Algorithms: Choosing Strategies Strategically

Cynthia Hockman-Chupp

I developed a new appreciation for the power of Bridges strategies when a third grader approached me for help on a worksheet he received in his (non-Bridges) classroom.The “Zero-Concept” worksheet included 36 problems with multi-digit subtraction, intended for practice with borrowing across zeros, solely using the standard algorithm. 

Algorithms: Choosing Strategies Strategically

Although the intent was practice with borrowing/regrouping, it quickly became obvious that several other strategies might produce more efficient results. The third grade standard 3.NBT.2 specifically calls for this:

“Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.”

One of the key words, “algorithms” is plural for a reason. We want students to develop fluency defined by accuracy, efficiency, and flexibility. In this standard, students demonstrate fluency using multiple, flexible strategies — strategies selected because of their strength with a particular set of numbers.

The very first problem, 420 – 115, seemed a good candidate for Partial Place Value Splitting.

Algorithms: Choosing Strategies Strategically

The student could mentally solve the problem using this strategy; he was surprised by how easy it was to break the subtrahend into manageable pieces and then subtract.

Another problem, 200 – 189, seemed ideal for Finding the Difference.

Algorithms: Choosing Strategies Strategically

Again, once he understood the strategy, the problem was easy to solve mentally. In comparison, the standard algorithm was very complex and inefficient, leaving a lot of room for error.

Algorithms: Choosing Strategies Strategically

The Removal Strategy worked for 500 – 333. The student also noted that this could be done mentally using Partial Place Value Splitting, taking away 300, then 30, then 3.

Algorithms: Choosing Strategies Strategically

Once again, borrowing across multiple zeros seemed unnecessarily complex with a high possibility of error.

A problem like 703 – 187 became a prime candidate for Constant Difference.

Algorithms: Choosing Strategies Strategically

He agreed that it was far easier to solve 716 – 200 than 703 – 187.

Looking over the worksheet, we noted that while the standard algorithm might be an efficient method for a handful of problems, for the majority it was not. But perhaps most surprising to my student: the number of problems that could be completely solved with mental math, using one of the above strategies. 

If we think of fluency in terms of accurate, efficient, and flexible thinking, students are best served when they have a variety of strategies from which to choose. By the time we were done, my young friend heartily agreed!