Hello!
In this article, I want to talk about MATH WORD PROBLEMS. They indeed give many children "problems", but that doesn't have to be!
Here's the outline of the article:
Have you ever noticed this kind of "recipe" for math lessons in many math books?
In other words, the word problems are usually in the END of the lesson, and just a few. But worse... if the lesson is about topic X, then the word problems are usually about the topic X too!
Children might be learning about multidigit multiplication, or subtraction, or dividing decimals. After the calculation exercises come some word problems, which oddly enough are solved by using the exact operation just practiced!
Worse YET... typically the word problems in elementary grades only have TWO numbers in them. So, even if you didn't understand a word in the problem, you might be able to do it. (The misguide to problem solving at the end of this article refers to this also.)
Just try: the following madeup problem is in FINNISH... and let's say it is found within a long division lesson. I assume now that you do NOT know Finnish — but can you solve it?
Drag your mouse over the white space below to see the translation (highlight it).
My thought is that over the years, when children are exposed to such lessons over and over again, they figure out that it's mentally less demanding to NOT even READ the problem too carefully! Why bother? Just take the two numbers, divide (or multiply, or add, or subtract) them, and that's it. They "learn" (intelligently) this unspoken rule:
It is no wonder that our students are not able to solve anything that's different from that routine!
How can you avoid this terrible situation?
Children simply need to be given good problems to solve. That way, they will get used to THINKING.
These good word problems should advance in difficulty over the grades, and foster children's logical thinking. ONEstep problems are adequate for 1st and 2nd grades, and then here and there mixed in with other types of problems.
In early elementary grades, one needs to mix up the (often onestep) word problems so that not all of them are solved by the operation just studied. I have done that often in my Math Mammoth books.
But children need to start solving MULTIstep problems as soon as they can, including even in 1st grade. (And yes, Math Mammoth books do include some twostep problems starting in 1st grade materials.)
For more word problems than in your current math curriculum, check the list of problem solving resources at the end of this article.
What about Math Mammoth and word problems? you might ask. I don't currently have a single book with only word problems (though we're hoping to make one, one day). Instead, word problems are simply present in ALL of the Math Mammoth materials.
Here's how to help a student with problem solving and word problems using Math Mammoth. Such a student could go through all of the Math Mammoth Light Blue or Blue series, starting from grade 1 materials, and do ONLY the word problems and related lessons.
Next, let's take a look at some PROBLEMS. First, look at this example problem from a Russian fourth grade book:
A similar problem is included in the 5th grade Singapore textbook:
Now, these are not anything spectacular. You can solve these types of problems for example by using a bar model (image from Math Mammoth grade 5 Light Blue series):
Or, you can solve them with arithmetic: simply take away the given difference from the total, then divide the remaining amount evenly, and lastly add the given difference to one person's part.
Yet in the U.S., these kind of problems are generally introduced in Algebra 1 — in ninth grade(!), AND they are only solved using algebraic means.
Here is another example, of which I remember feeling aghast, found in a modern U.S. algebra textbook:
Thirdgrade children 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 a bigger number and not 42. (Try 3422. ) Don't such simple problems in algebra books just encourage students to forget common sense and simple arithmetic?
Another example, a 3rd grade problem from Russia:
You can draw a boy and a girl, draw two pockets for the boy, and one pocket for the girl. This visual representation easily solves the problem.
Here is an example of a Russian problem for grades 68:
(You can solve this in many different ways — and that is a often the case withe good word problems.)
So, once again, the only true way of learning to solve word problems is .... TO SOLVE LOTS OF THEM. They don't have to be reallife or involve awkward numbers (such as occur in real life). Realistic, complex problems might be good for a "spice", but not for the "main course". "Fantastic" (unreal) problems are fine.
One of my girls used to cherish the word problems in my Math Mammoth books MORE than the other kinds of problems. And the general consensus of the feedback I hear from Math Mammoth users is that once children get used to solving a variety of word problems, it's no longer a problem!
Some children, if not used to solving word problems, may need to start from the beginning — with a collection of simple onestep word problems (mixed up). But that's okay. They will learn!
Lastly, a joke by Lynn Nordstrom:
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 16 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.
Math Stars
Free newsletters with the purpose of challenging students with good problems that inspire curiosity. Math Stars are in sets for Grades 18 and include commentaries for teachers.
Problem Solving Decks
A free collection of problems for grades 18, originally published by the North Carolina State Board of Education.
Math Puzzles and Brainteasers, Grades 68
A book with over 300 delightful puzzles and problems that teach math and problem solving.
Ray's Arithmetic
Ray's Arithmetic is an old math book, full of word problems, starting from the simplest. It basically uses word problems to teach arithmetic. It is available FREE online.
3rd Grade Math Word Problems Worksheets
Look especially for the MIXED word problem worksheets in this collection. The site offers word problem worksheets for all of the grades 15 (look in the menu).
Word Problems in Russia and America
An article by Andrei Toom. Contains some good example problems from Russia!
StepbyStep Problem Solving, Grade 4
A problemsolving workbook that provides stepbystep instruction, problem pages at varied levels of difficulty, a math strategies overview, and a complete answer key. The book utilizes bar models just like Math Mammoth and Singapore Math. Other grades available also.
Mixed operations word problem worksheets
These are all onestep problems but useful for grades 34 students who need such practice.
Challenge Math For the Elementary and Middle School Student
For children in grades 47. Contains 20 chapters with instruction and problems at three levels of difficulty.
Problem solving resources
A long list of yet MORE problem solving resources! Lots of these are totally free!
By Maria Miller
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