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

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.

Memorizing the Math Facts

Central City Times Tables[Photo by dsb nola via flickr.]

The most effective and powerful way I’ve found to commit math facts to memory is to try to understand why they’re true in as many ways as possible. It’s a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.

Actually, a close friend of mine describes this same experience: he couldn’t learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can’t remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.

Ben Blum-Smith
Comment on Math Mama’s post What must be memorized?

The entire discussion (article and comments) is well worth reading:

You may also enjoy:

2016 Mathematics Game

[Feature photo above from the public domain, and title background (below) by frankieleon (CC BY 2.0) via Flickr.]

2016-math-game

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.

Math Forum Year Game Site

Rules of the Game

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.

My Special Variations on the Rules

  • 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.

Click here to continue reading.

Teaching the Standard Algorithms

[Feature photo above by Samuel Mann, Analytical Engine photo below by Roͬͬ͠͠͡͠͠͠͠͠͠͠͠sͬͬ͠͠͠͠͠͠͠͠͠aͬͬ͠͠͠͠͠͠͠ Menkman, both (CC BY 2.0) via Flickr.]

Babbage's Analytical Engine

An algorithm is a set of steps to follow that produce a certain result. Follow the rules carefully, and you will automatically get the correct answer. No thinking required — even a machine can do it.

This photo shows one section of the first true computer, Charles Babbage’s Analytical Engine. Using a clever arrangement of gears, levers, and switches, the machine could crank out the answer to almost any arithmetic problem. Rather, it would have been able to do so, if Babbage had ever finished building the monster.

One of the biggest arguments surrounding the Common Core State Standards in math is when and how to teach the standard algorithms. But this argument is not new. It goes back at least to the late 19th century.

Here is a passage from a book that helped shape my teaching style, way back when I began homeschooling in the 1980s…

Ruth Beechick on Teaching Abstract Notation

“Understanding this item is the key to choosing your strategy for the early years of arithmetic teaching. The question is: Should you teach abstract notation as early as the child can learn it, or should you use the time, instead, to teach in greater depth in the mental image mode?

Beechick-EasyStartArithmetic

“Abstract notation includes writing out a column of numbers to add, and writing one number under another before subtracting it. The digits and signs used are symbols. The position of the numbers is an arbitrary decision of society. They are conventions that adult, abstract thinkers use as a kind of shorthand to speed up our thinking.

“When we teach these to children, we must realize that we simply are introducing them to our abstract tools. We are not suddenly turning children into abstract thinkers. And the danger of starting too early and pushing this kind of work is that we will spend an inordinate amount of time with it. We will be teaching the importance of making straight columns, writing numbers in certain places, and other trivial matters. By calling them trivial, we don’t mean that they are unnecessary. But they are small matters compared to real arithmetic thinking.

“If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more. And when she does get to abstractions, she will understand them better. She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.”

— Ruth Beechick
An Easy Start in Arithmetic (Grades K-3)
(emphasis mine)


howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.

2015 Mathematics Game

[Feature photo above by Scott Lewis and title background (right) by Carol VanHook, both (CC BY 2.0) via Flickr.]

2015YearGame

Did you know that playing games is one of the Top 10 Ways To Improve Your Brain Fitness? So slip into your workout clothes and pump up those mental muscles with the Annual Mathematics Year Game Extravaganza!

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.

Math Forum Year Game Site

Rules of the Game

Use the digits in the year 2015 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 2-0-1-5 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.

My Special Variations on the Rules

  • 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 NOT use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. Math Forum allows these, but I’ve decided I prefer my arithmetic straight.

Click here to continue reading.

Fraction Game: My Closest Neighbor

[Feature photo above by Jim Larrison, and antique playing cards below by Marcee Duggar, via Flickr (CC BY 2.0).]

I missed out on the adventures at Twitter Math Camp, but I’m having a great time working through the blog posts about it. I prefer it this way — slow reading is more my speed. Chris at A Sea of Math posted a wonderful game based on one of the TMC workshops. Here is my variation.

Math concepts: comparing fractions, equivalent fractions, benchmark numbers, strategic thinking.

Players: two to four.

Equipment: two players need one deck of playing cards, three or four players need a double deck.

Continue reading Fraction Game: My Closest Neighbor

World Maths Day 2013: Register Now

It’s time to register for World Maths Day, which will take place on March 6, 2013. Last year, more than five million students from all around the world combined to correctly answer nearly 500 million math problems.

Would you like to help break the record this year? Register now so you can practice in advance!

About World Maths Day

  • Play with students from schools all around the world. Individuals and homeschoolers are welcome, too.
  • The competition is designed for ages 4-18 and all ability levels. Teachers, parents and media can also register and play.
  • It’s simple to register and participate. Start practicing as soon as you register.
  • And best of all, it’s absolutely free.

Continue reading World Maths Day 2013: Register Now

PUFM 1.5 Multiplication, Part 2

Poster by Maria Droujkova of NaturalMath.com. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

Multiplication is taught and explained using three models. Again, it is important for understanding that students see all three models early and often, and learn to use them when solving word problems.

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

I hope you are playing the Tell Me a (Math) Story game often, making up word problems for your children and encouraging them to make up some for you. As you play, don’t fall into a rut: Keep the multiplication models from our lesson in mind and use them all. For even greater variety, use the Multiplication Models at NaturalMath.com (or buy the poster) to create your word problems.

Continue reading PUFM 1.5 Multiplication, Part 2

Build Mathematical Skills by Delaying Arithmetic, Part 4

To my fellow homeschoolers,

While Benezet originally sought to build his students’ reasoning powers by delaying formal arithmetic until seventh grade, pressure from “the deeply rooted prejudices of the educated portion of our citizens” forced a compromise. Students began to learn the traditional methods of arithmetic in sixth grade, but still the teachers focused as much as possible on mental math and the development of thinking strategies.

Notice how waiting until the children were developmentally ready made the work more efficient. Benezet’s students studied arithmetic for only 20-30 minutes per day. In a similar modern-day experiment, Daniel Greenberg of Sudbury School discovered the same thing: Students who are ready to learn can master arithmetic quickly!

Grade VI

[20 to 25 minutes a day]

At this grade formal work in arithmetic begins. Strayer-Upton Arithmetic, book III, is used as a basis.

[Note: Essentials of Arithmetic by George Wentworth and David Eugene Smith is available free and would probably work as a substitute.]

The processes of addition, subtraction, multiplication, and division are taught.

Care is taken to avoid purely mechanical drill. Children are made to understand the reason for the processes which they use. This is especially true in the case of subtraction.

Problems involving long numbers which would confuse them are avoided. Accuracy is insisted upon from the outset at the expense of speed or the covering of ground, and where possible the processes are mental rather than written.

Before starting on a problem in any one of these four fundamental processes, the children are asked to estimate or guess about what the answer will be and they check their final result by this preliminary figure. The teacher is careful not to let the teaching of arithmetic degenerate into mechanical manipulation without thought.

Fractions and mixed numbers are taught in this grade. Again care is taken not to confuse the thought of the children by giving them problems which are too involved and complicated.

Multiplication tables and tables of denominate numbers, hitherto learned, are reviewed.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

Continue reading Build Mathematical Skills by Delaying Arithmetic, Part 4

Build Mathematical Skills by Delaying Arithmetic, Part 3

To my fellow homeschoolers,

How can our children learn mathematics if we delay teaching formal arithmetic rules? Ask your librarian to help you find some of the wonderful living books about math. Math picture books are great for elementary students. Check your library for the Time-Life “I Love Math” books or the “Young Math Book” series. You’ll be amazed at the advanced topics your children can understand!

Benezet’s students explored their world through measurement, estimation, and mental math. Check out my PUFM Series for mental math thinking strategies that build your child’s understanding of number patterns and relationships.

Grade IV

Still there is no formal instruction in arithmetic.

By means of foot rules and yard sticks, the children are taught the meaning of inch, foot, and yard. They are given much practise in estimating the lengths of various objects in inches, feet, or yards. Each member of the class, for example, is asked to set down on paper his estimate of the height of a certain child, or the width of a window, or the length of the room, and then these estimates are checked by actual measurement.

The children are taught to read the thermometer and are given the significance of 32 degrees, 98.6 degrees, and 212 degrees.

They are introduced to the terms “square inch,” “square foot,” and “square yard” as units of surface measure.

With toy money [or real coins, if available] they are given some practise in making change, in denominations of 5’s only.

All of this work is done mentally. Any problem in making change which cannot be solved without putting figures on paper or on the blackboard is too difficult and is deferred until the children are older.

Toward the end of the year the children will have done a great deal of work in estimating areas, distances, etc., and in checking their estimates by subsequent measuring. The terms “half mile,” “quarter mile,” and “mile” are taught and the children are given an idea of how far these different distances are by actual comparisons or distances measured by automobile speedometer.

The table of time, involving seconds, minutes, and days, is taught before the end of the year. Relation of pounds and ounces is also taught.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

Continue reading Build Mathematical Skills by Delaying Arithmetic, Part 3