The different types of problems for each operation are not subtle variations: They are major differences. Also, for each operation, the conventional algorithm for that operation reinforces only one of the types at the expense of the other: The standard method for addition reflects “combining,” that for subtraction, “take away,” that for multiplication, “grouping/repeated addition” and that for division, “grouping/repeated subtraction.” This is one of the reasons many elementary school students have difficulty with arithmetic word problems that involve the other types: Unwittingly, they’ve been “taught” not to understand them. Another reason, and perhaps the most compelling, is that eight different types of problems is too many to keep straight in one’s mind.
The Combiners and its companion book, , are about a new way of teaching arithmetic word problems, a way that reduces the eight types of problems to two broad categories, “combining” or “separating,” and then how it’s done, “just” or “neatly,” where “just” means not necessarily in equal amounts and “neatly” means by or into 2s, 3s, 4s .... The Combiners does so for addition and multiplication featuring Motley Crab Adder and Sir Crab Multiplier, respectively. The Separaters does so for subtraction and division spotlighting Scruffy Scrap and “Neat” Nic, respectively. The actions of the characters are then examined using the following 2-step heuristic, in particular, the 2nd step, since, as the titles of the books suggest, we already know what’s happening in The Combiners, combining, and we already know what’s happening in The Separaters, separating.
1. What’s happening, combining or separating?
2. How’s it happening, “just” or “neatly” (by or into 2s, 3s, 4s…)?
As the diagram on the back cover shows, the heuristic transforms combining/separating actions into operations: If the action is combining and “just” combining, it’s addition. If combining and “neatly” (by 2s, 3s, 4s…), multiplication. If separating and “just” separating, subtraction. If separating and “neatly” (into 2s, 3s, 4s…), division. ALL arithmetic word problems are combining or separating problems and then “just” or “neatly” problems. ALL may be solved with the 2-step heuristic.
The following illustrate how each of the two types of addition, subtraction, multiplication and division problems are combining or separating problems and may be reduced to “just” or “neatly” problems.
Addition as “Just” Combining
A tent pole is stored in three sections. The bottom section is 2 feet long, the middle one 4 feet long, and the top one 3 feet long. How tall is the pole?
Using snap cubes for feet, a 2-stick, 4-stick, and 3-stick would be joined or “just” combined to show the height of the pole.