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
I spent many untold hours of pain and suffering which were shared with my parents, trying to learn multiplication tables.
"Here is an arbitrary matrix of numbers which you must learn, and some have relationships to each other that you might discover by accident because you're smart, but which we won't tell you because we don't value the meaning, we value your performance."
So, staring at the giant mechanical multiplication game my parents bought for me, I realized that any number times 5 is half the number times 10. Or 3 and 6, or 4 and 8, and thus I only really had to memorize parts of it.
I never made it to 'higher math' classes, because of my poor math grades. But what do I do for fun now? I script building tools, and make objects that simulate movement and vehicular behavior in 3D worlds. No doubt this would be a real job for me if I'd made it over the threshold into 'higher math' back in school. But now at the tender age of fortysomething, my brain seems to be reluctant to remake itself after the fashion of Issac Newton.
So here's my advice: Start with philosophy. Teach kids about logical systems. Teach them how to understand a provable statement and how to spot a fallacy. Then say, "We're going to now apply this same set of rules about philosophy to math." Then teach algebra. The details of arithmetic will then follow, imbued with purpose and meaning.
Teaching kids to memorize a multiplication table in order to understand math is like teaching kids how to make a yard sign in order to understand government.
That's fabulous. "Start with philosophy. Teach kids about logical systems. Teach them how to understand a provable statement and how to spot a fallacy. Then say, "We're going to now apply this same set of rules about philosophy to math." Then teach algebra. The details of arithmetic will then follow, imbued with purpose and meaning."
As always, you have given me so much to think about. And what a pleasant surprise to see my own name in this space that I rely on as a voice of sanity in the oft-crazy world of special education. You are too kind.
I suppose that part of what is driving my strong urge not to do more of the same with these students is my own son’s experience. As a toddler, my husband and I were amazed at his keen observational skills and how he could identify, tabulate, categorize, integrate, and generalize information he picked up from the world around him. But long about third grade somebody decided he was “bad” at math. So, he was forever “bad at math.” It almost kept him from graduating high school. The teacher who tutored him and got him through Algebra II (bless her) told me he had natural math ability (no surprise to us) but that somewhere along the line something had convinced him he didn’t. As a teacher, it’s sad to me that I know what did that to him: school.
So I see the logic in the real world approach you suggest. It also terrifies me. No textbook? No teacher edition? No curriculum guide? Lions and tigers and bears, oh my!
I look forward to hearing responses from others to your post. I would love to hear from math teachers, too, and those who have taught students with learning disabilities using real world math.
Thank you, Ira. You continue to inspire me to be a better teacher.
Alice Mercer on Classroom 2.0 pointed out your blog to me and as soon as I read your comment, I knew I was in the right place.
We are confusing arithmetic a rote, easily automated branch of mathematics, with the whole field of mathematics. We spend 4 years teaching basic computation and if a student has a problem they are labeled as Math Disabled. Suppose we made arithmetic invisible, how much more in terms of Math concepts can we teach first grade children?
It is important to know basic computation at least how to estimate accurately, but basic arithmetic should not be a requirement for other branches of mathematics. In the real world, estimation is often all we need and if precision is essential, the I would prefer a calculator or computer to an arithmetic "expert". Would you choose an accountant who bragged they did all the arithmetic "in his or her head"?
My personal preference would be to start students in grade 1 on Excel and never look back.
There is too much to say in this area and I would enjoy discussing it with you.
I am a retired professor of psychology and directed over 100 ph.d. dissertations. I believe you are a grad student and I am offering my help with the dissertation process. In addition, I can help with statistics if that is an issue for you.
Another brilliant post Ira! I love your garbage truck story! I would have loved to see your teacher's reaction.
Several years ago I was teaching an 8th grade computer applications class. And of course, that meant teaching students about spreadsheets. Once we learned a few basics, I gave them this assignment: find out how much your favorite car costs online. Dream big-- if you want a Ferrari, go for it. One girl went for a used car because that was what she knew she could someday afford. Well, the point is they were WAY more interested in learning about spreadsheets because now they could figure out car payments, how interest rates and down payments affected their monthly payment, whether to take the rebate or the lower interest rate. What 13 or 14 year old doesn't dream about having a car in a couple of years?!? They understood why spreadsheets were so powerful and why they needed to know about them for the "real" world.
One thing which might make it less scary, the kids will begin to supply the curriculum. They all do have "real world" math needs, time, money, et al. So let them bring the content.
My email is on the way to you.
Perfect. This is, of course, cross-curricular. A high school librarian I know begins 9th grade lessons in internet search with cars as well. Always works with American kids. Why fight student interests when we can leverage them?
- Ira Socol
Thank you for writing this post. As a teacher on a mission to make math a joyful learning experience, I applaud your ideas and passion.
There are small groups of people throughout the country who are trying to change the way math is taught in schools. We all share a desire to toss out the textbooks and teach math in relevant and meaningful contexts.
One such movement, currently underway online, is being driven by a passionate group of (primarily) high school math teachers. Based on Dan Meyer's What Can You Do With This? blog series, this group hopes to move problem solving away from the contrived situations in textbooks to real world scenarios presented through digital media. Kate Nowak, publisher of f(t), is using Diigo to organize the materials. There's not much there now but we do expect it to grow quickly over the next few months. I would like to see this reformation make its way to the elementary grades as well.
I'm including some links to relevant information in case you'd like to learn more.
Dy/Dan's WCYDWT archive:
Diigo WCYDWT bookmarks:
f(t)'s paradigm shift blog:
LearningInMathland's revolution outline:
Ira, as you know, I *am* a math teacher, and have taught at the university level as well. I totally agree with everything you have said. I suggest you have a look, if you haven't yet, at Darren Kuropatwa's blog A Difference (I think it is at adifference.blogspot.com). I hope Darren will keep up with it in his new job!
Be well, Hadass.
There were 2 main differences between myself and many maths teachers I worked with: (1) I was new because (2) I previously worked as an (IT) professional - main point being not that I was in IT but that I worked in the "real world" for a significant time.
These differences meant that my approach and philosophy to teaching maths were often in question. Blogging became an integral part of my practice.
The most recent post is a case in point. I wanted to take maths beyond the set curriculum which I (dare I say it?) found boring to teach, much less learn.
In my brief experience as a maths teacher (3 years), I realise that students need context for learning maths - doesn't have to be the "real world" but something they can anchor their learning to. I have had a couple of students who were awesome thinkers but so bad at calculation and notation - made a point to tell them this and the importance of the latter (yes the latter are important, too).
My blog shows how I fumble through all this, as well as some successes. However, I'm still searching....
I don't fancy myself a great teacher nor do I necessarily aspire to be. However, it is gratifying to see changes in students who have typically struggled in maths - and I don't mean grades. I still count one of my top achievements as having most of my C-students say "I loved Algebra".
Sorry - this has turned out so long!
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