What's New with Problem Solving in Common Core?

In addition to content standards, Common Core includes 8 practice standards that are the same for all grade levels. These standards are designed to improve students' problem-solving abilities. The first five practice standards are very similar to the process standards that NCTM has advocated for years. However, there are a few notable differences in Mathematical Practices 6, 7, and 8. I feel that the wording of these is advanced for elementary grades, but the ideas are critical to helping students become proficient with solving problems.

MP6: Attend to precision.

Precision involves using appropriate vocabulary, measurement units, graph labels, and symbols. It also includes adjusting answers to be appropriate for the context of the problem. For example, if the answer to an area or volume question has more nonzero digits than the given information, digits that don't make sense should be rounded. In summary, students need to "pay attention to details" when writing or showing answers to problems. For my students, I extend this to include neatness in written work.

MP7: Look for and make use of structure.

Proficient students can discern patterns and structure in problems. In both geometric and numeric problems, students should be encouraged to break apart or combine parts as they look for patterns. It may also be helpful to sort or rearrange information. An easier summary statement for this standard is to "break apart problems."

MP8: Look for and express regularity in repeated reasoning.

With elementary students, I call this the "look for shortcuts" standard. Students should notice similarities between problems and look for general methods and shortcuts. In the past, some math teachers have expected students to show every step in calculations or solving equations. But to be proficient in math, it is important to recognize and apply shortcuts. For example, students in Grades 4-8 may be able to solve an equation such as 500 + x = 560 by mentally thinking of the missing addend. Because of the general relationship between subtraction and finding a missing addend, students should be encouraged to solve this mentally rather than writing a step of subtracting 500 from each side. The use of shortcuts should be rewarded, not penalized.

Overall, the goal of the Standards for Mathematical Practice is to help students learn habits and processes for solving problems. By encouraging students to pay attention to details, break apart problems, and look for shortcuts, you will help them become better problem solvers.

I have developed a set of 12 posters called "Be a Star" that have checklists and an icon for each practice standard written in language that is easier for elementary students to understand. You can find these on the following site:
http://www.mathpaths.com/practice.html

Steps for Solving Equations

In CCSS, the description of Standard 8 for Mathematical Practice says, "Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts."

This week, I was tutoring an 8th grade student who was very bogged down when solving equations. Her teacher, like many others, was requiring students to "show all of the work." Here is an example of work that the student was showing to solve the equation 7x – 17 = 11. Notice that her first step was to write +17 under each side and draw lines to add. My concern is that this step detracts from key idea of equation solving, which is to write equivalent equations that become simpler and simpler until the solution is clear.

In most textbooks, the second line above would be shown as an equation:
7x – 17 + 17 = 11 + 17
This is OK since it is an equivalent equation, but the goal should be for students to do more steps mentally, as follows (in black text):

When I encourage students to use shortcuts and show only key steps, they usually respond positively. They are glad to try to do more of the work mentally, and it is easier for them to see the progression to simpler and simpler equivalent equations. As a result, students become more confident and their homework papers look neater!