By Derek Stoll
Math teacher at Syracuse City School District
We knew we had to do things differently at Expeditionary Learning Middle School (ELMS) in Syracuse, NY if we wanted different results from our intervention efforts in mathematics. First, we needed to position every student as a mathematician and set expectations that every student has the right to have their thinking honored and stretched outside of Tier I instruction. Second, we needed to raise awareness that math interventions were as critical to our students as interventions for English Language Arts and organize the people and time to make it so.
In the first part of this twopart series, we look at how we used strengthbased approaches to advance conceptual understanding and the importance of the three F’s of intervention: focused, flexible, fluid.
Taking a strengthsbased approach to addressing mathematical gaps and advancing student conceptual understanding
Focus the intervention on what a student “does or knows” rather than focus on what they “can’t do or doesn’t know”. Often as classroom teachers and interventionists, we start with what a student doesn’t know. “This student can’t do multiplication!” I hear that from teachers all the time. But what does that really mean? One of the most important things I have learned from my intervention work at ELMS is to “Let the students’ work lead.” The types of tasks/problems we give kids has a direct correlation to the kind of work they show or don’t show us. For example, consider these two types of tasks.
Task A 
Task B 

John has 34 points in math. Jeanne has 6 times as many points than John. How many points does Jeanne have? 
34 X 6 
What types of things might be learned about students by asking Task A rather than the Task B? Or conversely, what types of conceptual thinking might be missed by only asking students to complete Task B? An intervention has to start with what students do and know or we, as teachers, risk assuming our students can perform at grade level and allowing them to advance with major areas of unfinished learning that we should have addressed within the intervention.
So, at the start of the year, we screen every student with three question screeners nested within the domains of Operations and Algebraic Thinking with a focus on addition/subtraction, Operations and Algebraic Thinking with a focus on multiplication/division, and Number and Operations – Fractions with questions that look like task A. The outcome is to better understand students as “problem solvers” and how they use their mathematical reasoning, rather than “question answerers”. This not only allows for determination of their strengths and areas of need, but also the way they reason, model, and problem solve in different contexts. For example, when I gave task A, consider the student work below.
Graphics of student work:
Look at the variety of student responses. What can be discovered about students’ problem solving from this work? What do students understand related to multiplication? What are some strategies that students are comfortable using within this domain? And how effective were those strategies to solving the problem? These questions become the vehicle in which a classroom teacher or interventionist can target conceptual understandings or misunderstandings/overgeneralizations in addition to filling any areas of unfinished learning.
(For more about assetbased approaches to mathematics, check out these articles.)
Creating interventions that are focused, flexible, and fluid
After the initial screening comes the fun part—how to use the information to create focused, flexible, and fluid interventions for EVERY student. Often with intervention programs, math teachers get focused on the pointbypoint trajectory for a group of students. For example, a group of students might be identified as “2 grade levels behind in math”. So, a teacher decides that they can’t fluently multiply fast enough which affects division, fractions, ratio and proportions, geometry etc. However, I have found at ELMS that this is usually not the case. I can think of a whole group of students that might struggle with multiplication or division fluency but have awesome ways of using multiplication or division in different settings that is far above grade level. There is no debate that mathematical fluency plays a role in the success of student performance. However, don’t let that focus be the center of mathematical intervention. Use the taskbased or openended questions for screening to determine what students know and can do, and where there are spaces of unfinished learning that need to be addressed. Once there is an established natural starting point for the intervention, use a more targeted “progress monitor” of conceptual based prompts to really dig in on what the targeted focus of the intervention will be moving forward. This process can be seen below:
The progress monitoring and adjusting steps are exceedingly important to teachers. Often with interventions, they can end up being used as a “tracking system” where students incrementally make progress but all as one group. That might work for a student or two, but often in this model, students stay in one group for a very long time and the intervention loses focus which creates boredom and apathy. That is why flexibility is important. During each progress monitoring stage, use the student work to guide the intervention and possible adjustments made. Let the student work guide the nature of the next intervention cycle. If a student is not making growth, look at the type of intervention being given; maybe there is a different area of need that should be addressed first. If a student is making growth, then move them along, even to a different group, if necessary, because the student should still be challenged within that domain. This conversation is not only important for the student, but also for the teacher. Use the student work to better understand the “why” a student has a gap of unfinished learning, which will better address the “how” to create focused subgroups for intervention.
At ELMS in 6th grade, here is an example of how I do this work within the domain of Ratio and Proportional Relationships after the tier 1 instruction has been taught in the beginning of their 6th grade year. This happens around November in math tutorial (an Academic Intervention Services block), with any student that has been determined to have baseline conceptual understanding and procedural fluency within the domain of multiplication and division.
After the progress monitoring stage, you need to let the student work help you assess whether the student is ready for a different intervention group. Some students may move to a deeper conceptual group, others may maintain that current group, and others might move to a previous subgroup, but the adjustments are key to ensure the intervention works for all students. The intervention is also built within the domain. So, students might be receiving intervention that is below, at level, or above their currently grade level, but constantly adjusting based on the progress monitors and work within that stage. This makes the intervention span multiple grade levels and can act as a support for some students and an advancement for others.
After the progress monitoring stage, you need to let the student work help you assess whether the student is ready for a different intervention group. Some students may move to a deeper conceptual group, others may maintain that current group, and others might move to a previous subgroup, but the adjustments are key to ensure the intervention works for all students. The intervention is also built within the domain. So, students might be receiving intervention that is below, at level, or above their currently grade level, but constantly adjusting based on the progress monitors and work within that stage. This makes the intervention span multiple grade levels and can act as a support for some students and an advancement for others.
Intervention should strengthen conceptual and procedural knowledge to close an existing gap so that students can move smoothly to and make connections with other mathematics. The longterm goal of intervention should be to help students gain independent strategies and take responsibility for their own learning. This approach to intervention leads to an emphasis on bigger ideas in mathematics and their applications so that important skills do not become trivial, isolated, or fragmented.
National Council of Teachers of Mathematics. (2011, July). Intervention: A position of the National Council of Teachers of Mathematics. Reston, VA: Author. Retrieved from http://www.nctm.org/uploadedFiles/About_NCTM/Position_Statements/Intervention.pdf#search=%22intervention%22
By letting the students’ work determine what intervention group each student goes to, the focus is what each student needs, and not a onesizefitsall model. It is about maximizing the time that students get the right next instructional step for them rather than sitting in a classroom doing more work on a concept they already understand while waiting to get to the areas of math they still need to work on.
Read the next article to find out more about the structures that the school put in place and the results that the intervention programming showed in their benchmark assessments!
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Derek Stoll is a 6th grade math teacher and grade level math interventionist at the Expeditionary Learning Middle School (ELMS) in the Syracuse City School District, in Syracuse, NY. Derek is passionate about crafting experiential learning opportunities for students that are interdisciplinary and help engage students to become leaders of their own learning. He also supports students grades 68, as a CREW leader, supporting their SEL, as an academic advocate and character building. An avid golfer, and Buffalo Bills fan, he is supported and encouraged by his wife, Aliza and four children, Duncan (5 years old), Beth (3 years old), Olivia (over a year and a half) and another boy due in January.
SPECIAL THANKS: This wonderful work would not be possible without the support of our 6th grade ELMS intervention staff, including Jamie Erickson, Amber Talev, Karen Boyle, Carol Coles, Kate Taddeo, and huge administrative support from Jill Znaczko and Kevin Burns and the other supportive staff at ELMS that make such wonderful place for our students!