Years ago, I served as a math specialist at a small progressive pre-K–8 school. The classes were generally small, and during one year in particular, the fifth grade population happened to be predominantly male and included a few precocious boys who tended to dominate math class. The parents of the girls in this grade banded together and asked that the girls be taught math separately to make sure their developing mathematicians’ minds weren’t squashed by the vocal majority. So I was called in to teach this experimental class.

I got to teach a wonderfully small class of nine students. Yet as small as the class was, it was full of big personalities. Fifteen years later, I can still remember every child and can describe each one as a math learner. I can describe what kind of work they liked, the attitudes they brought to math class each day, what their notebooks looked like, how they would respond to different problems—indeed everything. The class was a teacher’s dream: small enough to get to know every student well, to give thoughtful feedback on their work, to take them on excursions where they could apply their math learning, and to communicate with their homeroom teachers and advisors easily and often if something came up.

Even though the class was half the size of most other math classes, the typical range of student strengths and challenges still existed. Some had historically excelled in math and gobbled up every challenge; others were still learning multiplication facts while having sub-zero confidence in themselves as math learners; and a handful of students were in between. To meet their different needs, I provided differentiated classwork and homework as often as possible. We’d investigate the same grade-level material in class, but I offered assignments with three levels of “spiciness,” to borrow a term used by Peter Liljedahl in Building Thinking Classrooms. Students could pick which level worksheet they took, with my guidance and oversight. The worksheets differed mainly in how difficult the numbers were to work with.

This work had more differentiation than a lot of teachers are able to provide each day, and I could only do it because my primary role was as a math specialist, which meant I wasn’t managing classes and students for the other seven hours of the school day. I had time to think carefully about this class and make three versions of the same homework.

However, in spite of providing three levels of differentiation, I still found that the work wasn’t exactly right for any one student. And herein lies the problem with averages: the entire concept of an average is built on no single person being average. (There’s a fantastic 99% Invisible podcast episode on this.) Even though I had a class of nine, with three levels of worksheets, I was still “averaging” the math content into what I thought the most novice, the most middle-level, and the most accelerated learner might need for that topic. The reality was that for any given assignment, one student might benefit from a graphic organizer while another needed differently worded directions. One student might have done well having a multiplication chart available while someone else could have used an extension beyond the “spiciest” level I provided. What I found was that three levels of differentiation for nine students wasn’t providing enough differentiation.

This is where you as a teacher might start feeling your heart race and wonder, “Is she suggesting that teachers attempt to tailor every single assignment for each student? If this was barely sustainable, or effective, in a class of nine, how on earth can it be done for 60, 90, or 120 students?” Don’t worry, I am not suggesting this approach.

What I found was that certain problems allowed students to work to whatever level they were comfortable. These weren’t leveled problems, and they weren’t problems from our textbook. Instead, they were problems I found on the (now defunct) Math Forum or on other websites. They tended to be longer than a traditional math problem to offer multiple entry points and ways of solving. They were often classic problems that mathematicians have wrestled with for generations, but that involve skills students learn in elementary school. I began calling these problems “self-differentiating problems.”

Why “self-differentiating”? Because these types of problems seemed to meet each student where they were and enabled them to work to whatever level the student was comfortable with within the problem. Even my three levels of differentiated worksheets couldn’t differentiate this well as each student was limited by the problem on the worksheet. If I urged someone to take a “mild” worksheet (the most straightforward/easiest access level, in Liljedahl’s terminology), that student’s understanding of the material was limited by the work on the page. They couldn’t access the more challenging math because it wasn’t on the page in front of them.

You may be more familiar with the term Jo Boaler has popularized, which is “low-floor/high-ceiling” problems. According to HMH, a low-floor/high-ceiling problem is one “where all students can find their way in, then they can stay engaged for however much time they have.” The low floor means there are no barriers to access—a student can engage with the problem no matter where their understanding is at that moment. A high ceiling means there is a lot of mathematical depth to the problem—the task can be extended easily and also lead to other meaningful math explorations.

Here’s an example of a classic self-differentiating problem and how you can use it in a classroom.

Let’s say you teach fourth, fifth, or sixth grade and want students to understand multiples and factors and how to find them, as well as prime versus composite numbers. Maybe you also want them to be able to identify perfect squares. We might give students a problem set asking them to list the factors or multiples of each number, to identify the perfect squares, and to label numbers as prime or composite. If we want mild, medium, and spicy versions of this, we’d probably just change the numbers to be higher or more difficult ones to work with. A problem set like this provides rote practice, which is sometimes necessary, but it doesn’t call for any creativity, problem-solving, or exploration. Some of your students will struggle, and some will finish in five minutes and declare that it was easy. These problems include no higher math concepts for students to explore, should they so choose.

So what if, instead, we provided a problem like “The Locker Problem,” as I did with my fifth-grade class?

Your school has a long hallway with 100 lockers lining the walls. All the locker doors are open at the beginning of the day, until Student #1 walks through the hallway and closes each one. When Student #2 walks down the hallway, they open locker #2, as well as every other locker after that. When Student #3 comes down the hallway, they change the state (either to an open or closed position) of locker #3 and every third locker after that. Student #4 enters the hallway and changes the state of locker #4 and every fourth locker after that. This pattern continues until 100 students have walked through the hallway. After all 100 students have walked through, which lockers are open and how do you know?

If students figure out which lockers are open, we can follow up with more questions: Which lockers are closed? What number of students touched lockers 36 and 48? How many lockers were touched twice? Which locker was touched the greatest number of times? What if there were 200 students? 1,000 students?

There are a myriad of tools and approaches that can help someone approach this problem; the hardest part for most is figuring out where to start. I’ve seen students try to keep track of which of the 100 lockers are open using interactive virtual lockers, hundreds charts, and drawings on whiteboards. One of my fifth graders decided to create a cardboard model of lockers with functioning doors! Students could even act out the problem in a hallway of lockers. The low floor here is that anyone can access the problem once they understand the scenario. The high ceiling is…well, I won’t tell you in case you haven’t worked on this problem and want to. If you’re really curious, you can read Illustrative Math’s commentary on it. But there is some great math exploration to be done.

Like any good math problem—and, I would argue, any self-differentiating problem—it calls on all eight Standards for Mathematical Practice, as quoted below. When students work on “The Locker Problem,” they have to:

1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning

I understand that you can’t give this kind of problem every day. Just as there is a place for exploration, there is also a place for explicit instruction and for skill practice. But in moments where you have identified exploration and discovery as the goal, see if you can find a self-differentiating task that will meet every student where they are. If you’re not sure where to find these, here are some places to start:

• Open Middle
• Nrich
• Problem of the week bank at NCTM
• Rich Tasks at MathforLove
• Activities and Tasks at YouCubed^®

References

Blankman, R. (2023, July 13). Low floor, High ceiling Math tasks. HMH.

Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. Jossey-Bass/Wiley.

Liljedahl, P. (2020). Building thinking classrooms in mathematics, grades K–12: 14 teaching practices for enhancing learning. Corwin Press.

University of Texas at Austin, Charles A. Dana Center. (n.d.). Mathematical practice standards. Mathematical Practice Standards | Inside Mathematics.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

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