Early Algebra and Mathematical Generalization
By David W.Carraher, Mara V.Martinez, Analu´cia D.Schliemann
In this article, the authors investigated how 15 grade 3
students generalized on algebra questions. The questions and exercise were
given in a sequence of lessons. The previous 33 lessons were used to introduce
some algebra concepts such as variables. The study focused on two table seating
capacity questions. In a warm-up class, separated tables were used to introduce
a relatively basic expression of p=4t in the practice. The study then moved to the
“harder” algebra questions: how many guests could sit when tables are jointed.
The authors argued that “many students scan output values of (n) as n increases,
conceptualizing the function as a recursive sequence which means “students identify
recursion(‘‘keep adding two’’)as the principle that generates successive values
in the output column” (p.5).
This is actually an interesting article which links to
our previous readings in discussion of abstraction and empirical experiences. Bobby’s
written response to the question demonstrated that some students were able to
identify the pattern and express it in an algebraic manner with several
trail-and-errors. In this sense, a good mathematics learning experience should
be one which can eventually lead to mathematical generalizations and
abstraction; rather than focusing on memorizing multiplication table by rote
drills.
However, I found that the authors’ notion on patterns needs
further discussion. They argued that “a pattern is not a mathematical object”(p.4)
and moving towards pattern will create frustration since it lacks rigorous inference.
I appreciate their stance of preferring function over pattern but for younger
students from K-3, I found it is more acceptable if I start with pattern
identification.
My question is whether or not you are supportive to provide
algebra lessons for elementary students as young as grade 3?
