
06439681
a
2015d.00672
Boester, Timothy
Lehrer, Richard
Visualizing algebraic reasoning.
Kaput, James J. (ed.) et al., Algebra in the early grades. London: Routledge (ISBN 9780805854725/hbk; 9780805854732/pbk). Studies in Mathematical Thinking and Learning Series, 211234 (2008).
2008
London: Routledge
EN
H10
G40
C30
E40
algebraic reasoning
visualization
geometry
generalization
spatial structure
research
design studies
grade 6
lower secondary
metarepresentational competence
representational competencies
graphical representations
coordinates
modes of representation
complicated sorting
comparing the slopes of lines
ratio
patterns in tables
similarity
early education in algebra
From the text: According to {\it A. D. Aleksandrov}, {\it A. N. Kolmogorov}, and {\it M. A. Lavrent'ev} [Mathematics, its content, methods, and meaning. Cambridge, MA: MIT Press (1969)], ``Arithmetic and geometry are the two roots from which has grown the whole of mathematics" (p. 24). Algebra is generally understood as having derived from the arithmetical root. In Chapter 9, the authors highlight algebra's indebtedness to the geometric root of mathematics, noting that ``spatial structure serves as a potentially important springboard to algebraic reasoning, but also that algebraic reasoning supports coming to `see' lines and other geometric elements in new lights." Their argument is not historical but rather psychological: ``Visualization bootstraps algebraic reasoning and algebraic generalization promotes `seeing' new spatial structure".