By:
**Erica Litke, Ph.D.**, * Assistant Professor, School of Education*, University of Delaware

In its current form, school algebra serves as a gatekeeper to higher-level mathematics. Researchers and policy makers have pushed to open that gate—providing more students access to algebra, focusing in particular on those students historically denied access to higher-level mathematics. For example, states and districts have enacted policies aimed at both enrolling more students in this key course and eliminating bias in who is permitted access (Remillard et al., 2017). Similarly, reformers have focused on the timing of the course, aiming to enroll students as early as possible to open pathways to calculus and to diversify access to higher level mathematics. One district in North Carolina, for example, developed a policy that targeted students likely to be successful in algebra in 8^{th} grade for advancement in the early middle school years. The state of California attempted to do this by shifting algebra content into 8^{th} grade standardized assessments (e.g., Domina et al., 2015; Dougherty et al., 2017).

Together, efforts like these *have* been successful in increasing access—more students are enrolling in algebra and enrolling ever earlier in their academic trajectories (Remillard et al., 2017; Stein et al., 2011). Yet we have not seen equal advances in achievement (National Center for Education Statistics, 2019). Overall NAEP mathematics proficiency rates have remained stable in the past decade in both 8^{th} and 12^{th} grades. Even more worrisome have been persistent demographic gaps in achievement, showing that despite the focus on access, inequitable mathematics education persists. Thus, improving access is alone insufficient to remedy racial inequities in mathematics education.

At the same time, school algebra itself has been in the crosshairs, with some calling for its reform and others its abolition (e.g., Berry & Larson, 2019; Levitt, 2019). Some argue that algebra needlessly holds students back and is only minimally relevant for future careers (e.g., Hacker, 2012). Others argue that school mathematics more generally is out-of-date and out of touch. This has led to recent calls to replace the emphasis on algebra in schools with courses that center on quantitative literacy, data fluency, or statistical investigation (e.g., Boaler & Leavitt, 2019). However, absent from the discussions of access, timing, and relevance of school algebra has been a focus on *instruction*.

**Let’s Not Be So Quick to Give Up on Algebra**

**Arguments that suggest we do away with school algebra frame algebra itself as the problem and leave unexamined how and in what ways that algebra is taught.** This is a problem because it absolves us of having to think critically about *changing how we* *teach* algebra to create more equitable classroom spaces that support students in learning algebraic ideas.

Large-scale examinations of mathematics classrooms at the middle and secondary level find evidence that instruction remains largely teacher-centered and procedurally focused (Boston & Wilhelm, 2017; Hiebert et al., 2005; Jacobs et al., 2006; Litke, 2020). Furthermore, these classrooms rarely include a focus on supporting students’ mathematical identity (Gutiérrez, 2012) or include teaching that affirms students’ cultural and linguistic backgrounds. Mathematics classrooms that do not sufficiently support students’ learning—whether they center algebra or statistics—will not support equitable outcomes for students.

Regardless of what mathematics courses students take in school, they will learn essential algebra content and use algebraic reasoning (National Council of Teachers of Mathematics, 2018). For example, students will need to understand growth and change, explore and generalize patterns, and connect across representations of mathematical ideas. This necessitates that we turn our attention to how to support students in learning algebraic ideas. First, if we wish to make algebra classrooms more equitable and affirming spaces, it is critical to enact such practices as positioning students as mathematically competent, attributing mathematical authority to students, and connecting students’ home and cultural contexts to the mathematics classroom, among others (e.g., Wilson et al., 2019). Doing so supports both students’ learning and success and works to make mathematics classrooms less marginalizing spaces.

In addition, in order to make sure that students are provided with equitable opportunities to be successful in algebra and beyond, it is important to consider how to teach algebra in a way that leverages instructional practices that research shows support student learning. While school algebra was once considered generalized arithmetic and has traditionally included a good deal of symbolic manipulation of expressions and equations, it has evolved to incorporate a focus on functions and relationships, patterns and generalizations, and application. Research and professional organizations suggest that algebra teaching integrate conceptual understanding, procedural fluency, mathematical reasoning, and critical thinking (Berry & Larson, 2018; National Council of Teachers of Mathematics, 2018), but teachers are often left unsure about how to do this or unsure what this looks like in the algebra context.

**Key Instructional Practices to Support Students’ Opportunities to Learn Algebra**

Drawing on what we know about instructional practices that support student learning in algebra (e.g., Booth et al., 2017; Chazan & Yerushlamy, 2003; Kieran, 2013; Rittle-Johnson & Star, 2009; Sleep, 2012; Star et al., 2015), I have studied over one hundred algebra lessons to understand how and in what ways teachers might better support students’ learning opportunities. Lesson videos were collected as part of the Measures of Effective Teaching (MET) study and came from 9^{th} grade algebra classes from five metropolitan school districts. In one study, I worked with a research team to understand and describe the ways in which teachers supported students’ learning opportunities in algebra. Through an iterative process of watching videos, consulting the research literature, categorizing and discussing our observations, and returning to the video, I identified five key instructional practices that teachers engaged in that help to integrate algebraic concepts with algebraic procedures and support students in making critical connections within and across algebraic ideas (Litke, 2020).

**. Students can make sense of procedures by making meaning of the individual steps of a procedure or the solution generated by a procedure. Students can also be supported to attend to the purpose or goal of a procedure or consider the mathematical properties underlying the procedure.**

*give meaning to algebraic procedures***. Developing procedural flexibility means noticing and using algebraic structure to know which procedure to use, under what conditions, and in what ways. This can happen by having students note multiple pathways through a procedure or attend to key conditions for steps within a procedure. Students can also attend to the applicability conditions of a procedure, knowing what must hold for a procedure to be employed. Instruction can also include opportunities for students to compare multiple procedures for the same problem type for their efficiency, their affordances, or their limitations.**

*develop flexibility within and across procedures***, understanding how different representational forms relate to one another and how they show similar algebraic ideas.**

*make connections across representations***, situating the content in any given lesson within the storyline of algebra or within the broader domain of mathematics. Teachers might do this by connecting what they are teaching to previously learned content or future topics. Doing this helps students to see the ideas in algebra as connected to one another, rather than as discrete and unrelated topics.**

*build connections across topics***. One reason that algebra is difficult for students is that it is more abstract than prior school mathematics. Teachers can leverage numeric examples, concrete representations, or even analogies to support the students’ understanding of abstract algebraic concepts, formulas, or notation. For example, teachers might do this by using numbers to develop generalized rules or properties, relate an abstract concept to its analogous numerical idea, or use concrete examples or manipulatives to introduce or illustrate abstract concepts.**

*make connections between numeric and abstract algebraic ideas*These five instructional practices are specific to algebra in that they support students’ algebraic learning opportunities, but they are not unique to algebra and could be leveraged in other mathematics content domains as well.

**Implications for Preparing Effective Algebra Teachers**

In another study using video data drawn from the MET project (Litke, 2020), I looked at 108 9^{th} grade algebra lessons from 30 teachers across the five districts using an observational tool called the Quality of Instructional Practice in Algebra (QIPA) to measure the frequency and depth of these five instructional features. I found that the vast majority of algebra lessons in the sample included a large focus on teaching algebraic procedures and instruction was largely teacher-directed with only a modest degree of student participation and engagement with mathematical ideas.

However, within this context, I found that teachers enacted the five instructional practices to varying degrees, though not always in great depth. For example, over one-third of all lessons included times where teachers and students gave meaning to algebraic procedures to a modest extent and in half of the lessons, teachers supported students to develop procedural flexibility to some degree. Other practices occurred less frequently—close to two thirds of lessons in the sample included no instances where teachers worked to build connections across mathematical topics. However, these glimmers of promise show that teachers may be aware of these practices and with attention and development, can enact them more frequently when appropriate and in greater depth.

My research suggests that integrating these five instructional practices represent small adjustments to existing practice that could pay large dividends in terms of supporting students’ learning of algebra content. But it also suggests that teachers and those learning to become teachers may need additional support in developing their teaching practice along these lines. I am not suggesting we abandon the larger push for student-centered or conceptually focused instruction, but that building on what teachers already feel comfortable doing can serve as a bridge to more ambitious teaching practices. Improvement efforts should not focus on enacting each practice in great depth at all times, but rather on understanding *when* instruction would benefit from a focus on a given practice and on identifying opportunities to deepen instruction.

Teacher educators, instructional coaches, and others involved in instructional improvement efforts can support teachers to focus on these practices in a number of ways:

- Make explicit the need to integrate conceptual understanding and procedural fluency, choosing one or more of these five instructional practice as a focus in mathematics content or pedagogy courses in teacher preparation.
- Support pre-and in-service teachers to enact specific instructional routines that align with these instructional practices. Instructional routines are a way to leverage predictable teaching structures to support students in engaging with key mathematical content and ways of thinking. For example, Jon Star, Bethany Rittle-Johnson, and Kelly Durkin’s Compare and Discuss Multiple Strategies routines use side-by-side worked examples alongside reflection questions to support students to give meaning to algebraic procedures and develop procedural flexibility. Another example from Star et al., (2015) focuses on teaching strategies for improving algebra knowledge in middle and high school students. Similarly, Amy Lucenta and Grace Kelemanik’s Connecting Representations Routine provides a structure for students to develop their ability to make connections between and among algebraic representations like graphs and equations.
- Work with pre- and in-service teachers to select one of the instructional routines to be a focus of professional learning. Support teachers to develop questions to guide students and design their own instructional routines tied to their focus.
- Record videos of teaching and watch them through the lens of one or more of the five teaching practices—where are instructional moments where these practices occurred? What would it look like if a given practice was enacted at a deeper level? For example, when watching the teaching of an algebraic procedure, the teacher educator or coach might encourage the teacher to include explanations of why a procedure works, what the solution means, or how individual steps connect to mathematical properties. A teacher might also be encouraged to consider whether there are other procedures that would generate a solution and what it would look like to include a discussion comparing multiple procedures for their affordances and limitations.

My research team and I are developing instructional routines tied to each of the five algebra-focused instructional practices listed above. We anticipate a November release of the tool, Supporting Algebra Teaching. Please email me at litke@udel.edu for more information.

**Closing**

It is true that too many students fail algebra courses, keeping them back from additional opportunities. To say that algebra is too hard or too irrelevant, however, implies that it is too hard for *some* students. This way of framing the problem wraps itself in a cloak of equity while avoiding issues of who is afforded access (and how) and the nature of the instruction that students experience when they are in algebra classes. Rather than turn our backs on school algebra or on those who teach it, we can shift toward thinking about how to improve the teaching of algebra content such that students’ personal and mathematical identities are affirmed and supported through the instruction they experience and such that they are provided instruction that supports the development of algebraic understanding.

## Leave a Reply