2 Introduction
relates to the confi dence and ease with which students work
with mathematics concepts.Accuracy, effi ciency, fl exibility, and fl uency are goals built
upon one another within each math concept. Accuracy comes
fi rst because there is no point in having any of the other goals
without exactness. We do not want students to increase speed
if they are getting the answers wrong. Likewise, there is little
need to be fl exible and fl uent with incorrect answers. Educators
should focus on accuracy fi rst, even when it takes a long time for
students to come up with the correct answer. As soon as students
have established accuracy, we need to move right on to focus
on effi ciency. Our goal is to help students learn how to maintain
accuracy while increasing speed. As soon as the students quickly
obtain the right answers, we want to focus on fl exibility by
providing new situations that require changes in their thinking.
The fl uency aspect of mathematics can be addressed during any
of the three previous goals. We can build students’ fl uency while
increasing accuracy, effi ciency, and fl exibility.Think about this scenario. A student is trying to solve the
following division problem: There are 67 boxes of cookies.
The boxes of cookies are packed in cartons. Each carton holds
eight boxes of cookies. How many cartons are needed to pack
all the boxes of cookies? The student tackles this problem using
a guess and check strategy. She starts with 100 cartons and fi gures
out ten cartons hold 800 cookies. Because 800 are way too many,
she tries 50 cartons. Then she tries ten cartons followed by
nine cartons. She tries eight cartons and realizes there are some
leftover boxes of cookies, so she tries nine cartons again. The
strategy worked because her answer is correct; therefore, she
has accuracy. Yet, the guess and check strategy was quite time