Developing math fact fluency requires more than memorization.
GUEST COLUMN | by Dennis Pierce
For years, students have been taught to memorize their basic math facts for multiplication and division. But next-generation learning standards require a deeper understanding of what that really means.
If students don’t have a mental picture for what “9 x 3” represents, and they’re just trying to remember that the answer is 27, then what are they going to do if they forget this fact—and how are they going to apply this knowledge to larger and larger numbers?
Students need visual models to give them a conceptual understanding of these strategies.
Getting to Fluency
Fluency requires more than just memorizing isolated math facts. Students need to see how these facts are connected, says math education expert James Burnett—and they also need strategies for solving these problems.
“If we give students a strategy, and they forget an answer, then they have a way of recreating it,” says Burnett (pictured below), who is the president, co-founder, and senior curriculum author for Origo Education, a developer of standards-based elementary math education, curriculum and digital products. “And if students have a thinking strategy, they can use that same strategy to do calculations with greater numbers.”
For instance, one strategy would be for students to use their knowledge of tens facts to figure out facts involving five. Because five is half of 10, then the product of any number and five would be half of that same number when multiplied by 10. “If you know ten threes is 30, then five threes must be half of 30, or 15,” Burnett explains.
Once students know their tens and fives math facts, they can use a strategy known as “build up” or “build down” to learn their sixes and nines facts. Students can derive their nines facts by building down from a known fact involving 10.
For example: “9 x 3” is the same as “10 x 3” minus three; 30 – 3 = 27. Similarly, students can derive their sixes facts by building up from five: When students know that “3 x 5” is 15, then they also know that 3 x 6 is just one more three, or 18.
What’s more, students can learn their twos, fours, and eights math facts by doubling numbers once (for twos), twice (for fours), or three times (for eights).
In other words, “two times three” is the same as “double three,” or six. “Four times three” is the same as “double double three,” which is also “double six,” or 12.
Armed with these kinds of strategies, students can apply them to solve multiplication problems involving greater numbers.
Try This On For Size
“Say you have a T-shirt that costs $19,” Burnett says. “You want to buy three shirts. How much does that cost? ($57). How do you figure that out in your head?
“Well, 19 is one less than 20, so if you know 20 threes is 60, then 19 threes must be three less than sixty (57). In this way, we are ‘building down’ from a known fact involving 20.”
And once students have mastered their basic multiplication facts, learning division becomes easier as well.
“The number one strategy for teaching division is to think multiplication,” Burnett says. “If you are faced with ‘30 divided by five,’ and you know five sixes are 30, then you also know 30 divided by five is six.
“Use your understanding of multiplication to figure out your division facts. Again, having an understanding of the connection between the operations assists with the learning of the facts.”
But just talking about these strategies isn’t enough.
Visual Models for Conceptual Understanding
“Students need visual models to give them a conceptual understanding of these strategies,” Burnett notes.
“If you just say, ‘Well, you don’t know your fives, but you know your tens, so all you have to do is multiple by 10 and half it,’ then all you’ve done is given the students another procedure for which they don’t understand why it works. You’ve got to give them visual pictures so they understand why the strategy works. For this particular strategy, it is good to picture an array of dots.”
In a Nov. 28 webinar hosted by edWeb, Burnett will demonstrate easy-to-make visual aids and games that can help students master these basic math fact strategies for multiplication and division.
During the webinar, Burnett will show attendees how to introduce each strategy using something concrete or pictorial; reinforce the thinking associated with each strategy through games and activities; practice each strategy to help students develop automatic recall of basic multiplication and division facts; and extend each strategy by applying it to numbers beyond the number fact range.
“That’s the sequence of steps we take students through within our curriculum materials. Every good program will do exactly that,” he says.
Dennis Pierce is the former managing editor of eSchoolNews where he worked for 17 years. As a veteran education technology journalist, he has a unique ability to see the big picture regarding technology’s role in education. Contact him through LinkedIn.