Why teach using a spiralled math curriculum?
Twenty years ago, Battista (1999) noted the need to change the way math is taught, by highlighting the lack of retention of math facts. This alone does not argue for spiralling as the necessary change; however, Schutte et al. (2015) note, “[t]he concept of distributing practice over time versus massing practice into one time period has been studied since the early days of psychological research…[and] the collection of basic research has identified very robust findings regarding the benefit of distributed practice over massed practice for learning and memory” (p.150). While Schutte et al. (2015) are discussing the benefits of distributed (or spiralled) practice in general, there is ample evidence of the benefits remaining true for math learning and retention (Foster et al., 2019; Rohrer, 2009; Schutte et al., 2015; Taylor and Rohrer, 2009).
BENEFITS OF A SPIRALED CURRICULUM ON STUDENT LEARNING
CONCERNS WITH CURRENT MATHEMATICS EDUCATION
Wolfram (2010) identified a problem in mathematics education: “At one end we’ve got falling interest in education in math and at the other end we’ve got a more mathematical world, a more quantitative world than we’ve ever had.” Over a decade before him, Battista (1999) noted another problem in mathematics education: “Though the same topics are taught and retaught year after year, the students do not learn them” (Battista, 1999). Indeed, garnering interest in mathematics and the retention of mathematics knowledge are both well-studied areas. However; it is work of Lockhart (n.d.) that clearly identifies an overarching necessity: “Math teachers need to understand math, be good at it and enjoy doing it” (p. 11).
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In the information era of the 21st-century, there are numerous resources available to educators to inform their practice. Missing in many of these helpful resources is a proper explanation as to why the resource was created. Houchens and Keedy (2009) speak to the importance of “reflective practice” and how it “may contribute to higher levels of student achievement” (p. 54). An essential part of reflective practice is asking the question, ‘Why am I doing this this way?’ If teachers are unaware of the research and reasoning behind the spiraled approach, they will remain unable to answer this question beyond a base level. Similarly, without knowing the benefits of spiraling, teachers will be unaware of the intended goals and thus unable to ensure they are realizing those benefits.
The more commonly-used ‘massing’ or ‘blocked practice’ strategy, teaching one concept at a time, is often viewed as more effective. In truth, massing does lead to higher performance in short-term scenarios, such as a unit test, but spiraling yields better results in long-term retention (Everyday Mathematics, n.d., Schutte et al., 2015). Rohrer (2009) states,“whereas blocked practice requires students to know how to perform a procedure, it does not require them to know which procedure is appropriate” (p. 10), which becomes evident and troublesome when students have to write an exam or standardized test that covers a wide range of concepts.
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Spiraling allows for the teaching of similar concepts together so that students may draw similarities and differences, and have to actually think about what they are supposed to be doing for a word problem rather than being able to assume it is the same strategy or concept they were reviewing the entire time as one would find with blocked practice (Phillips, n.d.). Taylor and Rohrer (2009) theorize that since “interleaving [spiraling] requires participants to repeatedly switch between different kinds of tasks, they must learn how to pair each kind of task with its appropriate formula” (p. 845). This is superior to blocked teaching, as “[w]hen practice problems are blocked, […] students can successfully solve a set of practice problems without learning how to pair a problem with the skill […] students are effectively relying on a crutch” (p. 846).
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The research on spiraled learning supports a spiraled approach to mathematics teaching as a superior method that is proven to yield better retention rates for years. In fact, “[t]his effect has been replicated over many disciplines since 1885!” (Phillips, n.d.). According to a review of research into the effects of spacing provided by Rohrer (2009), “spaced [spiraled] practice is being grossly underutilized in mathematics instruction” (p. 9), despite the fact that research has shown "those studying with greater spacing intervals were found to retain more mathematical rules" (Schutte et al., 2015).
WHY MUST WE KNOW WHY?
The goals for spiraling can thus be summed up as follows:
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1) Spiraling for better long-term retention of mathematical knowledge
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2) Spiraling to aid students to draw similarities and differences between problems,
deepening their mathematical understanding and problem-solving skills.
Literature Review
GOALS OF A SPRIALED MATHEMATICS PROGRAM
Petrina (2004) discusses two main questions: “What should be learned? How should it be organized for teaching?” (p. 81). If we are going to shift mathematics to a spiraled approach, then we are making a change to how mathematics is organized for teaching. We are also implicitly changing what is the focus of learning - a shift from isolated skills to teaching through connections. A change of this magnitude will require ‘teacher buy-in’. While research has discussed the importance of involving teachers in early stages of curriculum design to assist with ‘buy-in’ (Broom, 2016; Northouse, 2016), what about the teachers that come in after the fact and are being asked to or are interested in implementing this teaching strategy?
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Just as curriculum design is led by the two guiding questions from Petrina (2004), the implementation of curriculum changes are guided by two general concerns:
Why? - What is the rationale behind this curriculum change? What is the intended goal?
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How? - What resources/supports are available? Where can I access them and where do I start?
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Teachers need to be able to answer both of these guiding questions before they can begin.