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!

The (New) Way to Learn Math – Part 1: A Little History

This is the first in a series of posts about teaching math.

There was a time (not very long ago) when we taught math procedurally:

1. We taught students a procedure for solving a certain type of problem (for example, subtraction with regrouping).

2. We had students practice the procedure over and over again until they were very good at using the procedure to solve that type of problem.

3.  We emphasized speed.

4.  The student scored well on a test of that procedure.

5.  We considered ourselves done.

Image courtesy of Pixabay

There turned out to be some problems with that teaching:

1.  Students had trouble knowing what procedure to use if we didn’t tell them because procedures were taught in the context of a certain type of problem.

2.  When students forgot the procedure, they couldn’t recreate it from their math knowledge.

3.  We often saw answers to problems that just didn’t make sense.

4.  Students didn’t see any room for creativity in math.

5.  Students were bored and turned away from math.

6.  Timed tests in particular created math anxiety in some students.

Image courtesy of Pixabay

So, thanks in part to the new Common Core, we have a new focus in math:

1.  We are asking students to really look at the numbers in a problem to try to make sense of them.

2.  We are asking students to be creative in solving problems and to share their solutions with others.

3.  We are exposing students to numerous strategies for solving problems to deepen their thinking and make it clear that there are multiple ways of looking at problems and solving them.

4.  We want students to see that math doesn’t have to be boring.

We teach differently:

1.  We talk about math myths and debunk them.  

2.  We do regular math talks with the class in which students are asked to solve a problem mentally, explain how they solved it and consider multiple other ways of solving it.

3.  We practice estimating.  This is one way for students to see whether or not an answer makes sense.

4.  We play games that strengthen mathematical thinking.

5.  We focus on the ways we use math every day.  We want students to see the connection between math and their world.

Image courtesy of Pixabay

We want student to know that math can be creative, interesting, useful and fun!

Our Global Read Aloud pick for 2015

Last year was my first year to participate in the Global Read Aloud and I'm hooked.  We read the Miraculous Journey of Edward Tulane, by Kate DiCamillo and I like it so much that I bought a class set to use in my classroom this year.  We blogged with an amazing third grade teacher and her class in Atlanta, Georgia.  The kids learned from each other and I learned from Mrs. Fisher.

This year, the two choices for younger readers are The Year of Billy Miller, by Kevin Henkes and Fish in a Tree by Lynda Mullaly Hunt.  Just knowing the authors, I knew we couldn't go wrong with either book.  Kevin Henkes wrote one of my all time favorites, Lily's Purple Plastic Purse, as well as one of my first week staples, Chrysanthemum, and many, many others.  Lynda Mullaly Hunt wrote One for the Murphys, a favorite from when I taught 5th grade and Lynda participates in TeachersWrite!, something I like to participate in during the summer.

So, I ordered both books and read them the first weeks of summer.  I started with The Year of Billy Miller.  Billy Miller, the character, piqued my interest right away because he's a boy and it seems so many younger grade books feature girls.  He also had a big accident during the summer and he's afraid that his head bonk will affect his school work.  This seems like just the kind of believable problem that our students might have, and the book continues on with realistic events that stretch Billy in second grade.  He deals with a classmate he doesn't like and he has a sometimes bratty younger sister.  I really liked the gentle way that the book progresses with a hero who isn't a superhero.

Next up was Fish in a Tree.  Fish in a Tree also begins with school worries.  Ally is supposed to write something for her teacher, but she finds writing about as easy as "climbing a tree using only her teeth".  Since school is hard for her, Ally spends way too much time with the principal until her new teacher, Mr. Daniels, comes along.    Ally not only gets help conquering her school woes from Mr. Daniels, but also from her multi-faceted friends, Keisha and Albert.  The three classmates are funny, sincere and brave.

I try not to write in my books so that I can pass them straight on to students, but as I read Fish in a Tree, I kept seeing topics ripe for blogging during the Global Read Aloud.  I have notes, and notes about my notes.  Ally has a sketchbook of impossible things - we should share our sketches of impossible things with our blogging buddies.  Mr. Daniels plays a math game called You're the Bus Driver with his students.  We can play that game with our blogging buddies.  My notes continue.

For this year, the clear winner for our third grade read aloud is Fish in a Tree, but that doesn't mean I didn't like The Year of Billy Miller.  I will have a couple of copies of that book in my classroom library for students to read themselves.

Developing Number Sense Is More Important Than The Math Facts (But You Can Master Both)

One of the rites of passage of third grade is learning the basic math facts for multiplication and division.  Depending on your child, the thought of this may create excitement or inspire fear.

First,

In order to dispel any possible fear, I want to explain a little bit about the relative importance (or unimportance) of math facts.

1 - Everyone can learn their math facts.
2 - Some kids will learn them more quickly than others, BUT
3 - Faster isn't necessarily better, BECAUSE
4 - Understanding how numbers work is the real goal.

Studies repeatedly show that the best math students are not necessarily the fastest at doing math problems, but the students who have number sense.  They understand how math "works", they know about patterns in math, and they know that there are multiple ways to solve a problem.  Developing number sense will serve your child much better than just memorizing math facts.

We spend weeks on multiplication so that students can develop an understanding of what they are doing when they multiply.  It's not just magic answers pulled from their rote memorization, it's working with numbers.

However,

It is faster to multiply if students know their math facts, and learning the facts is a stated requirement for third grade.  The good news is that there are easier ways than others to learn the multiplication facts.

If you follow six simple steps, you can help your child with a strategy for learning the multiplication tables that should demystify the experience and demonstrate that instead of 100 facts to learn, there are really only 7 difficult facts to learn.

1.  Learn to count to 10 (done) and by 2 and 5 (probably done).

You are down to 36 facts.

2.   Learn to count by 3.

You are down to 25 facts.

3.  Master a 9s Trick.

You are down to 16 Facts.

4.  Realize that 6 facts are the same (ie: 6x4=24, 4x6=24).

You are down to 7 "toughies".

5.  Have fun and games learning these 7.

6.  You are a multiplication facts wizard!

Kahoot!

Thanks to @swampfrogkids I just learned about Kahoot! I saw it on her blog, and decided to check it out with my morning cup of tea.

I decided to create a Getting to Know You survey that I will use on the first day of school.  I picked 9 questions, typed them in, added 2 - 4 possible answers for each, dropped in some free clipart and my survey was done in less than 15 minutes.

Then, to see how it really works, my daughter and I logged in to Kahoot.it on our phones.  We typed in the game pin and took the survey.  It was easy, it was fun, and the most impressive feature was that I was able to print out a report that showed our individual answers to the survey.

This looks like a great way to get quick, individual answers to surveys or beginning of the year assessments.  I'm looking forward to using Kahoot!

Popplet app

One of our favorite apps to use is Popplet.  This app allows students to create mindmaps of linked boxes or "popples".  Here they are using it for a vocabulary lesson.  There are popples for the vocabulary word, the definition of the word, a sentence using the word and a drawing that shows they know what the word means.

More about electrical circuits - Squishy Circuits

Our holiday cards were so much fun that Katie came in to teach the class more about electrical circuits.  We used squishy dough to make holiday creations that lit up.

Here are some photos so you can share in the fun.

Holiday Postcard Exchange

This holiday season, our class participated in a really fun project organized by Projects by Jen.  We made holiday cards for more than 20 schools and received cards back in return.  It was a fun way to find out more about the 50 states.  We also got one card from Canada.  I am thinking of extending this project next year and organizing some Mystery Skype sessions after we have made our card map.

Maybe the most fun was making our own cards though.  We learned about how electrical circuits work, then made cards with circuit stickers in them so we could make little LED lights light up.  I think these cards were very festive!