We’ve examined how our vision of mathematical success shapes our children’s learning. Do we think math is primarily a tool for solving problems? Or do we see math as a web of interrelated concepts?

*Instrumental understanding* views math as a tool. *Relational understanding* views math as an interconnected system of ideas. Our worldview influences the way we present math topics to our kids. And our children’s worldview determines what they remember.

In the past two posts, we looked at different ways to understand and teach rectangular area and fraction multiplication. But how about algebra? Many children (and adults) believe “math with letters” is a jumble of abstract nonsense, with too many formulas and rules that have to be memorized if you want to pass a test.

Which of the following sounds the most like your experience of school math? And which type of math are your children learning?

Every mathematical procedure we learn is an instrument or tool for solving a certain kind of problem. To understand math means to know which tool we are supposed to use for each type of problem and how to use that tool — how to categorize the problem, remember the formula, plug in the numbers, and do the calculation.

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.

Each mathematical concept is part of a web of interrelated ideas. To understand mathematics means to see at least some of this web and to use the connections we see to make sense of new ideas.

The concept of rectangular area has helped us understand fractions. Let’s extend it even farther. In the connected system of mathematics, almost any type of multiplication can be imagined as a rectangular area. We don’t even have to know the size of our rectangle. It could be anything, such as subdividing a plot of land or designing a section of crisscrossed colors on plaid fabric.

We can imagine a rectangle with each side made up of two unknown lengths. One side has some length *a* attached to another length *b*. The other side is *x* units long, with an extra amount *y* stuck to its end.

We don’t know which side is the “length” and which is the “width” because we don’t know which numbers the letters represent. But multiplication works in any order, so it doesn’t matter which side is longer. Using the rectangle model of multiplication, we can see that this whole shape represents the area .

But since the sides are measured in pieces, we can also imagine cutting up the big rectangle. The large, original rectangle covers the same amount of area as the four smaller rectangular pieces added together, and thus we can show that .

With the FOIL formula mentioned earlier, our students may get a correct answer quickly, but it’s a dead end. FOIL doesn’t connect to any other math concepts, not even other forms of algebraic multiplication. But the rectangular area model will help our kids multiply more complicated algebraic expressions such as .

Not only that, but the rectangle model gives students a tool for making sense of later topics such as polynomial division. And it is fundamental to understanding integral calculus.

*To be continued. Next up, Understanding Math Part 7: The Conclusion…*

**CREDITS:** “Math Workshop Portland” photo (top) by US Department of Education via Flicker (CC BY 2.0). This is the sixth 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…

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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!

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The monthly ** Math Teachers at Play** (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

**Don’t procrastinate:** *The deadline for entries is this Friday, January 22.* The carnival will be posted next week at **mathematicsandcoding**.

Thank you so much to the volunteer bloggers who have stepped up to carry this MTaP math education blog carnival through the years! I would never be able to keep the carnival going on my own.

If you’d like to join in the fun, we have plenty of openings for 2016. Read the instructions on our **Math Teachers at Play page**. Then leave a comment or **email me** to let me know which month you’d like to take.

While you’re waiting for next week’s *Math Teachers at Play* carnival, you may enjoy:

**
**

- Browse past editions of the
*Math Teachers at Play*blog carnival - Carnival of Mathematics
- Carnaval de Matemáticas

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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.

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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?

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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?

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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?

*Click here to read Part Three…*

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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.

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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 +, -, x, ÷, sqrt (
**square root**), ^ (**raise to a power**), ! (**factorial**), and**parentheses, brackets, or other grouping symbols**. - 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.

- 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.

As usual, we will need every trick in the book to create variety in our numbers. Experiment with decimals, two-digit numbers, and factorials. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it might be more helpful.

**Use the comments section below to share the numbers you find.**

But please don’t spoil the game by telling us how you made them! You may give relatively cryptic hints, especially for the more difficult numbers, but be careful. Many teachers use this puzzle as a classroom or extra-credit assignment, and there will always be students looking for people to do their homework for them.

**Do not post your solutions. I will delete them.**

There is no authoritative answer key for the year game, so we will rely on our collective wisdom to decide when we’re done. We’ve had some lively discussions the last few years. I’m looking forward to this year’s fun!

As players report their game results below, I will keep a running tally of confirmed results (numbers reported found by two or more players). I’ve been fighting a really nasty flu for the past several weeks, however, so I won’t spend much time at my computer. And if I ever get to feeling better, I was hoping to do some traveling. So this tally will usually lag behind the results posted in the comments.

Percent confirmed: 0%

Reported but not confirmed: 90%

1-75, 77-82, 84-85, 87, 90, 92, 94, 96, 99-100

Numbers we are still missing: 10%

76, 83, 86, 88-89, 91, 93, 95, 97-98

Students in 1st-12th grade may wish to **submit their answers to the Math Forum**, which will begin publishing **student solutions** after February 1, 2016. Remember, Math Forum allows double factorials but will NOT accept answers with repeating decimals.

Finally, here are a few rules that players have found confusing in past years.

**These things ARE allowed:**

- You must use each of the digits 2, 0, 1, 6 exactly once in each expression.
- 0! = 1. [See Dr. Math’s
**Why does 0 factorial equal 1?**] - Unary negatives count. That is, you may use a “−” sign to create a negative number.
- You may use (
*n*!)!, a nested factorial, which is a factorial of a factorial. Nested square roots are also allowed. - The double factorial
*n*!! = the product of all integers from 1 to*n*that are equal to*n*mod*2*. If*n*is even, that would be all the even numbers, and if*n*is odd, then use all the odd numbers.

**These things are NOT allowed:**

- You may not write a computer program to do the puzzle for you — or at least, if you do, PLEASE don’t ruin our fun by telling us all the answers!
- You may not use any exponent unless you create it from the digits 2, 0, 1, 6. You may not use a square function, but you may use “^2”. You may not use a cube function, but you may use “^(2+1)”. You may not use a reciprocal function, but you may use “^(−1)”.
- “0!” is not a digit, so it cannot be used to create a base-10 numeral. You cannot use it with a decimal point, for instance, or put it in the tens digit of a number.
- The decimal point is not an operation that can be applied to other mathematical expressions: “.(2+1)” does not make sense.
- You may not use the integer, floor, or ceiling functions. You have to “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals.

**Mathematics Game Worksheet**

For keeping track of which numbers you’ve solved.

**Mathematics Game Manipulatives**

This may help visual or hands-on thinkers.

**Mathematics Game Student Submissions**

For elementary through high school students who wish to share their solutions.

For more tips, check out **this comment** from the 2008 game.

**Heiner Marxen** has compiled hints and results for past years (and for the related Four 4’s puzzle). **Dave Rusin** describes a related card game, Krypto, which is much like my **Target Number game**. And **Alexander Bogomolny** offers a great collection of similar puzzles on his **Make An Identity page**.

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

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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.

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.

*To be continued, probably after the holidays. Next up, Understanding Math Part 6: Algebraic Multiplication…*

**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…

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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

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 (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…

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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.

*Click to continue reading Understanding Math, Part 4: Area of a Rectangle.*

**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…

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**Easier activities for elementary and middle school.** When you get to the Nrich website, click a number to go to that day’s math.

**Activities for middle and high school.** When you get to the Nrich website, click a number to go to that day’s math.

**“Wild Maths” puzzles and articles for teens and up.** When you get to the +Plus Magazine website, you can tell which links are live by the drop shadow under the picture. One link becomes live each day — so come back tomorrow and discover something new!

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My November “Let’s Play Math” newsletter went out early Monday morning to everyone who signed up for Tabletop Academy Press math updates. This month’s issue focused on figurate numbers, and it also included the latest updates on the *Let’s Play Math* paperback edition (coming in early 2016).

If you didn’t see it, check your Updates or Promotions tab (in Gmail) or your Spam folder. And to make sure you get all the future newsletter, add “Denise at Tabletop Academy Press” [Tabletop Academy Press @ gmail.com, without spaces] to your contacts or address book.

And if you missed this month’s edition, no worries—there will be more playful math snacks in the next issue. Click the link below to sign up today, and we’ll send you our free math and writing booklets, too!

As a Bonus: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.

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