I said, "get them out of school." Figuratively if not literally.
Let me back up and start with this story. I have perceived myself as terrible with math all of my life. And surely almost every teacher I've had would agree with that perception. But I'm really only bad at math in school.
My son, who has a Bachelors Degree in Mathematics, tells this story. "I never could figure it out," he says, "Ira [he has always called me by my first name] can't subtract a two digit number from a three digit number on paper. I've seen him end up with a bigger number than either trying to do that, but when he coached my baseball team he'd know everybody's batting average as each at bat happened." And I tell this story. It was only as my son broke past the standard high school math, and got into conceptual stuff, that what he was talking about made sense to me. "How do you get this stuff but not division?" My kid would ask. "Division is just too hard," I'd answer.
But here's the thing. I can't subtract on paper, but I could do really well in architectural engineering courses. I can figure out sports statistics. I understand the mathematical concepts behind statistics well enough that I can tell you why most statistical analyses used in the social sciences is fiction. How's that work? And how might that help Nancy's students?
Math (or Maths, depending on your side of the Atlantic) is a series of ideas. These ideas are important. In fact, not understanding them can be disastrous in many ways. Arithmetic, on the other hand, is simply a tool set for expressing some math concepts. Arithmetic is to math as forming letters is to writing. Traditionally, a tool set you needed, but perhaps not anymore.
The problem is that we use arithmetic as a gatekeeper stopping kids from getting to math, just as we use alphabetic decoding as a wall keeping kids away from reading. "Math" thus becomes, in the minds of many kids, a nightmarish battle with a bizarre symbolic code, just as reading does. They never get to what's important, what's useful, or surely, what's fun.
The Telling Time Joke starts at about 8:30 in
Telling Time and Negative Garbage Trucks
All right, I'm no math teacher, but I know a few things. One of the things Ms. Stewart told me is that one of these boys "can't tell time." She immediately followed that up by saying that this didn't matter, he has his phone in his pocket.
Issue 1: Problems of decoding shouldn't be mistaken for problems of concept.
Ms. Stewart knows that this boy does indeed know "how to tell time." His phone has a clock on it. He can look at that clock and, apparently, know what point it is in the day. So, he can tell time. What he can not do is interpret a strange antique system for displaying time. While it might be nice if he could, there is no reason to waste a minute of school time on this. It would be nice if I could tell the month by watching Stonehenge, but I can't, and its no big deal - I have ready alternatives.
Which brings me to the question of how we teach everything to do with math. We all too often create fake issues, fake circumstances, fake problems - which strip all motivation from the subject. We say, "you need to learn to tell time," when we are already quite good at that. The "clock face" of yore is a fake issue. And fake issues drive kids away.
Issue 2: Reality or Not.
When I finally passed "college algebra" I initially really struggled with the course. In an early week we had to graph this problem about finding the optimum number of garbage trucks for a community - you get the idea, productivity goes up as you start to add trucks but then it drops off if you add too many. I did the graph.I got 50% of the credit. "You didn't do the left half of the graph," the professor told me. "The left half?" "Yes," she replied, "the negative garbage trucks."
"What the f--- is a negative garbage truck?" I yelled. "Does it come and dump sh-- on your lawn early in the morning?"
This brought up the biggest issue with much of math education - dumb story problems. Don't mention garbage trucks if you're not dealing with reality. Don't send the train east from Tokyo to collide with the submarine headed west from Denver. Nothing drives kids away faster than this kind of nonsense.
If you want to use real world examples, go outside. Make them real. Make them relevant.
Real World Math
What's your batting average? Your on base percentage? Your earned run average? Your goals against average? Your yards per carry? Your shooting percentage? What's the mean of batting averages for your team? What's the range? Whatever the sport that gets your kids going, if you can't teach a world of math ideas through that sport you may not be trying.
I laughed last year when I watched a bunch of educators struggle with this simple question: There are 65 teams in the NCAA Basketball Tournament. It's single elimination. How many games in the tournament? I asked, and people pulled out pencils and papers. They were trying to do arithmetic, and I was asking for a math concept.
What's great about sports is, you need to grasp very specific statistical rules, rules, in this case, which the kids usually know. What are the rules of batting average? This matters because every system of math is based in a set of rules which allow it to work. Change the rules, or misunderstand them, and your answer will change. That's such a basic idea. Two apples plus two apples equals four apples only if we accept the rule that every apple counts as "one" no matter what size or quality. So sports stats teaches that rule idea clearly. Once you've got the rules, you need a formula - in batting average we've got hits divided by at bats. If you know how to find the rules and where to find the formula, you've got it. The rest is punching numbers into a calculator.
Issue 3: The Calculator.
Of course you use calculators. We're humans. We use tools. As I mentioned on Twitter, if you ban calculators you should probably require mittens as well. Don't want them counting on their fingers.
How much will it cost to buy a car? Buy that guitar? That drum set? How do I figure interest? If I spread the payments out over 12 months what will it cost? What do those "cost per unit" stickers in the grocery store mean? How do I know my gas mileage? If I spend this much driving to school when gas is $1.92 a gallon (or £0.94 per liter) and it goes up 30% how much will it cost to drive to that concert?
Money matters to kids. Money is real. Most teachers know that when understanding decimals gets hard, we just need to put a currency sign in front of it (we'll not deal with Great Britain or Ireland before 1971). Why not start with that currency sign. Money gives you so many real examples of the need to find unknowns from knowns (how many payments, how much per payment, how much interest are they charging) that you could run with this for years.
If you are building a roof, how do you know what size lumber to use? How many bricks do you need to build that wall? Can you carry this many pieces of concrete in that truck? Construction takes you from the simple math of area and quantity to the complexity of bending moment and shear diagrams. But unlike the way these math skills are usually taught, these are real - even get dirty - issues which attract kids. I once coached an Odyssey of the Mind team, most who struggled in math, to a medal in the structure competition.
How far did you run? How fast can you drive? Why do different gears on a bicycle switch the distance traveled per pedal turn? How long will it take to get there? Distance and speed, when connected to real life, are essential to kids. Throw out those stupid story problems - your students have their own.
If Henry V's longbow archers at Agincourt could shoot 17 arrows per minute while the French with their crossbows could fire only four times per minute, and the French had 5,000 archers and the English just 1,000, who had the advantage? Integration of math into everything is a huge part of the solution. How far did that book's character walk? (Google Earth, Google Maps) What does a "marginal tax rate" mean? What does "4gb" of memory mean?
The biggest problem with most math in school is that it is taught as a disconnected skill. No wonder no one is interested. Math is really part of everything we do, and if we demonstrate that, we will motivate our students.
Issue 4: Solve non-math problems.
So kids can focus on the learning. If writing the symbols is a problem, use Equation Editor (in Microsoft Word) and stop writing. If arithmetic gets in the way, drop it. Pick your cognitive loading carefully.
A few tricks:
Use calculators which integrate with taking notes and recording answers. Graph-Calc is free and everything can be copied to Microsoft Word or Google Docs.
Beware using b, q, d, p as symbols, beware of using Greek symbols too close to our alphabet. Many of the "math disabilities" I see are really reading disabilities. Switch to distinct symbols and use those consistently. Math textbooks like to switch things around, but that just drives students crazy.
If you need to, eliminate the reading (see paragraph above), if a student can not get to the question because of reading problems, they can not demonstrate what they know about the concept.
Those are my thoughts, but I am not a math teacher, so I'd love to hear yours.
- Ira Socol