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?

New Crutches

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.

Mental-Math-Goal


Photo 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)

Click for details about Let's Play Math bookThis 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.


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Dreams for our Children

Don’t you love this quotation?

For our children, we dream that mathematics…

… makes sense
… is more than just arithmetic
… is joyous
… makes them strong
… is meaningful
… is creative
… is full of fascinating questions
… opens up many paths to solutions
… is friendly
… solves big problems and makes the world better
… is a powerful tool they can master
… is beautiful
… lets them learn in their own ways
… is connected to their lives
… asks “why” and not just “how”
… opens the world

Avoid Hard WorkFrom the upcoming new book Avoid Hard Work by James Tanton and the Natural Math team.

Join the crowdfunding campaign and reserve your copy today!



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FAQ: Trouble with Worksheets

Worksheet problems make my daughter’s brain freeze. Even simple things such as “2 + ___ = 2” confuse her. What can I do?

Can your daughter do math if you put away the worksheet and ask her a real-life problem: “I have a lunch sack. I put two cookies into the sack, and then I give it to you. When you look into the sack, you see two cookies there. Can you tell me what was in the sack at the beginning, before I put my cookies in?”

Or can she solve problems when the answer isn’t zero? Could she figure out how many you started with if she saw four cookies when she looked in the sack?

The idea of having a number for “nothing” can seem strange to young children.

Worksheet Calculations Are Not Math

Can your daughter think mathematically, without calculations?

The symbols on the worksheet are not math. They are just one way of recording how we think about number relationships, and not a very natural way for children. Mathematics is a way of thinking — paying attention to the relationship between ideas and reasoning out connections between them. Encourage your daughter to notice these relationships and wonder about them.

Try watching Christopher Danielson’s video “One is one … or is it?” together, and then see how many different examples of “one” she can find around the house.

The Power of Story

Many kids at this age have a hard time with abstract number math — then their brains will grow up, and they’ll be able to do it. Development varies from one child to another.

When I do worksheets with young children, I turn each equation into a little story. Like the “cookies in a lunch sack” story above.

Sometimes we use blocks or other manipulatives to count on, but often the mental picture of a story is enough. Having something solid to imagine helps the child reason out the relationships between the numbers and symbols.

FAQworksheets


Quote photo: Carl Vilhelm Holsøe ‘Interior with a mother reading aloud to her daughter’ 19th Century. Image from Plum Leaves via Flickr. (CC BY 2.0)

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Playing with Math Shapes

Playing-with-shapesI love it when a plan — or rather, a series of math thoughts — comes together.

On Monday, Emily Grosvenor (author of the Tessalation! picture book) asked me how parents who are insecure in math could help their children learn through play, and I responded with this quote from my Let’s Play Math book:

If you are intimidated by numbers, you can look for patterns of shape and color. Pay attention to how they grow. Talk about what your children notice.

But I wasn’t entirely satisfied with that answer. So many adults have come away from their own school experience thinking math is only numbers. Even with shapes, isn’t it the numbers about them — how many sides, what size of angles, calculate the the area or perimeter — that are important? That’s what school math tends to focus on.

Those of us who are comfortable with math know that there are many more things to notice and think about than just numbers. We know that it’s this noticing, thinking, and wondering that is at the heart of math. And that just playing with shapes can build a powerful foundation for future math learning.

And then yesterday, Malke Rosenfeld posted a beautiful article about a paper manipulative created by Paula Krieg. Which included this video:

The ability to create, and maintain, and manipulate shapes mentally — that’s the goal. Just like kids who can put numbers together in their heads, kids who can rotate, flip, and think of how shapes fit together in their heads have a powerful tool to analyze not only simple shape puzzles, but dividing up an area that’s a more complex room shape … to look at a piece of artwork … or look at a building … For these kids, all the world around becomes a playground to use mathematical ideas.

— Doug Clements
Problem Solving Development: Composing Shapes

Of course, pattern blocks are good for much more than just filling in worksheet pictures. But I love this peek into how a child’s understanding grows, in bits and spurts — without any numbers at all — until the world itself becomes a playground for mathematical ideas.

Want more?

You know what? Children like mathematics. Children see the world mathematically … When we do a puzzle, when we count things, when we see who’s got more, or who’s taller … Play and mathematics are not on opposite sides of the stage.

— Doug Clements
Why Early Childhood is the Right Time to Start Learning Math


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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.

FAQ-Lifelong-Learning

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.

Frayer-Model

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.

Always Ask Why!

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.


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Let’s Play Math FAQs: Introduction

I’ll let you in on a secret about teaching: there is no place in the world where it rolls along smoothly without problems. Only in articles and books can that happen.

—Dr. Ruth Beechick
You Can Teach Your Child Successfully

Learning math is an adventure into the unknown. The ideas we adults take for granted are a wild, unexplored country to our children. Like any traveler in a strange land, they will stumble over rocky places and meet with unexpected detours.

Whenever I visit a parenting forum, I feel compassion for the families who are struggling with math. No other school subject elicits such depths of frustration and despair:

  • I’ve explained until I’m hoarse, and she still doesn’t get it. Help!! I want to pull my hair out.
  • My child is not a mathy person at all. Now he’s convinced that he’s “dumb.”
  • She says she can’t do it. She says she hates math. She says she can’t think. She hits her head and pounds her fists in frustration. I am so tired of fighting over math Every. Single. Day.
  • The problem is not him … It’s me. I am a failure at math.
  • I am sooooo struggling to teach my daughter math. Please, does anybody else deal with this? I will try anything!

Let's Play Math FAQs: Introduction

Yes, There IS Hope!

Solving the problems of math education is not easy. Situations have built up over years, so they will take time to resolve.

But children are resilient, so improvement may not take as long as you fear.

No matter how much your family has struggled, there is hope. If children can get over the “I’m no good at math” mental block, they can learn all of elementary arithmetic in one school year of determined study.

Does that seem unbelievable? Consider Daniel Greenberg’s experience:

Math as a Second Language

If math feels like a strange and dangerous wilderness to your children, you may need an experienced guide to lead you through the rough spots. For arithmetic, try Herb Gross’s Math as a Second Language webpages:

For upper-level math topics, explore Murray Bourne’s Interactive Mathematics pages or take a look at Kalid Azad’s Better Explained site:

About the Let’s Play Math FAQ Series

The questions in this blog post series will be based on actual forum discussions, though I always change the details, removing anything that might identify the families involved.

We’ll look at a variety of struggles with math, such as:

  • Lifelong Learning for Parents
  • Primary Level Problems
  • Middle Grade Mishaps
  • The Agonies of Algebra
  • Gaps and Standardized Testing

The questions will cover a wide range of common frustrations that resonate with anyone who has tried to explain an abstract idea to a confused child. Some questions apply specifically to homeschool math, yet non-homeschooling families can use many of the resources I recommend to supplement their children’s schoolwork or to keep skills sharp over the summer.

Special Cases

In my FAQ post answers, I will assume you are working with children of normal intelligence, facing the mental strengths and weaknesses that are common to us all. The human brain is not designed for working with abstraction, so most people find math difficult.

But some face additional hardship because their minds are unable to process numbers and related concepts. If you suspect one or more of your children may struggle with a learning disability, please have them tested and get advice from someone who can help you learn to deal with their special circumstances.

Auditory or vision problems, undiagnosed food allergies, and a family crisis or other emotional strain may also affect a child’s concentration. Sometimes, the best way to help your children learn math is to let it go and deal with other issues first.

To be continued . . .


Click for details about Let's Play Math bookThis 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.


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Did You Get Your Playful Math Snacks?

Circumferències de FordMy April “Let’s Play Math” newsletter went out early this morning to everyone who signed up for Tabletop Academy Press math updates. This month’s issue celebrated the 200th anniversary of the Farey Sequence.

The Farey Sequence was described in 1816 by English geologist John Farey, who was disparaged by the famous mathematical snob* G.H. Hardy as “at the best an indifferent mathematician.”

“I rather like the idea that the Farey Sequences are named after someone who noticed a pattern and asked a question — and not even the first person to notice the pattern, ask the question, or provide the answer. As math teachers, we teach plenty of indifferent mathematicians who wake up when they experience the joy of discovering something that is new to them, not necessarily new to the whole world.”

— Debra K. Borkovitz,
Farey Fraction Visual Patterns

If you’re a subscriber but didn’t see your newsletter, check your Updates or Promotions tab (in Gmail) or your Spam folder. And to make sure you get all the future newsletters, add “Denise at Tabletop Academy Press” [denise.gaskins @ tabletop academy press .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 next month. Click the link below to sign up today, and we’ll send you our free math and writing booklets, too!
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As a Bonus: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.



* See A Mathematician’s Apology Revisited by W.W. Sawyer.