The end of word problems

I will consider my career a success when my students no longer tell me that they are bad at (or good at) word problems.

The direction of K-12 mathematics curriculum in the United States in recent years has been towards giving students a sense of how people actually use mathematics. There are many people who use abstract mathematical ideas on a regular basis outside the mathematics classroom-physicists, statisticians, mathematicians and the like. There are many more people who regularly use mathematics to solve practical problems that matter. (See for instance, this recent New York Times article on airline bumping). But no one solves word problems such as the following unless they are in math class:

The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2409. What are the two numbers? (from Introductory and intermediate algebra: full citation below)

This is a classic word problem of the sort that I hope to eradicate from mathematics instruction (at least from my own).

Why eradicate them? I have already presented my first argument-that they do not represent the ways anybody actually uses mathematics. At their very very best, word problems are intriguing puzzles and perhaps like a clever lyric or a piece of a melody they provide a bit of satisfaction to the aesthetic soul. But few word problems meet this standard. Instead, they represent to many students roughly half of what mathematics is (most the other half is abstract symbolic manipulation). Students who find these problems silly come to believe that mathematics has nothing to offer them.

I believe students are right in asking When am I ever going to use this? I can honestly tell students that they will not ever be in a situation where they know the sum of two addresses, and where they know that the addresses are consecutive, but where they do not know either address.

Occasionally I let myself believe that American math teaching has made substantial strides since the release of the 1989 NCTM Standards for School Mathematics and the revised Principles and Standards 2000.

I recently had an experience that set me straight. I was in my College’s Math Center, where the developmental math courses are housed (i.e. those courses not bearing college credit because they cover remedial topics). The apartments problem cited above comes from the textbook we use for these developmental courses.

Things were slow so I browsed a small selection of books on a shelf. At least one had been culled from the college library recently. It was titled, How to solve word problems in algebra: A solved problem approach by Mildred Johnson (full citation below). I became intrigued by the Table of Contents, which closely matches the kinds of problems in our text: Numbers; Time, Rate and Distance; Mixtures; Coins; etc. Of course the problems in this book match those in our text as well. Consider:

The sum of three consecutive integers is 54. Find the integers. (p. 13)

This problem, at least, is more honestly posed as a puzzle. The apartment problem puts the same idea into an unrealistic setting. But How to solve word problems commits the same crime. Consider:

Mrs. Mahoney went shopping for some canned goods which were on sale. She bought three times as many cans of tomatoes as cans of peaches. The number of cans of tuna was twice the number of cans of peaches. If Mrs. Mahoney purchased a total of 24 cans, how many of each did she buy? (p. 14)

Seeing the similarities between the perspectives of our textbook and this supplementary book, I noticed the Preface.

There is no area in algebra which causes students as much trouble as word problems…Emphasis [in this book] is on the mechanics of word-problem solving because it has been my experience that students having difficulty can learn basic procedures even if they are unable to reason out a problem.

And here is the crux of the matter. I have already argued that the very nature of word problems is such that people’s actual experience has no bearing on solving them. But in this preface is the rarely stated truism that we can train students to work these problems even when we cannot teach them to think mathematically. Entire sections of textbooks are devoted to the translation of word problems into algebraic symbols and Ms. Johnson has written the book on it.

While I appreciate Ms. Johnson’s efforts to help students through the arcane world of word problems, I am saddened by the uncritical approach. There is no discussion of why students should be forced to learn to solve artificial word problems, nor a questioning of whether there might be better uses of their (and their teachers’) time.

I began to wonder how recently the book had been written, given the similar perspectives of our text with this supplement. The copyright is 1976. In thirty-four years, we have truly made no progress.

Back to work.

References

Bittinger, M. & Beecher, J. (2007). Introductory and intermediate algebra: Third edition. Boston: Pearson.

Johnson, M. (1976). How to solve word problems in algebra: A solved problem approach: Updated first edition. New York: MacGraw Hill.

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22 responses to “The end of word problems

  1. I have a further objection against word problems: They make it unnecessarily hard to get at the actual underlying mathematical problem (in particular when the formulation is ambiguous). Take the hotel problem: what is actually asked for is “2x + 1 = 2409. What is x?”—yet time is wasted trying to deduce this equation.

    I agree that there is some justification to doing this, so the students learn to extract the right facts, but like with most low-level math: Either one gets it fast or not at all. It is more a question of innate intelligence than of training.

    The above examples may be too trivial to illustrate my point, so:

    I once studied a puzzle book (possibly by Samuel Loyd) where each puzzle consisted of a one to two page anecdote of irrelevant blurb with the pertinent facts hidden through-out—and where I grew so bored of sifting for these facts that I eventually threw the book away… A puzzle book should contain puzzles—not boring anecdotes. A math exercise should be an exercise in math—not reading comprehension.

    • RE: Michael Eriksson’s response

      Michael, what do you believe the role of the teacher to be if, “Either one gets it fast or not at all. It is more a question of innate intelligence than of training.” is your frame of reference?

      Christopher makes an excellent point of curriculum’s propensity for having word problems solely for the sake of learning how to solve them (often for standardized test practice). There is no doubt good mathematics should be at the heart of the problem AND we can do a better job of choosing those problems from the functioning world around us. Wouldn’t that be a suitable role for a teacher? What skills should a teacher possess in order to assuage this matter? OR is it as simple as some people will get it and some people won’t as you propose Michael?

  2. The role of the teacher is a complicated question where I have no ready-made answer. However, I strongly feel that today’s class room education is highly inefficient and often ineffective—and that the teacher is often a direct hindrance for those even border-line gifted. (Something born out by the experience of gifted students and their parents.) Some thoughts on the issue are present on http://www.aswedeingermany.de/50Humans/50IssuesRelatingToEducation.html

    To expand on the related thoughts about “learning for the sake of learning”:
    I have some doubts as to whether it is justified to force those not interested in learning to study, after a basic skill and knowledge set has been met. Learning for the sake of learning is something that I strongly believe in for myself, but which others see as an imposition and a waste of time. Considering the enormous amounts of time stolen from these people by school, the resources spent by society, and the adage “you can lead a horse to water, but you cannot make him drink”, the entire system may need rethinking: Give the students the opportunity to learn when they actually want learning (even if past the normal school age)—do not wainly try to force learning down their throats when the government thinks it would be appropriate.

    Looking at the issue of getting it: My experiences so far are indeed that either one gets it or not. To some degree, additional work or better teachers/text-books can compensate, but without working miracles. Notably, I have repeatedly heard claimed about mathematics that this principle applies in general, that even college or grad school students eventually land on a level of math that they simply do not get—despite working hard. My own experiences are similar: There have been a few advanced courses that had me spend an entirely disproportionate amount of time (compared to less advanced courses) studying to get through them.

    As a final note, the above quote “Emphasis [in this book] is on the mechanics of word-problem solving because it has been my experience that students having difficulty can learn basic procedures even if they are unable to reason out a problem.” is potentially highly relevant: Students can learn to mechanically solve a problem that they do not understand; however, because they do not “get it”, they have to work harder, lack a deeper understanding of what they do and why they do it, and are likely to forget the method used (and thereby lose the skill) in a comparatively short time.

    • Michael, thank you for your challenging perspective. I think I will just highlight a few points of agreement and disagreement and only elaborate on a few that best relate to Christopher’s blog. I think I would continue the discussion on the site you included –thank you for providing that.

      Points of agreement:
      1. Classrooms can be inefficient and even ineffective, no doubt we have both been a part of each.
      2. Gifted students are poorly represented in schools and the discussion about improving the system.
      3. Students need to understand what is taking place in a word problem.

      Points of disagreement:
      1. One gets it or they don’t. I would hate to think what would happen to some if, at first they did not succeed. My experience has awarded persistence over natural inclination.
      2. Believing students should learn when they want to learn (good intent there). This leaves out too many circumstances where someone is effectively forced into a certain form of study and the value of it becomes immeasurable in their later lives.

      In keeping with Christopher’s original intent, I think the point of emphasis is on the sense-making of the problem which, I would claim, would best be fixed if the inclination to understand the problem (its elements and what is being asked) were cultivated at an earlier age. If, right from when they could read, mathematics problems in words were introduced then the need to isolate them as special “word” problems would disappear; they would simply be good mathematical problems (with assurance that mathematics was at the heart of the problem). Helping students develop strategies for problem solving (problems written in words included) has proven fruitful in my experience, including, but not limited to the use of tools, graphics, symbolic representation etc. “Not getting it” can be a factor of not having been introduced to an appropriate framework yet don’t you think?

      Finally, I think you are on to something, Michael, with learning for the sake of learning. I would contend that early elementary students do just that, and any time spent with an infant or toddler makes it clear that play and learning are synonymous — nobody is twisting their arm to learn. That also makes a case for your thoughts on giving students opportunities to learn what they want to learn (very Montessori in spirit). What is the limit of this potentially anarchical approach to mathematics?

      • Neil, I apologize for giving a late answer. My Internet connection was severely malfunctioning for some time around your reply and I have been lazy with the resulting backlog.

        Anyhow:

        Point of disagreement 2: You are right in that a too dogmatic approach would probably bring many problems of its own. Certainly, I see a major part of such a system involving the schools actively creating interest in the students, making education less dull, and otherwise making the students study through evoking thirst (rather than leading them to the water and detaining those who do not drink).

        However, I am not certain about your counter-example: How often would you say that this actually happens? I do not really see pre-college classes having such an effect—and the element of being forced is rarer and weaker in college.

        Your penultimate paragraph:

        To some part, I agree about the framework’s role in “getting it”. In an extreme, developments in the field at hand can cause radical changes. (For instance, from what I have heard, division was considered very, very tricky when Roman numerals were used. Today, with arabic numerals, even children can learn how to do it. Trigonometry, vectors, and calculus have moved some extremely hard problems to the high-school math-club level. Etc.) The point I intended (if I still remember correctly…) was rather that the current mode of teaching, with repetitions of variations of the same explanations and examples, do not help those who did not get it early on. The area in between is tricky: Would better examples and explanations help? Here the answer will likely depend on the exact circumstances, including whether a certain quality base-line had already been exceeded by the original explanations, but I suspect that there would still be a significant component of either-getting-it-or-not. (The practical problems of making the teachers understand and adapt the new methods aside…)

        Looking at early introduction of word problems, I cannot rule out that you are right. However, the way I see these occurring, they are not merely word formulations of mathematical problems—but often deliberate obfuscations. Further, focus on one skill at a time may be needed to give sufficient value for the time the student invests. Further yet, learning to abstract and to think in terms of “math” is vital for those who wish to proceed beyond arithmetic and elementary algebra.

  3. Some of those seemingly ridiculous word problems are good as part of developmental practice. A few of them really ARE practical. In the real world, problems needing solutions come in the form of combined observations and countable and measurable quantities. Those impractical word problems are just part of the path students take while reaching great practical problem-solving ability.

    • I actually disagree quite strongly. The claim, “Those impractical word problems are just part of the path students take while reaching great practical problem-solving ability,” while appealing on its surface and often repeated among math teachers, has no evidence to support it. In fact, there is quite a bit of evidence that students learn to see “word problems” as meaningless, silly and useless, and that they project these impressions onto the field of school mathematics more broadly as a result.
      The peaches and tuna cans problem in the post exists solely so that students can translate it into a system of three linear equations in three unknowns. If that’s an important topic, we ought to be honest with students and either (1) tell them that these are important mathematical objects which have intellectual merit and beauty just as a poem does, or (2) find plausible contexts that students can believe someone might actually encounter.
      Consider the contrast to compound interest problems. People actually do deal with compound interest; they actually do see APY and APR in life. Their intuitions and questions can be used to make mathematical progress.
      I am not opposed to giving students artificial puzzles involving cans of tuna and peaches; I am opposed to the way these problems dominate the mathematics curriculum and to our refusal to point out the difference to students.
      Fosnot and Dolk have written a lovely series of books titled Young Mathematicians at Work in which they make a distinction between “word problems” and “contexts”.
      A context causes students to “interpret their lived world on the basis of mathematical models.” This should be what we are after with our story problems, I think, not “trivial, camouflaged attempt[s] to elicit school mathematics.”

  4. You have possibly found an overabundance of those Algebra exercises which use trivial, camouflaged attempts to represent applications. What I wrote on Oct. 11, comes directly from my personal experience learning and then using Algebra in the real world. So many academic exercise problems could be along a range of impractical word problems to useful practical word problems; but deciding where each fits along that range is difficult.

    • Who uses algebra in the real world? Off the top of my head, I cannot recall the last time I did. (Apart from some border-line cases like e.g. a simple proportionality, which could be formulated as an algebraic problem.)

      • Months later Michael, I am back and failed to mention I really appreciated your October 22nd post. As I continue work with early elementary mathematics a central tenet of our work is developing understanding in a seemingly inverted fashion – students work in the abstract prior to concrete. If you are interested in reading more about it there is a link included below. The development of algebraic thinking is also one of our focuses. The traditional definition of algebra, solving multi-step equations really is useless to the vast majority of us. I am defining algebraic thinking as the recognition of patterns or phenomena in numbers that ultimately can be generalized in some way, most typically in a mathematical model. I really like Christopher’s reference to Young Mathematicians at Work, where context helps students “interpret their lived world on the basis of mathematical models.” Could you see giving real data and structures to students and asking of them to make sense of those via mathematics? Would that be an “algebra” useful in the world we live in? Again, thank you for your thoughts and avocation for the gifted.
        Link: http://www.nctm.org/news/content.aspx?id=12326

  5. I just left the comment below on Dan Meyer’s Blog dy/dan about this post.

    I’m not sure if you are using “progress” ironically. This is a good example of the inertia in the textbook industry. I’ve been thinking about that a bit because I’m working with a small group in AMATYC about online educational resources. Your article gave me some good things to think about.

  6. Stumbled in via the Dan Meyer blog, hope you don’t mind. I am one of the authors of an NSF-funded high school program and we dealt with these issues in designing our Algebra 1 book (see link for more info).

    Something connecting the comments that followed the main post is the algebraic purpose of these made-up pretext word problems. I feel these problems can be useful in developing strong connections between arithmetic and algebra — to use the Common Core language, “look for and express generality in repeated reasoning”.

    Take the example cited: the sum of a room number and its neighbor is 2409. What is the room number? If the room number was known, the sum is easy to calculate — this is a much easier question.

    So, an algebraic tactic we teach in our text is to guess at the answer, not in a “guess-and-check” sense to track down the answer, but to determine a generic guess-checker. Here are some guesses and checks:

    Guess: 42. Next room: 43. Total: 85, wrong, supposed to be 2409.
    Guess: 912. Next room: 913. Total: 1825, wrong, supposed to be 2409.
    Guess: 1408. Next room: 1409. Total: 2817, wrong, supposed to be 2409.
    Guess: N. Next room: based on the work, it’s N+1. Total: N + (N+1), supposed to be 2409.

    Therefore an equation to solve is N + (N+1) = 2409, where N is the first room.

    This method works on almost every type of crappy word problem! And it has to do with the connection between arithmetic and algebra, and the nature of equation solving! It actually works in practice, too, and reduces the perceived difficulty level of these types of problems.

    This wouldn’t be enough by itself to use the method. However, the same method can be used to determine the equations of lines, circles, and other shapes, by thinking of the equation as a “point tester”, determining whether or not points make it true until a generalization emerges (Common Core cites this specifically in Mathematical Practice #8). It becomes a tactic students can use on a wide variety of problems, including silly paint-the-house junk.

    Early in my teaching career, I remember asking kids to write three consecutive odd numbers starting with X as the first one (because it was in the book). No surprise: most of them wrote X, X+1, X+3. That year, I drummed it out of them with memorization, which worked in the short term. But once I figured out I could have them generalize from some numeric examples, they were much more successful, and they could explain why X+2 was odd by relating it to the case when X was 41.

    Still at issue is the value in using obviously artificial problems. And we can and should do better — real contexts should be used when possible, such as the monthly payment on a car loan. But there is still algebraic “meat” in these puzzle problems, and methods that can be applied equally to real contexts, fake pretexts, and other mathematical topics. My opinion is that these problems can have mathematical value, but the methods suggested in the Johnson text (and, truly, in most current American textbooks) ignore the mathematical potential in favor of short-term mechanical methods.

    This doesn’t just happen with word problems. There are lots of topics in real mathematics where textbooks seem to ignore the real, interesting, deep mathematics of the topic and dive straight into procedure and mechanics. If you’ve taught FOIL or weeks of y=mx+b you might agree with me!

    Sorry for charging in late to the discussion. That 1976 book is really amazing. I remain hopeful this sort of “mathematics” can work its way as quickly as possible out the door of the classroom. Thanks for the thoughtful post and comments, everyone, and best of luck with your teaching.

    – Bowen Kerins, CME Project

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  8. Mind, Bowen? Of course not; I’m delighted that you and others have found your way here.
    You make an important point that the crappiness of the word problem is one dimension for analysis of curriculum, but not the only. The intended solution methods-i.e. the mathematics we are using the problems to teach-constitute another important consideration. Crummy problems may get us some useful mathematics, just as a great problem can be used for ill purposes.
    But I would argue that for far too long, the field has been content to keep recycling the same crummy problems, and that we do so at risk of alienating large numbers of students.
    I used a mildly crummy context to start the study of exponential and logarithmic functions in my College Algebra last week. But I talked with my students about why we were starting there-we needed a simple context with whole numbers and a physical component that we would be able to refer to when the contexts get more real and the numbers and relationships messier. I think we can be honest with students about the usefulness of our word problems, and we need to be prepared to defend our choices when those choices when they are not satisfying to our students.

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  14. I just had to laugh at the absurdity of the Mrs. Mahoney word problem (which is typical of most)- Why not look in her cart, or ask her? LOL.

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