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Improving Mathematics Problem Solving Skills for English Language Learners with Learning Disabilities
The Problem
Not all students with learning disabilities struggle in mathematics. They do however have some characteristics in common. By definition, the term "specific learning disability" means a disorder in one or more of the basic psychological processes involved in understanding or in using language, spoken or written, which may manifest itself in imperfect ability to listen, think, speak, read, write, spell, or do mathematical calculations. Such term includes such conditions as perceptual disabilities, brain injury, minimal brain dysfunction, dyslexia, and developmental aphasia. Such term does not include a learning problem that is primarily the result of visual, hearing, or motor disabilities, or mental retardation, or emotional disturbance, or of environmental, cultural, or economic disadvantage. (IDEA amendments of 1997, P.L. 10517, June 4, 1997, 11 stat 37 [20 U.S.C. §1401 (26)]).
Students with learning disabilities tend to experience academic difficulties, yet they have average to above average intelligence (Friend, 2005).
The academic language difficulties that are characteristic of students with learning disabilities are very similar to those of students learning a second language. For example, both may have difficulty retrieving words. In the case of a student with a learning disability, this may be due to a perceptual, processing or memory disorder. With English Language Learners (ELL), it is more a matter of learning a new word. In both cases comprehension may be slower due to the effort taken to remember words and their concepts. What strategies are useful for a teacher of mathematics whose student has a true learning disability and is also learning English as a second language?
In this article, we will specifically address the area of problem solving because of the strong emphasis it has been given by the National Council for Teachers of Mathematics (2000). With the strong demand for language and conceptual development in problem solving, we will highlight aspects of the teacher and student's roles and the importance of discourse. We will provide some strategies for working through mathematical problems, questioning, and assessment. We are making the assumption that the determination of a learning disability was made using best possible assessment practice. We are assuming that the learning disability exists in both languages and that the student is being provided with a rich native language development curriculum.
Implications for Teacher Practice
Teaching Standards
The Teaching Standards are an integral part of the Professional Standards for Teaching Mathematics (NCTM, 1991). The first three standards include items that support language development for EL students who also have learning disabilities.
Worthwhile Mathematical Tasks
The teacher of mathematics should pose tasks that are based on
 Sound and significant mathematics
 Knowledge of students' understandings, interests, and experiences
 Knowledge of the range of ways that diverse students learn mathematics
and that
 Engage students' intellect
 Develop students' mathematical understandings and skills
 Call for problem formulation, problem solving, and mathematical reasoning
 Promote communication about mathematics
 Represent mathematics as an ongoing human activity
 Display sensitivity to, and draw on, students' diverse background experiences and dispositions
 Promote the development of all students' dispositions to do mathematics
Teacher's Role in Discourse
The teacher of mathematics should orchestrate discourse by
 Posing questions and tasks that elicit, engage, and challenge each student's thinking
 Listening carefully to students' ideas
 Asking students to clarify and justify their ideas orally and in writing
 Deciding what to pursue in depth from among the ideas that students bring up during a discussion
 Deciding when and how to attach mathematical notation and language to students' ideas
 Deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with a difficulty
 Monitoring students' participation in discussions and deciding when and how to encourage each student to participate
Students' Role in Discourse
The teacher of mathematics should promote classroom discourse in which students
 Listen to, respond to, and question the teacher and one another
 Use a variety of tools to reason, make connections, solve problems, and communicate
 Initiate problems and questions
 Make conjectures and present solutions
 Explore examples and counter examples to investigate a conjecture
 Try to convince themselves and one another of the validity of particular representations, solutions, conjectures, and answers
 Rely on mathematical evidence and argument to determine validity
Tips for Teachers
Working Through Mathematical Problems
One approach to planning lessons involving mathematical problem solving is to plan in three parts: before, during and after (Raborn, 1991; Van De Walle, 2004).
Observing how a student approaches a problem can provide important information for teachers. Not all students will need to do each of the following steps every time they approach a mathematics problem. However, the skills that are listed below can help students prepare mentally for comprehending and solving problems. The following inventory also provides information for teachers to use in order to identify strategies that can be taught to develop and strengthen the language and concepts of mathematics. We have included an Inventory of Student Skills for Mathematics Problem Solving in Appendix A that can be used as an observational checklist for collecting assessment data. Appendix B provides lesson plan Web sites for mathematics instruction.
Before Solving Mathematical Problems
Student Skills 
Teacher Strategies 

Previews knowledge of the topic involved in the problem 
KWL 
Previews pictures, print, and concrete materials to generate ideas 
Direct Instruction 
Is familiar with the mathematics materials and is clear about expectations on how to use and care for them 
Direct Instruction 
Identifies and requests any additional materials that may be needed to solve the problem 
Direct Instruction 
Accesses and applies background knowledge to understand the gist of the problem, the vocabulary and the mathematical concept involved in the problem 
GIST 
Analyzes the QuestionAnswer Relationships to examine where to find the answer 
QA R 
Sets purpose – demonstrates understanding about what he/she needs to figure out (the problem and end product) 
Direct Instruction 
While Solving Mathematical Problems
Student Skills 
Teacher Strategies 

Brainstorms ideas for solving the problem 
Listen. 
Uses concrete materials to manipulate ideas and to test solutions 
Have materials readily available for use. 
Monitors own comprehension 
Provide hints only when necessary. 
Integrates new concepts with prior knowledge 
Break down the concept and introduce a simpler version of the problem first. 
Looks for patterns 
Encourage students to express ideas with peers or represent ideas in drawings, writing, or models. 
After Solving Mathematical Problems
Student Skills 
Teacher Strategies 

Summarizes and explains problem and solution 
Questioning 
Makes a symbolic/graphic representation to record solution 
Prior Direct Instruction and Guided Practice 
Makes a table or chart to show findings 
Prior Direct Instruction and Guided Practice 
Evaluates ideas from solving problem (ideas from self and peers) 
Questioning 
Makes applications to student's own life 
Questioning 
Connects ideas from problem and solution to broader community and societal issues 
Questioning 
Teaching Strategies
KWL and PREP are two similar strategies that activate students' prior knowledge and provide the teacher with information on what students already know about the topic.
KWL
With KWL, teachers ask students to identify:
 What the students already Know (K)
 What the students Want to learn (W)
 What the students have Learned (L) after the lesson or unit of study
Generally, the class constructs a chart on the chalkboard or on poster chart paper. The chart is divided into three sections with ample room for students to contribute to each section.
K 
W 
L 


PREP
Pre Reading Plan (PREP) is a strategy that helps the teacher to determine the prior knowledge and vocabulary that students have on a given topic (Langer, 1982). The teacher asks the students to tell her or him everything they know about the topic. There are two ways this can be done. Either the students give information verbally in an open discussion, or they contribute first in writing and then verbally. Some teachers distribute three postable notes to each student. Students write one fact they know about the topic on each postable note and place them in a basket. The teacher collects the notes and reads them one at a time with the whole class group. In either case, the teacher uses the student information to develop a chart of what students already know. As students are discussing what they know, the teacher can prompt further discussion with questions such as "What made you think of. . . ?" "Do you want to add or change your first response?" Some teachers organize the chart by subtopic and then keep the chart posted in the room for students to review as they continue to learn about the major topic of study. Many teachers find it helpful to make a chart that organizes the extent to which students are familiar with the topic:
Much Knowledge 
Some Knowledge 
Little Knowledge 


GIST
GIST is a strategy for identifying the most important aspects of a story problem. First, the teacher defines 'gist' to the students as the main idea without excessive details. The teacher draws a chart on the board for each paragraph of the text, or uses a premade overhead transparency with charts already outlined. Each chart has 20 boxes. The class can work in groups to capture the main idea of each paragraph in 20 words or less (one word for each of the 20 boxes). If the text includes more than one paragraph, students read and 'gist' the first paragraph completely before proceeding to the second paragraph. Then the students incorporate the information from both the first and second paragraphs in just 20 words. It is recommended that text with no more than three paragraphs be used in this exercise.
GIST Chart for Word Problems














QuestionAnswer Relationship
QuestionAnswer Relationship (QA R) strategies assist students in examining information provided by the author of the text (Raphael, 1986). Comprehension tasks now require students to answer both explicit and implicit questions. Students are taught to read the text carefully to determine whether answers are in the text or whether they will have to draw from their own knowledge to find answers to questions. Although this strategy is useful with reading of books and longer text, it can be used in problem solving to help students identify what information is indeed provided in the problem and what they will need to draw from their own knowledge. QA R uses four categories. When the answer is in the text, it is either stated directly "Right There" in one part of the text, or the student can "Think and Search" to put the answer together from information found in several parts of the text. When the answer is drawn from student knowledge, it is either between the author and reader (the reader must consider what the author is providing in the text and fit it with what the reader already knows), or the reader can answer the question without even reading the text (no part of the answer is in the story, the reader must draw totally from their own experience and prior knowledge).
Questioning Strategies
Current standards emphasize the importance of mathematics as communication. Many students with learning disabilities and many ELL students will benefit from the use of language to elicit, support and extend mathematical thinking. We have included a table of strategies and corresponding examples of questions to use to help students process and conceptualize solutions to mathematical problems. Yeatts, Battista, Mayberry, Thompson, & Zowojewski (2004) and Fraivilling (2001) recommend the strategies included in the table below. The questions were adapted from Carin, Bass & Contant (2005).
Strategies to Elicit Student Thinking 
Questions to Elicit Student Thinking 

Elicit many solution methods for one problem 
Did anyone find a different way to solve this problem? 
Wait for, and listen to, students' descriptions of solution methods 
What did you do? 
Encourage students to elaborate and discuss 
What surprised you about . . . ? 
Use students' explanations as a basis for the lesson's content 
What are some things you noticed about the. . . . ? 
Convey an attitude of acceptance toward students' efforts 
I see . . . So you're saying that . . . 
Promote collaborative problem solving 
Ask directly, "How did you work together to solve this problem?" 
Strategies to support students' thinking 
Questions to support students' thinking 
Remind students of conceptually similar problems 
How was this like the problem we solved last week using raisins? 
Provide background knowledge 
What objects do you see here? 
Lead students through "instant replays" 
What happened first? 
Write symbolic representations of solutions when appropriate 
Let's write out all the solutions we found. 
Strategies to extend students' thinking 
Questions to extend students' thinking 
Maintain high standards and expectations for all students 
Make sure that each student has an opportunity to participate and respond as a valued member of the group and class.

Encourage students to make generalizations 
How does this relate to what your family thinks about...? (topic of discussion) 
List all solution methods on the board to promote reflection 
Looking at all of our work, how would you state the problem? 
Resources
Appendix A: Inventory of Student Skills for Mathematics Problem Solving
Preparing to Solve Mathematical Problems 
With Teacher Support 
Without Teacher Support 

Previews knowledge of the topic involved in the problem (KWL, PREP) 


Previews pictures, print, and concrete materials to generate ideas 


Is familiar with the mathematics materials and is clear about expectations on how to use and care for them 


Identifies and requests any additional materials that may be needed to solve the problem 


Accesses and applies background knowledge to understand the gist of the problem, the vocabulary and the mathematical concept involved in the problem (GIST) 


Analyzes the QuestionAnswer Relationships to examine where to find the answer (QA R) 


Sets purpose – demonstrates understanding about what he/she needs to figure out (the problem and end product) 


Solving Mathematical Problems: 

Brainstorms ideas for solving the problem 


Uses concrete materials to manipulate ideas and to test solutions 


Integrates new concepts with prior knowledge 


Looks for patterns 


Draws a picture or uses some type of graphic representation to record findings so that they can be reviewed later by self or others 


Summarizes and explains problem and solution 


Makes a symbolic/graphic representation to record solution 


After Finding a Solution to the Problem: 

Summarizes and explains problem and solution 


Makes a symbolic/graphic representation to record solution 


Makes a table or chart to show findings 


Evaluates ideas from solving problem (ideas from self and peers) 


Makes applications to student's own life 


Connects ideas from problem and solution to broader community and societal issues 


Appendix B: Lesson Plan Web sites
http://www.coled.org/cur/math.html
http://www.awesomelibrary.org/math.html
http://mathforum.org/alejandre
Click the "References" link above to hide these references.
IDEA Amendments of 1997, PL 10517, 20 U.S.C. &#sect;1400 et seq.
Carin, A.A., Bass, J.E., & Contant, T.L. (2005). Teaching science as inquiry. (10th ed.). Upper Saddle River, NJ: Pearson Prentice Hall.
Friend, M.P. (2005). Special education: Contemporary perspectives for school professionals. Boston, MA: Allyn and Bacon.
Langer, J. (1982). Facilitating text processing: The elaboration of prior knowledge. In J. Langer & J.T. SmithBurke (Eds.), Reader meets author/Bridging the gap. Newark, DE: International Reading Association.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
Raphael, T.E. (1986). Teaching questionanswer relationships, revisited. The Reading Teacher, 39(6), 519.
Raborn, D. (1992). Cooperative learning and assessment: A viable alternative for language minority and bilingual students. Cooperative Learning: The Magazine for Cooperation in Education, 13(1), 911.
Van de Walle, J.A. (2004). Elementary and middle school mathematics: Teaching developmentally. (5th ed.). New York, NY: Addison Wesley Longman, Inc.
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