Gattegno’s cryptomorphisms: Modeling algebraic understanding in the early years

Room 102, Jordan Building, Stanford University

Caleb Gattegno was a pure mathematician and educationalist. A sometime associate of Jean Piaget and Jean Dieudonne, Gattegno pioneered a radical reconceptualization of pre-university mathematics, including advocating the study of algebra before arithmetic. Gattegno approached teaching the integer and rational number systems through student investigations using sets of ideograms of color coded cuboids (“Cuisenaire rods”) in a variety of configurations together with the equivalence relations between them. He wrote, "All mathematical discoveries of importance can be traced to a dynamic alteration within our mind of existing organized images, or ideas."(1) He elaborated a theory of learning to account for this process. A central idea was that, in learning to listen and speak, or to see and move, a baby has already developed the capacity to reason algebraically. Educationists have recently rediscovered the merits of "early algebraization."(2,3) Similarly his learning model forshadows "dual process" psychological theories of higher cognition.(4) To Gattegno, all mathematical reasoning was grounded in mental imagery that required a conscious “awareness” of algebraic structure for its manipulation. His proposal for the new (Cui) curriculum challenged the learner with a series of progressively more complex “rod worlds.” Elements and actions in the rod world are mirrored in virtual actions on imagery suggested by these actions. Color codes and notation are used to give names to these elements and actions. Learners are encouraged from Year 1 (aged 5) to read and write expressions and equations in simple formal languages. They learn to move fluently between these representations of mathematical ideas so that the elements and operations or actions of one structure can be substituted for the elements and operations or actions of the other: a “cryptomorphism.” In this seminar we will explore exercises drawn from Gattegno's text-books, and illustrate these activities with examples of students’ work.

Participants may wish to download virtual rod resources to their laptop, iOS or android devices: links to follow can be found

here.

(1) C. Gattegno, Thinking Afresh About Arithmetic, Arithmetic Teacher, v6, 1, 1959, p30-32

http://www.jstor.org/stable/41184118

(2) For an overview see Chapter 1: Treating the Operations of Arithmetic as Functions, David Carraher, Analúcia D. Schliemann and Bárbara Brizuela

Source: Journal for Research in Mathematics Education. Monograph, Vol. 13, 2005, NCTM

http://www.jstor.org/stable/30037729

(3) Early algebraization : a global dialogue from multiple perspectives, Jinfa Cai and Eric Knuth, 2011

(4) Dual process theories of higher cognition: Advancing the debate. Johnathan St. B. T. Evans, Keith E. Stanovich, Perspectives on Psychological Science, 3, 223--241