126 ◆ Modeling Thinking
Charles states that bar diagrams are a visual approach to teaching
word problems. He notes that
problem solving is grounded in reasoning. Quantitative reasoning
involves identifying the quantities in a problem and using reason-
ing to identify the relationship between them.
(http://assets.pearsonschool.com/asset_mgr/current/
201218/MatMon110890Charles_SWP_Revise_eBook.pdf p.5)For example,
John has 25 green marbles in his collection. He has 10 more multicol-
ored marbles than green ones. How many multicolored marbles does John
have? How many does he have altogether? (See Figure 7.32.)
The quantities in this word problem^ are:
- The number of green marbles (a known value 25)
- The number of multicolored marbles (an unknown value)
GM2525 +10??Figure 7.32As Diezmann and English note, “a diagram can serve to ‘unpack’
the structure of a problem and lay the foundation for its solution” (cited
in Charles, Monograph 24324). Nickerson found that the ability to use
diagrams is integral to mathematics thinking and learning (cited in
Charles). Other researchers found that
training children in the process of using diagrams to [meaningfully
represent and] solve [mathematical word] problems results in more
improved problem-solving performance than training students in
any other strategy.
(Yancey, Thompson & Yancey, cited in Charles)Charles emphasizes that the most important thing is to get our students
thinking about the relationships between these quantities. Often, students
rush to the operation without fully understanding the problem. They are