Is the Standard Algorithm Always Best?

I always see comments from parents on the way their children learn math. The way math is taught now should look rightfully different. The way math was taught in the past focused too much on memorizing procedures and steps. Conceptual understanding and application often took a back seat for procedural fluency. All three are equally important in math, but conceptual understanding and application should be prioritized first in my opinion. If someone understands those two pieces well, the procedural piece will likely come easily... or even on its own without instruction on it. The standard procedure we were all taught isn’t even always the best way.

An example... borrowing. Who likes borrowing in subtraction? It’s annoying. If someone has number sense, a conceptual understanding of how numbers work, it’s unnecessary. Take the example below. 1000 - 843. The whole borrowing from zero thing that was always taught is cumbersome. That’s a lot of borrowing and time to subtract with the standard algorithm. Let’s ponder the idea of differences for a moment. 10 - 6 = 4, 9 - 5 = 4, 8 - 4 = 4, 7 - 3 = 4. What do we notice? When both numbers are reduced by 1, the difference is still 4. Shifting both numbers by the same amount preserves the difference! So forget borrowing; use that number sense. Instead of 1000 - 843, let’s reduce both by 1 and do 999 - 842 instead. Much faster! 157!

Also, those arrow strings you may see elementary students doing. We in math education call that “counting on.” It is a way to visualize subtraction using all the place value they worked with in Kindergarten and first grade, and it’s also how the brain can easily process subtraction mentally. Part of the goal is to build better mental math. Now it looks like a lot on paper because children need to write it out and visualize it at first. But over time they can REALLY quickly in their heads do... “843 plus 7 is 850. 50 more is 900. 100 more is 1000. So, 100 + 50 + 7 is 157.” That seems like a lot written out, but that takes mere seconds in one’s head.

The “standard algorithm” isn’t always the best approach. Conceptual number sense is key.