Introduction: About Word Problems

Part A: Why Word Problems are considered Difficult

Word problems are considered the most difficult Math problems to solve for the following reasons:

1. Reading Comprehension: Before you even figure out how to solve a problem, you have to really understand its description, the facts, and what you are supposed to solve for. It is not always easy since sentences can be complicated and not all problems are well written. In fact, in my experience, many problems are not clear and have ambiguous information that can be understood in different ways. There are also many students who do not read well; reading comprehension is known to be a big issue. Further more, to really understand a problem you need to have a mental model of it, which is not always easy to do EVEN if you understood the parts.

1. With this book you will learn how to break complicated descriptions to small parts, how to understand those parts, and how to see the relationships between the parts. You will also learn to use a set of tools that will help you understand and create Mental Models of the problem.

1. Translation: After you read and understood the problem, which does not mean you know how to solve it, you have to translate it to Mathematics. Mathematics is a language, and while it is not fundamentally difficult to translate human speech to Mathematics, the principles of translation are not really taught in school and students are expected to figure it out by themselves. While some students figure it out, most students only do so partially, and therefore get stuck at this stage. Even when the principles are taught (mostly not explicitly even then), they are taught with unintuitive symbols and methods (all those x and y).

1. With this book you will learn how to systematically translate Word Problems to Mathematics in a simple and intuitive way.

1. Problem Solving Strategy: This is the least understood, and the least taught (if at all) element of problem solving. It is also the most critical one, because even if you have mastered steps 1 and 2, you can get stuck without a good strategy. The few text books that do talk about “strategy” do not provide a strategy but are actually discussing the tools I mentioned in (1). Strategy is an overall approach that applies to most, if not all, problems. A tool is a way to solve a specific problem, or a one type of problem, but does not fit other types. It is a very important difference that underlies the problems in how Word Problems are taught. Note that you never study “word problems” as a topic, but only as part of a specific subject.

1. With this book you will learn a strategy that applies to ALLproblems, not just word problems. It is a general strategy to solve any type of problem, not just in Math but also in life. Use this strategy for all the problems you encounter and you will see how powerful it is.

1. Mental Block: Due to all these problems mentioned above (and the one mentioned in 5), people (not just kids) often get stuck when they try to solve problems, either math or non-math ones. It is a normal and natural human reaction – freezing in the face of complexity and ambiguity. No one teaches how to overcome such mental blocks, except maybe say “have confidence”, which is not very helpful when you are not confident and cannot see what is going on. What is more important is for you to have a way to solve problems EVEN if you do not fully understand them, and you do not have “confidence”.

1. Intuitive and Quick Students. Another for a mental block is the presence of students who have an intuitive understanding of math and can quickly see the path from the problem description to the solution. When the teacher asks a question in class and such a student jumps up with the answer, the rest of the students, who are not as quick, think that they must be “bad in math”, because they could not solve it as quickly. Of course, that is not true. It is only that some students are exceptionally intuitive and quick about math. Those students do not have a “magic wand” in their head; they do all the same steps required to solve the problem, but they do them so quickly that it seems like they “jumped to the solution”. With the right explanations and methods EVERYONE can be good in Math. Maybe never as quick and intuitive as those special students, but good enough to solve every problem and get an A.

1. The strategy you will learn with this book helps you solve problems even if you don’t have confidence. It also guides you in how to make sense of the problem and how to solve it EVEN IF you do not fully understand it!

1. Knowledge: If you do not have the “technical” knowledge required for a given problem, you cannot solve it. That is a requirement that cannot be circumvented by having a good strategy, or solution tools. If a problem requires knowledge of addition, you must know addition facts and methods; if a problem requires knowledge of fractions, you must know fractions and how to use them.

This is why I write the other types of books about specific subjects (such as Fractions): to make sure students know (really know) what they need to know about a specific subject.

Part B: Common Wisdom about Difficulty Levels

The Common Wisdom

1. The Difficulty level is determined by the number of steps required for solving a problem.

1. A 1-step problem means that AFTER translating the problem to math only one step is required to move from problem to solution (for example: x + 4 = 10 is the translation and one step of subtracting 4 from both sides would get us to x = 6). Similarly, a 2-step problem would mean that two steps [mathematical or logical] are required to move from problem to solution AFTER translation.

1. More steps = More difficult.

Why so?

The reason problems are categorized by difficulty levels is that to solve a problem without the 7 Principles (7Ps), you have to digest the problem as a whole and “see” the solution. Therefore, in your mind’s eye, you need to be able to see in your mind’s eye the path from the beginning of the problem to the solution, even if the path requires many steps that are not obvious.

However, if you could break the problem to a series of 1-step problems, then a 2-step problem would not really be more difficult than a 1-step one, and a 4-step is not more difficult either. It would be just four 1-step problems.

The 7 Principles Difference

The 7Ps provide a method to simplify EVERY problem to a series of 1-step problems. In fact, the principles provide a method to move one step at a time EVEN if you cannot see all the required steps, which is the most common reason why we get stuck while solving problems.