### Students with Disabilities Learning Math – Part 1

In this first part of a two-part series on teaching math to students with disabilities the focus is on the needs of *students* learning math and how teachers can provide instruction to meet these needs. Part 2 (CORE Academic Quarterly January 2019) will focus on the resources teachers need for teaching math in special education settings.

Students with disabilities are required to receive a rigorous education. In *Endrew F. vs Douglas County School District RE-I (2017) *the United States Supreme Court ruled that for any student the “educational program must be appropriately ambitious in light of his[/her] circumstances” and “every child should have the chance to meet challenging objectives” (p. 14). At all levels appropriate math instruction means developing mathematical proficiency along five strands as described by the National Research Council in *Adding It Up* (Kilpatrick, Swafford, & Bradford, 2001).

- Productive disposition
- Conceptual Understanding
- Adaptive Reasoning
- Strategic Competence
- Procedural Fluency

These five strands are interconnected in developing mathematical understanding, fluency, and the ability to apply math. Mastery of any single strand by itself does not represent proficiency in math. Fluency without conceptual understanding is a dead-end, and conceptual understanding without fluency is a wobbly foundation at best. Students must learn to reason with math and think strategically about how to solve problems, make sense of the math they are learning, know when to use procedures, and know how to adapt procedures and strategies to a given situation. To have a suitably challenging math education, students with disabilities must learn math along all five strands in ways appropriate to each student’s circumstance. Resources from the CEEDAR Center (McLeskey et al., 2017), Institute for Educational Sciences (Gersten et al., 2009), the National Council of Teachers of Mathematics (NCTM, 2014), and the National Mathematics Advisory Panel (NMAP, 2008) provide recommendations and insights for teachers on how best to focus and deliver instruction (see references at end of this article). What follows is a description of the math learning needs of students with disabilities within the framework of the five proficiency strands and ways teachers can meet these needs.

### Productive Disposition

Students benefit from a productive disposition, which means students see math as sensible, useful, and something that with effort they can learn (Kilpatrick et al., 2001). Part of a productive disposition is having a growth mindset. Students should reflect on learning, reflect on attitudes and habits, and notice that with hard work they are succeeding (Hattie, Fisher, & Frey, 2017). Making sense of mathematics means students see why procedures work. Students are making sense of equivalent fractions when they recognize why 1/5 and 2/10 are the same value. Students recognize that 2/10 is simply the result of cutting each part of 1/5 into two pieces without actually changing the whole or overall amount taken.

Telling students to think positively and believe in their own ability to learn is not enough. Actual success with learning goes a long way towards building confidence, which in turn, feeds greater success. Students with disabilities have the greatest chance for success when teachers habitually use effective teaching techniques. Research recommends using explicit instruction, focusing on the most important concepts and skills, building fluency and number sense in short daily practice sessions, scaffolding lessons appropriately, providing timely and specific feedback, and utilizing both massed and distributed practice (Gersten et al. 2009; Hattie et al., 2017, McLeskey et al. 2017). Using these effective practices helps build a productive disposition.

### Conceptual Understanding, Strategic Competence and Adaptive Reasoning

Conceptual understanding is the foundation for mathematical proficiency. For example, students understand fractions conceptually when they see that 2/5 is a whole divided into five equal parts and we are considering two of these parts. Adaptive reasoning refers to the capacity to think logically about the relationships between concepts and situations. Adaptive reasoning is recognizing that adding fractions is like adding whole numbers. We change fractions to the same denominators because we can only add like units together. With whole numbers the units are ones, tens, hundreds, etc. With fractions the units are halves, thirds, fourths, tenths, etc. Seeing the connections between why we change fractions to the same denominator and how we add whole numbers is an important step in developing proficiency with fraction addition and subtraction. Strategic competence means being able to think strategically how to apply math to solve problems. This includes word problems and other real-world applications that require students to see which concepts and procedures to apply, when, and how.

Students with disabilities can understand, reason about, and apply math concepts. To believe otherwise would severely limit their educational potential. If it were true that most students with disabilities could not understand the concepts underpinning basic math procedures, then these students would also not be able to understand the concepts taught in upper elementary, middle and high school math. However, research shows that no such generalizable upper limit exists for students with disabilities. With *Endrew* the Supreme Court ruled we must provide the greatest appropriate level of challenge for each student. Consequently, students with disabilities must have the opportunity to learn concepts, connections, and applications in mathematics.

Research provides guidance on how to help students reach conceptual understanding, improve reasoning, apply math, and develop procedural fluency. Teachers must make math ideas explicit by connecting visual models to abstract models, providing clear problem-solving strategies, and by having students discuss math with each other in order to make sense of math and think productively. Students with disabilities need more explicit instruction, fewer strategies taught to them, more practice, and more frequent specific feedback (Gersten et al., 2009; Hattie et al., 2017). McLeskey et al. (2017) explain that the use of content enhancement scaffolds, such as advance organizers, mnemonic illustrations, cue cards, guided notes, and strategy sheets have positive effects on both factual and conceptual learning for students with disabilities (pp. 79-80).

### Procedural Fluency

Procedural fluency is an important aspect of developing mathematical proficiency. Research recommends short daily practice building fluency with basic facts and skills (Gersten et al., 2009). Research also tells us that the most important areas of focus for students struggling with math are work on operations with whole numbers and with rational numbers (Gersten et al., 2009; NMAP, 2007). One of the most helpful ways to assist special education teachers with building fluency is bringing more robust fluency activities to their classrooms – activities that are engaging for students and build number sense, reasoning abilities, and fluency simultaneously.

In a special education class last year I modeled using KenKen Puzzles. Two examples of these puzzles are below.

*At KenKenPuzzle.com you can create puzzles within a range of difficulty levels using one or more of the four basic operations (+, −, ×, ÷). Puzzle sizes range from 3 by 3 to 9 by 9 grids. The numbers used to fill in the cells on the puzzle are determined by the puzzle size. On a 3 by 3 puzzle only the numbers 1, 2, and 3 can be used to fill in the cells. With a 4 by 4 puzzle, only the numbers 1 2, 3, and 4 can be used to fill in the cells. Each row and each column must end up with a full range of numbers. Cells linked together with a bold outline have a result that equals the amount shown in the upper left corner of the linked cells.*

When I introduced these puzzles to the special education class, students were excited by the idea of solving puzzles. I modeled explicitly how the puzzles work, completed one with the class, and began a second puzzle (harder puzzle) with the class. Then I released students to work independently with a packet of puzzles that began with a very basic puzzle and increased in difficultly through small increments to other puzzles. Students worked enthusiastically for about 40 minutes and all students were successful in solving at least three puzzles. Students practiced basic math facts, engaged in strategic thinking, and learned to be flexible with numbers. All students felt challenged, empowered mathematically, and were highly motivated with math for an extended time.

This is one example of how to engage and challenge students with math not only in special education classrooms but in all math classrooms. Hands on materials, games, charts, and visuals such as number lines are effective tools for developing fluency. These types of activities require students to think, explain, and practice using numbers and procedures. It is also worth noting that many good activities exist that are simply focused on building fluency. Sometimes just practice focused on one or more isolated skills is the best work for a student. However, it is important to remember that fluency with facts and procedures should not be the only outcome for math instruction. All students can learn to reason about and with mathematics.

### Conclusion

Students with disabilities often struggle with learning math. Indeed, this struggle may be related to their disability. However, research and experience tell us these students are capable of high-level learning. They can understand concepts, think strategically, reason abstractly, become fluent with facts and procedures and feel confidence with and make sense of mathematics. The difficulty is finding the appropriate level of challenge for students and providing the necessary scaffolds and techniques to enable success.

Part 2 of this series (CORE Academic Quarterly January 2019) will focus on the resources teachers need so that they can provide appropriately challenging math instruction in special education settings.

**References**

Endrew F. v. Douglas County School District, RE-1, 580 U.S. (2017).

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). *Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools* (NCEE 2009-4060 ed.). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from https://ies.ed.gov/ncee/wwc/PracticeGuide/2

Hattie, J., Fisher, D., & Frey, N. (2017). *Visible Learning for Mathematics*. Thousand Oaks, CA: Corwin

Kilpatrick, J., Swafford, J., and Bradford Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Research Council, National Academy Press.

McLeskey, J., Barringer, M-D., Billingsley, B., Brownell, M., Jackson, D., Kennedy, M., Lewis, T., Maheady, L., Rodriguez, J., Scheeler, M. C., Winn, J., & Ziegler, D. (2017, January), *High-leverage practices in special education*, Arlington, VA: Council for Exceptional Children & CEEDAR Center.

National Council of Teachers of Mathematics. (2014). *Principles to Actions Ensuring Mathematical Success for All*. Reston, VA: Author.

National Mathematics Advisory Panel. (2008). *Foundations for Success: The Final Report of the National Mathematics Advisory Panel*. Washington, DC: U.S. Department of Education. Retrieved from https://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf