I created this list based on a request for a Ph.D. reading list for a student interested in critical and Disability Studies approaches to mathematics education. Figured I should share with more, as there are so many wonderful resources on this list. Updated May 2018. Please comment if you have recommendations to add!
One of the most important agents of change in special education is paraprofessionals. These dedicated professionals are often the mediators between classroom teachers and our students with disabilities. They help create access for the child to the curriculum. Unfortunately, paraprofessionals are almost never given planning time with teachers, and are not usually allowed access to professional development. In mathematics, this can mean that the paraprofessional and the mathematics teacher have different ideas about what and how the student should be engaged in mathematics.
Judy Storeygard (author of several excellent texts on mathematics teaching for students with learning differences and/or special education services (Count Me In and My Kids Can) has been working with colleagues on addressing the issue of access for paraprofessionals. Funded by the National Science Foundation, they have provided a pilot program for paraprofessionals in the Boston Public Schools, designed to increase access for paraprofessionals to high-quality professional development in mathematics. I would love to see more projects like this!
Check out this video: http://stemforall2018.videohall.com/presentations/1095
1) Paras need opportunities, over time, to immerse themselves in mathematics activities that help them to develop their mathematical thinking. Often, they have not had positive experiences learning math themselves, and so as part of this process, they are developing their math confidence. Many have become quite excited about doing math!
2) Our paras (and other paras we have spoken with) can identify the ways in which math instruction is quite different than what they experienced as students–but since many haven’t been involved in a teacher education program, they aren’t always sure about what it means to instruct with an inquiry-oriented curricula or resources. Understanding the goals of such curricula and experiencing inquiry approaches within their PD have been important to them. Also, they have been eager to learn and try out instructional strategies in their classrooms (e.g., how to listen for student thinking, asking questions that provide students with new challenges within an activity, asking students to re-tell the story problem), and then return to PD to debrief with each other and us.
3) Follow-up activities have also been helpful. Through our project, the paras have had opportunities to plan with their teachers and reflect on their students’ math learning and struggles.
4) And last, but not least, the paraeducators have created a math learning community where they provide each other with support, encouragement, and new ideas. They are incredible resources for each other!
Over one million para educators currently assist in classrooms, and another 100,000 are likely to be added in the next ten years. Para educators are often required to teach content, such as mathematics, but there are few efforts to provide them with the knowledge or supervision they need to be effective when working with a range of students, including those with disabilities and for whom English is a second language. While professional development will enable paras to make a greater difference in the classroom it may also increase their access to continuing education and workplace opportunities. Our project is designed to develop, pilot, study, and refine PD, that focuses on developing the confidence, mathematics knowledge, and teaching strategies of para educators, grades K-3 in the Boston Public Schools, as well as providing support for their collaborating teachers.
This post below presents the story of a student with dyslexia who fell in love with science through an inquiry-based classroom. I love how the writer highlights how his teacher’s relationship was the core support for learning. When a teacher is attuned to students, anything is possible, and students (like this one) can transform.
In a related search, I found this on leveraging neurodiversity in maker spaces. I so agree- my neurodiverse students and colleagues can thrive in maker spaces and project-based classrooms, where they can be led by their considerable curiosity and drive.
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 (@heidifessenden) 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!
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!
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.
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:
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!
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.
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.
Franke 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.
- 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.
So excited to discuss a new article from my colleague Paulo Tan, who is a professor at the University of Tulsa and studies mathematics education from a critical disability studies perspective. His new article describes his personal experience advocating for his son in Individual Education Plan [IEP] meetings, which are the meetings in which teachers and families gather to collectively plan the educational progress of the child. These meetings can be contentious affairs, however, in which families are not given a equal voice (Valle & Aponte, 2002). As I mentioned in a previous post, students with disabilities are often given IEP goals that narrow their mathematics education, focused on procedures and memorization. Dr. Tan writes,:
“Personal experiences promoting inclusive mathematics education for my own child have mostly been met with staunch resistance on the part of educators, and a resulting breakdown in collaborative efforts during individualized education program (IEP) meetings. However, I found that utilizing certain strategies and introducing innovative mathematics education resources during the IEP meeting have contributed to a more collaborative and productive meeting toward inclusive practices beyond mathematics. In this article, I describe these strategies, resources, and related processes to guide effective IEP practices and future research.”(Tan, 2017)
Dr. Tan makes it clear— even for a mathematics education researcher focused on disability— IEP meetings are challenging because his child is described and understood through his deficits, rather than his strengths. Dr. Tan has to strongly advocate for his child, pushing back against these deficit conceptions. He suggests three strategies:
(1) Reframing mathematics
Dr. Tan suggests beginning the IEP meeting by discussing mathematical mindsets (Boaler, 2015), purposefully starting the conversation with a larger, more humane understanding of what mathematics is and can be. Using resources from YouCubed, Dr. Tan presents his child’s strengths in the context of a mathematics of meaning.
“In my experience, the process of engaging with this resource and spending a substantial amount of time thinking through and writing down my own child’s strengths, preferences for learning, and sources of knowledge helped to navigate a more productive conversation grounded in powerful mathematics minds. Qualities such as “He’s a good problem solver”, and “He possesses extensive multicultural knowledge” combined with ways that math is framed such as “Math is about creativity and making sense” (Youcubed, 2014, p. 1), positions students with disabilities as crucial members of the mathematics learning community.”(p. 33).
(2) Developing goals that support understanding
Next, Dr. Tan works with the educators at the meetings to create goals that support understanding. Too often IEP goals in mathematics are focused on a narrow set of skills. These goals then dictate instruction, creating a feedback loop that rewards narrow instruction on limited computational goals. Dr. Tan pushes the group in the IEP to consider goals that are based on the Mathematical Practices. He uses this resource, available here.
(3) End the discussion by insisting that math needs students with disabilities.
Dr. Tan ends the meeting by articulating the larger vision for their work together. Mathematics education scholar of equity Rochelle Gutiérrez states that while the focus tends to be on how kids of color need math, but math NEEDS kids of color (Gutiérrez, 2013). Those that have been traditionally excluded from math bring new ideas, new perspectives, and new solutions to the puzzles of the field. Dr. Tan describes how he focuses the conversation not just on what math his child needs, but that the world needs his child, and others like him, to become empowered mathematically.
And best of all, full-text of his article is available here:
Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68.
Tan, P. (2017). Advancing Inclusive Mathematics Education: Strategies and Resources for Effective IEP Practices. INTERNATIONAL JOURNAL OF WHOLE SCHOOLING, 13(3), 28–38.
What do we know about teaching meaningful mathematics to students with learning disabilities, particularly those with profound difficulties learning to read, otherwise known as dyslexia?
Here is a great place to start: a blog post by a mathematician with dyslexia who lists some of the many strengths of people with dyslexia. Here is a quote from the blog:
As I said before, dyslexia comes not only with weaknesses, but with strengths*. Some of these strengths play well into mathematical thinking, such as:
- Spatial reasoning. Dyslexics tend to be good at thinking relationally in three dimensions. This is great for many areas of math. Topology is one of those. My PhD is in topology.
- Seeing connections. Progress in mathematical research is made by drawing connections between disparate subfields. Dyslexics often have a strength here also.
- Thinking in narratives. The way to predict how well a child will do in math is by how complicated a story they can tell. See this ScienceNews article. It makes sense: proofs are the bread-and-butter of math. Proofs are really a lot like stories. Dyslexics are usually good narrative thinkers.
I really didn’t like math at all up until I hit geometry in ninth grade. That’s when math became much easier for me. It may be confusing that moving into more advanced mathematics actually made math easier for me. The key to remember is that my brain works differently from the brains that school curricula were designed for.
*I’m stealing this list of strengths from The Dyslexic Advantage by Eide and Eide.
This epitomizes the concept of neurodiversity, focused on strengths rather than deficits, looking for evidence in the experience of experts, people with disabilities. It is hard to recognize how very ill-suited traditional mathematics education has been for people with dyslexia. For example, I hear frequently about difficulties with retaining memorized facts or procedures. When a child has difficulty in this area, all instruction stops until they are “remediated,” or fixed. The assumption is that you cannot move on unless the facts are mastered. This assumption is dangerous because the student has no access to higher level mathematics, no access to the meaning making of mathematics that might finally connect for that student. No wonder that kids with learning disabilities tend to stall out at about a 5th grade level.
In this post from the website the Dyslexic Advantage, Fernette Eide writes about the four core cognitive advantages demonstrated by dyslexic learners. How does these four strengths matter for learning mathematics?
The Four MIND-Strengths
In our book The Dyslexic Advantage (2011) we described the results of this investigation, and the four patterns of dyslexia-associated strengths it revealed. With a little tweaking, we used the acronym MIND-Strengths to describe these strength patterns, which are:
M-Strength for Material Reasoning, which is primarily reasoning about the position, form, and movement of objects 3D space
I-Strengths for Interconnected Reasoning, which is primarily the ability to spot, understand, and reason about connections and relationships (e.g., analogies, metaphors, systems, patterns)
N-Strengths for Narrative Reasoning, which is primarily the ability to reason using fragments of memory formed from past personal experience (i.e., using cases, examples, and simulations rather than abstract reasoning from principles)
D-Strengths for Dynamic Reasoning, which is the ability to accurately predict using patterns derived through experience the future or the unwitnessed past
Building on the blog post above, what might these strengths mean in mathematics? How can we leverage these strengths?
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?
How do you learn more about the history of math ed and special ed?
A great intro to this topic:
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
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 https://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keywords=Dyslexic+Journey+peel
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.