Mathematics Education:
What's the Point of a Word Problem?
From Common Knowledge, Volume 11, Number 3, 1998.
© 1998 Core Knowledge Foundation. Not to be copied or reproduced without permission from the Core Knowledge Foundation, 801 E. High Street, Charlottesville, VA 22902.
Obviously no one is arguing against higher order thinking skills in mathematics. The question is what does it mean? In the last year,there has been a great deal of hand-wringing about the nation's performance on the Third International Mathematics and Science Study (TIMSS), especially over the decline that the U.S. experienced between the 4th and 8th grades,and now to the final — and most damning of all — 12th grade results,coming out just a couple of weeks ago. William Schmidt [National Research Coordinator for U.S. TIMSS and a professor of education at Michigan State University] has been going around the country — including California's State Board of Education — telling people that the reform movement in mathematics education in this country is close to what the high scoring Asian countries of Singapore, Korea,and Japan do. In the loosest possible sense, I suppose, he may be correct, but in any practical sense, he is doing more harm than good.
The real truth is seen in the video taping project of TIMSS led by Jim Stigler of the UCLA Psychology Department. The reform movement classrooms of the U.S.are nothing like the classrooms of Japan. Prof. Stigler wrote a very informative article on the issue published in the September 1997 issue of the Phi Delta Kappan, that is still available at their website.
Casual glances at the textbooks used by the Japanese and by the Singaporeans tell the real story. They don't look anything like our multicolored, fluff-filled wonders. My favorite is Japanese Grade 7 because it looks so much like Saxon's Algebra 1/2, the book my daughter's school uses in seventh grade. Everybody in her school is competent at the Algebra 1 level a year later. The Japanese problems are generally not “real” problems, they are problems that reinforce, and extend the mathematical ideas that have been built before. The Korean or Japanese equivalent of “how many widgets?” is perfectly OK if there is underlying mathematics to be used and, more importantly, to be extended in the day's lesson.
Our current fad mandates that students will be more interested in mathematics, understand it better, and pursue it to deeper levels if it is based on actual data from real world models and, above all, hands-on, “I did it myself ” kind of thinking. The superiority of such an approach is simply not borne out in the numbers. It is supported mostly by religious conviction. The greatest problems with “real-world ” problems are that either the mathematics is not sophisticated enough or it is much too sophisticated to be appropriate given the knowledge base of the students, and the numbers tend to be “nasty”; they haven't been rigged to allow focus on the mathematics itself. Artificial problems can be — in fact,traditionally are — tailored to fit the mathematical concepts being taught. The result of the current fad has been to keep the content at the low end. For example,the primary assessment device used to evaluate Core-Plus, a National Science Foundation-sponsored program out of Western Michigan University, is the ATDQT test of the ITED. By their own admission, the mathematics content of this exam is at the junior high level. Incidentally, that same philosophy held for the TIMSS study:the mathematics for the twelfth-grade study was the level of seventh grade. Even the calculus students didn't have to know algebra,let alone standard applications of calculus, or, God forbid, more innovative and conceptual ideas using more advanced mathematics. Under these parameters, the Asians declined to participate. That's OK;we looked bad enough as it was.
The great myth that I already mentioned,extending back even to the New Math era,is that machine-scored tests do not measure anything. That they do not measure everything is undeniable but irrelevant. It is impossible to measure everything and they do measure an awful lot. Do have students submit their work fully and clearly presented but not because it will be mandated by some external evaluation; do it because that's how it should be done. You want to train your students to work the way it should be done. Neatly, systematically, powerfully. That will last a lifetime.
I lied a little bit about my own math experience 50 years ago. There was,even for that era, a bit too much emphasis on computational speed and accuracy for awkward numbers, not for the algorithms, they are important, but for the speed and accuracy of “ugly” computations. Machine computation, with real machines, though not with the electronic wizardry of today, was already the way of life in offices and serious business situations. But there were plenty of word problems of a routine nature as well. To do well, you not only drilled arithmetic, you solved word problems. They are critical in their own right but they are also critical for the reading skills that they develop. There is more than one way to read. You do want to be able to scan lightly and get the author's message in some kinds of reading but in many math and hard science subjects, actually understanding a single page can be a major achievement. Inability to read that kind of literature means a dead-end for a large class of academic endeavors and insufficient exposure to word problems makes learning to do that kind of reading very difficult. The world of ideas is communicated in words, and making correct and standard symbolic representation for those ideas is a critical part of the learning process in any common human endeavor. Mathematics, particularly its algebraic representation, is the language of that process for a substantial part of several important academic disciplines.
And it goes beyond the math-based academically inclined to the general population as well. Traditional word problems of elementary school are the stuff of life. “Percent” as a concept in isolation is never used, but something as a percent of something else is an everyday occurrence. “The interest rate is falling again; should I refinance?” Nor is it just percent. Even really simple things such as, “the temperature drops 26 degrees in 4 hours.” Or, “Can I toss this on-sale ham into the shopping cart or did I not bring enough money.” A politician on the evening news anticipates next year's budget surplus at 3 billion dollars. Much, perhaps a majority, of this country's citizenry turns off all mental switches as soon as numbers pop up in the way that I just described and they pop up that way all the time. Usually that's from ignorance rather than stupidity. Practice in handling word problems competently will develop the sense of confidence that there is really not that much there. So what if it involves some numbers? This is trivial. By “standard ”word problems, I do not mean problems that are set up to be so mechanical that students do not have to read,but simply follow some kind of formula. In fact, legitimate criticism of word problems is leveled at the idea that key words determine some, almost rote, strategies. Some of that is not bad;it is a confidence builder and learning to do routine things routinely is an important part of life. However, it must not take the place of reading to see what is actually being said, no matter how contrived or artificial it might be.
John is 8 and has $23,Mary is 11 and has $22, Rosy is 10 and has $18. How much money do the two younger children have together? How much older is Mary than John? If John and Mary put their money together, how much more will they have than Rosy? Questions like that,early in the grades, is what I have in mind. Plenty of them and of deepening sophistication as the children develop mathematically. A bit later in the curriculum: There is two-thirds of a pie left and four children wish to share it equally. How much of the pie should each child get?
Are these really practical? Really, only by stretching quite a bit can they be interpreted as practical. Who does a fraction problem in order to divide a chunk of a pie evenly into four equal parts? Nobody. You just do it. That is irrelevant,just as is the fact that every “how many widgets would it take to …” is irrelevant. The important thing is to be able to read it and to know what it said. Not to read at novel speed, nor to have the answer in a flash, but to be able to analyze it carefully and do what it implies needs to be done That's all that's necessary. But it's the old “how to lift a bull” idea.You know,start with a newborn calf and just keep doing it every day. Of course,that is as ridiculous as the idea that everyone will be going off to MIT and become an engineer or physicist. Interest and ability will guide people in many different directions but that is also irrelevant. All who should be considered educated should be able to handle certain trivial things trivially.
Furthermore, and now my fundamental belief structure will be showing, if it calls for an easy algebraic solution, that should be no problem either. There has been so much nonsense written in support of a thing called “algebraic thinking” that the math ed industry has almost lost sight of the fact that it is genuine algebra that opens science doors for kids,not its avoidance.
Let me conclude with an example of this phenomenon from the November 1996 issue of Teaching Mathematics in the Middle School, the NCTM journal for teachers at the level where algebra readiness, if not algebra itself, should be the goal. Instead of that simple idea — focusing toward algebra readiness and algebra competence — the goal always seems to be to make it a “real,” hands-on experience if at all possible. The reality is that the symbolic manipulation is important,but not for its own sake. The power is in the ability to use it comfortably when a problem situation described in words makes that the natural tool. Here is the problem:
“A man competing on a game show ran into a losing streak. First he bet half of his money on one question and lost it. Then he lost half of his remaining money on another question.Then he lost $300 on another question. Then he lost half of his remaining money on another question. Finally he got a question right and won $200. At this point he had $1200 left.How much did he have before his losing streak began?”
Not mentioned in the suggested solutions that were worth writing up in this leading national journal was the standard algebraic solution: Let x be the unknown amount. Then the description itself gives:
(½ -[½ -½ (x)]-300)+200 = 1,200
at least it does as soon as one notices that x -(½)x = (½)x; that is, losing half your money is the same as keeping half your money. This is nothing but a linear equation with one variable that simplifies to
1/8(x)-150 = 1,000
and is solved trivially to obtain x = 9,200.
Compare that with “the most exciting solution ”given by a student. Let me try to read the solution verbatim while manipulating the piece of paper. “Take a sheet of paper, which will represent what the man started with. Tear it in half to represent the first loss. With the half left, tear it in half, representing the second loss. [At this point you are actually holding one-fourth of the original piece of paper.] Then tear off a piece [any size] to represent the loss of $300. Label this piece ‘$300.’ You are now left with a small piece of what you started with. Tear this in half to represent the last loss. We know that this represents $1,000, since winning $200 on the next question would produce $1,200. Label this piece $1,000, the next to last piece is also $1,000. Adding the $300 piece to the two $1,000 pieces gives us $2,300. These three pieces are actually the same as the next-to-last piece, making that piece also $2,300. You now have $4,600. These four pieces together are the same as the first piece we got when we tore the original sheet in half, making it also $4,600. We now have $4,600 plus $4,600 for a total of $9,200.”
The summary comment is truly enlightening: “This solution really opened everyone's eyes to the power of hands-on work, particularly for those who might not be able to see abstract solutions.” I submit that any teacher who avoids giving an unknown quantity a variable name when it is natural to do so,and fails to proceed accordingly,is also avoiding development of the algebraic foundation, the very intellectual capital, to use Hirsch's term from The Schools We Need and Why We Don't Have Them, needed to proceed in all of the sciences,mathematics,and engineering. This is not “the power of hands-on work.” It is immorality committed against innocent children.
WAYNE BISHOP is a founding member of Mathematically Correct, a group that pressed for the reforms to math education in California that its State Board of Education recently adopted.
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Last updated: Fri, March 14 2008
