To provide a flavor of the differences among programs, a small content area will be illustrated. The specific topic used in this is example is the placement of the decimal point in decimal multiplication by summing the number of decimal places in the factors. This example is based on the treatment in the fifth grade programs.

It should be stressed that this example will NOT address the meaning of decimal multiplication as can be illustrated in various ways, or any of the other content related to decimal multiplication. Only the one content element is addressed here.

Explicit Methods

In an explicit presentation, the textbook explains the process clearly and in sufficient detail to be understood by students.

For example, in the Saxon program the process is first illustrated with an area model and fractions, and then the following instructions appear:

When we set up a decimal multiplication problem, we do not try to line up the decimal points. That rule is only for adding and subtracting. When we multiply, we just set up the problem as though it were a whole-number problem and multiply. To place the decimal point in the answer, we first count the total number of digits to the right of the decimal point in both factors. Then we place the decimal point in the answer so that there is the same total number of digits to the right of the decimal point in the answer. Copy and study the following examples and solutions.

These instructions are followed by three examples, and these are followed by further reiteration, "... we count the total number of decimal places in the factors. Then, starting from the right side of the answer, we count over that many digits to the left and mark the decimal point."

Similarly, the SRA program provides an illustration with fractions first, and then says:

To multiply two decimals, multiply as though there were no decimal points. Then place the point in the answer as many places to the left as the points are to the left in the two factors. This is easier to understand with examples.

Two examples are then given. In each the number of places past the decimal is indicated for each factor, the addition of these two values is indicated, arrows indicate the correct placement of the decimal point in the answer, and the answer is checked by approximation.

Less Explicit Methods

Other programs require more careful work or a greater level of inference by students to make sure they grasp this simple procedure. In other words, these presentations run a greater risk of confusion or ineffective learning.

In the McGraw Hill program, a careful student will notice that the beginning of the lesson says:

In the last lesson, you used models to multiply decimals. In this lesson, you will use estimation and counting decimal places.

Without further instruction or explanation, examples follow immediately. In each example, the instructions are "Estimate the product ... Multiply as with whole numbers ... write the decimal point in the product."

In each example, the number of decimal places in each factor and in the answer are indicated, but placement of the decimal by summing these two counts is not explicit. In the first example, a note to the side states, "Note: You can use the estimate or count decimal places."

In the Scott Foresman - Addison Wesley program the method is also given minimal treatment. Again, the lesson has no instruction beyond the examples. There are three examples. The number of decimal places in the factors and the product are indicated in each example. In step 2 of the first example, the text does say, "The number of decimal places in the product equals the sum of the decimal places in the factors." Careful students will note and learn this point, but others may miss the point. Later in a "Talk About It" section, the text says, "Give a rule for placing the decimal point in the product."

In these less explicit treatments, careful and skilled teachers will give more attention to this small element of the curriculum to make sure students learn this material, but there is a risk that this might not happen.

Deficient Methods

Even worse than the Less Explicit Methods are those cases where this topic is not even covered. This is the case in both the Everyday Learning program and in the Dale Seymour program.

There are clear differences among programs in terms of how thoroughly this curriculum element is addressed, and corresponding differences in student learning might reasonably be expected for these different approaches.