essay on Mathematics Education

[Teacher Educator]

Healing Math Learners

George W. Gagnon, Jr.

This essay is about the stagnant condition of mathematics education in
the United States. In 1957 Admiral Hyman Rickover, father of the
nuclear navy, called for improved math and science learning in
response to the launch of Sputnik by the U.S.S.R. I was in elementary
school then, and have watched math teaching remain in turmoil. In five
decades since Sputnik we have lived through old math, new math, math
reform, math standards, and math wars. Given the current political
climate, national defense may provoke another educational "call to
arms" in response to a perceived worldwide threat. International
competition in the global economy wasn't a sufficient wake up call in
the previous decade.

But we have a moral obligation to address the growing inequity in math
learning between students in communities of wealth and communities of
poverty, between suburban and urban students, and between white and
Asian students and other students of color. Whatever the provocation,
educators must change the way we have taught math for two generations
for the majority of learners to actually learn math and attain
quantitative literacy (Ellis, 2001).

Mathematics teaching and learning today is still narrowly focused on
skills, is rarely integrated with science or other subjects, and does
not connect theoretical understanding with real world applications.
Mathematics instruction now excludes many learners. About a third of
our students succeed in learning math to the level of algebra, and
less than 30 percent of high school graduates have studied advanced
math (National Science Board, 2006). Most of the others, including
many teachers, have been wounded by their experiences trying to learn
math. They believe that learning math is difficult and subtly
communicate this attitude to their students. Many parents were also
wounded and a fair number convey the impression to their children that
learning math really doesn't matter very much in life. "We don't use
algebra in our daily lives anyway, so why learn it?" This is an
American phenomenon. Students in other countries are more successful
in learning math through calculus and often complete advanced math or
science degrees in this country.

For the last fifteen years I have been working with students in urban
schools and coaching their math teachers. Together we are asking the
question, "Why are some students learning algebra while others are
not?" After a decade of supporting math education as a teacher
educator, classroom coach, and university researcher, I am reporting
my observations. Many math students were seriously wounded by their
early math teaching and don't obtain the most basic levels of
education required to become quantitatively literate. If we don't do
something different, yet another generation may be lost.


Synopsis of Conditions in Math Learning
In this section, I offer six observations about the current condition
of math teaching and learning.

1. Only on third of students learn enough math to enter science and
technology oriented careers. We teach math as a pen on overhead,
marker on board, or paper on pencil procedure for individuals rather
than a team-based process of making meaning for groups of students.
About a third of the students get most of it, about a third get some
of it, and about a third get little of it. Less than a third of
students in urban schools are learning enough math to complete STEM
(Science, Technology, Engineering, Mathematics) majors in college
although only a third of these successful students actually enroll in
those majors. Another third of the students understand enough to move
to the next grade level and generally avoid advanced courses in
science or math. The remaining third are lost to STEM careers or to
school. "Of 659,000 minority high school graduates in 2003, only
26,000 had the requisite preparation in science and mathematics to
qualify for admission to study engineering or technology at the
college level" (Slaughter, 2005).

2. Only students who learn the way we traditionally teach succeed in
math. Pencil and paper manipulation of algorithms has changed little
since the advent of common schools a century and a half ago. Students
who can look at examples in textbooks and than replicate patterns in
their imagination are making sense of math the way schools now teach
it. Students who think best through communication with others now
struggle with math instruction, as do those who think best with their
bodies and feelings since few patterns are taught this way.

We have put aside math discourse and math models in favor of
textbooks, examples, and practice problems or worksheets. Some
teachers believe only a few students are mentally qualified to learn
math rather than expecting most students to learn math through
calculus as 95% do in Japan. We expect that almost two-thirds of our
students will enter non-technical fields that involve only reading,
writing, and relating so they "don't need to learn" much math. The
seduction of consumer marketing, lottery fantasies, and inattention to
and retirement planning are evidence enough that a lack of numeracy
does matter (Ellis, 2001).

3. What schools teach and students learn are algorithms and
properties. We teach algorithms and properties in a vacuum of
conceptual understanding about such fundamentals as place value,
number relationships, and basic operations, particularly practical
applications in science or social studies. The current culture of
testing facts, figures, and formulas does not emphasize these
fundamentals. Elementary schools have few science classes and middle
schools do not integrate math into science since making the equations
and calculations is beyond many students' functional math level. A
respected veteran eighth grade science teacher reports that most of
his students in an urban school cannot:
• Easily weigh 5 grams of salt on a balance beam scale in a paper cup
that weights 9 grams.
• Calculate how much 50 milliliters of water weighs if one milliliter
weighs one gram.
• Given a ruler with inches on one side and centimeters on the other,
calculate how many centimeters are in an inch.
Each of these examples demonstrates that student lack conceptual
understanding and practical applications for the operations of
addition, subtraction, multiplication, and division.

4. Teachers teach what they know and teach math the way they were
taught. Most elementary math teachers have little preparation beyond
college algebra and a two-credit teacher education course in math
methods or about twenty-four classroom hours. Middle school math
teachers frequently have a major in a field other than math.

Despite the advent of an increasingly technological world, few math
classrooms use computer technology, multi-media data display, or
wireless communication. Most schools and math classes look like they
did fifty years ago with the possible exception of overheads or
whiteboards replacing chalkboards and graphing calculators replacing
slide rules and tables. One of the most difficult dilemmas is use of
calculators to do simple computations. Many teachers advocate
calculators and encourage students to do so. Others discourage
calculator use and ask students to rely on mental math skills to
estimate or make accurate computations. In general, the more students
can do without calculators the better their math skills are. ( I
encourage students to use place value and number relationships to
ground mental calculations in meaning rather than procedures.)

5. Students are being wounded by current elementary math teaching.
These are not self-inflicted injuries but rather wounds that result
from poor curriculum, poor instruction, and poor assessment in
arithmetic. Many students come from elementary school wounded by their
math teaching. Most of these students have given up and are lost by
the sixth grade before they leave high school in frustration. Few of
those who drop out or are incarcerated have more than rudimentary
arithmetic skills and no understanding of algebra. As a mathematics
education community, we must reach into the elementary schools to
support the teachers, create a healthy math learning culture and
staunch the wounds. This can only be done by first healing the math
wounds of current elementary teachers and improving the quality of
preparation (Kenschaft, 2005).
6. Teachers in middle school try to triage these math wounds. Like
interns in hospital emergency rooms, middle school teachers "triage"
or sort students into three categories – non-urgent, emergent/urgent,
and untreatable. Visit many urban middle schools and you will find
three levels of math classes: college preparatory classes for those
students who are on track and do their homework; grade level classes
where students take the same math curriculum but for twice as long,
either over two years rather than one or in double period classes;
remedial or "academy" classes where students have a regular class and
a second class to revisit concepts they should have learned
previously. Effective middle school math teachers report that most of
the wounded students need to experience some success in mathematics to
begin healing their math wounds. Many students are so deeply wounded
they no longer care if they experience success and behave accordingly.

What Will Heal Math Learners?
Current grants from the National Science Foundation and many private
foundations are putting funds into secondary math reform rather than
math learning by elementary teachers. However, the need for healing
math wounds in middle school will continue until the math education
community assumes responsibility for the professional development of
elementary teachers. This must not be a trivial in-service effort to
familiarize them with textbooks, standards, and curriculum objectives
as legislation now requires. Elementary teachers must improve their
understanding of mathematics and study the way students think and
learn math.

This learning by elementary teaches will require elementary math
coaches, professional learning communities, and commitment to
curriculum reform focused on practical applications of operations with
understanding. Until that happens as a matter of course, elementary
teachers will learn with and from one another and gain confidence in
their own conceptual understanding of math, but at a slower rate and
at the expense of their students. Steen (2003) observes the lack of
progress in improving math performance as an indicator "of widespread
underestimation of the depth of understanding and intensity of effort
required to teach mathematics effectively. A lack of respect for the
complexity of the problem encourages quick fixes (smaller classes,
higher standards, more tests, higher teacher salaries) that do not
yield greater disciplinary understanding or pedagogical skill" (p. 3).
With appropriate support, elementary teachers can move beyond the
formal algorithm and testing approach to teaching math and focus on
the informal diagnosis and assessment of math learning that leads to
individual and conceptual understanding of math operations.

Teachers Can Heal Math Wounds Through Diagnosis and Treatment
The current educational system has almost snuffed out the art of
individual diagnosis and treatment that is the foundation of healing.
Schools approach teaching quite narrowly as a function of explanation,
practice, and testing primarily through textbooks, workbooks, and
worksheets completed in class or as homework. Evaluation is based on
standards, achievement testing, and high stakes exit exams in most
public schools. Evaluating the condition of patients does not heal
their wounds. Diagnosing the extent and nature of individual wounds
and treating each patient accordingly is a long-term and expensive
process but produces reliable results. The most effective math
teachers know this and work with students at lunchtime and after
school to improve their understanding of math concepts.

Curriculum Reform Must Focus on How Students are Thinking and Learning
Math
The math education research community is ponderous and moving very
slowly in the direction of healing. Studies are so tightly focused on
single problems or specific skill sets that the big picture is very
diffuse and cloudy-- like an impressionist painting that only looks
real from a distance. We need to mount a serious effort to determine
how children think about and learn mathematics as they make meaning
individually and collectively. Programs such as Cognitively Guided
Instruction and Developing Mathematical Ideas have been successfully
engaging elementary math teachers in healing math wounds and
conceptually understanding math for two decades or more.

We must support mathematics learning throughout the educational
enterprise. All teachers deserve full support so they can engage in
more powerful levels of math learning for themselves and their
students. Efforts in the East Bay Area around UC Berkeley such as
Diversity in Math Education, Leading for Excellence and Equity in Math
and Science, and Pre-Engineering Partnerships academic acceleration
academies are demonstrating that secondary teachers can heal math
wounds more effectively.

Classroom math coaching and professional learning communities are at
the core of full support for professional development of elementary
and secondary teachers. As teachers learn to understand and teach math
conceptually, then pre-school children will learn patterns and
counting, elementary students will learn place value and operations,
middle school students will learn algebra and geometry, and high
school students will learn calculus. Then the majority of students,
not only the privileged few, will succeed in understanding
mathematics.

Educational Policy Must Include Numeracy as Well as Literacy
Education and legislative policy makers must realize that the future
of our ecology, technology and economy depend upon a highly educated
workforce that can solve complex problems, relate with one another,
and communicate with colleagues throughout the world. Math literacy
must be elevated to the status of language literacy by showing each
other and our children real world applications and working with adult
learners in math the way we enroll newcomers in English classes. This
approach would require policies that make math literacy a foundation
of learning the way oral and written communication are.

Parents and teachers would be supported to study math as adult
learners. Many parents themselves do not know algebra, so they can't
assist their children with homework or convey the attitude that math
has been important in their lives or careers. For our educational
system to succeed in making mathematics truly accessible, parents must
also have access to adult education in math so they can heal their
wounds as well.

Concluding Comments
We risk losing another generation of urban youth to the street culture
where they fight, injure, and waste each other at an enormous psychic,
spiritual, and financial cost to our cities, indeed, to our whole
society. When only the upper and middle class youth have access to
high quality education, we all suffer the consequences. Further, math
will continue to be the gatekeeper preventing a majority of public
school students from equal access to an adequate, much less higher,
education.

Despite some rising test scores, American math education is wounding
far to many students. Far more fail than succeed, and most developed
countries of the world do a more effective job of teaching their
youth. Mathematics learning can only be improved if we start to help
elementary students conceptually understand number relationships and
applied operations. Their teachers need full support to heal
themselves and their students.

New metaphors of math learning are being embraced in the academy and
can influence all levels of education. Visions of math learning would
contain creative and useful diagnostics so students and teachers could
more powerfully create specific learning opportunities. Healing math
wounds requires a curricular, community, and cultural commitment to
math education. We face a complex problem that requires a simple
solution-- change the way we have been teaching math for 150 years and
fund our entire educational system appropriately!

References

Ellis, W. (2001), Numerical common sense for all. Chapter in
Mathematics and democracy: The case for quantitative literacy. Ed. pp.
1-22.

Kenschaft, P. (2005) Racial equity requires teaching elementary school
teachers more mathematics. Notices of the AMS. 52(2), 208-212.

National Science Board (2006) Science and Engineering Indicators.
Retrieved 6-21-06 http://www.nsf.gov/statistics/seind06/c1/c1h.htm

Slaughter, J. (2005) National Action Council for Minorities in
Engineering, Inc. NCAME Symposium 2005 Keynote address. Retrieved
6-27-06 from: http://www.nacme.org/symposium/index.html

Steen, L. A. (2003) Math education at risk. Issues in Science and
Technology, Summer 2003, 61-65. Woodrow Wilson National Fellowship
Foundation. Retrieved 6-20-06 http://www.maa.org/Ql/061-66.pdf

About the author



George W. Gagnon, Jr. is the Pre-Engineering Partnerships (PEP)
Director of the Center for Underrepresented Engineering Students in
the College of Engineering at the University of California, Berkeley.
He works to remove barriers for admission to Berkeley Engineering for
underrepresented students in the East Bay region through academic
acceleration in mathematics and pre-engineering curriculum.
gwga...@berkeley.edu