Category: intervention

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.


Beyond Differentiation

The following post includes links from a presentation on October 17, 2016.

A blog post explaining neurodiversity

A great book to learn more about neurodiversity in classrooms: Armstrong, T. (2012). Neurodiversity in the classroom; Strength-based strategies to help students with special needs succeed in school and life. Alexandria, VA: ASCD.

Universal Design for Learning by CAST

Research on links between dsylexia and spatial processing

Blog post by a dyslexic mathematician

The Dyslexic Advantage website


PDF of Differentiation Talk (draft)

Memoirs written by people with learning disabilities referenced in the presentation

Abeel, S. (2005). My Thirteenth Winter: A Memoir. New York, NY: Scholastic.

Arrowsmith-Young, B., & M.D, N. D. (2013). The Woman Who Changed Her Brain: How I Left My Learning Disability Behind and Other Stories of Cognitive Transformation (Reprint edition). Simon & Schuster.

Jr, Jackson. J. T. (2010). Shhhhhhh, I Have Something to Say: The Joe Thomas Story. S.l.: PublishAmerica.

Peel, R. (n.d.). My Dyslexic Journey.  Retrieved from

Rodrigues, J. (2013). High School Dropout to Harvard: A Dyslexia Success Story. CreateSpace Independent Publishing Platform

Young, K. (2012). Smart on the inside: A true story about succeeding in spite of learning disabilities. Highland Park, IL: Writers of the Round Table

Young, S. (2011). How I Learned.

Collections of narratives written by individuals with learning disabilities

Connor, D. J. (2007). Urban Narratives: Portraits in Progress; Life at the Intersections of Learning Disability, Race, and Social Class. New York: Peter Lang Publishing.

Rodis, P., Garrod, A., & Boscardin, M. L. (2001). Learning disabilities and life stories. Allyn and Bacon.


How to help kids use more complex strategies in multiplication? Give them harder problems.

Zhang, D., Xin, Y. P., & Si, L. (2013). Transition from Intuitive to Advanced Strategies in Multiplicative Reasoning for Students with Math Difficulties. Journal of Special Education47(1), 50–64. I love this article because it uses constructivism to understand the development of three kids with disabilities in learning multiplication, particularly drawing from the work of Siegler on how kids use multiple strategies over their course of their development with a new mathematical operation. Take-away- kids only switched strategies from counting when the numbers got big enough. Lesson to be learned is to stop giving kids only easy problems, because such problems actually encourage them to continue using less sophisticated strategies. Continue reading “How to help kids use more complex strategies in multiplication? Give them harder problems.”


A great introduction to mindset from Jo Boaler.

Mindset is incredibly important for kids with disabilities in mathematics.  Too often, kids with disabilities are given the impression that they lack the ability to do complex mathematics.  And, perversely, when kids are labeled LD, they are often told that they are “smart” but their brains are not “wired” to read easily, or do mathematics easily.  In this sense, we are constructing fixed mindsets for kids.  We tell them that kids with LD are smart, and that their abilities in some areas are fixed at a high level, and their abilities in other areas are fixed at a low level. When we know how destructive fixed ability mindsets can be, why do we purposely create them in kids with LD? Continue reading “Mindset”

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”

Designing intervention in operations

In Models of Intervention in Mathematics, edited by Cathy Fosnot and published by NCTM in 2010, Susan Jo Russell wrote a piece called “The habit of reasoning: Uncovering competence through focused instruction.” I love this piece because it gives a structure to how teachers can plan intervention with kids that builds on strengths. Continue reading “Designing intervention in operations”