The Mock Turtle Does Math

The Mock Turtle went on. ‘We had the best of educations … Reeling and Writhing, of course, to begin with, and then the different branches of Arithmetic: Ambition, Distraction, Uglification, and Derision.’  Carroll, Alice Through the Looking Glass

 Knowledge is context-bound. Zull  (2204) states: “Don’t explain”.  He reasons that because people’s brains are uniquely organized, students need to construct knowledge from their own experiences. The constructivist approach has taken hold in American mathematics instruction in much the same way that implicit approaches have overtaken reading instruction over the last 30 years.

The National Council of Teachers of Mathematics content standards have been religiously adopted throughout the country. These standards clearly call for a constructivist model in the teaching of mathematics despite little evidence for the validation of these methods in the context of actual instruction (Carnine, 2004; Goldman & Hasselbring, 1997).

Constructivist approaches are characterized by the building of responsive environments, child-directed choices, and facilitated discovery of basic principles (Carnine, 2004; Anderson, et al, 2009).  The model is grounded in Piagetian theories of underlying cognitive processes that unfold in a dialectical, hierarchically integrated fashion. Constructivism is also heavily influenced by Vygotski’s ideas regarding a “zone” of proximal development. For example, the NCTM problem-solving strategies include:

  • making a list
  • drawing a diagram
  • trial and error
  • looking for patterns
  • acting out a problem
  • guess and check

Everyday Math is one of the most popular constructivist math programs in use today. The program strongly advocates for the construction of general principles through exploration, extensive use of cooperative learning, and significant time spent playing games. The group emphasis may make it difficult to determine whether a child is arriving at his or her own answers until assessment time.  Moreover, the extensive use of group work may actually reduce the number of opportunities to respond per student per lesson. Effortful engagement, brisk pacing, and frequent response opportunities are important factors in maximizing progress in learning (Fuchs & Fuchs, 2001; Bos & Vaughn, 2008).

The “research-base” section of the Everyday Math website is sorely limited. It cites Marilyn Suydam’s 1985 summary of literature, which indicated that short periods of intense review are more effective than long periods, and that games are an effective review technique. The site also references Ebbinghaus’ 1885 work on distributed practice, and specifically states “Everyday Math is designed to take advantage of the Spacing Effect…to ensure multiple exposures to important concepts and skills, spread over two years.”

The first grade Everyday Math teacher’s manual explains the concept of distributed practice as follows:

They may not initially remember it, but with appropriate reminders, they will very likely recall, recognize, and get a better grip on the skill of concept when it comes around again in a new format or application—as it will!”

These skills are referred to as “developing” in Everyday Math lingo, and students do not need to “know” them as the program will eventually spiral back to them.

It has been well established that separating learning events with appropriately spaced periods of review has a positive effect on recall (Cepeda, et al, 2006; Dempster, 1988). However, optimal ratios of learning to review have not been consistently established and were not even studied systematically until more recently (Cepeda, 2006).

There are a number of issues with the spiral approach as outlined in the Everyday Math program. For example, true spiral approaches systematically and frequently (eg. weekly) review concepts within the context of small increments of new learning. Concepts are spiraled after they have been brought to mastery, and not as surface or exposure level skills. Recall requires more than review of concepts previously introduced. The learner must actually do something with the information in order to anchor it for later recall (Goldman & Hasselbring, 1997). Because children have not actively used the “developing” concepts they are subject to rapid decay (Bransford, et al, 2000 p. 20). The two year spiraling cycle ensures the information will be lost. It is unlikely that “appropriate reminders” will generate meaningful recall of useful information, and so the extensive instructional time spent on the introduction of these concepts may be better spent in mastery learning of foundational skills. Research based spiral approaches, such as Orton-Gillingham, are analogous to a corkscrew. Everyday math is more like a planetary orbit.

Another concern about the program, despite a more recent update, is the de-emphasis on calculation skills. Use of the calculator is strongly encouraged, so there is no need for automatic or efficient declarative processes underlying procedural knowledge. There is also a de-emphasis on working symbolically with formulas (Wang, 2001). Without procedural knowledge, students rely on inefficient processes such as tally marks, drawing pictures or making lists.

While it is common practice to use manipulatives to teach math concepts, brain research underscores the importance of kinesthetic and spatial representation of number in the development of numeracy skills.  The mapping of spatial concepts linguistically is also necessary for us to access our mental number line.  Consider the following example:

Everyday Math

This exercise, and many others like it, is included in the first grade Everyday Math program.  Clearly, the spatial language is incorrect, and the vertical orientation of the number line adds to the confusion.  At the same time, horizontal number lines, hundreds charts, and thermometers are introduced. The orientation is inconsistent with our spatial representation of number, and is presented at a time when these concepts are just developing in children’s language acquisition.  Children who are delayed in spatial and temporal concepts are doubly disadvantaged. There is significant evidence that children need to develop solid and fluent linguistic representations of both spatial and count sequence concepts in order to access number.





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