Cognitively Guided Instruction, Lesson 5:
Writing addition and subtraction word problems for students at the direct modeling stage.
The problem types we have discussed so far are the ones that children can most readily solve by direct modeling. When you create your own word problems for children to solve who are at a very concrete, direct modeling stage, there are other considerations also that can make the problems easier or harder for children to sovle and understand.
The points I wish to make here are the same ones that are discussed on page 11 in the book Childrens Mathematics. In the previous lessons you have seen examples and discussions of what makes a problem fall into one type rather than another, that some problems are easier for children to solve by directly modeling them. In addition, within a given problem type, there are ways to write problems that make them easier or harder for children to understand and solve by direct modeling. Most often in the early grades, our goal is to write problems that are easily understood, so the goal I am most concerned with is that you be able to write problems that are as easy to understand as possible.
Action order: In Join and Separate problems, the problems are more easily understood if the quantities are presented in the order in which they happen. In these problem types there is a start amount, an amount that the set is changed by, and a resulting amount, if those quantities are presented in that order, it makes the problem easier to solve. For example:
| Easier | Harder |
| JRU: Janet had 5 mini-erasers. Her mom bought 2 more mini-erasers for her. How many mini-erasers does Janet have now? | Janet's mom bought her 2 more mini-erasers. Before she got the new erasers, she had 5 mini-erasers. How many mini-erasers does Janet have now? |
| SRU: Sam had 8 carrot sticks. He ate 5 of the carrot sticks at lunch time. How many carrot sticks does Sam have left? | Sam at 5 carrot sticks at lunch time. Before lunch he had 8 carrot sticks. How many carrot sticks does Sam have left? |
| JCU: Kelly had 5 stickers this morning. Her teacher gave her some more stickers, and now she has 9 stickers. How many stickers did her teacher give her? | Kelly's teacher gave her some stickers, and now she has 9 stickers. She had 5 stickers this morning. How many stickers did her teacher give to her? |
Summary: Having the wording follow the action sequence makes it easier for children to understand and act out what is happening in the problem so they can use direct modeling to solve the problem.
How many are needed: Join change unknown problems are easier for children to understand and solve if they are asked in a "how many more are needed" question than if the problem is presented as having happened in the past:
| Easier | Harder |
| JCU: John has 6 Pokemon cards. How many more cards does he need to have 8 Pokemon cards? | John had 6 Pokemon cards. His friend gave him some more cards. Now he has 8 cards. How many Pokemon cards did his friend give him? |
Summary: Planning ahead/wanting to get more questions are more familiar and understandable to children than are similar problems where an unknown action took place in the past.
Discrete vs. continuous: One more way of looking at problem difficulty is in what is being counted in the problem. Problems where you are working with discrete (separate, individual) objects are easier to visualize and model than sets where you are working with a continuous, measured quantity. This is true for all problem types. Eventually, when children are learning about measurement, it becomes desirable to give problems about measured quantities, but childrens early experiences with addition and subtraction should be with problems about discrete objects:
| Discrete objects | Measured quantities |
Pencils |
Ounces (of juice) Inches (of ribbon, or height, or string) Pounds (of sugar, or weight of a person or animal) Pints (of milk) Feet (of sidewalk, or string) |
| Easier | Harder |
| JRU: John had 5 toy dogs. His aunt gave him 4 more toy dogs. How many toy dogs does John have now? | John's puppy weighed 5 ounces when he John first got him. A month later, his puppy had gained 4 more ounces. How much does John's puppy weigh now? |
| SRU: Candice had 7 cookies. She gave 2 of them to her little brother. How many cookies does Candice have now? | Candice had a ribbon that was 7 ft long. She cut off 2 feet of her ribbon, and gave it to her little brother. How long is her string now? |
| JCU: Sally had 7 bean plants. She planted some more beans, and more bean plants grew. Now she has 10 bean plants. How many more bean plants grew? | Sally's bean plant was 7 inches high last week. Her bean plant grew, and now it's 10 inches high. How much did the bean plant grow? |
| CDU: Mary has 5 Webkinz, and Tasha has 8 Webkinz. How many more Webkinz does Tasha have than Mary? | Mary's bean plant grew 5 inches last week. Tasha's bean plant grew 8 inches last week. How many more inches did Tasha's bean plant grow than Mary's? |
In each case, making the abstraction from discrete objects (toy dogs, cookies, bean plants, Webkinz) to representing them with counters is less of a leap than representing a continuous measurement (ounces, feet of ribbon, height of bean plant, growth of bean plant) to a representation with discrete counters. Translating inches to counters is a bigger abstraction than translating cookies to counters, and the last example is even more abstract since it isn't even comparing the heights, but rather the amount of growth. Introducing problems with continuous measurements needs to be done gradually, and should follow having spent rather a long time solving problems with discrete sets.
Summary: problems that ask questions about discrete, easily countable objects are easier to direct model and solve than questions about continuous, measured quantities.