Knowing vs Using "Basic Math Facts"

If you pay attention to social media and other media outlets, you’ll see a steady stream of angsty posts, fear-based articles, and memes like this one bemoaning how behind our kids are in mastering basic math facts.

But this storyline itself is a miscalculation. Calling out a simple mathematical error made by a politician in the spotlight makes for a catchy headline (so-and-so doesn’t even know the right answer to 7x8!). And the issue seems real enough to a parent raised on rote memorization of times tables. For many of us, this was the essence of mathematics. This automatic recall of facts is part of what can be defined as mathematical “fluency,” and while it has its place, it is just one aspect of true mathematical fluency.

In recent years, the US has moved toward deemphasizing the rote memorization of what we at MathTrack call the Foundations of Mathematics through the Common Core Standards (CCS) initiative and other similar initiatives at state levels. We’ve written about mathematical cycles, and this historical cadence is similar to what we traced as the patterned changes of “new math.” But, alas, even these progressive standards rely on  “fluency,” which teachers often misinterpret as the continued need to focus on rote memorization. It’s worthwhile to unpack this a bit further.

In a previous blog, I shared that while I have a Ph.D. in Mathematics Education, I don’t consider myself a “math person” in the way you might think. Specifically, over time, I developed a robust set of skills that allowed me to execute arithmetic quickly, but those skills were learned and were not something I was born with. Truthfully, many of those skills were learned through teaching others. This sentiment is something that many teachers regularly share about how they learned mathematics. Consider how many people are excluded from being “math people” if your narrow definition is based only on one’s ability to calculate quickly. I hold fast to my belief that every person is a math person by being human.

The theme of “basic facts'' continues the “math person myth.” When rote memorization is emphasized in the classroom, the basic facts are considered the essence of the discipline. The ability to calculate quickly in your head or recall facts instantaneously becomes what it means to be a “math person.” This builds a significant amount of anxiety, angst, and then disengagement with mathematics for many students (Ashkraft & Krause, 2007). You might be one of them, especially if you’ve heard yourself saying,  “I’m not a math person.” But you are. Let me explain a bit more.

Finding Fluency

Memorization has its place, and I’m not advocating that people should stop using exposure to memorization of foundational mathematics over time. But fluency in its truest sense comes from having the opportunity to work through a series of what I call efficiency cycles. As I also believe mathematics is a language, let’s use that as a metaphor. You cannot become fluent in a language by memorizing words. Someone fluent in a language can easily string words in a followable rhythm into coherent thoughts. The language flows from the speaker with relative ease — the enacted rules of the language become implicit. A similar fluency is sought in mathematics, where language is a mathematical habit of mind and creative problem-solving. Research like cathy fosnot and Maarten Dolk's book, Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction (2001), was seminal to my understanding of these ideas. In this book, the authors defined mathematical fluency as understanding how to build and take apart numbers so you can utilize those insights as tools to flexibly solve problems with increasing efficiency over time.

Our Hands Are Natural Calculators

Everyone finds fluency differently, and that also includes at different paces. What does that look like when learning the Foundations of Mathematics? I’ve worked with thousands of students learning the Foundations of Mathematics. When it comes to multiplication, for example, exposing students to certain fluency cycles helps them learn at their own pace. It’s about students building Mathematical Habits of Mind (MHM) through physical and mental tools to approach a problem or a task. If you have fingers, toes, or aardvarks, use them!

Thematically, this plays out when students are trying to hide their hands under their desks so the teacher doesn’t “catch” them counting with their fingers. The student experiences anxiety in this setting when they have a method for solving the task, but the adult or power structure in the classroom does not let them use it. Our base 10 number system, which is ingenious but rife with issues, is based on ten because we have ten fingers. Use them!

Multiplication Is Patterns, Not Magic

The next part of the fluency cycle for multiplication is learning ways to hunt for the patterns embedded within them. Better yet, thoughtful coaching and teaching of mathematics allow students to hunt on their own. This approach requires patience. If you are a guide and have been up the mountain a thousand times, you may be bored with its beauty—but a new traveler is simultaneously tired, confused, and, at times, awe-struck. Some of the patterns of multiplication are repetitive in nature, and some of them are deep within number theory but very accessible to even a 9-year-old. Here is an example of some of the multiplication facts with ways of thinking about them that will build fluency:

• Multiplication by 2s (multiplier) is about doubling. If a child can add the number (multiplicand) being multiplied by 2 to itself, then all of the basic multiplication facts for 2 can be created with doubling addition.
• Multiplication by 9s has a really interesting pattern. All of the products have digits that add up to 9. For example, 9x3 = 27; if you look at the digits of the product (answer), they are 7 and 2. Add those together, and you get 9. A fun fact question is, which number/s doesn’t this work for between 1 and 20?
• Multiplication by 11s reveals interesting patterns for kids to learn as tools. Of course, when looking at 1-9, it's just two of those numbers for the product. For example, 8x11= 88. But what about when you get to tasks like 11 x 13 = 143? If you take the 13 and “spread it out,” and then to find the middle number you add the 1 and the 3, which equals 4. I know, ridiculous—try it with 17! 17x11 = 187. You are a new adventurer.
• Multiplying by 10 is taking the number and moving it one place value higher. People like to say, “just add a zero,” but I think that hinders the robustness of understanding what you could build with this idea. If you are asking a child to find 10 x 6 = ?? you could think about this as, “What if 6 was in the tens place, what would we call that?”
• When multiplying by 12, you combine the robustness of multiplication by 2 and by 10. Whatever number times 12 you are trying to find, you can double it, and you take the original number and put it in the tens place (add a zero), then add those two newly found quantities together. For example, 12 x 6= (6 x 2) + (6 x 10) = 12 + 60 = 72. This is possible because of the distributive property, one of foundational mathematics' most powerful properties. But you probably remember it as something called FOIL. Yeeeesh.

There are fluency tips like this for all levels of arithmetic. They aren’t mathemagic, and they aren’t cheating. They are patterns that students can find comfort in using as they move towards cycles of greater efficiency and fluency. That takes time. A parent and teacher can further their understanding by learning why these patterns work or exist. In Malcolm Gladwell’s 2008 book Outlier, he posits that mastering something can take around 10,000 hours. However, he later elaborated that a significant chunk of time is more a symbol than a specific magic number to work toward. Fluency requires consistent, continuous effort.

One key is that you, as the learner, need a mentor, peer, or coach constantly modeling good MHMs for building mastery, giving good feedback, and being patient with your growth. While continuous growth requires hard work, it should have as little anxiety as possible. No one will pursue something as long as fluency takes to acquire if they feel anxious or ineffectual the entire time. Not the proverbial hitting kids' knuckles with rulers because they use their naturally born physical tool sets to figure out an arithmetic task.

Let’s Build Better Mathematical Habits of Mind — Together

Some of our favorite researchers on these topics, like Dr Jo Boaler, Elena Silva, and Taylor C. White, and many others, have extensively written and researched how we can build fluency in mathematics without fear. Our approach, MathTrack Institute, is built from these researchers but focuses on training the educator and including parents in the education process. If we can consistently improve teachers’ (parents’, too) ability to model high-quality MHMs for their students, we can increase the probability of more equitable mathematics learning in the US.

So the next time you find yourself working with a young person and wondering why they don’t know a basic fact, think, “How can I model mathematical habits of mind for this student?” Emphasizing habits of mind over rote memorization of facts will reduce the anxiety and fear for the student, teacher, and parent. We may also observe something even more profound and unexpected emerging from this collective approach—something I will write about in the future.