Author: Bon Crowder

When to Give a Kid A Calculator

When we teach kids how to drive, we give them a few months in the classroom so they can learn the basics of driving and the rules of the road. Nobody in their right mind puts a teenager behind the wheel and says, while flying down the road, “Now, the brake pedal is the one on the left.”

Not only is it safer, but it makes more sense to teach them outside of the car first. After they pass a competency test then they’re allowed to use the technology (car).

We drop a calculator into the hands of teenagers and ask them to learn math at the same time. There isn’t a safety factor here, but the principle is the same.

There’s a different challenge in learning which buttons to press than learning the reasons behind why you press those buttons. We bring technology into the classroom thinking we’re in service of the children, and instead do them a disservice. We double the concepts and think that one is helping the other. It isn’t.

Begin by teaching user’s manuals.

The use of a calculator, a program or web-based application can be easily taught by teaching children how to read a user’s manual or follow instructions. It’s a device, a tool.

Before they start up their new John Deere riding lawnmower, they should read the user’s manual. Likewise, before they turn on their Hewlett-Packard 32sII, they should bend the spine of its little book.

Math classes and home schools can incorporate user’s manual reading in their curriculum. It will prepare students to learn and understand technology, including calculators and applications.

Introduce the calculator after they have mastered the concept.

Teaching children to do math through calculator use can destroy their sense of confidence in doing it themselves and also make them uncomfortable with the tool. Allow them to fully understand an entire concept in mathematics before giving them technology.

If you want them to learn to graph on a calculator, make sure they can do it with pencil and paper first. Allow a few years between the initial introduction of a concept and learning to make it work on a calculator. This should provide ample time for them to practice it and build their confidence that they can do it without the crutch.

When they are ready, give them the user’s manual to the calculator. Have them do problems by hand on one side of their paper. Have them write the page numbers from the user’s manual and keystrokes for the calculator in a column next to it.

When they have completed this, let them confirm their keystrokes are correct by doing it in the tool. This reinforces the connection between what they have learned, and can do on paper, with what’s being done in the machine.

Do you give your kids a calculator to learn on? Will you continue to do so? Share your thoughts and ideas in the comments.

March 19, 2011
3 Places to See the Coordinate Plane in Action
I was browsing through the book Family Math and found a nifty game called Hurkle. It goes a little something like this:
1. Someone picks a place on the coordinate plane (in secret) for Hurkle to live.
2. Others guess Hurkle’s home by giving coordinates and pointing out those coordinates.
3. When they are wrong, they are given a hint of which way to go (from their guess) to find Hurkle.
The coordinate “space” is around too.

This made me think of the conversation I had with a neighbor the other day while walking our children to the park. When Scotty beams someone in Star Trek, he needs coordinates. But those coordinates must be measured from some origin (0,0,0) in the universe to make any sense. So where’s the origin?

A student of mine long ago found that the origin was Earth. My neighbor argued it should be Vulcan – apparently they started the United Federation of Planets.

And we can find coordinates at work in board games.

With thoughts of these two in my mind, I wonder what ever happened to the game of Battleship. It’s a great game of coordinates along with logic. Once you hit a ship, you have to go in each direction to determine how big the ship is and in what direction.

I’m excited about looking around my world today for more coordinate systems. Where do you see them in your world? And how do you apply them in your teaching? Leave your thoughts and ideas in the comments.
March 18, 2011
1 Word to Eliminate from Your Teaching Vocabulary: Just

Yep – that’s the word: “just.”

That little devil does so much damage to a kid’s math-esteem. His cousin is also a bad guy: “all you have to do is…”

It says to a kid, “It’s so easy, and… what? you can’t do it? GOSH!”

Get rid of it. Let your students charge you a quarter every time you say or write either one.

Say instead: “I find that doing this helps me…”

Or: “How would you feel about doing it this way…?”

If you find math easy, great. Give the kiddos a chance to work through the discovery process, too. And allow them to fail and struggle while supporting them. Just don’t say, “just.”

March 16, 2011
How To Teach Algebraic Closure

Thanks to Kellie D for this question: “How do I know if a set is closed under the rule of addition, same question for multiplication?”

Closure under addition or multiplication is a concept from Abstract Algebra that’s taught in not-so-abstract Algebra classes. It can be intuitive, if taught properly.

You first have to start with two things: a set and an operation. A set is a bunch of things. Okay, the definition is fuller than that, but for general purposes, it’s a basket full of goodies.

An operation is a way to “combine” any two things in that basket.

You can’t speak of closure without having both of these together. In Abstract Algebra, this pair of things is called an Algebraic System. That’s a fancy math term that isn’t necessary to teach, but might be helpful if you need to look things up.

We say a set is closed under an operation if you can combine any two things and the result is back in the set. Notice that the definition has BOTH the words set and operation. You can’t just be closed under an operation – you gotta have the set to put it in context.

Example 1 – Kitchen Goodies

Suppose your basket of goodies is this: {flour, sugar, chocolate chips, baking powder, egg, butter}.

And your operation is this: “mix two things together in a bowl with a spoon.”

It isn’t mathematical, per se, but it’s a set and an operation.

So if you combine two things from the basket by mixing them together, what do you get? Yummy goodness for sure, right? But the result of the blend is not found in the basket as one of the things!

Mix sugar with butter and then look in the basket. Is it there as a single blob of buttery sweetness? Nope. So the set of goodies is not closed under the operation of mixing.

Note: the opposite of closed is “not closed.” The word “open” isn’t used here.

Example 2 – Math Stuff

Now you have another basket of goodies: all the even numbers (including positives and negatives and zero). We can write this in the fancy math way like this: { x | x is an even integer }

The operation is addition.

If you combine (add) two things from the basket (any two even numbers), what do you get?

8 + 24 = 32

Is it back in the basket, i.e. is the answer an even number? YES!

Try some more to make sure. Technically you should prove it, but usually your intuition is good enough – especially in a high school or undergraduate class.

So we write “the set of even numbers is closed under the operation of addition.” Again, notice that we have BOTH set and operation involved.

Example 3 – More Math Stuff

Here’s your set: all the integers (positive, negative and zero whole numbers).

And the operation: Division.

If you combine (divide) two things from the basket (integers), what do you get?

27 ÷ 3 = 9

Looks good, right? 9 is an integer. But what about 27 ÷ 2 = 13.5?

13.5 isn’t an integer because it has a decimal piece. Some bits (integers) when combined (divided) turn out to be in the basket (another integer). But not all do. And they ALL have to work in order for it to be called closed.

So this is NOT a closed system. Or we can write, “the set of integers is not closed under the operation of division.”

Example 4 – Even More Math Stuff

Set: {1, 5, 6, 7, 14}

Operation: addition.

Is it closed?

1 + 5 = 6 check!

5 + 6 = 11 oops! 11 isn’t in that set.

This set is not closed under the operation of addition. You might notice that a finite set (a set with only so much stuff) of regular numbers won’t ever be closed under addition – just add the two biggest numbers and you “bust” out of the set.

Going Beyond…

Your sets can be just about anything: real numbers, rationals (fraction-like numbers), whole numbers. They can even be bunches of matrices, polynomials, imaginary numbers, complex numbers or functions! The operations would be one of the normal operations that you would use on things in that set anyway.

Questions about how to teach this? Questions on how to learn it? Leave them in the comments.

March 15, 2011
Confessions of a Calculator Addict

I remember being allowed in Jr. High to use the calculator to “check my work.” Soon after I learned that the books in High School had the answers in the back! It was like condoned cheating!

How could I go wrong with the magic box and the answers given to me straight from the publisher?

And then I became addicted.

Sometime after Geometry I lost my multiplication facts. I wasn’t just checking my work on the calculator.

Subconsciously I figured there was no reason to trust my potentially faulty memory of math facts if I had the absolute sure thing right there next to me.

For years I stopped doing arithmetic.

And my dad chastised me. Every time some quick calculation came up in the kitchen, garage or grocery store, I would stare at him blankly. Then I would reach for my calculator.

The way he looked at me, you’d think I had reached for a bong, ripped off my bra, sang Kumbaya and spat on the pope.

I ignored him.

For years.

Until one day I realized that I had absolutely no memory of $8 \times 7$ . Yep – $8 \times 7$ was what did it. I started watching myself. I always did simple arithmetic (even addition of single digits) on the calculator!

Then I watched other people. I saw the clerks in the grocery store reach for the magic box to figure out 10% off something. I saw an older man at McDonald’s send the girl into a tizzy because he modified his cash payment after the girl had already typed it in.

“There’s a problem here,” I thought. Maybe Paps was right.

I put down the magic box. Cold turkey.

I started using prime factors to help me remember my old multiplication facts. I re-engineered subtraction so I could actually do it. I read Dead Reckoning: Calculating Without Instruments. And then I refused to allow students to use the “devil box.”

I put it on my syllabi that calculators were strictly prohibited (unless expressly invited by me – in the case of probability and statistics). I growled at anyone who reached for one.

And I taught them arithmetic.

And we were all better off.

Are you a calculator addict? Share your story in the comments.

March 10, 2011
How to Know When a Kid is Confused

When Cassy over at Singapore Math Source commented on my post 7 Ways to Wrangle a Word Problem, it made me think.

Her question to me was, “Why wouldn’t you just rewrite the problem to focus on the appropriate concept?” She suggested that having students restate the problem in their own words without numbers would have them demonstrate that they know what is being asked of them.

Here is the original problem:

The Beebo bird lives in two places in the world. Some live in Texas and some live in Greece. Greek Beebos are about 20 inches high and weigh around seven pounds. There are about thirty-nine thousand Greek Beebos. The total weight of all the beebos in the world is 500,000 pounds. How much do the Texas Beebos weigh altogether?

Here is her suggested rewrite without numbers:

There are only two types of Beebos in the world, Greek and Texan. I know the weight of one Greek Beebo and I know how many Greek Beebos there are in the world. I need to find out how much the Texan Beebos weigh altogether.

On the outset, this seems great. If your kid does this:

It’s more likely your student will do this (especially if they’re struggling or you’re a hired tutor):

Notice the struggle and strain? And notice that both videos show the same thing – the “student” (me) just reading the problem and replacing the numbers with “I know how much…”

Watch students carefully. Listen to their intonations, watch their faces, watch their bodies. Whether you’re in a classroom or one on one, watch! If they got it, you can see it. If their little foreheads are wrinkled and they are tense – stop. They don’t have it. They are guessing. Go back. Try something else.

See? Let me know what you think in the comments.

March 9, 2011
7 Real “How to Succeed in Math” Tips
I bought a handful of math texts at Half Price Books this weekend. I opened up a Basic Mathematics text and the first thing that caught my eye was the intro titled “To the Student: Success in Mathematics.”

“Really?” thought I. As I read it, I grew more and more agitated.

Have these folks spent any time inside a math classroom? Did they pay attention to the students? If so, they should know that the likelihood of a student to do what they suggested is downright ridiculous. So why do we tell students to do it? Why can’t we give them tips that they can and will do? Like these:
1. If you feel comfortable asking questions in class, do it. If not, write your questions down to ask later. You don’t have to ask the instructor, especially if he or she is intimidating. Find a tutor or go to the school’s math lab instead. You don’t have to work with someone you’re uncomfortable with.
2. Read the stuff inside the gray boxes. We know it’s likely you’ll not read the text, but the things inside the gray boxes are really helpful.
3. Before you start on your homework assignment, do something physical or something you can do well. Run a mile, do a load of laundry or play tennis for a half hour. This will remind you of the things you are good at and get your endorphins flowing. It will help you be confident during your homework time.
4. Absorb the lectures, don’t copy them. If you can do it, try to just watch. See how the teacher thinks through a problem. You will gain more from this than from frantically trying to copy everything.
5. If you do take lecture notes, don’t dwell on notes that you can’t figure out. Many times you mis-copy or mis-write things the teacher wrote or said. If it doesn’t make sense, move on.
6. Tear out the back of this book (the part with all the answers) and burn it. It is important that you build your confidence. Checking your work with the magical back of the book just gives you a crutch. And don’t use a calculator to “check your work.” That’s just another crutch.
7. Do the first two problems in every section and subsection. If you can do those, do the last two. If you can do those too, continue to the next subsection. Math isn’t a spectator sport, but it isn’t an elliptical machine either. Do all the problems if you need the practice. And if you have it down, move on.
Give it a shot. Let the students know that what they want to do is okay to do. Let’s quit giving them the B.S. that’s been passed down to us over the last few decades. It’s time to go Math Book 2.0.

Whatcha think? What’d I miss? Let me know in the comments.
March 8, 2011
How to Use Prime Factors to Cancel Fractions

I previously posted about prime factors and about using them in multiplication. Now we’ve got fractions to handle.

Before we get in too deep, first let me note the main reason why we do this with arithmetic. In algebra, students will be asked to take a rational expression, factor it and reduce it like this:

If they are familiar with how factoring works with numbers, this will be normal to them.

I also find that reducing factors in this neat and clean way helps a lot. In this video I showed the first few steps of reducing the fraction a sloppier way after I did it the “neat” way:

Here’s one that’s a little more complex:

Will be helpful to show your kids? Is there something I’m missing? Share with us in the comments.

March 2, 2011
How to Remember Exponent Rules without Brain Ooze

This post is by request from @corrincross on Twitter.

Exponent rules are hard. Well, they are hard to remember, anyhow. But there’s an easy way that won’t make your brain ooze – and that’s doing a mini-experiment each time you have to deal with them.

First remember what an exponent really is. It’s a shortcut for multiplication. Check it out:

Notice that there are really only two rules that get you screwed up: when to multiply the tiny buggers and when to add them. Here’s how I teach this (and how I remember it myself!):

So the new rules are $x^3 \cdot x^2 = x^{3+2}$ and $(x^3)^2 = x^{3 \cdot 2}$ .

And then what about those negative exponents? Again, you can’t remember a rule unless you remember what they heck is really going on. We go back to adding in this video to explain the similarity between the two shortcuts of multiplication and exponents:

Can you use this in your classrooms? Share your thoughts in the comments.

Thanks to Corrin Cross (@corrincross on Twitter) for requesting this post. Corrin is a Secondary Ed student with a math major and music minor at the University of Regina in Saskatchewan, Canada.

February 25, 2011
How to Use Prime Factoring to Win Friends and Influence People

Alright, you might not win friends with this, but the more you can do arithmetic in your head… well… okay, you can’t influence people with it either. Regardless, it’s handy to know and helps with multiplication.

Yesterday I posted a couple of videos about factoring numbers into primes. Now it’s time to put those to work.

How about a rather simple one to get us started:

Here’s one that’s a little more challenging. The key is that you can combine the bits to something you’re more comfortable doing. I don’t like multiplying nines, so I avoid those.

Can you use this? Why or why not? Ha ha, just kidding – no essay questions here! – but if you do find a use for this, or if this annoys you, let me know in the comments.

February 24, 2011