a*****g
发帖数: 19398 | 1 我一直说 common core 出的问题，现在终于有其他人出文章了
http://www.theatlantic.com/education/archive/2015/11/math-showi
Common Core-era rules that force kids to diagram their thought processes can
make the equations a lot more confusing than they need to be.
Rogelio V. Solis / AP
At a middle school in California, the state testing in math was underway via
the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to
the problem on the computer screen and asked “What do I do?” The proctor
read the instructions for the problem and told the student: “You need to
explain how you got your answer.”
The girl threw her arms up in frustration and said, “Why can’t I just do
the problem, enter the answer and be done with it?”
The answer to her question comes down to what the education establishment
believes “understanding” to be, and how to measure it. K-12 mathematics
instruction involves equal parts procedural skills and understanding. What
“understanding” in mathematics means, however, has long been a topic of
debate. One distinction popular with today’s math-reform advocates is
between “knowing” and “doing.” A student, reformers argue, might be able
to “do” a problem (i.e., solve it mathematically) without understanding
the concepts behind the problem-solving procedure. Perhaps he or she has
simply memorized the method without understanding it and is performing the
steps by “rote.”
One distinction is between “knowing” and “doing.”
The Common Core math standards, adopted in 42 states and the District of
Columbia and reflected in Common Core-aligned tests like the SBAC and the
Partnership for Assessment of Readiness for College and Careers (PARCC),
take understanding to a whole new level. “Students who lack understanding
of a topic may rely on procedures too heavily,” states the Common Core
website. “But what does mathematical understanding look like?” And how can
teachers assess it?
One way is to ask the student to justify, in a way that is appropriate to
the student’s mathematical maturity, why a particular mathematical
statement is true, or where a mathematical rule comes from.
The underlying assumption here is that if a student understands something,
he or she can explain it—and that deficient explanation signals deficient
understanding. But this raises yet another question: What constitutes a
satisfactory explanation?
While the Common Core leaves this unspecified, current practices are
suggestive. Consider a problem that asks how many total pencils there are if
five people have three pencils each. In the eyes of some educators,
explaining why the answer is 15 by stating, simply, that 5 x 3 = 15 is not
satisfactory. To show they truly understand why 5 x 3 is 15, and why this
computation provides the answer to the given word problem, students must do
more. For example, they might draw a picture illustrating five groups of
three pencils. (And in some instances, as was the case recently in a third-
grade classroom, a student would be considered to not understand if he or
she drew three groups of five pencils.)
Consider now a problem given in a pre-algebra course that involves
percentages: “A coat has been reduced by 20 percent to sell for $160. What
was the original price of the coat?”
A student may show the solution as follows:
x = original cost of coat in dollars
100% – 20% = 80%
0.8x = $160
x = $200
Clearly, the student knows the mathematical procedure necessary to solve the
problem. In fact, for years students were told not to explain their answers
, but to show their work, and if presented in a clear and organized manner,
the math contained in this work was considered to be its own explanation.
But the above demonstration might, through the prism of the Common Core
standards, be considered an inadequate explanation. That is, inspired by
what the standards say about understanding, one could ask “Does the student
know why the subtraction operation is done to obtain the 80 percent used in
the equation or is he doing it as a mechanical procedure—i.e., without
understanding?”
In a middle school observed by one of us, the school’s goal was to increase
student proficiency in solving math problems by requiring students to
explain how they solved them. This was not required for all problems given;
rather, they were expected to do this for two or three problems in class per
week, which took up to 10 percent of total weekly class time. They were
instructed on how to write explanations for their math solutions using a
model called “Need, Know, Do.” In the problem example given above, the “
Need” would be “What was the original price of the coat?” The “Know”
would be the information provided in the problem statement, here the price
of the discounted coat and the discount rate. The “Do” is the process of
solving the problem.
Students were instructed to use “flow maps” and diagrams to describe the
thinking and steps used to solve the problem, after which they were to write
a narrative summary of what was described in the flow maps and elsewhere.
They were told that the “Do” (as well as the flow maps) explains what they
did to solve the problem and that the narrative summary provides the why.
Many students, though, had difficulty differentiating the “Do” section
from the final narrative. But in order for their explanation to qualify as
“high level,” they couldn’t simply state “100% – 20% = 80%”; they had
to explain what that means. For example, they might say, “The discount rate
subtracted from 100 percent gives the amount that I pay.”
An example of a student’s written explanation for this problem is shown in
Figure 1:
Figure 1: Example of student explanation.
For problems at this level, the amount of work required for explanation
turns a straightforward problem into a long managerial task that is
concerned more with pedagogy than with content. While drawing diagrams or
pictures may help some students learn how to solve problems, for others it
is unnecessary and tedious. As the above example shows, the explanations may
not offer the “why” of a particular procedure.
Under the rubric used at the middle school where this problem was given,
explanations are ranked as “high,” “middle,” or “low.” This particular
explanation would probably fall in the “middle” category since it is
unlikely that the statement “You need to subtract 100- 20 to get 80” would
be deemed a “purposeful, mathematically-grounded written explanation.”
While drawing diagrams or pictures may help some students, for others it is
unnecessary and tedious.
The “Need” and “Know” steps in the above process are not new and were
advocated by George Polya in the 1950s in his classic book How to Solve It.
The “Need” and “Know” aspect of the explanatory technique at the middle
school observed is a sensible one. But Polya’s book was about solving
problems, not explaining or justifying how they were done. At the middle
school, problem solving and explanation were intertwined, in the belief that
the process of explanation leads to the solving of the problem. This
conflation of problem solving and explanation arises from a complex history
of educational theories. One theory holds that being aware of one’s
thinking process—called “metacognition”—is part and parcel to problem
solving. Other theories that feed the conflation predate the Common Core
standards and originated during the Progressive era in the early part of the
20th Century when “conceptual understanding” began to be viewed as a path
to, and thus more important than, procedural fluency.
Despite the goal of solving a problem and explaining it in one fell swoop,
in many cases observed at the middle school, students solved the problem
first and then added the explanation in the required format and rubric. It
was not evident that the process of explanation enhanced problem solving
ability. In fact, in talking with students at the school, many found the
process tedious and said they would rather just “do the math” without
having to write about it.
Related Story
When Parents Are the Ones Getting Schooled by the Common Core
In general, there is no more evidence of “understanding” in the explained
solution, even with pictures, than there would be in mathematical solutions
presented in a clear and organized way. How do we know, for example, that a
student isn’t simply repeating an explanation provided by the teacher or
the textbook, thus exhibiting mere “rote learning” rather than “true
understanding” of a problem-solving procedure?
Math learning is a progression from concrete to abstract. The advantage to
the abstract is that the various mathematical operations can be performed
without the cumbersome attachments of concrete entities—entities like
dollars, percentages, groupings of pencils. Once a particular word problem
has been translated into a mathematical representation, the entirety of its
mathematically relevant content is condensed onto abstract symbols, freeing
working memory and unleashing the power of pure mathematics. That is,
information and procedures that have been become automatic frees up working
memory. With working memory less burdened, the student can focus on solving
the problem at hand. Thus, requiring explanations beyond the mathematics
itself distracts and diverts students away from the convenience and power of
abstraction. Mandatory demonstrations of “mathematical understanding,” in
other words, can impede the “doing” of actual mathematics.
Advocates for math reform are reluctant to accept that delays in
understanding are normal and do not signal a failure of the teaching method.
Students learn to do, they learn to apply what they’ve mastered, they
learn to do more, they begin to see why and eventually the light comes on.
Furthermore, math reformers often fail to understand that conceptual
understanding works in tandem with procedural fluency. Doing a procedure
devoid of any understanding of what is being done is actually hard to
accomplish with elementary math because the very learning of procedures is,
itself, informative of meaning, and the repetitious use of them conveys
understanding to the user.
Mandatory demonstrations of “mathematical understanding,” in other words,
can impede the “doing” of actual mathematics.
Explaining the solution to a problem comes when students can draw on a
strong foundation of content relevant to the topic currently being learned.
As students find their feet and establish a larger repertoire of mastered
knowledge and methods, the more articulate they can become in explanations.
Children in elementary and middle school who are asked to engage in critical
thinking about abstract ideas will, more often than not, respond
emotionally and intuitively, not logically and with “understanding.” It is
as if the purveyors of these practices are saying: “If we can just get
them to do things that look like what we imagine a mathematician does, then
they will be real mathematicians.” That may be behaviorally interesting,
but it is not mathematical development and it leaves them behind in the
development of their fundamental skills.
The idea that students who do not demonstrate their strategies in words and
pictures or by multiple methods don’t understand the underlying concepts is
particularly problematic for certain vulnerable types of students. Consider
students whose verbal skills lag far behind their mathematical skills—non-
native English speakers or students with specific language delays or
language disorders, for example. These groups include children who can
easily do math in their heads and solve complex problems, but often will be
unable to explain—whether orally or in written words—how they arrived at
their answers.
Most exemplary are children on the autism spectrum. As the autism researcher
Tony Attwood has observed, mathematics has special appeal to individuals
with autism: It is, often, the school subject that best matches their
cognitive strengths. Indeed, writing about Asperger’s Syndrome (a high-
functioning subtype of autism), Attwood in his 2007 book The Complete Guide
to Asperger’s Syndrome notes that “the personalities of some of the great
mathematicians include many of the characteristics of Asperger’s syndrome.”
And yet, Attwood added, many children on the autism spectrum, even those who
are mathematically gifted, struggle when asked to explain their answers. “
The child can provide the correct answer to a mathematical problem,” he
observes, “but not easily translate into speech the mental processes used
to solve the problem.” Back in 1944, Hans Asperger, the Austrian
pediatrician who first studied the condition that now bears his name,
famously cited one of his patients as saying that, “I can’t do this orally
, only headily.”
Writing from Australia decades later, a few years before the Common Core
took hold in America, Attwood added that it can “mystify teachers and lead
to problems with tests when the person with Asperger’s syndrome is unable
to explain his or her methods on the test or exam paper.” Here in Common
Core America, this inability has morphed into an unprecedented liability.
What testing does is measure “markers”of learning and understanding.
Explaining answers is but one possible marker.
Is it really the case that the non-linguistically inclined student who
progresses through math with correct but unexplained answers—from multi-
digit arithmetic through to multi-variable calculus—doesn’t understand the
underlying math? Or that the mathematician with the Asperger’s personality
, doing things headily but not orally, is advancing the frontiers of his
field in a zombie-like stupor?
Or is it possible that the ability to explain one’s answers verbally, while
sometimes a sufficient criterion for proving understanding, is not, in fact
, a necessary one? And, to the extent that it isn’t a necessary criterion,
should verbal explanation be the way to gauge comprehension?
Measuring understanding, or learning in general, isn’t easy. What testing
does is measure “markers” or byproducts of learning and understanding.
Explaining answers is but one possible marker.
Another, quite simply, are the answers themselves. If a student can
consistently solve a variety of problems, that student likely has some level
of mathematical understanding. Teachers can assess this more deeply by
looking at the solutions and any work shown and asking some spontaneous
follow-up questions tailored to the child’s verbal abilities. But it’s far
from clear whether a general requirement to accompany all solutions with
verbal explanations provides a more accurate measurement of mathematical
understanding than the answers themselves and any work the student has
produced along the way. At best, verbal explanations beyond “showing the
work” may be superfluous; at worst, they shortchange certain students and
encumber the mathematics for everyone.
As Alfred North Whitehead famously put it about a century before the Common
Core standards took hold:
It is a profoundly erroneous truism … that we should cultivate the habit of
thinking of what we are doing. The precise opposite is the case.
Civilization advances by extending the number of important operations which
we can perform without thinking about them. |
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