It is tempting as a teacher to give students shortcuts for doing math, but most teachers know that, in the long run, these tricks and rules often backfire, and students are left bewildered, either because the rule no longer applies in the later grades, or because they have forgotten how the rule or trick works. Too often, students then discover they don't understand the underlying math concepts, and they are unable to solve a problem in front of them.

As part of a book study on

As part of a book study on

__Making Number Talks Matter__, I read an article by Karen Karp, Sharon Bush and Barbara Dougherty entitled "Thirteen Rules That Expire" and the book, Nix the Tricks, by Tina Cardone and the on-line math community known as MTBoS. Here are a few rules and tricks these sources mention that appear in the elementary grades. A mantra that I particularly liked from__Nix the Tricks__is "math makes sense." It is up to teachers to be sure their students are sense-making when in math class. If they learn these rules and tricks, it should only be after they understand the underlying math concepts.
1.

**The "=" sign can only have one number on the right as the "answer**."
In the last few years, there has been an increased emphasis
on "=" meaning that the quantities on either side of the "=" are the same. However, it is still common for students to be stumped
by problems such as 5 + ___ = 3 +
4.

We need to be mindful that we do not imply that = always
means "find the answer."

**2. Addition always makes numbers bigger.**

It’s tempting to oversimplify early addition. Even in the elementary grades this rule is
not always true because 6 + 0 = 6.
Students who believe that addition always results in a bigger number may struggle with integers. For example,
-2 + -8 = -10, a sum which is smaller than either addend.

When teaching simple addition, we can let our students know that this rule is true when
adding positive whole numbers greater than 0.

**3. Subtraction always makes numbers smaller.**

This is a good rule of thumb for early elementary math (except when subtracting 0), but
not in the later grades. Again, integers
prove an exception to the rule: -3 – (-9) = 6.

Students should be advised that this rule is true when dealing with positive
whole numbers greater than 0, but it will not always be true.

**4. You cannot take a bigger number from a smaller number**

This is another often-used shortcut to help students learn
subtraction with regrouping. It is very
common for elementary students, when solving a problem such as 25 – 19, to
start off by saying, “Well, I can’t take 9 from 5, so I have to go next door to
the 2 and borrow . . .” In fact, this was the
way I was taught to subtract.

It is important to use proper math terminology, or students
may feel betrayed when they find out about the existence of negative numbers. They
are not actually trying to subtract 9 from 5, they are trying to subtract 19
from 25, a problem that allows for the possibility of regrouping. It would be
better to have students thinking “I cannot subtract 9 ones from 5 ones, but
need to regroup the tens and ones so that the 25 is restated as 1 ten and 15
ones." In this way, students begin to understand that they can only subtract digits with the same place value.

5.

Students who follow one of the many rounding rules often have no idea that rounding means asking what is the closest number at the next larger place value. Then often cannot transfer the rule from the 10s to the 100s to the 1,000s etc.

It is more helpful to teach rounding using number lines so students can actually see which 10, 100 or 1,000 is closer to the number being rounded.

I didn't know until I read

5.

**Rounding can be taught using a rhyme (5 or more, let it soar)**Students who follow one of the many rounding rules often have no idea that rounding means asking what is the closest number at the next larger place value. Then often cannot transfer the rule from the 10s to the 100s to the 1,000s etc.

It is more helpful to teach rounding using number lines so students can actually see which 10, 100 or 1,000 is closer to the number being rounded.

I didn't know until I read

__Nix the Tricks__that the rule to round 5 up to 10, 50 up to 100 etc. is the accepted convention in elementary school, but not in the world of science. Scientists round down if there are even digits before the 5 and round up if there are odd digits before the 5.**6. Multiplication makes numbers bigger.**

It’s easy to think this rule is true in third grade when
multiplication is always about even groups
of things or even rows of things. The
product in those cases will be greater than the factors (except in the cases
when a factor is 0 or 1). However, when
fractions are introduced, the rule is no longer true: ½ x ¼ = 1/8. The product is smaller than the factors.

Again, when teaching multiplication, we can add the caveat that the product will be greater than the factors when multiplying whole numbers greater than 1.

**7. When you multiply a number by 10, just add a 0 to the number.**

It is tempting to teach students this shortcut when they are
learning their multiplication facts.
It’s so easy, just add a 0 to the number you are multiplying by 10 and
you know the 10s facts. The problem is
that the rule does not apply to decimals. If
you multiply 6.8 X 10, the answer is not
6.80.

A better shortcut would be to have students skip count by
10s to reinforce that they are dealing with multiples of 10. Later, when multiplying decimals, it would be
helpful to have students add 6.8 ten times so that they could also see that
6.8 x
10 is indeed 68.0.

**8. Division always makes numbers smaller.**

Again, this rule works for positive whole numbers, as when
third graders are determining how many groups of 3 are there in 12. In later grades, when students are determining
how many 1/2s there are in 12, the quotient 24 will be the largest number in the
equation.

If we reinforce that we are talking about a unit of measure when we divide, students may make the transition to dividing fractions more easily.

Many elementary teachers, including me, learned math as a series of rules and algorithms. It is time for us to re-examine these procedures to determine the number sense (and nonsense) underlying them. Then, we can be sure to explain operations in a way that students understand the math they are doing and are prepared for the math to come.