If You Want to Learn Algebra, You Need to Have Automaticity on Basic Arithmetic

by Justin Skycak on

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.

Cross-posted from here.

What skills do algebra teachers wish their students had mastered before taking algebra?

Automaticity on basic arithmetic.

That is, the ability to carry out basic arithmetic effortlessly without conscious thought.

Automaticity goes beyond capability, proficiency, and even fluency. Simply recalling information or executing a skill isn’t enough. It must be instantaneous, flawless, and effortless.

Why is automaticity so important?

Because it frees up mental resources for higher thinking.

Think of a basketball player who is running, dribbling, and strategizing all at the same time. If they had to consciously manage every bounce and every stride, they’d be too overwhelmed to look around and strategize.

The same goes for math, especially algebra. 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.

Case study illustrating the importance of automaticity

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 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 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 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.

The big takeaway

This case study demonstrates 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.

And by the way, these results compound. If you don’t develop automaticity on prerequisite skills, that’s going to prevent you from learning and developing automaticity on new skills, and that’s going to compound into a massive “learning debt.” Lack of automaticity is like a high-interest loan that, over time, compounds into a massive financial debt.

How does one build automaticity?

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:

  1. Start with conceptual understanding. (This does not have to go super deep -- the student just has to understand the intuitive / concrete meaning of the symbols they are working with.)
  2. Once a baseline amount of conceptual understanding has been established, move on to practice activities in an untimed setting.
  3. After a student can successfully and consistently execute a skill in an untimed setting, have them practice in a timed setting. (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.)

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.

But what about creativity?

Some people think that because automaticity requires repeated practice, it turns students into mindless robots, and to leverage the power of human creativity, one needs to break free from that robotic mindset. But this is a false dichotomy.

In reality, automaticity is a necessary component of creativity. 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.

As a concrete example, consider what is typically considered one of the most creative activities: writing. Effective writing requires a frictionless pipeline from ideas in one’s mind to words on paper. If a writer had to consciously think about spelling, grammar, word definitions, transitions between sentences, when to make a new paragraph, etc., they would become bogged down in low-level robotic tasks and would have no mental bandwidth to think about high-level creative details like vivid imagery, logical cohesiveness, and emotions evoked by various phrases and ideas.

Further reading

I’ve written extensively on this. See Chapter 14: Developing Automaticity and Chapter 19. The Testing Effect (Retrieval Practice) in the working draft here for more info and plenty of scientific citations to back it up.

Addendum: defense against misinterpretation

  • This post in no way suggests to put fluency before conceptual understanding. You should know what multiplication means before you memorize multiplication facts.

    But for the vast majority of students, especially those who don't enjoy thinking about math outside of class (which is virtually everyone), fluency doesn't naturally happen after conceptual understanding, and you can't create fluency by beating conceptual understanding to death.
  • This post in no way advocates for giving up on students who haven't developed automaticity on basic skills.