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Topic: Engineering Good Math Tests
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Jerry P. Becker

Posts: 16,576
Registered: 12/3/04
Engineering Good Math Tests
Posted: Oct 7, 2012 3:37 PM
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From Education Week [American Education's Newspaper of Record],
Wednesday, October 3, 2012, Volume 32, Issue 6, p. 23. See

Engineering Good Math Tests

By Hugh Burkhardt

Narrow math tests inevitably drive down real standards because
accountability pressures principals and teachers to teach to the
test. Conversely, well-engineered tests of the math we actually want
people to study and learn raise standards. It may not surprise you
that high-performing countries such as Singapore have better
mathematics - as defined in the Common Core State Standards - on
their tests than the United States does. More surprisingly, good
tests are less expensive in real terms.

There are worrying signs that the actual common-core assessments will
be too close to "business as usual," albeit computerized. If so, most
U.S. students and future citizens will be condemned to further
mediocrity in mathematics.

The need for better tests is accepted by business, industry, and
government. In 2009, President Barack Obama called on "our nation's
governors and state education chiefs to develop standards and
assessments that don't simply measure whether students can fill in a
bubble on a test, but whether they possess 21st-century skills like
problem-solving and critical thinking, entrepreneurship, and

Since then, the states have led the development of common standards
in mathematics that embody this broader vision, and two consortia of
states, the Smarter Balanced Assessment Consortium, or SBAC, and the
Partnership for Assessment of Readiness for College and Careers, or
PARCC, have been funded to develop assessments aligned with the
standards. Much progress has been made.

Everyone accepts that, when used as the linchpin of accountability,
tests are not "just" measurement, but often direct the efforts of
school employees and dominate what is taught in classrooms. The SBAC
"content specification" for the common-core math assessment (which I
helped write) features problem-solving and modeling with mathematics,
reasoning, and critiques of reasoning, alongside the concepts and
skills needed to make these possible.

SIDEBAR: "To find out if students can do mathematics, we need to
find how well they can create, critique, and explain substantial
chains of reasoning."

Crucially, it also includes many examples of assessment tasks that
show how these principles have been realized in math examinations in
the United States and around the world. Examples are harder to
misinterpret than descriptions. Teachers, students, and citizens
understand that items on the tests represent the types of tasks
students must learn to do.

The feedback to SBAC on this content specification has been
overwhelmingly positive. So what is the problem?

A strong undertow of fear appears to be pulling the system back to
the familiar. This is a test of our courage - a test our tests may

There is growing concern that test implementation will be a
third-rate realization of the common core - that the design and
"engineering" will not be good enough. The problems seem to be caused
by a mixture of fear and lack of experience and by a decision making
structure unsuitable for innovation. State assessment directors are
fearful of cost and litigation if their well-oiled testing systems,
already sometimes controversial, have to change. High-quality
examinations that cover the common core and meet international
standards are outside their experience and their zone of comfort.

In high-performing countries, mathematics-curriculum experts have
final say on the problems and scoring of the examinations.
Psychometricians are technical advisers. In the United States, the
practice has been turned upside down: Psychometricians too often have
the final say on the items in a test, while the mathematics experts
play a secondary role. SBAC and PARCC continue this upside-down
tradition that values technical measurement above accountability for
teaching and learning the core mathematics in the standards.

What is the problem with current tests? Multiple-choice tests and
their latest variant - computer-adaptive tests - measure with many
very short items. The grain size of these items is much smaller than
the basic concepts of mathematics. The items have a very indirect
relationship to the targets of instruction: the math in the
standards. In mathematical reasoning and problem-solving, the whole
is more than the sum of the parts. This is recognized in
English/language arts, where we assess substantial pieces of reading
and writing.

To find out if students can do mathematics, we need to find how well
they can create, critique, and explain substantial chains of
reasoning. Multiple-choice tests cannot handle this, nor can their
computer-based variants. When you look at the "technology-enhanced
items" designed to assess "depth of knowledge," you find that
potentially rich tasks have been broken into sequences of short
items. This ignores the real target: chains of student reasoning that
may take diverse paths and be expressed with words, sketch diagrams,
and symbols in diverse ways. Mathematics is not treated as a coherent
body of mathematical content and practices, but as fragments
indirectly related to the target knowledge. This makes a test that
defines the targets of instruction invalid.

It is easy to do better. You ask students to tackle tasks that
represent the kinds of performance that you really want them to be
able to do, not proxy tasks that are easy to assess. As with writing,
you have them scored by trained human beings using specific rubrics
for each task that award points for the core elements of performance.
You audit the process to ensure reliable scoring. This is the way
examinations are run in other advanced countries.

A well-made test matches the depth and balance of the learning
targets. This involves selecting an appropriate balance of short
items and substantial performance tasks so that teachers who teach to
the test, as most teachers will, are led to deliver a balanced
curriculum that reflects the standards. This needs a "mathematics
board," a body whose members are experts in math education and
mathematics. The consortia should establish such panels for task
selection and test balancing.

Where will the tasks come from? Designing accessible assessment tasks
that demand substantial chains of reasoning is a challenging area of
educational design. Test vendors have little experience, and the
skills do not come quickly. However, there is a large international
literature of well-engineered tasks across this range that can be
licensed for use in tests. (Disclosure: A project in which I am
involved-the Mathematics Assessment Resource Service, or MARS-is one
not-for-profit source.)

And what of cost? Vendors charge a dollar or two for traditional
tests, and they only need a class period of testing time. People are
rightly concerned at "wasting" teaching and learning time. Yet ask
teachers how much time they spend on otherwise unproductive test
preparation. Typical responses are that test prep for state tests
takes 20 days a year. That's more than 10 percent of teachers' time
and, worse, more than 10 percent of the students' learning time. This
is the real cost of aiming at a cheap target.

Good tests cost a bit more than computer-based tests. How much
depends on how you manage them. One inexpensive model is to make
scoring training and actual scoring part of each teacher's job. This
is high-quality professional development, showing teachers what is
valued in math performance and what other students can do. If this
takes two days a year, you are still well ahead on the test-prep
clock, with many more days for real teaching and learning than with
artificial tests.

What about test prep for good tests? With "tests worth teaching to,"
that is something you want. Test tasks are valuable learning
experiences. The test itself is not a waste of learning time; it is
instead exactly the task for which teaching prepares you.
Hugh Burkhardt has, since 1982, led a series of assessment projects
with test providers in the United States and the United Kingdom who
sought to align their mathematics tests with learning goals. He is
based at the University of Nottingham's Shell Center in England,
where he works with the Mathematics Assessment Project of the
Mathematics Assessment Resource Service, and the University of
California, Berkeley. He founded the International Society for Design
and Development in Education and chairs the advisory board of its
e-journal, Educational Designer.
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244

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