id: 05495527
dt: j
an: 2009f.00496
au: Amit, Miriam; Neria, Dorit
ti: “Rising to the challenge”: Using generalization in pattern problems to
unearth the algebraic skills of talented pre-algebra students.
so: ZDM, Int. J. Math. Educ. 40, No. 1, 111-129 (2008).
py: 2008
pu: Springer, Berlin/Heidelberg
la: EN
cc: H23 C33
ut: algebraic thinking; elementary algebra; pattern problems; generalization;
educational research; lower secondary
ci:
li: doi:10.1007/s11858-007-0069-5
ab: Summary: This study focuses on the generalization methods used by talented
pre-algebra students in solving linear and non-linear pattern problems.
A qualitative analysis of the solutions of three problems revealed two
approaches to generalization: recursive-local and functional-global.
The students showed mental flexibility, shifting smoothly between
pictorial, verbal and numerical representations and abandoning additive
solution approaches in favor of more effective multiplicative
strategies. Three forms of reflection aided generalization: reflection
on commonalities in the pattern sequence’s structure, reflection on
the generalization method, and reflection on the “tentative
generalization” through verification of the pattern sequence. The
latter indicates an intuitive grasp of the mathematical power of
generalization. The students’ generalizations evinced algebraic
thinking in the discovery of variables, constants and their mutual
relations, and in the communication of these discoveries using invented
algebraic notation. This study confirms the centrality of
generalizations in mathematics and their potential as gateways to the
world of algebra.
rv: