According to the National Mathematics Advisory Panel (2009), there are three important types of knowledge that students need to become proficient in mathematics. The first is conceptual understanding, or understanding of meaning. Conceptual understanding means not just knowing how to do calculations but also why a particular calculation process works. The second is procedural proficiency, or the ability to carry out a particular mathematical procedure with accuracy. The third type of knowledge—the one I want to discuss here—is factual fluency.

**What Is Factual Fluency?**

Factual fluency is not only the ability to recall math facts quickly and accurately, but it is a mixture of knowing some answers from memory and others by recognizing patterns or using a non-counting strategy. Let’s take the addition of 9 + 4, for example. In Japan and Singapore, students do this calculation by decomposing 4 into 1 and 3, adding 1 to 9 to make 10, and then adding 10 and 3 to make 13.

This may seem overly complicated to some people, but it is actually a very efficient and effective method that is common in high-performing East Asian countries but rarely, if ever, taught or learned in the US. If you give a child two ten frames and ask them to place 9 counters on one and 4 on the other, it is easy for the child to think of moving 1 from the 4 to the 9 to make 10.

This method, however, is contingent on knowing how to decompose numbers, knowing the addition facts with sums up to 10 by memory, and knowing how numbers from 11 to 19 are composed of 10 and some more. Understanding this method develops algebraic thinking in young children because it is based on the associative property of addition: 9 + 4 = 9 + (1 + 3) = (9 + 1) + 3.

**The Benefits of Facts Fluency**

According to cognitive psychologist Daniel Willingham, “…automatic retrieval of basic math facts is critical to solving complex problems because complex problems have simpler problems embedded in them.” (Is It True that Some People Just Can’t Do Math, 2009). To carry out the procedure for long division, for example, a student needs to know multiplication and subtraction facts. MRI imaging shows that math facts are held in the working memory section of the brain (Beilock, 2011). Willingham points out that being fluent in math facts frees up working memory, helps avoid errors, and is associated with better performance on complex math tasks. This is especially important for struggling students.

**More than “Drill and Kill”**

The Common Core Progressions documents state: “By the end of the K–2 grade span, students have sufficient experience with addition and subtraction to know single-digit sums from memory. This is not a matter of instilling facts divorced from their meanings, but rather as an outcome of a multi-year process that heavily involves the interplay of practice and reasoning.” If students rely solely on memory, it is difficult for them to reconstruct a fact if they forget it. It is not uncommon even in adults to forget some multiplication facts. If a student understands that 7 x 8 means 7 groups of 8 and these 7 groups can be split into 5 groups and 2 groups, they can add the results of 5 x 8 and 2 x 8 (both easy to remember) to get the product of 7 x 8. In addition, if they learn this method, they will understand the distributive property of multiplication: 7 x 8 = (5 + 2) x 8 = 5 x 8 + 2 x 8.

It has become a popular notion that knowing math facts is not important in the 21st century. I could not disagree more. Learning math facts by memorizing some and understanding how to compose and decompose numbers to do others makes it easier for students to develop conceptual understanding, procedural proficiency, and understand the properties of arithmetic.

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