Strategies Are Not Algorithms

  • Strategies Are Not Algorithms

    By Ian Whitacre and Donna Wessenberg, posted December 5, 2016 —

    Let’s just say it: Strategies are not algorithms. In the era of the Common Core, believing that dramatic reforms are happening in mathematics education may be easy to do. So is believing that research-based recommendations have finally broken through to the mainstream and are making a difference.

    The evidence of change is all around us. A dad on Facebook is upset that he doesn’t understand his kid’s homework. Rather than put in a little effort to make sense of it, he spends time to create a meme mocking the “new way” that kids are being taught to subtract. He gets 15 minutes of fame. Meanwhile, his kid still needs help with his homework. This kind of story is familiar. Just do a Google search for “Common Core math” and see for yourself. So, we must be in an era of dramatic change—we must be experiencing a renaissance in mathematics education—right?

    We wish we could say that the dad on Facebook is looking at an invented subtraction strategy and mistaking it for an algorithm. The dad remembers being taught one way of doing subtraction. He sees a different way in his kid’s homework. So, he assumes that this is the one “new way” of doing subtraction, and he doesn’t like it because it’s different. That would be a tragic story but a simple one. In that version of the story, there really is something new and special happening in mathematics education—kids are being given opportunities to solve problems in novel ways that make sense to them and to learn from each other by engaging in mathematical discussions.

    Unfortunately, the real story is more complicated. It’s not just the dad mistaking a strategy for an algorithm. It’s the curriculum materials, too. The idea of “invented strategies” has been distorted to such a degree that strategies are being treated like algorithms in many textbooks and classrooms across the country. So, in this light, the dad is not so wrong after all. The difference is that now kids are being taught half a dozen different algorithms for performing the same operation. Then they’re supposed to choose the one they like best. (By the way, we don’t know where that line of reasoning originated, but it is inconsistent with research-based recommendations.)

    In 1998, Tom Carpenter and his colleagues documented grades 1–3 students’ use of invented strategies and standard algorithms. The vast majority of students in the study used some invented strategies. The researchers found that students who used invented strategies before learning standard algorithms showed better understanding of place value and properties of operations than those who learned standard algorithms earlier. This powerful study helped to put invented strategies on the map.

    The Common Core State Standards for Mathematics (CCSSM) specify that students are expected to use “strategies based on place value, properties of operations, and/or the relationship between addition and subtraction” (CCSSI 2010, p. 16) years before they’re responsible for knowing the standard algorithms. Great, right? Studies like the one described above helped to highlight the value of meaningful, invented strategies. Eventually, research findings and the recommendations of NCTM and the National Research Council influenced policy. Now (at least in theory) teachers are encouraged to support students’ development of invented strategies before presenting standard algorithms. The catch? Strategies are not algorithms.

    When students invent their own strategies, they have sensible reasons for manipulating numbers in the ways that they do, and they’re unlikely to make the kinds of errors that we see when students use algorithms that they don’t understand (e.g., 201 – 199 = 198). But what happens when students experience direct instruction in the use of various “strategies”? Carpenter and his colleagues (1997) warned about the possibility that strategies could come to be treated like algorithms:

    We hypothesize that direct instruction could change the quality of children’s understanding and use of invented strategies. If these strategies were the object of direct instruction, there would be a danger that children would learn them as rote procedures in much the way that they learn standard algorithms today. (p. 19)

    Many of the viral criticisms of Common Core math are based on the assumption that any computational method is an algorithm. People see a “new way” of performing subtraction and assume that it is replacing the “old way.” Rarely do these criticisms consider the actual content of the standards or recognize that both strategies and algorithms are included in the standards. On the other hand, the unfortunate reality in many classrooms is that strategies are being treated like algorithms. Some popular textbooks take the strategy-of-the-day approach, presenting a new way of performing an operation and instructing all students to follow examples of this strategy. As a result, students are positioned to mimic these methods, regardless of whether they understand them.

    Let us be as clear as possible: Invented strategies are strategies invented by kids. That’s what invented means. It does not simply mean nonstandard. If “invented strategies” are consistently coming from the textbook or from the teacher, those are just new algorithms.

    To learn to use invented strategies, students need the opportunity to invent their own strategies! That requires time, patience, and a consistent commitment to pose problems to students and to give them the chance to reason through those problems. It requires the realization that the long run matters more than the short run—that giving students opportunities to make sense is a worthwhile investment.

    Classrooms in which students engage in problem solving and have the opportunity to invent strategies do exist. Those classrooms are beautiful, but they seem to be few and far between.

    2016_12_05_Whitacre_WessenbergIan Whitacre is a faculty member in the School of Teacher Education at Florida State University in Tallahassee. He studies how children think about math, and he collaborates with teachers to improve mathematics teaching and learning. Donna Wessenberg is a mother of three and a kindergarten teacher at English Estates Elementary. She has spent two years teaching fifth grade and eight years teaching kindergarten. She has a Masters of Elementary Education from the University of Phoenix and a Bachelors of Fine Arts from the University of South Florida.


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