Introduction 3consuming. If we teach this student to use repeated subtraction,
her speed will increase. Beginning with 67, she subtracts eight,
then eight more, then eight more, and so on until she has three
boxes left. She knows these boxes also need a carton and comes
up with the correct answer to the problem. She has maintained
accuracy and increased her speed. To help this student move
to fl exibility, we can encourage her to consider fact families
and inverse operations. She knows 8 x 8 = 64 and therefore can
apply this knowledge to the division situation. Furthering her
fl exibility, she compares 64 to 67 and accurately and effi ciently
fi nds that nine cartons are needed to hold 67 boxes of cookies.
Her confi dence and ease have increased based upon the
successful experiences.
Timing is very important. Sometimes well-meaning math
educators jump too fast to effi ciency and fl exibility. They
impose all of these rules and procedures with intentions
of helping students see effi cient and fl exible ways to solve
problems. However, teaching rules and procedures is not the
same as teaching concepts. Understanding what to do is not
enough. We must teach students why a procedure works so
that they better understand the concept. If we only teach rules
and procedures, struggling students typically achieve random
accuracy (if they remember the rules and procedures) and
rarely achieve fl exibility or fl uency because they do not truly
own the knowledge. Math educators need to teach concepts to
help students build foundations so that they can understand
the math. When concepts are ignored and the focus is solely
on rules and procedures, struggling students often develop
misconceptions and learning gaps. However, if students
understand concepts, it is appropriate to help them increase
speed and fl exibility by teaching rules and procedures, but we