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Frequently Asked Questions
Q: Would Early Algebra education
help a child's performance on the MCAS?
A: We are looking forward to answering just that
question this year. For the most part, the MCAS does not
test algebra skills until the 8th grade. This year, one
group of students with whom we previously worked in
elementary school is entering the 8th grade. We do plan
to compare their scores with the average scores from
their schools, communities, and Massachusetts as a whole
to look at the long term effects of Early Algebra
lessons.
Our preliminary data so far is promising, when fifth
grade early algebra students were presented with algebra
questions from the 8th grade MCAS test, they
occasionally outperformed their older Boston
counterparts.
Q: What do you look for in your research?
A: We conduct our research in classrooms, with the
cooperation of teachers, with whom we implement early
algebra lessons. We evaluate the results of our
intervention through analysis of students’ participation
in video-taped lessons and through their performance in
written assessments given before and after the
intervention, comparing their results with those of
students who did not participate in the lessons.
Our approach to algebra in elementary school is based on
the premise that a deep understanding of arithmetic
requires mathematical generalizations and understanding
of basic algebraic principles. We view the introduction
of algebraic activities in elementary school as a move
from computations on particular numbers toward thinking
about relations among sets of numbers. Central to our
approach is the use of contextualized problems and use
of multiple representations, namely, natural language,
line segments, function tables, Cartesian graphs, and
algebra notation.
We have conducted three longitudinal studies with the
aim of documenting how children’s algebraic thinking
evolves as they are introduced to algebra principles and
representations over rather long periods of time (one to
three years).
In the first study we developed 16 lessons over the year
in a third grade classroom of 18 children.
 The second study was implemented in four classrooms (69
children), with one weekly 90 minutes early algebra
lesson from the second semester in second grade to the
end of fourth grade.
In the third study we worked with 26 students as they
progressed from 3rd through 5th grade. In 3rd and 4th
grades, the children participated in two 60 minutes
lessons per week (total of 50 lessons in 3rd grade and
36 in 4th grade), each one followed the next day by a 20
minute homework review; in 5th grade they participated
in one 90 minutes lesson per week (18 lessons total)
followed the next day by 30 minutes of homework review.
The students in all three studies come mainly from
minority and first generation immigrant families located
in the Boston area.
In our classroom work we seek to generate a teaching and
learning environment that is conducive to children’s
presentation of their own perspectives, ideas, and ways
of representing the problem. Children’s first verbal
reactions to the problem are brainstormed and they are
asked to show, on paper, their ideas about the problem
and their suggested solutions. The children in the front
of the class then share the notations they produced. The
instructor then guides them towards the development of
other mathematical notations.
Our goal is to provide children with opportunities to
explore and represent relationships between sets of
numbers, to deal with variables and functions, and to
solve verbal problems using the tools of algebra.
Q: If early algebra is a good idea, why isn't
it already part of our educational system?
A: First, our current system tends to follow a
progression in mathematics, in which certain topics are
addressed first and others follow behind. This has
traditionally been true within elementary school (first
addition, then subtraction, then multiplication…) as
well as across the K-12 curriculum, in which arithmetic
is mastered first and then algebra, geometry, calculus
and other topics are introduced in succession. So
tradition plays a role in the current structure of
mathematics education.
Second, working with algebra means introducing the idea
of a "variable", often some letter, like "n", that we
use to stand for some number we don't know. Some
researchers and theorists believe that students of
elementary school age are not able to think about an
abstract concept like this. However, students in our
studies have been able to work with a variable quite
readily, and when they use a variable they often explain
to us that they're using it to stand for something they
don't know.
Third, historically it was thought that a mastery of
arithmetic was needed to deal with the operations used
in algebra. However, we believe that algebra can be used
to generalize arithmetic. In doing so, students may be
able to deepen their understanding and increase their
mastery of arithmetic itself! For example, the current
system emphasizes arithmetic operations such as 1+3,
2+3, 3+3, 4+3 and so on. But we can introduce the
student to the idea of n+3, where the n could be any
number! This gives the student a view of the pure idea
of "plus 3" -- an idea that is part of a deep
understanding of arithmetic itself! This sort of work
may help to ease the transition to algebra in middle
school or high school, and remove the stigma of algebra
as a completely separate kind of math.
Q: What can I do to bring Early Algebra
education into schools?
A: To Teachers, Mathematics Coordinators, and
School Principals:
As teachers already know, the National Council of
Teachers of Mathematics proposes that algebra should be
introduced in elementary school. If you are a teacher,
math coordinator, or school principal, and would like to
find out how this could be done, Materials section
displays a wealth of early algebra lesson plans that
could be implemented in your school. You should also
read some of our research papers: they describe how we
implemented some lessons and show how the students
responded to them. A short account of our approach is
found in the paper by Carraher, D.W., Schliemann, A.D.,
& Brizuela, B.B. (2001).
Algebra in the Early Grades.
Hands On!, Spring 2001, Vol. 24, No. 1, 8-11. Other
paper that may interest you are:
-
Early
Algebra is Not the Same as Algebra Early
Carraher, D.W., Schliemann, A.D. & Schwartz, J. (2007).
Early algebra is not the same as algebra early.
In J. Kaput. D. Carraher, & M. Blanton (Eds.),
Algebra
in the Early Grades. Mahwah, NJ, Erlbaum, pp. 235-272
-
Ten-year-old
Students Solving Linear Equations
Brizuela, B.M.,& Schliemann, A.D. (2004).
Ten-year-old students solving linear equations. For the Learning of
Mathematics, 24, 2, 33-40
- Algebra in Early Mathematics: A Longitudinal
Intervention
Carraher, D.W., Schliemann, A.D., & Brizuela,
B.B. (2008). Algebra in Early Mathematics: A
longitudinal Intervention. Paper presented at the
11th International Congress on Mathematical
Education. Monterrey, Mexico, July, 6-13.
-
Algebra
in Elementary School
Schliemann, A.D., Carraher, D.W., Brizuela, B.M.,
Earnest, D., Goodrow, A., Lara-Roth. S. & Peled, I.
(2003). Algebra in elementary school. In N. Pateman, B.
Dougherty, & J. Zilliox (Eds.) Proceedings of the 2003
Joint Meeting of PME and PME-NA. CRDG, College of
Education, University of Hawai'i: Honolulu, HI, Vol. 4,
pp. 127-134.
Finally, if you would like to have one of us visiting
your school, we will be happy to give a presentation on
our work at your school.
To Parents:
If you would like to help your children develop a better
understanding of the arithmetic they are learning at
school, our lesson plans include problems that you could
ask them to work on. You should also read the paper by
Carraher, D.W., Schliemann, A.D., & Brizuela, B.B. (2001).
Algebra
in the Early Grades. Hands On!, Spring
2001, Vol. 24, No. 1, 8-11: It describes how we
implemented some lessons and show how the students
responded to them. This will help you in discussing the
problems with your children. Our Frequently Asked
Questions link may also answer some of your questions.
To Policy Makers and
Administrators:
The best way to decide what is best for schools is to know about the results
of classroom interventions that carefully monitors students’ progress. Our
research shows how young children’s achievements after they participated in our
intervention program. Of interest to policy makers are the following papers:
- Algebra in Early Mathematics: A Longitudinal
Intervention
Carraher, D.W., Schliemann, A.D., & Brizuela,
B.B. (2008). Algebra in Early Mathematics: A
longitudinal Intervention. Paper presented at the
11th International Congress on Mathematical
Education. Monterrey, Mexico, July, 6-13.
-
Algebra
in Elementary School
Schliemann, A.D., Carraher, D.W., Brizuela, B.M.,
Earnest, D., Goodrow, A., Lara-Roth. S. & Peled, I.
(2003). Algebra in elementary school. In N. Pateman, B.
Dougherty, & J. Zilliox (Eds.) Proceedings of the 2003
Joint Meeting of PME and PME-NA. CRDG, College of
Education, University of Hawai'i: Honolulu, HI, Vol. 4,
pp. 127-134.
To Curriculum Developers:
Our Materials section displays a wealth of early algebra lesson plans that
could be inspire the development of elementary mathematics curricula. Our
research papers in the Publications section will provide the foundations for
early algebra curricula.
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