Author: Rachel Lambert

Resources for Access talk (CGI UCLA)

Deep Accessibility by Star Ford

Feeling the Weight: Some Beginning Notes on Disability, Access and Love Mia Mingus 2012

Mia Mingus Keynote Disability and Intersectionality Summit 2018

My webpage on disability and math:

A post on CGI and special education that links to great research on kids with disabilities engaged in meaningful mathematics

An article I wrote about inequitable access to meaningful mathematics for students in special education “Indefensible, Illogical, and Unsupported”; Countering Deficit Mythologies about the Potential of Students with Learning Disabilities in Mathematics

A great book about disability and mathematics– Humanizing Disability in Mathematics Education by Paulo Tan, Alexis Padilla, Erica Mason and James Sheldon.

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: Teaching Math to SwD during Emergency Remote Teaching

Just uploaded a preprint (before peer review, so not final!) of a study I did this spring with Rachel Schuck, a doc student at UCSB in Special Education. I was working on a research study on UDL with some exceptional special educators. After schools were closed, some of the teachers consented to a series of interviews about their experience teaching during Emergency Remote Teaching (ERT), which is not distance learning, since ERT was unplanned and during a crisis. This first article is focused on teaching math from the perspective of a special educator teaching in a self-contained 3-5th grade classroom. We have a second article which is focused on the experiences of teachers whose students have significant support needs.

Here is the abstract:

“This paper presents a case study of the experiences of a highly experienced special educator named Ms. Z teaching standards-based mathematics during Emergency Remote Teaching (ERT) during the spring of 2020. Pre-COVID, Ms. Z provided her students, all of whom had an IEP for Specific Learning Disabilities and/or Attention Deficit Hyperactivity Disorder, daily opportunities to tackle challenging mathematical problems and taught self-regulation strategies for students to better understand themselves as learners. After the shift to ERT, Ms. Z described “the wall between us” as various digital barriers that made teaching online far more challenging than in person. Challenges included supporting students with productive struggle when not present with them, engaging students in mathematical talk, and creating accessible multi-modal materials. Another barrier was supporting student self-regulation. We analyze Ms. Z’s experience using Universal Design for Learning as the important themes spanned emotional and cognitive engagement, as well as strategic self-understanding. We include recommendations for engaging students with LD and/or ADHD in meaningful mathematical problem solving in ERT. Despite the “walls” in ERT, teachers must create meaningful relationships with students, provide opportunities for students to engage in mathematical talk, create accessible multi-modal materials, and support students to better understand themselves as learners.”

Please read the article for more. Here is just one juicy quotes by this exceptional teacher (fake name Ms. Z), with some of my thoughts that are not in the article. She describes how her first few Zooms in April were taken over by kids:

The first week I started off very just like, the kids wanted to talk to each other. It was hilarious. In one screen, you see somebody having their little cars zooming by and another one, somebody is holding up like five Pokemon cards, just switching through them. And another one, somebody’s got like a family picture. And I’m just like, you guys, you’re supposed to be listening. But they were just so excited to see each other. So we were just doing share outs.

You can see from what Ms. Z says that her students, all of whom have disabilities, wanted time and space to connect with their peers. So much so that they took over the Zoom for an old-fashioned share time. It really bothers me that so many students with disabilities are not being given equal access to synchronous learning on platforms like Zoom. I hear districts and teachers saying that they are not allowed to use Zoom because someone might see a student with an IEP on zoom and know that they have an IEP. Yes, privacy is a concern, but it is inequitable to deny students with disabilities the chance to socialize during a pandemic that is resulting is widespread social isolation and SIGNIFICANT mental health issues. Students with disabilities are already more likely to have anxiety or other emotional disabilities. We need to control privacy risks without discriminating against students with disabilities.

Ms. Z describes “the walls between us” as the barriers that distance and technology have put between her and her students. What walls are you dealing with? Your students dealing with?

Paper citation:

Lambert, R., & Schuck, R. (2020, September 2). “The wall now between us”; Teaching math to students with disabilities during the COVID spring of 2020 PREPRINT.

Accessibility and captions on WordPress

I advocate for accessibility, yet I have had videos on one of my websites for 2 years without captions. I truly believe that it is necessary to make everything in our society accessible to all of us. Inaccessibility creates distance, lack of communication, lack of access, lack of equity. When I realized I needed captions of videos of me teaching a number string on my other website, I thought it would be fairly easy. I started a year ago, ran into problems, tried other solutions, tried other websites, tried captioning in YouTube, tried it in Amara, tried in in Transana, forgot my passwords, forgot about it, remembered, and started again but forgot how to use all the sites again. I finally figured it out and thought I would share for anyone having a hard time, like me. And also I will forget next time and now I can just look at this post! It still uses 3 separate webpages, but it is the simplest way I found.

  1. Transcribe the video. I ended up paying at to do the transcriptions/captions. It was 26 dollars to do about 25 minutes of video.
  2. Check the captions before you download them. You can easily check the captions in and change them before you download them. The ones from were great, including nice descriptions of student’s sounds, etc. You want to watch the video and make sure who is speaking is clear, so I added somethings like [student off camera] to make it a little clearer for those using captions.  I took out student names (although I had permissions from all the families of the kids on this video).  I also had to fix some of the dialogue.
  3. Download the captions as a vtt file
  4. Go to Vimeo. I have a free account to do what I did, although they are constantly trapping me in pages that are trying to make me upgrade. Upload the videos.
  5. In Vimeo for each video, go to Settings, then Distribution, and then Subtitles.
    Upload your vtt file. Make sure to move the little toggle switch over so it lights up blue.
  6. Then you can embed the video into the WordPress site. I had to figure out the new Blocks system to do this.

Note: The captions for the first video I did in Amara (free). It took me a long time. And I couldn’t figure out how to connect the captions to the video. But now it seems much easier in Amara- I just downloaded my original captions as a vtt file and added them to Vimeo using the rest of the steps. I think all this technology is getting better. And we are getting better. I am sorry that it took me so long.

Happy captioning!

NCSM Presentation- quick links

I wanted to share the entire essay by Lynn Pelkey, Lynn Pelkey’s Story

  • Pelkey, L. (2001). In the LD bubble. In P. Rodis, A. Garrod, & M. L. Boscardin (Eds.), Learning disabilities and life stories. Boston, MA: Allyn and Bacon.

This article is a summary of lots of what I talked about today, but more organized!

  • Lambert, R. (2018). “Indefensible, Illogical, and Unsupported”; Countering Deficit Mythologies about the Potential of Students with Learning Disabilities in Mathematics. Education Sciences, 8(2), 72.

This article is a summary of the problems with math ed research compared to special ed research in math, and how that relates to problem-solving

  • Lambert, R., & Tan, P. (2017). Conceptualizations of students with and without disabilities as mathematical problem solvers in educational research: A critical review. Education Sciences, 7(2), 51.

And here is the PDF of all the slides



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.

Intervention in Participation

On Saturday, 11/5/16 I did a presentation at PME-NA in Tucson about a research review I recently did on increasing the participation of students with LD in mathematical problem solving and discussion.

The basic idea is this: why is intervention in mathematics always focused on content. Could we also design intervention in participation?

When a learner is having difficulty in class, we assess that learner. We try to figure out what the learner knows, and what they don’t know. This is an assumption, however, about mathematical knowledge: that it exists independently in the brain, and those who know can recall it without assistance. It also assumes that we are focused on the content standards, rather than the practice standards. While understanding knowledge as individual, as memorized, is important for certain aspects of mathematics, it is not all of mathematics. Secondly, focusing on content as the basis of intervention focuses on the individual as the problem. But what about the classroom? Why didn’t the student “get it” the first time? Yes, we can “reteach” but perhaps we ought to take a closer look at what is happening in the classroom to make this content accessible, or not, to all learners.

In general, students with disabilities have less access to standards-based mathematics, since they all too often segregated from the general education population into self-contained classrooms. These separate classrooms tend to focus on more procedural mathematics (Jackson & Neel, 2006) and more basic content (Kurz et al., 2010).

In an article I wrote with Trisha Sugita, we looked at some studies of the engagement of students with learning disabilities (referring to specific learning disabilities, which includes dyslexia and dyscalculia) when these students are included in standards-based mathematics. In a study of the participation of students in a standards-based mathematics curriculum, students with LD did not participate in whole group mathematics discussion, and tended to focus on non-mathematical tasks such as materials management during small group work (Baxter, Olsen & Woodward, 2001). In another study, students with LD were called on fewer times than other students, and were less involved in small group work (Bottge, Heinrichs, Mehta, & Hung, 2002).

These studies suggest that just including kids with LD in standards-based mathematics is not enough. We need to deepen engagement and participation for these students. We found a small number of studies (n=7) that looked at the participation of students with LD in standards-based mathematics and found that particular teacher moves appeared to increase engagement and participation.

The design of curriculum may be critical. Students were supported in standards-based mathematics classrooms in which they were presented with tasks that offered multiple solution paths (Bottge et al., 2007; Foote & Lambert, 2011; Moscardini 2010). Students were supported by curriculum that was responsive to student thinking, so that teachers designed the tasks based not on a predesigned sequence but on the current understandings of their students (Carpenter et al., 1999/2014). Students were supported when mathematics curriculum provided multi-modal representations of content, and multiple ways of engaging in mathematical activity (Baxter et al., 2005; Bottge et al., 2007; Foote & Lambert, 2011).

These studies also suggested that teacher moves are critical in increasing student participation. While some studies documented significant differences in participation between students with LD and those who were not (Baxter et al., 2001; Bottge et al., 2002), there were classrooms in which participation was equalized (Foote & Lambert, 2011). In these classrooms, teachers supported student problem solving through strategies such as supporting student participation in discussion. Other strategies included rewording problems during problem-solving (Moscardini 2010) and supporting equity in small group work (Bottge et al., 2007). Teachers supported the equal participation of students with LD by giving equal status to presentations that included the notebook and manipulatives as supports (Foote & Lambert, 2011). Researchers noted that consistency in classroom routines may have increased the participation of students with LD.

Research in the mathematical learning of students with learning disabilities must focus attention on intervention in participation. This review included only a small number of studies that suggest a future direction for research in this area. Such research could provide a much needed focus on what learners who are labeled LD can do within standards-based mathematics, rather than what they cannot do. With increasing evidence that such students can construct effective strategies on their own (e.g. Peltenburg, Heuvel-Panhuizen, & Robitzsch, 2012), educators can no longer assume that standards-based mathematics will not work for these students based on perceived deficits. Nor can we assume that simply including students with learning disabilities in standards-based mathematics classrooms will lead to higher achievement. Instead, we must learn more about how to support all learners for full participation in standards- based mathematics classrooms. These supports are likely to be helpful for a wide range of learners, assisting teachers in making standards-based mathematics more equitable for all.

My favorite slide is this:


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.



Rehumanizing mathematics for students with disabilities- a special issue on critical perspectives on disability and mathematics

A couple of years ago, James Sheldon and Kai Rand started a Working Group at the Psychology of Mathematics Education North America conference. This group, now called Critical Perspectives on Disability and Mathematics, is made up of mathematics education scholars who work at the intersections of disability and equity. We seek to create new discussions about disability and mathematics, discussions that move beyond deficit discourses. One of our projects was a special issue, which was just published in Investigations in Mathematics Learning. This special issue brings together scholars from mathematics education and disability studies in education. We are so pleased with the range of scholarship in this special issue. Check out the editor’s introduction here for a introduction to all the articles in special issue: Lambert, Tan, Hunt & Candela 2018

New article: “Indefensible, Illogical, and Unsupported”; Countering Deficit Mythologies about the Potential of Students with Learning Disabilities in Mathematics

For over two years, I have had a word document on my computer entitled, “Myths in Teaching Mathematics for SwD.” I kept adding bits of writing, particularly when I encountered another myth. Imagine my excitement when Jo Boaler sent out a call for a special issue of Education Sciences on Myths in Mathematics Education. I am so proud to have a paper in this issue, which is amazing and available for free online. (I particularly recommend this amazing piece on dyscalculia by my colleagues Katherine Lewis and Dylan Lane)

My paper:

I decided to focus the paper on students with Learning Disabilities (or specific learning disabilities in reading, writing or math, otherwise known as dyslexia, dysgraphia or dyscalculia). While I wanted to write about a wider range of disabilities, the best research evidence was on this group of learners. I also picked two myths to focus on:

  1. Students with LD ONLY benefit from explicit or direct instruction.
  2. Students with LD cannot create their own strategies in math, and cannot handle multiple strategies.

The first is a major myth that I hear all the time, and the second is a kind of a sub-myth. The assumption that students with LD cannot construct strategies is so pernicious that I decided to include it as a separate myth.

I structured the paper around two things: first a quote written about students with disabilities. This was published in a prominent special education journal in 1998:

“The premise that secondary students with LD will construct their own knowledge about important mathematical concepts, skills, and relationships, or that in the absence of specific instruction or prompting they will learn how or when to apply what they have learned, is indefensible, illogical, and unsupported by empirical investigations.”.
(Jones, Wilson, & Bhojwani, 1998, p. 161)

This quote still shocks me. Having known, taught, been a friend to and a family member or so many people with various permutations of LD, the idea that such learners cannot “construct knowledge” is exceptionally bigoted and wrong. This particular article described constructivism as “ideology” rather than a valid approach to teaching math. In the paper, I try to describe why these myths are themselves “indefensible, illogical and unsupported.” I do not ignore the strong empirical evidence from special education mathematics that students with LD can benefit from explicit instruction, but I present evidence that suggests inquiry instruction as also effective. We also need to consider why we teach mathematics- it is not just to make students into effective computers, but to help them develop life-long identities as mathematical thinkers and explorers. The myth emerges from the assumption that there exists sufficient evidence that inquiry mathematics is NOT effective for students with LD, or that explicit instruction is the only method that is evidence-based. As the National Mathematics Advisory Panel states, “it is important to note that there is no evidence supporting explicit instruction as the only mode of instruction for students [with LD]” (2008, p. 1229).

As I was writing this piece, I checked Twitter and found this tweet:

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Thank you Abby. This tweet inspired me to keep writing, and keep poring through research. If you are more interested in understanding the research divide between math ed and special ed, I would check out another article I wrote with Paulo Tan in Education Sciences (


  1. What myths still need to be unpacked?
  2. What kind of research would you like to see around students with disabilities and mathematics? What specific questions have emerged from your work?