Mathematics Teacher Education: Singapore Approach: CPD on Singapore Maths in London. The difference between instrumental and relational understanding. I recently found this article written by Richard Skemp that Gary Davis (@republicofmath) highlighted on his blog (update: link broken). I recommend reading the whole article. Skemp describes the difference between instrumental and relational understanding, and how the word understanding is used by different people to mean different types of understanding. He also makes the observation that what we call mathematics is in fact taught in two very distinct ways. Skemp uses an analogy to try and explain the difference between relational and instrumental knowledge which I would like to explore. Imagine you are navigating a park, and you learn from someone else some specific paths to follow in the park. Imagine that instead of navigating the park by specific paths shown to you, you get to wander all over the park. Here's Richard Skemp's description of the analogy.
Unfortunately, our system tends to favour instrumental understanding too much. Jean Piaget | Cognitive Theory. By By Saul McLeod 2009, updated 2015 Piaget's (1936) theory of cognitive development explains how a child constructs a mental model of the world. He disagreed with the idea that intelligence was a fixed trait, and regarded cognitive development as a process which occurs due to biological maturation and interaction with the environment. Piaget was employed at the Binet Institute in the 1920s, where his job was to develop French versions of questions on English intelligence tests.
He became intrigued with the reasons children gave for their wrong answers to the questions that required logical thinking. Piaget (1936) described his work as genetic epistemology (i.e. the origins of thinking). What Piaget wanted to do was not to measure how well children could count, spell or solve problems as a way of grading their I.Q. Piaget (1936) was the first psychologist to make a systematic study of cognitive development. Piaget's Theory Differs From Others In Several Ways: Schemas Stages of Development. Vygotsky. Bruner - Learning Theory in Education. By Saul McLeod published 2008,updated The outcome of cognitive development is thinking.
The intelligent mind creates from experience "generic coding systems that permit one to go beyond the data to new and possibly fruitful predictions" (Bruner, 1957, p. 234). Thus, children as they grow must acquire a way of representing the "recurrent regularities" in their environment. So, to Bruner, important outcomes of learning include not just the concepts, categories, and problem-solving procedures invented previously by the culture, but also the ability to "invent" these things for oneself.
Cognitive growth involves an interaction between basic human capabilities and "culturally invented technologies that serve as amplifiers of these capabilities. " The aim of education should be to create autonomous learners (i.e., learning to learn). In his research on the cognitive development of children (1966), Jerome Bruner proposed three modes of representation: Bruner's Three Modes of Representation Enactive. Growth vs Fixed Mind-set. Mindset affects learning. Stanford Report, February 7, 2007 When psychology Professor Carol Dweck was a sixth-grader at P.S. 153 in Brooklyn, N.Y., she experienced something that made her want to understand why some people view intelligence as a fixed trait while others embrace it as a quality that can be developed and expanded.
Dweck's teacher that year, Mrs. Wilson, seated her students around the room according to their IQ. The girls and boys who didn't have the highest IQ in the class were not allowed to carry the flag during assembly or even wash the blackboard, Dweck said. "She let it be known that IQ for her was the ultimate measure of your intelligence and your character," she said. Asked what seat number Dweck occupied during that memorable year, the professor paused, and silently raised her right index finger. From that experience, Dweck became fascinated with intelligence, convinced that IQ tests are not the only way to measure it. "What was important was the motivation," Dweck said.