Critical Thinking Through Manipulatives: A Staff Development Intervention for Middle Grades

Date of Completion


Document Type

Open Access Capstone

Degree Name

Master of Arts (MA)

First Advisor

Judith Collison


This thesis proposes a curriculum development project for mathematics education in the middle grades. I intend to provide theory and to contribute practical applications, both intended for future in-service staff development and teacher workshops. The rationale explores the causes and effects of the lack of manipulative materials in current mathematics classrooms. This exploration results from my interest in designing, constructing and implementing instructional aids, from my experience as staff developer in the Amigos Bilingual Program of Cambridge, and from my graduate studies at the University of Massachusetts Boston. My objective is to review the history of the teaching of thinking skills and how that history relates to the use of instructional aids in mathematics. I will examine this topic from three different perspectives, philosophical, psychological and pedagogical, that have shaped school practices. The philosophical perspective that I most use in this work is perhaps best exemplified by Lev Vigotsky's cultural evolution theory based on historic materialism. I use in addition, Robert H. Ennis' approach, which proposes that teaching of critical thinking be emphasized as learning how to think and what to know. I refer to developmental psychology researchers like Jean Piaget and Jerome Bruner as representatives of the constructivist trend within cognitive psychology. Both Piaget' s and Bruner' s work have greatly influenced the study of learning, motivation, perception and educational psychology. The inclusion of their ideas is important when considering any initiative of curriculum or staff development that attempts to improve teaching methods and materials. The contribution of these developmentalists to my own understanding of thinking and reasoning processes have greatly influenced my work here. For the section on pedagogy, my conviction of the need of various sensory modes to represent thinking led my search to Bruner's spiral curriculum and later Lesh' s model for the translation of representational thoughts. From these two models we may develop an instructional method in which the teacher moves deliberately, in gradual steps, from concrete to symbolic modes of thinking. Some facts, however, give us pause when considering a total manipulative approach. The first is that the learning outcome of a mathematics curriculum is almost totally symbolic, particularly from the middles grades up. We want our students to be able to perform, eventually, at an abstract level with numbers, operation signs, parentheses and equations. Nevertheless, current findings suggest a schism between this symbolic form of mathematics dexterity and the desirable manipulative methods of good mathematics instruction. I believe that manipulative aids provide for both the improvement of teaching practices and, consequently, for better students understanding of the covered concepts. The use of such devices as realia, pictures and games represent a major benchmark in the paradigm shift from transmission of knowledge practices to student-centered practices. Indeed, this shift provides for the various representational modes of thinking: intuitive, concrete, pictorial and abstract. Consequently, with proper activities, these representations will also facilitate growth toward more complex mental modes and operations: generalizations, making connections, problem solving and the like--the modes where critical thinking resides.


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