Math Interventions

Once a student is able to use a Direct Modeling strategy to reliably solve a particular problem type, he is ready to start using a Counting On/Back strategy. During this developmental stage, a student is able to think of a given number and count up or back from that number by using his fingers, concrete objects, pictures, or a number line to keep track of his counts. This is a more advanced technique because the student only needs to represent new information in the problem with objects, rather than the entire problem (For example, when adding 5 + 2, the student can hold the number 5 in her head, and then add two more counters to make 6, and finally, 7). Read on to learn about how to support your student's ability to count on/back to solve problems using addition, subtraction, multiplication, or division.

Math Story Problem Types. (n.d.). Retrieved from: http://www.teachertipster.com/CGI_problem_types.pdf

This page includes intervention strategies that you can use to support your student's ability to count on/back to solve join, separate, compare, or part/part/whole problems. As you read, consider which of these interventions best aligns with the particular problem type with which your student is having difficulty.

Explicit Instruction

If you are intervening to support your students' ability to use Count On/Back to solve join, separate, compare, or part/part/whole problems, you should start by explicitly teaching the skill. This sounds like:

• Explain the Skill/Concept. Define Counting On/Back, and explain activity. ("Counting On means counting forward, which means that the numbers are getting bigger. Counting Back means counting backwards, which means that the numbers are getting smaller" "Today, we are going to learn strategies that will help us count on and count back to solve story problems." )
• Model Skill with Examples. Think aloud about how to count on (or count back) to solve a problem.  ("The problem is: The boy had three apples. The girl had two apples. How many do they have all together? I'm going to count on to solve this problem. I'm going to start by writing the number three, since the boy had three apples. Now, I'll count on and add two more apples, so I'll lift two more fingers and count them as I add them to the three I already have. (Touch the three on the paper) Three, (finger) four, (finger) five. Five apples all together.")
• Model Skill with Non-Examples. Think aloud about how to count on (or count back) ineffectively. ("Now what if I used my fingers like this: The boy had three apples. [Teacher writes the number 3.] The girl had two apples. [Teacher holds up two fingers, but starts counting at one.] One, two. Wait, that count doesn't match how many apples there are all together. I have to make sure I start counting with the three I already know about!)
• 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 give you a problem...")

Activity A: Count On
If your student is able to solve Join (Result Unknown) problems and Part-Part Whole (Whole Unknown) problems (6 + 3 = __ ) using direct modeling, teach him to Count On. In this strategy, a student represents the first number (using the larger) in his head, and then uses counters or fingers to count up.  

Counting On in Action 
In this video, Ama Awotwi explicitly models how to use a counting on strategy to join two quantities together. 

As you watch, consider: How does Ama's model support her student's execution of this strategy?

Counting On To in Action 
This example refers to the following problem.

Jose had 14 teddy bears. How many more teddy bears does he need to get to have 21 teddy bears?

Teacher: To solve this problem, I'm first going to say the number of teddy bears that Jose already has, 14. Now, I'm going to count additional cubes until I reach my goal number 21. So, remember, I'll start with 14, and count up: 15, 16, 17, 18, 19, 20, 21. Now that I've reached my goal number, I'll count how many cubes I added to my original number, 14, to get to 21: 1, 2, 3, 4, 5, 6, 7. I added 7! So, that means Jose needs 7 more bears to get to 21.  

Activity C: Counting Down 
If your student is able to solve Separate (Result Unknown) problems (8 - 5 = __ ) using direct modeling, teach him Counting Down (Carpenter et al., 2015). In this strategy, a student starts with the given number, and then uses objects or fingers to count backwards a certain number of times to find the result.

Counting Down in Action

In the video below, Emily Art explicitly models how to count down to solve a subtraction problem.  

As you watch, consider: How does Emily teach the student to use multiple models?

If your student is able to solve Separate (Change Unknown) problems (8 - __ = 3)  using direct modeling or struggles to solve Part/Part/Whole (Part Unknown) problems (8 - __ = 3), teach Counting Down To (Carpenter et al., 2015). This strategy teaches a student to start at a large quantity and count backward using cubes or fingers to keep track of his count until he gets to the result provided in the problem.

Counting Down To in Action

In the video below, Emily Art shows the student how to hold a number in her head to count down.

As you watch, consider: How is this approach similar and different to counting down?

Carpenter, T. P., Fennema, E., Franke, M.L., Levi, L., & Empson, S. B. (2015). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Forbringer, L., & Fuchs, W. (2014). RtI in Math: Evidence-Based Interventions for Struggling Students. Hoboken: Routledge Ltd.

Math Story Problem Types. (n.d.). Retrieved from: http://www.teachertipster.com/CGI_problem_types.pdf

Counting strategies for multiplication and division usually involve skip-counting (Carpenter et al., 2015). This page includes intervention strategies that you can use to support your student's ability to count on/back to solve problems related to multiplication and division. As you read, consider which of these interventions best aligns with the particular problem type that your student is having difficulty with.

Explicit Instruction

If you are intervening to support your students' ability to Count On/Back to solve multiplication or division problem types, you should start by explicitly teaching the skill. This sounds like:

• Explain the Skill/Concept. Define skip-counting and explain activity. ("Skip-counting means that you count up or back by a number other than one." "Today, we are going to use skip-counting to help us do multiplication and division problems." )
• Model Skill with Examples. Think aloud about how to skip-count.  ("The problem is: The boy had 4 sandwiches that each had 5 pieces of cheese. How many pieces of cheese did he have in all? To figure this out, I can skip-count by 5s, four times. That means, each time I hold up a finger, I'll count by multiples of 5. Watch: 5, 10, 15, 20. I skip-counted by 5, because there were 5 pieces of cheese on each sandwiches, and I counted 4 fingers, because each had four sandwiches. So, the total number of pieces of cheese was 20!")
• Model Skill with Non-Examples. Think aloud about how to skip-count ineffectively. ("Now what if  I skip-counted like this: 5, 10, 20, 21. That's not the proper way to skip-count because you have to make sure you are counting by the same amount every time you put up a finger. This helps us multiple and divide.")
• 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 give you a problem...")

Activity A: Skip Counting - Multiplication and Division

If your student is able to use direct modeling to solve basic multiplication and division problem, teach him Skip Counting.  In this strategy, a student uses an object or fingers to represent a group of numbers and skip-counts by a given amount in order to find the total.

Skip Counting in Action
This example refers to the following problem.

Dionne has 3 boxes of donuts. There are 6 donuts in each box. How many donuts does Dionne have in all? 

Teacher: I'm going to skip-count by 6 to find the answer. Each of my fingers will represent a box of donuts, and each finger will be worth six, because there are six donuts in each box. I know I have three boxes, so I'll put up three fingers. [Teacher counts by 6 for each finger she has up.] 6, 12, 18... All in all, he has 18 donuts!

Activity B: Skip Counting - Measurement Division

If your student is able to use direct modeling to solve measurement-division problems, teach him Skip Counting for Measurement Division (Carpenter et al., 2015). This strategy teaches a student to solve division problems by skip-counting on his fingers by a known amount up to a known quantity and then counting the number of groups (which are represented by the number of fingers he put up when skip counting).

Measurement Division in Action

This example refers to the following problem.

There are 18 donuts that need to be boxed. 6 donuts need to go into each box. How many boxes does Dionne need for his donuts?

Teacher: I know that there are 6 donuts in each box and 18 donuts all together. I can skip-count by 6 to 18 to see how many boxes I need: 6, 12, 18. [Teacher extends one finger every time she skip-counts by 6.] Now, I'll count the fingers I have up. 3. That means Dionne needs 3 boxes for his 18 donuts.

Activity C: Skip Counting - Partitive Division

If your student is able to use direct modeling to solve partitive division problems, teach him Skip Counting - Partitive Division (Carpenter et al., 2015). This strategy teaches a student to solve division problems by skip-counting on his fingers by a known amount up to a known quantity and then counting the number of objects in each group (which are represented by the number of fingers he put up when skip-counting).

 Partitive Division in Action

This example refers to the following problem.

There are 18 donuts in boxes. There are 3 boxes total. How many donuts are in each box?

Teacher: I know that there are three boxes and 18 donuts total. First, I'll skip count by 3 to 18: 3, 6, 9, 12, 15, 18 [Teacher extends one finger every time she skip-counts by 3.] Now, I'll count how many fingers I have up: 6. That means, there are 6 donuts in each box. 

Carpenter, T. P., Fennema, E., Franke, M.L., Levi, L., & Empson, S. B. (2015). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Math Story Problem Types. (n.d.). Retrieved from: http://www.teachertipster.com/CGI_problem_types.pdf

Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for counting on/back. This example refers to the following problem.

Jesus had 8 toy cars. He gave 5 toy cars to Rosalee. How many toy cars does Jesus have left?

     Teacher: "Show me how can count back to get your answer."
     Student, holding up eight fingers: "8, 9, 10...10?"

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 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.