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.

 

 

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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: 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:

  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 (http://www.mdpi.com/2227-7102/7/2/51).

Questions:

  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?

 

Resources for teachers on math and disability from a strengths-based perspective

After I made the list of academic papers, I realized I should do something similar for writings that are aimed at teachers that take a non-traditional point of view on disability and teaching mathematics. Updated May 2018. Please comment if you have items to add!
Behrend, J. L. (2003). Learning-disabled students make sense of mathematics. Teaching Children Mathematics, 9(5), 269–73.
Chick, C., Tierney, C., & Storeygard, J. (2007). Seeing Students’ Knowledge of Fractions: Candace’s Inclusive Classroom. Teaching Children Mathematics, 14(1), 52–57.
Fosnot, C. (2010). Models of Intervention in Mathematics: Reweaving the Tapestry. National Council of Teachers of Mathematics.
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. https://doi.org/10.3390/educsci8020072
Lambert, R., & Sugita, T. (2016). Increasing engagement of students with learning disabilities in mathematical problem-solving and discussion. Support for Learning, 31(4), 347–366. https://doi.org/10.1111/1467-9604.12142
Lewis, K. E., & Lynn, D. M. (2018). Against the Odds: Insights from a Statistician with Dyscalculia. Education Sciences, 8(2), 63. https://doi.org/10.3390/educsci8020063
Lynch, S. D., Hunt, J. H., & Lewis, K. E. (2018). Productive Struggle for All: Differentiated Instruction. Mathematics Teaching in the Middle School, 23(4), 194–201.
Moscardini, L. (2009). Tools or crutches? Apparatus as a sense‐making aid in mathematics teaching with children with moderate learning difficulties. Support for Learning, 24(1), 35–41. https://doi.org/10.1111/j.1467-9604.2009.01395.x
Moscardini, L. (2010). ‘I like it instead of maths’: How pupils with moderate learning difficulties in Scottish primary special schools intuitively solved mathematical word problems. British Journal of Special Education, 37(3), 130–138. https://doi.org/10.1111/j.1467-8578.2010.00461.x
Storeygard, J. (2009). My Kids Can: Making Math Accessible to All Learners, K-5 (Pap/DVD). Heinemann.
Storeygard, J. S. (2012). Count Me In! K-5: Including Learners With Special Needs in Mathematics Classrooms (1 edition). Thousand Oaks, Calif: Corwin.
Storeygard, J., & Tierney, C. (2005). Including all students in meaningful mathematics: the story of Darrell. Teaching Exceptional Children Plus, 1(3).
Tan, P. (2017). Advancing Inclusive Mathematics Education: Strategies and Resources for Effective IEP Practices. International Journal of Whole Schooling, 13(3), 28–38.
Zager, T. (2017). Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms. Portland, Maine: Stenhouse Publishers.

Bibliography: Critical Approaches to Mathematics and Disability

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!

Behrend, J. L. (2003). Learning-disabled students make sense of mathematics. Teaching Children Mathematics, 9(5), 269–73.
Borgioli, G. M. (2008). A critical examination of learning disabilities in mathematics; Applying the lens of abelism. Journal of Thought, 43(12), 131–147.
Bottge, B. A., Stephens, A. C., Rueda, E., LaRoque, P. T., & Grant, T. S. (2010). Anchoring Problem-Solving and Computation Instruction in Context-Rich Learning Environments. Exceptional Children, 76(4), 417–437.
Chick, C., Tierney, C., & Storeygard, J. (2007). Seeing Students’ Knowledge of Fractions: Candace’s Inclusive Classroom. Teaching Children Mathematics, 14(1), 52–57.
de Freitas, E., & Sinclair, N. (2016). The cognitive labour of mathematics dis/ability: Neurocognitive approaches to number sense. International Journal of Educational Research, 79, 222–230.
de Freitas, Elizabeth, & Sinclair, Natalie. (2014). Mathematics and the Body: Material Entanglements in the Classroom. Cambridge University Press.
Fernandes, S. H. A. A., & Healy, L. (2013). Multimodality and mathematical meaning-making: Blind students’ interactions with Symmetry. International Journal for Research in Mathematics Education, 3(1), 36–55.
Foote, M. Q., & Lambert, R. (2011). I have a solution to share: Learning through equitable participation in a mathematics classroom. Canadian Journal of Science, Mathematics and Technology Education, 11(3), 247–260. https://doi.org/10.1080/14926156.2011.595882
Fosnot, C. (2010). Models of Intervention in Mathematics: Reweaving the Tapestry. National Council of Teachers of Mathematics.
Healy, L., & Fernandes, S. H. A. A. (2011). The role of gestures in the mathematical practices of those who do not see with their eyes. Educational Studies in Mathematics, 77(2–3), 157–174.
Hunt, J. H. (2015). Notions of equivalence through ratios: Students with and without learning disabilities. The Journal of Mathematical Behavior, 37, 94–105. https://doi.org/10.1016/j.jmathb.2014.12.002
Hunt, J. H., Tzur, R., & Westenskow, A. (2016). Evolution of unit fraction conceptions in two fifth-graders with a learning disability: An exploratory study. Mathematical Thinking and Learning, 18(3), 182–208.
Hunt, J. H., Westenskow, A., Silva, J., & Welch-Ptak, J. (2016). Levels of participatory conception of fractional quantity along a purposefully sequenced series of equal sharing tasks: Stu’s trajectory. The Journal of Mathematical Behavior, 41, 45–67.
Hunt, J., & Tzur, R. (2017). Where is Difference? Processes of Mathematical Remediation through a Constructivist Lens. The Journal of Mathematical Behavior, 48, 62–76. https://doi.org/10.1016/j.jmathb.2017.06.007
Jackson, H. G., & Neel, R. S. (2006). Observing mathematics: Do students with EBD have access to standards-based mathematics instruction? Education and Treatment of Children, 29(4), 593.
Lambert, R. (2015). Constructing and resisting disability in mathematics classrooms: a case study exploring the impact of different pedagogies. Educational Studies in Mathematics, 89(1), 1–18. https://doi.org/10.1007/s10649-014-9587-6
Lambert, R. (2017). ‘When I am being rushed it slows down my brain’: Constructing self-understandings as a mathematics learner. International Journal of Inclusive Education, 21(5), 521–531. https://doi.org/10.1080/13603116.2016.1251978
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. https://doi.org/10.3390/educsci8020072
Lambert, R., & Sugita, T. (2016). Increasing engagement of students with learning disabilities in mathematical problem-solving and discussion. Support for Learning, 31(4), 347–366. https://doi.org/10.1111/1467-9604.12142
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. https://doi.org/10.3390/educsci7020051
Lambert, R., Tan, P., Hunt, J., & Candela, A. G. (2018). Rehumanizing the Mathematics Education of Students with Disabilities; Critical Perspectives on Research and Practice. Investigations in Mathematics Learning, 0(0), 1–4. https://doi.org/10.1080/19477503.2018.1463006
Lewis, K. E. (2014). Difference not deficit: Reconceptualizing Mathematical Learning Disabilities. Journal for Research in Mathematics Education, 45(3), 351–396. https://doi.org/10.5951/jresematheduc.45.3.0351
Lewis, K. E. (2017). Designing a Bridging Discourse: Re-Mediation of a Mathematical Learning Disability. Journal of the Learning Sciences. Retrieved from http://www.tandfonline.com/eprint/A5wfenur9ZCNTn4atCtD/full
Lewis, K. E. ., kelewis2@uw. ed., & Fisher, M. B. ., mfisher4@uw. ed. (2016). Taking stock of 40 years of research on mathematical learning disability: Methodological issues and future directions. Journal for Research in Mathematics Education, 47(4), 338–371.
Lewis, K. E., & Lynn, D. M. (2018). Against the Odds: Insights from a Statistician with Dyscalculia. Education Sciences, 8(2), 63. https://doi.org/10.3390/educsci8020063
Lynch, S. D., Hunt, J. H., & Lewis, K. E. (2018). Productive Struggle for All: Differentiated Instruction. Mathematics Teaching in the Middle School, 23(4), 194–201.
Magne, O. (2001). Literature on special educational needs in mathematics: a bibliography with some comments. Retrieved from https://dspace.mah.se/handle/2043/6043
Moscardini, L. (2009). Tools or crutches? Apparatus as a sense‐making aid in mathematics teaching with children with moderate learning difficulties. Support for Learning, 24(1), 35–41. https://doi.org/10.1111/j.1467-9604.2009.01395.x
Moscardini, L. (2010). ‘I like it instead of maths’: How pupils with moderate learning difficulties in Scottish primary special schools intuitively solved mathematical word problems. British Journal of Special Education, 37(3), 130–138. https://doi.org/10.1111/j.1467-8578.2010.00461.x
Moscardini, L. (2013). Primary special school teachers’ knowledge and beliefs about supporting learning in numeracy. Journal of Research in Special Educational Needs, v15 n1 p3747(n1), p37-47. https://doi.org/10.1111/1471-3802.12042
Peltenburg, M. (2012). Mathematical potential of special education students (Dissertation). Utrecht University.
Radford, L. (2013). Perceiving with the eyes and with the hands. International Journal for Research in Mathematics Education, 3(1). Retrieved from http://www.luisradford.ca/pub/2013%20-%20Perceiving%20with%20the%20eyes%20and%20with%20the%20hands.pdf
Sheldon, J. (2013). Erasing Queer Subjects, Constructing Disabled Subjects: Towards a Queering of Mathematics Disabilities. Retrieved from https://eric.ed.gov/?id=ED560838
Sheldon, J. (2017). Problematizing Reflexivity, Validity, and Disclosure: Research by People with Disabilities About Disability. The Qualitative Report, 22(4), 984–1000.
Sheldon, J., & Rands, K. (2017). Whatever Will Be Will Be: Queering Disabled Subjects’ Temporality. Philosophical Inquiry in Education, 24(4), 368–378.
Storeygard, J. (2009). My Kids Can: Making Math Accessible to All Learners, K-5 (Pap/DVD). Heinemann.
Storeygard, J. S. (2012). Count Me In! K-5: Including Learners With Special Needs in Mathematics Classrooms (1 edition). Thousand Oaks, Calif: Corwin.
Storeygard, J., & Tierney, C. (2005). Including all students in meaningful mathematics: the story of Darrell. Teaching Exceptional Children Plus, 1(3).
Tan, P. (2017a). Advancing Inclusive Mathematics Education: Strategies and Resources for Effective IEP Practices. International Journal of Whole Schooling, 13(3), 28–38.
Tan, P. (2017b). TOWARD INCLUSIVE MATHEMATICS EDUCATION FOR” INFERIOR STUDENTS WITH NO BRAINS:” A CASE STUDY OF A STUDENT WITH AUTISM AND HIS PEER. Journal of Ethnographic & Qualitative Research, 11(3).
Tan, P., & Alant, E. (2018). Using peer-mediated instruction to support communication involving a student with autism during mathematics activities: A case study. Assistive Technology, 30(1), 9–15.
Tan, P., & Kastberg, S. (2017). Calling for Research Collaborations and the Use of Dis/ability Studies in Mathematics Education. Journal of Urban Mathematics Education, 10(2), 25–38.
Truman, J. V. (2017, August 8). Mathematical reasoning among adults on the autism spectrum: Case studies with mathematically experienced participants (Thesis). Education: Faculty of Education. Retrieved from http://summit.sfu.ca/item/17501
Woodward, J., & Montague, M. (2002). Meeting the challenge of mathematics reform for students with LD. Journal of Special Education, 36(2), 89–101.

Paraprofessionals and Mathematics!

One of the most important agents of change in special education is paraprofessionals. These dedicated professionals are often the mediators between classroom teachers and our students with disabilities. They help create access for the child to the curriculum. Unfortunately, paraprofessionals are almost never given planning time with teachers, and are not usually allowed access to professional development. In mathematics, this can mean that the paraprofessional and the mathematics teacher have different ideas about what and how the student should be engaged in mathematics.

Judy Storeygard (author of several excellent texts on mathematics teaching for students with learning differences and/or special education services (Count Me In and My Kids Can) has been working with colleagues on addressing the issue of access for paraprofessionals. Funded by the National Science Foundation, they have provided a pilot program for paraprofessionals in the Boston Public Schools, designed to increase access for paraprofessionals to high-quality professional development in mathematics. I would love to see more projects like this!

Check out this video: http://stemforall2018.videohall.com/presentations/1095

One of the presenters, Karen Mutch-Jones, wrote the following in a comment on the video linked above. She was describing preliminary findings about what was important in designing this kind of professional development for paraprofessionals.

1) Paras need opportunities, over time, to immerse themselves in mathematics activities that help them to develop their mathematical thinking. Often, they have not had positive experiences learning math themselves, and so as part of this process, they are developing their math confidence.  Many have become quite excited about doing math!

2) Our paras (and other paras we have spoken with) can identify the ways in which math instruction is quite different than what they experienced as students–but since many haven’t been involved in a teacher education program, they aren’t always sure about what it means to instruct with an inquiry-oriented curricula or resources. Understanding the goals of such curricula and experiencing inquiry approaches within their PD have been important to them.  Also, they have been eager to learn and try out instructional strategies in their classrooms (e.g., how to listen for student thinking, asking questions that provide students with new challenges within an activity, asking students to re-tell the story problem), and then return to PD to debrief with each other and us.

3) Follow-up activities have also been helpful.  Through our project, the paras have had opportunities to plan with their teachers and reflect on their students’ math learning and struggles.

4) And last, but not least, the paraeducators have created a math learning community where they provide each other with support, encouragement, and new ideas.  They are incredible resources for each other!

Here are more details about their grant:
Over one million para educators currently assist in classrooms, and another 100,000 are likely to be added in the next ten years. Para educators are often required to teach content, such as mathematics, but there are few efforts to provide them with the knowledge or supervision they need to be effective when working with a range of students, including those with disabilities and for whom English is a second language. While professional development will enable paras to make a greater difference in the classroom it may also increase their access to continuing education and workplace opportunities. Our project is designed to develop, pilot, study, and refine PD, that focuses on developing the confidence, mathematics knowledge, and teaching strategies of para educators, grades K-3 in the Boston Public Schools, as well as providing support for their collaborating teachers. 

Project based learning, science, maker spaces and dyslexia

This post below presents the story of a student with dyslexia who fell in love with science through an inquiry-based classroom. I love how the writer highlights how his teacher’s relationship was the core support for learning. When a teacher is attuned to students, anything is possible, and students (like this one) can transform.

http://composeourworld.org/blog/2016/01/20/accessible-project-based-learning/

In a related search, I found this on leveraging neurodiversity in maker spaces. I so agree- my neurodiverse students and colleagues can thrive in maker spaces and project-based classrooms, where they can be led by their considerable curiosity and drive.

https://www.edutopia.org/blog/encouraging-neurodiversity-in-makerspace-classroom-patrick-waters