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.
Ian 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.