graphic Dept. of Education, Paige Hall, Tufts University, Medford, MA 02155 | Tel: 617.627.2934 | Fax: 617.627.3901
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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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