Math Interventions

Once a student has been able to effectively create a mathematical model and use it to solve a problem, it's time for him to interpret the results. To do so, a student must be sure include the relevant units in his answer, and reflect on whether or not his results make sense. As you read, consider how to implement the intervention below so that it 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 interpret the results while modeling, you should start by explicitly teaching the skill that you are about to show the student. This sounds like:

• Explain the Skill/Concept. Define interpreting results. ("After we have created a mathematical model and used it to solve a problem, we need to interpret the results. This means that we go back to the original problem and make sure we've included the information we need, and that the solution makes sense." "Today, we are going to learn a strategy that helps us interpret the results." )
• Model Skill with Examples. Think aloud about how to interpret the results.  ("Let's take the problem: The boy had one apple. The girl had two apples. How many apples do they have all together? If the answer I got was 3 apples, I'd go back to the problem to see if it includes the information I need, and makes sense. Let's see: the problem is How many apples do they have all together? and I wrote 3 apples, so that answers the question. Now, I'll see if it makes sense: The boy started out with 1, and the girl had 2, and my answer is 3. So, the answer I ended up with is larger than the two numbers in my problem, so 3 apples makes sense.")
• Model Skill with Non-Examples. Think aloud about an ineffective way to interpret results. ("Now what if I wrote 1? I wouldn't have answered the question, because it asked me how many apples, and I just wrote 1. If I didn't go back to the problem, I wouldn't know that this answer didn't answer the question, because it doesn't contain the unit the problem is asking for, apples, and it doesn't make sense because that answer is the same as the number of apples the boy had, which wouldn't account for how many he and the girl had together. That's why it's important to always interpret the results.")
• 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.")

Activity A: Interpret the Solution

If a student needs additional support with interpreting his answer, teach him to Interpret the Solution by connecting it back to the problem situation. This strategy can be helpful to teach a student who doesn't remember to include units in his answer or a student who uses an appropriate strategy to solve the problem but reports the wrong information when stating his solution.

Interpret the Solution in Action 
Point to the solution that the student got, and then walk him through the following steps to connect the solution back to the problem situation. The below script 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: First, I’m going to read the problem to make sure that I really understand what I’m trying to figure out. [Read the problem aloud]. I’m trying to figure out ‘how many birds were on the wire before any flew away’

Teacher: Then, I’m going to write an answer sentence that will remind me what I’m trying to figure out “____ birds were on the wire before any flew away”

Teacher: Then, I’m going to solve the problem and fill in the answer sentence so that I’m relating my solution back to the original problem. “__+ 9 = 12; I know that 3 + 9 = 12. 12 were on the wire before any flew away.”

Teacher: Finally, I’m going to ask myself ‘Does that make sense?’ to make sure that my answer is reasonable. ‘I remember that there were three birds that flew away, and 9 were still on the wire. So, the number of birds that were there originally should be greater than nine, which it is. Yes! This makes sense! 

Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for interpreting his results. This example refers to the following problem.

Ronald has 2 kittens. Lupita has 5 kittens. How many more kittens does Lupita have than Ronald?

     Teacher: "What is your answer?"
     Student: "3."
     Teacher: "Can you interpret your results to make sure they make sense?"
     Student: "Ok. Well, 3 makes sense." 

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.