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2
The Basic Framework
CGI—Starting with Structure
All state standards use some type of schema-based framework for story
problems. A schema is a way of organizing word problems by type. There
has been a great deal of research on the effectiveness of schema-based
word problems in teaching and learning (Willis & Fuson, 1988; Jitendra &
Hoff, 1996; Fuchs, Fuchs, Finelli, Courey & Hamlett, 2004; Griffin & Jiten-
dra, 2009). One of the most informative professional development books
to teach schema-based problem solving is Children’s Mathematics: Cogni-
tively Guided Instruction (Carpenter, Fennema, Franke, Levi & Empson,
2014). The research from the book frames problem solving around getting
students to understand the different problems conceptually so that they
can reason, use efficient strategies and have procedural fluency.
Furthermore, the research makes the case that the key word method
should be avoided! Students should learn to understand the problem types
and what they are actually discussing rather than “key word” tricks. The
thing about key words is that they only work with simplistic problems,
and so as students do more sophisticated work with word problems, the
key words do not serve them well. They may actually lead them in the
wrong direction, often encouraging the wrong operation. For example,
consider this problem: John has 2 apples. Kate has 3 more than he does. How
many do they have altogether? Many students just add 2 and 3 instead of
unpacking the problem. Here is another example, given this problem: Sue
has 10 marbles. She has 2 times as many marbles as Lucy. How many marbles
does Lucy have? Oftentimes, students just multiply because they see the
word times, instead of reading and understanding the problem.
Introduction to the Types of Problems
There are four general categories for addition and subtraction problems.
In third through fifth grades, the students should be able to solve all of
the problem types using larger whole numbers, fractions and decimals.