What This Means for Instruction

Here are some tips to help educators plan instruction on operations:

• Involve students as active learners, encouraging them to find multiple ways to answer problems and talk with their classmates about what they are learning.
• Engage in extensive exploration of numerical relationships with manipulatives prior to introducing paper-and-pencil activities.
• Beginning in kindergarten, provide students with encounters with the four basic operations in a variety of problem situations. Meaning for the operations must be developed through modeling using physical models and must be embedded in real-world problem situations. Activities should link to students' language and informal knowledge.
• Give students opportunities to solve problems from all of the problem situations listed below for each of the operations:

• Addition:
• "Join together" - combine two separate or disjoined groups. (combining)
• Include two separate groups. (static)
• Subtraction:
• "Take away" - remove a subset of the original. (take away)
• Find the difference between two sets of objects. (comparing)
• If a part is missing and the whole is known, determine how much more must be added to the part to make the whole. (missing addend)
• Multiplication
• Repeatedly combine sets of equal size. (repeated addition)
• Determine the number of possible pairings between elements of two sets. (combinations)
• Solve a row-by-column problem. (array)
• Division
• "Measure" or separate a group of objects into equal sized groups of a specific size until all objects are distributed. The number of the groups is the answer. (measurement)
• Share a group of objects evenly into a specific number of groups. The number of objects in each group is the answer. (partitive)
• Engage in activities in which students investigate the relationships between operations, as well as the various problem structures for each operation.
• Help students develop "operation sense:"
• understanding the properties and relationships for each operation;
• understanding the relationship between operations;
• understanding the relative effect of operations on numbers.
• Developing "operation sense" should precede basic fact mastery and computational proficiency.