Algebra as a Gatewaykeeper to Post-Secondary Success
In her recent Education Week blog post, Susan Fairchild explains the system forces that create off-track students and the system forces that make it hard for students to get back on track. She suggests that the unidirectional nature of the educational system, the cumulative nature of learning, and the cascading effects that occur when students who have not reached grade-level proficiency are promoted and passed on to teachers who may be unprepared to deal with such learning deficits, make the educational system well designed to generate off track students. Rethinking the structure of our schools is urgently needed if we are to address this problem. Here, we’ll take a detailed look at how this problem manifests in high school mathematics and consider how schools can better deploy resources to both remedy past weaknesses and support future success.
How students lapse into cycles of failure in school mathematics
For secondary students, Algebra I has long been viewed as a potential gateway to postsecondary success. However, for others, Algebra I often serves as a gatekeeper, becoming another obstacle to success in high school and beyond. In “Algebra and the Underprepared Learner,” Timothy Stoelinga and James Lynn explain that “Algebra 1 has historically represented an important transition point in the learning of mathematics, requiring the use of generalized models, mathematical abstractions, and understandings of variables and symbols, all of which are particularly challenging for many students...Research has also indicated that many eighth and ninth grade students who are required to take Algebra 1 are also underprepared and need more support to succeed because of weak foundations in prerequisite concepts.” (p.3)
The students struggling in Algebra I are not an insignificant population. Looking at the performance of freshmen in New Visions schools during the 2013-2014 school year (N= 13,288 total students studied), we saw a significant number of students struggling with Algebra with an unsurprisingly lower course passage rate compared to all other core-courses.
Students who struggle when first attempting Algebra I often continue to do so when channelled into historically unsuccessful, post-course remediation. This repeated, reinforcing struggle arises when students are asked to take the same course taught with the same approaches despite the evidence that these approaches have not succeeded with these students. These “reinforcing patterns of failure” can lead to students believing that they will not be able to earn a high school diploma, and can, in turn, be a driver of drop-out rates in the absence of effective interventions.
How can high schools best deploy resources to meet the challenge of Algebra I?
High schools therefore face a double-challenge when deploying resources to meet the needs of underprepared Algebra I students: they must both remedy the lack of mathematical skills brought from middle school and teach a very-crowded high school curriculum—and, in New York State at least, both challenges must be met in anticipation of high-stakes mathematics exams required for graduation. There are two prongs to a successful solution: first, school mathematics must be grounded in richer mathematical problems that offer multiple entry points to students as well as actionable formative information for teachers. Second, curricula that allow students to engage in algebraic thinking, even with pre-algebraic methods, must be used alongside the traditional, standards-driven high school curricula.
For the last three years, New Visions has tackled the challenge of supporting student success in high school mathematics through Accessing Algebra through Inquiry (a2i), a federally-funded math program aiming to increase students’ achievement in Algebra I (as well as Geometry and Algebra II) by supporting instructional practice grounded in rich performance tasks, such as Illustrative Mathematics, and opportunities to employ mathematical thinking in conversations with their teacher and their peers. Through a2i, teacher inquiry teams, supported with in-school assistance and ongoing professional development, use curriculum units developed by the a2i staff, research-based assessments, and carefully designed and learner-responsive formative assessment lessons (FALs) in order to improve student achievement (as measured by course passage and scores on state math exams) and to prepare them for success in postsecondary study.
While the rich tasks found in a2i’s curriculum provide students with multiple entry points and support students in using elementary and middle school math strategies to solve algebraic problems, feedback indicates that teachers need resources more finely targeted to support students who were struggling with foundational math concepts, those often unable to construct even basic, below-grade-level approaches to these algebraic tasks. In order to better support these students, New Visions reached out to the Education Development Center (EDC) to pilot a newly developed course, Transition to Algebra, as a supplement to a2i Algebra I.
Transition to Algebra is a classroom resource that approaches algebra instruction differently. Instead of focusing on the traditional algebra algorithms, the course uses logic puzzles, problems, and explorations to help teachers uniquely build students' mathematical conceptions. The developers of the curriculum argue that while success in Algebra I requires arithmetic skills, students should not wait to learn new algebraic content until they have mastered arithmetic. As Goldenberg, Mark, Kang, Fries, Carter and Cordner explain, “putting off algebraic learning while redoing and redoing arithmetic further delays students who are already behind, further convinces them that they are making no progress and are slow learners, and (typically) employs methods that have already failed. In fact, algebraic ideas often help students see previously unnoticed patterns in arithmetic and use those ideas to become successful even at the arithmetic.” (Making Sense of Algebra p.51).
A number of studies have shown that intentionally sequencing developmental supports with grade-level mathematics, and providing students with a “double dose” of Algebra, has long-lasting positive impacts on their academic careers. Studying the impact of a double period of Algebra on students in the Chicago Public School system, Kalena Cortes, Joshua Goodman and Takako Nomi found “positive and substantial longer-run impacts of double-dose algebra on college entrance exam scores, high school graduation rates, and college enrollment rates.” This approach mirrors the co-requisite model advocated by Complete College America in order to improve the numbers of students who complete college-credit math courses: “Co-requisite developmental education enrolls students in remedial and college-level courses in the same subject at the same time. Students receive targeted support to help boost their understanding and learning of the college-level course material.”
The combination of these two strategies–addressing algebraic thinking through non-traditional models and increasing the time students participate in school math courses–offers the best combination of resources for students who have struggled with Algebra I. Starting this Fall, a2i will recommend that schools with historically high rates of first-time attempt failure on the Algebra Regents Examination (a state examination) offer two periods of math to 9th grade students through a2i Algebra I and Transition to Algebra. While it requires a shift in resources to provide 9th grade students with an additional math period, research shows that it is a more effective use of resources in schools where the majority of students will have to be programmed for repeated algebra courses in subsequent years. The a2i project team will build and support the integration of the two courses, so that teachers can bridge the habits of mind developed in Transition with the traditionally-examined content explored in Algebra I. The combination of additional resources and strategic instruction will better support identified students who, based on their middle school test scores or diagnostic exams, would most likely struggle with algebra, giving them the support they need to be successful in high school mathematics and beyond.