id: 06675837
dt: a
an:
au: Tall, David
ti: Long-term effects of sense making and anxiety in algebra.
so: Stewart, Sepideh (ed.), And the rest is just algebra. Cham: Springer (ISBN
978-3-319-45052-0/hbk; 978-3-319-45053-7/ebook). 43-62 (2017).
py: 2017
pu: Cham: Springer
la: EN
cc: C30 C20 H20 H30
ut: sense making; anxiety; supportive met-befores; problematic met-befores;
level reduction; crystalline concepts
ci:
li: doi:10.1007/978-3-319-45053-7_3
ab: Summary: This chapter offers a framework for the long-term development of
sense making and anxiety for mathematics in general and algebra in
particular. While many may see the development of algebra building from
the basic ideas of arithmetic and generalizing to algebraic techniques
for formulating and solving problems, over the long-term increasingly
subtle changes of meaning may give pleasure to some yet become
problematic for others. The symbol “$-2$” starts off as an
operation “take away 2” but later represents the concept of a
negative number, “minus 2.” The algebraic symbol “$-x$” however
only represents a negative number if $x$ is positive, and takes on the
new meaning as the ”additive inverse” of $x$. While some students
find algebra a source of pleasure and delight as it grows in
sophistication, others find it problematic and seek to rote-learn
techniques in ways that lack meaning in more sophisticated contexts.
Here we consider how successive experiences that individuals encounter
effect long-term learning. Sometimes experiences that are supportive in
one context may become problematic, leading to negative emotional
reactions. The chapter considers how various visual and symbolic
approaches involve specific supportive and problematic aspects.
Sometimes curriculum design that reduces the level of difficulty can
give short-term success yet inhibit long-term sense making. On the
other hand, by reflecting on profound underlying structures
(“crystalline concepts”), mathematical ideas may be constructed and
connected in ways that offer long-term flexibility.
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