the QAR categories though modeling. In subsequent lessons the teacher will ask the students to fill in the QAR categories and describe how they knew which category the question represented. Finally, students will be given comprehension questions and the QAR categories and expected to determine the category and answer the questions cor-rectly.
Material Scaffolding
The student will be provided with a QAR prompt card (Figure 9.4) to use with the com-prehension questions at the end of each reading passage given. Initially, this prompt will serve as a guide to remind students of the question–answer relationships and how those relationships can help them answer comprehension questions. Students will be provided multiple opportunities to practice using their QAR strategy until they are able to do so independently and successfully answer comprehension questions.
Stage 6 for Summarization: Independent Performance
At this stage the student is ready to use the strategy independently. The teacher’s main task is to monitor the student’s performance and check on proper and consistent use of the strategy. However, it is important to keep in mind that our main goal is improved academic performance. Teachers must evaluate whether or not the strategy is being used and whether or not academic performance has improved. Generalization may
also be a concern. Students will not always generalize strategies to appropriate situa-tions; they will need to be prompted and encouraged to do so. Note that activities to promote generalization should start very early in the process. For example, in the dis-cussion stage (Stage 2) teachers and students will often list situations where a strategy could or should be used.
To promote generalization, students should be encouraged to use the strategy in other content areas where they are required to read. All team teachers should be informed about the use of the strategy, the language, the prompts, and what is required at each step. All team teachers should be given a wall chart to hang up in their room as a reminder for students to use the Summary Writing Guide strategy when appropriate.
Students should be assessed regularly using the comprehension questions at the ends of their texts. These scores should be recorded and tracked for trends in progress.
The goal is improved academic performance, and if performance isn’t improving a reteaching of the strategy may be necessary. Once students are successful using the strategy and answering comprehension questions their performance should be moni-tored periodically. Even when students have reached the independent performance stage, they should be monitored to ensure proper use of the strategy. If students deviate from the given summarization strategy, performance should be evaluated, but action should only be taken if performance is no longer improving.
FINAL THOUGHTS
Reading is one of the most important skills students gain in school. No student can suc-ceed without well-developed comprehension skills. Improving a student’s comprehen-sion of text can have a positive and lifelong impact on learning outcomes. In closing, we should stress that instruction in comprehension strategies should begin early and continue through a student’s academic career.
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Guilford Press.
Strategies in Mathematics
M
athematics is a core academic competency. Mastering basic mathematic skills (e.g., knowledge of basic facts, problem solving) is critical for successful functioning in society. We are constantly confronted with problems that involve mathematics skills.These might range from “Do I have enough money to pay for these three items I want to purchase?” to I have to go 30 miles to a meeting that takes place in 30 minutes. How fast do I have to drive to get there on time?” Mastering the skills and strategies neces-sary to solve these problems is critical for functioning effectively in society (Patton, Cronin, Bassett, & Koppel, 1997). Our level of mathematics achievement is a matter of national concern, as research suggests that American children have serious deficits in many areas (Jitendra & Xin, 1997; Xin & Jitendra, 1999).
For purposes of this chapter, we limit our discussion to three major areas of mathe-matics: basic facts, computation procedures, and word-problem solving. Knowledge of basic math facts is a foundational for more advanced operations. Fluency with mathe-matics requires students to learn and store basic facts (e.g., 3 + 5 = 8; 9× 7 = 63) in long-term memory, and then rapidly and accurately recall and apply them. Computational procedures are also fundamental. Students must master procedures involved in com-putations such as long division, place value, and regrouping. They must also remember and apply these procedures correctly while simultaneously recalling and utilizing basic facts. Finally, students are required to develop and utilize word-problem-solving skills.
Here students are confronted with word problems that require them to identify impor-tant information, conceptualize how to solve the problem, define appropriate equa-tions, and solve equations. Again, knowledge of basic facts and computational proce-dures must also be accessed and utilized.
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PROBLEMS FOR STUDENTS WITH LEARNING DISABILITIES
Problems with underachievement in mathematics are particularly pronounced for stu-dents with learning disabilities (Geary, 2003). Studies suggest that 5 to 10% of all school-age children have some type of serious deficit in mathematics and that difficul-ties in mathematics are common among children with LD (Geary, 2003; Rivera, 1997).
Difficulties experienced by children with LD span all three areas (basic facts, computa-tion procedures, and problem solving). As we’ve noted previously, children with LD are extremely heterogeneous; however, there are some problems that commonly occur (Geary, 2003). These include:
• Poor understanding of concepts underlying mathematical procedures
• Problems with counting and using counting to solve problems
• Use of immature or inappropriate strategies
• Difficulty with fact retrieval
• Difficulty coordinating and monitoring steps in computation
• Difficulty in word-problem solving
On the whole, children with LD are only slightly behind typically achieving peers in terms of development of number concepts (Geary, Hamson, & Hoard, 2000). But some commonly occurring problems, such as difficulty with counting concepts, have been demonstrated (Geary, Bow-Thomas, & Yao, 1992; Geary, Hoard, & Hamson, 1999).
For example, children may believe that nonsequential counting (i.e., skipping some items and counting them later) will result in an incorrect count (Geary, 2003). Another example would be failure to grasp the commutative principle. Students who did not understand this principle would not understand that 8× 5 is the same problem as 5× 8. Poor understanding of number concepts can inhibit the use of advanced strategies and may affect ability to detect procedural errors (Geary, 2003). Counting problems are a serious concern because counting serves as an important preskill in the mastery of basic addition facts. Difficulty counting can inhibit the development of basic addition facts. Children with LD may be developmentally delayed in the use of counting to solve arithmetic problems. They may also fail to perceive the link between counting and problem solving. Many see counting as simply a rote activity (Geary, 2003).
Use of appropriate strategies is also a concern. Many children either fail to develop or fail to utilize more advanced strategies. For example, many children with LD don’t gradually switch from counting to direct retrieval of math facts (Geary, Widaman, Lit-tle, & Corimer, 1987). Thus, these children use an inefficient means of retrieval. Mem-ory problems are a major factor in poor mathematic performance. Children with LD commonly produce more errors in math fact retrieval than their peers. Research sug-gests that children with LD may have problems in storing basic facts in long term mem-ory and accessing them readily. Errors in retrieval may be due to problems inhibiting irrelevant information and/or associations (Geary et al., 2000). For example, with a problem such as 6 + 2, many will answer 7, the next number after 6. For complex prob-lems (i.e., those that involve a number of operations), children with LD frequently make procedural errors. They may omit steps (e.g., fail to regroup) or have difficulty
performing steps in the correct sequence. Procedural errors in computation appear to be due to problems monitoring and coordinating the sequence of problem-solving steps (Russell & Ginsberg, 1984). Problems with working memory are thought to be related to procedural difficulties.
Solving word problems is often extremely difficult for children with LD. The diffi-culties with word-problem solving go well beyond problems with decoding. Parmar, Cawley, and Frazita (1996) found that students with disabilities had difficulties repre-senting problems, identifying salient information, and choosing the appropriate arith-metic operations. Word problems may contain irrelevant information, and solving them may require multiple operations or steps that can also cause serious problems (Fuchs & Fuchs, 2003). Thus, word problems place a number of demands on children’s cognitive processing. These problems are exacerbated by the fact that mathematics in-struction for children with LD often is focused on memorization of facts and computa-tion skills at the expense of applicacomputa-tion of mathematics in problem-solving situacomputa-tions (Bottge, 1999; Parmar, Cawley, & Miller, 1994).
PREREQUISITE SKILLS
A list of all the possible prerequisite skills for mathematics would be beyond the scope of this chapter. There are, however, three major areas of preskills that are extremely important for successful development of mathematical skills: number sense, basic mathematics principles, and mathematic basic facts rules. These prerequisites are criti-cal to the development of skill with basic facts, computation procedures, and problem solving. In this section we discuss each preskill.