Teaching and learning algebra has undergone a critical transformation in the US over the last two decades. Recognizing that historical paths to algebra have been largely unsuccessful in terms of students’ achievement in algebra, mathematics educators have increasingly advocated that algebra be re-conceptualized in school mathematics as a longitudinal, grades K-12 strand of thinking. In particular, mathematics educators advocate that students have long-term, sustained algebra experiences in school mathematics, beginning in the elementary grades, that build their natural, informal intuitions about structure and relationships into formalized ways of mathematical thinking. Yet research has not systematically addressed how the development of children’s algebraic thinking effects their understanding of core algebraic concepts and practices in comparison to students who receive more traditional arithmetic-based instruction. Indeed, a fundamental assumption of early algebra education is that it will increase children’s understanding of algebraic concepts and practices and, ultimately, improve their success in the study of more advanced mathematics—particularly algebra—in secondary grades. To date, however, this premise is largely untested. The projects that comprise the Project LEAP portfolio (described below) seek to address this premise.
In our first project (NSF, 2009-2013), we constructed an Early Algebra Learning Progression [EALP] consisting of a curricular framework and progression developed by coordinating research, curricular, and mathematical perspectives; a Grades 3-5 instructional sequence based on the framework and progression; associated assessments; and levels of sophistication describing strategies observed in children’s mathematical work. We also conducted a preliminary study to examine the impact of our EALP’s instructional intervention as measured by our assessments and found that students who experienced the early algebra intervention outperformed comparison students on our early algebra assessment and were more apt by posttest to use algebraic strategies to solve problems.
In our second LEAP project, we used the instructional sequence and grade-level assessments developed in the first project to conduct a small-scale longitudinal study to measure the impact of our comprehensive, sustained Grades 3–5 early algebra intervention on students’ algebra understanding in elementary grades and their algebra readiness in middle school. The project used a quasi-experimental design to compare the performance of students who received our early algebra intervention to students who receive more traditional elementary grades instruction. We found that intervention students outperformed control students on post-assessments at each of Grades 3–5 and were more able to flexibly interpret variables in different roles and use variable notation in meaningful ways in different mathematical contexts.
The Impact of a Teacher-Led Early Algebra Intervention on Children’s Algebra-Readiness for Middle School (LEAP 3)
We scaled up our work in LEAP 2 to a much larger number of schools (approximately 46 schools) and students (approximately 3200 students), and are studying the effectiveness of our EALP’s instructional intervention when implemented by regular classroom teachers. We are comparing the performance of students who received the early algebra intervention in Grades 3-5 to students who received more traditional elementary grades instruction, and are following both sets of students (intervention and control) into Grade 6.
The goal of this project is to extend our Grades 3–5 work into the early grades by designing and testing a Grades K–2 early algebra learning progression. This work will include developing an instructional sequence and interview assessments and testing the impact of the instructional sequence on students’ learning. Two classrooms of students will be followed throughout Grades K–2 in a researcher-led intervention. In Year 4, a cross-sectional teacher-led effectiveness study will take place, with one experimental and one control classroom at each grade level.
Building a Grades K-2 Early Algebra Learning Progression Prototype for Diverse Populations (LEAP 4.2)
The goal of this project is to extend our Grades 3–5 work into the early grades by designing and testing a Grades K–2 early algebra learning progression and paying particular attention to the needs and experiences of the diversity of learners, including those with math difficulties or disabilities. This work will include developing an instructional sequence and interview assessments, and testing strategies to help a diverse range of students, including examining how mathematical tools mediate students’ algebraic thinking. A cross-sectional researcher-led intervention will take place in one classroom in each of Grades K–2.
Identifying Effective Instructional Practices that Foster the Development of Algebraic Thinking in Elementary School (LEAP 5)
In this project, we are revisiting the classroom video collected in The Impact of a Teacher-Led Early Algebra Intervention on Children’s Algebra-Readiness for Middle School (2014-2018) with the goals of developing an early algebra classroom observational instrument and identifying instructional practices that are associated with students’ early algebra learning.
Stephens, A., Stroud, R., Strachota, S., Stylianou, D., Blanton, M., Knuth, E., & Gardiner, A. M. (In press). What early algebra knowledge persists one year after an elementary grades intervention? Journal for Research in Mathematics Education.
Blanton, M., Stroud, R., Stephens, A., Gardiner, A., Stylianou, D., Knuth, E., Isler, I., & Strachota, S. (2019). Does early algebra matter? The effectiveness of an early algebra intervention in grades 3–5. American Educational Research Journal, 56(5), 1930-1972.
Blanton, M., Isler-Baykal, I., Stroud, R., Stephens, A., Knuth, E., & Murphy Gardiner, A. (2019). Growth in children’s understanding of generalizing and representing mathematical structure and relationships. Educational Studies in Mathematics, 102, 193–219.
Stylianou, D., Stroud, R., Cassidy, M., Gardiner, A., Stephens, A., Knuth, E., Demers, L. (2019). Putting early algebra in the hands of elementary school teachers: Examining fidelity of implementation and its relation to student performance. Journal for the study of education and development/Infancia y Aprendizaje (Special Issue: Early Algebra).
Fonger, N. L., Stephens, A., Blanton, M., Isler, I., Knuth, E., Gardiner, A. (2018). Developing a learning progression for curriculum, instruction, and student learning. Cognition and Instruction, 36(1), 30–55.
Stephens, A., Fonger, N., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Gardiner, A. (2017). A learning progression for elementary students’ functional thinking. Mathematical Thinking and Learning, 19(3), 143–166.
Knuth, E., Stephens, A., Blanton, M., & Gardiner, A. (2016). Building an early foundation for algebra success. Phi Delta Kappan, 97(6), 65–68.
Blanton, M., Stephens, A., Knuth, E., Gardiner, A., Isler, I., & Kim, J. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87.
Stephens, A., Blanton, M., Knuth, E., Isler, I., & Gardiner, A. (2015). Just say YES to early algebra! Teaching Children Mathematics, 22(2), 92–101.
Isler, I., Marum, T., Stephens, A., Blanton, M., Knuth, E., & Gardiner, A. (2014). The string task: Not just for high school. Teaching Children Mathematics, 21(5), 282–292.
Stephens, A., Knuth, E., Blanton, M., Isler, I., Gardiner, A., & Marum, T. (2013). Equation structure and the meaning of the equal sign: The impact of task selection in eliciting elementary students’ understandings. Journal of Mathematical Behavior 32(2), 173–182.