By Martha Motley, posted September 14, 2015 –

During a conversation I had with teachers about fact fluency versus memorization, a veteran teacher shared that if students hadn’t memorized their facts by the end of third grade, they would have difficulty progressing into fourth-grade content. Another teacher contended that students could have fluency with their facts and enough flexibility with numbers to allow them to be proficient math students.

I have heard this argument again and again, and I must confess that at one time I also believed strongly that I could not work with fourth graders on fractions, multiplication, and division if they had not yet memorized their multiplication facts up to ten. I used to give out flash cards and encourage students to practice at home and on the computer. I gave weekly timed multiplication quizzes and graded them. I am ashamed to admit that these grades were often factored into quarterly averages. Now I know that the ability to memorize multiplication facts has very little to do with the understanding a child has about the structure of multiplication.

A few months ago, I was in a third-grade classroom in my role as a mathematics coach. The class was beginning a unit on multiplication. On this first day of the unit, the class would be exploring things that come in equal groups. As I entered the classroom, one student had a set of flash cards in his hand. He proudly told me that his class was starting multiplication, but that he already knew all his facts. He quickly flipped to 6 × 7 and 8 × 9 and stated the products. As part of the lesson, students were asked to think about things that come in groups of 3, 4, 5, and 6. Around the classroom were charts for the numbers through ten, where students were able to record their thoughts. I saw one group write yogurts on the 6 chart and pudding cups on the 4 chart. One child wrote wheels on a tricycle on the 3 chart and sides on a square on the 4 chart. Later, students looked at which charts had more recorded “things” and discussed why that might be. The follow-up activity had students choose one of the items from the chart and think about what would happen if they had a certain number of those groups. For example, four tricycles with three wheels on each tricycle is equal to twelve wheels. Students then represented their expression, 4 groups of 3 = 12, and wrote the equation 4 × 3 = 12.

The student who had the flashcards seemed very confused by this activity. He had a hard time thinking about things that come in groups. His partner helped him along. When we came to the point where he was asked to think independently about a situation with a number of groups and write an expression, he struggled. This child had an extremely difficult time finding the relationship between the activity we were working on and his multiplication facts. He was comfortable with the facts. They were memorized and recited without thought. Ours, however, was a very different activity. What I found was that although he had memorized the facts, he did not know what multiplication really was. This child had truly learned to recall the product, but he did not understand the structure of multiplication.

I returned to this classroom throughout the course of the unit. This student gained understanding but struggled more than his peers. He had a preconceived notion about what multiplication was, which made it difficult for his teacher to reach him, because he did not want a context. He wanted facts, naked numbers, without an understanding of the relationship those numbers had to the product.

This experience was so eye-opening to me. I spent years believing that if students just memorized their facts, then I could do all my teaching and make them into great little math students. What I did not consider was that students need to understand the relationship between those numbers. They must understand what equal groups are and how to solve problems in a context. Students need to be taught to think. This child wasn’t thinking. He was spouting facts. It took more work to get him beyond the facts because he didn’t want to think about what the numbers might represent in his everyday world. This child would have been able to perform quite well on the timed multiplication quizzes I gave each week in years past. He would have performed, but he may not have understood. That quiz would have given me absolutely no information about what he had learned about multiplication.

Jo Boaler, a professor of mathematics education at Stanford University, wrote an article, “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts.” This article, along with the recent work we have been doing in my district with Kathy Richardson’s Assessing Mathematics Concepts series (Mathematical Perspectives 2003), has given me a lot of insight into the need for students to build fluency. Many children are not good at memorizing, but through using their facts and understanding the relationships between numbers, students can become fluent with their facts. The research these ladies have conducted shows that students who are quick memorizers are not necessarily great math students. Students need opportunities to work with the numbers. Giving facts and timed quizzes can cause anxiety in students and give teachers a false sense of students’ abilities. Timed fact quizzes give no information on student understanding.

Students should become more fluent with their facts as they progress through school. We certainly don’t want students drawing tallies or circles with dots in fifth and sixth grade. Students do need to understand the structure of multiplication. They need to have time to understand the relationships between doubling factors and what happens to products. They need time to learn to use properties flexibly. Students also need to have constant opportunities to think about the math in context. Instead of asking a student, “What is seven times eight?” ask “How many hot dogs are in seven packs if there are eight hot dogs in each pack?” If a student tells me, “I know that five packs would have forty hotdogs, and two more packs would have sixteen, so that is fifty-six,” I would know that this child has understanding of multiplication and the distributive property. Would I ask that third grader to define the distributive property? No, but the child clearly has an understanding of the property and has used it to decompose and then compose the numbers. This is fluency, and it does give me insight into the child’s ability to think and understand equal groups in multiplication.

Next Steps

Take time to read Boaler’s article, “Fluency Without Fear.” You can find it on her site, youcubed.

Pose problems in context. Allow students to think about the math and to build number relationships. Take time to listen to students’ explanations about their mathematical thinking and use this information to guide your instruction.

Your Turn

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Martha Motley is a K–grade 6 instructional math coach in the Kannapolis City Schools in North Carolina. She spent almost fifteen years in the classroom, teaching exceptional children and regular education second and fourth grades before beginning her role as a coach. Motley has always enjoyed problem solving in mathematics. During the past few years, as her knowledge of mathematics and best math practices has grown, she has enjoyed teaching and sharing strategies with teachers to bring positive change to math instruction in her school district.