Invented Strategies

By Saler Axel, RME Research Assistant 

In traditional classrooms, students are often taught one or two strategies for whole-number computation. They memorize rules to compute different operations. At first glance, teachers may mistakenly think their students “get” how to compute. What is often the case though is that students may be able to compute using a tried and true method, but cannot explain why it works. When students attempt cousin items that do not read exactly the same as what they are used to seeing, they may struggle or may not correctly calculate an answer.

When students learn how to compute algorithms, but do not learn the concepts behind them, they miss important stepping stones. By teaching students how to invent strategies, they learn what methods work best for them and which will better serve them in the “real-world.”

According to John Van de Walle in Elementary and Middle School Mathematics: Teaching Developmentally, invented strategies positively impact students’ academic success.

1. Students make fewer errors. When students compute with strategies they understand, they make fewer errors. When students make errors and do not understand the concepts behind their actions, they may have a far more difficult time fixing their efforts.
2. Less re-teaching is required. Teaching conceptual understanding is time consuming, but worth the effort! Not only can students gain the strategies necessary to be more successful in mathematics, the time spent teaching them is meaningful. When students know the “how” of computation but not the “why,” more re-teaching is necessary to help students develop computational skills.
3. Students develop number sense. Students’ development and use of algorithms provide a deeper understanding of the number system.
4. Invented strategies are the basis for mental computation and estimation. Mental computations are invented strategies. When students are taught how to use invented strategies, they are being taught mental computation. There is therefore little need to provide direct lessons in other computational formats or how to do mental math.
5. Flexible methods are often faster than the traditional algorithms. Van de Walle provides the following example to clarify: Consider the product 64 x 8. An invented strategy may be to calculate 60 x 8 = 480 and 8 x 4 = 32. Then find the sum of 480 + 32 which is 500 + 12 which equals 512. A student that uses a traditional algorithm will likely spend more time than someone that uses an invented strategy such as the one above.
6. Algorithm invention is itself a significantly important process of “doing mathematics.” When students invent successful computation strategies, their confidence in mathematics is strengthened. Younger students have been traditionally taught to compute algorithms without understanding why they work or being given the latitude to create their own methodologies. Van de Walle suggests that by opening the door to invented strategies, elementary students gain a valuable view of “doing mathematics.”

Van de Walle gives some examples of invented strategies with multiplication, such as useful visual representations, complete-number strategies (23 x 6 = 23 + 23 + 23 + 23 + 23 + 23 = 138), partitioning strategies, compensation strategies, and using multiples of 10 and 100.

By shifting your practice from teaching students traditional methods to increasing students’ awareness of how computation works, you can provide a solid foundation to enabling the use of invented strategies in mathematics.  As you teach, remember that more math drills is not the answer. Find which facts the student is struggling with and what current strategies they are using on the facts they do know. Break students into teams and challenge them to come up with multiple ways to solve a problem while always explaining how they got the answer. Being able to explain how students came to their answer is essential.

Consider your students. What strategies will you use to best encourage their mathematical thinking and “doing?”

Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, NJ: Pearson Education, Inc.