Introductory note: For the past year, I've been working as a K-5 Math Coach. Not surprisingly, I have learned a lot. I hope the following blog post expresses just a bit of the wonder of the past year...
I developed a new appreciation for the power of strategies taught in Bridges 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.
Although this was the intent (and yes, the way many of us were taught!), 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, one of several strategies explored in Bridges.
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.
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.
The Removal Strategy (using a Number Line) worked for 500-333. He also noted that this could be done mentally using Partial Place Value Splitting, taking away 300, then 30, then 3.
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. Here it's illustrated on a number line:
He agreed that it was far easier to solve 716-200 than 703-187. And it's so simple to get there. Just add 13 to both the minuend and the subtrahend.
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 the 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!