# How to Respond to Significant Gaps In Mathematical Understanding

Whether you are a classroom teacher or an interventionist working primarily with small groups you may have encountered a student or group of students who have significant gaps in mathematical understanding.

Determining the next steps can feel overwhelming when they seem to need “everything” but we can simplify the process for determining the next steps so that your students are primed to make significant progress in the school year ahead!

You absolutely do not need a fancy assessment or screener to get this done. The purpose of this assessment is so that you know where to start with your student. Nothing more and nothing less. This assessment can be as simple as sitting down with your student with a white board and a pile of linking cubes and just rapid-fire asking questions to get a sense of what they know vs. what they still need to learn.

Prioritize learning what your students know about numbers and what they know about operations rather than focusing on the gaps in their mathematical understanding.

1. Can your student recognize and name numbers to 5? 10? 20?
2. How high can your student count?
3. Can your student count a set of objects to 5? 10? 20?
4. Given 2 single digit numbers can your student add them together?
• How do they do adding numbers within 10? 20?
• If your student has difficulty adding within 10, can they do so if you provide linking cubes or a marker and whiteboard to draw on?
• Do you notice them using counting all, counting on or related fact strategies?
5. Can your student subtract within 10? Within 20?
• Do you notice them using counting all, counting up, or related fact strategies?
6. Given a 2-digit number and a 1-digit number does your student have a strategy for putting these numbers together? How about for any sums within 100?
7. Given a 2-digit number can your student represent that number using a tool such as base ten blocks?
8. Given two 2-digit numbers can your student tell you which is larger? Which is smaller?

Feel free to use the questions above to drive your interview but, in general ask yourself:

1. Does this student understand numbers to 10? Can they apply operations to 10?
2. Does this student understand numbers to 20? Can they apply operations to 20?
3. Does this student understand numbers to 100? Can they apply operations to 100?

You are searching for the numbers your students is comfortable working with and then whether or not they have strategies to work flexibly with those numbers. These checklists may also be a helpful tool.

Don’t be overwhelmed by the fact that these interviews will take time. You do NOT need to interview every single student in your class. This is simply a fact-finding mission that will help you determine the next steps for your most at-need students. You can read more about intervention assessment strategy here.

## Notice What is Happening in Tier 1

Ideally, you would be able to meet your students exactly where they are and work on exactly the skill that *they* need next. Unfortunately, your students are being exposed to tier 1 instruction daily. This exposure is helpful for some students but in other cases this exposure can be less than productive.

For example, consider a student who doesn’t recognize numbers to 10 or understand the meaning of addition but is sitting in a first-grade room where the class is working on counting up strategies for subtraction and is working on the example 8 – 3. This student is not able to access the example at hand because they do not recognize or understand the number 8. Further, they don’t have a concept of addition or subtraction so counting up to subtract may be compounding their confusion around operations that they are barely grasping to begin with.

In this case, where would you start with this student in math intervention? You may be tempted to spend more time recognizing and developing number sense around numbers up to 10. And if this student was in a kindergarten classroom this would be a very appropriate response. But, in the context of a first-grade classroom we might need to consider taking what the student knows (numbers to 5) and spending some time on addition and subtraction to 5 so that they are building a strong foundation in these operations rather than becoming more confused and frustrated in the tier 1 setting.

## Develop a Plan for Filling In Gaps in Mathematical Understanding

Once you have determined what your students know, and you have considered what is happening in the tier 1 setting that might impact your choice around the next steps, you need to develop a specific and targeted plan for filling in the identified math skill.

Given the example in the previous section, you might develop the following goals

1. Student will add numbers to 5 using a count-all strategy and hands-on materials with 85% accuracy.
2. Student will subtract numbers within 5 using a count-all strategy and hands-on materials with 85% accuracy.
3. Student will identify numbers to 10 with 100% accuracy.

For the most part, take your goals one at a time. In this case I would likely start a unit on adding numbers to 5 and then I would also add in a quick fluency activity at the beginning or end of my math block that continues to give my students exposure to recognizing and identifying numbers to 10.

My math block might look something like this:

• 3 minutes Jack Hartman Counting to 10 Song– Especially if my students are coming from an academic block where they have been sitting, starting with a song gives your kids a chance to move before you begin!
• 7 minutes Instructional Activity on Adding to 5 (Teacher Led Activity)
• 10 minutes Independent Practice Activity on Adding to 5
• 3 minutes Exit Ticket/Quick Assessment to determine student’s understanding of the day’s lesson
• 3 minutes Number Line to 10 Game

## Continually Assess for Progress

As your students are working toward their goals, continually assess. Assessment includes observation of daily lessons, a quick check-in at the end of each lesson or a more formal assessment when you think you are approaching the end of a unit and your student has met their targeted goal.

In any case, constantly look for and document progress in your students. This will help you to continue to push ahead with your students rather than getting “stuck” in a topic and it will also provide the documentation that you need should you pursue further supports for a student.

If you are using my math intervention units, these free assessment documents are a helpful tool for documentation.

## Related Materials

If you are looking for short and systematic math intervention units including pre-and post-assessment, instructional activities, independent practice, and daily quick assessments, check out these full-year math intervention bundles.

These units make it incredibly easy to target a specific skill in a systematic way while keeping your students engaged as they make progress!