Ahh, word problems.
They too often evoke dread in math students. Just as frequently, math teachers — particularly if they’re new to the business — have trouble understanding why.
Ben Orlin, an Oakland-based high school math teacher and writer, describes this phenomenon in his popular blog Math With Bad Drawings: “I was shocked to find how fervently my students despised the things they called ‘word problems,’” he writes in the aptly titled post “The ‘Word Problem’ Problem.” “They treated ‘word problems’ as some exotic and poisonous breed.” His students felt that word problems “had nothing to do with the main thrust of mathematics, which was apparently to chug through computations and arrive at clean numerical solutions.”
After pointing out that most of the world’s practical math, in fact, aims to solve problems in finance, science, or design, Orlin recasts the role of these mathematical objects of loathing: “Word problems aren’t an invasive species. They’re the whole biosphere.”
To be fair to Orlin’s aggravated students (and to be fair to Orlin, who acknowledges this), the word problems in far too many textbooks and standardized tests seem very disconnected from students’ lives.
Why, exactly, would a student care about trains passing each other at a certain time in Bethesda? Or how far a man has to walk away from a light post before his shadow grows to a certain length? For that matter, who would buy a bunch of material to build a fence without knowing the size of the yard being enclosed?
As Charles Schulz’s Sally Brown remarks in one Peanuts strip, “Only in word problems can you buy 60 cantaloupes and no one asks what the heck is wrong with you.”
This was not always the case. According to Penn State emeritus professor Frank Swetz’s fun read, “Mathematical Expeditions: Exploring Word Problems Across the Ages,” the oldest-known word problem — from the former Sumerian city Shuruppak — dates back six millennia, and focuses on shares for local grain distribution. In the ensuing centuries, word problems inspired a range of math problem–solvers, from middle-class British women to Indian schoolchildren, to engage with challenging subject matter in an approachable way. And yet, in recent decades, the genre has earned a bad reputation.
Why are word problems, which are meant to demonstrate the real-world applications of mathematical concepts, so disconnected from actual life?
“The root of the issue is that word problems are often both developed and implemented in a way that is contrary to what mathematics is really about. It's easy to forget that mathematics is a human creation that was developed to meet human needs,” says Katie Rich, a curriculum developer for the University of Chicago’s Center for Elementary Mathematics and Science Education and author of the occasional word problem herself. For a long time, she notes, math was “taught as an abstract discipline full of numbers and procedures that have little relationship to anything outside of math class.” To bring math into students’ frames of reference, word problems were (sometimes haphazardly) appended to the ends of lessons. And, according to Rich, students perceived these “as math problems that require extra interpretation steps” meant to add a “purposeless layer of difficulty to mathematics.” The problem, Rich concludes, is that “word problems [in textbooks] aren't applications.” Instead, they are retrofitted to content and curricula that have already been written. While they’re meant to serve as practical applications, “they really are just computation practice with an added layer of complexity — just like students think they are.”
But Rich has hopes that the way professionals write word problems is changing. For one thing, authors are aiming to make them more inclusive of all genders and ethnicities. In addition, curricula are increasingly introducing applications alongside new mathematical concepts. Rich also points to technology as an answer to making the word-problem process more thought-provoking. “The most interesting problems tend to be the most complicated, so interesting applications get left out of curricula when there isn't enough time to address all the component skills. With technology, though, we can build custom tools that allow students to see the big picture while focusing on particular aspects of a problem.” (Rich is currently exploring this very use of technology through her project Number Stories, which offers free access to mathematics application problems.)
The Common Core — both famously and infamously — is also affecting the development of word problems. "The standards themselves made attempts to change the perception and teaching of word problems," Rich says. For instance, the standards for mathematical practice are geared toward contextualizing problems within the realm of math. Similarly, a whole range of tasks is aimed at helping kids think practically about mathematical operations — like, for instance, the fact that “putting together” involves addition.
Despite these laudable goals, the standards have not exactly earned a stellar reputation among math problem–solvers. For starters, the implementation of the standards has required teachers, school leaders, and test developers to take an inventory-like approach, in which they literally tally up how many of each problem type is on every assessment. “This leads to the big ideas being atomized into tiny pieces, and unfortunately this usually means the big ideas get lost altogether by the time they get to students," Rich points out.
While students today fiercely battle word problems, many of their parents are already scarred. But there is hope for them — and all of us — yet.
Po-Shen Loh, an associate math professor at Carnegie Mellon University, creates word problems that defy the stereotype. His enthusiasm is clear — and contagious. “I think word problems are a fascinating topic!” says Loh, who last year coached the first U.S. team to win gold at the international Math Olympiad in 21 years.
Math is a cross between art and law, he adds. “The word problem then specifically trains that art of translating the world into quantitative language. It also provides motivation which hopes to explain to the general public why math is useful.”
Each week, Loh puts a set of five word problems on his site, expii. You may not be able to solve them all, but you should be able to work your way into them; they’re presented in order of increasing difficulty. “The first problem in this set each week is targeted at the general public (anyone with a middle-school pre-algebra background or higher, all the way to retirees).” From there, “the difficulty rapidly ramps up”; the fifth problem is for experts.
The problems are often fun and engage with real-world goings-on. For example, Oscar lovers can figure out how many sequins are on a famous actor’s gown as she walks the red carpet. And no, Loh adds, you won’t get as much information as you need, and you will have to do some guesswork about how many yards of fabric the dress may have. (Thinking about cylinders may help.)
“I think that real-world motivation is extremely important, because if a person decides that they really want to learn something, then they will,” Loh says. “This is how so many people in the world can toss a basketball into a hoop, even though the amount of money we invest as a country in publicly funded basketball education is minimal compared to math education.”
With the passage of the Every Student Succeeds Act, educators will now have new opportunities to create word problems that — like Loh’s — get students excited about math. Whether they will seize this opportunity remains to be seen.
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