Math Interventions

Another way to support your student's understanding of quantity is through composing and decomposing whole numbers. This skill will help your student solve simple math facts in his head or use invented algorithms to solve multi-digit addition and subtraction problems. This page includes intervention strategies that you can use to support your students in this area. As you read, consider which of these interventions best aligns with your student's strengths and needs in the whole-learner domains.

Explicit Instruction

If you are intervening to support your student's ability to compose and decompose numbers, you should start by explicitly teaching the skill. This sounds like:

• Explain the Skill/Concept. Define composing and decomposing numbers, and explain activity. ("Composing numbers means putting parts together to make a new number. Decomposing numbers means breaking numbers down into their subparts." "Today, we are going to learn strategies for composing and decomposing numbers." )
• Model Skill with Examples. Think aloud about how to compose or decompose a number.  ("Let's say that we read the problem 'How much is 3 tens and 1 one?' In this problem, I need to compose, or put numbers together. I know that three tens is 10 + 10 + 10, or 30. Then, I can add the 1 to get 31. So, 3 tens and 1 one is 31. Now, if I were to decompose this number, the problem might be: How many tens and ones are in 31? Well, first I have to think about how many tens are in 31. I see that there are 3 tens in the tens place to make 30. I also see that there is 1 one in the ones place. So, there are 3 tens and 1 one. I've decomposed 31 into its 10s and 1s.")
• Model Skill with Non-Examples. Think aloud about composing or decomposing a number ineffectively. ("What if I tried to solve the problem without composing or decomposing numbers, or without breaking them down into their parts? I might think that 3 tens and 1 one is 4 because I just add all the numbers as though they were equal. Or, I might say there are 31 tens in 31. I need to remember to properly examine each part to compose or decompose numbers.")
• Practice the Skill. Engage in one or more of the activities below to practice the skill with your student, providing feedback as necessary. ("Now you try. I'm going to show you...")

Activity A: Finger Games
If your student is struggling with composing basic numbers (up to 10), teach him this skill by using Finger Games (Clements & Sarama, 2009). In this intervention strategy, the teacher helps support a student's understanding by asking him to make numbers with his fingers. Here are a series of activities the teacher can use to support a student's ability to compose basic numbers with his hands:

• Show Me. Ask child to show 4 with his fingers. Tell your partner how you did it. Now show me a different way. Tell your partner. Now make 4 with the same number on each hand. Ask child to show 5 with his fingers and discuss responses (Did he use one hand or two? Can he do it a different way?) Ask him to show another way to make 5, using both hands. Repeat the task above, but "you can't use thumbs." 
• Bunny Ears. In this modification, have the child make the numbers as "bunny ears" holding his hands above his head to make the numbers 1-5 in different ways.
• Up and Down. In another session, ask the child to show 4 on one hand. Ask how many fingers are up and how many are down (all on one hand only). Repeat with 0, 1, 2, 3, and 5 across several days

Finger Games in Action
Teacher: We are going to learn to compose, or make, numbers with our hands! First, I want you to make a 1 with your fingers! (Student makes a 1). Great! Now, make a one using a different finger. (Student makes one with pinky.) Great! Okay, now let's make a 2.

Activity B: Make a Number 
If your student can compose numbers up to 10, but needs to build fluency with composing numbers up to 20, teach him to Make a Number (Clements & Sarama, 2009). In this strategy, the student builds numbers by using a deck of cards. First, the teacher takes out all the cards that state the number the student is trying to compose (such as 7) or greater out of the deck (for example, if the teacher was trying to get the student to compose the number 7, she would remove all cards at the number 7 or greater out of the deck). This will leave her with cards numbered 2, 3, 4, 5, and 6, which the student can use to compose the number 7 (such as 5 + 2, 4 + 3, etc). Then, she takes turns with the student to draw cards, flipping them over every time one is drawn. With each turn, the teacher and student try to make that number (such as 7) by combining it with any face up card. For example, if they have a 4 flipped over, they would need a 3 to make 7 (4 + 3 = 7). If they can make the target number, they keep both cards. The game continues this way until all cards in the deck have been used. The person with the most pairs wins. 

Note: An alternative way to teach this strategy is through Memory. Instead of having the student draw cards to make that number, have her flip cards over and match the cards to make the target number. Here, the same rules apply as before: all cards at, or of a greater value than, the target number are removed from the deck, and the person with the most pairs at the end of the game wins. 

Make a Number in Action 
Teacher: We are going to play a card game to see who can make the number 7 out of his cards. (Teacher explains directions) It's my turn first, so I'll draw a card. Okay, I got a 5. I can't make a 7 yet. Now, it's your turn. Flip a card over to see if you can make 7. 

Student: I got a 2! 5 + 2 = 7! So I get to keep the two cards. Now, I'll draw one. I got a 3.

Teacher: I got a 2. 3 + 2 is 5, which isn't 7. We'll flip these over and try again to make 7 with a new pair.

Activity C: Composing 10s and 1s
If your student struggles with place value, Composing 10s and 1s (Clements & Sarama, 2009) will help him learn how to understand two-digit numbers as tens and ones. In this strategy, the teacher shows the student connecting cubes (such as 4 tens and 3 ones) for 2 seconds, and then hides them under a cloth. Then, she asks him how many there are, and they discuss how he knows. The teacher then repeats this activity with new amounts. Then, the teacher adds an additional layer of challenge: she tells the student that she has 2 tens and 17 ones, and then asks him how many there are in all. Once he tells her, he can count the cubes. Continue this activity until the student is fluent in converting ones to tens. 

Composing 10s and 1s in Action
Teacher: I'm going to show you a group of connecting cubes, and I want you to tell me how many there are. Ready? (Teacher shows 20 cubes or 2 ten-rods). How many?

Student: 20.

Teacher: Okay, you can check your answer by counting the cubes.

Student counts to 20.

Teacher: Let's try again. How many cubes? (Teacher shows 2 ten-rods and 5 ones)

Student: 20 and um....1, 2, 3, 4, 5. 25!     

Clements, D.H. & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.

Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for composing and decomposing numbers. The example refers to the following problem.

     Teacher: "Show me how you use the number line to add 25 + 30."
     Student: "Um, well I start at 25. Now, I'll go up to 30, which is 5 more up. 25 + 5=30. So,
     I think it's 30?"

In such a case, what might you do? 

When you are planning your lessons, you should anticipate that your student will make errors throughout. Here are a series of prompts that you can use to respond to errors. Keep in mind that all students are different, and that some students might respond better to some types of feedback than to others.

If your student struggles to meet your objective, there are various techniques that you might try in order to adjust the activity so as best to meet your student's needs.