V.2 #3 Mathematics - Algebraic Learning and Students with Learning Disabilities
Algebraic learning constitutes a challenging goal for students with Learning Disabilities (LD), as it presupposes the possession of skills in the performance of which students with LD often encounter considerable difficulties. Some of these skills are: the recognition of operational signs and numerals, the performance of arithmetic operations, the understanding of mathematical laws, the application of procedures for deriving the value of unknown variables, and the use of symbols and mathematical expressions to represent relations among numbers or information from word problems (Foegen, 2008).
Although in the case of many students it may seem that they started failing in mathematics when they first encountered algebra, the respective research shows that the real sources of their difficulty are rather: a) these students’ dysfunctional relationship to arithmetic, which deprives them from the general mathematical underpinnings of algebraic learning (e.g. number relationships, notation, mathematical laws), and b) their insufficient preparation for regarding algebra as a continuation and extension of arithmetic that allows them to express familiar arithmetical notions in a generalized and abstract way, using variable notation (Ketterlin- Geller, Jungjohann, Chard, & Baker, 2007).
The relatively scant but, nevertheless, informative research on teaching algebra to students with LD has identified some principles and some techniques that may effectively help teachers prepare their students for successfully coping with algebra. Examples of such principles and techniques are:
Provide students with ample opportunities to explore and understand the different kinds of numbers and their relationships. Algebra is a more abstract and more general version of arithmetic, focusing on properties that are common to all the numbers of a number system (e.g. integers, negative numbers). Studying the different kinds of numbers (e.g. whole numbers, fractions, decimals) through multiple experiences allows students to approach the meaning of quantities, magnitudes, functional relationships, and properties of numbers. Thus, students can gradually accept that there are analogies and common elements between the number systems, and when they are introduced to variables, they can understand that algebraic expressions satisfy the properties they are already familiar with (Wu, 2001).
Implement a graduated instructional sequence to help students advance to abstract levels of algebraic understanding following grasping of underlying concepts at concrete and semi-concrete level of knowledge representation. For example, students can start their exploration of the commutative property of addition by manipulating groups of concrete objects (to see and show that changing the order of the groups does not change their sum). Then they can present an example of the same property using pictures of objects, lines or circles. Finally, they can present examples of symbolic expressions of the property, using both numerals (e.g. 8 + 6 = 6 + 8) and variables (e.g. 8 + x = x + 8).
Make sure students possess the necessary prerequisite knowledge, especially basic computational skills and symbol manipulation. Fluency with computation gives students a feeling of comfort when it comes to mathematical activities (it creates a positive mental disposition toward the subject) and, at the same time, it allows them to concentrate on the specifics of the algebraic learning. Efficient symbol manipulation is a prerequisite for the proper isolation of the variables and the finding of the solutions (Witzel, Smith, & Brownell, 2001).
Use authentic problem types that connect mathematics to students’ real life, and teach systematically strategies for converting word problems into algebraic expressions. Attaching personal meaning to the content of a lesson helps students memorize algorithms, understand concepts, and develop motives for learning. Authentic problems support the development of personal meaning to the lessons, and may also help in translating the problem into an algebraic statement, as they refer to situations familiar to the student. Converting word problems into algebraic expressions may require breaking the problem into separate parts, and then translate each part into an appropriate expression. Problem–solving strategies that have been taken up in the 4th column of this series may also help in this perspective (Maccini, & Hughes, 2000).
Use think-aloud techniques and other procedures of explicit instruction. Modeling problem-solving (e.g. explicitly showing to the student how to identify the unknown component in a problem or combine like variables in an equation) helps students with LD better understand the procedures. In addition to hearing teacher’s verbal descriptions and explanations, students may share their own verbal explanations and bring their own examples. Appropriate examples and sound explanations from the side of the students constitute signs of successful learning.
Employ sound instructional methodology and monitor systematically student progress. Sufficient guided practice followed by focused independent practice, and systematic monitoring of student progress with appropriate feedback, create favorable conditions for students to build their own proper understanding of how to apply the procedures for correctly dealing with algebra.
By following these guidelines teachers may successfully prepare students with LD for algebraic learning, and support them in their effort to understand the beauty and the utility of mathematics.
Foegen, A. (2008). Algebra progress monitoring and interventions for students with learning disabilities. Learning Disability Quarterly, 31 (Spring), 65-78.
Ketterlin- Geller, L., Jungjohann, K., Chard, D., & Baker, S. (2007). From arithmetic to algebra. Educational Leadership (November), 66-71.
Maccini, P. & Hughes, C. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15(1), 10-21.
Witzel, B., Smith, S., & Brownell, M. (2001). How can I help students with learning disabilities in Algebra? Intervention in School and Clinic, 37(2), 101-104.
Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2), 10-17.
Ioannis Agaliotis, Ph.D. is Assistant Professor of Instructional Methodology for Students with Special Educational Needs in the Department of Educational and Social Policy of the University of Macedonia of Thessaloniki, Greece. Dr. Agaliotis is co-editor of the journal Insights on Learning Disabilities: From Prevailing Theories to Validated Practices, published by LDW®. He has presented at national and international conferences and has published articles and books on inclusive education, assessment and instruction for students with mild disabilities, mathematics for students with special needs, and academic and social support for students with learning disabilities.