Category: differentiation

Getting real about the challenges of differentiation

I thoroughly enjoyed reading this post by   (@ablinstein) about a challenging class she is teaching. I love a post that begins with a real challenge, a problem that needs to be solved. She writes about a high school class that includes multiple grades, skill levels, and previous experiences with math. While typically successful in focusing her classroom around collaborative group work (using Thinking Classrooms), this class challenges her. She notices that the kids are disengaging from group work, and hears complaints that the class is moving both too fast and too slow.

Anyone who has taught has faced this challenge. This challenge is particularly common when we teach based on lectures, but that is not the case here- Anna has designed her classroom around best practices of group work and rich tasks. Other classes are working with a range of learners, but this one is not. While she is not writing specifically about students with disabilities, this post is a test case of the “good teaching is good teaching” myth. I believe that both “students need to be tracked, students with disabilities need different math” and “just put them together, good teaching is good teaching” belie how challenging it is to teach a wide range of learners.

So what does she do? According to this post, she does a tremendous amount, some of which is supported by already existing research, and some of which is innovative and should be researched. She writes

Some suggestions that I implemented that seemed to make a difference:

  • Taking a break from random groups to help students regain their trust that the class would meet their needs; doing some work in pairs designed to foster productive collaboration; allowing students choice as to who to work with while also asking them to work with different students at times; being explicit when the goal of a task was to build collaborative skills

I love that she writes here about trust, knowing that her job is to gain the trust of kids, not just in her as a teacher, but in her ability to organize a class that will work for them. She offers more options for engagement for kids in this class, individual, pair, and group, offering additional choice in whom they work with. She makes it clear that goals are not always content-focused. If this sounds a lot like UDL (Universal Design for Learning), you are right. She is building flexibility and choice into her classroom.

Structuring activities so there was time at the start for individual exploration before asking students to share their thinking with others thus giving more processing time for students who worked more slowly; circulating and helping some students get started; building more optional challenge into tasks for students who worked very quickly or who had already had prior experience with a topic; creating tasks that could be approached with a greater variety of methods and building more writing into tasks so that different ways of thinking mathematically could be valued

This paragraph is a master class in creating an inclusive classroom in math. Thinking time is super important, particularly as for many neurodiverse learners, processing time is perhaps the fundamental difference from their peers. If class rushes past them, they don’t have the time to engage to their potential. Building in an optional challenge is a great way to engage students who don’t need that time- I think we should spend more time talking about how to do that so that students who work faster are offered opportunities to think deeply, not just move on to the next topic. She writes about creating rich, multi-leveled tasks, and basically notes that she had to be more careful about the tasks being multi-leveled as her class was more heterogeneous in terms of skill level. She looks to her tasks, rather than to deficits in the kids, as a point of change.

Meeting students where they were to regain trust and buy-in; this included at times splitting the class into two groups (students chose which group to join) – a more free-form exploratory group with more open and challenging problems and a more structured group where students would get some problems to activate prior knowledge and smaller, more concrete problems that would build over time to greater generalization and abstraction and more teacher guidance and reassurance that they were on the right track

This is a great strategy, and very UDL, as it is focused on students making the choices about how they learn, rather than a teacher doing the categorizing and sorting into differentiation groups. I would love to hear more about how these groups worked, and how students responded to them. Also, did students always pick the same groups? Did it vary by topic?

Noticing struggling students’ successes and highlighting them publicly; selecting which students would share their thinking to make sure that different voices could be heard over time

Here we see the important insights from Complex Instruction about status treatments, particularly important when working to establish status for kids with less status in heterogeneous environments.

Make sure to leave time for synthesis and practice problems (at different levels) during class – this helped address student concerns that they were leaving class with lots of questions and feeling unsettled about the concepts they had explored that day

Giving students more feedback during class about their understanding of a topic rather than relying more heavily on groupwork and self-assessment for students to know how they were doing and what might be helpful next steps

Here, she attends to making sure that students leave the class feeling secure in new knowledge. She attends to synthesis. Feedback is a critical element of learning, with kind of an unfortunate name that suggests a computational model of learning. But feedback is really getting human interaction around your work, to see what you think mirrored in another, and can be particularly important when coming from the teacher, since our kids (mostly!) value what we think about what they think. It makes our kids feel valued.

Providing more problems at different levels and helping students navigate which problems might be more helpful for them to do during/after a particular lesson – here is an example of a tiered homework problem set.

I love this example of a tiered homework problem set. It begins by laying out the essentials, then gives problems that are Important, Interesting, and Challenging. I love that these are not reducible to Easy, Medium and For The Smart Kids. The first set is Important, which is why you should do them.

This post helps us build on the important work of Complex Instruction, layering in practices that allow kids to learn in their Zone of Proximal Development, making choices about their own learning, which leads to increased meta-cognition.

Yes, as Anna notes, this is an unsustainable amount of work.  Yes, this is a tremendous amount of work, but the structures that you put into place are repeatable. That lovely homework, for example, is now made! Can departments, can the #MTBOS share these kinds of differentiated assignments?  UDL is meant to solve problems, and then to help save you work, to build flexible supports into classrooms as a key aspect of design, not afterthought.

And this post is a perfect example of UDL in action. Here, she views curriculum and pedagogy as the problem, not the kids. She takes up a problem-solving attitude, working to redesign the classroom around the edges. This classroom redesign is not about the mythical middle, but working to make the class work for those who are on the edges, both needing more time and less, etc. The redesign builds in choice and flexibility as the primary tool to accomplish that.



Counting Collections and Inclusion

I love Counting Collections. As a classroom teacher, I would always have my students count everything in the classroom, differentiating based on what number set they needed. We would work on representing numbers, and we would use charts to represent groups of ten, pushing understanding of place value. We didn’t call it Counting Collections, but a similar practice. I always saw it as a great activity for a class where kids were learning to count very different number sets, since I could easily differentiate within the context of counting. But now I don’t have to write that post, since Heidi Fessenden () recently wrote the best one ever. It is a beautiful post about her work with counting collections and students with disabilities and inclusion. Read it! It is beautiful!

Counting Collections: One Nearly-Perfect Answer to Inclusion

Her post highlights several critical features of great mathematics teaching for inclusion. First, throughout the piece, she frames her own work as a problem-solver, not as a expert, not as someone who has already figured it out. Heidi writes how difficult it has been to figure out how to include students with autism, who need additional scaffolds and support, into a general education mathematics routine. That attitude, having a problem-solver approach, is how she IS able to include all children in beautiful, collaborative learning. She tries, and if she fails, she thinks, she problem-solves, and she tries again. She provides kids with partners, and understands that both partners will learn, not just the child with the disability. Heidi incorporates collaboration, broadening the concept of learning beyond the individual. She provides scaffolds, like a visual schedule for the counting collection. I also love how she learned to not micromanage! Students with disabilities, particularly autism, are often micromanaged to within an inch of their lives. They are offered very little space to think, to grow, and to make choices. Heidi gave them space, gave them scaffolds, and let them count!



Our Kids Are Not Swiss Cheese!

Our Kids Are Not Swiss Cheese!

This blog post was co-written by Andrew Gael and myself, and available on both our blogs.

A couple of years ago, Andrew Gael and myself were talking about how students in special education are conceptualized. We were sick of hearing about the “gaps” and the “holes” in our student’s learning. One of us, (probably Andrew!), blurted out – ” Our students are not Swiss cheese!” We laughed, since this summed up for us how learners with disabilities are both over-analyzed and under-educated, always seen as the sum of their deficits, not their strengths. What follows are two recent experiences we’ve each had in which the idea of thinking of students as swiss cheese has resurfaced.

Rachel’s Story

During a professional development session recently in southern California, I asked the gathered special education teachers to describe their questions in working with students with disabilities in math. Many of their answers referenced “holes” and “gaps.” They were concerned about what their students did not know. Yet this thoughtful group of educators also shared puzzling contradictions about what their students did know. One teacher described how her student with a significant disability, who was not able to reliably recognize numerals, was able to do addition when presented with a CGI story problem and manipulatives. Another noted that she had a child who had difficulty with addition and subtraction under 20, but could do multiplication with multi-digit addends. Are these gaps? Holes? Or do these kids, with a unexpected patterns of development, actually help us question the tired metaphors of knowledge that we use in education?

What do I mean by that? This metaphor of “holes” and “gaps” presupposes that real mathematical knowledge is a filled-in, completely solid progression. These two students are not playing along with this metaphor; they know more advanced topics first and struggle with the more basic ones. And they are not alone. I have met many students with disabilities who learn topics in a turned-around order, who challenge typical linear learning trajectories. I also know many “typical” learners with plenty of “holes” and “gaps.” Maybe it is not the learners; maybe it is the way that we conceptualize learning . . . .

One phrase used to name these student’s unique learning trajectories was “splinter skills.” This is a phrase often used in relationship with autism, and tends to suggest that even when a person with autism knows some higher-level content, such as advanced mathematics, because that person has other challenges (usually with lower level content), those skills are “splinter,” unconnected, unimportant. Attention should be paid to what the person cannot do- otherwise how will we fix them?

Are there other metaphors for learning, not so filled in, not so linear? I prefer “landscape of learning,” used by Maxine Green (1978) (pioneer in arts education), and in mathematics education, by Cathy Fosnot and Martin Dolk in their series, Young Mathematicians at Work (2002). In this way of thinking, there are multiple paths to mathematical understanding. In this image, I have reconstructed a landscape of learning multiplication with a group of teachers:

[blackboard with many index cards, with math learning terms around multiplication such as doubling and halving, partial quotients]
This also makes sense in my (Rachel’s) experience teaching students, both with and without disabilities. Students learning multiplication sometimes rely on skip-counting, some-times repeated addition, sometimes move easily between the two.  Learning is complex, multi-leveled, and no one is all the way “filled in.” I asked participants at the workshop to think about themselves as learners—are they are the way filled in? I know that I am not. I am constantly learning, and it is not like filling in a coloring book. Instead, one thing I know leads to another, to another, like paths in a landscape.

If a child is having difficulty with multiplication, and we assess them, we might find that they typically use skip counting, and not repeated addition. Do we address the gap, teaching them repeated addition? Or do we build on what they know, skip-counting, to move them towards the real goal: conceptual understanding and procedural fluency with multiplication?

When we conceptualize learning as linear, students with disabilities, who tend to learn “differently,” will be conceptualized as incomplete. When we conceptualize learning as a process, as having multiple paths, we can understand all learners as in movement, as in process. We can focus on guiding them along their path, rather than remediating. After all, our kids are not swiss cheese!

Andrew’s Story

This summer I spoke at the Cognitively Guided Instruction conference held at the University of Washington. During the conference, Megan Franke, co-author of the CGI series of books, gave a pertinent talk titled, “No More Mastery, Leveraging Partial Understanding.” Here is the description of the talk from the CGI Seattle program:

How do we notice and use what students DO know to support them to make progress in their thinking? Partial understandings provide great opportunities. This session will focus on seeing how we can use partial understandings to support students’ mathematical learning and thus challenges our common notions of mastery.

[Megan Franke standing in front of slide, which pictures a child with their head in their hands and the text, No More Mastery: Leveraging Partial Understandings]
Franke spoke about how teachers can not only uncover individual student understanding by focusing on “partial understanding,” but they can also leverage this understanding in a strengths-based method to push students further, and not to remediate their deficits.

IMG_1298Franke ended her talk by revealing she was not happy with the term “partial understanding” because it inherently carries deficit connotations. She asked that we try to conceptualize a different term; one which could bring the positive strengths-based allusions she was hoping for teachers to utilize. The NCTM publication, The Impact of Identity in K-8 Mathematics: Rethinking Equity Based Practices offers another view of Franke’s “partial understanding.” The authors suggest “leveraging multiple mathematical competencies” as way to structure student collaboration so that student strengths are maximized and this desire to “fill gaps” is minimized. Students are strategically grouped in ways that allow students to learn from each other and push each other forward in their learning. Utilizing this viewpoint, I believe teachers do not need to fill gaps or holes, because all students are able to contribute their mathematical competence to the community of learners.  

Screen Shot 2017-11-09 at 5.21.41 PM 


  • Let’s move beyond tired metaphors of learning that assume that learning is linear and predictable, towards a landscape of learning
  • These overused metaphors of “gaps” lead to conceptions of students as “swiss cheese”
  • Instead, let’s build on what students know, which always provides a path forward
  • And help students see that in when they collaborate in groups they can leverage multiple mathematical competencies.


Mathematics and Dyslexia Part II

What can we learn from mathematics educators who have dyslexia, or whose children have dyslexia? In a serious of posts a year ago, Paula Beardell Kreig wrote some thoughts about working with her children, who have dyslexia. I recommend reading through both posts.

Here is the first:

And here is the second

What are the implications of what she suggests? How can we build on this work in the classroom?

Designing intervention on the landscape of learning

Designing intervention on the landscape of learning

In 1999, I was working as a special educator in an inclusive elementary school. My ideas about mathematics instruction were old-fashioned.  A few of my fifth-grade students had particular difficulties in multiplication, and it was hard to see what they could do—I was so focused on what they couldn’t do.   That summer I attended the Summer Institute at Mathematics in the City at the City College of New York.  One (of many) ideas that transformed my work was the idea of the landscape of learning.  Continue reading “Designing intervention on the landscape of learning”