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:
I thoroughly enjoyed reading this post by Anna Blinstein (@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.
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
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: http://www.mdpi.com/2227-7102/8/2/72
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:
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:
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 (http://www.mdpi.com/2227-7102/7/2/51).
I created this list based on a request for a Ph.D. reading list for a student interested in critical and Disability Studies approaches to mathematics education. Figured I should share with more, as there are so many wonderful resources on this list. Updated May 2018. Please comment if you have recommendations to add!