## Kiss those Math Headaches GOODBYE!

### Let STUDENTS make the math Problems, for a change

If you want students to look at you like you’re crazy — and have fun because you know you’re doing a good thing — try this.

Tell students it’s their turn to make up a math problem.

Math Meeting Board and Lesson (Photo credit: Old Shoe Woman

Yes, they’ll give you that look like, what are you talking about? But it’s o.k. Persist. Not only that … tell them to make up a word problem just like one in the textbook or on the worksheet. And tell them to make it relevant to their own lives.

For example, if you’re doing problems on rate, time and distance, suggest that students make up a skateboarding problem. One of my students came up with this:

You want to skate over to Ted&Tom’s (a local hangout), and you need to get there by 2:15 pm. If you’re 3 miles away and you leave at 1:30, going 4 mph, will you get there in time? [Answer:  You’ll get there right on time, not a minute too soon or too late.]

See how easy it is? Not really hard.

Or, let’s say that you’re doing ratio problems. Suggest that students do a problem on price comparisons. Another one of my tutees came up with this:

Lip gloss is on sale, 4 tubes for \$7. At that rate, can you buy 12 tubes if you have exactly \$20? [Answer:  No, since you won’t get the special if you have only \$6 for the last set of lip gloss tubes.]

The benefits for students are many.

1)  Students start to see that math problems are “all around them.” i.e., They start to see math in their everyday situations. And they start to realize that they can actually use the math you’ve been teaching them to figure out  real-life problems.

2)  By developing their own problems, students grasp the concepts in the problems more deeply. In the same way that we teachers learn by teaching, students learn by making (and solving) their own problems.

3)  Making problems is a creative activity, and once students see they can pull their problems from real life, they start to enjoy the activity. And because this involves creativity, this exercise engages the “creative types” who often feel like math does not “speak to them.”

4)  If you take the activity one step further, you can help students build their critical thinking skills. The one step further is: require that students get a whole number answer for their problem. This requirement forces students to think about how the numbers in the problem affect the value of the answer. And when they need to fine-tune those problem numbers to get out a particular kind of numerical result (like a whole number answer), they learn about the “innards” of the problem. They learn how the problem works more deeply than they would if they only were solving a problem someone else gave them.

5)  If you make the solving process cooperative, you can add even more fun to the process. By this I am suggesting that after students make the problems, they give them to other students to solve them. This way two students can exchange problems. I’ve seen students really get into this. They start making problems harder until they are just at the level that makes their partner “sweat.” But they enjoy this process, and it helps them get to know each other. I’ve found that this is a good way to get some fun socializing into a math class.

One last nice thing:  I’ve found that students cannot actually make up problems if they don’t know how to solve the problems. That means that this exercise tells you, the teacher, which of your students do understand the problem. And if they don’t get it, you can help them get it by helping them make the problem. It’s a nice, indirect way to teach.

So give it a try in your class or teaching situation, whatever that may be. I have a hunch you’ll find it as helpful and enjoyable as I have found it to be.

### Rubik’s Slide: play your way to geometric knowledge

A toy that educates … could it be a dream?

I recently found something that fits that category, educating students in concepts of GEOMETRY.

It’s called the Rubik’s Slide, created by Techno Source. I bought this Rubik’s Slide a few months ago because I needed another puzzle to keep my tutoring clients entertained while I grade their work, which I often do at the start of sessions.

Rubik's Slide Logo

### Times Tables, Learning the Threes

What’s more important for early math than knowing the times tables?

Not much, right?

Since the times table facts are so fundamental, and because many students struggle with them, I’d like to share a strategy I came up with today for
learning the 3s. This technique works particularly well with students
who struggle with memorizing apparently random facts. (We know these
facts are not random, but if learned with nothing more than flash cards,
they can appear random.)

The strategy involves three stages, each stage bringing the child closer
to being able to QUICKLY access the desired multiplication facts. Here are the stages, in order they should be taught.

STAGE ONE:  “Patty-Cake Threes”

What I do here amounts to a “patty-cake” approach to learning the threes, which works like this.

The student and I sit facing each other with our hands up. We hit our right hands together and say “one,” then hit our left hands together and say “two.” Then we
hit BOTH HANDS TOGETHER and say, “THREE.” When saying the “one” and “two,” we utter the numbers quietly. But when we say “THREE” and all successive multiples of three, we say these numbers loudly, almost (but
not quite) shouting.

After three, we continue:  “four, five, SIX … seven, eight, NINE, ten,
eleven, TWELVE … ” and so on. So this gives children a fun way to
hear — and get a feel in their body for — the multiples of three, in the proper
order.

Image by davie_the_amazing via Flickr

STAGE TWO:  “Finger-Drumming”.

After the child has the rhythm of the number three, from the “patty-cake” approach, we do “finger-drumming.” To “finger-drum” the multiples of 3, the child makes a fist with one hand, and shakes it, saying with each shake, “one, two, THREE!” And when saying “THREE,” the child extends one finger from the fist. The child continues: “four, five, SIX,” and at “SIX,” he extends another finger, so he has two fingers out.

Then you ask the child, for example, “What is three times two?” Answer: the number he just said, “six.”

In this way, the child can “finger-drum” out all of the multiples of three. To
reinforce the times tables as you go, ask questions like:  “What is 3 x 4? What is
3 x 5? etc.” Each time you ask, the child must “finger-drum” till s/he gets the
correct answer. This flows very nicely from the “patty-cake” approach as it
builds on the rhythmic feel for counting in threes.

STAGE THREE:   “Finger-Skip-Counting”   The third stage follows “finger-
drumming.” To begin finger-skip-counting, the child must have done enough “finger-drumming” so s/he is quite familiar with the multiples in the correct order.

To “finger-skip-count,” 3 x 4, for example, the child holds out a fist and
runs through the multiples of 3, like this:  “Three (extending one finger), Six (extending two fingers), Nine (extending three), Twelve (extending four fingers).”  You ask, “So what is 3 x 4?” And the child answers:  “3 x 4 equals 12.”

I found it helpful to first just challenge the child with the multiples from 3 x 1 through 3 x 5. Once s/he develops competence there, proceed to “finger-skip-counting” the multiples from 3 x 6 through 3 x 10. Finally do 3 x 11 and 3 x 12.

Put all together, these three stages offer a fun and rhythmic way for children
to learn their multiples of three. I’m curious to find out if I can use a similar
approach for the 4s, and I’ll find out soon.

I can’t be sure, but it seems like children could probably learn their 4s
by jumping rope, or doing other activities with a rhythmic nature.

If any of you have used an approach like this one for learning the times
tables, feel free to share it.

### Memorizing those Times Tables

Is there any area of elementary math more fraught with stress and anxiety, save, perhaps, long division? Probably not. But for good reason.

Despite what a tiny minority of conceptual-learning purists might say, the times table facts ARE critical. Let’s face it: you really DON’T want your children to spend the rest of their lives reaching for the calculator to figure out 6 x 7; a certain amount of math simply needs to become automatic, to allow students to succeed at higher math skills and and to gain higher math concepts. Not only that, but knowing the times tables is widely recognized as a crucial milestone in children’s elementary math development.

In my work as a tutor, I’ve used many approaches to teach the times tables over the years, and each of them has one benefit or another. But I’ve settled on one technique as my “old-faithful” approach. This technique combines elements of both play and discipline, and it also melds both the “conceptual” approach and the “pure memorization” approach.

This technique relies on a three-step process, and it’s easy to learn and teach.

The first step is to simply isolate a particular times table fact set you’d like your child to work on, for example, the 4s. This act of isolating itself is critical. The child knows that she or he is required to memorize a limited set of facts for now (not the entire times tables), and that narrowing of the task decreases anxiety.

Once you’ve settled on the fact set, the second step begins, and it can be quite fun. In this second step there should be no mention even made of the times tables. All you’re doing in this step is laying the foundation for times tables facts. What you do here is work with your students/children to help them learn to first COUNT UP by the number you’re dealing with. So for example, if you’re teaching the 4s, you simply teach children how to COUNT UP by 4s. What that means is that you teach your children how to think their way through knowing and saying the following with speed and ease:  0 – 4 – 8 – 12 – 16 – 20 – 24 – 28 – 32 – 36 – 40 – 44 – 48.

I’ve found that most children take well to this learning process if you approach it in the spirit of a game. You might, for example, start by saying 0 and then throw your child a ball. She or he will then say 4 and throw the ball back to you. You then would say 8, and then throw the ball back to your child. Keep going till you hit the peak number, 40, 48, or wherever you decide to stop.

Another way to make this into a game for young children is to make it into a game like “patty-cake.” Make up a set of hand gestures to which you, very quietly, say:  1-2-3, and then clap hands and loudly say “4!” Then use the same hand gestures to quietly say:  5-6-7, and then clap again and loudly say: “8!” There are many ways to make this process of counting by 4s game-like. And if you’re short on ideas, ask your children/students what would make it fun for them.

In any case, once your children can accurately COUNT UP by 4s, work with them in the same fashion to COUNT DOWN by 4s. Same idea, but now you start by saying 48, or 40, and then help them count DOWN:  44 – 40 – 36 – 32 –  28 – 24 – 20 – 16 – 12 – 8 – 4 – 0. This takes a bit more time, but it can be done — and more easily than you might imagine.

Once your child can count both up and down, she or he has the mental “scaffolding” on which the times table facts are hung, as it were.

And so the third step involves combining this “scaffolding” with the actual times tables. Here’s how.

Have your children memorize what I call THE THREE KEY MULTIPLICATION FACTS:
x 1,  x 5, and x 10.

For example, when learning the 4s, these key facts would be:
4 x 1 = 4
4 x 5 = 20
4 x 10 = 40

Once children memorize those three key facts, help them see that to find 4 x 2 and 4 x 3, they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 1 = 4. Similarly, to find 4 x 6 and 4 x 7 they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 5 = 20. And to find 4 x 11 and 4 x 12, they just COUNT UP by 4 once or twice beyond the key fact of 4 x 10 = 40.

Work on this first, and have them master it before proceeding.

Once a child knows these facts, she or he has 9 of the 13 key facts (going from 4 x 0 through 4 x 12).

To learn the four other facts, help children see that to find 4 x 4 and 4 x 3, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 5 = 20. And to find 4 x 9 and 4 x 8, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 10.

By breaking the process of learning the times tables into these steps, you make the process less daunting for children. By teaching students how to COUNT UP or COUNT DOWN by the number you’re learning, you help children develop many rich aspects of number sense. And by connecting the process of COUNTING UP or DOWN to the times tables, you help children learn these critical facts both solidly and with understanding.

My advice:  try it. I guarantee that you’ll like it.

Happy Teaching,

—  Josh