Saturday, April 2, 2011

Curricular Spiraling and Distributed Practice

While researching the scientific evidence supporting the Reform Math (Everyday Math) claims on curricular spiraling and discovery learning, Mr. Ze’ev Wurman was kind enough to select a few quotes from the most important publications and reports available and to highlight how Everyday Math misinterprets the research. I post his explanations, quotes, references, and Mr. Wurman’s bio (at bottom).

Mr. Ze’ev Wurman and I are members of the American Math Forum (AMF). We are thankful for the opportunity AMF has provided in making this Q & A session possible.

Curricular Spiraling and Distributed Practice:

How Everyday Mathematics Misinterprets the Research.

Reading Everyday Mathematics justification for distributed practice[1], which is also frequently called spaced learning, one reads:
From the beginning, accordingly, Everyday Mathematics was designed to take advantage of the spacing effect. An explicit attempt was made to ensure multiple exposures to important concepts and skills, spread over two or more years. As the First Grade Everyday Mathematics teacher’s manual states, “If we can, as a matter of principle and practice, avoid anxiety about children ‘getting’ something the first time around, then children will be more relaxed and pick up part or all of what they need. They may not initially remember it, but with appropriate reminders, they will very likely recall, recognize, and get a better grip on the skill or concept when it comes around again in a new format or application—as it will!”

Yet distributed practice in cognitive science literature is not about children "pick[ing] up part or all of what they need" over "two or more years." It is about distributed practice rather than distributed instruction, and the distribution, or spacing, should occur over period of weeks rather than years[2].
For example, this is what the National Advisory Mathematics Panel[3] wrote about distributed practice:
Distributed practice should naturally occur as students progress to more complex topics. However, if basic skills are not well learned and understood, the natural progression to complex topics is impeded. (p. 4-xxv)
At all ages, there are several ways to improve the functional capacity of working memory. The most central of these is the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory. … For other types of information, including much of mathematics that is taught in school, automaticity is achieved only with specific types of experiences, including practice that is distributed across time (e.g., Cooper & Sweller, 1987). (p. 4-5)
The fast and efficient solving of arithmetic combinations and execution of procedures requires considerable practice that is distributed over time. (p. 4-39)
Research on learning in general … indicates a benefit for practice that is distributed across time, as contrasted with the same amount of practice massed in a single session. Pashler, Rohrer, Cepeda, and Carpenter (2007) provide a recent review, including discussion of one area of mathematics. Initial experimental studies with mathematics, specifically teaching probability, are consistent with the more general literature. ... The course work results suggest that the distributed review and integration—which was likely to have occurred more consistently for algebra than geometry—of the material across years contributes to the retention of the material throughout adulthood. (p. 4-87)
Retention of algebraic skills into adulthood requires repeated exposure that is distributed over time. This occurs as core procedures and concepts are encountered across grades. In much of mathematics, distributed practice should naturally occur as students progress to more complex topics. However, if basic skills are not well learned and understood, the natural progression to complex topics is impeded. This is because students will continue to make (and potentially practice) mistakes. As an example, procedures for transforming simple linear equations are embedded in more complex equations and thereby practiced as students solve them. The practice will not be effective, however, if students incorrectly transform basic equations, as they often do. (p. 4-89)
Here is what the IES Practice Guide has to say on the subject of spacing learning over time:
To help students remember key facts, concepts, and knowledge, we recommend that teachers arrange for students to be exposed to key course concepts on at least two occasions—separated by a period of several weeks to several months. Research has shown that delayed re-exposure to course material often markedly increases the amount of information that students remember. The delayed re-exposure to the material can be promoted through homework assignments, in-class reviews, quizzes (see Recommendation 5), or other instructional exercises. In certain classes, important content is automatically reviewed as the learner progresses through the standard curriculum (e.g., students use single-digit addition nearly every day in second grade math), and this recommendation may be unnecessary in courses where this is the case. This recommendation applies to those (very common) course situations in which important knowledge and skills are not automatically reviewed.[4]

What research indicates is that once a topic has been taught, a distributed exposure to it, over period of weeks and perhaps few months and in the form of practice or various quizzes and test, is conducive to fluency and long term retention. And that this is particularly important for topics that are not routinely practiced already in the process of subsequent learning. The research does not find that topics should not be taught to a full understanding and mastery when instruction occurs. Rather, it says that fluency and retention will be better achieved by distributed practice. Contrast this with the interpretation that Everyday Mathematics offers, where it instruct teachers not to expect mastery when taught and it re-teaches essentially the same content over period of 2-3 years. That is exactly what the National Advisory Mathematics Panel warned against: “Any approach that continually revisits topics year after year without closure is to be avoided[5].”

In summary, Everyday Mathematics repeatedly misinterprets the research about distributed practice and applies it in support of distributed instruction in their spiral curriculum, a practice that is not supported by research.

[4] Organizing Instruction and Study to Improve Student Learning: A Practice Guide. Institute for Education Sciences, 2007, page 5.
[5] First recommendation of The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education, 2008.

Mr. Ze'ev Wurman worked over 30 years in the high tech industry, most recently as the Chief Software Architect with Monolithic 3D, a semiconductor start-up in the Silicon Valley. He has a long involvement with mathematics standards and assessment in California and served on the 1997 Mathematics Framework Committee and on the STAR Mathematics Assessment Review Panel since its inception in 1998. He was a member of the 2010 California Academic Content Standards Commission that evaluated the suitability of Common Core’s standards for California. He was a member of the Teaching Mathematics Advisory Panel to the California Commission on Teacher Credentialing. Between 2007 and 2009 Wurman served as a Senior Policy Adviser to the Assistant Secretary for Planning, Evaluation, and Policy Development in the U.S. Department of Education. Wurman has B.Sc. and M.Sc. degrees in Electrical Engineering from the Technion, Israel Institute of Technology.


  1. Excellent article. Mr. Wurman makes my point exactly! Note that this does not refer to constructivism - but rather the misguided philosophy of EDM (and Saxon Math, for that matter).
    The National Math Panel Report actually supports the theory of constructivism - applied appropriately.

  2. Sally,

    Here, you say you agree with the post and correctly state that the post does not refer to constructivism. But then you say the NMAP report endorses the theory of constructivism. What’s the point? You seem to be an ardent defender of “constructivism” and are obviously trying to distance it from various reform math programs. Would you mind revealing more about yourself?

    Saxon does not use spiraling. Saxon applies “distributed practice” and encourages mastery at instruction. EDM uses spiraling and discourages mastery at instruction.

    My reading of the NMAP report does not indicate that it supports the theory of constructivism per se. It found strong points in both methods and suggested both should be used. However, it does support “direct instruction” for “struggling learners” and rejects EDM-type of 'not-teaching-to-mastery' spiraling.

  3. I think Sally is wrong in claiming that Saxon math is misguided regarding distributed practice (if that is what she meant).

    In fact, a superb implementation of distributed practice is one of the hallmarks of Saxon math. Their practice sets are very carefully sequenced to take advantage of spaced reinforcement and in this regard is the best I have ever seen in *any* textbook series. The flip side of that is that with Saxon one cannot jump around the textbook at will, or this careful sequencing will be destroyed.