Math Rules and Tricks - Not the Best Teaching

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 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.  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.

Busting math myths

Image courtesy of Lišiak, Wikimedia

During the first month of school students took a True or False math quiz designed to bust some math myths.  Can you get all the answers right?

 Here are the answers:

1.  False, research shows that boys and girls can do equally well at math.  Unfortunately, girls often hear the message that girls aren't good at math, and too often they believe it.

2.  False, the language you speak has nothing to do with how well you can do at math.  Again, lots of Hispanic children hear that Spanish speakers aren't as good at math, but research shows that's just not true.  Everyone can do well at math.

3.  False.  There's no such thing as having a "math brain."  Everyone can do well at math, some people may just have had better experiences with math or more practice thinking mathematically.

4.  False.  Math is really a creative activity.  There are many ways to arrive at the answer.  This week students were asked how they arrived at the answer to 12 + 7.  In less than 15 minutes, 11 students explained their 11 different approaches.

5.  False.  I am not always right!  Students often have fun pointing out my mistakes in class.

6.  False.  Students are often surprised to find out that getting the wrong answer to a problem doesn't mean they got the whole problem wrong.  They may have done a lot of things right along the way.  I am more interested in their thinking process than just getting the answer right.  (Hint:  the more students show their work, the more credit they are likely to get.)

7.  False.  Mistakes aren't something to be embarrassed about, they are to be investigated and celebrated.  Would you believe that making mistakes and correcting them is one of the best ways to grow your brain?  It's true.

8.  False.  Your brain doesn't do much growing if you are just doing something you already do well. You need a bigger challenge.

9.  False.  If you have a question, you are probably thinking and stretching your brain.  Please ask it, someone else may have the same one!

10.  False.  Everyone has questions, just not everyone is willing to speak up.

11.  False.  Learning the algorithms in math doesn't actually mean you understand math.  The best way to learn math is by developing "number sense."  Some ways to do this are by doing number talks and really thinking about what we are doing when we do math.

12.  False.  Studies have shown that students who talk to each other about math do better.  Talking about math exposes students to the many different approaches to problem solving.

13.  False.  Often the students who finish first rushed through what they were doing without stretching their brains.  Sometimes what they were working on was not challenging enough for them.

14.  False.  Faster isn't necessarily smarter.  In fact, there are some very famous mathematicians who work slowly.

15.  False.  You can be a poor test taker or be given poor tests.  In either case, tests won't tell what you really know about math.

16.  True.  This year students will be given self-assessments to let me know how well they think they know what I want them to know in each chapter.  This will give them a chance to go back and ask questions about things they are uncertain about before a test.

17.  True.  Students will have the opportunity to practice something that was hard for them and show me that they have mastered it.

18.  True.  Sometimes making sense of math takes time, but that's all right.  It is so much fun when the light bulb goes on and a student "got it."

19.  False.  You use math every day even though you may not realize it.  You use it when you estimate how much time before bed.  You use it when you put just the right amount of milk on your cereal.

20.  False.  Although you would know a lot of math if you took math every year until you graduated from college, there is more math out there to be discovered.  No one knows it all!

If you found this quiz and the answers interesting, I suggest you check out the Brain Science link at youcubed.org.  Maybe you too will become a youcubian!