A high-schooler was struggling with the answer: what is 6 times 7. He said “I know 6 times 6 is 36, so 36 is 6 more.” “Good, so what is the answer?” I asked. Then he counted his fingers, “37, 38, 39, 40, 41, 42. Forty-two!” Gulp. Correct answer, but apparently he lacked number facts and numerical fluency, and it is especially concerning for a grade-11 student.

The Importance of Number Facts and Numerical Fluency

“Basic number facts” includes all addition, subtraction, multiplication and division problems – in which students are commonly expected to recall the answers.

Students must know the basic facts of each operation (the times table, for instance, are “facts” for multiplication) before they can be successful with the more advanced computational algorithms (multi-column addition, subtraction, and multiplication, as well as long division).

Why many middle-schoolers and high-schoolers have troubles with basic number facts?

The root of this problem lies in the way they learn number facts in earlier grades.

(1). Finger counting. A dad came to us when dropping off his son, and asked us to show his son how to do multiples of 9. “At school they teach him to count with fingers,” he said, a bit concerned. While we’re not sure this dad understood clearly on how the school taught the students, we are definitely not a fan of finger counting – especially this student is already in grade 4. If you don’t “kill” this habit while they were young, the first illustration above – where a 11^th grader did finger counting – would happen to this young guy. Finger counters frequently get the wrong answer because either they count too many or too few, not to mention it will hinder a student to do more complex calculation down the road. It’s like you can’t run because you rely on crutches to walk.

(2). Another issue is putting too much emphasis on memorization in the kindergarten through fourth grade. Not only is this a painful process for everyone involved (kids, teachers, parents) but in many cases it also does not result in the desired outcome. And there might be another unwanted outcome as the result of memorization: a student might look like they have great computation skills, but when presented a math problem that requires analysis and some steps to solve it, they’re struggling because they lack creativity thinking and problem solving.

(3). Too much emphasis on procedural approach in solving math problems would result students rely heavily on one method: algorithm. It’s very common to see sixth grader using algorithm to solve half of 90 to get to 45:

It is the correct answer, but I hope you see there’s a problem if your child relies on algorithm all the time and make it as their first instinct to solve math problems albeit a simple one. In this case, they can use an approach that makes more sense instead: because they know half of 80 is 40 and half of 10 is 5, so altogether is 45. Or when asked what is 165 minus 163, instead of right away answer 2, they stack the two numbers together, and finally come up with 2 as the answer. There’s no sense of magnitude of the numbers and how they relate to each other. It’s like you’re able to read a paragraph of a story word by word, but you don’t understand the whole story because you don’t know how these words related to each other.

(4). Another common possible cause is: letting students use calculators too early. I’m still surprised with experts who say it’s ok for elementary kids to use calculators (I can understand if parents say that, but “experts”??). Letting them use a calculator before they are mastering basic facts is like training them to do data entry jobs. When students become reliant on calculators, their fluency and number sense actually decline over time. No wonder some highschoolers do not know 6x7! The best calculator actually lies between their ears! Using Calculators – Yes or No?

What to do, then?

Instead of attempting to have students memorize the number facts, or letting them to rely on algorithm way or finger counting, it is best for them to be taught to develop their number facts and numerical fluency. Real mathematical growth takes place when students are led through a process of building the concepts themselves. In addition to learning the number facts, the students will also acquire the basic structures of multi-step problem solving.

Start with the basic, start with single digit numbers, then expand to bigger numbers, and then apply to real life problems like counting change when shopping.

We will show you how to do addition and subtraction using reliable, quick and knowable ways to handle single-digit addition and ultimately the basic facts for all four arithmetic operations in another article. So stay tuned!