Many times, carrying out the plan is the exclusive focus of the math classroom. Students are often not responsible for the earlier steps of defining and organizing the plan for solving problems. When they do set out a plan for themselves it becomes a higher priority for them to find an answer. Actual calculation is a multi-step process. In order to calculate, a translation occurs between actual data and their abstract representation as symbols. It is the manipulation of the symbols that is the calculation. After the calculation, the translation occurs again as the results are interpreted not simply in the abstract symbols but in the context of the original problem.
The more complex the calculations, the more time is required to be spent in the symbolic area of thought. The anchors of context must be put aside to be picked up later. Here students can rely on their plan, and the tools that they have collected to perform the calculation. When the object of calculation is understood, students will correctly substitute within formulas and correctly interpret calculated values.
In order that this process is not continually frustrating and exhausting, some parts of the process need to pass from being true inquiry to being habit. Dewey describes phases of reactions to stimuli. There is an open or excitation phase, as the organism experiences a stimulus, and reacts. As the organism becomes familiar and "learns" about the stimulus, it may enter a closing or integration phase. When a student experiences a similar stimulus repeatedly, it may begin to form habits. These habits may behave as intellectual shortcuts to understood knowledge.
Let's use ratios as an example. The more times a student considers a problem with ratios, the more solid is their conceptual ability in that arena. Initially, a student may learn concretely what is meant by one half, as in one half of an apple. This may move to abstractions like shaded blocks and eventually to a symbolic notation as in: 1/2. Repeated trips through these thought pathways may form habits of comprehension, as a student begins to see two halves of an apple are a whole and 1/2 + 1/2 = 1
The problem 1/2 x 1/2 = 1/4 is much more difficult conceptually than 1/2 + 1/2 = 1.
The application of 1/2 x 1/2 = 1/4 to the problem of conditional probability is still more difficult. Without a mental construction of why and how these symbols operate, students fail to recognize similarities among problems. Once they see patterns in the process they can gain a repertoire of known approaches.