Written by LouAnn Lovin—James Madison University
(News Bulletin, January/February 2006)
In U.S. mathematics curricula, basic addition facts are commonly taught by using the following:
1. Fact families
(e.g., 2 + 3 = 5, 3 + 2 = 5, 5 – 3 = 2, 5 – 2 = 3)
2. Facts for one number
(e.g., 4 + 0 = 4, 3 + 1 = 4, 2 + 2 = 4, 1 + 3 = 4, 0 + 4 = 4)
3. Thinking strategies
(e.g., solving 5 + 4 = ? by recognizing that 4 + 4 = 8, and
that 5 + 4 must be one more, or 9)
Research does not suggest that any one of these approaches is more effective than the others for teaching basic facts. Moreover, Principles and Standards in School Mathematics (NCTM 2000) suggests using a variety of approaches. Even so, some states’ standards can lead teachers to emphasize one approach more than the others.
Take the following Virginia standard for example: “The student, given a simple addition or subtraction fact, will recognize and describe the related facts which represent and describe the inverse relationship between addition and subtraction
(e.g., 3 + ___ = 7, ___ + 3 = 7, 7 – 3 = ___, 7 – ___ = 3).”
Because the standard shows a fact family, some teachers believe that fact families are the preferred way of teaching basic facts.
Let me show you what I mean. Figure 1 is an exercise developed by a textbook publisher to prepare students for Virginia’s third-grade standardized test. Steven, a second grade student, is routinely given sheets with multiple-choice items such as this to prepare for the state assessment. He identified (b) as the answer. He reasoned that if you take 2 from 12 and 2 from 7, you get 10 – 5, which he knew was 5. Therefore, the answer, according to Steven, must be 5 + 2 = 7 because it was the only one that had a 2, 5, and 7. The teacher marked the item wrong, explaining that Steven was supposed to use fact families to respond to the question.
|Figure 1. An item based on a Virginia state standard and generated by a textbook publisher to prepare students for a state-level standardized test.
Twenty-six second-grade teachers were asked to respond to the task in Figure 1 and to indicate all the addition facts that could be used by a student to solve 12 – 7. All 26 teachers chose (d) as at least one of their choices because it was part of the appropriate fact family. And, like Steven, 14 of the 26 teachers indicated at least one of the other choices and provided a variety of explanations of how that choice could help determine 12 – 7.
The teachers were then asked how they would respond to Steven. Most indicated that if they knew what his reasoning was they would offer praise for “good mathematical thinking,” but several teachers maintained that Steven “needs some help” in seeing the right way to get the answer.
Here we can see how the state standard led the textbook publisher to produce a test-prep item that focused on fact families and led Steven’s teacher to do the same. However, a second look at the state standard shows there are actually two important areas that need to be assessed: 1) computation skills and 2) an understanding of the relationship between addition and subtraction. An open-ended response item such as the following would assess a student’s mastery of computational skills more effectively: “Find 12 – 7. Explain how you determined the answer.” And the following would reveal whether or not a student understands the relationship between addition and subtraction: “Johnny had 12 cards. He hid some cards so that you can see only 7 cards. How many cards did he hide? Write a subtraction equation for this situation. Write an addition equation for this situation.” Teaching to these test items would allow students to develop computational skills by nurturing number and operation sense.
In short, if the assessments and test-prep items that we use are thorough, then teaching to the test might be a good thing. But if we are merely preparing students to work through poorly designed assessments items, then we may be nurturing another generation that believes mathematics is a subject of static rules and procedures. We need to be alert and determine whether the standards and tasks to which our students are exposed can lead to a rich understanding of mathematics. If not, then we must teach what we know is important and advocate for practices that actually serve the best interests of our students—just as Cathy Seeley suggests in the current President’s Message.