Busy Learning

Many of my students attend a private K-8 school that offers a very traditional math curriculum. Students in grades 5, 6, and 7 spend many months studying shopkeepers' math. The main focus is percentages - discounts, sales tax, tip, simple and compound interest, commissions, annuities, etc. Each of these variations is taught as a series of formulas to be memorized. The vocabulary is beyond the comprehension of most 12 year olds. Even the numbers themselves are quite tedius. Students spend hours calculating 17.25% of $12,650.85 by hand.

A group of 7th grade boys came in with homework from their new math unit - an introduction to linear relationships. They each had a worksheet with about 15 tiny grids upon which lines were drawn. The students were asked to calculate the slope of each line. That was it. No context and no explanation. That was their starting point.

"We don't really know what we're supposed to do," said one student.

"Yeah, it's just a bunch of lines," said another.

"We're lost," said the third member of the group.

The other two solemnly nodded in agreement.

Maybe it was the fact that I had just watched Stand and Deliver over the weekend and wanted to be like Jaime Escalante but I grabbed the worksheet, pretended to scrutinize it, and delivered the following line with exaggerated astonishment and admiration.

"Whoa!!! You guys are doing this? Wow! I can't believe it."

"What do you mean? What is this stuff?," they asked, almost in unison.

I walked to the door, checked to see if anyone was around, then closed it.

"Alright, I'll show you," I said.

One of the advantages of working in a defunct science center is access to a closet full of physics toys. I grabbed a few matchbox cars, ramps, and books and set up an impromptu math lab. What ensued over the next 45 minutes was a spirited look at independent and dependent variables, data plotting, best fit lines, and, of course, the meaning of slope.

At the end of the class, one of the fathers came in to pick up the students. I was putting away supplies but I heard him ask his son if he got his homework done. The boy said no. The father, clearly upset, asked his son for an explanation. The boy said....

"Dad, I can do my homework later. I didn't do it here because I was busy learning."

Yeah!

A Math Problem Revisited

One of the advantages of teaching in a learning center rather than a classroom is that I often get to see the same student at various stages of his or her academic life. A student who is in one of my precalc classes today was the very first student I ever taught. Back then it was number facts and place value. Today it's trig identities and parametric equations. While I haven't seen this student every year since first grade, we've reunited for key math events. We were together for whole number operations, met again for fractions, and stayed close through algebra and geometry. The opportunity to observe students over the entire length of their K-12 careers has been an incredible learning experience. It has given me a unique perspective regarding how children learn math and how various math programs succeed or fail in preparing students for each new stage of concept development.

One of the math classes I'm teaching combines problem solving with computer programming. The group is made up of 8th graders I had taught a few years back. They had participated in a math olympiad course. One type of problem that appeared frequently dealt with simultaneous equations. For example,

At the movie theatre, tickets cost $6 for adults and $5 for children.
A total of 24 people went to the movies and paid $128 for tickets.
How many children went to the movies?

At the elementary level, the students were encouraged to create a chart to keep track of the number of adults and children, and total cost. The process is simply an organized version of guess and check. The efficiency of this method is dependent on the extent of number sense developed by the students and the number of potential solutions.

I thought it would be interesting to revisit this type of problem four years later from a different angle. If we removed the constraint of the total number of people, would we be able to find the exact number of reasonable solutions to the problem? In this case, only positive integer values would be sensible since the problem deals with whole people.

We had done some hand graphing of linear equations, so I led the group in that direction. The students identified the variables and came up with an equation to graph, 6x + 5y = 128. They determined that there were 4 combinations of adults and children that would result in a total cost of $128. The graph showed that while there were an infinite number of solutions to the equation, only 4 solutions made sense in the context of the problem.

We then talked about other situations that would result in an equation of this type and developed the general equation, ax + by = c. Next, I challenged them to create a computer program that would find all positive, integer solutions of this generalized equation. From the graph, the students knew that we needed to test all x values from zero to some equation dependent maximum value. The students were familiar with programming loops and agreed that this would be the best approach. Here's the pseudo-code version of what the group invented.

var a:Number;
var b:Number;
var c:Number;
var x:Number;
var y:Number;
var nMax:Number = c/a rounded to the next highest integer;

for(all values of x from 0 to nMax)
{
y = (c-ax)/b; //rearranged equation in terms of y
if(we divide y by 1 and get a remainder of zero)
{
then y is an integer
this xy pair is a solution
store these values
}
else
{
y is not an integer
this xy pair is not a solution
}
}

So far so good. My students were very proud that they were able to work out the logic in this program. They had to figure out a way to calculate the maximum value of x, determine if a number was an integer, and decide what to do when y was an acceptable value. Each of these areas led to rich discussions and deeper connections. Our next step is to find a way to display our results in both graph and chart form. 

If I'm fortunate enough to teach these students a couple of years from now, perhaps I could challenge them to develop an algorithm based on parametric equations.

Fun For Whom?

I drive my students crazy.
I know I do.
And this is why.

Student: Is this right?
Me: What do you think?
Student: I don’t know.
Me: Well, how did you get your answer?
Student: I just did what you showed me before.
Me: Ok, but what does that look like for this problem?
Student: (utterly exasperated) Why can’t you ever just say yes or no?

Me: (to myself) Because saying yes or no is the easy route and you deserve better than that from me. Because doing so discourages analytical thought and makes you dependent on the approval of others. Because this is the one thing I can do to ensure you develop confidence in your own problem solving abilities. Because this tiresome process is my gift to you; you just don’t know it yet.
 

Me: Oh, it’s much more fun this way!

Save the Zogs

"Four frightened Zogs have left the safety of their planet and are floating around in space. The Duplicators, a band of space travelers with the ability to imitate others, have infiltrated the floating Zogs. This is making the rescue mission very difficult.

Fortunately, the Zogs are very clever. They can assemble themselves along a straight line path. The Duplicators cannot exist on this path. If the rescue team can determine the equation of this line, then the Zogs will be saved. The Duplicators will be left behind.

To rescue the Zogs, you need to learn as much as possible about linear equations and the lines they create. What happens when the slope is zero? What effect does the y intercept have on the position of the line? The more you know, the more Zogs you can save."

The game begins with simple horizontal and vertical lines, builds to equations with non-zero slopes passing through the origin, and ends with equations that have both non-zero slopes and y-intercepts. A tracking tool enables students to rotate and move a line into position to help them visualize whether or not the line passes through the correct Zogs. Once the visual is in place, students must then identify the equation the describes the line.

What middle school student can resist the plight of the Zogs?

Middle Zeros

I had an opportunity to spend some time with a very enthusiastic group of third grade students at a local school. They were subtracting three digit numbers by regrouping and were doing quite a spectacular job. That is, until we reached the dreaded Middle Zero.

The problem was 601 - 347. The following is an actual transcript of the dialogue that ensued.

Me: Does anyone know how to do this problem?
Student 1: You can't take away 7 from 1 so you have to make the zero a 9.
Me: You mean I can just turn a 0 into a 9? How can I do that?
Student1: Well, you make the 6 into a 5 and then the zero becomes a 9.
Me: I still don't understand how the zero becomes a 9. And why does the 6 become a 5?
Student2: There aren't any tens so you have to borrow a ten from the hundreds column.
Me: Ok, that makes sense. How many tens are in the hundreds column?
Student2: 6

and so on...

It was fairly obvious that they had been introduced to the mechanics of regrouping but they had not achieved a true understanding of the process. The beauty of mathematics and the amazing "aha!" moments are revealed only when we investigate the math beneath the surface.

Third grade students are generally very competent when it comes to regrouping tens to ones in the case of 85 - 29 or 371 - 158. Problems arise when there are no tens in the tens column. This requires a visit to the hundreds column whereupon one of the hundreds is regrouped as 10 tens.

For example, in the problem:

402 - 138

the number 402 would have 3 hundreds, 10 tens and 2 ones. Now it's possible to regroup one of the tens into 10 ones and add those to the ones column. The final version of the number 402 is 3 hundreds, 9 tens, and 12 ones.

While this is no doubt trivial and rather obvious to you as a teacher or parent, third grade students find this incredibly confusing. In fact, I'm convinced that even with the benefit of base ten blocks and dozens of practice problems, students are still not really sure what's going on here.

What if we approach problems with middle zeros a little differently. First, help your students see numbers in terms of tens. If you use the number 80, ask how many tens that is. If the number is 527, help them see not only that there are 5 hundreds, 2 tens, and 7 ones but that there are 52 tens. Once that is understood, you can explain regrouping numbers with middle zeros more easily.

Let's revisit the problem, 402 - 138. A ten is needed to make 12 ones. Turn this into a single step process by having the students think "40 tens". Now they can use one of those tens to change the number of ones from 2 to 12 and they'll have 39 tens left. It's much simpler, there's less crossing out, and students really get it.

 

This method works with numbers containing more than three digits as well.


























How do you teach regrouping with middle zeros?

Comparing Fractions

My fourth grade students are learning how to compare fractions. They've mastered this concept for very specific types of comparison problems, for example, when the denominators are the same and when the numerator is 1. Now I'm trying to teach them how to decide if a fraction is less than or greater than 1/2. They seemed to really grasp the idea that there are many ways to express the quantity 1/2 using fractions - 2/4, 3/6, 4/8, etc. The students recognized that the denominator is always double the numerator. In fact, one particularly clever student suggested the fraction 1/8 / 1/4, remembering the relative sizes of the fraction pieces we had used during the beginning of the unit. Despite this, my students are having difficulty determining whether a fraction such as 3/7 is more or less than 1/2. 

To help students visualize more difficult fractions, I designed a simple mathlet that draws two partitioned rectangles whose shaded sections represent the fractions typed in by the students. My students set one fraction to 1/2 and then alter the other fraction for comparison. I think they are starting to develop a better sense of relative sizes among fractions. Feel free to try it with your own students.

Invert and Multiply

When it comes to the dividing fractions, I have to admit that I graduated from the school of "Don't Ask Why, Just Invert and Multiply". Were you taught something similar?

There are so many things wrong with that approach. It reinforces misconceptions that students may have about the mysterious and magical nature of math. Is dividing 1/4 by 1/3 really so incomprehensible that we shouldn't bother to explore its mind-boggling complexities? Fortunately for the current generation of students, that sort of teaching disappeared along with transistor radios and hoola hoops.

Or did it?

I had the opportunity to teach a 5th grade class recently. They had been immersed in a unit on fractions for several weeks and were just wrapping up a section on division. To find out what the students had learned so far, I placed the following problem on the board:

A lively conversation ensued.

Me: Does anyone know how to solve this problem?

Student1: Make it a multiplication problem.

Me: Like this?

Student1: No, you have to flip over the fractions.

Me: Oh, you mean like this?

Student1: No, they don't both get flipped. Only one of them.

Me: Which one?

Student1: I'm not sure. I forget which one.

Me(to myself): Uh huh. Where have I heard that before?

Me: Does anyone have a different approach?

Student2: Well, half of 2/5 would be 1/5 so that's the answer.

Me(to myself): Ok, there's some number sense here but we're not quite there yet.

Me: Do you think the problem is asking us to divide 2/5 into 2 equal parts?

Nearly the entire class nods a self assured yes.

Me: If that was true, wouldn't the problem read: 2/5 divided by 2?

The class expresses a collective look of bewilderment.

Me: Actually, the problem asks you to figure out how many halves there are in the fraction 2/5. It works just like whole numbers. When you are given a problem such as 40 divided by 8, you are being asked to figure out how many groups of 8 are in the number 40. So how many halves are in the fraction 2/5?

Lots of blank stares. I better try another approach.

Me: Who thinks there's at least one 1/2 in 2/5?

Way too many hands go up. Ok, we've got a number sense problem here. The division lesson is over for now. Where's that giant number line? We've got some work to do!

The old invert and multiply mentality is alive and well today and continues to ensure that students have absolutely no idea what is going on with division by fractions. While invert and multiply is a tried and true method for dividing fractions (provided one remembers which fraction to invert), it is not the only way to divide fractions.

Suppose you were given the following problem:

You don't even need to invert and multiply in this case. You can actually just divide the numerators and the denominators independently and get a more direct answer.

Students should be taught to assess each fraction problem to determine whether or not it would be easier to just simply divide. Perhaps then, students wouldn't automatically apply a poorly understood algorithm to every problem. Such a skill would serve them well when faced with more advanced math down the road.

How do you teach division of fractions to your students?

Animated Word Problems

Engaging students has always been a considerable challenge for teachers. Fortunately, there are a vast amount of resources and technologies available to help meet that challenge. One resource for upper elementary and middle school students is a catalog of animated word problems that guide students through the process of solving multi-step problems. The narrator and animated host is a young woman named Infinity Quick, or IQ for short. The introductory animation shows IQ receiving a new word problem on her computer. The movie is paused, giving students a few minutes to tackle the problem on their own. They may then proceed to watch the problem solving unfold.

The solution to each problem is presented with diagrams, charts, and calculations. The video solutions may be paused or replayed as many times as needed. Some of the video solutions contain interactive components, allowing students to further connect with the math lesson. After the video presentation, students are invited to try a similar problem on their own. They are then taken to the practice room where a new interactive problem is presented. If help is needed, there are a series of hints available to the student at the press of a button. The hints make use of illustrations that are familiar to students who have previously viewed the video. To further assist students, problem solving tools such as a calculator and a sketch pad are also available.

There are currently 19 problems available covering topics such as fractions, ratios, percentages, probablility, geometry averages and algebra.

Learning in Mathland

The book that influenced my approach to teaching mathematics to children is Mindstorms by Seymour Papert. Subtitled Children, Computers, and Powerful Ideas, this book provided a glimpse into a world of which I knew little but eagerly wished to learn more. As promised, powerful ideas emerged quickly and continued to be revealed at every turn. The most compelling idea for me, the one that figuratively jumped off the page, was the author's concept of Mathland. Papert envisioned a learning environment designed to make the development of mathematical concepts a more natural process. This novel vision captured my imagination and laid the groundwork for what would later become our after school math center.

So many questions arose. Could math achievement be improved by weekly trips to an authentic Mathland? What would such an environment look like? What did we want students to learn and how would they learn it? After a few false starts, we eventually succeeded in making our version of Mathland a reality. Each week students were transported to a world of numbers, mathematical ideas, and meaningful challenges. Gear Math was a hands-on, problem solving program designed to teach multiplication, division, ratio, and proportion to upper elementary students. Students programmed robots to traverse number lines and coordinate systems and designed containers based on volume and surface area restrictions. Most importantly, the residents did very well in school math despite the fact that curriculum objectives rarely overlapped.

How was that possible? Students learned to think and act and approach problem solving much like a mathematician would. This was accomplished by immersing students in credible learning environments and by providing challenges that were not only engaging but also demanded intellectual investment. Getting the right answer was important but it wasn’t the only goal. Making sense of those answers was critical. Explanations were reworked and refined. Multiple representations were encouraged. Our students developed a familiarity with math that all the practice sets and timed drills in the world could never accomplish. And, most importantly, it laid the foundation for future success in math. As our students moved from elementary to middle school, high school, and beyond, they always carried a piece of Mathland with them.

An Introduction

Welcome to the Math Playground blog. My name is Colleen and I've been deeply immersed in teaching and learning math for the past 15 years. It all began when my husband and I co-founded a math learning center in the Boston area called Math Advantage. Over the years, we've taught math to thousands of K-12 students. Some of our students come to review math concepts they're studying in school; others explore topics they have not yet seen. My main role at the learning center is curriculum development and I spend much of my time finding new and creative ways to teach math concepts. Games, computer programming, and real world math projects are some of the ideas we've tested. I also work directly with students, helping them move forward in their study of mathematics. 

In addition to teaching, I publish a math website for children called Math Playground. I developed the site in 2002 as a way to provide additional practice for my students outside of class. Since then it has grown into a trusted and respected math resource for thousands of educators throughout the world. On Math Playground, you'll find practice games, thinking puzzles, learning activities, math worksheets, videos, and problem solving challenges.

The purpose of this blog is to share all that I've discovered over the years about mathematics, teaching, and learning. I will highlight math resources you can use in the classroom immediately, invite you to step into my own classroom to view my teaching style, and discuss issues relevant to all of us as math educators. I hope you'll find my experiences both educational and inspiring.