Category: theories

What is different in math research on students with and without disabilities; an article that took 8 years to get published!

What is different in math research on students with and without disabilities; an article that took 8 years to get published!

To call this a new article is a bit of a stretch. I first started writing this article in 2013. I planned to write a Research Commentary for JRME (a fancy math ed journal) on what I saw as the stark differences between 1) math ed research and 2) special ed research on math. I noticed that the methodologies were different, the pedagogical theories used were different, and the kind of mathematics studied were different. I realized that I did not want to just assume these differences existed; I wanted to study them empirically. So I took one year (2013) and began analyzing every article I could find on math teaching and learning, comparing those that included disability to those that didn’t. I presented my findings at NCTM in 2014 and that was how I met Paulo Tan, who became my collaborator on this project.  Together we added 2014, then 2015. We were rejected by one fancy journal in 2015, another in 2016. In 2017 we submitted it to another fancy journal and were rejected after several rounds of revisions including adding 2 additional years of analysis (2016 and 2017).  We submitted to Mathematics Education Research Journal in 2019 and it was accepted. The editors wanted to add it into a special issue on inclusion and mathematics in the Australasian, and so we added additional analysis on the Australasian context. Here is a link to the article at the journal site– I think there are some free downloads. Also here it is in preprint (free earlier draft) Lambert Tan 2020_Preprint Draft_MERJ

We learned a lot through this process, including what the word Australasian means. First, it takes a long time for articles to get published. It is a long slog. Second, I learned that peer reviewers want you to stick close to your findings in an article, and do not allow exciting, bold pronouncements. We had to cut many of those over the years. But I also feel we faced some unfair criticisms. We were told that our paper was clearly biased towards qualitative research over quantitative when we were simply noting that most research in special ed is quantitative (over 80%). One reviewer said that the implications of our findings were clear- math ed is broken and needs far more quantitative research.  We were told by a reviewer that adults over 18 with disabilities were not adults, and we should include them and compare them to nondisabled children (?!what the unholy ableism?!). We were told that we had an agenda that was anti-special education, that this was an ideological crusade. That may be true, but that is exactly why we engaged in such an extensive empirical review- we did not want to make unsubstantiated pronouncements about the difference between educational research for kids with and without disabilities. We wanted to be able to show what was actually happening, with data backing up our claims.

Please see the paper itself for lots on the methods, but here they are very briefly- we gathered as many articles on math teaching and learning as we could find for the years 2013 – 2017 (5 years) from 88 separate journals (over 2400 articles). We coded all those articles for the methodology (how the research was done), the age of the participants, the math that was studied, and the theoretical orientation of the research project. Using the titles, abstracts and keywords of each article, we separated these articles into those that focused on Disability (including an explicit focus on disability and/or math difficulties), calling this group the Disability Set. We had 408 articles in this set. The No-Disability Set were articles that had no clear focus on Disability (based on just the title, abstract and keywords). We had 2069 articles in that set. So keep in mind that the No–Disability Set is larger. We then analyzed these two different sets using the coded categories.

So, what do you notice and wonder about this graph? How is the method of research different for students with disabilities?

Graph of research types comparing Disability Set to No-Disability Set. Research in the No-Disability Set was 44% quantitative, 41% qualitative, 6% mixed methods and 9% undetermined. Research in the Disability Set was 81% quantitative, 13% qualitative, 3% mixed methods and 3% undetermined.

While math ed research is about evenly split between quantitative and qualitative research, students with disabilities are far more likely to be included in quantitative studies. Disability is understood through numbers and big data, while other math learners are understood through close analysis and description. Why are these methods so different? What kinds of implications do you think this has for teachers? For kids?

Who gets studied in this research? We found that research in the Disability Set was far less likely to focus on the teacher. While 36% of research in the No-Disability Set was focused on the teachers, only 9% of research in the Disability Set was. The actions of the teacher, the art and science of teaching mathematics, is much less a focus in the Disability Set.

We found that some disabilities are studied more than others. This graph shows the number of articles in this 5 year period that had an explicit focus (in the title, abstract or keywords) on a particular disability category. General means that the article focused on “special education students” or “students with disabilities” without a specific focus. Learning Disabilities in general was the category that was most studied, with 48 of those 116 articles focused on Mathematical Learning Disabilities/dyscalculia. 83 studies were on students who were labeled as having Math Difficulties or other euphemisms for students who were having difficulty specifically in math. The number of studies we found on students with Intellectual Disabilities, Autism, Emotional Disabilities, ADHD, and students with hearing and visual impairments were few.

Graph on the number of articles in each disability category. 64 articles were general, 116 were about Learning Disabilities, 38 were about MD/Dyscalculia, 83 about Math Difficulties, 30 about Intellectual Disabilities, 29 about Anxiety, 24 about Autism, 9 about Emotional Disabilities, 8 about ADHD, 6 about Other Health Impaired (excluding ADHD), 6 on Deafness, 2 on Orthopedi

And here is my favorite graph from our paper. How is the research different by pedagogical perspective? Students with disabilities are studied through medical perspectives (studies of cognitive domains, neuroscience), behavioral (direct instruction, explicit instruction, training) and information processing (working memory, metacognition, generalized strategies). Students without disabilities are far more likely to be included in studies that focus on constructivism (studies of how students think about mathematical topics) and sociocultural theories of learning (studies of the community of practice within a math classroom, studies of identity and talk). This is the most profound difference. We know a lot less about how students with disabilities THINK about mathematical topics. Not because those students don’t think (THEY DO!) or can’t think (THEY CAN!), but simply because their thinking is not a subject of research. We argue that this sends a message to educators that students with disabilities cannot think, a deficit mythology (Lambert, 2018) that pervades how we imagine we can teach to students with disabilities.

Graph on the percentage of articles in different theoretical categories. The No-Disability set was 5% medical, 4% behavioral, 5% information processing, 33% constructivist, 21% sociocultural, 8% sociopolitical, and 6% social psychology. Disability set was 22% medical, 27% behvaioral, 15% Information Processing, 10% constructivist, 6% sociocultural, 7% sociopolitical, and 9% social psychology.

What do you notice and wonder about this graph? What topics are studied for students with disabilities? What topics are not? Paulo and I wrote a paper about how “problem-solving” was defined differently for students with disabilities (students with disabilities were given “word problems” and students without disabilities were conceptualized as problem posers).

Fig. 4 MERJ revised

The question is, for working teachers and teacher educators, how does this matter? Does it matter that the research is quite different? Different methods? Different mathematics? Different theories of learning? Is this because the kids are so different, or because the research traditions are so different? Let me know your thoughts!

Research Breakdown: My article on how disability in mathematics is political, relational, emotional and complexly embodied (Lambert, 2019)

Part of my intention with this blog is to provide access for a wide audience on research into disability in the context of mathematics. Research is too inaccessible; hard to get the articles and hard to understand them even when you do get your hands on them! Today I want to return to an article I published a year ago in January 2019, which takes up some themes I have been thinking about since my dissertation in 2010. This paper took me nine years to write. If you can’t get access to ZDM, a European math ed journal, here is my final draft of the paper: Lambert 2019 ZDM_authorcopy

Basically, in this paper, I argue that-

  1. Emotions and relationships are central to mathematics learning and identification for everyone, but particularly so for marginalized groups in math. I focus particularly on students of color with disabilities. When you are marginalized in math class, made to feel lesser over time, you have feelings about that. Those feelings are part of how we construct our identities as math learners, as emotion layers over math experience. It affects every aspect of our mathematical learning. There is no learning outside emotion.
  2. Disability Studies (and Disability Studies in Education) are academic fields that study disability in terms of social construction, ableism, and a social justice standpoint towards disability. The most important thinking on disability has emerged from this tradition, shaped particularly by disabled scholars and activists. In this paper, I use the ideas of Tobin Siebers (2008) on complex embodiment and Alison Kafer (2013) on the political relational model of disability to conceptualize disability as complex, relational, political, and always embodied and felt.  Using these theoretical tools, I argue that neither the medical model nor the social model of disability (UPIAS, 1977) is complex enough to help us understand disability as experienced in schools.
  3. I use one young woman’s narratives, Desi (a pseudonym) about ADHD and Anxiety, as well as her label of Learning Disability, to explore how complex a kid’s construction of disability is. Her narratives increase our understanding of the theory, and vice-versa. I often use the word kid instead of student, because kid is more multi-dimensional.
  4. I argue in this paper that we need to include emotion in mathematical embodiment. For some reason, math ed scholars rarely do. Embodiment is the body, but not the mind. Using the idea of bodymind (Price, 2009), I argue that this is an artificial distinction and unhelpful.

Here is how I introduce Desi:

Twelve years old at the beginning of the study, Desi identified herself as a “girl” and “from the Dominican Republic.” Desi is bilingual in Spanish and English. Desi was a powerful social and moral force in both her sixth and her seventh-grade classrooms. In her eighth- grade year, I observed Desi delivering a bilingual poem that touched on her identity as Dominican, her sense that adults tried to control her in school, and her rejection of “labeling.” Her peers applauded her loudly, and I heard one boy say, “Desi is the best poet in the school.” Desi identified most clearly as a poet and an activist, one who did not see mathematics with the same passion as she did literacy. (Lambert, 2019, p. 284)

I hope you can feel in that description how absolutely amazing Desi was. It has taken me a long time to write about her narratives, mainly because I worried about portraying her point of view accurately. She is a complex human, including contradictions (like all of us!). This paper will not focus on how Desi makes sense of how her culture, gender, and languages matter in mathematics class, but I am currently working in a research collaborative to develop that analysis using new work on intersectionality and math ( Bullock 2018; Hernández-Saca et al., 2018). I found that I could not do that important work alone.

This particular paper focuses on how Desi makes sense of herself in terms of her disability identifications. This paper is not about whether she “has” ADHD, but how she thinks about ADHD, particularly in terms of her math learning. What is ADHD from the student perspective?

Tobin Siebers, a scholar of Disability Studies, introduces the idea of Complex Embodiment. I describe it here:

Siebers rejects both the purely social model, as lacking in attention to embodiment, and the medical model, which defines disability as individual and requiring medical intervention. Siebers defines disability as “a social location complexly embodied”(p. 14). In this formation, disability is not solely within the body, within impairment. Nor is disability a social construction. It is both, from the beginning. He pro- poses new ways of integrating social and bodily aspects of disability, particularly by proposing the concept of complex embodiment.

Complex embodiment allows for a particular kind of situated knowledge, one that “adheres in embodiment”(p. 23). Embodied knowledges are them- selves produced through cultural processes; language to describe our bodies does not spring from a neutral source. Thus when we describe a body, we use concepts formed in social worlds, which in turn shape our bodies. He calls for critique which maps the construction of ideology onto bodies: “precisely because ideologies are embodied, their effects are readable, and must be read, in the construction and history of societies”(p. 32). (Lambert, 2019, p. 282)

Basically— to understand how disability is embodied, we need to consider social worlds. How we talk about our bodies maps onto how we understand our own bodies. Before I go on, think about ADHD. How have you heard it described? By media, doctors, teachers? How do you think kids with ADHD come to understand their own difference? I hope this makes you wonder immediately how absolutely horrific it is that we make ADHD seem so negative. We provide the language that kids use to make sense of themselves. I have learned tons from those in my inner circle, my work settings, and my students who have taught me new ways to think about ADHD, ones that are not focused on negative ways of understanding oneself. But back to how this connects with math and Desi—Pay attention to how Desi describes ADHD in math class:

At one point her sixth-grade math class, Desi told me, “Normally I would be zooming out. We zoom out” during class. That was “why most of us like to sit next to the window,” cause then “I am in my own little planet” and “in my world.” Here we have a collective, embodied description of those who “zoom out.” That group, according to Desi was herself and a group of boys, who also had IEPs. Her description of zooming seemed entirely internal, as Desi didn’t move around during class, and her body was unusually still. Desi’s description of zooming as related to attention challenges normative concepts of ADHD, which stress an excess of energy or hyperactivity. (Lambert, 2019, p. 285)

Desi and her friends have made their own word- Zoomers. This feels much more positive (and descriptive) than other words they could have taken up, and describes how Desi talks about her own difference. Desi is not the kid that needs to move a lot, she stays pretty still in class. But her mind is always active, and she has to work hard to stay “on task” and with the rest of the class. Her brilliant mind, in other words, is busy zooming.

The problem is not the zooming, but how this works in her math class. Desi told me that her problem in math was paying attention the way teachers demanded it. I asked her “Who stands out in your math class?” (a great question to ask kids to learn what they value in other students). Desi named certain students as “good.” She told me how these students learned math:

Desi: They pay attention [gestures hands moving straight forward]. It is like they are a movie, or a computer, and they just suck it all in [gestures with hands around her brain], like a sponge, until they absorb every little piece of it [gestures grabbing tiny pieces of something in front of her]. (Lambert, 2019, p. 285)

Notice in the previous narrative how closely she relates ability to attention, beginning with “they pay attention.” The primary characteristic of the “good” math student is attention, and then memory, or being a “sponge.” Her final gesture suggests that the kids are sucking in things—understanding mathematical knowledge as isolated bits and pieces.

For Desi, ADHD is a way of being in the world, of zooming, which makes it really hard for her in math class because she sees math as memorization.  I didn’t really explore this in the academic article, but I wonder what Desi would have said if she had had more experience with meaningful mathematics, particularly connected to social justice, something she cared very much about. Her teacher, at the end of her 7th-grade year, told me that Desi needed a class where she could connect math to what matters, and then she would soar.

This is an example of complex embodiment because Desi constructs ADHD in the context of math learning using the materials of culture—the language that kids use to describe their learning is flavored by the language we use with them. Yet she also feels how feels to be a “zoomer” by the windows. Disability is both her own personal embodied experience, how she makes sense of that experience (using terms and concepts she has heard), and how her environment reacts to her disability (having trouble in a math class all about paying attention). So it is always both socially constructed and also real. Both. Always. Complex embodiment takes embodiment, experience, seriously.

The second theory I use to understand Desi’s narratives is the political relational theory, developed by Alison Kafer (2013). Kafer uses queer theory in tandem with disability theory to move beyond the binary of medical and social models of disability. Here is how I explain her theory of political relational, and how it relates to special education.

Disability is produced in interaction, always relational and political. Kafer inserts the political into her definition to speak back to the pervasive construction of disability as operating outside politics. We depoliticize disability when diagnosis is assumed to be scientific truth, even as these diagnoses shift over time, and if social issues are excluded from analysis of disability. For example, traditional special educational research tends to label DS as “ideological,” and its own work as “pragmatic” (Brantlinger 1997). Instead of understanding all constructs of disability as embedded in political contexts, special education claims a non-ideological position, outside of culture. However, disability is always political and implicated in relationships of power; Kafer asks, “How is the category of disability used to justify the classification, supervision, segregation and oppression of certain people, bodies, and practices?”(p. 9). Special education is a system designed to do just that: classify, remediate and segregate students based on particular conceptions of disabilities (Linton 1998). With the word relational, Kafer reminds us that the political is not situated in impersonal institutions, but interactions with other people. The work of the special education system is done by individuals whose role is to notice and report disability. (Lambert, 2019, p. 283)

This is pretty radical stuff. Kafer is suggesting that ableism is carried out by us, through our participation in an unjust system, including how we routinely rank and sort kids as math teachers. Disability is both interpersonal and systematic. And we as teachers operate within those systems, and our relationships with kids are impacted by our participation in an ableist system.

Desi breaks that system down. Here, she is explaining to me how low expectations impact her:

Desi: It’s like they feel like, you have to be able to be this or that, and even if you have a disability like, cause, I have ADHD or something, some people say, that they are amazed at the fact that I can actually learn and pay attention and try to pay attention when it is, like, hard for me. (Lambert, 2019, p. 286)

Desi tells her story as if she is talking to “them”- some group of anonymous people who saw success in mathematics as innate: “you have to be able to be this or that.” In the middle of the narrative, she used her own ADHD as an example. These unnamed individuals seemed to be surprised that Desi can achieve in school. Desi suggested that these unnamed critics believe that ability is fixed and that those who have a disability are incapable of learning. She strongly disagreed, and then expanded on this theme, again emphasizing the role of effort.

Desi: And then many people are always just like, has to do with abilities that you have and it has to do with the fact that you have to be like, if you are not good at this you are not good at it, and if you are not good at it at all then you have to be like in special ed or something and I am like, no, that’s a lie. You can do it, it’s just that you are not putting in the effort.

Desi referred to a narrow conception of mathematical ability: either you are “good at it” or not. She suggested that some believe that mathematical ability is static, creating a binary between those who are good, and those “in special ed.” For Desi, separation into special education is tied to the notion of innate ability in mathematics. Both times that Desi used formal disability discourse in this narrative she added, “or something,”(“ADHD or something” and “special ed or something”), suggesting distance from medical terms for disability. Desi critiqued the theory of innate ability in mathematics, taking up an alternative voice from her classroom teacher—effort alone determines academic success. Desi appears to take issue not with the naming of disability, but the use of such categories to separate learners in categories of capable and not.

These narratives attest to the relational construction of disability. Desi never posited that ADHD does not exist, she questioned the low expectations and segregation that accompanied labels. She critiqued not disability, but institutional structures that seek to separate out those with disabilities from the rest of the students. For Desi, these arguments are deeply relational, as she told them through stories of an argument with a friend, animated by voices from multiple points of view. This politics of disability is embodied, lived, felt through relationships. (Lambert, 2019, p. 287)

I hope these two stories, and Desi’s powerful narratives, have helped you think about how complex disability is. We can’t think of it just through the medical model- just a problem within Desi. But neither can we think about just with a social model of disability- it is not just the system doing this to Desi. It is more complicated than that. A simple binary will never work. Disability is both socially constructed and real- felt powerfully by people. This is why I argue for emotion to be considered within embodiment— we need to understand not just Desi’s attentional differences, but how those feel to her, and how she is made to feel by math classrooms that invalidate her experience or unique set of strengths.

We have a lot of work to do in math education to better understand how disability matters in our classrooms. I think we need to 1) listen to kids, particularly kids of color, about their own experience, 2) learn from fields like Disability Studies, which offer a great deal in terms of theory from the perspective of disabled people, and 3) constantly pay attention to emerging voices, particularly those that build an understanding of disability from non-white perspectives. I am working now in a research collaborative to develop our understandings of how intersectionality affects the emotional experience of math class, including the development of math identities in the context of race, language use, genders, and more. I am hoping there will be a Desi Part II which presents her narratives in these areas. Again, this kind of analysis takes time, and in this case, it will take collaboration.

One final note: Desi had participated in a unit on disability studies in her sixth-grade year. It seemed to help her develop this political orientation to disability. I would urge schools and teachers to teach disability, not through a medical model, not through simulations (which devalue the experience of disabled people), but through sharing the stories and perspectives of disabled people, through the history of social activism by people with disabilities, and new movements like Disability Justice (Sins Invalid).

Read the full article for more!

Cites and related readings:

Brantlinger, E. (1997). Using ideology: Cases of nonrecognition of the politics of research and practice in special education. Review of Educational Research, 67(4), 425–459.

Bullock, E. C. (2018). Intersectional Analysis in Critical Mathematics Education Research: A Response to Figure Hiding. Review of Research in Education, 42(1), 122–145.

Hernández-Saca, D. I., Kahn, L. G., & Cannon, M. A. (2018). Intersectionality Dis/ability Research: How Dis/ability Research in Education Engages Intersectionality to Uncover the Multidimensional Construction of Dis/abled Experiences. Review of Research in Education, 42(1), 286–311.

Kafer, A. (2013). Feminist, Queer, Crip. Indiana University Press.
Price, M. (2009). “Her Pronouns Wax and Wane”: Psychosocial Disability, Autobiography, and Counter-Diagnosis. Journal of Literary & Cultural Disability Studies, 3(1), 11–33.
Siebers, T. (2008). Disability theory. University of Michigan Press.

Concrete Representational Abstract (CRA) in mathematics

In response to a Twitter inquiry, I decided to write up some longstanding thoughts on the Concrete Representational Abstract (CRA) sequence that is popular particularly in designing instruction for learners with disabilities. First, what is CRA? Here, from a researcher who done several studies on CRA with students with disabilities in mathematics: Continue reading “Concrete Representational Abstract (CRA) in mathematics”

Universal Design for Learning (UDL)

Universal Design for Learning (CAST, 2011) was inspired by Universal Design in architecture.  If you design for people with disabilities before you built the house, it can be more accessible, less expensive, and more beautiful.  UDL applies that theory to learning.  Beginning with the premise that variability is what all learners share, curriculum should be designed to work for the widest variety of learners possible.  Continue reading “Universal Design for Learning (UDL)”

Quick Theory Post #2: Neurodiversity

Neurodiversity grew out of the work of autistic self-advocates, or autistic people who identify as activists. Here is a definition from an interview with a neurodiversity activist you can read in full here.

Neurodiversity, the word, simply means the whole variety of different brain wirings people have…from the different kinds of normal to the different kinds of not so normal. Then there’s Neurodiversity, the movement which is the shocking idea that people with non standard wiring are human and deserve to be treated as such without being “fixed” first.

Continue reading “Quick Theory Post #2: Neurodiversity”

Quick Theory Post #1: Disability Studies in Education

Disability Studies in Education (DSE) developed out of Disability Studies (DS), which is an interdisciplinary academic field that questions what meaning is made of difference.  Simi Linton, a DS scholar, writes,

Disability studies takes for its subject matter not simply the variations that exist in human behavior, appearance, functioning, sensory acuity, and cognitive processing but, more crucially, the meaning we make of these variations. The field explores the critical divisions our society makes in creating the normal versus the pathological, the insider versus the outsider, or the competent citizen versus the ward of the state. (Linton, 1998, p.2)

Continue reading “Quick Theory Post #1: Disability Studies in Education”