## Making Sense of Arithmetic

Homeschoolers have an advantage in teaching math: As our students grow, our own understanding of math grows with them because we see how the ideas build on each other.

This is especially true for those of us with large families. We pass through the progression of concepts with each student, and every pass lays down another layer in our own minds.

If you’d like to short-cut that process, check out Graham Fletcher’s Making Sense of Elementary Math video series. He’ll walk you through the topics, showing how manipulatives help build early concepts and gradually give way to abstract calculations.

“Understanding the vertical progression of mathematics is really important in the conceptual development of everyone’s understanding. This whole Making Sense Series has truly forced me to be a better teacher.”

— Graham Fletcher

## FAQ: He Won’t Stop Finger-Counting

“My oldest son has somehow developed the horrid habit of counting on his fingers. We worked on the math facts all summer. He knows the answers in simple form, such as 9 + 4, but if it’s in a bigger problem like 249 + 54, he counts up to add or counts down to subtract, all using fingers. My younger children have no problem with mental math, but he can’t seem to get it. Are there any tips or tricks to stop this?”

Counting on fingers is not a horrid habit, it is a crutch. Please think for a moment about the purpose of crutches. The blasted things are an uncomfortable nuisance, but there are times when you can’t get anywhere without them. And if you need them, it does you no good for a friend to insist you should crawl along on your own.

That is how your son feels right now about his fingers. He is struggling with something his younger siblings find easy, and he can tell that you are frustrated. His confidence is broken, in a cast, and needs time for healing. So he falls back on what he knows he can do, counting up the answer.

Think positive: this means he still believes that math ought to make sense — that to understand what he is doing is more important than to guess at an answer. You want him to value sense-making, because otherwise he will try to memorize his way through middle school and high school math. That is the road to disaster.

“Schools spend a lot of time working with young children to get these facts memorized, but many children aren’t ready for that task yet. They’ll count on their fingers, and may be reprimanded for it.
“What happens when a person becomes embarrassed about counting on their fingers? If they still want to think, they’ll hide it. That’s the better option. The worse option that way too many students choose? They start guessing. When math becomes too incomprehensible, or not living up to someone else’s expectations becomes too painful, many students give up on math, and then they just guess.
“We count on our fingers as part of a thinking process. Perhaps the thing I want to figure can be memorized. But if I haven’t memorized it yet myself, the most efficient way to figure it will likely involve fingers.

—Sue VanHattum
Philosophy

### The Problem of Transfer

What you describe is called the problem of transfer, and it is one of the huge, unsolved problems of education.

We can train someone to do a simple, limited task such as answering flash cards. But how do we get that knowledge to sink in, to become part of the mind, so they can use it in all sorts of different situations?

No one has figured that out.

There is no easy solution. It requires patience, and providing a variety of experiences, and patience, and pointing out connections, and asking the student to think of connections, and lots more patience.

### Some Things to Try

It might help to do fewer math problems in a day, so you can take time to work more deeply on each one. Talk together about the different ways you might solve it. Make it a challenge: “Can we think of three different ways to do it?”

In math, there is never just one way to get a solution. Thinking about alternatives will help your son develop that transfer of skills.

Or pick up some workbooks that target mental math methods. The Mental Math workbook series by Jack Hope and Barbara and Robert Reys will help him master the techniques your younger kids learned without effort. It may still take him longer to do a calculation than what you are used to with the other children, but these books will give him a boost in recognizing the types of mental tools he can use.

Here are a few of my previous blog posts that include mental math tips:

Or perhaps encourage him to keep using his fingers, but to switch to a more efficient system, such as Chisenbop. According to math education expert Jo Boaler, research shows that finger-counting supports mathematical understanding.

### Mental Math: A Battle Worth Fighting

Jumping into mental math is hard for an older child who wasn’t taught that way. I believe it’s a battle worth fighting, because those mental math techniques build understanding of the fundamental properties of numbers.

But the main goal is for him to recognize his options and build flexibility, not to do each calculation as fast as possible.

And be sure he no longer needs those crutches before you try to take them away.

CREDITS: “Stryde Walking To School on his New Crutches” by Jim Larrison and “Silhouette of a boy” by TimOve via Flickr. (CC BY 2.0)

This post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, as are many of the articles in my Let’s Play Math FAQ series.

## New Book: Avoid Hard Work

I’ve loved James Tanton’s How to Be a Math Genius videos for years. He offers great problem-solving tips like:

• Visualize: think of a picture.
• Use common sense to avoid grungy work.
• Engage in intellectual play.
• Think relationally: understanding trumps memorization.
• Be clear on what you don’t know — and comfortable enough to admit it.

Seriously, those are wonderful videos. If you haven’t seen them before, go check them out. Be sure to come back, though, because I’ve just heard some great news.

### Natural Problem-Solving Skills

Tanton has joined up with the NaturalMath.com team of Maria Droujkova, Yelena McManaman, and Ever Salazar to put together a book for parents, teachers, math circle leaders, and anyone else who works with children ages 3–10.

It’s called Avoid Hard Work, and it takes a playful look at ten powerful problem-solving techniques.

### Join the Crowdfunding Campaign

For more details about Avoid Hard Work, including a 7-page pdf sample with tips and puzzles to enjoy, check out the crowdfunding page at Natural Math:

Read the questions and answers. Try the activities with your children. And donate to support playful math education!

## FAQ: Lifelong Learning for Parents

“I’m so tired of being ignorant about math. I can memorize rules and do calculations, but if I miss a step the numbers make no sense at all, and I can’t spot what went wrong. Another struggle I have is keeping everything organized in my mind. When I learn a new concept or strategy, I easily forget it. My son is only a toddler now, but as he grows up, I don’t want to burden him with my own failures. Where should I start?”

As a first step, convince yourself that math is interesting enough to learn on its own merits, because parental guilt will only carry you so far. Start with Steven Strogatz’s “Elements of Math” series from The New York Times, or pick up his book The Joy of x.

As a next step, reassure yourself that elementary math is hard to understand, so it’s not strange that you get confused or don’t know how to explain a topic. Get Liping Ma’s Knowing and Teaching Elementary Mathematics from the library or order a used copy of the first edition. Ma examines what it means to understand math and to clearly explain it to others.

Don’t rush through the book as if it were a novel. There are four open-ended questions, each at the beginning of a chapter, after which several possible answers are analyzed. When you read one of these questions, close the book. Think about how you would answer it yourself. Write out a few notes, explaining your thoughts as clearly as you can. Only then, after you have decided what you would have said, read the rest of that section.

Don’t worry if you can’t understand everything in the book. Come back to it again in a couple of years. You’ll be surprised how much more you learn.

### Books for Parents and Teachers

To build up your own understanding of elementary arithmetic, the Kitchen Table Math series by Chris Wright offers explanations and activities you can try with your children.

If you want more detailed guidance in understanding and explaining each stage of elementary mathematics, you can pick up a textbook designed for teachers in training. I like the Parker & Baldridge Elementary Mathematics for Teachers books and the Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction series. The two series are completely different, but they complement each other well. Check out the sample chapters from the publishers’ websites to see which one you prefer.

Discover more great books on my Living Math Books for Parents and Other Teachers page.

### Focus on Relationships

As you learn, focus on how the math concepts relate to each other. Then the more you learn, the easier you will find it to connect things in your mind and to grasp new ideas.

You might want to keep a math journal about the things you are learning. When you write something down, that helps you remember it, even if you never look back at the journal. But if your mind goes blank and you think, “I know I studied that,” the journal gives you a quick way to review. Make it even easier to flip back through by writing the topic you are studying in the top margin of each page.

When you run into a new vocabulary word, draw a Frayer Model Chart and fill in all the sections. The Frayer Model provides a way to organize information about a new vocabulary word or math concept.

And if you read something that’s particularly helpful, you may want to turn to the back page of your journal and start a quick-reference section.

Find a fellow-learner to encourage you on your journey. Bouncing ideas off a friend is a great way to learn. You might want to join the parents and teachers who are learning math together at the Living Math Forum.

And here is the most important piece of advice I can offer. Your slogan must be the one used by the Chinese teachers Liping Ma interviewed: “Know how, and also know why.”

Always ask why the rules you learn in math work. Don’t stop asking until you find someone who can explain it in a way that makes sense to you. When you struggle with a concept and conquer it, it will make you free. You don’t have to be afraid of it anymore.

Know how, and also know why.

This post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, as are many of the articles in my Let’s Play Math FAQ series.

## Multiplication Is Not Repeated Addition: Update

[Photo “Micah and Multiplication” by notnef via Flickr (CC-BY 2.0).]

Some Internet topics are evergreen. I noticed that my old Multiplication Is Not Repeated Addition post has been getting new traffic lately, so I read through the article again. And realized that, even after all those words, I still had more to say.

So I added the following update to clarify what seemed to me the most important point.

I’d love to hear your thoughts! The comment section is open down below . . .

## Language Does Matter

Multiplication: multiplier × multiplicand = product. The multiplier and multiplicand have different names, even though many of us have trouble remembering which is which.

• multiplier = “how many or how much”
• multiplicand = the size of the “unit” or “group”

Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

## A New Type of Number

In multiplication, we introduce a totally new type of number: the multiplicand. A strange, new concept sits at the heart of multiplication, something students have never seen before.

The multiplicand is a this-per-that ratio.

A ratio is a not a counting number, but something new, much more abstract than anything the students have seen up to this point.

A ratio is a relationship number.

In addition and subtraction, numbers count how much stuff you have. If you get more stuff, the numbers get bigger. If you lose some of the stuff, the numbers get smaller. Numbers measure the amount of cookies, horses, dollars, gasoline, or whatever.

The multiplicand doesn’t count the number of dollars or measure the volume of gasoline. It tells the relationship between them, the dollars per gallon, which stays the same whether you buy a lot or a little.

By telling our students that “multiplication is repeated addition,” we dismiss the importance of the multiplicand. But until our students wrestle with and come to understand the concept of ratio, they can never understand multiplication.

## For Further Investigation

If you’re interested in digging deeper into how children learn addition and multiplication, I highly recommend Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.

To learn about modeling multiplication problems with bar diagrams, check out the Mad Scientist’s Ray Gun model of multiplication:

And here is an example of the multiplication bar diagram in action:

## Understanding Math: Conclusion

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?; Understanding Math, Part 4: Area of a Rectangle; Understanding Math, Part 5: Multiplying Fractions; and Understanding Math, Part 6: Algebraic Multiplication.

Earlier in this blog post series, I gave you three middle-school math rules. But by exploring the concept of rectangular area as a model of multiplication, we discovered that in a way they were all the same.

The rules are not arbitrary, handed down from a mathematical Mount Olympus. They are three expressions of a single basic question: what does it mean to measure area?

There was only one rule, one foundational pattern that tied all these topics together in a mathematical web.

Many children want to learn math instrumentally, as a tool for getting answers. They prefer the simplicity of memorizing rules to the more difficult task of making sense of new ideas. Being young, they are by nature short-term thinkers. They beg, “Just tell me what to do.”

But if we want our children to truly understand mathematics, we need to resist such shortcuts. We must take time to explore mathematics as a world of ideas that connect and relate to each other in many ways. And we need to show children how to reason about these interconnected concepts, so they can use them to think their way through an ever-expanding variety of problems.

Our kids can only see the short term. If we adults hope to help them learn math, our primary challenge is to guard against viewing the mastery of facts and procedures as an end in itself. We must never fall into thinking that the point of studying something is just to get the right answers.

We understand this in other school subjects. Nobody imagines that the point of reading is to answer comprehension questions. We know that there is more to learning history than winning a game of Trivial Pursuit. But when it comes to math, too many parents (and far too many politicians) act as though the goal of our children’s education is to produce high scores on a standardized test.

### What If I Don’t Understand Math?

If you grew up (as I did) thinking of math as a tool, the instrumental approach may feel natural to you. The idea of math as a cohesive system may feel intimidating. How can we parents help our children learn math, if we never understood it this way ourselves?

Don’t panic. Changing our worldview is never easy, yet even parents who suffer from math anxiety can learn to enjoy math with their children. All it takes is a bit of self-discipline and the willingness to try.

You don’t have to know all the answers. In fact, many people have found the same thing that Christopher Danielson described in his blog post “Let the children play” — the more we adults tell about a topic, the less our children learn. With the best of intentions we provide information, but we unwittingly kill their curiosity.

• If you’re afraid of math, be careful to never let a discouraging word pass your lips. Try calling upon your acting skills to pretend that math is the most exciting topic in the world.
• Encourage your children to notice the math all around them.
• Search out opportunities to discuss numbers, shapes, symmetry, and patterns with your kids.
• Investigate, experiment, estimate, explore, measure — and talk about it all.

### The Science of Patterns

Patterns are so important that American mathematician Lynn Arthur Steen defined mathematics as the science of patterns.

As biology is the science of life and physics the science of energy and matter, so mathematics is the science of patterns. We live in an environment steeped in patterns — patterns of numbers and space, of science and art, of computation and imagination. Patterns permeate the learning of mathematics, beginning when children learn the rhythm of counting and continuing through times tables all the way to fractals and binomial coefficients.

— Lynn Arthur Steen, 1998
Reflections on Mathematical Patterns, Relationships, and Functions

If you are intimidated by numbers, you can look for patterns of shape and color. Pay attention to how they grow, and talk about what your children notice. For example, some patterns repeat exactly, while other patterns change as they go (small, smaller, smallest, or loud, louder, loudest).

Nature often forms fractal-like patterns: the puffy round-upon-roundness of cumulus clouds or broccoli, or the branch-upon-branchiness of a shrub or river delta. Children can learn to recognize these, not as a homework exercise but because they are interesting.

### Math the Mathematician’s Way

Here is the secret solution to the crisis of math education: we adults need to learn how to think like mathematicians.

For more on what it means to think about math the mathematician’s way, check out my Homeschooling with Math Anxiety blog post series:

As we cultivate these characteristics, we will help our children to recognize and learn true mathematics.

CREDITS: “Frabjous 01” photo (top) by Windell Oskay and “Back to School” photo (middle) by Phil Roeder via Flicker (CC BY 2.0). “I Can Solve Problems” poster by Nicole Ricca via Teachers Pay Teachers. This is the final 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.