Imagine a scene twenty years in the future. One of your children invites you to work for lunch and to meet their coworkers. You relax, meet their friends, have fun, and listen with half an ear to stories about what their children are getting up to at home. During a break in the conversation, you happen to overhear a couple of people talking in one of the next rooms over. It sounds like they are dealing with some kind of minor emergency. Because that amazing ham-and-cheese wrap you just ate is making you feel kinda drowsy, you listen in but don’t worry much about what it all means. You certainly don’t try to follow the details. Which conversation do you think you are more likely hear?
Conversation 1 (with Henry):
- Boss: Henry, we have an emergency!
- Henry: What is it, boss?
- Boss: The board of directors just called, and they need to know the value of 118 times 53 right now!
- Henry: No problem, I’m on it. Do they need anything else?
- Boss: Well, they did give me a list of other problems that they need solved. Here’s a list. (Hands Henry a sheet of multiplication problems.)
- Henry: Wow, that’s a long list. I’d better get it done quickly.
- Boss: No kidding! The more you get done correctly in the next hour, the more money the company will make.
- Henry: Can I use a calculator?
- Boss: No, that’s cheating.
- Henry: Huh? But if we have a computer do it, we can make a lot more money!
- Boss: Don’t ask me, I don’t make the rules.
Conversation 2 (with Sally):
- Boss: Sally, we have an emergency!
- Sally: What is it, boss?
- Boss: I just learned some bad news. The Initech trucker union is striking! They deliver all our Widgets to the stores!
- Sally: Wow, that’s bad. OK, let’s see. What other ways to we have for delivering Widgets?
- Boss: We already have a contract with Unimove to deliver Thing-a-ma-bobs. We don’t use them for delivering Widgets because they cost more. Thing-a-ma-bobs need their special trucks, though, so we pay it.
- Sally: Yeah, but Widgets go bad in less than a day. We have (Sally types on computer) 1180 Widgets to deliver today! At $53.00 per widget that’s (thinks) roughly $60,000.00 a day we’re going to lose. I think we can pay extra to get everything delivered on time.
- Boss: (Pauses.) I think you’re right. Let’s see…Unimove’s deliveries cost $80 dollars per delivery and Initech’s cost $100.
- Sally: I thought that Unimove cost more!
- Boss: Unimove trucks can hold only 75% as many widgets as Initech trucks can.
- Sally: Oh, right! How many Widgets can each Unimove truck hold?
- Boss: Lessee…
- Mumbles to self while looking using calculator: Each Initech truck holds about 50 widgets, so .75 times 50 is…37.5.
- Looks up: Let’s say 36.
- Sally: Right, so with 1180 widgets that’s…(pause at computer) about 330…no, too large (Sally corrects mistake)…33 loads or so for a single day. At $80 per load, it comes to (another pause as Sally enters numbers into the computer) $2,622 or per day to deliver our widgets. Yeah, we need to contact Unimove now! Even if they charge double, we need to get our Widgets moving.
- Boss: I’ll get in contact with Unimove right now.
- Sally: And this is going to affect our budget! I need to work out how much money extra money we’re going to have to spend, depending on how long the strike lasts. I’ll make up a spreadsheet to work it all out.
- Boss: Thanks, Sally! Knew I could count on you.
I don’t know about you, but if I overheard the first conversation, I’d be thirteen types of confused. First, I’d wonder what kind of madmen were running the business. Second, I would probably start thinking of ways to convince my kid to find a different job. The only place we ever hear conversations like the one between Henry and his boss are inside of schools. Even in schools, they generally stop early in college, especially if the students are following a STEM track. They never take place in businesses, government offices, or research labs. They certainly never happen in advanced mathematics, although this hardly matters for most people.
If I heard the second kind of conversation, though, I’d probably forget that it ever happened. At best, I would remember that they had a small emergency but that they solved it fairly quickly. I would probably then turn my attention back to my kid and ask whether they were still planning on having that barbecue at their place on Saturday. Conversation 2 is perfectly normal for any kind of job, especially a small business which has to move quickly. Which is kind of sad, because Sally and her boss both understand arithmetic very well, which is why they were able to solve their problem so quickly. This deeper and very useful understanding of mathematics is what is called mathematical fluency [1], and it is the ambitious goals of modern math education.
What is mathematical fluency?
The name mathematical fluency may be a bit confusing to some people. After all, most of the time we use the word fluency to represent how well we use the common human languages like English or French. When we are fluent in a language, we can quite literally think using the words of the language. Talking with other people in that language comes naturally and we hardly remember we did it. We are most obviously fluent in our native language, which often forms the largest part of the world around us. How does this relate to mathematics?
Well, mathematics can be considered a type of language. Mathematics came out of out of ideas which were first written down using the common languages of the day. Consider, for instance, such simple words as one, two or three. Those words are both mathematical ideas and words in common use. Math sometimes needs to be very careful, though, in places where regular languages are not. That’s why the language used to describe mathematics and the language of every-day life slowly split. This split has been going on for at least three thousand years and has become much faster over the last four centuries or so. Nowadays, math and the common languages have split so far it is sometimes hard to see that they were ever even related.
From this perspective, then, mathematical fluency is the ability to use the language of math as easily and naturally as we use our native languages. In means that when we have an idea or problem, we can translate it into or out of math correctly. We can mix mathematical ideas with regular common-sense and common-language ideas and still have everything work. It means making math what is has always intended to be: a tool good enough to solve the problems we face. It often means knowing which problem to solve as much as it means knowing how to solve the problem.
In the above dialog, Sally and her boss show several examples of mathematical fluency. Their conversation is mostly in English, but include several words from mathematics. These include several numbers (1180, 53.00), units (dollars per widget, widgets per load) and several mathematical operations (percent, multiplication and comparing the sizes of different numbers.). The skills they show include knowing not just how, but why, they need to use math. Several times they know that they aren’t going to get an exact answer, and they do the best they can—which is good enough.
There is one other part of mathematical fluency which is not so easily seen in the conversation between Sally and her boss, and that is the idea of flexibility. The reason we can’t see it is because it is all behind the scenes, in Sally’s head. When Sally realizes she needs to solve the problem 1180 times 53, this is how she actually solves it:
- 1180 is the same as 1200-20.
- 53 is the same 50+3.
- So 1180 times 53 is the same as (1200-20)x(50+3).
- From long experience, Sally knows that this is the same as 1200*50+1200*3-20*50-20*3.
- 1200×50 is (using the standard algorithm) 60,000. All the other numbers in the sum work out to be smaller and don’t make much difference.
- So the answer will be very close to 60,000.
You may be wondering why Sally used this complicated way to solve the problem. After all, the standard method gets the exact answer every time! Why did she need this flexibility to change how she solved it? Why not just work it out the right way? After all, the answer can be calculated to be $62,540.00.
There are two reasons Sally chose to do it this way. The first one is that Sally likes and understands this way of solving the problem. She is faster at it than she is at the standard method, and since it works, why not use it? Her boss doesn’t grade her based on how she did the math; he cares only about whether the answer is right. The second is that Sally knows that, in this case, the exact answer isn’t all that useful. The number of widgets they sold yesterday isn’t the exact same number as how many they are going to sell today, and tomorrow will be different again, so there is no point in working out the price for exactly 1180 widgets. Her approach gives her about the right answer in many fewer steps, allowing to get on to the next part of the problem. But she does know the standard algorithm (and how to use a calculator), so when she needs the exact number, she does the extra work and gets it.
Things which aren’t mathematical fluency
It is sometimes just as important to know what something isn’t as to know what it is. In the case of mathematical education, this is doubly true because of how very easy it for a student to assume that they are bad at math when they are really just bad at something that looks like math, but isn’t really useful.
Here are some things that math fluency is not:
- Speed. You can’t be too slow, obviously, but there is no point in becoming Mr. Micromachine. In fact, raw speed often means you miss important details. If you really need a lot of problems solved as quickly as possible, a computer is your best bet.
- Specific Techniques. If you don’t know some techniques for solving common problems, you can’t actually solve the problems, so you’d better figure something out. Once you get out of school, though, nobody cares how you solve problems, just that you solve them correctly. Finding something that works efficiently is all that matters.
- A Competition. Everything important nowadays gets done with large groups of people working together. Math is a language. It is meant for communicating. Math competitions are useful only in as much as they are fun and encourage people to learn math.
Are you saying its more important to know how math works than it is to know how to do math?
No!
Just in case you missed it, the answer to this question is NO!
The key is in the term fluent. What kind of person could possibly be considered fluent in English if they didn’t know how to use words like “a” or “the”? If they always had to stop and think in order to remember the correct word for something common, like “car” or “lunch”? If they used sentences like “She am coming to town yesterday.”? That can’t be considered fluency in English. And math is far more exact than English, often having only one correct solution to a properly phrased problem. No, you cannot be considered fluent in math if you consistently get problems wrong, if it takes you a long time to do problems which other people can solve quickly, or if you can’t tell other people what you did.
True fluency in math means you solve the correct problems, correctly and efficiently. It also means that you can use the math you just did to tell your answer to somebody else. Thus fluency in math includes the ability to solve basic math problems, such as 5*13 or 2+2, quickly and easily and in a way other people can follow. It is vital for any child who is fluent in math to understand at least one standard method to solve the most common math problems, both so that they can be confident that they have the right answer, and so that they can tell other people what they did. However, this part of math, while vital, is only a part. We do not expect our children to stop studying English just because they can read their first-grade primer. Nor do we think that just because they read all the words correctly in that primer, that they understood the story. Math is so much more than the words we use to talk about it! But we certainly need to know those words.
There is one thing about fluency that people sometime miss, though. Someone who is fluent in a language knows more than one way to say the same thing. What’s more, because they know different ways of saying the same thing, they often say the same thing in different ways depending on who they are talking to. For instance, if you get into a car accident you are going to tell your friends about it in very different ways that you are the policeman who is called to the scene. When you talk to the insurance company, you will describe it in a different way again. Mathematical fluency is the same thing. If you are mathematically fluent, you can describe the same problem in different ways. (Sometimes very different ways.) Because of this, you are less likely to get stuck solving a problem which looks hard when written down one way, but easy written down another. It means that when you explain it to other people, you aren’t stuck hoping they learned how to do it exactly the same way you did; instead you can keep trying to communicate until you find something that works. It makes you faster and a better communicator, and best of all, when faced with a problem, you now know lots of ways to solve it and can choose the best.
Summary
Math is many things, but for most uses math is two different things: a language and the tricks we use to say things correctly in that language. Fluency in math means understanding and using the language of math as naturally as we use the languages we grew up with. Because of the great depth math has developed over the centuries, this fluency includes learning the tricks we have found actually work over the past several centuries, but it means so much more. It means that rather than using math like a phrasebook to deal with common situations which come up, we can slip into and out of math as the problem requires. It means knowing how to express and use math in different ways, so that we are the master of the math, rather than math telling us what we can do with it.
Footnotes
1. Mathematical fluency is a common idea in modern math education circles, as seen in a simple Google search. My description in this article is my own, although I originally got the term from the title of an article by Susan Jo Russell.
The Math Dragons 