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. 105-17, 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 led, 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 counterexamples 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 PREP |
| 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 Question-Answer Relationships to examine where to find the answer | Q-A 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. Give students time and space to discuss ideas. Make behavioral expectations clear |
| Uses concrete materials to manipulate ideas and to test solutions | Have materials readily available for use. Observe use of materials |
| Student 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 What we Know | W What we Want to Learn | L What we Learned |
|---|---|---|
| |
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 pre-made 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
Question-Answer Relationship
Question-Answer Relationship (Q-A 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. Q-A 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 |
|---|---|
| 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? What happened when you…? |
| Encourage students to elaborate and discuss | What surprised you about… Were all of the groups' solutions the same? How were they different? Why do you think there was a difference between your groups' answers? |
| 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…You may need to rephrase student response to clarify for student what was said. Write out student solutions for all to see, or have students contribute to group display. |
| Promote collaborative problem solving | Ask directly, "How did you work together to solve this problem?" or after one student has described the answer to a question, redirect the question to another member of the same problem solving group, "Rosa, would you like to add anything else?" |
| 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? (Explicitly teach vocabulary of new and unknown items) What do you see that is new to you? Provide clear and vivid examples/stories or act out any actions required to solve the problem. |
| Lead students through "instant replays" | What happened first? In what sequence/order did things happen? |
| Write symbolic representations of solutions when appropriate | Let's write out all the solutions we found. Is there a way to organize these answers? (Organize solutions into tables or graphs) |
| 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. Simulate situations that require the particular problem solved. If you go to the doctor's office, you will expect them to know how to solve this type of problem. Your life could depend on it. Today, you are going to show me what you know by being my doctor. You will explain to me how you have solved this problem… |
| Encourage students to make generalizations | How does this relate to what your family thinks about…? (topic of discussion) As a consumer, how would this help you make a choice about the products you will buy? |
| List all solution methods on the board to promote reflection | Looking at all of our work, how would you state the problem? What are the alternative choices? What personal and societal values are reflected in this problem? Which choice do you think is the best choice for you? |
Resources
| 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 Question-Answer Relationships to examine where to find the answer (Q-A 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 | ||
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