Tova Brown explores the second part of Hilbert’s Hotel Paradox. What’s infinity plus infinity?

Running an infinite hotel has its perks. Even when the rooms are full you can always find space for new guests, so you proudly welcome everyone who appears at your door.

When two guests arrive at once, you make room. When ten guests arrive, you accommodate them easily. When a crowd of hundreds appears, you welcome them all.

Tova Brown’s introduction to Hilbert’s Hotel Paradox, a riddle about the nature of infinity…

Once upon a time, there was a hotel with an infinite number of rooms. You might be thinking this is impossible, and if so you’re right. A hotel like this could never exist in the real world.

But fortunately we’re not talking about the real world, we’re talking about math. And when we do math we can make up whatever rules we want, just to see what happens.

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza. Please join us!

For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.

Use the digits in the year 2016 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

You must use all four digits. You may not use any other numbers.

Solutions that keep the year digits in 2-0-1-6 order are preferred, but not required.

You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.

You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

My Special Variations on the Rules

You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.

You MAY use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.

The carnival is short this month, but full of treats. Enjoy!

“So here we are on December 23, 2015, the 93rd edition of Math Teachers At Play! As per tradition, what’s so fascinating about the number 93?

“First, it’s a prime number! No. Wait, that’s clearly false. So 93 is not a prime number. But that’s not very fascinating. Aha! But 93 is a semi-prime! since 93=31×3. Even more interesting is that 94 and 95 are semi-primes. So a question is, is there another triplet of positive integers that are also semi-primes? It’s a good question to ask your students! …”

In this post, we consider the second of three math rules that most of us learned in middle school.

To multiply fractions, multiply the tops (numerators) to make the top of your answer, and multiply the bottoms (denominators) to make the bottom of your answer.

Instrumental Understanding: Math as a Tool

Fractions confuse almost everybody. In fact, fractions probably cause more math phobia among children (and adults) than any other topic before algebra.

Children begin learning fractions by coloring or cutting up paper shapes, and their intuition is shaped by experiences with food like sandwiches or pizza. But before long, the abstraction of written calculations looms up to swallow intuitive understanding.

Upper elementary and middle school classrooms devote many hours to working with fractions, and still students flounder. In desperation, parents and teachers resort to nonsensical mnemonic rhymes that just might stick in a child’s mind long enough to pass the test.

Relational Understanding: Math as a Connected System

Now let’s zoom in on our rectangle. Imagine magnifying our virtual grid to show a close-up of a single square unit, such as the pan of brownies on our table. And we can imagine subdividing this square into smaller, fractional pieces. In this way, we can see that five-eighths of a square unit looks something like a pan of brownies cut into strips, with a few strips missing:

But what if we don’t even have that whole five-eighths of the pan? What if the kids came through the kitchen and snatched a few pieces, and now all we have is three-fourths of the five-eighths?

How much of the original pan of brownies do we have now? There are three rows with five pieces in each row, for a total of 3 × 5 = 15 pieces left — which is the numerator of our answer. And with pieces that size, it would take four rows with eight in each row (4 × 8 = 32) to fill the whole pan — which is our denominator, the number of pieces in the whole batch of brownies. So three-fourths of five-eighths is a small rectangle of single-serving pieces.

Notice that there was nothing special about the fractions 3/4 and 5/8, except that the numbers were small enough for easy illustration. We could imagine a similar pan-of-brownies approach to any fraction multiplication problem, though the final pieces might turn out to be crumbs.

Of course, children will not draw brownie-pan pictures for every fraction multiplication problem the rest of their lives. But they need to spend plenty of time thinking about what it means to take a fraction of a fraction and how that meaning controls the numbers in their calculation. They need to ask questions and to put things in their own words and wrestle with the concept until it makes sense to them. Only then will their understanding be strong enough to support future learning.

CREDITS: “School Discussion” photo (top) by Flashy Soup Can via Flicker (CC BY 2.0).This is the fifth post in my Understanding Math series, adapted from the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…

Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

In this post, we consider the first of three math rules that most of us learned in middle school.

Area of a rectangle = length × width

Instrumental Understanding: Math as a Tool

The instrumental approach to explaining such rules is for the adult to work through a few sample problems and then give the students several more for practice.

In a traditional lecture-and-workbook style curriculum, students apply the formula to drawings on paper. Under a more progressive reform-style program, the students may try to invent their own methods before the teacher provides the standard rule, or they may measure and calculate real-world areas such as the surface of their desks or the floor of their room.

Either way, the ultimate goal is to define terms and master the formula as a tool to calculate answers.

Richard Skemp describes a typical lesson:

Suppose that a teacher reminds a class that the area of a rectangle is given by A=L×B. A pupil who has been away says he does not understand, so the teacher gives him an explanation along these lines. “The formula tells you that to get the area of a rectangle, you multiply the length by the breadth.”

“Oh, I see,” says the child, and gets on with the exercise.

If we were now to say to him (in effect) “You may think you understand, but you don’t really,” he would not agree. “Of course I do. Look; I’ve got all these answers right.”

Nor would he be pleased at our devaluing of his achievement. And with his meaning of the word, he does understand.

As the lesson moves along, students will learn additional rules.

For instance, if a rectangle’s length is given in meters and the width in centimeters, we must convert them both to the same units before we calculate the area. Also, our answer will not have the same units as our original lengths, but that unit with a little, floating “2” after it, which we call “squared.”

Each lesson may be followed by a section on word problems, so the students can apply their newly learned rules to real-life situations.

Relational Understanding: Math as a Connected System

In contrast, a relational approach to area must begin long before the lesson on rectangles.

Again, this can happen in a traditional, teacher-focused classroom or in a progressive, student-oriented, hands-on environment. Either way, the emphasis is on uncovering and investigating the conceptual connections that lie under the surface and support the rules.

We start by exploring the concept of measurement: our children measure a path along the floor, sidewalk, or anywhere we could imagine moving in a straight line. We learn to add and subtract such distances. Even if our path turns a corner or if we first walk forward and then double back, it’s easy to figure out how far we have gone.

But something strange happens when we consider distances in two different directions at the same time — measuring the length and width of the dining table automatically creates an invisible grid.

In measuring the length of a rectangular table, we do not find just one point at any given distance. There is a whole line of points that are one foot, two feet, or three feet from the left side of the table.

And measuring the width shows us all the points that are one, two, or three feet from the near edge. Now our rectangular table is covered by virtual graph paper with squares the size of our measuring unit.

The length of the rectangle tells us how many squares we have in each row, and the width tells us how many rows there are. As we imagine this invisible grid, we can see why multiplying those two numbers will tell us how many squares there are in all.

That is what the word area means: the area of a tabletop is the number of virtual-graph-paper squares it takes to cover it up, which is why our answer will be measured in square units.

Making Sense of Mixed Units

What if we measured the length in meters and the width in centimeters?

With a relational understanding of area, even a strange combination of units can make sense. Our invisible grid would no longer consist of squares but of long, thin, rectangular centimeter-meters. But we could still find the area of the tabletop by counting how many of these units it takes to cover it.

Square units aren’t magic — they’re just easier, that’s all.

CREDITS: “Framed” photo (top) by d_pham via Flicker (CC BY 2.0).This is the fourth post in my Understanding Math series, adapted from the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…

Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

From the outside, it’s impossible to tell how a person is thinking. A boy with the instrumental perspective and a girl who reasons relationally may both get the same answers on a test. Yet under the surface, in their thoughts and how they view the world, they could not be more different.

“Mathematical thinking is more than being able to do arithmetic or solve algebra problems,” says Stanford University mathematician and popular author Keith Devlin. “Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns.”

And our own mathematical worldview will influence the way we present math topics to our kids. Consider, for example, the following three rules that most of us learned in middle school.

Area of a rectangle = length × width.

To multiply fractions, multiply the tops (numerators) to make the top of your answer, and multiply the bottoms (denominators) to make the bottom of your answer.

When you need to multiply algebra expressions, remember to FOIL: multiply the First terms in each parenthesis, and then the Outer, Inner, and Last pairs, and finally add all those answers together.

While the times symbol or the word multiply is used in each of these situations, the procedures are completely different. How can we help our children understand and remember these rules?

Over the next three posts in this series, we’ll dig deeper into each of these math rules as we examine what it means to develop relational understanding.

Many people misunderstand the distinction between Instrumental and Relational Understanding as having to do with surface-level, visible differences in instructional approach, but it’s not that at all. It has nothing to do with our parenting or teaching style, or whether our kids are learning with a traditional textbook or through hands-on projects. It’s not about using “real world” problems, except to the degree that the world around us feeds our imagination and gives us the ability to think about math concepts.

This dichotomy is all about the vision we have for our children — what we imagine mathematical success to look like. That vision may sit below the level of conscious thought, yet it shapes everything we do with math. And our children’s vision for themselves shapes what they pay attention to, care about, and remember.

CREDITS: “Math Workshop Portland” photo (top) by US Department of Education via Flicker (CC BY 2.0).This is the third post in my Understanding Math series, adapted from the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…

Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.