Improving Fluency and Number Sense with Simple Number “Stretching”

People often ask for tips about striking the perfect balance between computational fluency (think: basic facts and algorithms) and number sense (think: comparing, ordering, and estimating values). Those in the know seem to agree that number fluency is as important as number sense, and that each one depends, in part, on the other. But the question remains: can we teach both in harmony, without favoring either one?

For me, the answer is captured in the notion of “flexible numeracy.” As a child, I remember reading an article in a children’s magazine about teenaged gymnasts in the Cirque du Soleil. The article featured a photograph of a gymnast relaxing in front of the television, her feet resting comfortably atop her shoulders. The caption observed that these young gymnasts stretched almost continuously throughout the day to keep their bodies limber for performance.  

As a math teacher, I try to think of my students as number gymnasts. I want them to be able to slip effortlessly between and around exercises; do hand springs and back bends with operations; bounce from strategy to strategy like they’re jumping on trampolines. Just as real-world gymnasts stretch every day, with purpose and dedication, students who wish to be flexible with numbers must stretch their numerical skills daily. My role, for five minutes a day, becomes that of a fitness coach, guiding my students enthusiastically in a brief stretching exercise, to keep them nimble with numbers.

The exercises themselves are simple: since the objective is strictly to practice manipulating numbers, there is no need to come up with real-world contexts. (Problem-solving is embedded throughout the rest of the math lesson.) But don’t let the simplicity of the exercise fool you: as they contemplate solutions, students’ minds flow through a range of interrelated number concepts: the relationship between addition and subtraction, the pairs that make ten, ordering numbers on the number line, conservation of number, and plenty more. They are building fluency while exercising number sense; they are accessing number concepts while practicing basic facts. In short, they are becoming numerically flexible.

A Sample Problem

Give students a minute or two to solve this exercise mentally.  Flexible students will come up with a variety of accurate and efficient strategies that could be used to solve the problem without touching pencil to paper. If you’re up for a challenge, stop and brainstorm before reading on: how many student-friendly strategies can you come up with?

Here are some of my favorites:

1.  Subtract 20 instead of 18. 33 — 18: hard.  33 — 20: less hard! Of course, now you’ve taken away 2 too many... so add 2 back, to get 15.

2.  Add 2 to both numbers. Wouldn’t you rather solve 35 — 20 than 33— 18? Or, subtract 3 from each, for 30 — 15. You probably know how far 15 is from 30. (Is it easier if you think in minutes?)

3.  Find the distance between 18 and 33 on a number line. Counting up one-by-one is a perfectly valid entry point for students who are not ready for chunking. More advanced students will start with a jump of 10 (from 18 to 28), then add 5 more, one-by-one, for a total of 15.

4.  Subtract 18 from 30, instead of 33. If you’ve mastered your pairs to ten, getting from 18 to 30 is a breeze: it’s 12. Then add the 3 you set aside, and presto: 15!

5.  Break 33 into 20 + 13, then subtract 18. This one’s tricky. First, subtract the 18 from the 20, then combine the resulting 2 with the 13. According to mathematician and children’s author, Greg Tang, “Breaking numbers apart in smart ways is the key to being good at arithmetic.”  

6.  Use the traditional algorithm. The algorithm is a valid method, and it will always be the first choice of some students, but for this particular problem, it’s just not efficient. Also, when depending too much on an algorithm, many students slip into a “number coma” and lose touch with the meaning of the problem they’re solving. If your students bring it up, validate their reasoning, and then seize the opportunity to remind them that mathematicians seek efficient solutions, not just “right answers”.