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COGNITIVE ROOTS
The notion of cognitive root has developed in my writing over the years. I first talked about it (Tall 1989e) as “an anchoring concept which the learner finds easy to comprehend, yet forms a basis on which a theory may be built.” At the time I was developing a cognitive approach to the calculus and gave the example of a cognitive root as the notion of “local straightness” in calculus.
Later, following an idea of Tony Barnard, the notion of ‘cognitive unit’ was formulated as “a piece of cognitive structure that can be held in the focus of attention all at one time,” together with its immediately available cognitive connections (Barnard and Tall, 1997, p. 41). The power of a cognitive unit “lies in it being a whole which is both smaller and greater than the sum of its parts — smaller in the sense of being able to fit into the short term focus of attention, and greater in the sense of having holistic characteristics which are able to guide its manipulation.” (Barnard, 1999, p. 4).
I then realised the fundamental idea of a cognitive root was a special type of cognitive unit that related to the fundamental human knowledge familiar to the student who is beginning a new conceptual development. Tall, McGowen and DeMarois (2000e) proposed the following:
Definition: A cognitive root is a concept that:
(i) is a meaningful cognitive unit of core knowledge for the student at the beginning of the learning sequence,
(ii) allows initial development through a strategy of cognitive expansion rather than significant cognitive reconstruction,
(iii) contains the possibility of long-term meaning in later developments,
(iv) is robust enough to remain useful as more sophisticated understanding develops.
This was further discussed in the plenary “biological brain, mathematical mind and computational computers” given in Thailand in December 2000.
Papers showing the development of cognitive roots include the following:
1989e Concept Images, Generic Organizers, Computers & Curriculum Change, For the Learning of Mathematics, 9,3 37-42. [the first paper to define cognitive roots.]
2000e (with Mercedes McGowen and Phil DeMarois), The Function Machine as a Cognitive Root for building a rich concept image of the Function Concept, Proceedings of PME-NA, 1, 247-254.
2000d (with Mercedes McGowen and Phil DeMarois), Using the Function Machine as a Cognitive Root for Building a Rich Concept Image of the Function Concept, Proceedings of PME-NA, 1, 255-261.
2000h Biological Brain, Mathematical Mind & Computational Computers (how the computer can support mathematical thinking and learning). In Wei-Chi Yang, Sung-Chi Chu, Jen-Chung Chuan (Eds), Proceedings of the Fifth Asian Technology Conference in Mathematics, Chiang Mai, Thailand (pp. 3–20). ATCM Inc, Blackwood VA. ISBN 974-657-362-4.
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