Math Interventions

When a student keeps plugging away at an inappropriate approach to problem solving, or reports a solution that does not make sense in the context of the problem, he may need support with either monitoring his progress to evaluate whether the strategy he chose is effective, or reflecting on the solution to evaluate whether or not his answer makes sense. This page includes intervention strategies that you can use to support your students in each of these areas. 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 students' ability to monitor progress or reflect on the solution, you should start by explicitly teaching the skill. Note: For the sake of clarity, the explicit instruction below focuses on a student who needs support reflecting on his solution to make sure that it makes sense. This sounds like:

• Explain the Skill/Concept. Define reflecting on the solution, and explain the activity. ("Once we've solved a problem, it's important to make sure that our answer makes sense in terms of the problem that we're trying to solve. We call this part of the problem-solving process Reflecting on the Solution." "Today, we are going to do some activities that will help us reflect on the answer we got and make sure that it makes sense and answers the question being asked." )
• Model Skill with Examples. Think aloud about how to reflect on the solution.  ("I solved this problem and found that the kids had three apples. When I reflect on the solution, I go back and reread the original question. Then, I try to fit in the answer I got to make sure it answers the question posed and makes sense. Watch me reread this question: The boy had one apple. The girl had two apples. How many apples did they have together? Since I got the solution of 3, I would answer the question by saying "They had 3 apples together." Then, I need to ask myself: Does this make sense? Well yes, I know that the boy had less than three apples and the girl had less than three apples, so it makes sense that they might have three all together.")
• Model Skill with Non-Examples. Think aloud about what happens if this step is skipped. ("If I didn't reflect on my solution, I might end up with an answer that is incorrect, or only partially correct. What if I solved this problem quickly, got the answer "2," and just submitted this number as my answer? There would be two problems: First, the person reading my work wouldn't know if I was talking about two apples or two aliens! Also, I wouldn't realize that I made a mistake (because if you are adding something to two, your answer should be bigger than two).")
• 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: Elicit Student Thinking 

If a student has only completed part of the problem, or has completed the problem incorrectly, use this strategy to prompt the student to self-monitor progress toward the end goal or to reflect on his solution. Elicit Student Thinking (Carpenter et al., 2015) uses prompts to ask students to go back and reflect on their process and their solution. This strategy will lead students through the thought process that you eventually want them to go through on their own when self-monitoring progress or reflecting on their solutions.

Elicit Student Thinking in Action 

Read the following PDF to see how to elicit student thinking in different ways. As you read, consider: Which of these questions are best suited to a student who is not adequately monitoring progress and may need to change course while solving a problem? Which of these questions are best used after a student has solved the problem and needs support evaluating his solution?

Elicit Student Thinking.pdf

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

Activity B: Discuss the Strategy

If a student needs additional support with evaluating the answer that he got, teach him to Discuss the Strategy (Carpenter et al., 2015) by reflecting upon additional strategies that he could have used to solve the problem, and identifying the best strategy for solving that particular problem. This strategy can be helpful to teach a student who always chooses the same strategy to use, or doesn't demonstrate flexible thinking when solving problems. In this strategy, the teacher engages the student in considering other's ideas for solving problems and discussing how the strategies used to solve the problem are the same or different. If the teacher is teaching this strategy in a 1:1 context, she can refer and/or share student work that she has seen in other classes. In this intervention, the student shares the way he has solved a problem, and then the teacher shares the way that another student solved the problem. Then, she asks the student to explain if he solved the problem the same way, or a different way. If the student solved it differently, she asks him to explain what he did.

Discuss the Strategy in Action 
This example refers to the following problem.

There were some birds on the telephone wire; 3 flew away, and now there are 9 on the wire and 3 on the tree. How many birds were on the wire before any flew away?

Teacher: Joelee, how did you solve this problem?

Student: I solved it by drawing three circles and then drawing nine circles and then adding them up to get 12. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Teacher: I'm going to share another way a student solved a problem. Can you tell me how the way you solved the problem is similar? Or, can you explain how you solved this problem differently?

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

Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies to monitor progress and reflect on his solution. This example refers to the following problem.

There were some birds on the telephone wire; 3 flew away, and now there are 9 on the wire and 3 on the tree. How many birds were on the wire before any flew away?

     Teacher: "Does your answer make sense?"
     Student: "I got 6. I think that makes sense." 
     Teacher: "Let's reread the problem, and then reflect: Should your answer be greater or
     less than 9?"
     Student: "Well, less than 9, because I got 6, and that's less than 9!"

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.