What is So Great About Automaticity?

Every other week, our founder and CEO, John Tapper, shares what's on his mind, from his thoughts on math education to what's been inspiring him lately. This is your chance to hear directly from the person who started it all. We believe great ideas are worth sharing, and Founder's Corner is our way of bringing you closer to the heart of All Learners Network (ALN) and the passion that drives everything we do.

I was meeting with one of my third-grade intervention students who was having a hard time remembering what 6x7 was. “The sevens are so hard!” he confided. “My mom and I practiced them, but I can’t always remember.”

“You know your doubles, right?” I asked.

“Sure,” he answered.

“OK, what’s 6x6?”

My student knew right away that it was 36.

“Well,” I said, “if six sixes is 36, what’s seven sixes?”

He screwed up his face. “Wait a minute. You mean 6x7 is SEVEN SIXES???”

“Yep,” I said.

“That’s just six more than six times six!”

It’s interesting to think about knowing the times tables without knowing what they mean. This happens more often than you think in school. Automaticity built out of understanding and automaticity drilled in are two different things that look identical on a timed test, and schools treat them as interchangeable because the test cannot tell them apart.

When Automaticity Isn't Enough 

Many (most?) math teachers have a belief that having math facts and procedures memorized represents a high level of math thinking. I’ve always found that having math facts (like times tables or addition facts) quickly at hand makes mental computation quicker and often helps find interesting strategies. But, automaticity also presents a problem. When you’re doing something automatically you’re not thinking about it very much.

I got back into thinking about this as I was reading Ellen Langer’s excellent book, The Power of Mindful Learning. A Harvard researcher, she has spent her career exploring how mindfulness and expectation shape learning and health. She doesn’t use the term mindfulness in the way that it’s most commonly used. She isn’t talking about meditation. Rather, she treats mindfulness as a kind of “notice and wonder” on steroids. She has quite a bit to say, supported by her own research, about the way mindfulness improves learning, and automaticity can tend to constrain learning:

“(Basics)… are not useful for everyone… across all situations. If they are mindlessly overlearned, they are not likely to be varied even when variation would be advantageous. Perhaps one could say that for everyone there are certain basics, but that there is no such thing as the basics.”

“We can learn a skill by accepting at face value what we are told about how to practice it or we can come to an understanding over time of what the skill entails.”

The discussion about the relative value of memorization and drill has been going on a long time. In the 1920s Edward Thorndike created the psychological justification for drill in learning. A decade later William Brownell answered Thorndike’s theory with Psychological Considerations in the Learning and Teaching of Arithmetic (1935), arguing that arithmetic taught as bonds and habits produces students who compute without knowing what they are doing. Almost 100 years ago, educators (and psychologists) were having this same discussion.

A Century-Old Debate

Eddie Gray and David Tall’s 1994 paper, Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic, argues that strong mathematical thinking comes from treating symbols like 3+2 as both a process to carry out and a concept that can be mentally manipulated. Their work came at a time when proponents of memorization were making claims that automaticity “freed working memory,” allowing for deeper thinking. The pedagogical arguments for automaticity came from John Sweller’s work on cognitive load. Gray and Tall were pushing back against the idea that automation is enough, especially when the automatized behavior is a hollow procedure with no conceptual content.

Sweller's cognitive load theory rests on a finding that has held up well. Working memory can hold only a few new things at once, but it handles material already stored in long-term memory almost for free. A student who has to reconstruct 6x7 while also tracking a multi-step problem is spending capacity that a student who simply knows it can spend on the problem. That is not a small point, and any teacher who has watched a child lose the thread of a word problem while counting on their fingers knows it is true.

But Sweller says it isn't repetition that’s doing the work. It's the schema, an organized piece of knowledge that lets you treat many elements as one. Repetition is a way to build one. It is not the only way, and it is not a reliable one. My third grader had practiced the sevens. The practice had not built anything he could use. What he found in about fifteen seconds was a schema, and the fact came along with it.

The Real Constraint is Teacher Load

From my perspective, the real constraint is not cognitive load for students. It is teacher load. To demonstrate procedures and then provide worksheets for students to practice them is, by far, the lowest amount of teacher work. If a teacher knows the math they’re teaching, talking about it and then giving students prepared materials for practice is straightforward, predictable, and familiar. Yet, since we’ve collected data on math learning we’ve known that this “think like me” approach leaves out the majority of students.

The alternative pedagogy, helping students develop deep mathematical understanding for themselves, is a more complex, subtle, and time-consuming endeavor. Providing mathematical access for all students requires far more than telling students what to think and handing them a practice sheet.

Constance Kamii, in one of the books that shaped my thinking as a new teacher (Young Children Reinvent Arithmetic) shows us that teaching standard algorithms early doesn't just fail to build understanding, it actively unteaches the place-value thinking children arrive with. In our work with many school districts, we find that they can considerably reduce the number of students receiving math intervention by avoiding teaching the standard algorithm, or procedure. Like Kamii, we find that when children “learn” a procedure without understanding they let go of the good thinking they bring with them. Worse, they learn that their own thinking must always take a back seat to whatever the teacher tells them. They learn not to look for a concept to make sense mathematically. They simply wait for the teacher to tell them whether they are right or wrong and move on.

I need to make my position on learning math facts clear, so that any reader won’t go away with the notion that, “John is against learning math facts.” I’m absolutely not. I think they can be very useful. But I would agree with Arthur Baroody (a longtime math education researcher) who says that relations are more important than repetition when it comes to learning math facts. As Ellen Langer suggests, what we notice and connect builds deeper learning than anything we simply try to automate.

What Now? 

1. Leveraging Launch and Games for Multiplication Fact Fluency — See how a well-designed launch and the right game can build the kind of fact fluency that actually sticks, rooted in understanding rather than drill.
   
   
2. Three in a Row ALO Multiplication Game — Put it into practice today. This free game builds multiplication fluency the way John describes, through strategy, relationships, and engagement rather than repetition
   
   
3. Computational Fluency: What is it? — Fluency is more than speed and accuracy. This article breaks down what computational fluency actually means and why the distinction matters for every student in your classroom.
   
   
4. Bring ALN to your school or district! — Contact us to explore embedded professional development. 

All Learners Network is committed to supporting pedagogy so that all students can access quality math instruction. We do this through our online platform, free resources, events, and embedded professional development. Learn more about how we work with schools and districts here.