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Cognitively Guided Instruction is "a professional development program based on an integrated program of research on (a) the development of students' mathematical thinking; (b) instruction that influences that development; (c) teachers' knowledge and beliefs that influence their instructional practice; and (d) the way that teachers' knowledge, beliefs, and practices are influenced by their understanding of students' mathematical thinking". [1] CGI is an approach to teaching mathematics rather than a curriculum program. At the core of this approach is the practice of listening to children's mathematical thinking and using it as a basis for instruction. Research based frameworks of children's thinking in the domains of addition and subtraction, multiplication and division, base-ten concepts, multidigit operations, algebra, geometry and fractions provide guidance to teachers about listening to their students. Case studies of teachers using CGI have shown the most accomplished teachers use a variety of practices to extend children's mathematical thinking. It's a tenet of CGI that there is no one way to implement the approach and that teachers' professional judgment is central to making decisions about how to use information about children's thinking.
The research base on children' mathematical thinking upon which CGI is based shows that children are able to solve problems without direct instruction by drawing upon informal knowledge of everyday situations. For example, a study of kindergarten children [2] showed that young children can solve problems involving what are normally considered advanced mathematics such as multiplication, division, and multistep problems, by using direct modeling. Direct modeling is an approach to problem solving in which the child, in the absence of more sophisticated knowledge of mathematics, constructs a solution to a story problem by modeling the action or structure. For example, about half of the children in a study of kindergartners' problem solving were able to solve this multistep problem, which they had never seen before, using direct modeling: 19 children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit three to a seat, and how many can sit two to a seat?
Students may solve this problem by counting down from eleven or by counting up from six. With the use of manipulatives students would be able to represent their thoughts for this problem multiple ways. For instance, they might make a row of six counting blocks next to a row of eleven counting blocks and then compare the difference.
The CGI philosophy is detailed in Children's Mathematics which is co-authored by Thomas Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan Empson.
Educational psychology is the branch of psychology concerned with the scientific study of human learning. The study of learning processes, from both cognitive and behavioral perspectives, allows researchers to understand individual differences in intelligence, cognitive development, affect, motivation, self-regulation, and self-concept, as well as their role in learning. The field of educational psychology relies heavily on quantitative methods, including testing and measurement, to enhance educational activities related to instructional design, classroom management, and assessment, which serve to facilitate learning processes in various educational settings across the lifespan.
Instructional scaffolding is the support given to a student by an instructor throughout the learning process. This support is specifically tailored to each student; this instructional approach allows students to experience student-centered learning, which tends to facilitate more efficient learning than teacher-centered learning. This learning process promotes a deeper level of learning than many other common teaching strategies.
In psychology, cognitivism is a theoretical framework for understanding the mind that gained credence in the 1950s. The movement was a response to behaviorism, which cognitivists said neglected to explain cognition. Cognitive psychology derived its name from the Latin cognoscere, referring to knowing and information, thus cognitive psychology is an information-processing psychology derived in part from earlier traditions of the investigation of thought and problem solving.
Situated learning is a theory that explains an individual's acquisition of professional skills and includes research on apprenticeship into how legitimate peripheral participation leads to membership in a community of practice. Situated learning "takes as its focus the relationship between learning and the social situation in which it occurs".
Procedural knowledge is the knowledge exercised in the performance of some task. Unlike descriptive knowledge, which involves knowledge of specific facts or propositions, procedural knowledge involves one's ability to do something. A person doesn't need to be able to verbally articulate their procedural knowledge in order for it to count as knowledge, since procedural knowledge requires only knowing how to correctly perform an action or exercise a skill.
Problem-based learning (PBL) is a student-centered pedagogy in which students learn about a subject through the experience of solving an open-ended problem found in trigger material. The PBL process does not focus on problem solving with a defined solution, but it allows for the development of other desirable skills and attributes. This includes knowledge acquisition, enhanced group collaboration and communication.
A cognitive tutor is a particular kind of intelligent tutoring system that utilizes a cognitive model to provide feedback to students as they are working through problems. This feedback will immediately inform students of the correctness, or incorrectness, of their actions in the tutor interface; however, cognitive tutors also have the ability to provide context-sensitive hints and instruction to guide students towards reasonable next steps.
Constructivism is a theory in education which posits that individuals or learners do not acquire knowledge and understanding by passively perceiving it within a direct process of knowledge transmission, rather they construct new understandings and knowledge through experience and social discourse, integrating new information with what they already know. For children, this includes knowledge gained prior to entering school. It is associated with various philosophical positions, particularly in epistemology as well as ontology, politics, and ethics. The origin of the theory is also linked to Swiss developmental psychologist Jean Piaget's theory of cognitive development.
Cognitive apprenticeship is a theory that emphasizes the importance of the process in which a master of a skill teaches that skill to an apprentice.
Constructivist teaching is based on constructivist learning theory. Constructivist teaching is based on the belief that learning occurs as learners are actively involved in a process of meaning and knowledge construction as opposed to passively receiving information.
Discovery learning is a technique of inquiry-based learning and is considered a constructivist based approach to education. It is also referred to as problem-based learning, experiential learning and 21st century learning. It is supported by the work of learning theorists and psychologists Jean Piaget, Jerome Bruner, and Seymour Papert.
Math wars is the debate over modern mathematics education, textbooks and curricula in the United States that was triggered by the publication in 1989 of the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM) and subsequent development and widespread adoption of a new generation of mathematics curricula inspired by these standards.
Mathematical anxiety, also known as math phobia, is anxiety about one's ability to do mathematics.
Reform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics (NCTM). The NCTM document Curriculum and Evaluation Standards for School Mathematics (CESSM) set forth a vision for K–12 mathematics education in the United States and Canada. The CESSM recommendations were adopted by many local- and federal-level education agencies during the 1990s. In 2000, the NCTM revised its CESSM with the publication of Principles and Standards for School Mathematics (PSSM). Like those in the first publication, the updated recommendations became the basis for many states' mathematics standards, and the method in textbooks developed by many federally-funded projects. The CESSM de-emphasised manual arithmetic in favor of students developing their own conceptual thinking and problem solving. The PSSM presents a more balanced view, but still has the same emphases.
Early Algebra is an approach to early mathematics teaching and learning. It is about teaching traditional topics in more profound ways. It is also an area of research in mathematics education.
Ann Elizabeth Fennema was an American educator specializing in the teaching of mathematics.
Embodied design grows from the idea of embodied cognition: that the actions of the body can play a role in the development of thought and ideas. Embodied design brings mathematics to life; studying the effects of the body on the mind, researchers learn how to design objects and activities for learning. Embodiment is an aspect of pattern recognition in all fields of human endeavor.
In U.S. education, deeper learning is a set of student educational outcomes including acquisition of robust core academic content, higher-order thinking skills, and learning dispositions. Deeper learning is based on the premise that the nature of work, civic, and everyday life is changing and therefore increasingly requires that formal education provides young people with mastery of skills like analytic reasoning, complex problem solving, and teamwork.
Arthur "Art" J. Baroody is an educational psychologist, academic, and an expert in mathematics education research. He is a Professor Emeritus of Curriculum and Instruction at the University of Illinois at Urbana-Champaign, and a Senior Research Fellow in Morgridge College of Education (COE) at the University of Denver.
Susan Baker Empson is an American scholar of mathematics education whose work includes longitudinal studies of children's mathematical development, the use of Cognitively Guided Instruction in mathematics education, analyses of childhood understanding of the concept of fractions, and research on the professional development of mathematics educators. She is a professor emerita in the Department of Learning, Teaching, and Curriculum at the University of Missouri, where she held the Richard Miller endowed chair of mathematics education.