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And you keep going. No fancy pictures, no marbles or blocks, no sidebars about careers in math, just practice, practice, practice.
Divide:
2/3 ÷ 4/3
7/8 ÷ 2/5
11/4 ÷ 33/8
Solve for x:
x2 + 3x – 1 = 0
2x2 – 5x + 2 = 0
4x2 + 8x = 5
Now I'd catch regular heck from the scholarly literature (and occasionally from kids sharp enough to get a whiff of the education trends) that would say, "Oh, you've got to get away from boring old problems. Kids need hands-on, real-world examples to really learn math.
Maybe... or maybe not. A new study by Jennifer Kaminski of Ohio State University finds kids learn math better when they are given abstract problems rather than real-world examples like story problems.
I love math, but story problems still throw me for a loop, whether I'm teaching them or doing them (that's why my graduate stats class is kicking my can -- it's all story problems, no formulas!).
Back to the expert:
"We're really making it difficult for students because we are distracting them from the underlying math," said Jennifer Kaminski, a research scientist at Ohio State University, whose study appears in the journal Science.
The findings cast doubt on the widely used practice among elementary and middle schools in the United States and elsewhere of using friendly, concrete examples to teach abstract math concepts.
For example, a teacher might use a bag of colored marbles to explain probability, or teach a formula about distance with the classic example of two trains departing from different cities and traveling at different speeds.
"The danger with teaching using this example is that many students only learn how to solve the problem with the trains," Kaminski said [Julie Steenhuysen, "Hold the Marbles: Abstract Approach Best for Math," Reuters via Yahoo News, 2008.04.28].
I remember one of my happiest math experiences was learning trigonometry on my own when I had the mumps in 10th grade. I opened my textbook to one assignment and found a list of 100 problems. The only instructions, the only words on the page: "Solve." Glorious!
Answers: 1/2, 35/16, 2/3, (–3 ± √13)/2, 2 or 1/2, 1/2 or – 5/2.
The beauty of story problems is that they test whether a student has actually mastered the concepts. Can they take the concepts they have learned and apply them to solve real world problems?
ReplyDeleteNicholas Nemec
BS Mathematics 1980 USD
Absolutely! But Kaminski's research suggests those real-world applications are the end goal, not the best teaching tool. To ensure that we can apply math to specific real-world situations, we first have to understand the abstract forms, the formulas and procedures independent of the twists and turns of the real world.
ReplyDeleteCAH
BA Math/History 1994 SDSU :-)
I got the fractions, but guess it's been too long since my high school math classes to get the algebra! I loved math - my favorite subject. Never could understand good ole Shakespeare, but give me a math problem and I was happy.
ReplyDeleteI was going to help my nephew with math when he was in middle school several years ago, and I was completely lost with all the abstraction and craziness in the problems. So was he, BTW. I'ts much simpler just to teach the basics, and I agree that the kids learn it better and understand it better.
Nonnie
I'm currently in the process of writing two math books that approach the subject from an abstract point of view. These books are of, by, and for the math geek.
ReplyDeleteThere are plenty of weird situations in the real world, but none can compare in weirdness to some of the things that can happen in pure mathematics.
In what other universe could you have a container that holds a gallon of paint that will cover an infinite surface area?
If you want heavy-duty abstration and mind-expanding theory, I'm busy creating some for you. I don't want to turn this post into an advertisement, but I'm happy to read that in fact, my hunch (that people want to know the "why" as well as the "what") is shared by at least some others.
Math geeks, take heart! I have not forgotten you. Meanwhile, I recommend "A Mathematician's Apology" by G. H. Hardy for starters.
Sounds like a good text, Stan! Just load us up on practice problems!
ReplyDeleteFor many people math is a black box with a few symbols sticking out of it.
ReplyDeleteI don't think the problem is story problems, it is poorly written story problems more aimed at tricking the student than in educating the student.
The other problem is that algebra should begin before or at the same time as arithmetic. Solving the Tom has 3 apples and John has three
apples, how many does Jane have if she has twice as many as Tom and John together type problems are easier if the fundamental idea of algebra solving for unknowns is presented.
For most people, the beauty of math systems is irrelevant. They need to know how to use math as a tool and do it without introducing absurd errors. Think about the real world, what math do you actually use if you are not a hard sciences or social scientist or engineer?
You sure need to add and subtract, but calculators make that almost too easy. A couple of trig formulas might be handy.
Geometry might help any person think logically and now and then be useful.
For people growing up in rural areas and on farms, abstractions may be interesting only if they relate to reality.
Esoteric math may be exciting to some and the math formulated may not even be currently shown to relate to any process or part of the real world right now. Perhaps later, but does all this need to be beaten into the heads of kids in the form of abstraction?
My guess after taking too many math classes..some too many times, and also having taught math classes is that most math classes not aimed precisely at math majors need to show how the math can be useful to real people in the real world and not just interesting to math wizards who float in a world all their own.
If accused of "floating in a world of my own," I would have to plead "No contest."
ReplyDeleteThere sure are enough different types of math texts out there. Some kids really are into the abstract part of math, and into it as a "world of its own," and they should not be neglected, any more than those who need math for more practical purposes should be neglected.
This brings to mind a theory of multi-valued numbers that I toyed with in college. At that time, I thought I was "breaking new ground." I had never heard of such things, so I invented a whole new algebra for them. You might have a number that could take all the values between, but not including, 0 and 1, for example. How would this add to, subtract from, multiply by, or divide by another such multi-valued number that takes all the values between 1 and 2? All of this came about because I had a still more rarefied goal: defining division by zero, or more accurately, the ratio of a number to zero.
Little did I know that someone had done this back in the 1930s, and their world was buried in the literature, and lost in obscurity. As mine has been; I never did become that math prof I dreamed of becoming (thank God). So the whole abstract theory still lies in yellowed old spiral notebooks more than 30 years old. I did scan them onto CDs ...
Now for the punch line. Someone dredged up that theory from the 1930s and employed it for the purpose of programming guided missiles to more effectively and consistently reach their targets.
G. H. Hardy sang the praises of the "uselessness" of pure mathematics. I used to say to the co-eds in college when they asked why on earth I would major in math, "I hope it will be equally useless for all practical purposes." I actually got some dates out of that one.
Little did I know that a theory just like mine, devised as I floated in that detached world of the study carrels in the fraternity at the U of Minn, would be used to refine our methods of killing people.
So you see, even the most abstract mathematics may, someday, become "useful" in the "real world." In this case, I would argue that the theory would better have remained utterly useless, except for "math wizards."
Well it has been an awful long time since I have been in any kind of school. Back when I was in high school I only had a smattering of algebra and the same applies to when I was in college, taking up el. ed. Therefore I didn't get any of the algebra and I got two of the fractions the last one stumped me.
ReplyDeleteI have two grandaughters who have dropped out of high school in Sioux Falls, because they failed Algebra twice, and decided it was hopeless. If they had been my kids they would have stayed in school until they graduated, but my daughter thinks I don't know anything.