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Now a new study by Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State, reported in today's NYTimes, suggests that this kind of hands-on math is bad for everyone.
Quoting the Times: "The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems."
As the Times also remarks: "Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler did something relatively rare in education research: they performed a randomized, controlled experiment."
14 comments:
From this article: The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.
“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”
Oh boy, have I seen that with my child who has attention issues. She’s likely to remember the colors of the manipulatives or some other details, but misses the underlying math lesson.
On the other hand, my child who processes information quickly often gets bored with the manipulatives.
It’s a lose/lose situation for us.
. . . they performed a randomized, controlled experiment.
Maybe this will start a trend.
I have to get Vicky to read this!
She told me something very similar - she's thought this for a long time.
Unfortunately, the notion of research is also heavily politicized within ed schools....
Here's a passage from A guide for the perplexed: Scientific educational research, methodolatry, and the gold versus platinum standards by D.C. Philips:
The US Congress was on the verge of positioning itself at the far right of the continuum (one of the few times a political elected body has taken a stand on research methodology), by considering restricting research funds in education to scientifically rigorous research, as defined narrowly by the use of randomized experimental or field trial designs – the so-called RFT, the “gold standard” methodology – a step that it has since taken.
The paper is actually interesting & makes a useful point...but I hate to see controlled experiments characterized as "far right."
Here's more:
In this chapter, we explore the development of scientism in education by examining the advancement of human science through the work of seminal figures such as David Hume, Sir Francis Bacon, Auguste Comte, and Herbert Spencer. We also introduce the foundational principles on which empirical research in education is based by reviewing the basic tenets of logical positivism, the epistemology on which human science is constructed.
There is some belief that the shift toward more qualitative research practices undermines the stranglehold that logical positivism holds over education research. In response to this belief, we examine in Chapter 2 whether postpositivist research paradigms and practices constitute an actual departure from positivist assumptions. After briefly discussing the philosophical critiques of human science emerging from existentialism and pragmatism, we conclude the chapter by outlining the current neo-liberal context in which education research and policy development take place.
Scientism and Education
Empirical Research as Neo-Liberal Ideology
Emery J. Hyslop-Margison and M. Ayaz Naseem
I suspect that if we hunted around the AERA web site we'd see some grousing about logical positivism and its stranglehold on education research.
Of course, one man's stranglehold is another man's complete and utter absence of anything resembling a stranglehold....
This discussion may be worth reading:
We really are poorly served by this gold standard terminology. I think that when you use randomized experiments, which I am basically going to defend in this context, they are much like what Winston Churchill once said about democracy. He said, ‘It’s the worst form of government except for all the others that have been tried from time to time.’ I do not think this is the gold standard. I think that for impact assessment randomized experiments are the worst methodology except for some of the others that have been tried from time to time. That is pretty much my theme here.
Determining Causality in Program Evaluation & Applied Research: Should Experimental Evidence Be the Gold Standard?
oh, heck
I can't get Science on Bobcat
I was going to pull the article
wow
I just read it.
Incredible.
One could argue that there is no such thing as a "real world" math problem. There are real world gardening problems, grade problems, accounting problems, pipe problems, engineering problems, and physics problems, in other words these problems require math to be used as a tool in their solution. But the problems themselves are not mathematical. A math problem has as its object of interest a mathematical object, which by definition do not exist in the physical world.
Gardening problems do indeed obscure math problems.
But it seems to me (as I argue this week in my problem of the week problem on OILF) that a key skill in algebra is translating a word problem into math.
Good word problems go to the heart of a mathematical concept, without the elaborate set-up, excess detail, & atrocious prose.
Here's one I cite in my post:
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A gold and copper bracelet weighs 238 grams. The volume of the bracelet is 15 cubic centimeters. Gold weighs 19.3 grams per cubic centimeter, and copper weighs 9 grams per cubic centimeter. How many grams of copper are mixed with gold?
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Incidentally, I wonder how many of today's algebra students could solve this one?
Interestingly, our daughter has the reverse problem, she does better at word problems than regular problems!
I just realized it today, normally she works on her own and reads her word problems silently, but today Dad was working from home and watching little brother, so I sat next to her and watched her and had her read the word problems aloud.
She came up with the answers for the word problems lightning quick--as soon as she finished reading them. (These are not train problems, but addition or subtraction, and sometimes addition of 3 numbers.) Her regular math problems took anywhere from approx. half a second to 20 seconds longer to answer.
We have no idea why.
"The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols."
Maybe there's something here related to synthetic phonics vs. analytic phonics vs. whole word.
Synthetic phonics is the most abstract. (And, the most direct method, as are the abstract symbols in the above math explanation.)
"
A gold and copper bracelet weighs 238 grams. The volume of the bracelet is 15 cubic centimeters. Gold weighs 19.3 grams per cubic centimeter, and copper weighs 9 grams per cubic centimeter. How many grams of copper are mixed with gold?
---
Incidentally, I wonder how many of today's algebra students could solve this one?"
It's a good problem, but I also see Myrtle's point -- it isn't really a math problem, or not a pure math problem anyway. It is a good setup for an important science concept, of a weighted average, and when I've taught a summer bootcamp for incoming first-generation college students, I teach them how to solve that kind of problem. However, I am figuring that most of them need to review math so they can do well in science or economics classes, not in math (beyond calculus anyway).
I think that most people agree that learning to do word problems is useful, but I do think that starting with word problems before the basics are mastered helps more than it hurts. There are a lot of unique skills in word problems, notably recognizing the type of word problem. Most students who are still working on the basic algebra just get overwhelmed with the additional steps.
"weighted average"
I distinctly remember having difficulty with mixture problems and the meaning of "pure". I had the idea that if water was added to something else, then the mixture could still be pure, because water was, well, pure. If the teacher approached the problem as a weighted average math problem first, It would have been much easier for me.
My difficulty had to do with too many new pieces of information in the word problem. I wasn't sure exactly what they implied. I also remember that we didn't start getting into word problems until seventh grade and the problems took a big jump in difficulty, from simple add and subtract word problems you could do in your head to non-trivial mixture problems.
On top of that, we were introduced to all sorts of techniques to solve different types of problems. For mixtures, I remember grids (boxes) that we had to fill in. unfortunately, they would not work for all variations of mixture problems. When I finally was able to convert all of these techniques into one sysematic approach using algebra, I was so much happier.
That's why I'm a big advocate for introducting simple algebra for word problems in the early grades, even if you don't really use variables. For example,
If Mary has 25 marbles and gives 7 to Tom, How many marbles does Mary have left?
25 - 7 = ?
These are usually done in your head without writing anything down. There is little practice (other than guess and check) with easy problems. Then, when you get to algebra, you start getting word problems that you can't do in your head or with guess and check, but you don't have any skills to break the problem down into basic steps.
If you use too many crutches just to avoid variables and algebra, then once you do have to learn algebra, the word problems seem way too difficult. I think that's why so many have difficulty with word problems. They jump from simple problems that you can do in your head (or with guess and check) to those that require algebra, but none of your thinking or guess and check skills are useful anymore.
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