Cognitive Development Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching?

This article provides an overview of some perspectives about special issues in classroom mathematical teaching and learning that have stemmed from the huge explosion of research in children’s mathematical thinking stimulated by Piaget. It concentrates on issues that are particularly important for less-advanced learners and for those who might be having special difficulties in learning mathematics. A major goal of the article is to develop a framework for understanding what effective mathematics teaching and learning is, because doing so is so important for struggling studentsand for researchabout them.Piaget’s researchhada fundamental influence on the on-going tension between understanding and fluency in the classroom, supporting efforts toward increasing understanding. But in some countries, misinterpretations of Piaget led to practices that are counterproductive for children, especially struggling learners. Such misinterpretations are identified and a more balanced approach that also draws on Vygotsky is described—a learning-path developmentally-appropriate learning/teaching approach. © 2009 Elsevier Inc. All rights reserved. A major goal of this article is to develop a framework for understanding what effective mathematics teaching and learning is. Doing so is extremely important for students having difficulty learning mathematics. The framework develops out of a major historical tension in mathematics education between understanding and fluency. In the USA this tension has been so intense that it has been termed “the math wars.” It, and variations in the relative emphases on understanding and fluency, has also influenced mathematics education in European countries, although these histories vary in E-mail address: [email protected]. 0885-2014/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.cogdev.2009.09.009 344 K.C. Fuson / Cognitive Development 24 (2009) 343–361 the timing and intensity of the tension. Piaget’s research had a fundamental influence on this tension, supporting efforts toward increasing understanding. But in some countries (perhaps especially in the USA), misinterpretations of Piaget led to practices that are counterproductive for children and especially for struggling learners. This article focuses on early numerical learning (up through multi-digit addition and subtraction) because that is the focus ofmost of the research on students with learning difficulties and because it is especially important for such students to gain competence in these early topics. The article necessarily simplifies complex issues as it summarizes some broad themes and a few more particular points. It is not a review of the literature, although some major researchers are mentioned along the way. For more detail and fuller references, see research reviews by Clements and Sarama (2007), Fuson (1992a, 1992b), Verschaffel, Greer, and DeCorte (2007), and the research summarized in the NRC Adding It Up report, 2001, and the NRC early childhood math report, 2009. 1. The powerful influence of Piaget on research about children’s mathematical learning Many researchers of my generation probably remember the first time they read a book by Piaget on children’s mathematical knowledge. In 1965 as amathmajor at Oberlin College, I was interested in children’smath learning. Amath professor sentme to look at Piaget’s books, and I sat stunned, reading more andmore vignettes that revealed how children thought about mathematical ideas. They did not think like adults! And there were learning progressions in their thinking—their ideas developed to become adult ideas. And you could invent wonderful tasks (or at least Piaget could) that would reveal how children did think differently and then progressed in their learning. It is difficult to explain to current researchers how revolutionary these ideas were at the time or the pervasive extent of the explosion of creative research they spawned, all aimed at understanding children’s mathematical cognition in more detail. Because of this explosion of research, we know hugely more about children’s mathematical thinking than before Piaget’s work became widely known. Many of the research studies frommathematics education,developmentalpsychology, andcognitivedevelopment summarized in the two latestHandbooks of Research onMathematics Education (Grouws, 1992; Lester, 2007) are influenced in someway byPiaget’s ideas that children construct their ownconcepts and that these ideasdevelopalong learning paths. Later researchers adapted and extended Piaget’s theory. Neo-Piagetians like Case cast Piagetian stages into a neater organization through studies that added successive steps in problem situations (Case, 1991; Case & Okamoto, 1996). He and his students also did important educational research in several mathematical domains (e.g., Griffin, 2005; Kalchman & Koedinger, 2005; Moss, 2005). These studies often used bridging contexts that were carefully chosen to be clear examples of the class of situations for the mathematical domain that could support student understanding. Vergnaud (2009) developed the frameworkof conceptual fields that extendedPiaget’s notionof scheme to include goals, rules to generate activity, conceptsand theorems-in-action, and generative possibilities of inference. Concepts involve a set of situations and a set of linguistic and symbolic representations as well as the operational invariants contained in schemes. Theseextensions in the conceptualfield frameworkallow the specific characters of variousmathematical domains to enter the analysis, as is necessary, and shift the focus from the general logical operations studied by Piaget to specific mathematical operations. Piaget’s theories and the subsequent research about children’s understandings were especially important to educators and researchers who were concerned about the lack of mathematical understanding resulting from traditional rote-teachingmethods. Far too often the teacher stood at the front and did a problem or two with no or minimal explanation and then the students were to imitate (or there even was no initial demonstration at all). These rote-teaching methods failed to reach many children, especially those less-advanced than their peers and thus all children with mathematical learning difficulties. Such traditional approaches emphasized fluency and not understanding, and the opponents of such approaches might cleverly have termed them teaching without learning. In this context, the idea that children could and did think, and had their own mathematical ideas and approaches, was very refreshing and encouraging. But Piaget’s focus on the concrete operational period in children’s thinking and on the centrality of children’s interactions with objects during this K.C. Fuson / Cognitive Development 24 (2009) 343–361 345