An Introduction to Mathematics

[Teacher Tools] [Case Studies]

Teaching mathematics to students with special needs can be an exciting and rewarding experience for both teachers and students. Many innovative practices are emerging that are both making the learning of mathematics more enjoyable for students and making what students are learning more meaningful. The National Council of Teachers of Mathematics (NCTM) Illuminations Internet site provides links to examples of a variety of mathematics instructional practices:

NCTM Illuminations: http://illuminations.nctm.org/

A brief review of the learning standards for mathematics developed by the National Council of Teachers of Mathematics (NCTM) demonstrates the complexity of this content area and the far-reaching implications mathematics has for helping our children and youth develop critical life skills:

10 NCTM Standards

- Number & Operations
- Algebra
- Geometry
- Measurement
- Data Analysis & Probability
- Problem Solving
- Reasoning & Proof
- Communication
- Connections
- Representation

While such an emphasis has begun to impact mathematics instruction at a general level, the impact for students who have special needs has been nonexistent. Compared to other content areas such as reading, mathematics has received considerably less emphasis as it relates to teaching students with special needs. Research in the area of mathematics and students with special needs is greatly limited in comparison to research in reading. Making meaning from text (i.e. reading) is obviously important. Similarly, making meaning from both the symbols of mathematics and the meaning of mathematics is also important. Nonetheless, students with special needs consistently perform at lower levels than their peers without special needs (Cawley, Parmar, Yan, & Miller, 1996).

There are a variety of reasons for the lack of emphasis on mathematics instruction for students with special needs and a variety of barriers that result in low performance outcomes for these students (e.g., curriculum pacing guides that prevent students with special needs from mastering mathematics concepts and skills; instructional practices and curriculum that do not meet their learning needs; learning characteristics that result from disabilities that make learning mathematics difficult). Some of these barriers are not ones which teachers can address. However, there are other barriers that teachers and other educators can directly address to make the mathematics learning both more emphasized in schools and a more successful experience for students with special needs.

As can be seen by examining the NCTM standards, mathematics is much more than working with numbers and symbols and doing arithmetic. While understanding how to manipulate numbers and symbols to do arithmetic is certainly important for all students, there is much more that students must understand and be able to do mathematically in order for them to apply mathematics in ways that will enhance their future lives. Students must understand the meaning of mathematics in order for them to truly benefit long-term from mathematics instruction. Mathematical understanding is vital if students with special needs are to successfully apply mathematics in their day-to-day lives, from childhood through adulthood. It is our duty as educators who work with students with special needs to value mathematics, to value the importance it holds for the future success of students with special needs, and to value developing our abilities to teach mathematics in ways that make mathematics meaningful for our students.

This may sound a bit optimistic to you, particularly given the struggles you and your students face every day. However, there are very direct actions you can take to make such a goal become increasingly possible for you and your students:

Value Mathematics Yourself

The first thing you can do is to evaluate your own values toward mathematics. If you are like me, your experience learning mathematics was probably less than meaningful except for that special teacher or two you had who could make anything seem exciting and learnable. Sometimes, our own values toward the subjects we teach can impact how we teach that subject. If we value something, we tend to emphasize it more; we spend more time learning about it; as teachers, we tend to spend more time addressing it when we teach. It's human nature to do this! If you have less than fond feelings about mathematics, examine why this may be the case. Perhaps you had one or more bad experiences with mathematics while you were in school. Perhaps you just kind of went through the motions (like me!) when you were in your mathematics classes, not really seeing the relevance of mathematics; it was a subject that was not meaningful to you because it was presented to you in mostly abstract ways and memorization was the primary mode of "learning." Perhaps mathematics was a wonderful learning experience for you. Whatever your experience, spend some time thinking about how much mathematics is embedded in your life today. Remember that the same will be true for your students, and probably to an even greater extent given the advancements continually being made in the areas of mathematics, science, and technology. By valuing mathematics, you will undoubtedly express this to your students in both subtle and direct ways, providing them a wonderful model for valuing mathematics themselves!

Enhance Your Own Level of Mathematical Understanding

We can all benefit from continual professional development. Elementary education and special education teachers, in particular, receive relatively little preparation in mathematics education. One course in mathematics teaching methods is the norm for most teacher preparation programs. Most of us rely on our own mathematical understandings as a basis for teaching our students. For those of us who learned mathematics by virtue of good memorization skills (memorizing procedures for solving equations, formulas and the like), or who rely primarily on an abstract level of mathematical understanding, we will find it difficult to effectively teach our students with special needs how to successfully learn and do mathematics. Professional development is something all of us as educators must embrace. To make the most of our professional development it is helpful to take a structured approach. In terms of your professional development in teaching mathematics for students who have special needs, this is especially true. Professional development should have at least two purposes. Professional development that focuses on pedagogy; that is the teaching of mathematics for students with special needs, is an important focus, one that is the primary purpose of this module. However, professional development that focuses on your understanding of the mathematics that you teach is also essential, particularly for those of us who do not have an extensive background in mathematics and mathematics education. This second focus will be one that you must undertake on your own, at least to a larger degree than the first. If you are to teach your students mathematical understanding in meaningful ways, then you must understand it yourself. You must posses all three levels of understanding for the mathematics concepts, skills, and standards you teach: concrete, representational/semi-concrete, and abstract (See the Teacher Tool section for additional information about these three levels of understanding and why they are important con siderations for students with special needs). A multi-level basis of understanding on your part is essential if you are going to effectively work with students who have special needs. Depending on the level of understanding of your student, you need to provide learning experiences commensurate with that level of understanding that accurately depict the mathematics concept/skill they are learning. In doing so, you will provide your students a solid foundation for building their conceptual understandings of the mathematics you teach. Possessing such understanding on your part will open up a multitude of possibilities for both you (instructionally) and for your students (learning). The better you understand what the different mathematics concepts/skills you teach are about and how they apply to real life situations, then the more adept you will be at providing your students with meaningful mathematics learning experiences. When you then apply effective instructional practices, you will become a powerful mathematics learning conduit for your students. Also of critical importance is for you to understand the scope and sequence of mathematic concepts and skills before, during, and after the grade levels you teach. You cannot be effective if you cannot recognize the prerequisite concepts and skills your students do not posses when you are teaching concepts and skills that build on those prerequisite concepts and skills. Moreover, you must know what concepts and skills your students will be exposed to after they leave your class, so you can prepare them for what they will learn in the future.

Apply Instructional Practices That Work For Students with Special Needs

In order for students with special needs to make meaning of mathematics and be able to do mathematics in ways that they can apply to their lives, they will require instruction that meets their particular learning needs. While math educators and special educators have often disagreed about what mathematics instruction should be, such disagreement is not helpful for the students we serve. A similar statement can be said for the area of reading and literacy instruction. Students with special needs benefit from and deserve mathematics instruction that integrates "best practice" from both disciplines. Like general education literacy and reading educators, general education mathematics educators have typically advocated more student-centered learning practices (e.g. discovery learning, inquiry-based learning). Special educators have tended to advocate more direct instruction practices (explicit) applied to discrete mathematics learning objectives. Students with special needs can benefit from instruction that incorporates essential aspects of both student centered and teacher directed mathematics instructional practices (Mercer, Lane, Jordan, Allsopp & Isli, 1997; Mercer, Jordan & Miller, 1996). Research can inform us on how to take the best of both disciplines to improve the mathematics learning outcomes for students with special needs. General conclusions derived from the research that can guide instruction practices for students with special needs include:

- Direct instruction where the teacher provides students a high level of support, structure and guidance while they learn basic mathematics concepts and skills is more effective for students with special needs compared to instructional practices that are primarily student centered in nature (Kroesberg & Ban Luit, 2003; Miller, Butler & Kit-hung Lee, 1998; Miller & Kit-hung, 1998; Swanson, 1999 ).
- Instructional practices that emphasize the teaching of strategies for problem solving are more effective for students with special needs compared to more conventional types of instruction such as basal mathematics programs (Miller & Kit-hung, 1998; Miller, Butler & Kit-hung Lee, 1998; Owen & Fuchs, 2002; Swanson, 1999).
- Instruction that emphasizes the development of mathematics computation and problem solving skills through use of concrete level instruction positively impacts their development of these skills (Miller & Kit-hung, 1998). Moreover, continuing concrete level instruction through representational/semi-concrete and abstract level instruction helps students to transfer their concrete understandings to the abstract level (Miller, Butler & Kit-hung Lee, 1998; Miller & Mercer, 1993; Miller, Mercer & Dillon, 1992; Miller & Mercer, 1997).
- Monitoring student performance, communicating to students their progress, and reinforcing their success on a continuous basis result in increased mathematics learning outcomes (Miller & Mercer, 1997).
- Teaching students self-regulating behaviors such as goal setting, verbalizing their thinking as they solve problems, teaching students simple ways to self-monitor their learning, and teaching them to apply strategies to problem solving situations all promote mathematics success (Miller & Mercer, 1997; Miller, Butler & Kit-hung Lee, 1998).
- Peer-mediated instruction (i.e. peer tutoring and cooperative learning groups) can be effective ways for students with special needs to enhance their mathematical understandings and skills through practice when peer-mediated instructional activities are well-planned and highly structured (Rivera, 1996; Miller, Barbetta, Drevno, Martz & Heron, 1996 ).
- Students with special needs can develop mathematical understandings beyond simple rote repetition of mathematics algorithms and procedures (Parmar & Cawley, 1991; Woodward & Baxter, 1997).

The teacher tools described in this module derive from this research base and emphasize both relevant teacher directed (explicit) instructional practices and student centered (implicit) instructional practices. Each teacher tool can be applied to develop both procedural (e.g. computations) and conceptual mathematical understandings.

Developed by: David Allsopp, Ph.D., University of South Florida