Elementary Problem Solving: The Tools
[This article begins a series rescued from my old blog. Moving has been a long process, but I'm finally unpacking the last cardboard box! To read the entire series, click here: elementary problem solving series.]
Most young students solve story problems by the flash of insight method: When they read the problem, they know almost instinctively how to solve it. This is fine for problems like:
There are 7 children. 2 of them are girls. How many boys are there?
As problems get more difficult, however, that flash of insight becomes less reliable, so we find our students staring blankly at their paper or out the window. They complain, “I don’t know what to do. It’s too hard!”
We need to give our students a tool that will help them when insight fails.
Three Steps to Solving Word Problems
In the process of solving a word problem, the student must work his way through three steps:
- Translate the words into a mathematical calculation or algebraic equation.
- Do the calculation or solve the equation.
- Interpret the resulting number in the context of the original problem.
When our students struggle with solving problems, most of the time it is step one that gives them trouble. They do not know how to translate the problem from English into “mathish.” If we want to help our students with their math problems, we need to teach them how to do this sort of translation.
One approach teaches key or signal words. For instance, we can tell our students that a problem asking, “How many more…?” is probably going to require subtraction. The question asks for the difference between two quantities, and difference is the answer when you subtract. Unfortunately, this technique only works for the most simplistic story problems, and some textbook (or test) writers enjoy using key words in ways that will mislead a careless student.
If you would like to teach key words to your students, here is a list of the most useful ones:
I do teach a few key words to my students. I particularly like the translation “of” = “multiply” when dealing with fraction and percent problems. But I also want my students to read a math problem and analyze what is happening, no matter what words are used to describe the situation. For this, they need a more powerful tool than key words.
Tool #1: Algebra with Words
- efficient, getting to an answer quickly
- flexible, applicable to almost any situation
- useful in higher mathematics
- abstract, thus difficult for children to understand at a glance
- abstraction may lead to mental block for some students
- takes plenty of practice to learn to use well
We can introduce young children to algebra by using whole words, not letter variables. For instance, for the problem in the first paragraph above, we might write:
Children = 7
Girls = 2
Boys = Children - Girls = ?
With older children, we proceed to using initials, but it is important for the student’s understanding that we still say the whole word as we write:
C = 7
G = 2
B = C - G = ?
This is an easy approach when the relationships are simple, as in this situation. As the relationships between quantities in the problem get more complex, the algebraic equations must get correspondingly more complex. Our students must learn to use some basic but important rules to use as they work through their equations:
The Rule of Inverse Operations
If a mathematical operation has been done on some quantity, you can “undo” it by using the inverse of that operation. For example, if you know that N + 5 = 8, then you can “undo” the addition by subtracting 5.
The Balance Rule
Whatever you want to do to one side of the equation, you must also do the same thing to the other side, to keep the equation in balance. So in the example N + 5 = 8, if you want to subtract 5 from the “N + 5″ side of the equal sign, you must also subtract 5 from the 8.
The Rule of Substitution
If one quantity is equal to another quantity, then it may be substituted for the other quantity in any equation. Thus, if you have determined that N = 3 in this particular problem, you can then use that information in any other equation within the same problem.
The Rule of Parentheses (Distributive Property)
If you add (or subtract, or multiply) a group of quantities, usually set apart from the rest of the equation by parentheses, you must add (or subtract, or multiply) everything within that group, one at a time. And you must do this very carefully, especially when subtracting, because this is one of the easiest places to make a mistake in algebra.
Tool #2: Bar Diagrams
- pictorial representation of abstract relationships
- fewer (and less abstract) rules to learn
- intuitive, easy to understand by students who have played with blocks
- flexible, applying in many situations
- diagrams eventually serve as a lead-in to algebra, giving students a way to visualize equations
- pictorial approach may not be intuitive to some students (or teachers)
- may not work in every situation
- takes plenty of practice to learn to use well
In bar diagrams, quantities (both known and unknown) are represented by block-like rectangles. The student imagines moving these blocks around or cutting them into smaller pieces in order to find a useful relationship between the known and unknown quantities. In this way, the abstract puzzle of the word problem becomes a 2-D visualization puzzle: How can we fit these blocks together? For many students, this pictorial approach makes the problem much easier to understand.
In the above problem, we would draw a rectangular bar to represent all the children. Then we would divide it into two parts, representing the boys and the girls:
As problems get more complex, the bar may be split into more than two parts. Also, the parts may be related to each other in ways that require a more involved diagram. However complicated the story, though, you usually begin by drawing a long bar to represent one whole thing and then dividing it into parts.
Again, the student must learn some basic but important rules:
The Whole Is the Sum of Its Parts
All bar diagrams descend from one very basic diagram showing the inverse relationship between addition and subtraction: The whole is the sum of its parts. If you know the value of both parts, you can add them up to get the whole. If you know the whole total and one of the parts, you subtract the part you know in order to find the other part.
In a picture:
Simplify to a Single Unknown Part (called a “unit”)
You cannot solve for two unknown numbers at once, so you must use the facts given in your problem and manipulate the blocks in your drawing until you can connect one unknown unit (or a group of same-size units) to a number. Once you find that single unknown unit, all the other quantities in your problem will fall into place.
Which Tool Should We Teach?
Because algebra is flexible and efficient in almost any problem-solving situation, it is our ultimate goal. We want all our students to learn to use it well. A friend has argued (on a now-defunct homeschooling forum) that this goal is best served by teaching algebra early and using it often. On the other hand, my students have done well with the Singapore Math approach, using bar diagrams in elementary school and making the transition to algebra in junior high.
I am not sure which of these tools will work best with your student, but I suggest you choose one of them and stick with it. Use the tool with your child over and over, until it becomes almost automatic. Start with simple story problems that are easy enough to solve with a flash of insight. Discuss how your chosen tool can model the situations, translating the English of the stories into math calculations or algebra equations—but don’t force your child to fully solve each problem. Just practice the tough part: the translation. Gradually work your way up to more challenging problems. If your math program doesn’t give your student enough problems to practice on, try the Challenging Word Problems series from Singapore Math.
In future posts, I plan to discuss typical word problems at different elementary grade levels and show how these problems can be translated into algebra and bar diagrams. I hope that seeing the two tools in action will help you decide which you prefer.