Mental Math Strategies

Development of mental math strategies is an important outcome of students' study of whole numbers for two reasons. First, use of these strategies helps students develop number sense and flexibility with numbers. Secondly, using mental math is often the most efficient means of solving problems, either for finding exact answers or for finding estimates when appropriate. Therefore, students need to have many opportunities to choose an efficient strategy for a given problem based on the numbers in the problems and the real-world context.

Students should have opportunities to develop strategies that make sense to them. Some strategies may include some mental calculations along with a few written notes. Representations may also support students as they develop these mental math strategies (e.g., coins, number lines, base-ten pieces). For example, to solve 345 + 205, students may visualize the number line but keep track of the hops by writing down numbers: 345 + 200 is 545 and 545 + 5 = 550.

Discussing their strategies and explaining them in writing will help students clarify their own thinking and provide new ideas for others. However, it is important that students have opportunities to practice strategies that emphasize their efficiency. That is, when they choose to solve a problem mentally, they should not have to explain their thinking every time. For these reasons, students will see different types of directions for solving mental math problems in the unit. In some cases they are asked to explain their thinking in writing. At other times they are asked to share their thinking with a partner in discussion or to choose one problem out of a set of problems to show their reasoning in writing.

Estimation Strategies

Estimation is found throughout the Math Trailblazer curriculum. In this unit, students discuss one tool that is frequently used when estimating the result of a computation—finding a "friendly" or "convenient" number that is close to a given number. For example, find the sum of 57 and 74 by thinking that 57 is close to 60 and 74 is close to 75, so the sum is close to 60 + 75. Students first work with base-ten pieces to develop their sense of numbers in relation to "friendly" numbers like multiples of 10 and 100. Estimation is discussed as a way of finding reasonable, close answers and a way of predicting or checking answers to computation problems. We introduce these ideas before students work on paper-and-pencil methods for exact answers so that they are encouraged to use mental strategies for estimates.

Paper-and-Pencil Strategies

In this varied collection of strategies, three are considered paper-and-pencil methods: expanded form, all-partials, and compact. All three algorithms are based on the organization of the base-ten number system. There are many opportunities for students to identify connections between these strategies. The compact method is the method traditionally taught in the United States. However, it is not the only efficient method and is sometimes not the most efficient.

In fact, the all-partials method is a very transparent method so it is easy to find errors. It is a method that helps support and develop number sense and therefore estimation strategies. The expanded form method helps students clearly keep track of the different partitions of numbers needed to solve a problem.

As students learn efficient paper-and-pencil algorithms, they should be encouraged to revisit base-ten pieces as necessary to establish the connection between the symbolic procedures and the concrete representation. While work with base-ten pieces should inspire work with symbols, the two processes are not identical in students' minds. One major difference between using the pieces and the algorithm is that with the pencil-and-paper approach, the original problem remains on the paper in some form, while with pieces, the original problem is not evident once the solution is found. It takes time to make connections between base-ten pieces and algorithms, and you will need to help students make the connections.

Because the link between the manipulative work and paper-and-pencil solutions is not always clear to students, some educators suggest that it is confusing to try to slavishly mimic manipulative work in a symbolic way, step-by-step, i.e., doing the two procedures in parallel. Rather, they recommend first working the problem with pieces and then working the problem symbolically. Parallels between the two methods can then be made.

Subtraction Facts

Daily Practice and Problems items in this unit can be used to assess students' fluency with the subtraction facts related to the addition facts in Group F (14 − 4, 14 − 6, 14 − 8, 14 − 10, 15 − 5, 15 − 6, 15 − 9, 15 − 10, 16 − 6, 16 − 7, 16 − 9, 16 − 10, 17 − 7, 17 − 8, 17 − 9, 17 − 10, 18 − 8, 18 − 9, 18 − 10). Counting-on, counting-up, and counting-back strategies are most commonly used to solve these facts.

"A fundamental goal of all our algebraic thinking work focuses on helping teachers understand and engage their students in relational thinking. By relational thinking we mean examining expressions and equations in their entirety rather than as a process to be carried out step by step. Doing so requires using fundamental properties of number and operation to transform the mathematical expression … Ideas of relational thinking fit well into the arithmetic curriculum because they often make computation easier and provide an opportunity to make explicit much of what students are doing when they operate on numbers using standard algorithms. We see this when students add multidigit numbers and use an invented algorithm, for instance, on the problem 25 + 37 a child may respond, "I know that 20 and 30 is 50. Then I added the 5 and the 7. That's like 5 and 5 is 10, and 2 more. So that's one more 10, so 60 and 2 is 62" (Franke, Carpenter, and Battey, 2008).

In this unit, students focus on just this type of relational thinking as they develop mental math strategies and multiple methods to add and subtract multidigit numbers.

Resources

• Franke, M., T. Carpenter, and D. Battey. "Content Matters: Algebraic Reasoning in Teacher Professional Development." In Algebra in the Early Years. James J. Kaput, D.W. Carraher, and M.L. Blanton, eds. Lawrence Erlbaum Associates, New York, NY, 2008.
• Fuson, K.C. "Developing Mathematical Power in Whole Number Operations." In A Research Companion to Principles and Standards for School Mathematics. J. Kilpatrick, W.G. Martin, and D. Schifter, eds. National Council of Teachers of Mathematics, Reston, VA, 2003.
• Fuson, K.C., and B.H. Burghardt. "Multidigit Addition and Subtraction Methods Invented in Small Groups and Teacher Support of Problem Solving and Reflection." In The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise. A.J. Baroody and A. Dowker, eds. Lawrence Erlbaum Associates, Mahwah, NJ, 2003.
• Hiebert, J., and D. Wearne. "Instruction, Understanding, and Skill in Multidigit Addition and Subtraction." Cognition and Instruction, 14 (3), pp. 251–283, 1996.
• Hiebert, J. "Relationships between Research and the NCTM Standards." Journal for Research in Mathematics Education, 30 (1), pp. 3–19, 1999.
• Hiebert, J., and D. Wearne. "Instruction, Understanding, and Skill in Multidigit Addition and Subtraction." Cognition and Instruction, 14 (3), pp. 251–283, 1996.
• Kamii, C., and A. Dominick, "The Harmful Effects of Algorithms in Grades 1–4." In The Teaching and Learning of Algorithms in School Mathematics L.J. Morrow and M.J. Kenney, eds. (1998 Yearbook of the National Council of Teachers of Mathematics, pp. 130–140). Reston, VA, 1998.
• National Research Council. "Developing Proficiency with Whole Numbers." In Adding It Up: Helping Children Learn Mathematics. J. Kilpatrick, J. Swafford, and B. Findell, eds. National Academy Press, Washington, DC, 2001.
• National Research Council. "Teaching for Mathematical Proficiency." In Adding It Up: Helping Children Learn Mathematics. J. Kilpatrick, J. Swafford, and B. Findell, eds. National Academy Press, Washington, DC, 2001.
• Principles and Standards for School Mathematics. The National Council of Teachers of Mathematics, Reston, VA, 2000.
• Verschaffel, L.B. Greer, and E. De Corte. "Whole Number Concepts and Operations." In Second Handbook of Research on Mathematics Teaching and Learning. F.K. Lester, Jr., ed. Information Age Publishing Inc., Charlotte, NC, 2007.
• Wearne, Diana, and James Hiebert. "Place Value and Addition and Subtraction," Arithmetic Teacher, 41 (5), pp. 272–274, 1994.