By Jon R. Star

*The education world has a bent toward large-scale reform. But that almost never works the way it should. Perhaps it’s better to ask teachers to make more modest changes to their current practices.*

The past decades have seen a steady push to improve K-12 mathematics instruction, culminating most recently in the Common Core State Standards and similar college- and career-readiness standards. By many accounts, mathematics achievement nationally is mediocre (Peterson, 2003), and many place the blame on subpar teaching (Kane & Staiger, 2012). There is optimism that the implementation of these new standards can improve both instruction and student achievement in mathematics (Schmidt & Houang, 2012).

New standards require teachers to substantially improve the rigor and coherence of the implemented curriculum, which, in turn, requires dramatic changes in teachers’ pedagogy. In particular, mathematics teachers are expected to sustain a focus on reasoning, communication, and real-life problem solving; to adopt more student-centered pedagogies, making greater use of small-group and whole-class discussion; and to consistently emphasize the conceptual rationale underlying mathematical procedures. Plans for expansive and expensive teacher professional development are underway in many districts.

Evidence from past efforts suggest that progress will be quite bumpy (Cuban, 1993; Tyack & Cuban, 1995). Certain instructional practices have proved to be deeply ingrained and difficult to change — teacher-led instruction, for example. Ambitious agendas for instructional change are often filtered through teachers’ belief systems, resulting in implementation that may not be well aligned with the reform vision. Also, organizational structures of classrooms, schools, and districts have worked to stymie change.

But instead of broad scale, it may be better to consider improving instruction incrementally by identifying small yet powerful changes that teachers can implement relatively easily in their instruction. These changes should be viewed by teachers as modifications or tweaks to what they are already doing in their classrooms — thus increasing the likelihood that teachers will carry out the reforms. Furthermore, these incremental changes should be strategically chosen by policy makers for their potential to positively affect student learning while opening a door within teachers’ practice for more ambitious instructional changes in the future

This idea is not entirely new; what is novel here is the suggestion that incremental change be explicitly sought as a policy goal, rather than as a consolation prize that results from incomplete attempts at transformational change. Also novel are two suggested strategies for designing instructional change initiatives: the selection of widespread practices that are typically implemented suboptimally and the use of research to identify mechanisms for improving those practices.

**Incremental instructional change**

Three instructional practices could yield small steps forward in mathematics instruction:

- Improving teacher-led instruction;
- Using worked examples judiciously; and
- Teaching multiple strategies.

There are likely other incremental changes in mathematics teaching that are worthy of attention by teachers and policy makers; these three were selected as convenient examples.

**Teacher-led instruction **

Teacher-led instruction places the teacher in control of the delivery of math content, as opposed to student-centered strategies such as cooperative learning, where students have greater opportunities for interaction and exploration. Criticisms of teacher-led instruction are numerous but center on the overuse of teacher talk and the absence of sufficient opportunities for student sense making (Weiss et al., 2003). Current policy recommendations suggest that teacher-led instruction be avoided as much as possible. Yet study after study finds that teacher-led instruction persists in classrooms at all levels (Hiebert et al., 2005). Given that many mathematics teachers bring experience and beliefs suggesting that teacher-led instruction is either appropriate or simply necessary given the needs of students and constraints of classrooms, reform messages that demonize this form of instruction require teachers to either transform their practice completely or reject reform recommendations entirely. An incremental step forward would allow teachers to continue to use teacher-led instruction but more effectively.

**Improving teacher-led instruction**

Effective teacher-led instruction is characterized by the presence of frequent teacher-posed questions, particularly questions that don’t have one right answer and require some thinking that prompts learners to generate explanations, a practice that has been found to positively affect learning and knowledge transfer (Rittle-Johnson, 2006). After posing such questions, teachers should carefully and consistently use appropriate amounts of wait time, providing students with ample opportunities to think. Teacher-led instruction is also enhanced when students have opportunities to talk to and ask questions of one or more peers, to assist in evaluating and monitoring their own knowledge (Kramarski & Mevarech, 2003). Finally, effective teacher-led instruction includes frequent opportunities for formative assessment to monitor students’ emerging knowledge (Black & Wiliam, 1998), including the regular use of short quizzes and tests (Rohrer & Pashler, 2010).

Improving teacher-led instruction makes sense as a strategy. First, it is by far the most prevalent instructional method in U.S. mathematics today, making up 90% of segments coded in a large and recent study of elementary mathematics instruction (Hill et al., 2014). Small improvements over such a large fraction of instruction would arguably yield more benefits than large changes for a small fraction of lessons. Second, choosing this strategy reflects the reality that many teachers see themselves in: Faced with conventional curriculum materials and pacing guides that require them to quickly move through material, they may see themselves as unable to undertake a complete overhaul of mathematics instruction. Finally, practices that would improve direct instruction — asking more questions, providing students with more time to think about problems — are also gateway practices into more extensive reforms of the kind called for by the Common Core.

**Use worked examples judiciously **

A second, related small step forward is the judicious use of worked examples in mathematics instruction. A worked example is “a step-by-step demonstration of how to perform a task or solve a problem” (Clark, Nguyen, & Sweller, 2006, p. 190). The use of worked examples continues to be in widespread use at all levels of mathematics instruction (Hiebert et al., 2005), whether a teacher demonstrates a procedure at the beginning of a lesson for students to subsequently practice or whether the teacher does the demonstration after students have had an opportunity to engage in exploration first. Typically, teachers demonstrate a correct solution procedure on almost identical examples immediately before assigning independent practice composed of more almost identical problems (Hiebert et al., 2005; Weiss et al., 2003). When used in this way, worked examples merely illustrate the one solution procedure that the teacher subsequently expects students to use on future problems. Students are expected to listen and take notes, asking questions only when they foresee difficulties in mimicking the teacher’s steps.

Research suggests several instructional practices that allow worked examples to be used more effectively. First, teachers can be more thoughtful about the selection of worked examples, with particular attention to using deliberate progressions of examples to build toward a lesson’s instructional goals. Systematic variation of presented examples (Watson & Mason, 2006) can help learners recognize patterns across examples, draw generalizations, and consider mathematical points that are broader than might be evident from a single problem. Second, teachers can move beyond prototypical (correct and complete) worked examples to include presentations of partially completed or even incorrect examples. Studying incorrect worked examples can help students become more aware of their own misconceptions as well as foster the growth of conceptual knowledge (Booth et al., 2013). And third, presenting multiple examples simultaneously can be used to stimulate conversations around similarities and differences of examples, which has been found to be quite productive for mathematics learning (Rittle-Johnson & Star, 2007; Star et al., 2015).

Rather than merely a demonstration of a method to be repeated, worked examples can become an instructional artifact that can be queried, problematized, compared, analyzed, extended, and discussed. Teachers will most likely perceive more effective use of worked examples as a relatively minor modification of their existing practice, opening the door to more creative, innovative, and potentially powerful uses of worked examples in the future.

**Teach multiple strategies **

Finally, a third incremental step forward in mathematics instruction is to teach multiple strategies for solving problems. Instruction should focus on multiple approaches to solving problems rather than exclusive emphasis on a single algorithm. Emphasizing multiple strategies has been part of the reform agenda in mathematics for some time, both at the elementary (Carpenter et al., 1989) and secondary level (Silver et al., 2005). Many teachers already know about teaching with multiple strategies (Lynch & Star, 2014a); students approve of multiple strategy instruction and indicate that they find this approach to be helpful (Lynch & Star, 2014b).

As with the previous recommendations, teaching with multiple strategies appears to be widespread. In a study using a nationally representative sample of K-12 mathematics teachers, Weiss et al. (2003) found that 62% of middle school mathematics teachers report comparing and contrasting different methods for solving problems at least once per week. Similarly, Lynch and Star (2014a) report that all 92 Algebra I teachers who volunteered to participate in a project related to teaching with multiple strategies reported that they already used this practice frequently in their instruction.

But it appears that this practice is not well enacted (Richland, Zur, & Holyoak, 2007). Hill and colleagues found high-quality implementation of multiple strategies instruction in only 12% of lesson videos that incorporated this practice (Hill et al., 2014). It seems that among teachers who use multiple strategies, this practice is often used merely for encouraging and validating student participation and can be devoid of any mathematical substance (Ball, 2001; Silver et al., 2005). One reason teachers may not be effectively teaching with multiple strategies is that current reforms have been overly ambitious in how this approach is conceptualized. In recent reform documents, teaching with multiple strategies is framed as part of several distinct instructional recommendations, including a focus on nonroutine, open-ended problems; a reliance on student-centered instruction; and an embrace of student-invented, nonstandard procedures (NCTM, 1989, 2000, 2014).

As an incremental step forward, teaching with multiple strategies can be disassociated from the constellation of associated reform practices and simplified. Instruction should focus on multiple strategies, including consideration of the relative merits of particular strategies and the rationale for choosing strategies under certain problem solving conditions. An emphasis on multiple strategies can shift the focus of teachers’ and students’ work from merely attending to the correctness of a strategy to a broader and richer consideration of how a problem was solved or could be solved. This would allow teachers and students to explore mathematically substantive questions: What similarities and differences exist between methods, and are there implications to these similarities and differences? Under what conditions are various methods applicable or preferable? When and why is one method better than another method, and what does “better” mean?

Teaching with multiple strategies may be viewed by teachers as a relatively modest change in their instruction, but one that has clear potential to improve learning. In addition, as teachers develop greater familiarity and comfort with teaching with multiple strategies, there is the potential that this practice can be reintegrated with other, more ambitious practices with which it is often associated.

In addition, as the research base on effective mathematics instruction continues to grow, other small steps forward could be identified. Furthermore, incremental changes in other content areas could also be identified.

**Incremental change can work**

Given the challenges faced by reformers in prior efforts to change instructional practice, one could reasonably wonder why incremental change would be any more effective. Although a majority of teachers express strong support for the ambitious reforms proposed by the Common Core (O’Brien, 2014), a variety of factors stand in the way of high-quality implementation. These include teacher reservations about wholesale changes in their practice (Tyack & Cuban, 1995), as well as the fact that teachers often make sense of reforms differently from researchers, thus adversely affecting the quality of implementation (Cohen & Ball, 1990; Spillane & Zeuli, 1999). Incremental change makes reform manageable for teachers, even when considering the at times constraining organizational structures of district, school, and classroom contexts under which they work. Incremental change does not require teachers to make sense of entirely new ideas about instruction but rather to understand small innovations in practices that they already rely upon heavily.

**Implications **

It is not typical for researchers and policy makers to argue for incremental changes, given the strong incentives in the field to find the newest, grandest, and most exciting ways to solve educational problems. Yet incremental change may actually be the most realistic approach and a way to hedge bets in the event that efforts at transformational change have limited success (as has frequently occurred in the past). Incremental changes are relatively easy to implement and have the potential to significantly improve classroom instruction. In essence, taking small steps forward seeks 10% improvement for 90% of teachers, rather than 90% improvement for 10% of teachers.

This approach is not uncommon in other fields. Innovation results because of radical transformative shifts, where a new and big idea emerges on the scene and sometimes leads to great changes. But, arguably, it is more common that change occurs because of very smart tweaks, where someone figures out how to make a transformative idea more usable, implementable, or palatable to users (Gladwell, 2011). Past educational reforms have failed in part because they did not meet teachers where they were, with expectations about major changes in practices that may have been unrealistic for many teachers even under ideal professional learning conditions. A focus on incremental change begins with the elements of instruction that teachers recognize, support, and/or perceive themselves to be already using and then leverages these instructional practices for future change. By taking small steps forward and meeting teachers where they are, our hopes for larger, more substantial, and long-lasting changes in instructional practice may be achievable.

**References**

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Spillane, J.P. & Zeuli, J.S. (1999). Reform and teaching: Exploring patterns of practice in the context of national and state mathematics reforms. *Educational Evaluation & Policy Analysis, 21* (1), 1-27.

Star, J.R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2015). Learning from comparison in algebra. *Contemporary Educational Psychology*, 40, 41-54

Tyack, D. & Cuban, L. (1995). *Tinkering toward utopia: A century of public school reform.* Cambridge, MA: Harvard University Press.

Watson, A. & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. *Mathematics Thinking and Learning, 8* (2), 91-111.

Weiss, I.R., Pasley, J.D., Smith, P.S., Banilower, E.R., & Heck, D.J. (2003). *Looking inside the classroom: A study of K–12 mathematics and science education in the United States.* Chapel Hill, NC: Horizon Research, Inc.

**JON R. STAR** is an associate professor of human development and education in the Harvard University Graduate School of Education, Cambridge, Mass.

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