Children’s first methods are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestors did. By trying to bypass the constructive process, we prevent them from making sense of arithmetic.
Fluency demands more of students than memorizing a single procedure does. Fluency rests on a well-build mathematical foundation that involves:
Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily.
Accuracy depends on careful recording, knowledge of basic number combinations and other important number relationships, and verifying results.
Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to choose an appropriate strategy for a specific problem.
Stages for Adding and Subtracting Large Numbers
Direct Modeling: The use of manipulatives or drawings along with counting to represent the meaning of the problem.
Invented Strategies: Any strategy other than the traditional algorithm and does not involve direct modeling or counting by ones. These are also called personal or flexible strategies or alternative algorithms.
U.S. Traditional Algorithms: The traditional algorithms for addition and subtraction require an understanding of regrouping, exchanging 10 in one place value position for 1 in the position to the left - or the reverse, exchanging 1 for 10 in the position to the right.
What do we mean by U.S. Traditional Algorithms?
Now that we have discussed some alternative methods for solving subtraction equations, let’s return to the problems we solved earlier. Go back and try to solve one or more of the problems using some of the ways on the subtraction handout. Try using a strategy that is different from what you used earlier.
Use story problems frequently. Example: Presents and Parcels picture problems from Grade 2 Bridges
Not every task must be a story problem. When students are engaged in figuring out a new strategy, bare problems are fine. Examples: Base-ten bank, work place games such as Handfuls of treasure and Scoop 100 from Grade 2 Bridges.
Suggestions for Using/Teaching Traditional Algorithms
Delay! Delay! Delay!
Spend most of your time on invented strategies. The understanding students gain from working with invented strategies will make it much easier for them to understand the traditional algorithm.
If you teach them, begin with models only, then models with the written record, and lastly the written numerals only.