Revised March 30, 2019. This resource used to be titled "Addition and Subtraction Clinical Interview Protocol."
The tasks in this protocol can help you explore these ideas about a child’s understanding of addition and subtraction. Here are three ideas that can be explored.
- Sets of objects can be changed by adding them together.
- A set of objects can be changed by separating them to subtract.
- Language and symbols can be used to show operations and relationships.
The resource Math Thinking Conversations provides guidelines for how to use these kinds of tasks to explore a child’s mathematical understanding.
The tasks that follow have been organized according to the questions listed above. Select video examples of assessment items are available in Addition and Subtraction Assessment Videos and Additional Operations Videos.
Adding and Subtracting Objects
- Detect and explain a change in number Show a child a number of objects, e.g. 4 bears, and say that you are going to do something special. Then hide the bears with a paper or cloth, and secretly add or remove one or two bears. You might say, “Close your eyes. I’m going to do something.” Then, uncover the bears so that the child can see the result of the addition or subtraction. Your goal should be to find out whether the child understands how the items were affected by what you did. “What do you think happened? Are there more bears now, or fewer, or just the same? Why do you think there are more (or fewer) now? How many bears were added (or taken away)? How many bears are there altogether now?”
- Add or subtract (with objects visible) Show the child two bears (or other small objects) and show another bear to the side. Ask: “How many bears will I have if I add one more? How did you know? How did you figure that out?” Have manipulatives ready in case the child wants to use them to solve the problem and show you the result.
- Number stories (with many objects visible and available) Put about ten bears (or other small objects), as well as chips, a marker, and paper on the table. Say, “I want you to figure out the answer to my questions any way you want. You can use your fingers, these chips, this paper and pencil, counting, or think in your head. My friend Tommy had one cupcake, and then his mother gave him two more.” If the child responds quickly, try larger numbers so that you will be able to probe for the child’s strategies. “How many does he have now? How do you know?” This can be done with subtraction as well.
- Number Stories (with no objects visible) Tell the child you will tell a number story. Say, “Make believe that there are 3 carrots. Then you get another 4 carrots. How many carrots are there all together?” See if the child can mentally find the sum. Do the same with a subtraction problem. Say, “Now there are 7 carrots. What would happen if I ate 2 of them? How many would be left?” Watch Magic Math Thinking Minute: Addition here.
- Number stories (check your work) Ask a child who has solved a number story in her head to check her answer using manipulatives. Notice how the child sets up the problem. For example, if the problem was 3 + 4, does the child create one set of 3 manipulatives, and then add a set of 4 to it, one at a time, to get the total? Does the child create a set of 3 and a set of 4 and then put them all together and count the new combined set? Watch Meredith: Number Story here.
Symbols for Operations
- Formal mathematical language (plus and minus) Ask, “How much is four plus two? Similarly, you can ask, “How much is five minus three?” If the child seems to be ready for it, present problems with sums greater than 10. For example, ask, “What is 8 plus 3?” Ask the child to explain how they know so you can figure out which strategies he used. For example, a child may say she counted on her fingers. A child may reason that 8 + 2 is 10, so 8 + 3 is 1 more than that. For a subtraction problem like 12 – 4, a child might start at 4 and count on until he reaches 12, showing 8 fingers. A child might also count backwards on his fingers or draw a picture. Watch Lulu: 8 plus 3 is 11 here.
- Mathematical symbols (plus and minus) Determine whether the child can use the standard written symbolism for problems by presenting problems such as 2 + 3 and 5 – 2. See if the child recognizes plus and minus symbols and understands their meanings. Watch Tasha: 3 + 1 equals 2 here and Chandra: 5 – 3 = 2 here.
- Mathematical symbols (equal sign) Determine whether the child understands that the equal sign shows equivalence. For example, if a child correctly writes that 1 + 2 = 3, see how the child responds to number sentences like 3 = 1 + 2, 1 + 2 = 2 + 1, or 3 = 3. Some children might say that the 3 has to be on the right of the equal sign because it refers to the result of addition or subtraction. In other words, they may see the equal sign as indicating that “you have to put the the answer after it.” Early math education should help the child to go beyond this narrow view of the equal sign. Watch Chandra: Equal Sign here and Chandra: 3 = 1 + 2 is wrong here.
- Analyze wrong answers (bugs) Sometimes children use flawed strategies, or bugs, to solve problems. In these cases, what the child is doing does make some sense, even though it does not yield a correct answer. Children enjoy getting to play teacher, so present a problem the child has correctly solved, such as 12 – 8, but say, “Suppose you were a teacher and a child wrote this: 12 – 8 = 16. Why might they do this?” Write the problem 12 – 8 = 16 vertically. See if the child can figure out how a child could have gotten that wrong answer. Another way to approach this is to present the problem and ask what mistake another child might make. For example, present 8 + 3 and ask what mistake a child might make. Examine how a child knows a particular answer is incorrect. For example, if you present 8 + 3 = 83, see how the child know the sum is incorrect. Do they say it is incorrect because they know the correct answer is 11 or because they just know that 83 is much too big to be the correct sum? (For more information about analyzing bugs, visit Analyzing the Thinking Underlying Wrong Answers.) Watch Angelo: 12 – 8 = 16 here and Lulu: 8 + 3 bug here.