Section 25.5 Going Deeper:Automaticity with Arithmetic
When it comes to arithmetic, faster is not always better. In fact, there a number of people who are not naturally fast at arithmetic, and yet have become very talented mathematicians. However, this does not mean that there is no value at all to being able to do so arithmetic moderately quickly and confidently.
It is easiest to understand this concept by using an analogy with reading. Consider the word cat. As an adult, you probably see the word all at once and can recognize that it's the word cat (as opposed to dog or catastrophe). But let's think about what goes into learning the word cat as a child. For young children, a common reading technique is to "sound out" the various letters. Rather than seeing cat all at once, it's seen as "c - a - t" (with each letter sounded out individually). And after making the sounds in faster and faster succession, the child will eventually understand that it is the word cat.
There's an interesting thing that happens to the brain with reading as people get better at it. It actually starts taking less and less brain power to read as you do more of it. The child that is sounding out the individual letters of the word cat is investing a significantly larger part of their brain than an adult that immediately recognizes it. Basically, our brains get used to seeing the word so often that we have a mental shortcut that allows us to identify it right away.
This skill doesn't make us speed readers, but it does allow us to read fluently. As sentences become longer and their meanings become more complex, the fact that our brains don't have to work hard to recognize words gives us more brain space to think about the ideas rather than using it all up just figuring out which words are on the page.
A similar thing happens with mathematics. Students that struggle with basic arithmetic and algebra are often unable to take in the larger mathematical ideas because their brains are bogged down in the calculations. But even a moderate level of mathematical fluency creates space for students to start to have mathematical ideas and make important connections.
The ability to immediately recognize basic arithmetic facts and fluidly perform simple algebraic manipulations is known as automaticity. Mathematical automaticity is the mathematical equivalent of sight words. It's when you can recognize that \(7 + 4 = 11\) and \(6 \cdot 9 = 54\) without needing to do a bunch of counting or other mental manipulations.
There is a thin line between the practice required to develop automaticity and what is often called "drill and kill" (the endless repetition of calculations that leads to the destruction of all motivation). To understand this, we can draw from another analogy, but this time with music. In the context of learning a musical instrument, there are a core set of exercises (scales and arpeggios) that students practice. These exercises are rarely an end for themselves. That is, nobody goes to concerts to listen to someone simply play scales. However, fluency with scales and arpeggios leads to greater skill in playing more complex music, as the complex music is built out of scales and the intervals found in arpeggios. Another way of saying this is that those exercises help to build the musician's musical vocabulary, so that they are better able to "understand" the music that they're learning.
The same is true for mathematics. The reason math teachers hope for students to reach a certain level of fluency with basic calculations is that our larger mathematical ideas are often built on those calculations. Basic arithmetic is required for almost every pursuit within mathematics, and almost every mathematical idea is built from some experience that can be grounded back in our experiences of basic arithmetic.
As hinted at above, the development of automaticity requires regular practice. Fortunately, the amount of practice is not very significant. Flash cards and online arithmetic practice programs are readily available, and there are cheap or free options. All it takes for many people is just a few minutes a day for a couple weeks to reach a reasonable level of proficiency.
What skills should you practice? Interestingly, it's the same basic arithmetic drills that we would use with elementary school children:
Addition: One-digit plus one-digit
Subtraction: One-digit or two-digit minus one digit
Multiplication: One-digit multiplied by one-digit
Division: The inverses of the multiplication problems
You are encouraged to keep a record of your progress. You might be quite surprised to see how much progress you make in a short period of time with just a little bit of practice. These exercise has been used with college students, and they have reported that their overall level of mathematical confidence has risen as a result of doing them. And in many senses, that is one of the most important obstacles that students who are struggling with mathematics can overcome. The pattern of mathematical self-doubt that a wide range of students bring with them to college is something that can hold them back from accomplishing their goals, and it's incredible how a simple exercise like this can reap positive benefits.
In Section 28.5, we'll talk about how to build larger mental arithmetic skills on the foundation of this basic arithmetic automaticity.