Category: math + disability article summary

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.

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:

Screen Shot 2018-07-17 at 1.01.33 PM

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?


CGI and special education

As I read the conference program for the CGI (Cognitively Guided Instruction) Conference  coming up this weekend in LA, I was reminded that while math education typically doesn’t include kids with disabilities, CGI has always been an exception. There are several great presentations this week that focus on kids with disabilities doing mathematics, including ones by Jeannie Behrend and Lio Moscardini whose research is discussed below. Here is a quick round-up of CGI research articles that included kids with disabilities (all these articles seem to focus on learning disabilities, which means that we need more work on CGI with other disability categories). Continue reading “CGI and special education”

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

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

New research on how learners in special education intuitively and accurately use indirect addition for subtraction

For far too long, the assumption has been that learners with disabilities cannot benefit from constructivist mathematics instruction.  This assumption, in my opinion, is based on the highly erroneous idea that kids with disabilities cannot think for themselves, but must be spoon-fed methods.  In my experience, this is not true.  First, kids with disabilities are a highly diverse group of learners, with different strengths and needs. Second, all learners construct knowledge based on their own experiences and knowledge. Unfortunately, many researchers in special education mathematics seem to misunderstand constructivist mathematics as discovery learning rather than the carefully designed and scaffolded instructional sequence that it is. Here is some new evidence to support that idea, particularly about subtraction. Continue reading “New research on how learners in special education intuitively and accurately use indirect addition for subtraction”