Reinventing Mathematics Intervention

A focus on unfinished learning in mathematics has brought intervention supports into the spotlight. This attention is long overdue. We know that our intense efforts to assist students who are significantly far behind are not resulting in desired outcomes. What follows is guidance on how to lead a productive change. 

The first step in transforming intervention is to embrace the idea that students who struggle in mathematics likely do so because there are foundational gaps in their conceptual understanding of prerequisite content knowledge. Those gaps widen as more content is addressed. The urge is often to “pre-teach” upcoming content so that when the learners who struggle encounter the content, they are more familiar—and ultimately more successful—in class. The flaw in this plan is that the pre-teaching is often necessarily procedural because the conceptual understanding is elusive without the foundational knowledge the students lack.

Confounding the issue is that many people who are tasked with providing the intervention lack access to professional development on best practices in mathematics teaching and learning. When intervention is provided in small groups, for example, teachers are using precious face-to-face instructional time with students but potentially not in a manner that will likely lead to desired outcomes. What do we do about it? We need to reinvent intervention.

Six features for reinventing intervention

The reimagining of intervention involves a multifaceted approach. In this article, I share six features that, when taken together, constitute a reinvention of intervention.

1. Focus on conceptual development
2. Connect concepts and procedures
3. Prioritize strategic content
4. Support discourse through engaging tasks and targeted questioning
5. Elicit and linger on common errors
6. Provide scaffolding just in time rather than just in case

These features are appropriate for all learners, regardless of their intervention needs, but also strategically cover ground that best support students who are struggling. Collectively, the features relate to emphasizing conceptual development, linking concepts to procedures, selecting content strategically, using productive discourse, uncovering and using important errors, and providing opportunities for students to develop perseverance.

Feature 1: Focus on conceptual development

The idea that mathematics is best learned by focusing on conceptual understanding prior to procedural fluency is not new (NRC, 2001). Focusing on understanding first provides a foundation to support students when they don’t remember when and how to apply specific rules and algorithms. Consider the multi-digit addition problem 376 + 258. It is important for students to understand why it is helpful to “line up” place values when adding. By making sense of the purpose of adding ones with ones, tens with tens, and hundreds with hundreds, students are preparing for success when adding problems like 3.76 + 25.8. Students who understand the role of place value in multi-digit addition will be more likely to avoid errors like this when adding decimals later:

In this case, the student overapplied a rule of lining up the addends regardless of the place values of the digits. They are aligning along the left or right, rather than lining up ones with ones, tenths with tenths, and so on.

Teaching for conceptual understanding is often more time consuming than teaching procedures. However, the time spent on foundational concepts should help students to access related on-grade level standards they encounter during tier 1 core instruction.

Feature 2: Connect concepts and procedures

Focusing on conceptual understanding prior to procedural fluency is not enough. To receive the return on the investment of time and energy required to teach conceptually, teachers must explicitly connect what is explored conceptually to what should be practiced procedurally. There is a video embedded in Making Sense of Mathematics for Teaching the Small Group (Dixon et al., 2019, p. 8) where students are guided to make the connection between adding multi-digit numbers using base ten blocks to adding with the standard algorithm. In this video, students first add 376 + 258 using base ten blocks and then they add it again using the standard algorithm. They are led to see that when they combine the 6 ones and the 8 ones to get 14 ones, the “1” they place over the digits in the tens column represents the 10 ones they exchanged for one ten using base ten blocks. For some students in the video, they are realizing for the first time the connection between base ten blocks and the standard algorithm. This connection allows students to better remember and apply the procedure on their own. The image below illustrates the connection between the “1” recording in the tens column and exchanging 10 ones for one ten.

Feature 3: Prioritize strategic content

Prioritizing key content is crucial when reinventing intervention. Multi-digit addition is an example of key strategic content for intermediate-grade students. It is strategic because it is foundational to adding decimals and performing multi-digit multiplication. 

It is not possible to address all the unfinished learning a student with significant struggles needs and have it all “stick.” When we try to reteach and pre-teach everything, the performance gap widens because there is not enough time to teach everything well. We need to be strategic.

According to the recently released joint position statement between the National Council of Teachers of Mathematics (NCTM) and the Council for Exceptional Children (CEC), a common goal is for students who struggle in mathematics to have access to and success with on-grade level content (NCTM & CEC, 2024). The challenge, then, is to determine what foundational gaps must be addressed to grant that access and success. What content provides the greatest return on investment of time, energy, and resources for students who struggle? As an author of Math 180, I helped create a course for students in Grades 3–5 where that strategic content is identified as addition, subtraction, and place value.

Feature 4: Support discourse through engaging tasks and targeted questioning

Once the content is determined, we can turn to teaching practices that support students in making sense of the content and connecting conceptual understandings they develop to procedural fluency. Key practices that support students in reasoning about mathematics include beginning with engaging tasks around important concepts, establishing discourse norms, and using questions to help students engage in those norms.

The identification and implementation of high-quality tasks during intervention is a sure way to make excellent use of instructional time. A helpful resource for identifying and implementing tasks, along with using productive questioning to support discourse, is Making Sense of Mathematics for Teaching to Inform Instructional Quality (Boston et al., 2019). Professional development on questioning during intervention should help with questions that focus on the mathematically important parts of tasks; this is typically where students make common errors.

Feature 5: Elicit and linger on common errors

Using errors as springboards for learning is a crucial aspect of successful intervention. Students who struggle with mathematics are likely to make common errors. Intervention is an excellent space to unpack the errors so that students can move past them. This is accomplished by using just the right tasks that we anticipate students will get wrong. When students make common errors, as they are likely to do, the teacher can then help the students to learn from them and be less likely to repeat them during independent work. The goal here is to deliberately elicit the errors and linger on them so that students can see the errors’ origins and avoid them in the future.

Feature 6: Provide scaffolding just in time rather than just in case

Eliciting errors from students is not typical behavior during intervention. Teachers are much more likely to provide scaffolding “just in case” students might need it so that their students will avoid making the errors and struggle less. While this teacher behavior comes from a place of caring for and nurturing students, it is not productive for learning.

The problem is that if scaffolding is provided just in case students might need it, students don’t have the opportunity to develop perseverance in a supportive environment. They are also less likely to make the errors that they would typically make during independent work. As a result, those errors are not used as springboards for learning. The goal is to provide scaffolding just in time, when the struggle becomes unproductive.

Putting the features together

It is likely that you are already implementing some of the six features discussed here. What features might you try next? Be sure to explore other posts that are available on Shaped to find ideas and strategies that work for you and your students.

Works cited

Boston, M., Candela, A. G., & Dixon, J. K. (2019). Making sense of mathematics for teaching to inform instructional quality. Solution Tree Press. 

Dixon, J. K., Brooks, L. A., & Carli, M. R. (2019). Making sense of mathematics for teaching the small group. Solution Tree Press.

National Council of Teachers of Mathematics & Council for Exceptional Children. (2024). Position statement on teaching mathematics to students with disabilities. Retrieved from https://www.nctm.org/Standards-and-Positions/Position-Statements/Teaching-Mathematics-to-Students-with-Disabilities/

National Research Council. (2001). Adding it up: Helping children learn mathematics (J. Kilpatrick, J. Swafford, & B. Findell, Eds.). The National Academies Press. https://doi.org/10.17226/10434

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The views expressed in this article are those of the author and do not necessarily represent those of HMH.

Looking to unlock mathematical learning in the students who need it most? Explore our math intervention programs for students in Grades 3–12.