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Topic: [ME] Different Teaching Methods, and the Results
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Jerry P. Becker

Posts: 16,576
Registered: 12/3/04
[ME] Different Teaching Methods, and the Results
Posted: Mar 30, 1999 2:45 PM
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Education Week on the web, March 31, 1999, Volume 18, Number 29, pages 52-30

Mathematics for the Moment, Or the Millennium?

By Jo Boaler

People in California are worried about the mathematical performance of
their children. So worried that debates about the "right" way to teach
mathematics have become so metaphorically bloody they have been termed the
"math wars." Much of this concern has stemmed from students' performance on
state and national tests, but nobody seems to have stopped at any point to
question the value of the type of knowledge assessed on these tests. I
would like to halt the debate for a moment in order to pose this question:
Is success on a short, procedural test the measure we want to adopt to
assess the effectiveness of our students' learning? In other words, do
these tests assess the sort of knowledge use, critical thought, and
reasoning that is needed by learners moving into the 21st century?

One of my concerns in this area is that the current debate about standards
assumes there to be one form of knowledge that is unproblematically
assessed within tests. This is despite the fact that a large
body of research from psychological and educational fields shows the
existence of different forms of knowledge. There is also increasing
evidence that students can be very successful on standard, closed tests
with a knowledge that is highly inert and that they are unable to use in
more unusual and demanding situations (such as those encountered in
the workplace).

Test knowledge, in other words, is often the sort of knowledge that is
nontransferable and is useful for little more than taking tests. To
demonstrate what I mean, I would like to describe the results of a research
project that monitored the learning of students who experienced completely
different mathematics teaching approaches over a three-year period. In
response to these different approaches, the students developed different
forms of knowledge and understanding that had enormous implications for
their effectiveness in real-world situations.

Two schools in England were the focus for this research. In one, the
teachers taught mathematics using whole-class teaching and textbooks,
and the students were tested frequently. The students were taught in tracked
groups, standards of discipline were high, and the students worked hard. The
second school was chosen because its approach to mathematics teaching was
completely different. Students there worked on open-ended projects in
heterogeneous groups, teachers used a variety of methods, and discipline was
extremely relaxed. Over a three-year period, I monitored groups of students at
both schools, from the age of 13 to age 16. I watched more than 100 lessons
at each school, interviewed the students, gave out questionnaires, conducted
various assessments of the students' mathematical knowledge, and analyzed
their responses to Britain's national school-leaving examination in

At the beginning of the research period, the students at the two schools
had experienced the same mathematical approaches and, at that time, they
demonstrated the same levels of mathematical attainment on a range of tests.
There also were no differences in sex, ethnicity, or social class between the
two groups. At the end of the three-year period, the students had developed
in very different ways. One of the results of these differences was that
students at the second school -- what I will call the project school, as
opposed to the textbook school--attained significantly higher grades on
the national exam. This was not because these students knew more
mathematics, but because they had developed a different form of knowledge.

At the textbook school, the students were motivated and worked hard, they
learned all the mathematical procedures and rules they were given, and they
performed well on short, closed tests. But various forms of evidence showed
that these students had developed an inert, procedural knowledge that they
were rarely able to use in anything other than textbook and test situations. In
applied assessments, many were unable to perceive the relevance of the
mathematics they had learned and so could not make use of it. Even when
they could see the links between their textbook work and more-applied tasks,
they were unable to adapt the procedures they had learned to fit the
situations in which they were working.

The students themselves were aware of this problem, as the following
description by one student of her experience of the national exam shows:
"Some bits I did recognize, but I didn't understand how to do them, I
didn't know how to apply the methods properly."

In real-world situations, these students were disabled in two ways. Not
only were they unable to use the math they had learned because they could
not adapt it to fit unfamiliar situations, but they also could not see the
relevance of this acquired math knowledge from school for situations
outside the classroom. "When I'm out of here," said another student, "the
math from school is nothing to do with it, to tell you the truth. Most of
the things we've learned in school we would never use anywhere."

Students from this school reported that they could see mathematics all
around them, in the workplace and in everyday life, but they could not
see any connection between their school math and the math they
encountered in real situations. Their traditional, class-taught mathematics
instruction had focused on formalized rules and procedures, and
this approach had not given them access to depth of mathematical
understanding. As a result, they believed that school mathematical
procedures were a specialized type of school code -- useful only in
classrooms. The students thought that success in regard mathematics
to be a thinking subject. As one girl put it, "In math you have to remember;
in other subjects you can think about it."

The math teaching at this textbook school was not unusual. Teachers there
were committed and hard-working, and they taught the students different
mathematical procedures in a clear and straightforward way. Their students
were relatively capable on narrow mathematical tests, but this
capability did not transfer to open, applied, or real-world situations. The
form of knowledge they had developed was remarkably ineffective. At the
project school, the situation was very different. And the students'
significantly higher grades on the national exit exam were only a small
indication of their mathematical competence and confidence.

The project school's students and teachers were relaxed about work.
Students were not introduced to any standard rules or procedures (until a
few weeks before the examinations), and they did not work through
textbooks of any kind. Despite the fact that these students were not
particularly work-oriented, however, they attained higher grades than
the hard-working students at the textbook school on a range of different
problems and applied assessments. At both schools, students had
similar grades on short written tests taken immediately after finishing
work. But students at the textbook school soon forgot what they had
learned. The project students did not. The important difference between
the environments of the two schools that caused this difference in
retention was not related to standards of teaching but to different
approaches, in particular the requirement that the students at the
project-based school work on a variety of mathematical tasks and think
for themselves.

When I asked students at the two schools whether mathematics was more about
thinking or memorizing, 64 percent of the textbook students chose
memorizing, compared with only 35 percent of the project-based students.
The students at the project school were less concerned about memorizing
rules and procedures, because they knew they could think about different
situations and adapt what they had learned to fit new and demanding problems.
On the national examination, three times as many students from the
heterogeneous groups in the project school as those in the tracked
groups in the textbook school attained the highest possible grade. The
project approach was also more equitable, with girls and boys attaining the
different grades in equal proportions.

It would be easy to dismiss the results of this study because it was
focused on only two schools, but the textbook school was not unusual in
the way its teachers taught mathematics. And the in-depth nature of the study
meant that it was possible to consider and isolate the reasons why students
responded to this approach in the way that they did. The differences in the
performance of the students at the two schools did not spring from "bad''
teaching at the textbook school, but from the limitations of drawing upon only
one teaching method. To me, it does not make any sense to set any
one particular teaching method against another and argue about which one is
best. Different teaching methods do different things. We may as well argue
that a hammer is better than a drill. Part of the success of the project
school came from the range of different methods its teachers employed and
the different activities students worked on.

Some proponents of traditional teaching want students to follow the same
textbook method all of the time. A few students are successful in such an
approach, but the vast majority develop a limited, procedural form of
knowledge. This kind of knowledge may result in enhanced performances on
some tests, but the aim of schools must surely be to equip students with a
capability and intellectual power that will transcend the boundaries of the

Jo Boaler is an assistant professor of education at Stanford University in
Stanford, CA. Her book Experiencing School Mathematics received the
Outstanding Book in Education award in the United Kingdom in 1997. She can
be reached by e-mail at <>
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Fax: (618)453-4244
Phone: (618)453-4241 (office)

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