Math is not about crunching numbers; true math is about pattern recognition and logic. This does not mean that mastering basic calculations is not important. Learning math facts is like learning the alphabet. A child who must sound out words, syllable by syllable is not able to comprehend a reading passage. Likewise, a child who struggles to complete basic computations is not able to recognize new mathematical relationships.
While the ability to memorize math facts is not related to mathematical reasoning skills required for higher level math it greatly influences the perception of a child’s math ability. Compare two children learning to add fractions. They are given the task of calculating 1/14 + 1/21. Both equally understand the concept and procedure to add fractions.
Sophia knows her times tables but uses skip counting for a few of the harder math facts such as 6 X 8 and 7 X 7. She remembers that a common denominator may be found by multiplying the two denominators. After performing long multiplication to find that 14 X 21 = 294, Sophia converts the fractions to 21/294 + 14/294 = 35/294. She then factors both 35 and 294, probably using long division twice to find that 294 may be factored to 2 X 3 X 7 X 7. After recognizing that 7 is a common factor, Sophia converts 35/294 to 5/42. While Sophia is able to eventually get the correct answer, this process required her to perform at least one three-step double-digit multiplication and probably three long division calculations.
Emma has instant recall of all of her math facts and quickly recognizes that both 21 and 14 are multiples of 7. Emma quickly converts the problem to 3/42 + 2/42 = 5/42. By avoiding the need to perform double-digit multiplication and long division Emma is able to finish at least five times as many problems as the Sophia.
Although Sophia’s mathematical reasoning skills matches Emma’s, Emma is perceived as the “better” at math. Since Sophia is the last to finish the math worksheet and has several errors due to simple calculation mistakes, she believes that she’s “not good at math.” These perceptions are reinforced by classmates, teachers and parents who also equate Emma’s speed and Sophia’s slowness to their math abilities.
Over time these perceptions become reality. Emma is able to complete more math problems because she is able to work faster. She is more likely to complete challenge problems, be selected for pull-out programs and eventually accelerated math. Sophia, on the other hand, does not have time to complete challenge problems or participate in enrichment math since it takes her so long to complete her regular assignments.
In reality, the difference between the two girls is the degree to which they learned their multiplication facts. Both understand the concept of multiplication. Both understand how to calculate a product through repeated addition of skip counting. But Emma recognizes multiples like sight words while Sophia needs to calculate a few of them like a child who must still sound out a few words.
While Emma and Sophia are fictional, I’ve worked with hundreds of students from algebra to calculus whose self-perceptions and actual ability to master higher level math have been limited not by their innate mathematical reasoning skills but their mastery (or lack thereof) of basic math facts.
One of my 4th graders, Lucas*, recently said to me, “I don’t need to learn my multiplication facts anymore because we’re done with that in school. Now I’m doing division instead.” In his mind, he was done with the task of multiplication and didn’t need it anymore.
Not only will Lucas need to know his multiplication facts for long division and working with fractions, but he won’t recognize the patterns formed by exponents, factorials, calculations of permutations and combinations, factoring polynomials or infinite series just to name a few. I often find it difficult to convince students and parents to invest the time and effort to memorize their math facts until they inevitably encounter the frustration of no longer being able to succeed in class.
Last month my 4th graders learned long division. All the students picked up the concept quickly and could easily dictate the next step required to complete a problem when we worked on group exercises. Completing problems independently was not as easy. One of the students, Jackson*, was visibly upset when he tried to use multiplication to check his answers and found that he didn’t get any of the correct. In each case, he found a multiplication error. Jackson’s an exceptionally bright boy who clearly understood the concept but didn’t know his multiplication facts well enough. He had been ignoring my admonishments to complete his math facts homework each week. His parents had also viewed learning math facts as less important than completing the other homework assignments.
Finally Jackson could see how not knowing his math facts was limiting his ability to advance. The next week, he brought in about 50 pages of math facts practice and was excited for math minutes. He finished his 50 multiplication facts up to 10 X 10 in only 40 seconds and was giddy waiting for it to be graded. He was also able to complete the long division exercises correctly.
Last week, Aiden*, one of my prealgebra students stayed late to talk to me after class. He’s another very bright student who doesn’t want to put in the work to memorize his math facts. His mother believes that Aiden isn’t good at memorizing and shouldn’t be excused from this requirement. As we talked, my fourth graders started to arrive. Jackson overheard the conversation. He told Aiden, “I was the same way for a long time but then I decided I was going to just DO IT. I did about 10 practice sheets every day for a week. It was hard, but now I know them. You can do it too.”
WHAT PARENTS CAN DO
Parents should accept the responsibility of making sure that their children are fluent in their math facts.
There is nothing complicated about teaching these to your children. It’s just a matter of time on task. While there it is impossible to completely avoid drilling of some sort, I recommend combining drills with fun games and activities.
Children must first understand the meaning of multiplication as repeated addition, skip counting and area before working on memorizing the math facts. After your child understands the concept of multiplication start working with them to learn small groups of simple multiplication facts, such as the two times tables. Gradually add more math facts as your child masters each set.
After your child is able to recall 50-60 math facts without assistance begin working on fluency or automatic recall. Your goal is to help the child transition from rapid skip counting to an automatic association. Provide math minutes worksheets starting with 30 multiplication facts up to 6 X 10. Instruct the child to complete as many of the problems as possible in 60 seconds. Specifically instruct the child to skip any “hard” problems. Pick three of the problems that the child skipped and focus on mastering those three math facts for the week. The child should write these three facts at least three times a day for a week. I tell my children to say the math fact while they write it. Continue working on mastering three math facts each week until the child is able to consistently complete the entire set in 60 seconds. Since my classes meet weekly, I require each child to correctly complete each level for three weeks before progressing to the next level. Parents who are working with their children daily may want to require their child to correctly complete a set every day for five consecutive days before progressing to the next level.
Gradually add more math facts and more problems per page until you child can consistently correctly answer at least 60 math facts at least up to 12 X 12.
Games and other Activities
While it is critical that students learn to study, it is equally important for them to love the learning process. Make sure that you balance drills with fun games and activities. Here are few that my students enjoy:
Prime Climb – Without a doubt the most popular board game with my students from 3rd grade through algebra. Students add, subtract, multiply and divide to be the first to reach 101. I remind students that they are not allowed to “count up” but need to calculate their next position. I often “think out loud” to introduce new concepts. For example, after rolling a 2 I may say “I can add 2 to 25 (my current position) to move to 27 OR I could multiply 25 by 2 and get to….humm, let’s see 25 X 2 is…” and let someone help me out. Great for learning math facts, the understanding order of operations, logic, and strategy. Will not promote math fluency because rapid recall is not important.
Super Math Genius – Players compete to quickly match multiplication problems with products. Although the manufacturer recommendation is for 1-6 players, I find that the children enjoy it most in groups of 2-4. Perfect for building fluency.
24 Game – Players race to be the first to find a way to combine 4 numbers to make 24. For example, the card shown may be solved as 7 X 3 + 4 – 1 = 24. There are different decks for different skill levels.
*Jackson, Aiden and Lucas’ names have been changed to protect their privacy.