### Word problems in Russia and in America

Recently I had the opportunity (HT Let's Play Math) to read an article "Word Problems in Russia and America" by Andrei Toom. It is an extended version of a talk at the Meeting of the Swedish Mathematical Society in June, 2005.

The article was comparing word problems in Russian and U.S. math books. As you can guess, the former were far more advanced than the latter.

I want to highlight a few things from the article. You're very welcome to download and read it too from the above link.

A problem from a Russian fourth grade book:
An ancient artist drew scenes of hunting on the walls of a cave, including 43 figures of animals and people. There were 17 more figures of animals than people. How many figures of people did the artist draw?

A similar problem is included in the 5th grade Singapore textbook:
Raju and Samy shared \$410 between them. Raju received \$100 more than Samy. How much money did Samy receive?

Now, these are not anything spectacular. You can solve them for example by taking away the difference of 17 or \$100 from the total, and then dividing the remaining amount evenly:

\$410 − \$100 = \$310, and then divide \$310 evenly to Raju and Samy, which gives \$155 to each. Give Raju the \$100. So Samy had \$155 and Raju had \$255.

A far as the figures, 43 − 17 = 26, and then divide that evenly: 13 and 13. So 13 people and 30 animal figures.

BUT in the U.S., these kind of problems are generally introduced in Algebra 1 - ninth grade, AND they are only solved using algebraic means.

I remember being aghast of a word problem in a modern U.S. algebra textbook:

"Find two consecutive numbers whose product is 42."

Third-grade kids should know multiplication well enough to quickly find that 6 and 7 fit the problem! Why use a "backhoe" (algebra) for a problem you can solve using a "small spade" (simple multiplication)!

I know some will argue and say, "Its purpose is to learn to set up an equation." But for that purpose I would use some some more difficult number and not 42. Doesn't using such simple problems in algebra books just encourage students to forget common sense and simple arithmetic?

(BTW, no matter what number you'd use − "Find two consecutive numbers whose product is 13,806" − I'd just take the square root and find the neighboring integers, and check.)

And this is what Toom also wonders greatly: why do U.S. instructors not teach children to solve many-step word problems using arithmetic only? It is as if the more complex word problems are extinct in the standard textbooks until algebra, and word problems in elementary grades are mostly reduced to one or two-step simple problems.

(I've written about that before, how the word problems found in lesson X are always solved using the operation taught in lesson X.)

Another example, a 3rd grade problem from Russia:

A boy and a girl collected 24 nuts. The boy collected two times as many nuts as the girl. How many did each collect?

You could draw a boy and a girl, and draw two pockets for the boy, and one pocket for the girl. This visual representation easily solves the problem.

Here's an example of a Russian problem for grades 6-8:
An ancient problem. A flying goose met a flock of geese in the air and said: "Hello, hundred geese!" The leader of the flock answered to him: "There is not a hundred of us. If there were as many of us as there are and as many more and half many more and quarter as many more and you, goose, also flied with us, then there would be hundred of us." How many geese were there in the flock?

(I personally would tend to set up an equation for this one but it can be done without algebra too.)

Toom talks about how "real life" word problems are emphasized in America, and "fantastic" problems that could not occur in reality are devalued. For example, a problem such as
"Sally is five years older than her brother Bill. Four years from now, she will
be twice as old as Bill will be then. How old is Sally now?
" may be deemed unfit since nobody would want to know such in real life.

However, like Toom argues, such problems do serve a purpose: that of developing children's logical and abstract thinking and mental discipline. One-step word problems won't do that!

In the U.S. word problems are perceived as "scary"; both students AND teachers tend to be afraid of them, and teachers might even omit solving them. This doesn't help, of course.

Here's a joke that Toom had included in his article, by Lynn Nordstrom:

### "Student's Misguide to Problem Solving":

• Rule 1: If at all possible, avoid reading the problem.Reading the problem only consumes time and causes confusion.
• Rule 2: Extract the numbers from the problem in the order they appear. Be on the watch for numbers written in words.
• Rule 3: If rule 2 yields three or more numbers, the best bet is adding them together.
• Rule 4: If there are only 2 numbers which are approximately the same size, then subtraction should give the best results.
• Rule 5: If there are only two numbers and one is much smaller than the other, then divide if it goes evenly -- otherwise multiply.
• Rule 6: If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem.
• Rule 7: If the rules 1-6 don't seem to work, make one last desperate attempt. Take the set of numbers found by rule 2 and perform about two pages of random operations using these numbers. You should circle about five or six answers on each page just in case one of them happens to be the answer. You might get some partial credit for trying hard.

I hope your students do not fit the above joke.

In my books, I've tried to avoid problems that would lead children to the above scenario. I do not claim to be perfect in this; I feel I have lots to learn. But I will keep striving to make problems that do require many steps and that do not "dumb down" our children, but that progressively get more difficult as school years go by.

mathmom said…
Just wanted to say, "Great Post!"

(Also, the joke formatting is wonky in my browser, so I can't see all the words on the right)
A boy and a girl collected 24 nuts. The boy collected two times more nuts than the girl. How many did each collect?

Regarding the above problem, since the boy collected two times MORE nuts than the girl, he actually collected three times AS MUCH as her. Hence, the boy collected 18 nuts and the girl 6.

Many verbal problems (including coin, age and distance problems) can be found in the Hands-On Equations Verbal Problems Book, which is used with the Hands-On Equations program. The Making Algebra Child's Play workshop for teachers shows them how they can introduce significant algebraic concepts at the grade school level, including solving equations such as 4x + 3 = 3x + 9, and applying these skills to solving verbal problems.

Here is one example taken from the book:

Tom buys 2 packs of football cards to add to the 5 cards a friend gave him. His mother then gives him 3 packs as a present. Now he has as many cards as Billy who owns 2 packs and 20 loose cards. If all the packs have the same number of cards, how many cards are in each pack?

The solution to problems such as this one using the game pieces (pawns and numbered cubes) may be found at the website www.borenson.com.

Thanks for referencing the book, Word Problems in Russia and America.

Henry Borenson, Ed.D.
Anonymous said…
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Maria Miller said…
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We have just begun an Algebra Verbal Problem of the Week for students in grades 3 and up. The first problem posted is noted below:

"Theresa could purchase four small gifts and a \$3 doll for the same price as three of the same small gifts and one \$5 doll. What was the price of each of the small gifts?"

This problem is designed for students already using Hands-On Equations. Otherwise, it would require a creative child with algebraic reasoning skills to solve the problem.

The posting of the problem can be found on:
http://hands-on-equations.blogspot.
Dave Marain said…
Maria,
Thanks to you, mathmom and others I have confidence that our students will one day return to "real" problem-solving as opposed to "real-world" problem solving! Keep up the great work and thanks for the link.

Encouraging elementary students to use their innate insight complemented by visual and physical representations is crucial for developing number and algebra sense long before the formalism of algebra is needed. I strongly agree with all of hte comments expressed in this post. Children need to be challenged with more sophisticated problems that develop insight (yes, one can develop it!) and computational fluency. IMO,"number sense" without numerical skill is analogous to a conceptual understanding of how to shoot a basketball without being able to put the ball in the basket...

Dave Marain
MathNotations
Anonymous said…
hey that is so cool what you did!!!
I think that the fact that the problem of finding two consecutive numbers admits a solution (taking the integer part of the square root) need not diminish the value of the problem, even if it was not the intended solution. This is just a perfect opportunity to illustrate the idea that most problems do not have a canonical solution, and even if they do, the canonical solution need not be the best or the shortest or the most intuitive. It's up to the teacher to exploit (or miss) such an opportunity.

I have written in a similar vein concerning a different word problem.