Opening the Middle of Special Education Math Tasks

  • Opening the Middle of Special Education Math Tasks

    By Andrew Gael, posted June 6, 2016 —

    For years, special education mathematics classes have focused on rote computation and money skills with steadfast tunnel vision (Browder et al. 2008). The meta-analysis by Browder and her colleagues reveals that about 93 percent of studies of mathematics classes for students with significant cognitive disabilities focused on the NCTM domains of Numbers and Operations as well as Measurement, specifically money and time. No wonder students with disabilities are “failing standardized tests” (Samuels 2016). Their classes focus on just one strand of mathematical proficiency: procedural fluency.

    However, greater importance has recently been placed on the other strands of mathematical proficiency: conceptual understanding, strategic competence, adaptive reasoning, and productive disposition. Why aren’t special education math classes also making this shift? This lack of progress is creating an inequitable landscape where students with disabilities are not given the opportunity to develop into mathematically proficient students. Small addresses this issue in her book Good Questions: Great Ways to Differentiate Mathematics Instruction:

    Although many teachers of language arts recognize that different students need different reading materials, depending on their reading level, it is much less likely that teachers vary the material they ask their students to work with in mathematics. The math teacher will more frequently teach all students based on a fairly narrow curriculum goal presented in a textbook (2012, p. 1).

    How can we do this? How can we broaden our scope for students with disabilities but still work on the skills that they need to be successfully independent adults? By opening the middle of math tasks in special education classes. Too often students with disabilities are left at the first level of Webb’s depth of knowledge (DOK). Problems like 44 + 27 =___ and “If you have 2 dimes and 3 pennies, how many cents do you have?” do not give students with disabilities the opportunity to develop into mathematically proficient students. Problems that Kaplinsky and Johnson refer to as open-middle problems will help develop mathematically proficient students. Open-middle problems usually have multiple solution pathways, a procedural look, a puzzle-like complexity, and a seemingly simple solution, but often a more complex optimal answer. Take this open-middle problem created by Domina and Kaplinsky to meet a second-grade standard:

    Make 47¢ using exactly 6 coins with either quarters, dimes, nickels, or pennies.

    This problem is considered DOK level 3 and allows students to use and develop all strands of mathematical proficiency. Students with disabilities can also express their knowledge and understanding in different ways. Here is an example from our presentation at this year’s NCTM Annual Meeting and Exposition in San Francisco:

    2016-06-06 fig1b

    Each of these students used a different form of expression (from left to right: algorithms, manipulatives, assistive tech, and drawings) to demonstrate their understanding of the problem. Because this problem had an open middle, students were not only allowed but also encouraged to explore different representations and tools to use to solve the problem.

    Often problems like this are considered “too hard” for “lower-level students,” but as Wees writes, these kinds of labels and expectations can have negative effects on our students, and this is especially true for students with disabilities. I am not advocating that special education math classes stop teaching computation and money skills, but if you do, please make sure that students have opportunities for to develop all the strands of mathematical proficiency. When you keep your expectations high, your students with disabilities just might meet them!

    What tasks that you already do in special education classes can be opened, in either the middle or the end, to encourage higher levels of depth of knowledge?


    Browder, Diane M., Fred Spooner, Lynn Ahlgrim-Delzell, Amber A. Harris, Shawnee Wakemanxya. 2008. “A Meta-Analysis on Teaching Mathematics to Students with Significant Cognitive Disabilities.” Exceptional Children 74 (4): 407–32.

    Small, Marian. 2012. Good Questions: Great Ways to Differentiate Mathematics Instruction. New York: Teacher’s College Press.

    2016-06 Gael aupic Andrew Gael has worked in education for more than ten years and as an educator of students with disabilities for the last six. He was born and raised in New York City, receiving his Master’s in Education from Brooklyn College. Gael now teaches math at the Cooke Center Academy, a school for students with developmental, learning, and physical disabilities in Manhattan. He has spoken nationally to advocate for equitable access to the highest quality math instruction for the typically under-served student population of students with disabilities. He writes about the intersection of math education, special education, and disability rights on his blog, The Learning Kaleidoscope


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    Drew Polly - 7/3/2016 9:47:20 PM
    Andrew- great point. Finding a happy medium is important especially looking for opportunities for open middle tasks. With students on the RTI Tier 2 and 3 there definitely is a need to consider how to promote and develop mathematical reasoning.

    Zachary Champagne - 6/23/2016 5:45:09 PM
    Thanks for bringing this point up Drew - and a very well stated response Andrew. I totally agree that there is space for both instructional techniques when and where they are appropriate for students.

    Andrew Gael - 6/21/2016 10:53:54 AM
    Hi Drew, thanks for the comment! Yes, the curriculum of Browder, et al. is very structured and explicit. Here's an example: I am definitely not saying we should abandon explicit instruction, especially if a teacher determines that is what their group of students needs to meet their mathematical goals. I am, however, advocating that we presume competence of our students with disabilities and encourage special educators to explore the implementation of mathematical tasks such as that of Marian Small and Open Middle when appropriate.

    Drew Polly - 6/19/2016 9:59:54 PM
    Great points Andrew. Browder and her colleagues here at UNC Charlotte are doing quite a bit of research on students solving some of the various problem types for addition and subtraction. Their work is very focused on explicit instructions with their population which is interesting to consider in light of your blog and Small's book which promotes more student centered inquiry. Interesting to think about the balance between the special education explicit instruction and the potential of students discovering mathematics concepts on their own.

    Beth Skipper - 6/7/2016 1:57:09 PM
    This is such an important topic. Thank you for initiating this discussion.

    Robert Kaplinsky - 6/7/2016 12:05:19 PM
    I love this Andrew. Thanks for the write up. I agree that open middle problems are good for students with disabilities, but only because they are good for all students. Keep on spreading the good word.