In this first article by R. Charles, we are presented with the two traditional teaching strategies for math word problems: key words and steps. Using key words teaches students to always use a particular operation whenever a word problem contains a certain word or phrase. This is problematic because key words can be misleading leading students to chose a wrong operation or nonexistant leading students to confusion. When assessment use word problems that don’t have any key words, it is not because they are trying to trick students but rather because everyday real-world problems are not usually asked in such a neat and tidy formula. Charles’ ultimate goal is for teachers to help students understand that word problems are a process and that it takes logic and reasoning to figure them out.

A new “Visual” approach to teaching word problems helps students identify the known and unknown quantities using a bar diagram and arranges them in such a way to show relationship. Using “meaningful representation” to understand word problems instead of just writing down an equation can be much more useful and help students understand a wider range of ways to solve problems. These bar diagrams can be used to help solve joining, separating, part-part-whole, comparison, joining equal parts, separating equal parts and comparison problems.

## Charles’ most convincing statement was: “One of the powerful attributes of this set of bar diagrams is that they are connected to parts and wholes. This consistency in visual relationships helps students see not only the connections between the diagrams but also connections between and among operations. An important part of understanding operations is to know all relationships between and among the four operations.”

Suggestions for teaching this method are:

- model bar diagram representations on a regular basis
- discuss and connect them to quantities in the word problem and to operations meanings
- use them to focus on the structure of a word problem, not surface features like KEY WORDS
- encourage students to use them to help understand and solve problems

In our second reading, Math Matters, the focus was on the importance of students attaching meaning to mathematical operations through the manipulations of concrete objects and then connecting their actions to symbols. As teachers, we need to give students a variety of word problems to work and gain experience. Though this article did focus on the same types of problems (join, separate, part-part-whole, and compare) they broke these specific types up further into subsets of each type of problem. This was done to further identify what the question is asking for and which quantity the student is asked to solve. I had never thought through addition and subtraction problems with this depth before. And, some of the problems that were put into the “join” category I would have put in the “separate” category because I would have changed information around.

For example: Laina had 4 dolls. She bought some more dolls. Now she has 6 dolls. How many dolls a did Laina buy? I absolutely see NOW how this is a “join” or addition problem, but I would have originally said it was a “separate” or subtraction problem because instead of writing 4 + __ = 6 I would have written 6 – 4 = __ to find the solution. I see the difference now and will need to be VERY CAREFUL when teaching younger students so that I do not confuse them. What do you think about my problem?

Classifying Addition and Subtraction Word Problems

- Carlton had three model cars. His father gave him four more. How many model cars does Carlton have now?
**Join Result Unknown: 3 + 4 = __** - Juan has nine marbles. Mary has six marbles. How many more marbles does Juan have than Mary?
**Compare Difference Unknown: 9 – 6 = __** - Janice has three stickers on her lunch box and four stickers on her book bag. How many stickers does she have in all?
**Join Result Unknown: 3 + 4 = __** - Catherine had a bag of four gummy bears. Mike gave her some more. Now Catherine has seven gummy bears. How many gummy bears did Mike give her?
**Join Change Unknown: 4 + __ = 7** - A third grader has seven textbooks. Four textbooks are in his desk. The rest of his books are in his locker. How many books are in his locker?
**Separate Result Unknown: 7 – 4 = __** - Vladimir had some baseball cards. Chris gave him 12 more. Now Vladimir has 49 baseball cards. How many baseball cards did Vladimir have before he received some from Chris?
**Join Initial Quantity Unknown: ___ + 12 = 49**