Fast, Correct Answers Do Matter in Mathematics
You gotta develop automaticity on low-level skills in order to free up mental resources for higher-level thinking!
Correct answers do matter in mathematics.
And furthermore, answers that are not just correct, but also reasonably fast.
Why?
Because you gotta develop automaticity on low-level skills in order to free up mental resources for higher-level thinking!
If you don’t, then low-level actions will constantly be interrupting your flow of thought and you won’t be able to see the forest for the trees.
Think about all the skills that a basketball player has to execute in parallel: they have to run around, dribble the basketball, and think about strategic plays, all at the same time.
If they had to consciously think about the mechanics of running and dribbling, they would not be able to do both at the same time, and they would not have enough brainspace to think about strategy.
The same thing happens in math.
For instance, solving equations feels smooth when basic arithmetic is automatic –
it’s like moving puzzle pieces around, and you just need to identify how they fit together.
But without automaticity on basic arithmetic, each puzzle piece is a heavy weight.
You struggle to move them at all, much less figure out where they’re supposed to go.
It’s the same all the way up the ladder of math: from arithmetic, to algebra, to calculus, to all sorts of university-level math and beyond.
If you try to advance to the next rung on the ladder without standing stable on the previous rung, then you’re going to fall off the ladder.
And even if you get saved by grade inflation and manage to make it up a step without completely falling off, it’s the next step that will get you.
Case Study
Here’s a case study illustrating why it’s crucial for math students to develop automaticity – that is, the ability to carry out lower-level skills effortlessly without conscious thought.
The case study will demonstrate that the more automaticity a student has on their lower-level skills,
- the easier they will find it to acquire new higher-level skills,
- the more quickly and independently they will be able to execute those skills,
- the better they will feel about the learning process as a whole, and
- the more excited they will be to continue learning more advanced material.
Students who develop automaticity will feel empowered, while students who do not will feel overwhelmed and defeated.
Okay, let’s jump into the case study.
Suppose that we have three different students – Otto, Rica, and Finn – whose names are chosen to represent their respective levels of automaticity.
- Otto has developed full automaticity on multiplication facts and procedures.
- Rica doesn't know her multiplication facts – she recalculates them from scratch. She is able to carry out multiplication procedures, but she isn't fully comfortable with them and has to proceed slowly, writing every single step down.
- Finn, likewise, doesn't know his multiplication facts – but he doesn't know his addition facts either, so he uses finger-counting for everything. He is not at all comfortable with multiplication procedures.
These students are each given a lesson on cubes of numbers.
After an explanation of what it means to cube a number, and a demonstration with a worked example, they’re each given a problem to practice on their own:
Compute $4^3.$
Let’s observe the thought processes (both reasoning and emotions) as each of these students solves the problem.
Otto’s Experience
Otto is so comfortable with his multiplication and addition facts that he solves the problem in 10 seconds in his head.
He feels it was easy, is excited to try another, and can’t wait for harder problems like cubing negative numbers, decimal numbers, and fractions.
“$4^3$ = 4 × 4 × 4. I know 4 × 4 = 16, easy, and then 16 × 4 = … well that’s 10 × 4 = 40 and 6 × 4 = 24, together making 40 + 24 = 64. Done, easy! What’s next?”
Rica’s Experience
Rica solves the problem in 2 minutes, but her answer is not correct.
She takes another 2 minutes to correct the mistake but gets tired and wants to take a break before moving on to the next problem.
She’s not looking forward to harder problems.
“$4^3$ = 4 × 4 × 4. What’s 4 × 4? I don’t know, let’s compute it.
4 × 4 is the same as 4 + 4 + 4 + 4, which is … well, 4 + 4 = 8, plus 4 is 12, plus 4 is 16.
Where was I? Oh right, 4 × 4 = 16 and then 16 × 4 = … ugh, gotta go through that multiplication procedure.
Put the 16 on top, then × 4 on bottom, and now we carry out the procedure. First 4 × 6 = 6 + 6 + 6 + 6, count that up to get 6 + 6 = 12, plus 6 is 18, plus 6 is 22. Write down 2, carry another 2. Then 4 × 1 = 4, add the carried 2, write down 6.
Done. Result is 62. Oh wait, the teacher says that’s close but not quite right. Fine, let’s try this again.”
(Rica repeats the entire procedure above and this time gets a result of 64.)
“Great, teacher says that 64 is right. I know there are more problems to do but that one was kind of hard and I’m tired. Teacher, can I take a break and do the next one later?”
Finn’s Experience
Finn takes 10 minutes to solve the problem, but his answer is not correct.
He tries again for another 10 minutes but makes a different mistake.
The teacher has to sit with him for another 10 minutes to carry him through the problem.
By the time Finn is done with the problem, it has almost been a full class period.
He is totally exhausted and overwhelmed and dreads doing the rest of the homework.
“$4^3$ = 4 × 4 × 4. What’s 4 × 4? I don’t know, let’s compute it.
4 × 4 is the same as 4 + 4 + 4 + 4, which is … ugh, gotta count all this up. This is annoying.
Start at 4, then 4 more is 5, 6, 7, 8.
Start at 8, then 4 more is 9, 10, 11, 12.
Start at 12, then 4 more is 12, 13, 14, 15.
Phew, that took a while, but now I have 4 + 4 + 4 + 4 = 15. Why was I doing that, again? Oh right, I was really doing 4 × 4 = 15.
Wait, we’re not even done yet. I did 4 × 4 = 15, but that was because I wanted to do 4 × 4 × 4. Okay so now I need to do 15 × 4. Ew, that’s going to be even harder. I don’t like this. But fine, let’s do it.
15 × 4 is the same as 15 + 15 + 15 + 15, and those are big numbers so I need to line it up on paper.
Put 15 at the top, then another 15 below, then another 15, then another 15.
Let’s add the right column:
Start at 5, then 5 more is 6, 7, 8, 9, 10.
Start at 10, then 5 more is 10, 11, 12, 13, 14.
Start at 14, then 5 more is 15, 16, 17, 18, 19.
Write down 9, carry the 1, then add down the left column: start at 1, then 1 more is 2, then 3, then 4, then 5. Write down the 5, we have 59.
Answer is 59. Glad that’s over. That took forever.
Oh wait, the teacher says that’s wrong. Noooo… do I have to do this whole thing over again?! This is way too much work.”
(Finn repeats the entire procedure above and this time gets a result of 66, which is still incorrect.
He is getting very noticeably frustrated and his teacher sits down with him to go through his work.
They find and fix several errors together and arrive at the correct result of 64.)
“I can’t do any more of this today. I’m too tired. I hate math, and my teacher gives me way too much work.
And the next problem looks harder, and there are even more on the homework! This is terrible.
Class is almost over so I’m just going to zone out until the bell rings.”
How to Build Automaticity
How does one build automaticity in math skills?
Practice. Lots of it.
Automaticity demands regular practice over time until skills are second nature.
But not just any practice. It’s a 3-stage process for each skill to be learned:
Start with understanding the meaning underlying the skill. (The meaning can be centered around a concept or procedure, and it does not have to go super deep -- the student just has to understand the concrete meaning of the symbols they are working with.)
Once a baseline amount of understanding has been established, move on to practice activities in an untimed setting.
After a student can successfully and consistently execute a skill in an untimed setting, have them practice in a timed setting, using the most efficient execution strategy. (Careful: if you time a student before they can even execute the skills untimed, all you're going to do is stress them out and make them hate math.)
Note that the most efficient execution strategy is sometimes detached from the meaning. For instance, in the case of fraction division, the most efficient strategy is "multiply by the reciprocal," NOT "convert both fractions to have the same denominator and then divide the numerators."
(And by the way, the most efficient strategy is sometimes memorization -- for instance, while a student should of course be able to compute 6 × 5 = 5 + 5 + 5 + 5 + 5 + 5 = 30, they absolutely need to memorize 6 × 5 = 30.)
Many heated arguments in math education ultimately amount to people focusing on their personal favorite part of this pipeline and deeming the rest of the pipeline unnecessary.
Come to think about it, many heated arguments in math education come from the false dichotomy of “which comes first, concepts or procedures,” so let me clarify my position before both sides of the debate accuse me of being on the other side.
Concepts and procedures build on each other as you go up levels of math. They have a mutually reinforcing, bidirectional relationship:
- low-level concepts support low-level procedures,
- low-level procedures support higher-level concepts, and
- higher-level concepts support higher-level procedures.
For instance:
- a student must be able to count to understand the basic concept of a number,
- a student must understand the basic concept of a number to carry out arithmetic procedures, and
- a student must be able to carry out arithmetic procedures to understand the concept of a variable or an algebraic equation.
There is no way to “teach all the concepts first,” or “teach all the procedures first.” They must be taught in tandem.
Concrete Example: Multiplication Facts
When to introduce flashcard practice with times tables?
Before a student is asked to instantly recall 6 × 5 = 30 flashcard-style, they should be able to do the following things on their own without assistance:
- work out the problem by computing 6 × 5 = 5 + 5 + 5 + 5 + 5 + 5 = 30 (or 6 + 6 + 6 + 6 + 6 = 21)
- draw a visual representation of the problem, i.e., a rectangle with sides of length 6 units and 5 units, and tell you that the number of unit squares inside the rectangle is 30 (or the equivalent formulation with groups of objects: 6 rows of 5 objects make 30 objects total)
If they can’t do these things first, then it’s going to be difficult for them to remember their flashcards because they won’t be able to leverage any macro-level structure to simplify the memorization process (e.g., understanding that 6 × 5 is the same as 5 × 6, and bigger than than 5 × 6).
Now, it’s easy to go way overboard with this and I’m not saying that a student has to perfectly explain these things to you word-for-word against a specified rubric.
I’m just saying that, in broad strokes, before a student moves onto flashcard-style practice with 6 × 5, the student should be able to
- compute 6 × 5 on their own, and
- draw up what 6 × 5 represents visually.
Why does memorization come last (when learning a particular skill)?
There’s a small group of educators who will disagree that memorization has to come last. For instance, why not have students learn 6 × 5 = 30 purely based on memorization, and then after they have done this purely rote exercise, learn what multiplication actually means?
I guess one could have a student memorize their times tables first and then only afterwards learn how to compute those facts using repeated addition, but I’d expect that the memorization process would take way longer.
That said, the question gets really interesting at higher levels of math. I would not say that students always have to derive things before memorizing them. For instance, I would agree with the usual approach of having students memorize derivative rules first and then later derive them from scratch.
Instead, I would say that a student should loosely understand what something means before memorizing it. This does not have to be a derivation / full computation – it’s just that in arithmetic, full computation turns out to be the most straightforward way to understand things.
In calculus, on the other hand, if you want to loosely explain what the derivative means, then full derivations/computations are not as straightforward, and instead it’s more straightforward to just explain that the derivative represents the instantaneous rate of change, or the slope of the tangent line to the curve. With that information, it makes sense that, e.g., the derivative of $x^2$ should get larger and larger as $x$ increases, and that the derivative of $\sin(x)$ oscillates up and down.
That’s the main thing I’m trying to get at – the reason why it’s helpful to “loosely understand what something means” before memorizing it is that it makes the memorization process a lot easier.
And I would say that the goal should always be to approach that baseline understanding in the most quick / easy / straightforward / simple way, whatever it may be.
Automaticity and Creativity
“If you practice skills repeatedly, won’t you turn into a mindless robot? Don’t you have to break free from that robotic mindset to unleash your human creativity?”
This is one of the biggest misconceptions about the role of automaticity in education.
In reality, repeated practice is a necessary component of creativity. It is the very thing that allows you to break free from robotic thought processes.
The whole purpose of automaticity is to reduce the amount of bandwidth that the brain must allocate to robotic tasks, thereby freeing up cognitive resources to engage in higher-level thinking.
If a student does not develop automaticity, then they will have to consciously think about every low-level action that they perform, which will exhaust their cognitive capacity and leave no room for high-level creative thinking.