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.

Continue reading FAQ: He Won’t Stop Finger-Counting

Multiplication Is Not Repeated Addition: Update

Multiplication Is Not Repeated Addition: Update[Photo “Micah and Multiplication” by notnef via Flickr (CC BY 2.0, text added).]

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

Addition: addend + addend = sum. The addends are interchangeable. This is represented by the fact that they have the same name.

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 fully understand multiplication.

For Further Investigation

nunes-doingmathIf 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:

New Hundred Chart Game: Odd-Even-Prime Race

Counting all the fractional variations, my massive blog post 30+ Things to Do with a Hundred Chart now offers nearly forty ideas for playing around with numbers from preschool to prealgebra.

Here is the newest entry, a variation on #10, the “Race to 100” game:

(11.5) Play “Odd-‌Even-‌Prime Race.″ Roll two dice. If your token is starting on an odd number, move that many spaces forward. From an even number (except 2), move backward — but never lower than the first square. If you are starting on a prime number (including 2), you may choose to either add or multiply the dice and move that many spaces forward. The first person to reach or pass 100 wins the game.
[Hat tip: Ali Adams in a comment on another post.]

And here’s a question for your students:

  • If you’re sitting on a prime number, wouldn’t you always want to multiply the dice to move farther up the board? Doesn’t multiplying always make the number bigger?

 
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This blog is reader-supported.

If you’d like to help fund the blog on an on-going basis, then please join me on Patreon for mathy inspiration, tips, and an ever-growing archive of printable activities.

If you liked this post, and want to show your one-time appreciation, the place to do that is PayPal: paypal.me/DeniseGaskinsMath. If you go that route, please include your email address in the notes section, so I can say thank you.

Which I am going to say right now. Thank you!

“New Hundred Chart Game: Odd-Even-Prime Race” copyright © 2015 by Denise Gaskins. Image at the top of the post copyright © geishaboy500 (CC BY 2.0).

Noticing Fractions in a Sidewalk

fraction-circle

My daughters didn’t want to admit to knowing me, when I stopped to take a picture of the sidewalk along a back street during our trip to Jeju. But aren’t those some wonderful fractions?

What do you see? What do you wonder?

Here is one of the relationships I noticed in the outer ring:

\frac{4 \frac {2}{2}}{20} = \frac {1}{4}

sidewalk

And this one’s a little trickier:

\frac{1 \frac {1}{2}}{12} = \frac {1}{8}

Can you find it in the picture?

Each square of the sidewalk is made from four smaller tiles, about 25 cm square, cut from lava rock. Some of the sidewalk tiles are cut from mostly-smooth rock, some bubbly, and some half-n-half.

I wonder how far we could go before we had to repeat a circle pattern?

Continue reading Noticing Fractions in a Sidewalk

Socks Are Like Pants, Cats Are Like Dogs

https://youtu.be/8cSaDlG_2f8&rel=0

Support This New Book from Natural Math

Socks Are Like Pants, Cats Are Like Dogs by Malke Rosenfeld and Gordon Hamilton is filled with a diverse collection of math games, puzzles, and activities exploring the mathematics of choosing, identifying and sorting. The activities are easy to start and require little preparation.

The publisher’s crowdfunding goal is $4,000. The book is almost ready to go to press, and I can hardly wait to see it!

Review Game: Once Through the Deck

Math Concepts: basic facts of addition, multiplication.
Players: one.
Equipment: one deck of math cards (poker- or bridge-style playing cards with the face cards and jokers removed).

The best way to practice the math facts is through the give-and-take of conversation, orally quizzing each other and talking about how you might figure the answers out. But occasionally your child may want a simple, solitaire method for review.

Continue reading Review Game: Once Through the Deck

Math Games with Factors, Multiples, and Prime Numbers

Students can explore prime and non-prime numbers with these free favorite classroom games:

For $15-20 you can buy a downloadable file of the beautiful, colorful, mathematical board game Prime Climb. Or pick up the full Prime Climb game box at Amazon.

Or you can try the following game by retired Canadian education professor Jerry Ameis:

Factor Finding Game

FactorFindingGame

Math Concepts: multiples, factors, composite numbers, and primes.
Players: only two.
Equipment: pair of 6-sided dice, 10 squares each of two different colors construction paper, and the game board (click the image to print it, or copy by hand).

On your turn, roll the dice and make a 2-digit number. Use one of your colored squares to mark a position on the game board. You can only mark one square per turn.

  • If your 2-digit number is prime, cover a PRIME square.
  • If any of the numbers showing are factors of your 2-digit number, cover one of them.
  • BUT if there’s no square available that matches your number, you lose your turn.

The first player to get three squares in a row (horizontal, vertical, or diagonal) wins. Or for a harder challenge, try for four in a row.

 
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This game was featured in the Math Teachers At Play (MTaP) math education blog carnival: MTaP #79. Hat tip: Jimmie Lanley.

This blog is reader-supported.

If you’d like to help fund the blog on an on-going basis, then please join me on Patreon for mathy inspiration, tips, and an ever-growing archive of printable activities.

If you liked this post, and want to show your one-time appreciation, the place to do that is PayPal: paypal.me/DeniseGaskinsMath. If you go that route, please include your email address in the notes section, so I can say thank you.

Which I am going to say right now. Thank you!

“Math Games with Factors, Multiples, and Prime Numbers” copyright © 2015 by Denise Gaskins. Image at the top of the post copyright © Jimmie via flickr (CC BY 2.0).

The Math Student’s Manifesto

[Feature photo above by Texas A&M University (CC BY 2.0) via Flickr.]

Note to Readers: Please help me improve this list! Add your suggestions or additions in the comment section below…

What does it mean to think like a mathematician? From the very beginning of my education, I can do these things to some degree. And I am always learning to do them better.

(1) I can make sense of problems, and I never give up.

  • I always think about what a math problem means. I consider how the numbers are related, and I imagine what the answer might look like.
  • I remember similar problems I’ve done before. Or I make up similar problems with smaller numbers or simpler shapes, to see how they work.
  • I often use a drawing or sketch to help me think about a problem. Sometimes I even build a physical model of the situation.
  • I like to compare my approach to the problem with other people and hear how they did it differently.

Continue reading The Math Student’s Manifesto

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)

2015 Mathematics Game

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

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.