asked me. So I promised to explain things using a real-world problem.

Thing is, you do these kinds of problems all day long, without even thinking of the order of operations. And that’s because you’re not writing out equations to solve problems. You’re simply using good old common sense.

Let’s say you’re going back-to-school shopping with your child. He’s chosen a pair of pants that are $15 and five uniform shirts that cost $12 each. But the pants are $5 off. What’s the total (without tax)?

You probably won’t write an equation out for this, right? (I wouldn’t.) Instead, you’d probably just do the math in your head or scribble some of the calculations on a scrap piece of paper or use your calculator. So here goes:

First the shirts: there are five of them at $12 each. That’s a total of $60, because 5 • 12 is 60.

Now for the pants: all you need to do here is subtract: 15 – 5 = 10. The pants total $10.

Finally, add the cost of the pants and the cost of the shirts: $10 + $60 = $70.

The above should have been super easy for most of us. And — surprise! surprise! — it used the order of operations. Here’s how:15 – 5 + 5 • 12 = ?The order of operations says you must multiply before you can add:

15 – 5 + 60 = ?

Then you can add and subtract:

10 + 60 = 70

There are other ways to set up this equation. In fact, I would use parentheses, simply because I want to keep the pants’ and shirts’ calculations separate in my mind:

(15 – 5) + (5 • 10) = ?

The result is the same, because the process follows the order of operations — do what’s inside the parentheses first and then add.

UPDATE: A reader asked if I’d also show how this problem can be done wrong. So here goes! When you do the operations in the wrong order, you won’t get $70.15 – 5 + 5 • 12 = ?10 + 5 • 12 = ?

15 • 12 = 180

That’s more than twice as much as the actual total!

Try this with your kid. You can make it more complex by figuring out the tax. And there are lots of different settings in which this works — from shopping to figuring the tip in a restaurant and then splitting the tab to dividing up plants in the garden.  Just about any complex math problem that involves different operations requires PEMDAS. And that’s something all kids need to know about.

When have you used PEMDAS in your everyday life? Did this example spark some ideas? Think about the math that you did yesterday — or today — and share your examples in the comments section.

*Have you liked the Math for Grownups facebook page yet? What’s stopping you? We’re having great conversations about the math in our everyday lives. And I ask questions of my dear readers. Come answer them!

When I started doing Math at Work Monday interviews, I thought of it as a little experiment. Would people I talk to actually recognize the math they do? Would they feel confident in their math skills? Would the the math they need to succeed in their careers get in the way? I had a theory: Most people don’t realize that they’re doing much of the math they need for an average day.

Now that I’ve got about a year of Math at Work Monday interviews under my belt, it’s a great time to take a closer look. Did my hypothesis stand up? Reading through these interviews again, I’ve noticed five interesting themes.

1. Everyone does math in their jobs. Okay, that’s a duh conclusion, right? But when you consider the number of school kids who ask, “When will I ever use this stuff?” it’s not necessarily a foregone conclusion. In other words, if kids think that by avoiding science, they’ll avoid math in their careers, they should think again.

Kiki Weingarten, a NYC-based executive, corporate and career coach uses math to help her clients understand the financial implications of a career change. Criminal profiler Mary Ellen O’Toole looked for patterns and used statistical analysis to help solve crimes.

2. Many folks don’t know that they’re doing so much math — until someone asks them about it. This has come up over and over again. I’ll ask someone to do a Math at Work Monday interview with me, and they’ll say, “Why would you want to talk to me? I don’t use math in my job.” But once they think about it — even a little — many of them are surprised by the sheer number of numbers in their jobs. From managing their business to practicing their passion, math is everywhere.

Painter Samantha Hand said that she didn’t realize how much math she uses, until we talked about it — then she started making big connections, including using proportions to help paint to scale. When I asked my sister, Melissa Zacharias to participate, she first said that she didn’t really use math. She soon discovered plenty of places that math is useful in her job as a speech therapist who works with adults.

3. Math is particularly prevalent in the visual arts. So much for the myth that people are either artistic or mathematically minded! In fact, math is required in a variety of different aspects of art, from working with materials to managing sales to envisioning the final design. That’s one of the reasons that I devoted an entire month to math in the arts. (And we didn’t even scratch the surface!)

From noted jewelry artist Shana Kroiz to glass artists Ursula Marcum and Beth Perkins, it became clear that the connection between math and art is undeniable. Even museum curator Ann Shafer uses math.

4. Using math tools is fine, but many people depend on their brains. I expected people to tell me that they depended heavily on computers or calculators to do the math they needed. But most folks admitted that they do a heck of a lot of mental math — from basic addition to finding percents.

Kim Hooper uses a calculator to check some figures, but as a copywriter, she also does “margin math,” a grownup version of showing her work. Executive vice president, Gina Foringer uses mental math to quote labor percents for new contracting jobs.

5. People generally like the math they do at work. Of course they don’t always think of the math they do as math (see #2), but the folks I interview feel confident in the skills they need to perform their jobs well. This includes those who say they didn’t do well in school math classes or that they feel like they’ve never really “gotten it.”

Costume designer, Katie Curry says that she doesn’t feel comfortable with math outside of the calculations she needs to draft a creative design (though she can balance her checkbook, of course). When hair stylist Nikki Verdecchia opened her salon a few years ago, she worried that the math would get in her way, but she quickly became comfortable with the calculations she needs to make her business work.

So there you have it — the unscientific results of my unscientific experiment. As I suspected, people don’t mind doing the math in their jobs, and that’s because they don’t even realize that they’re doing math. We’ll see if that trend continues in the upcoming year of Math at Work Monday interviews.

What about you? Do you like the math that you do at work? Are you now realizing that there’s more math than you originally thought? Share your ideas in the comments section.

Everybody loves a sale, right? The thrill of the hunt, the sense of accomplishment when landing a great deal.

But how many times have you reached the register and realized your purchase was more than you expected?  Or have you ever passed on a purchase because figuring out the discount was way too much trouble?

You don’t have to be afraid of the mental math that goes along with shopping.  (That goes for in-person and online sales.)  You also don’t have to be that giant geek standing in the sports goods aisle using your cellphone calculator to find 15% of $19.98.  Who has time for that anyway?

Believe it or not, figuring percents is one of the easiest mental math skills.  And it’s one of those things that you may do differently than your sister who may do differently than your boss.  In other words, you are not required to follow the rules that you learned in elementary school.  Now that you’re a grownup, you can find your own way.

Don’t follow?  Let’s look at an example.

Once again you’ve put off buying Mom’s gift.  It’s just about time to leave for her house, and you have literally minutes to find the perfect present for her — at the right price.  You’ve collected $40 from your brother and sister, and you can contribute $20.  Darn it, you’re going to scour the department store until you find something she’ll like that’s in the right price range.

And suddenly, there it is: a countertop seltzer maker, just right for Mom’s nightly sloe gin fizz. Bonus! It’s on sale — 40% off of $89.95.  But can you afford it?

There are a variety of different ways to look at this.  But first, let’s consider what you know.

The seltzer maker is regularly priced at $89.95.

It’s on sale for 40% off.

You can spend up to $60 ($40 from your sibs, plus the 20 bucks that you’re chipping in).

You don’t necessarily need to know exactly what the seltzer maker will cost.  You just need to know if you have enough money to cover the sale price.  And that means an estimate will do just fine.  In other words, finding 40% of $90 (instead of $89.95) is good enough.

Now you have some choices.  You can think of 40% in a variety of ways.

40% is close to 50%

It’s pretty easy to find 50% of $90 — just take half.

50% of $90 is $45

So, if the seltzer maker was 50% off, you could afford it, no problem.  But is 40% off enough of a discount?  You probably need to take a closer look.

40% is a multiple of 10%

It’s not difficult to find 10% of $90 either.  In fact, all you need to do is drop the zero.

10% of $90 is $9

What is 40% of $90?  Well, since 40% is a multiple of 10%:

There are 4 tens in 40 (4 · 10 = 40)


10% of $90 is $9


 4 · $9 = $36

It’s tempting to think that this is the sale price of the seltzer maker.  Not so fast!  This is what the discount would be.  To find the actual price, you need to do one more step.

$90 – $36 = $54

Looks like you can afford the machine. But there’s an even more direct way to estimate sale price.

40% off is the same as 60% of the original price

When you take 40% off, you’re left with 60%. That’s because

40% + 60% = 100%

Or if you prefer subtraction

100% – 40% = 60%

So you can estimate the sale price in one fell swoop.  Like 40%, 60% is a multiple of 10%.

There are 6 tens in 60 (6 · 10 = 60)


10% of $90 is $9


 6 · $9 = $54

The estimated sale price is $54, which is less than $60.  You snatch up the race-car red model and head for the checkout.

There are so many other ways to estimate sales prices using percents.  Do you look at these differently?  Share how you would estimate the sale price in the comments section.

Graham Laing is my brother, and I don’t think he’d be offended by my telling you that some of us in the family were a little worried that he might not amount to anything.  But that’s another story for another day.  Today, he’s a fish hatchery technician, which basically means he raises trout — “from eggs to eating size,” he says.  That means he moves truckloads of live fish from pond to pond (and raceway and stream) according to their size, and he treats them for parasites and other oogie things.  He also does a lot of weed whacking and mushroom hunting.  

You might not think that a guy who works outside all day long would use math, but Graham does.  And I think his approach is pretty unique.  As you read through this, see if you can figure out what he’s not doing.  I’ll share my thoughts at the end.

Graham wrestles a snapping turtle. Yes, this is part of his job (but I don’t think the turtle does math).

When do you use basic math in your job?

I use basic math every day. When we load the trucks in the morning, we’re told to load a certain amount of pounds of fish per tank on the truck. Since we can’t load all of the fish at one time, we’re handed a net of fish that usually weighs between 40 and 50 pounds. We have to keep track, in our heads, of how many pounds we have in each tank until it is loaded.

I also use basic math when we treat fish for parasites, using either salt or formalin. Salt baths depend on volume, so I find the volume of the tank in cubic feet and then multiply that by the number of gallons in a cubic foot–to get the total number of gallons to be treated. Then I have to multiply that by the number of pounds in a gallon of water to find the total number of pounds of water to be treated. Since we usually do a 5% salt bath, we find the number of pounds in 5% of the volume and weigh the salt.  Finally, we can mix the salt in the water.

When treating with formalin, we have to calculate a gallons-per-minute flow rate. We find this by counting the number of seconds it takes to fill a gallon and then divide that number into 60.  (There are 60 seconds in a minute.)  So if it takes 10 seconds to fill a gallon, the flow rate is 6 gallons per minute.  Since the treatment runs for an hour, I multiply by 60 and then multiply that number by 0.0036, which is the number of grams of formalin needed per gallon. Finally, I multiply by the parts-per-million needed for the treatment, which depends on the water temperature.

Do you use any technology to help with this math?  

I use calculators for sampling and for calculating the treatments. If we’re doing a lot of samples at one time, we plug the numbers into an excel spreadsheet that has the formulas we need. Calculators reduce error. One blown sample due to error could cause us to underestimate the number of fish in a raceway. Or it could cause us to underfeed a raceway, resulting in a large size-variation of the fish.

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

Graham is also a master at finding little critters like this toad. Click on the picture to see it up close. (Photo courtesy of Mary Bruce Clemons)

My whole job revolves around math.  Without math, the fish would die or become infected with parisites. We would not know how many fish we have on the farm, and we wouldn’t know if we were reaching our stocking goal set forth by the state.

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

I feel very comfortable with math and have since I was a very small child. When I got this job, I had all the skills I needed — it just took a little remembering to become adept at using them.

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

I took algebra, geometry, and trig.  I was forced to take trig, so I didn’t do so well in it. I slept through trig everyday and was still able to make 40s and 50s on the tests just by intuition.

Trust me.  If you met Graham you wouldn’t know he’s a math geek.  He doesn’t give a whit about calculus or abstract algebra or fractals.  He’s just really good at mental math.  

Here’s the interesting thing about Graham’s process: All of the math he describes above can be represented by formulas.  And when Graham uses a spreadsheet for the math, he has to use the formulas.  BUT when he uses math in the field, he unpacks each formula into a set of steps.  (First multiply, then divide, then multiply, etc.)  He doesn’t have to memorize a formula to do the work.  Instead, he thinks about the process, and he’s attached meaning to each step (“divide by 60 because there are 60 seconds in a minute”), so he doesn’t forget to do something.  This is the foundation of mental math — breaking up complicated problems into doable steps.

I’m betting many of you do the same thing.  Want to share that process in the comment section?  I sure hope you will!

And if you have questions for Graham — whether they’re about huge snapping turtles, tiny toads or wildlife management in general — post them, and I’ll be sure to get Graham to answer them.  (I am his big sister, so I can boss him around — a little bit.)

I’m on the right track, baby

I was born this way

–Lady Gaga

It was day two of my second year of teaching high school geometry, and already I had been called for a parent meeting in the principal’s office. I was a bit worried.  What on earth could a parent have issues with already?

Mrs. X sat with her 14-year-old son across the desk from the principal.  I shook her hand and took the chair next to her.  The principal handed me a copy of my geometry class syllabus that I’d sent home with all of my students during the first day of class.  Like every other class syllabus at this particular school, mine included class rules, the grading system, a list of general objectives and the obligatory notice that I’d be following all other relevant objectives outlined by the Commonwealth of Virginia.

“Mrs. X has some questions about your syllabus,” he said, turning the meeting over to her.

“I don’t understand what this objective is,” Ms. X said, pointing to her copy of the syllabus and then reading aloud: “‘Students will use their intuitive understanding of geometry to understand new concepts.’  What does ‘intuitive’ mean?  Are you going to hypnotize my son?”

I instantly relaxed.  Clearly, I was dealing with an over-zealous, perhaps under-educated parent, who had been listening to too much right-wing radio (which in the early 1990s was railing against witchcraft in the classrooms).  I might think she was crazy, but I could handle this.

I calmly explained that all students come into my class with a basic understanding of shapes and the laws of geometry.  I needed my students to tap into this intuitive understanding so that we could build on skills they already had.

In short: These kids already knew something about geometry, and as a professional educator, I was going to take advantage of that.

What I didn’t realize was that my heartfelt theory was not proven fact.  But in April of this year, the Proceedings of the National Academy of Sciences published a study that does just that.  Here’s the gist:

Member of the Mundurucu tribe of Brazil (photo courtesy of P. Pica)

French researcher, Pierre Pica discovered that members of the Amazon Mundurucu tribe have a basic understanding of geometric principles–even though they aren’t schooled in the subject and their language contains very few geometric terms. In other words, geometry is innate.

In fact, Pica found that French and U.S. students and adults did not perform as well on the tests as their Mundurucu brethren.  Turns out formal education may get in the way of our natural abilities.

“Euclidean geometry, inasmuch as it concerns basic objects such as points and lines on a plane, is a cross-cultural universal that results from the inherent properties of the human mind as it develops in its natural environment,” the researchers wrote.

Bla, bla, bla, and something about points and lines.

Not to toot my own horn or anything, but what this means is I was right all those years ago.  We may not have been born with Euclid’s brain, but we do, at the very least, pick up his discoveries just by interacting with our world, rather than sitting in a high school classroom.

Actually, the philosopher Immanuel Kant said as much when he was doing his thing in the 18th century, so this isn’t a new idea at all.  But many students (and parents) didn’t get that memo.

The bottom line: aside from uncommon processing and learning differences, there’s no reason that you can’t do ordinary geometry.  More than likely, any obstacles you face are rooted in fear or stubbornness.

And I, for one, won’t let you get away with that.

The more I talk to people about math, the more I hear this refrain: “I don’t like math, because math problems have only one answer.”


Okay, so it’s not such a crazy idea.  Most math problems do have one answer (as long as we agree with some basic premises, like that we’re working in base ten).  But math can be a very creative pursuit — and I’m not talking about knot theory or fractals or any of those other advanced math concepts.

I have a friend who is crazy good at doing mental math.  She can split the bill at a table of 15 — even when each person had a completely different meal and everyone shared four appetizers — without a calculator, smart phone or pencil and paper!  This amazed me, so I asked her how she does it.  And what I discovered was pretty surprising. She approaches these simple arithmetic problems in ways that I never would have thought of.  She subtracts to solve addition problems, divides to multiply.  And estimation? Boy howdy, does the girl estimate.  In other words, she gets creative.

(She also has a pretty darned good understanding of how numbers work together, which is probably the biggest reason she can accomplish these feats of restaurant arithmetic.)

While there may be one absolutely, without-a-doubt, perfectly correct answer to “How much do I owe the waiter?” there are dozens of ways to get to that answer.  Problem is, your fourth grade math teacher probably didn’t want to hear about your creative approach.

See, when we learn math as kids, we’re focused on computation through algorithms.  (In case you’re not familiar with the word, algorithms are step-by-step procedures designed to get you to the answer.)  You did drill after drill of multiplication, long division, finding the LCM (Least Common Multiple) and converting percents to fractions.  But nobody ever asked you, “How would you do it in your head?”

The good news is that now you’re all grown up.  There’s not a single teacher who is looking over your shoulder to see if you lined up your decimal points and carried the 2.  You can chart your own path!  And when people are given this freedom, they often find really interesting ways to solve problems.

Don’t believe me?  Try this out: Add 73 and 38 in your head.  How did you do it?  Now pose the question to someone else.  Did they do something different?  If not, ask someone else.  I will guarantee that among your friends and family, you’ll find at least three different ways of approaching this addition problem.

So, let’s do this experiment here.  In the comments section, post how you solved 73 + 38 without a calculator or paper and pencil.  Then come back later to see if someone else had a different approach.  If you’re feeling really bold, post this question as your Facebook status, then report the results in the comments section.

And while you’re at Facebook, be sure to visit and like the Math For Grownups Facebook fan page!