Modeling Thinking ◆ 121The Open Number line
After the students have had plenty of experiences with the number grid
and the number line, you want them to move to the open number line.
The open number line is a power tool—one that promotes powerful math-
ematical thinking. It helps children to show and explain their invented
strategies, builds flexibility with numbers and scaffolds the mental repre-
sentation of number and number operations to support mental arithmetic
strategies (Beishuizen, 1999; Fosnot & Uittenbogaard, 2007).
The open number line gives students a model for representing their
thinking. It requires that they be actively engaged in their explanations.
It is more cognitively demanding than either base 10 blocks or the 100
chart according to Klein, Beishuizen and Treffers, 2002 cited in Fosnot,
- It is great to use this strategy with the whole group, but if you really
want to have in-depth conversations, then you should do it in small guided
math groups so children can have the time to explain their thinking. In a
guided math group, the teacher would model the use of the open number
line and then give the students the opportunity to work on a problem
together with the open number line. Finally, the teacher would give each
student an opportunity to solve a problem using the open number line
as the model while explaining their thinking to the group.
Look at this problem: Johnny had 57 marbles. His brother gave him 26
more. How many does he have altogether now? On the open number line, the
child might jump from 57 and then break the 26 into two 10s and go 67
and then 77 and then go to the nearest 10, which would be 80 plus 3 more
to complete the 6, which would be 83.
57 67 77 80 83
To use the open number line, students draw a line, plot numbers on
it and count using a variety of strategies. For example, let’s take the prob-
lem 45 + 37 (see Figure 7.24).
45+30 +5 +275 80 82Figure 7.24The student starts at 45 and jumps 30 because they broke apart the
37 into 30 and 5 and 2. From 75, they jump 5 more to 80 and then 2
more to 82. Open number lines are a huge part of every state’s math
standards, and it is important to make sure that students are comfortable
using them.