How to support math growth with differentiation and scaffolding

It has been one year since COVID-19 flipped our world upside down. While hospital beds filled up, classroom desks emptied out. As educators, we were not prepared for an event like this. We were not prepared to teach our students from afar and to face challenge after challenge, uncertainty after uncertainty, in what is already one of the hardest jobs out there.

Yes, maybe some of us had dabbled in creating YouTube video lessons or experimented with a flipped classroom, but no K–12 educator was prepared to teach in a fully remote environment. And I don’t say that as a way to blame anyone or anything because there is no one to blame. I say it as a matter of fact. This is what happened, and this is our reality.

Learning to seize what’s in my control

COVID-19 has forced everyone to think—really think—about what we can and cannot control. In our current learning environments, there have been many aspects that we definitely can’t control: Who has reliable internet access. Who turns their camera on consistently. Computers, as they decide to restart moments before class begins.

Luckily there have been many things still in our control (thank goodness for small favors, am I right?). Though research shows learning gains have been uneven in math this past year, we can still rely on formative assessment to gauge how our students are doing, even if those formative assessment strategies look a bit different online. And we’ve still been able to differentiate and scaffold instruction. Schools that use MAP® Growth™ interim assessments have been using content providers that connect MAP Growth data to instructional content, or they’ve been using MAP® Accelerator™ to automatically build learning pathways in Khan Academy.

Differentiation refers to the separate tasks groups of students work on that are built to address their specific learning needs.

Reflecting on what’s within my control and what isn’t has done a lot to center me during this chaotic year. I’d like to share some strategies for differentiation and scaffolding for middle school math in the hopes that they will help you see what’s in your sphere of influence and what to let go of, too.

Differentiation and scaffolding in the math classroom

Differentiation refers to the separate tasks groups of students work on that are built to address their specific learning needs. Scaffolding is providing different levels of support to students, eventually removing those supports so kids can become self-directed learners. Often teachers provide scaffolds to students who need remediation, and it’s an excellent approach for students who need enrichment, too.

As an example, I am going to share a lesson well-suited for eighth-grade math. The Common Core Standards state that all students should be able to understand and apply the Pythagorean theorem by the end of the year.

When beginning work on this standard in my classroom, I would start by using data from my students’ most recent MAP Growth assessment, in addition to formative assessment and behavioral data, to get a more complete picture of where they were in their learning. This helped me find their respective zones of proximal development (ZPD), that sweet spot where a student has enough knowledge to approach a subject with context and confidence but not so much that they’re unchallenged by the material and get bored. Looking at the big picture all of this information created made it clear to me that my students usually fell into three distinct groups: kids needing support to access the grade-level content, kids right at grade level, and kids ahead of the curve who could benefit from some enrichment.

Scaffolding is providing different levels of support to students, eventually removing those supports so kids can become self-directed learners.

Tackling work on the Pythagorean theorem through small groups like these is a perfect example of differentiation. Because each group of students had different learning needs, temporarily separating them allowed me to tailor my instruction to each, making it a little different and better able to meet their needs. I would give each group a different task to complete.

For students below grade level

Here is an example of a task from Open Curriculum that is appropriate for that first group of students:

I often used this task, and my students were only allowed to use their meter sticks to measure the length, width, and height of the classroom. If this task is done correctly, students only need to apply the Pythagorean theorem two times: first to calculate the hypotenuse of the rectangular base (AC) and then to calculate the hypotenuse from opposite vertices of the classroom (CD). See the image below.

I chose this task for these students because it required them to think of a plan before jumping into the math. It also empowered them to use math in a physical manner—by literally getting out of their chairs and measuring things—and it encouraged the type of peer collaboration and communication that so often fosters learning. If students were struggling to start the task, I would then begin to scaffold my instruction by asking questions like “What do you know about a square?” and “Are there any right angles in squares?”

I made it clear to my students that the process they followed to figure out the problem was as important as the solution, so I wouldn’t give them the right answer right away. If they got stuck, I would scaffold further with questions like “What are the measurements you can take of the room?” and “If you draw a diagram of the room and label all the lines, are there any right angles in your drawing?” With about 5–10 minutes left to complete the task, I would share that ACD triangle and ask my students to think about how they could calculate diagonal CD. I would always ask them to share their answer and explain why they thought it was right.

(Side note: This task can easily be adjusted to a remote learning environment. In a breakout room, one student can find the length, width, and height of the room they’re in and share that data with the rest of the group. If they don’t own a measuring tool, they can use another object as a unit, like a notebook).

For students on grade level

For the second group of students, the ones who are right on grade level, this problem from Illustrative Mathematics is appropriate as it requires application of the Pythagorean theorem in real-world problems in two dimensions. It also pushes students’ problem solving and reasoning skills, because determining a strategy requires analysis of the details of the problem.

If students are struggling with starting the task you can scaffold instruction by asking a probing question, such as “Would it be easier to think of a strategy to solve if you had a diagram?” or “What information do you have and what are you trying to solve for?” Sometimes it is necessary to scaffold with redirecting questions, such as “Can you point to the shortest side of the tray in your diagram?” and “What is the size of the opening the manager is referring to?”

It is always important for any students below grade level to get exposure to grade-level content and, in this particular example, to the level of problem solving and reasoning skills required to solve this task. To do this, I recommend using this task as part of a whole-group discussion following small-group work.

For students above grade level

The third group of students can be given this task from the Mathematics Assessment Project:

The students given this task are ready for content above grade level, and it’s perfect for enrichment for a few reasons: The solution is not obvious or inferred. A strategy is required prior to beginning. There are many solution pathways but only one shortest pathway.

Similar to the fishing pole problem, this task allows for deep conversations about how students know the path they created is truly the shortest one. Because students in this group are performing above grade level, I encourage you to require that they explain their thinking in writing as well as out loud.

As with previous tasks, scaffolding should occur prior to students moving beyond productive struggle.  For example, I might ask, “Can you show a solution by drawing in one possible pathway?” and “What are some conditions for the shortest route?” I would also ask students who are finished, “How do you know this is the shortest route?”

You can do this

When we identify a student’s ZPD and use ZPD to guide our scaffolding and differentiation efforts, we create a space for students to take more ownership of their learning and to trust themselves and each other. Regardless of whether they’re learning online, in person, or in a hybrid format, students will benefit from differentiation and scaffolding.

As you’re reading this, maybe you’re worried that you can’t do this for your students. I understand your fear. This work takes time, and the unfinished learning many students are facing can feel daunting. But maybe believing is where it all starts. Believing you can do this for your class.

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