[Photo by By Willi Heidelbach via flickr.]

Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: 50 = 12 + 72 and 50 = 52 + 52. … I’m a teacher I have to ask: “So what’s the next bigger number to 50 that is the sum of two non-zero square numbers in two distinct ways?” …

There is always something to investigate in math. One of the major objectives of school math is to get students into this thinking habit without us telling them to do so but I’m digressing from my topic now.

Let’s get to the great posts submitted for this edition.

Go read the carnival post at Mathematics for Teaching.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

[Photo by Wellspring Community School. This article is an excerpt from my upcoming "Let's Play Math!" book series.]

My favorite playful math lessons rely on adult/child conversation — a proven method for increasing a child’s reasoning skills. What better way could there be to do math than snuggled up on a couch with your little one, or side by side at the sink while your middle-school student helps you wash the dishes, or passing the time on a car ride into town?

As soon as your little ones can count past five, start giving them simple, oral story problems to solve: “If you have a cookie and I give you two more cookies, how many cookies will you have then?”

The fastest way to a child’s mind is through the taste buds. Children can easily visualize their favorite foods, so we use mainly edible stories at first. Then we expand our range, adding stories about other familiar things: toys, pets, trains.

Math That Is Social

“Panther the barn cat went hunting in the field and caught two mice every day. How many mice did he catch in four days?”

Don’t limit your story problems to the child’s grade level. If she can make a picture in her mind, she will be able to work with it. You may encourage your child to count on her fingers: one finger for each mouse. Using fingers as symbols is a step into abstraction, paving the way for later algebra. If you do not believe in finger-counting, then teach your child to count on blocks or craft sticks, or make her a counting rope for working with bigger numbers than she can handle mentally.

“Panther went out to the woods and met a gray cat named Shadow. He invited her to come back to our barn and chase pigeons. There were 15 pigeons in the barn, and Panther chased six of them. He let Shadow have the rest. How many pigeons did Shadow chase?”

As you both get used to the game, occasionally throw in something harder: fractions, division with a remainder, an answer that comes out negative. See what your student can do with a tough problem. You might be pleasantly surprised — even a toddler has ideas about how to split three hot dogs between two people.

If your son is stumped, try not to give away the “right” answer. Instead, ask him to explain the problem back to you. As he puts the problem in his own words, he will often see a solution. Pretend to be Socrates, asking questions that guide him toward the answer.

“After Shadow came to live in the barn, we had two cats, and half of them were girls. But then Shadow had four kittens, and now 2/3 of our barn cats are girls. How many of the kittens were girl cats?”

Here is the most important rule of the oral story problem game: Take turns. If I ask my daughter a story problem, she gets to give me one. And I have to try to solve it, even if she uses made-up numbers like 80-hundred or a gazillion. This is playtime, not an oral quiz.

Things to Consider in Creating Story Problems

• Some quantities are discrete and countable, such as marbles or dinosaurs. Other quantities are continuous, like a pitcher of juice or a length of rope. Use both types in your problems.
• Addition and subtraction are often thought of as putting-together or taking-away sets of discrete items. But they can also be represented in stories by growth or comparison (how much more or less) or by classification of parts (separating sheep from goats).
• Multiplication and division are often thought of as counting or sharing out groups of items. But they may also be represented by growth or shrinkage (how many times as much) or by rates and ratios (cookies per child, hot dogs per package).
• Division of continuous quantities may lead naturally to fractions (sharing pizza or candy bars).
• Money provides an excellent way for children to begin thinking about decimal numbers.

Oral story problems are not just for young children. Students of all ages benefit from the practice of working math in their heads. As your children grow, let the stories grow with them: soccer games, horse stories, or space adventures will keep middle-school students figuring.

Bonus Resource

Make oral story problems a part of your daily bedtime routine with Bedtime Math. This website publishes a daily math problem (with answers) at three levels of difficulty: Wee ones, Little kids, and Big kids — approximately preschool to upper-elementary level.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• Kitten Poses a Puzzle
• Word Problems from Literature
• Story Problem Challenge Revisited
• Babymath: Story Problem Challenge III
• Word Problems in Russia and America

Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.

— W. W. Sawyer
Vision in Elementary Mathematics

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Most people think that mathematics means working with numbers and that being “good at math” means being able to do (only slower) what any $10 calculator can do. But then, most people think all sorts of silly things, right? That’s what makes “man on the street” interviews so funny.

Numbers are definitely part of math — but only part, and not even the biggest part. And being “good at math” means much more than being able to work with numbers. It means making connections, thinking creatively, seeing familiar things in new ways, asking “Why?” and “What if?” and “Are you sure?”

It means trying something and being willing to fail, then going back and trying something else. Even if your first try succeeded — or maybe, especially if your first try succeeded. Just knowing one way to do something is not, for a mathematician, the same as understanding that something. But the more different ways you know to figure it out, the closer you are to understanding it.

Mathematics is not just memorizing and following rules. If we want to teach real mathematics, we teachers need to learn to think like mathematicians. We need to see math as a mental game, playing with ideas. James Tanton explains:

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• The (Mathematical) Trouble with Pizza
• Quotable: What to Do When You’re Stuck
• More Than One Way to Solve It
• More Than One Way to Solve It, Again
• Quotations XXVI: On Teaching Math

photo by UggBoy

The Math Teachers at Play (MTaP) blog carnival is a monthly collection of tips, tidbits, games, and activities for students and teachers of preschool, primary, and secondary mathematics. Do you like to learn new things and play around with ideas? You’re sure to find something of interest in the carnival.

If you are a homeschooler or classroom teacher, student or independent learner, or anyone else who writes about math — now is the time to send in your favorite blog post for next week’s MTaP carnival. The deadline is this Friday night, and the carnival will be posted next week at Mathematics for Teaching.

• MTaP Blog Carnival — Submission Form
• MTaP Blog Carnival Information Page

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Enjoy!

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

photo by Chrissy Johnson1 via flickr

Our decimal system of recording numbers is ingenious. Once learned, it is a simple, versatile, and efficient way of writing numbers. … But the system is not obvious nor easily learned. The use of place value is subtle, and mastering it is the single most challenging aspect of elementary school mathematics.

Ironically, these challenges are largely invisible to untrained parents and teachers — place value is so ingrained in adults’ minds that it is difficult to appreciate how important it is and how hard it is to learn.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

In other words, we take place value for granted. I know this was true of me when I started teaching my kids. Every year, their textbooks would start with the obligatory chapters on place value, which seemed to me just busywork. I began to appreciate the vital importance of place value when I read Liping Ma’s book and saw how the American teachers were unable to properly explain subtraction or multi-digit multiplication.

Place value is the heart of our number system, the foundation on which all the rest of arithmetic must be built. Because of place value, “The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.”

Study Teaching Materials

[Experienced Chinese teachers] study how each unit of the textbook is organized, how the content was presented by the authors, and why. They study what examples are in a unit, why these examples were selected, and why the examples were presented in a certain order. They review the exercises in each section of a unit, the purpose for each exercise section, and so on.

— Liping Ma
Knowing and Teaching Elementary Mathematics

This lesson from our textbook covers topics taught in Singapore Primary Math (Third Edition) 3A pp. 6-17; 4A pp. 6-11; 5A pp. 6-9. If you have a different edition of Singapore math or a different textbook series, look for sections with titles like “Thousands, Hundreds, Tens and Ones” or “Number Patterns” or “Whole Numbers to 100,000″ or “Place Values”.

The following video shows how challenging it can be to tell for sure whether our children understand what we teach. When you first hear the girl in class, she sounds like she knows place value, doesn’t she? Yet under closer questioning, we see how fragile is her knowledge. Our students’ understanding of place value needs to become a foundation strong enough to build on, because so much of arithmetic depends on it.

This is one reason we must frequently ask our children, “How did you figure it out?” or, “Why does that work?” Besides, it can be fun to hear what they say.

Teaching Place Value

Craft sticks (popsicle sticks) and rubber bands make a cheap and helpful set of manipulatives for bundling tens and hundreds, as shown in our textbook on page 7. I’ve also heard of people using coins and dollar bills to teach place value. The popsicle sticks show what’s really happening, because the student can see ten sticks bound together to make a ten, and ten sets of ten to make a hundred. Coins are more abstract, and for many young children, coin values don’t make sense.

My own children disliked using manipulatives of any kind, because it takes so much longer to count out the numbers with blocks or sticks than just to think through the problem. They have strong imaginations, and therefore the Primary Math textbook pictures communicated the concepts well enough for them.

There are three steps to counting and working with numbers in a place value system:

1. Form bundles to represent the numbers you want to count or add or subtract.
2. Rebundle if necessary.
3. Record the number of bundles in each appropriate position.

The second step, what the Chinese teachers call composing and decomposing tens, is what gives elementary students the most trouble. When talking to my children, I called this step making and breaking tens. It is not only tens, of course: In any place value column, 10 of that size will bundle together to make one of the position to its left. And in any place value column, you can take one of that size and break it apart into 10 of the position to its right.

I Love Funny Numbers!

On page 8, our textbook authors talk about the value of counting-by puzzles to cement a student’s understanding of place value. The “Number Patterns” pages in Singapore Math 3A (pages 14-17 in my edition) are well worth studying.

But then our authors claim, “The number after 39 is not ‘thirty-ten’.”

At our house, we use “thirty-ten” and “eighty-fourteen” and whatever other funny numbers come up. I want to separate and emphasize that rebundling step. (Or renaming, regrouping, composing and decomposing, carrying and borrowing, whatever terms you like.) We work through our math books orally, and we often give answers such as “27 + 35 = fifty-twelve” — then we make it a separate step to change “fifty-twelve” into the standard form of “62.”

You can see funny numbers in action and pick up a few other mental math tips in Mental Math: Addition.

Expanded Form

If our students thoroughly understand the expanded form of numbers, as shown in the middle of page 9, they will be well prepared for mental math. I don’t think our book goes quite far enough on this point.

For instance, 405 can also be thought of as 40 tens and 5 ones. 3784 might be 37 hundreds and 84 ones. How you look at a number depends on the problem you are trying to solve. Students must be able to take numbers apart and put them back together so they can work with them in flexible ways.

Here is an expanded form download you can print on cardstock and cut out:

• Teaching Expanded Numbers: A Hands-on Manipulative

Also, the tens combinations (number bonds) on page 11 are very important, though I seem to be in the minority among homeschoolers because I don’t stress memorizing math facts. I believe most math facts are best learned through continual use in solving problems — and through number bond games like Tens Concentration.

I do use focused memory work as a mop-up activity, after most of the math facts have been internalized through repeated use. For my children, however, such drill work has only been needed with multiplication, and even as we practice those facts I try to build a pre-algebra perspective of deeper understanding. (See my Times Table Series.)

Summary

As teachers, we have several tools to help our students master place value:

• Bundling (craft sticks and rubber bands)
• Coins (or number chips) in columns
• Expanded form in words (orally)
• Expanded form in numbers (number cards)
• Counting by ones, tens, hundreds
• Putting numbers in order (small to big, or big to small)
• What comes before, what comes next, number patterns of all types

Students need to be able to move back and forth between all these representations. Can you find all these represented in the Primary Math books — or in whatever elementary textbooks you use? Which seem the easiest to you? Which require the most thought?

Homework

This was a LONG homework set! I was getting finger cramp by the time I was half-way through. I will only comment on a couple of the questions:

#1c) This highlights the difference between our number system and those that came before: The order of the numbers is more important than the value of each individual digit.

#6) I especially liked this problem, with the idea of using pennies, nickels, and quarters to represent base five numbers. Too bad we don’t have a $1.25 bill for the “thousands” column! In general, to convert any number to any other base, we have to think about what the place values are in that base, and so we are forced to notice principles that normally we take for granted.

Coda: Letter to a College Student

Dear John,

I am sending you $50, as you requested. By the way, please remember that 50 is written with one zero, not with two.

Love, Dad

This post is part of the Profound Understanding of Fundamental Mathematics Series. [Go to the previous post. Go to the next post. Or start at the beginning.]

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• The Problem with Manipulatives
• Memorizing the Math Facts
• 20+ Things to Do with a Hundred Chart
• If Your Kids Like Sir Cumference…
• Kitten Poses a Puzzle

photo by goXunuReviews via Flickr

Are you a homeschooler? Are you happy with your current curriculum, or would you like to break out of the textbook mold and explore math through “living” books and activities? Whether you hope to replace your math program or just to supplement it, I can show you ways to turn math into a learning adventure for the whole family. Your children will build a stronger foundation of understanding when you teach math as a game, playing with ideas.

Nearly a year ago, I wrote:

This blog originally grew out of my books, and now it’s coming full circle: New, expanded editions of my long-out-of-print books are ripening on the vine, growing out of the blog. To bring them to harvest, I’m going to need your help.

It has taken much longer than I had hoped to whip the manuscripts into form. My new goal is to publish ebook editions, since I will be able to sell them for about half what the original books cost twelve years ago. I’m hoping that I can finish at least a couple of the ebooks by mid-summer.

The “Let’s Play Math!” Book Series

Here are the titles I’m working on:

Let’s Play Math: How Homeschool Families Can Enjoy Learning Math Together
Discover new ways to explore math as a family adventure, playing with ideas. You can lay a foundation for success in math with toys, games, and living books that help children of all ages experience the “Aha!” thrill of creative mathematical thinking, no matter what math curriculum you use.

Be a Math Ninja: Master the Tough Topics of Elementary and Middle School Math
If teaching fractions makes you frantic and you are paranoid about percents, you are not alone. “Math monsters” are those scary topics of textbook arithmetic, the things that go thump! in the mind. Find out how to use hands-on games and manipulatives to teach the real killers of elementary and middle school math.

Algebra at Any Age: Getting a Handle on Abstract Math
Children can begin to learn pre-algebra skills in preschool, and elementary students can master the first stages of algebra through hands-on exploration. Teenagers will understand how to multiply binomials and gain insight into the history of Greek geometric algebra. Step-by-step instructions and examples take you from solving simple equations to factoring quadratic polynomials.

Math You Can Play: Games That Build Your Child’s Number Skills
Forget flashcards and worksheets. Your children can practice their math facts by playing cards. Beginners will enjoy simple addition and place value games, while more advanced students will be challenged to master fractions and negative numbers. (This book is completely new, though several of the games have appeared on my blog. It had originally been planned as the fifth book in my earlier how-to-teach-homeschool-math series, but that first self-publishing experiment ended after book four.)

More Math You Can Play: Games That Build Your Child’s Logical Thinking Skills
Strategy games combine a relief from tedious textbook work with the adventure of creatively logical reasoning. When children play strategy games, they learn to enjoy the challenge of thinking hard. Introduce your family to a variety of games you can enjoy at home or in the car.

Would You Like to Help?

I’m looking for a few more beta-readers to preview the books and let me know about any problems they can find. No writer can see her own work with unbiased eyes, so I need volunteers to help me find weaknesses and places that need revision.

I need to know things like:

• Does the manuscript make sense?
  Which parts seem clearest, and which seem hard to understand? Where do I need to add more details or examples? Does anything feel intimidating or hard to understand?
• Do the chapters flow?
  Does the book seem well organized? Where are there gaps in my reasoning or seemingly-irrational jumps from one topic to another? Do any sections seem boring or off-topic?
• What else would you like to know?
  If we could sit down for an afternoon and chat over a pot of tea, what questions would you like to ask?

If you have writing or editing experience and would like to help with my books, please email me or leave a comment on this post.

What Do You Want in a Math Book?

Finally, if you’re homeschooling, then I’d love to know what sort of math book you think is most helpful. Please vote in my sidebar poll!

[Note: This is the same poll I ran last year, so if you already voted and haven't cleared your cookies recently, it may not let you vote again.]

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

photo by Iain Watson via flickr

Many things in mathematics need to be understood relationally — that is, in relationship to other concepts. But some things just need to be memorized. How do you know which is which? A homeschooling friend pointed out that one thing children definitely need to memorize is the counting sequence from 1-100 and beyond. While there are some patterns that make counting easier, one does just have to memorize which “nonsense sounds” we have attached to each number.

Another sort-of counting that young students should master is subitizing — recognizing at a glance how many items are in a small group. Children do this instinctively, but we can help them develop the skill by playing subitizing games.

[Aside: In writing this blog post, I ran into some nostalgia. Back when we first did these PUFM lessons, my daughter Kitten was only a toddler. I wrote, "I've tried to do lots of counting with my youngest, who hasn't quite gotten beyond, '...eleven, twelve, firteen, firteen, nineteen, seven,...' The numbers tend to start appearing randomly after she gets past 10." Ah, memories.]

Lesson 1.1 Counting

Mathematics is a language and a way of thinking. Early exposure is enormously beneficial.

— Parker & Baldridge
Elementary Mathematics for Teachers

I thoroughly agree with this. Don’t wait until your children reach school age to do mathematics with them. Begin early with oral story problems and hands-on enrichment activities.

Notice the two models for numbers: set model and measurement model. I would call these discrete and continuous quantities, but the idea is the same. Some things come as separate, countable objects, like fish or apples or children. Other things are continuous, such as applesauce or pudding or a child’s height. Be sure, when exposing your children to numbers or making up oral story problems, to include both types.

Our math textbook uses the term whole numbers, but this is not a well-defined math term. Some people mean the whole numbers to include zero, others mean to exclude it, and a few people use the term to mean all of the integers. If it matters (as in a middle-school math contest problem), you would be well-advised to state your definition or to use an unambiguous label, such as non-negative integers.

The Number Line

The number line is the only model that continues to work as elementary school mathematics progresses through whole numbers, fractions, and on to negative, rational, and irrational numbers.

— Parker & Baldridge
Elementary Mathematics for Teachers

A child’s mental number line starts out with an almost logarithmic scale. The small numbers are spread out, but the bigger the numbers get, the closer they crowd together. For very young children, “100″ is almost the same as “infinity”. As our students become more comfortable working with numbers, their mental number line becomes more linear.

A number line is a valuable tool for many mental math calculations. And don’t neglect negative numbers, even with young children.

Numerals in Other Cultures

I don’t think the foreign number systems in the chapter are intended for teaching to young children. Singapore Math doesn’t even teach Roman Numerals, let alone Egyptian ones. These are fun enrichment activities for older children, however, and they’re great for helping teachers understand the importance of place value.

The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.

— Ernest Renan

The main “fact” he means is how place value makes calculation easier, especially with large numbers. But as our textbook authors point out:

There is a price to pay for this convenience: decimal numerals are more abstract than Egyptian numbers. Place value is tricky to learn and causes many problems in the early grades.

— Parker & Baldridge
Elementary Mathematics for Teachers

Note on writing Egyptian numbers: An Egyptian scribe would never write more than four of any shape in a row. They stacked the rows to make bigger numbers. Five was a row of three and a row of two, and I think I remember that the longer row went on top. Six was three and three, seven was four and three, eight was four and four, and nine was three rows of three. These were all fit on the same line with the rest of the number, so the shapes got smaller as they were stacked this way, but it was MUCH easier to read a number at a glance than just putting all the shapes in one long row. So in Homework Set 1, problem 1b: 648 would be written as 3-and-3 squiggles, 4 humps, and 4-and-4 tally marks.

To learn more about Egyptian numerals and calculation, you might enjoy:

• Egyptian Math in Hieroglyphs
• Alex’s Puzzling Papyrus
• Egyptian Math Puzzles
• Another Egyptian Math Puzzle
• The Secret of Egyptian Fractions

Homework Set 1

As you do homework, bear in mind that the goal is not merely for you to do the problems, many of which are not hard. Instead, the goal is to think about problems from the perspective of a teacher. Teachers must be able to identify the key steps in solving a problem, so they can guide and prompt students. They must also be able to give clear, grade-appropriate presentations of solutions.

— Parker & Baldridge
Elementary Mathematics for Teachers

Normally, I will not be posting homework answers. But I enjoy story problems so much, I will make an exception for them. One of the things that most amazed me in Liping Ma’s book was how the Chinese teachers were able to come up with so many different stories to illustrate one calculation. I can see how keeping these models in mind would help that sort of creativity.

For number 4 of the homework, here are my stories:

• [set model] 7 ducks are swimming on a pond. 5 ducks fly in to join them. How many ducks are there now?
• [measurement model] Mom made punch. She mixed 7 cups of ginger ale and 5 cups of pineapple juice. How many cups of punch did Mom make?

For number 6 of the homework: I always let my children do problems like these orally, just reading the number to me rather than writing it out. I don’t require the written work until 4th or 5th grade. My kids take that long (some longer!) to get their fine-motor skills, which are required for handwriting, caught up with their math ability. To require extensive writing before their muscles are ready for it only leads to frustration.

That’s all I have for this section. I look forward to hearing your thoughts!

This post is part of the Profound Understanding of Fundamental Mathematics Series. [Go to the previous post. Go to the next post.]

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• Easy-to-Make Counting Rope
• Elementary Problem Solving: The Early Years
• The “Aha!” Factor
• How to Make Math Cards
• Square One TV: The Mathematics of Love

Profound Understanding of Fundamental Mathematics (PUFM) is a phrase coined by Liping Ma in her landmark book, Knowing and Teaching Elementary Mathematics, to describe the deep, broad, and thorough understanding exhibited by several of the Chinese teachers she interviewed.

You gain PUFM the hard way: by teaching. The Chinese teachers with PUFM were the ones who had taught for years, taught multiple levels, and studied intensively the materials they taught. I doubt there’s any other way to do it. Home schooling is great for developing PUFM because you teach for years and teach multiple levels. The problem is, by the time you really understand the stuff, the kids are grown. Here are a few hints to help speed up the process a little bit:

• Learn as much as you can, wherever you can, even when the topic doesn’t seem to relate to what your kids are studying now. Ask questions.
• Pick up library books on math (510-519 on the Dewey Decimal shelves), some of which you’ll find helpful and some will bore you to distraction. Read the helpful ones and return the others — but try to get through at least 10 pages of a math book before giving up. You’ll learn a lot that way.
• Always look for connections between topics. Think about how addition and subtraction are related, or addition and multiplication, or fractions and division. Think about how the different levels of understanding a topic are related. (Hint: Start by reading the lesson titles as well as the lessons themselves. Lay out a few years’ worth of math books and just read lesson titles, to see how it all goes together.)
• Work on picking up the math vocabulary (distributive property, factors, sum, numerator, etc.) yourself and using it as you teach. Having the right words will help you hold ideas in your mind.

Preview of Coming Attractions

We’re almost ready to dig into our lessons in how to understand and teach elementary mathematics. If you want to join us, these PUFM lessons will follow the textbook Elementary Mathematics for Teachers, by Thomas H. Parker and Scott J. Baldridge.

Do you read the front matter of your textbooks, or do you just dive into Chapter 1? I’m compulsive: If it’s in print, I have to read it. In the Preface of our textbook, the authors make several points that are worth mentioning.

In order to teach elementary mathematics, we need to understand much more than simply how to solve math problems. We need to know:

1. How to present the ideas clearly.
2. How each concept fits into the overall plan of mathematics.
3. And where our students are likely to struggle, so we can ward off trouble in advance (or at least recognize it when it happens).

And what is the overall plan of elementary mathematics? In Grades K-4, our students lay a solid foundation for the four basic arithmetic operations (+/-, ×/÷) and begin working with fractions and decimals. In grades 5-7, our goal is to develop deeper insights into all these topics and to encourage more abstract thinking, taking initial steps toward a proof-based approach to algebra and geometry.

[Measurement, geometry, probability, and statistics are covered in a separate textbook.]

Like a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps toward them at all levels.

— Richard Askey
quoted in Elementary Mathematics for Teachers

How is Math Different?

Math is different from many other topics that our students study, in that it builds on itself — topics naturally grow from each other in logical steps, each one building a stronger foundation for the next. While some of the “big ideas” of math can be studied at a surprisingly young age, it is still necessary to help our students build the foundations of math firmly, one stone at a time.

Also, math is logical. Math is consistent — it makes sense. In everything we teach, our goal must be that our students “Know HOW, and also know WHY.” That is, they must know the steps to use in solving a math problem, and they also must know why those steps make sense. A student (or teacher) who only knows HOW does not understand mathematics.

I like to ask my students, “How did you figure that out?” When students explain the reasoning behind their answer to a math problem, they are giving a mini-proof. And if they get used to doing this in elementary school, they will find it much easier to think through more abstract proofs later on.

Finally, math requires practice to make the basics automatic:

Repetition progressively frees the mind from attention to details, makes facile the total act, shortens the time, and reduces the extent to which consciousness must concern itself with the processes.

— E. B. Huey
quoted in Elementary Mathematics for Teachers

Think like a Teacher

In addition to our textbook, the PUFM course will refer to several books from the Singapore Primary Mathematics series. At times, we will work homework problems from the books, imagining how we would help our students understand. Other assignments will involve “studying the textbook” — that is, reading it carefully in order to see how the mathematics is presented.

As you read through the Singapore math books, think like a teacher. Notice how the ideas are presented in small increments and how the illustrations add to the student’s understanding. What are the key steps in solving each problem? How could you explain these clearly and simply? Remember, the longer the explanation, the more likely it will confuse the student.

Pay special attention to the “concrete → pictorial → abstract” approach. Topics are introduced through hands-on experiences and real-world items, so that children experience the ideas directly. Then the book moves to a pictorial/symbolic approach, using diagrams and models. The book does not rush to a purely-numbers approach to arithmetic, but gives the student plenty of time to build up to such abstraction.

Notice also the many word problems in these books. Never underestimate the importance of word problems. They serve as a sort of mental manipulative, helping students wrap their minds around ideas that would be much more difficult to understand as pure abstraction.

For example, coins, nuts and buttons are clearly distinct and countable and for this reason are convenient to represent relations between whole numbers. The youngest children need some real, tangible tokens, while older ones can imagine them, which is a further step of intellectual development. That is why coin problems are so appropriate in elementary school.

Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be (and in Russia are) common already in middle school. Trains, cars and ships are so widely used in textbooks not because all students are expected to go into the transportation business, but for another, much more sound reason: these objects are easy to imagine moving at constant speeds and because of this are appropriate as reifications of the idea of uniform movement, which, in its turn, can serve as a reification of linear function.

Thus, we can move children further and further on the way of de-reification, that is development of abstract thinking.

— Andrei Toom
Word problems: Applications vs. Mental Manipulatives

This post is part of the Profound Understanding of Fundamental Mathematics Series. [Go to the previous post. Go to the next post.]

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• Number Bonds = Better Understanding
• If It Ain’t Repeated Addition, What Is It?
• Quiz: Those Frustrating Fractions
• Subtracting Mixed Numbers: A Cry for Help
• Percents: The Search for 100%

I am excited to host the 49th Math Teachers at Play Blog Carnival this week! Did you know April is Math Awareness Month? That makes it a great time to learn more about the amazing thing all of these mathematicians are doing!

Since it is the 49th Carnival, here are some fun facts about the number 49 …

Go read the post at Teach Beside Me!

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Profound Understanding of Fundamental Mathematics (PUFM) is a phrase coined by Liping Ma in her landmark book, Knowing and Teaching Elementary Mathematics, to describe the deep, broad, and thorough understanding exhibited by several of the Chinese teachers she interviewed.

The Chinese teachers with PUFM didn’t get it automatically. It grew over many years of teaching several levels of elementary math and of studying their textbooks and teaching materials. They met weekly in teaching research groups to learn from each other’s experience, to find multiple ways to solve problems, and to broaden their mathematical understanding.

More than eight years ago, a group of homeschooling friends started a Yahoo “teaching research group” to discuss math in hope of deepening our own understanding and learning to better help our students. We had a good time, but the busy-ness of everyday life eventually won out. The group has mostly disbanded, though the archives remain. Now I’d like to bring that study to my blog, bit by bit, updated with things I’ve learned in the years since.

Would You Like to Study With Me?

If you want to follow along, these PUFM lessons will use Elementary Mathematics for Teachers, by Thomas H. Parker and Scott J. Baldridge.

You will also need five books from the Singapore Primary Mathematics series: textbooks 3A, 4A, 5A, and 6A, and the 5A workbook. You may get any edition (I have a mixture of the old Primary Maths and the U.S. Edition) — the page numbers may not line up with our text, but just look for the matching topic headings.

We use the Singapore math books to “study teaching materials intensively,” as Liping Ma described. If you don’t have the Singapore texts, then look up the lesson topics in whatever math book your children are studying and analyze how your book presents each concept.

Our PUFM lesson posts will be titled with lesson number and topic, as in the headline above, and you will be able to find them all at the PUFM archive page. Work through the lessons at your own speed, and please post any questions or comments. When one person asks a question we all learn a little more because it makes us think more deeply.

As you study, remember the PUFM goal: “Know HOW, and also know WHY.” Most of us already know the “how” of these topics, but we are constantly learning new things about the “why” (and the “how to explain”). In this, homeschool teachers may have an advantage over our public school counterparts, because we teach our way through the entire curriculum. We can see how the concepts develop from year to year. Watch for that development, and for the deep connections between math concepts.

One thing to keep in mind is that mathematics is a story and that teachers are a story tellers; the teaching and curriculum sequences are there to help you with the structure of the story. If you can bring the story of mathematics to life then you will have a much better chance of reaching all your students.

What is easier: memorizing the story of the three little pigs, or learning to tell the three little pigs story on your own? Which is more satisfying?

— Scott Baldridge
Department of Mathematics, Louisiana State University
comment on the ProfoundUnderstanding Yahoo group

What if You’re Not a Math Person?

Then you’re in good company. Most homeschool teachers are not “math people” — but we can still enjoy the adventure of learning!

Don’t let Liping Ma’s book discourage you. The main thing you want to get from that book is a vision for the goal of PUFM. You want to see what it really means to understand elementary-school math: how the teachers who have a deep understanding of math make connections between topics, and how they build their explanations on a strong foundation of basic concepts.

Then, once you have the goal in mind, you need to work toward it at whatever pace you can. The Parker textbook is an excellent resource for this, and I hope our discussions here will also help. As you learn and as you teach, you will see your ability grow. By this time next year, you will be able to answer your child’s questions from a much stronger conceptual base than you can now — and you’ll still be learning more every year.

This is the first post in the Profound Understanding of Fundamental Mathematics Series. [Go to the next post.]

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• Elementary Problem Solving: Review
• Diagnosis: Math Workbook Syndrome
• Do Your Students Understand Division?
• The Cookie Factory Guide to Long Division
• Reading to Learn Math

via The Math Plane

If you are a homeschooler or classroom teacher, student or independent learner, or anyone else who writes about math — now is the time to send in your favorite blog post for next week’s Math Teachers at Play blog carnival. The deadline is this Friday night, and the carnival will be posted next week at Teach Beside Me.

• Math Teachers at Play Blog Carnival — Submission Form
• MTaP Blog Carnival Information Page

If you haven’t written anything about math lately, here are some ideas to get your creative juices flowing:

• Elementary Concepts: As Liping Ma showed, there is more to understanding and teaching elementary mathematics than we often realize. Do you have a game, activity, or anecdote about teaching math to young students? Please share!
• Arithmetic/Pre-Algebra: This section is for arithmetic lessons and number theory puzzles at the middle-school-and-beyond level. We would love to hear your favorite math club games, numerical investigations, or contest-preparation tips.
• Beginning Algebra and Geometry: Can you explain why we never divide by zero, how to bisect an angle, or what is wrong with distributing the square in the expression ? Struggling students need your help! Share your wisdom about basic algebra and geometry topics here.
• Advanced Math: Like most adults, I have forgotten enough math to fill several textbooks. I’m eager to learn again — but math books can be so-o-o tedious. Can you make upper-level math topics come alive, so they will stick in my (or a student’s) mind?
• Mathematical Recreations: What kind of math do you do, just for the fun of it?
• About Teaching Math: Other teachers’ blogs are an important factor in my continuing education. The more I read about the theory and practice of teaching math, the more I realize how much I have yet to learn. So please, fellow teachers, don’t be shy — share your insights!

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.