Check out the new math education carnival at Tom Bennison’s blog. Games, puzzles, teaching tips, and all sorts of mathy fun:
If you enjoy this carnival, why not send in a blog post of your own for next month? We love posts on playful ways to explore and learn math from preschool discoveries through high school calculus.
Entries accepted at any time!
Tova Brown concludes her exploration of the Hilbert’s Hotel Paradox with a look at the cardinality of the real numbers.
You run a hotel with an infinite number of rooms. You pride yourself on accommodating everyone, even guests arriving in infinitely large groups — but some infinities are more infinite than others, as it turns out.
Tova Brown dives deeper into Hilbert’s Hotel Paradox, considering the difference between rational numbers and reals.
You run an infinitely large hotel, and are happy to realize that you can accommodate an infinite number of infinite groups of guests.
However, a delicate diplomatic situation arises when a portal to another universe opens, introducing a different kind of guest, in a different kind of group.
Can you make room for them all?
Tova Brown continues to examine Hilbert’s Hotel Paradox, pondering infinite sets of infinite sets.
As the proprietor of an infinitely large hotel, you pride yourself on welcoming everyone, even when the rooms are full. Your hotel becomes very popular among infinite sports teams, as a result.
Recruitment season presents a challenge, however, when many infinite teams arrive at once. How many infinite teams can stay in a single infinite hotel?
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.
Is there no limit to your hospitality?
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.
[Feature photo above from the public domain, and title background (below) by frankieleon (CC BY 2.0) via Flickr.]
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.
The December math education blog carnival is up for your browsing enjoyment:
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! …”
Click here to go read the whole post at Math Misery? blog.
CREDITS: “celebrate trees” photo (top) by Lauren Manning via Flicker (CC BY 2.0).
Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; Understanding Math, Part 3: Is There Really a Difference?; and Understanding Math, Part 4: Area of a Rectangle.
In this post, we consider the second of three math rules that most of us learned in middle school.
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.
Do you remember our exploration of the area of a rectangular tabletop?
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.
Click here to continue reading: Understanding Math Part 6, Algebraic Multiplication…
CREDITS: “School Discussion” photo (top) by Flashy Soup Can via Flicker (CC BY 2.0, text added). 
This is the fifth post in my Understanding Math series, adapted from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, available at your favorite online book dealer.
Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; and Understanding Math, Part 3: Is There Really a Difference?
In this post, we consider the first of three math rules that most of us learned in middle school.
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.
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.
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.
Click to continue reading Understanding Math, Part 5: Multiplying Fractions…
CREDITS: “Framed” photo (top) by d_pham via Flicker</a (CC BY 2.0, text added)>. This is the fourth post in my Understanding Math series, adapted from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, available at your favorite online book dealer.
Denise Gaskins' Let's Play Math 
