The 5 Practices: Looking at Differentiation Through a New Lens

By Catherine Castillo

Our district had seen a downhill trend in standardized test scores in mathematics. This forced us, as educators, to take an intentional look at our teaching practices.

The past few years have been an exciting time in math instruction. Research on brain plasticity and mindset have caused a shift in the idea of what it means to know and do mathematics.

The problems students are being asked to solve on our state assessments and in the workforce require interpretation, analysis, and resilience. We knew we had to change our instructional practice to reflect the rigor of the tasks they would be asked to solve—yes, on the test, but ultimately as future employees and citizens of our world.

Gone are the days of “I do, we do, you do.” That model never allowed for enduring understanding of conceptual mathematics. It was designed to promote recall and computation and it succeeded in that…for some students. However, we now know that we are preparing students for a world that no longer requires such skills. Our charge is to develop thinkers. Problem solvers. Grit.

Teachers wrestle with the conflict between the way they learned to teach and what research now shows as best practice.

As we began our implementation of 6–8 Illustrative Math, our teachers were struggling with that very issue. It was apparent that we had lots of work to do in supporting teachers in their new roles as facilitators.

The IM curriculum was designed for much of the time to be spent with students working in heterogeneous collaborative groups and partnerships while the teacher monitored and facilitated discussions. When grade level teachers met after implementing a few lessons in Unit 1, they discussed their struggle with what differentiation looked like in this model.

It became clear that we had had a very narrow vision of differentiation. This required us to look at differentiation in a new light: one that empowers students to be the mathematical experts in the room.

We looked deeper at The 5 Practices for Orchestrating Mathematical Discussions to begin that shift.

  1. Anticipate
  2. Monitor
  3. Select
  4. Sequence
  5. Connect

Teachers carefully plan and monitor student interactions, and then use student work to connect strategies and support students at each level of their learning.

IM allowed this to become common practice due to the problem-based structure of the lessons and the ample opportunities for formative assessment.


IM has structured the teacher support materials to offer possible student solutions for each of the warm ups and activities. Teachers can use these to plan what strategies to call attention to during class discussions. This was an invaluable support. It can be difficult for teachers to generate and anticipate all possible solutions on their own.

Monitor, Select, Sequence, Connect

During the warm up and activities, students approach the concept in different ways. The teacher monitors by watching students approach the problem and recording strategies.

The teacher then selects which strategies to highlight based on instructional goals for that lesson.

Imagine a grade 6 classroom where students are working on exploring ways to think about division in order to prepare them for division of fractions.

Students are encouraged to approach the problem in different ways. The task they have been presented:

During a field trip, 60 students are put into equal-sized groups.

  1. Describe two ways to interpret 60?5 in this context. 
  2. Find the quotient. 
  3. Explain what the quotient would mean in each of the two interpretations you described.

This task starts by asking students to reason about what the equation means in relation to the problem. The question highlights the importance of sense making by asking them to show two different scenarios for what the equation might represent. This is a critical concept for students as often we see students simply look for numbers in a problem and apply an operation if they aren’t given a reason to make sense of the situation.

As the teacher monitors students as they work, the teacher might notice that several different strategies are being used for part b. Here are some examples of student responses:

  • Student A uses repeated subtraction by taking 5 from 60 repeatedly and then counting the number of times it has been subtracted.
  • Student B uses multiplication to solve the division problem by stating that they know that 12 x 5 is 60 so $60\div5$ must be 12.
  • Student C uses a tape diagram to partition by 5’s until they get to 60.

The teacher then sequences the order in which students will share their work or thinking. Maybe Student A will share their repeated subtraction strategy, followed by Student C’s tape diagram. The teacher might ask connecting questions like, “What is the same about these two representations? What is different? The teacher might then have Student B share and ask, “What connections do you see between this strategy and those shared earlier? Which do you think was more efficient?”

This process could be used with several examples of student work depending on the goal for students and what the teacher sees as he or she monitors students as they work. If a majority of the class is using the repeated subtraction strategy, the teacher may choose to show this one and a connection to another piece of work as above. If most of the students are using the relationship between multiplication and division, the teacher may choose to share that strategy first.

In the past, it was difficult to structure lessons in this way due to the types of tasks used in the classroom: problems that focused on computation and procedural recall. The rich tasks posed in IM 6–8 Math allow for multiple entry points, and for the procedural fluency to build from conceptual understanding. Because of this, all students are positioned to have a voice in the mathematical conversation.

Teachers have shared that one of the greatest struggles in shifting to this pedagogy is having the right questions to ask. We use this monitoring sheet that includes a place to note possible questions that might be asked to connect strategies or dig deeper into student understanding. Teachers can brainstorm these questions after anticipating student strategies so that they are better prepared to support thinking during the lesson.

I truly believe that problem-based resources, like the curricula developed by IM, have the potential to not only support students in developing enduring understanding of mathematical ideas, but also to develop the capacity of teachers in becoming expert facilitators of student learning.

Next Steps

Our charge is to develop thinkers. Problem solvers. Grit.