I don’t like the word differentiation. Teaches assume it means “give students different work”, when I see differentiation as a set of tools used to make sure all students learn, regardless of academic background, prior experiences, or identity. I think setting ambitious goals for all students to work toward together is one of the best things about a classroom community. I think there are a range of strategies I can use to do that, and I don’t have a better word to lump these strategies together than “differentiation”.
Low-Floor, High-Ceiling Tasks
A task with a low floor and a high ceiling doesn’t need to be anything fancy. It can be a pretty humble task that has features to provide access and extension to more students. One example I like is a card sort. I often use a lesson from the Shell Center in a unit on exponential functions, and have students complete this card sort:
Students match graphs, tables, equations, and descriptions together, each describing a situation of either simple or compound interest. The sorting makes this task accessible for more students, as they can start to make associations even without much fluency with exponential functions. There are two blank cards for students to fill in missing equations, and blank spaces in the tables for students to fill in missing values. Some of the descriptions are also trickier than others, and lead to analysis comparing how often different situations take to double and other potential extensions to challenge students. The entire activity helps students to connect different ideas and make explicit the different ways that exponential structure plays out in compound interest while distinguishing exponential growth from linear growth. The lesson also includes thoughtful follow-up tasks for students to do more work with these ideas and apply them in different situations.
Teaching to Big Ideas
Teaching to big ideas means exposing students to key understandings early in a unit, and returning to them over and over again. I just taught a unit on conic sections, and one of the big ideas is that replacing every x with an x – 2 moves a graph two units in the positive x direction, and replacing every y with a y + 3 moves a graph three units in the negative y direction. More broadly, these transformations can be used to translate a graph in any direction.
Students first see these ideas with circles, ellipses, hyperbolas, and parabolas. I like to give students an unusual example to further emphasize the big idea. This equation creates an interesting graph:
While it’s not a conic section, it does behave in similar ways. I ask students to describe how to graph a general version of this equation using similar strategies to the ones we use with conic sections. Some students explore with vertical and horizontal translations; others get into different types of dilations, and some can be challenged to analyze changing the signs and exponents to see what happens. There’s lots of great structure here and connections to other ideas, and students can engage with the big ideas on multiple levels through the same problem.
One advantage of this approach is that a focus on this big idea extracts the essential mathematics from conic sections while putting less emphasis on things I care less about. It’s easy in that unit to get caught up in manipulating conic sections algebraically. While that can be a useful goal, it’s also one that requires much more background knowledge, and serves less of a purpose in the broader mathematics progression. Focusing on this big idea, important in conic sections and recurring in other topics, helps to make sure that what students are thinking about is the math that will be most helpful to them in the future.
Making Big Ideas Explicit
While the task I used above provides students with a useful opportunity to engage with a big idea, working with that big idea without making it explicit leads to a less equitable classroom. Many students will make connections on their own, and figure out that the structure of this function has a lot in common with other conic sections. But some won’t, and those who won’t are likely to be students who have struggled with math in the past, compounding gaps that already exist. Making the big idea explicit means pausing during the lesson to discuss what students notice and what strategies they are using, and reflecting at the end of the lesson on how these ideas are connected with ellipses, hyperbolas, and other objects. These discussions and reflections are great learning opportunities for everyone, whether students are practicing articulating their ideas using mathematical language, pausing to notice features they might have rushed past while focused on completing the task, or seeing a connection for the first time.
In the same way that I need to structure that task to make big ideas explicit, I need to make them explicit at every other step of the way as well, both leading up to this task and as this idea comes up in the future. It’s easy to forget that students see math differently than I do — that they don’t have the background knowledge I have and don’t see the broader structure of math as readily. I need to constantly remember to make essential ideas explicit so that every student has an opportunity to engage with them, rather than leaving it to chance. For more thoughts on big ideas, read David Wees’s piece here.
Aim for Relevance
I don’t want to assume that every student finds math worth learning. I want to do what I can to help all students feel a sense of ownership and enjoyment in my class. One way of doing that is offering multiple perspectives on why students should learn math. I try to capture moments of wonder and curiosity, and help students have those a-ha moments that will make math class enjoyable. I frame math as a series of puzzles to be solved, practicing skills that will help them solve more puzzles in the future. I dive into applications of topics like exponential growth with examples of financial advice for students to evaluate based on what they’ve learned. I give students a window into how different ideas are relevant to higher math that they might study in the future. And I talk about how some math is worth learning just because it’s fascinating — mathematicians studied prime numbers largely out of curiosity for thousands of years before they became the building blocks of today’s web encryption standards. None of these arguments for learning math is sufficient on its own — each appeals to different students at different times. And not every student will be engaged every day; I need to be realistic in my goals. But offering a range of ways for students to make meaning of math class helps more students to come to class motivated and ready to engage.
Scaffolds are probably the strategy I think about the most, but they are also the hardest to master. I need to provide scaffolds that help students to access content, while also taking them away when possible. One strategy I’ve found useful is scaffolding different parts of a task at different times. For instance, after introducing the different parts of rational functions, I might ask students to analyze the end behavior of a group of functions. Then, I’ll give them this task, graphing rational functions when already given end behavior, vertical asymptotes, and intercepts:
Then, I’ll ask students to find all relevant features of a function without graphing. Then they will put it all together. Varying the scaffolds helps students to focus their attention on different parts of the whole, and builds fluency in chunks that are manageable before attempting the entire task.
Building relationships is both important and subtle. A lot of relationship building happens in small ways, inside and outside of class. One way I approach relationships systematically is to pay attention to particular students that I know often feel bored in class, or often struggle and feel confused. Say I’m having students engage with an Illustrative Mathematics task in a unit on exponential functions and logarithms:
Every student will experience this task differently. While students are working in groups, I am monitoring for different strategies to share with the class and making sure students stay on track. I’m also paying particular attention to students who often feel bored or students who make be bored in that task, trying to offer them an extension and doing what I can to keep them engaged. I’m also paying particular attention to students who often struggle and feel confused. My goal is that every student engages with the big ideas of the task to an extent where, when we go over it, they will be able to engage and learn from others’ strategies and understanding of the math. By timing student work around these students, I can make decisions deliberately that help every student engage and help every student feel like they can be successful. There’s no perfect solution, but by keeping tabs on those students’ experiences, I can make those decisions a little better.
These strategies are the result of a paradigm shift for me. One paradigm is that students come into math class with different levels of knowledge and skill, and I need to offer them different experiences to meet them where they are. A different paradigm is that students come in with different levels of knowledge and skill, and if I look at all of my instructional decisions through that lens I can provide meaningful, common experiences that help every student engage with mathematical thinking.
None of these strategies alone solves the problem of students with different backgrounds and different experiences trying to learn the same concepts. I still think that is a goal worth working toward, and I think that these strategies together can make a difference in moving in that direction. None are easy to implement — I’m still getting better at all of them. But I’m optimistic that that improvement will make me a better teacher for the students who most need it.