Changing Gears 2012: maths are creative, maths are not arithmetic

(1) ending required sameness     (2) rejecting the flipped classroom     (3) re-thinking rigor     (4) its not about 1:1      (5) start to dream again     (6) learning to be a society (again)     (7) reconsidering what literature means      (9) changing rooms     (10) undoing academic time     (11) social networks beyond Zuckerbergism     (12) knowing less about students, seeing more     (13) why we fight

"In Essays in Humanistic Mathematics, Philip Davis likens mathematics to literature. Like literature, mathematics has metaphor, ambiguity, paradox, and mystery. It has history. Mathematics has contributed mightily to philosophy. It has a sense of outcome, a feeling of rightness, a sense of catharsis... Like music, mathematics has harmony and dissonance." from Cut the Knot by Alex Bogomolny

I am really tired of schools chasing students away from mathematics. And I am really tired of schools confusing arithmetic - a mechanical grammar of numbers - with the field of mathematics. We need kids to get interested in maths. Mathematics is something essential in our society, and in our future, and we just cannot afford to continue to chase the bulk of kids away from the possibilities which come with math skills, interests, and capabilities. We cannot continue to either allow the assumption that there are "math kids" and "non-math kids" (as if math is a magical gift), or to separate students into "creative types" and "math types."

Because the "next world" - the jobs of the future which are, as we speak, constructing the world we will live in tomorrow -  is being built by the creative mathematicians of the world, and our students will either be part of that, or they won't. They will be able to develop their own solutions and have power over their world or they will be helpless consumers locked into their "App Store Education" (Will Richardson must read) and "App Store Existence." They will be participants or bystanders. And largely, that is for us, as educators, to decide.

"Why is this exciting? Why do I want to tell you this story?"
"Mathematics is not about 3+3..."

Let's begin here, with a quote from the end of the clip above, "Adding in clock notation, all of computer science begins when you say 1+1 = 0. It's not that you were wrong when you said 1+1 = 2, its just a different way of seeing it."

If this is over your head in some way, and you teach math in school, we need to talk. We need to talk now, because step eight of Changing Gears 2012 is re-understanding what mathematics are, and how we bring kids and mathematics together.

Two primary issues which lead to the bigger ones, no matter what age kids you teach. First, maths are creative, they are imaginative, they are powerful, and they are fascinating. Second, arithmetic cannot continue to be your gateway, your filter, blocking children from mathematics.

For the first, what have you done with Fibonacci lately, just as a first question? How does a maths idea shape how students perceive the spaces they are in? For the second, well, lets go back a month to a post I wrote, "In a math lesson a day later I watched a seventh grader, a kid who really struggled to divide 64 by 2 in his head, or 32 by 2, or, for that matter, 16 by 2, work diligently to explain to his disbelieving teacher how he knew - and he knew instantly - how many games are in the NCAA basketball tournament. He knew, because math is about rules and logic, and his logic was perfect and his understanding of the rules I had described was perfect, and because math is not arithmetic, no matter how much our poorly educated national and state leaders think it is. He and his classmates also understood, almost instantly, that the question - no calculators or paper or Google allowed - "If the temperature in Detroit, Michigan is 50 degrees what is the temperature likely to be in Windsor, Ontario? was about (a) culture, and then (b) understanding comparable scales, and then (c) order of operations."

If we get past these two ideas, we can begin to bring students into what mathematics is...

Pulling two quotes from a mathematics discussion board begins to get at the issues, the question being that old classic, does two plus two always equal four...
"I think this discussion goes right back to Aristotle (or another Greek of similar vintage). The question is pretty much: Three clouds; three pebbles; three goats; three thoughts; three olives; when you take away clouds, pebbles, goats, thoughts, olives, then what do you have left? The concept of threeness! Each such ....ness is an integer, and there is a reasonably obvious rule to move between such concepts. This rule permits of repetition, and thus establishes the countable numbers."

however...

"For a cook, 2 apples + 2 apples might well accurately equal 5 apples if those 4 apples are larger than normal. The mathematician would argue that 2 large apples + 2 large apples must equal 4 large apples. Correct. That’s the mathematical axiom Jon Richfield is talking about. The trouble is, in reality no apple is the same size as another, so the mathematician’s axiom is limited somewhat to arithmetic theory. So why should mathematicians get the final say? The cook’s application is commonsensical and thus more accurate and fair, so in real life 2 + 2 doesn’t always equal 4. Using the equation 2 + 2 = 5, the apple pie turns out normally, as intended. Nothing meaningless about that."

10 candies? Can these be evenly divided in half?
Or are these all completely different things?

And here we have established the arithmetic conundrum which pulls kids away from mathematics. It should never be taught in a reductionist form which removes the possibility of creativity.

Every child knows that not every apple, every piece of cake (even if the same size you have those differences in frostings), every student, is the same (a fact those who work in quantitative educational statistics have been trained to forget). Thus, the question about two apples plus two apples, as suggested above, becomes one we can argue and debate, even with five-year-olds.

That is not a path to nowhere, it is, rather, the path to understanding, and to bringing students into mathematics. We have to help them learn that mathematics is a set of systems which we can apply when helpful, or rethink and re-imagine if not helpful. A long time ago I wrote a piece called "Real World Math" and one of the things I talked about was why I loved sport statistics in school maths. You cannot compute a batting average in baseball without knowing the rules about what an "At Bat" is and how that differs from a "Plate Appearance." You need to know the difference, in football, between a "Shot" and a "Shot on Goal." You need to know, in American football, how a quarterback "sack" is counted in "run yards" even if that quarterback was tackled while running forward. So these statistics do not just connect maths to a kid's interests, they explain how mathematical systems work, and how a slight change in the rules which govern that system, would change the answer.

At the grocery, sometimes 2+2=4,
sometimes not...

Here we go...

"A particularly vexing problem is comparing players from different eras. One complicating factor is that the baseball rule book has changed every year since the first rule book for the National League was issued in 1877.

For example, did you know that prior to the 1930 American League season, and prior to the 1931 National League season, fly balls that bounced over or through the outfield fence were home runs! All batted balls that cleared or went through the fence on the fly or that were hit more than 250 feet in the air and cleared or went through the fence after a bounce in fair territory were counted as home runs. After the rule change the batter was awarded second base and these were called "automatic doubles" (ground-rule doubles are ballpark-specific rules) and are covered by rule 6.09(d)-(h) in the MLB Rule Book."

Change the rules, change the results. Could you add fruits as 2+2? Or just the same kind of fruit?  Three clouds; three pebbles; three goats; three thoughts; three olives; when you take away clouds, pebbles, goats, thoughts, olives, then what do you have left? The concept of threeness! Each such ....ness is an integer..." But an "integer" is an idea, it is a "construct," which students should learn to decide is either useful or not useful. Do we count "the number of people on the earth" (US Census is at odds with other counters) or measure the cumulative carbon footprint? (and what system of maths do we use to do that?).

Toss this into the mix... "three clouds"? the sky is full of water vapor, where does one cloud start and another stop? Is a three day old pygmy goat the same as an adult mountain goat? This "integer" idea, "threeness," what does it mean and how can we use it?

New York's Polo Grounds,
an interesting field made for
interesting stats.

Now, how many home runs did Babe Ruth hit? How many home runs did Lou Gehrig hit? But wait, 

"With the exception of a couple of months at the start of the 1920 season, from 1906 to 1930 the foul lines were "infinitely long": A fly ball over the fence had to land in fair territory (as determined by the infinitely long foul lines), or be fair when last seen by the umpire, in order to be a home run. In other words, a fly ball that went over the fence in fair territory but "hooked" around the foul pole (if there was a foul pole) was ruled a foul ball." How many home runs did Babe Ruth hit? How many home runs did Lou Gehrig hit?

So, the rules matter, and the rules are changeable - assuming you can make the right argument. And this is creative magic which infiltrates everything, from the music you listen to to how that classroom window frames the world beyond. Years ago I taught an Intro to Architecture course at Pratt Institute. I'd take my students to the corner of 53rd Street and Park Avenue in Manhattan, and we'd look. To the southwest was Charles McKim's 1916 Racquet and Tennis Club, to the northwest the 1952 Gordon Bunshaft Lever House, to the southeast Mies van der Rohe's incredible 1958 Seagram's Building - three absolute architectural masterpieces. The fourth corner, the northwest, is occupied by "399 Park Avenue," a 1961 structure by Carson Lundin, Kahn and Jacobs. It is an awful building, by just about anyone's standards.

We'd spend a long time standing on that corner trying to figure this out, and eventually, we'd get to maths and ratios and Fibonacci and the Golden Mean. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... Because it is that concept of ratio - embraced in three of those structures, ignored in the fourth - that so much of human comfort with proportion occurs. Europeans called it, "the divine proportion." Why? well, here you would seem to have a year long project which might carry your students anywhere and everywhere in mathematics.

Architects, artists, Wall Street traders, even, yeah, that student ID card or credit card
in your pocket... What makes this "the divine proportion"?

OK, that's one route. Another, hinted at near the top, is Coding. Coding is not just that mix of logic and creativity which is essential to maths, but it has a "real" feel. You don't get things "right" or "wrong," they work or they don't work.

Bring coding into your classrooms. Here's one simple free tool called Notepad++ which we have on our MITS Freedom Sticks. But better, take a look at student coding efforts around the world, from Mozilla's Hackasaurus to Ireland's CoderDojo and Scratch for Kinect which are all bringing kids into this in a big way.

As Stephen Howell (above) says, this is very different than working in that Steve Jobs iOS world where you buy the solutions you need in life. This is using the heart of mathematics to build your own world. Starting simply, kids get interested, they gain competence, they dig behind the curtains - something Jobs and Apple have never permitted - and they move deeper and deeper into what, eventually, begins to look like a much more engaging version of our curriculum. Eventually those Scratch programming kids will be building their own Lego blocks, teaching each other how to do it, challenging each other, and, yes, becoming the builders of that "next world."

So please, take the way you currently teach mathematics apart. Become a mathematics educator instead of a curriculum teacher. It might make all the difference in the future of your students.

- Ira Socol

next: changing rooms

Real World Math

Nancy Stewart, a teacher I have come to have great respect for, asked for help through Twitter. She has two students heading for eighth grade next year. Both are "Special Needs." Both are very far "behind" in their math skills. "Any ideas?" she asked the learning network that has sprung up in the world of 140 character microblogging.



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