The traditional models for mathematics education found within the revisions of the past century have been formulated within a perspective we are calling the procedural-formalist paradigm (PFP). The PFP holds that mathematics is an objective set of logically organized facts, skills, and procedures that have been optimized over centuries. This body of knowledge exists apart from human experience, thus making it inherently difficult to learn.
Positioning oneself within the frame of the PFP, one might reasonably believe the goal of school mathematics education should be for students to internalize a fundamental body of basic mathematical knowledge. In order to facilitate such learning, teachers must deliver carefully sequenced bits of mathematics to students through explanation and demonstration. Students repetitively practice these facts, skills, and procedures in an effort to memorize them and are then tested to discern what has been “learned.” Learning and assessment are structured around the notion that there is a unique, mathematically correct way to solve a problem. This set of assumptions guided the work of Thorndike and the back-to-basics advocates.
Alternatively, still operating within the PFP, one might try to show students forthright the logical structure of mathematics and hope they catch on. While many may not, they might at least catch a glimpse of the inherent beauty of mathematics. The students who do catch on to this structure will be well positioned to succeed in higher-level mathematics. This was the position taken by many of the New Math advocates. The later social efficiency progressives espoused yet another approach to school mathematics, but one still grounded in the PFP. Since most mathematics is outside of human experience, it is not relevant for students to learn. Any necessary practical mathematical skills can be learned within the context in which they might be used. This thinking led to the creation of consumer and vocational mathematics courses that offered the average student an “escape” from the rigors of such formal topics as algebra and trigonometry (reserving such classes for the mathematically elite). Importantly, the claim being posited here is not that there were no other possibilities being offered for reform during the past century. Rather, it is that the paths taken—those “reforms” that received strong support and were widely implemented (and the ways in which they were implemented)—were reflective of the PFP. Ideas that fell outside the bounds of this paradigm, though having localized effects, failed to gain wide acceptance, recognition, and/or support. However, that such efforts were not insignificant is acknowledged for these small-scale deviations from the paradigmatic ways of operating demonstrated to those involved that there was a fundamentally different way to conceptualize mathematics education.
It is the integration of new thinking about cognition and the greater acknowledgement of culture that has enabled mathematics educators to frame questions and conceptualize solutions in ways that were unlikely to develop from within the proceduralformalist paradigm. The unique perspective toward mathematics education that has come from the blending of cognitive psychological and (socio)cultural research has been made possible by the emergence of what we are calling the cognitive-cultural paradigm (CCP). The CCP takes mathematics to be a set of logically organized and interconnected concepts that come out of human experience, thought, and interaction—and that are, therefore, accessible to all students if learned in a cognitively connected and culturally relevant way.
The fundamentals of the cognitive-cultural paradigm lead to a radically different view of mathematics education than that of the procedural formalist. Many of the core beliefs of traditional paradigm are challenged. Emphasis is shifted from seeing mathematics as apart from human experience to mathematics as a part of human experience and interaction. This is not to imply that students must reinvent mathematics in order to learn it. Rather, for students to really understand mathematics they need opportunities to both a) share common experiences with and around mathematics that allow them to meaningfully communicate about and form connections between important mathematical concepts and ideas, and b) engage in critical thinking about the ways in which mathematics may be used to understand relevant aspects of their everyday lives. The challenge is no longer how to get mathematics into students, but instead how to get students into mathematics (Philipp, 2001). This implies a need for flexibility in how teaching is approached and how learning is evaluated, a move away from more static and deterministic models of the PFP.
In other words, new possibilities were envisioned and experienced, providing the seeds for later efforts at true reform. It would be profitable for further analyses to be brought to the specifics of the ways in which these “fringe” programs had rather important effects on the field.
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