This
study provides some insight into current school practices in relation to word problems,
which authors claim to result in a "suspension of sense-making". It
is suggested that the beliefs of students and teachers towards word problems significantly
contribute to the development of tendencies among children to disregard realistic
considerations in arithmetic word problem solving. The author suggests two main
purposes of word problems in schools: The first main purpose is practising solving
classes of problems that represent applications of mathematics to physical and
social phenomena, and the second, to be vehicles for reasoning about
mathematical structures, using imaginable manipulatives which often not
realistic. The chapter focuses on the first purpose, studying word problems
that describe practical situations arising in people’s lives. The authors claim
that students develop their beliefs towards word problems during the classroom
activities in which they are engaged:
“Students’ beliefs develop from their
perceptions and interpretations of the socio-mathematical norms that determine –explicitly
to some extent, but mainly implicitly – how to behave in a mathematics class,
how to think, how to communicate with the teacher, and so on.”
The
study further suggests that teachers are at fault for sharing the belief that
realistic considerations should not interfere with the "real"
mathematics principles that word problems are intended to relate to, which assumedly
impact on their actual teaching and on their students' learning processes and
outcomes.
STOP
I
find examples of beliefs specific to word problems quite convincing for the
argument that our practice of word problems is built on shaky ground. Let’s
look at two of those beliefs that students often hold:
There is a single,
correct, and precise numerical answer.
Violations of your knowledge or intuitions about the everyday
world should be ignored.
These
findings and many others suggest, indeed, that many children in elementary
school develop very specific beliefs that a word problem should be solved by simple
operations of numbers embedded in the text, leading to the confusion of many
teachers: “Tim did the same word question correctly yesterday but he made it
wrong today”. Maybe Tim just tried adding two numbers yesterday and he decided to
subtract them today, with no thought to the actual meaning of the problem.
The
experimental class introduced in the study also provides many constructive insights.
I would give it a try by replacing some “routine” or “classic” problems by some
non-routine problems designed to provoke awareness about the complexities involved
in taking into account realistic considerations such as “the fastest way
running from classroom to the gym”.
Question:
According
to the author, it is essential to establish a classroom culture by explicitly enhancing
mathematical norms about what counted a good solution; what resources and
support are available for school teachers?
Hi Ting, I agree that a change of mindsets is essential for the value of word problems to emerge more distinctly. I think the teachers may be constrained by the assessment and their responsibility to prepare their students to efficiently manage the assessment; hence it may be a tricky balance between setting up a classroom culture that explicitly enhances mathematical norms of what the authors may deem as a good solution for authentic problem-solving. There is definitely room to strike this balance, but it requires a concerted effort and dedicated time to recognising what the value is for authentic word problems and these corresponding good solutions, before being able to convince resources to be allocated in that direction. My conjecture is that, often times, this may be much bigger than the classroom or school, but rather the society at large and what it champions.
ReplyDeleteI think it's important to provide mathematical norms in a classroom. Math involves a language which mathematicians use to communicate, and students should be encouraged to learn these conventions (i.e., lining up equal signs, showing steps, etc.) Without these, mathematicians couldn't communicate with each other. Whether or not this idea should expand to word problems depends on whether we think it's important that students are able to solve word problems..
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