Take, for instance, counting as a means of adding, subtracting, multiplying and dividing:

“As an example of the very difficult mathematics that the below-average children were using, consider the strategy of counting back, which they frequently used with subtraction problems. For example, when they were given problems such as 16 – 3, they would start at the number 16 and count down 13 numbers (16-15-14-13-12-11-10-9-8-7-6-5-4-3). The cognitive complexity of this task is enormous and the room for mistakes is huge. The above-average children did not do this. They said, ‘16 take away 10 is 6, and 6 take away 3 is 3,’ which is much easier.”

At each interval, each up-count or down-count, there’s room for human error. Sustained focus is required. There are no benchmarks for comparison. However, when the question is broken up into derived facts and familiar numbers there are two places for error. The task is quicker. If the concept of derived facts is understood, the complex task is simpler.

Boaler extends this idea of simple = cognitively complex by using the metaphor of a ladder. If students keep using simple ideas to understand mathematics, then they’re faced with a lot of loosely connected steps when solving complex operations and problems. If students are unable to compress smaller concepts in to mathematical big ideas, then the task of solving problems (of climbing cognitive ladders) becomes a massive task. In her book Boaler references a study by Eddie Gray and David Tall who observed that “low achieving students were compressing less – they were so focused on remembering their different methods and stacking one new method on top of the next, that they were not thinking about the bigger concepts… like a never-ending ladder of rules, stretching up to the sky, with every rung of the ladder being another procedure to learn.”

I wonder if this is the case in other subjects? Are there other cases where simplicity actually makes work more difficult?