Math LD

$Math LD$

Imagery The Sensory-Cognitive Connection For Math;
Adapting Mathematics Instruction in the General Education Classroom for
Students with Mathematics Disabilities;
Learning Disabilities in Mathematics;
Mathematics and Dyslexia;
Larson, Hostetler,& Edwards Texts;
Math Study Skills K-12;
Math Study Skills HS-College

                        Here are a few web links to all kinds of math games!

     FREE EXCELLENT MATH GAMES FOR KIDS!
    MATH DOJO     FUNBRAIN   AAA Math   Algebra.com Fact Monster: Math

Four Function Quiz Google directory for math Math Baseball Math.com Math Forum

Math League Help Topics MathWorld Money Savvy Generation Puzzles.com

Roman and Arabic Numerals Yahooligans Math Visual Fractions

Calculators: Calculator Circle Solver Square Root Percent LCM or GCD (GCF) Lap Time

Sudoku logic puzzles Make and print sudoku puzzles
Clever Games for Clever People Interactive Fractals Site Interactive Fractals Site Mankala

Mr L's Functional Math Games Quizville - Online Math Skills Practice Games Curious and Useful Math

Einstein's Origami Math Cats Math Humor Math Mojo Homepage The Misadventures of Sailor Pi

Mudd Math Fun Facts Paper Folding Paul's Page of Pi AIMS Puzzle Corner Brain Teasers

Cut the Knot - Interactive Puzzles Diamond 16 Puzzle Easier Fibonacci Puzzles Jorge Rezende's Puzzle

Mathematics Puzzles - Thinks.com Nick's Mathematical Puzzles Puzzle Solvers at Cryptic Manor

Puzzles and Problems Tim's Interactive Puzzle Solution Center The Ultimate Puzzle Site

Enquiring Into Number Things Escot Problem of the Week (elementary through algebra & geometry)

Figure This - Math Challenges for Families Get Smarter Org - Math Quiz Famous Math Problems

Famous Math Problems OleMiss Problems of the Week Olympiad Math Madness Problem of the Week

Shack's Math Problems Word Problems for Kids Starship Math Create a Graph Math Playground

Math Zone Multiplication Hidden Picture ArithmAttack Make your dream house! Visualize Multiplication

Illuminations Function Machine Numeracy Games Measuring Angles Pizza Party Fresh Baked Fractions

Teaching Time - Clock Program Money Drills Money Central Station Exact Change, Please Perimeter Explorer

Area Explorer Shape Explorer Symmetry Geometry Center Ladybug Play with an abacus Probability (UK)

UK Interactive resources UK Interactive Resources - Tangrams Tangrams - Flash plugin. A story told with tangrams.

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Imagery The Sensory-Cognitive Connection For Math

Nanci Bell and Kimberly Tuley

OCTOBER 2003

Why can’t everyone think with numbers? Why do some children learn math readily, handle money and time concepts with ease, retain information from year to year, and think with numbers effortlessly? What cognitive processes do some have that others do not?

Mathematics is cognitive process-thinking-that requires the dual coding of imagery and language. Imagery is fundamental to the process of thinking with numbers. Albert Einstein, whose theories of relativity helped explain our universe, used imagery as the base for his mental processing and problem solving. Perhaps he summarized the importance of imagery best when he said, “If I can’t picture it, I can’t understand it.”

For the people who “get” math, the language of numbers turns into imagery. They use internal language and imagery that lets them calculate and verify mathematics; they “see” its logic.

Imaging is the basis for thinking with numbers and conceptualizing their functions and their logic. The Greek philosopher Plato said, “And do you not know also that although they [mathematicians] make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble…they are really seeking to behold the things themselves, which can be seen only with the eye of the mind?”

The relationship of imagery to the ability to think is one of the preeminent theories of human cognition. Allan Paivio, author of the Dual Coding Theory (DCT) and a cognitive psychologist, stated, “Cognition is proportional to the extent that mental representations (imagery) and language are integrated.” Research from the 1970s and into the 1990s has validated Dr. Paivio’s work as a viable model of human cognition and its practical, as well as theoretical, application to the comprehension of language (Bell, 1991). Dr. Paivio believes that in order to think and understand, humans must be able to simultaneously generate imagery and corresponding language to describe that imagery.

Mathematics is the essence of cognition. It is thinking (dual coding) with numbers, imagery and language; reading/spelling is thinking with letters, imagery and language. Both processes, often mirror images of each other, require the integration of language and imagery to understand the fundamentals and then apply them. Dual coding in math, just as in reading, requires two aspects of imagery: symbol/numeral imagery (parts/details) and concept imagery (whole/gestalt).

Numeral Imagery

Visualizing numerals is one of the basic cognitive processes necessary for understanding math. For example, we image the numeral “2” for the concept of two. When we see the numeral “3,” we know that it represents the concept of three of something: three pennies, three apples, three horses, three dots. If someone gives us two pennies for the numeral three, we have a discrepancy between our numeral-image for three and the reality (concept) of three. The first imagery needed for math is the symbolic (or numeral) imagery that represents the reality of a number concept.

What does numeral imagery look like? Here’s one example. Cecil was very good in math. He could think with numbers, arrive at answers in his head, and mentally check for mathematical discrepancies in finance or life situations easily. He explained this ability, “I just visualize numbers and their relationships. Certain numbers are in certain colors, and the number-line in my head goes specific directions.” Not only could Cecil visualize numerals and concepts, both types of imagery, but he also had an unusual talent for color imagery. He assigned colors
to specific numbers!

“What color is the number 14?” he was asked.

His eyes went up, and in all seriousness, he said, “Light blue.”

Similarly, number 3 was reddish pink and the number 88 “kind of a purple.” Quizzed again months later, Cecil assigned the same colors to the same numbers.

Chronological relationships appear in our minds on a number line, the days of the week, the months in the year. Imagery is our sensory systems’ way of making the
abstract real. It is a means to experience math.

Concept Imagery

While imaging numerals is important to mathematical computation, another aspect of imagery is equally important: concept imagery. Understanding, problem solving and computing in mathematics require another form of imagery--the ability to process the gestalt (the whole). Sometimes children or adults can visualize the numerals, the parts, but cannot bring those parts to a whole, just as they can sometimes visualize individual words but cannot bring those words to a whole to form concepts. Mathematical skill requires the ability to get the gestalt, see the big picture, in order to understand the process underlying mathematical logic.

“Concept imagery is the ability to image the gestalt (whole),” Bell (1991). Concept imagery is basic to the process involved in oral and written language comprehension, language expression, critical reasoning and math. It is the sensory information that connects us to language and thought.

The ability to create mental representations for mathematical concepts is directly related to success in mathematical reasoning and computation. However, because some children do not have this imaging ability, they are often mislabeled as not trying, unable to retain information, or having dyscalculia (the inability to perform arithmetic operations).

Manipulatives May Not Be Enough

Joanie’s second grade class covered a review of recognizing numbers, addition, subtraction, and even some multiplication. They worked a lot with concrete manipulatives and Joanie was doing well at the end of the year. But her third grade teacher complained that Joanie didn’t know anything about numbers.

Concrete experiences-manipulatives-have been used for many years in teaching math (Stern, 1971). However, like Joanie, many children and adults have often experienced success with manipulatives, but failure in the world of computation (NCTM, 1989; Moore, 1990; Papert, 1993). They have what has often been described as “application problems.”

Joanie’s second grade class had spent a lot of time with manipulatives. Some of the children moving on to third grade continued to “think with numbers.” Their experience with manipulatives became part of their mental deposit of imagery. Like a bank deposit, these images could be drawn upon at will. However, not all children create mental imagery as they work with concrete manipulative. For these children, the process of turning the concrete experience into imagery must be consciously stimulated.

On Cloud Nine® Math: Concrete to Imagery to Computation

Arnheim (1966) wrote, “Thinking is concerned with the objects and events of the world we know…When the objects are not physically present, they are represented indirectly by what we remember and know about them…Experiences deposit images.”

Numbers can be experienced and the relationships between them can be made concrete by using manipulatives. What appears abstract can be experienced and imaged to concreteness. Math’s roots are in the realm of the concrete, and imagery is the link to mathematical processing, retention, and application.

To develop concept and numeral imagery, the On Cloud Nine® math program (developed by the authors) integrates and consciously applies imagery to the cognitive process of computing and conceptualizing math and mathematical principles. As individuals become familiar with the concrete manipulatives, they are questioned and directed to consciously transfer the experienced to the imaged. They image the concrete and attach language to their imagery. The integration of imagery and language is then applied to computation. Individuals develop the sensory-cognitive processing to understand and use the logic of mathematics.

The program moves through three basic steps to develop mathematical reasoning and computation using: 1) manipulatives to experience the reality of math, 2) imagery and language to concretize that reality in the sensory system, and 3) computation to apply math to problem solving. On Cloud Nine® manipulatives serve two purposes: 1) to concretize numbers and mathematical concepts, and 2) to serve as a base for establishing imagery.

When asked to add the numbers 3 + 2, children who are drawing on their vault of images may see 3 apples and 2 more oranges to show 5 pieces of fruit. Others may draw on an image of a number line and place their mental finger on the 3 as a starting point. The “+” tells them to move forward and the “2” indicates how many places. They know the answer because they can “see it” in their mind’s eye. These children may look up as they access their images (defocusing).

Children who don’t seem to have a vault of images may say things like “I don’t remember that one.” They need explicit instruction in imaging the concrete and applying that imagery to the computation.

How does imaging as a conscious process work? The On Cloud Nine® math program begins with numbers in isolation—numeral imagery. A student is asked to view the written numeral, and then it is taken away. The student must demonstrate the “number” underlying the numeral by showing how many cubes represent that number. The student sees, says, and writes the number in the air. The goal is for the student, when she sees the numeral, to immediately create an image of the formation of that number and the value behind it.

The process continues with experiencing the number line, first as a concrete manipulative, then as a flexible mental image. “Show me where you see the number 15?” “What’s the number one step up from that?” “Is the 3 close to the 15 or quite far away?” “What number is closer to the 15 – the 10 or the 5?” Students develop a number line they carry with them in their vault of images. These students can access their vault of images at will. Conscious imagery and the ability to simultaneously create images and verbalize these imaging—dual coding—are continued as children are taught addition, subtraction, word problems, multiplication, division and more advanced math.On Cloud Nine® math integrates and consciously applies imagery to the cognitive process of computing and conceptualizing math and mathematical principles.
Children image the concrete and attach language to their imagery. The integration of imagery and language is then applied to every aspect of mathematical computation.

All children can develop the sensory-cognitive processing to understand and use the logic of mathematics. In every aspect of math, children can have access to what becomes an innate bank vault of imagery for memory and computation.

About the authors

Nanci Bell, owner and director of Lindamood-Bell Learning Processes®, is the author of two books on imagery as the base for language processing. Kimberly Tuley, the director of operations for Lindamood-Bell is a trainer and consultant in the application and refinement of Lindamood-Bell programs.

Bibliography

Aristotle. (1972). Aristotle on Memory. Providence, Rhode Island: Brown University Press.

Arnheim, R. (1966). Image and thought. In G. Kepes (Ed.). Sign, Image, Symbol. New York: George Braziller, Inc.

Bell, Nanci. (1991). Visualizing and Verbalizing for Language Comprehension and Thinking. Paso Robles: NBI Publications.

Moore, David S. (1990). On the Shoulders of Giants: New Approaches to Numeracy. Steen, L. (Ed.). Washington, D.C.: National Academy Press.

Papert, Seymour. (1993). The Children’s Machine: Rethinking School in the Age of the Computer. New York: Basic Books.

Paivio, Allan. (1981). Mental Representations: A Dual Coding Approach. New York: Oxford University Press.

Stern, Catherine and Stern, Margaret B. (1971). Children Discover Arithmetic. New York: Harper & Row, Publishers, Inc.
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Adapting Mathematics Instruction in theGeneral Education Classroom for                                            Students with Mathematics Disabilities
Robin H. Lock
The University of Texas at AustinLD Forum: Council for Learning Disabilities Winter 1996


Students with learning disabilities (LD) are increasingly receiving most of their mathematics instruction in general education classrooms. Studies show that these students benefit from general education mathematics instruction if it is adapted and modified to meet the individual needs of the learners (Salend,1994). Adaptations and modifications come in many forms. They can be as simple as using graph paper to help student with mathematics disabilities keep columnar addition straight or as complex as solving calculus equations with calculators. To ensure effective instruction, adaptations and modifications for instruction are necessary in the areas of lesson planning, teaching techniques, formatting content, adapting media for instruction, and adapting evaluation (Wood, 1992).

In general education classrooms, adaptations and modifications in mathematics instruction are appropriate for all students, not just students with LD. Teachers of mathematics will find that simple changes to the presentation of mathematical concepts enable students to gain a clearer understanding of the process rather than a merely mechanically correct response. Additionally, adapting and modifying instruction for students creates a more positive atmosphere that encourages students to take risks in problem-solving, which strengthens student understanding of the concept (McCoy & Prehm,1987).

For many teachers with limited or no preparation for working with students with LD, inclusion of students with mathematics disabilities may create concern. This article provides information on how to adapt and modify mathematics instruction to promote success and understanding in the areas of mathematical readiness, computation, and problem-solving for students with math disabilities. It also presents techniques that promote effective mathematics instruction for these students.

HOW CAN GENERAL EDUCATION TEACHERS FACILITATE THE LEARNING OF MATHEMATICAL SKILLS ?

Ariel (1992) stresses the need for all students to develop skill in readiness, computation, and problem-solving skills. As illustrated below, adaptations and modifications can be implemented to help students succeed in all three areas.

Readiness

According to Ariel (1992), students with LD must acquire (a) general developmental readiness, and (b) conceptual number readiness. General
developmental readiness includes ability in the areas of classification, one-to-one correspondence, seriation, conservation, flexibility, and reversibility. Knowledge of the student's level of general readiness allows the teacher to determine how adaptations and modifications must be enacted to allow for the student to progress. For some students, mathematics readiness instruction may need to include the development of language number concepts such as big and small and smallest to largest; and attributes such as color, size, or shape. Instruction, review, and practice of these concepts must be provided for longer time periods for students with mathematics disabilities than for other students.

Conceptual number readiness is essential for the development of addition and subtraction skills (Ariel,1992). Practice and review with board games or instructional software are effective ways to develop conceptual number readiness for students with mathematics disabilities. Manipulatives, such as Cuisenaire rods and Unifix math materials (e.g.,100 block trays) allow students with math disabilities to visualize numerical concepts and engage in age-appropriate readiness skills (see Lambert in this series for additional suggestions about manipulatives.

Computational Skills

Adaptations and modifications in the instruction of computational skills are numerous and can be divided into two areas: memorizing basic facts and solving algorithms or problems.

Basic Facts. Two methods for adapting instruction to facilitate recall of basic facts for students with math disabilities include (a) using games for continued practice, and (b) sequencing basic facts memorization to make the task easier. Beattie and Algozzine (cited in McCoy & Prehm, 1987) recommend the use of dice rolls, spinners, and playing cards to give students extra practice with fact memorization and to promote interest in the task by presenting a more game-like orientation. Further, McCoy and Prehm (1987) suggest that teachers display charts or graphs that visually represent the students' progress toward memorization of the basic facts. Sequencing fact memorization may be an alternative that facilitates instruction for students with LD. For example, in teaching the multiplication facts, Bolduc (cited in McCoy & Prehm, 1987) suggest,starting with the xO and x1 facts to learn 36 of the 100 multiplication facts. The x2 and x5 facts are next, adding 28 to the set of memorized facts. The x9s are introduced next, followed by doubles such as 6 x 6. The remaining 20 facts include 10 that are already known if the student is aware of the commutative property (e.g., 4 x 7 = 7 x 4). New facts should be presented a few at time with frequent repetition of previously memorized facts for students with LD.

Solving Algorithms. Computation involves not only memorization of basic facts, but also utilization of these facts to complete computational algorithms. An algorithm is a routine, step-by-step procedure used in computation (Driscoll, 1980 cited in McCoy & Prehm, 1987). In the addition process, McCoy and Prehm (1987) present three alternatives to the standard renaming method for solving problems, including expanded notation (see Figure 1 ) partial sums (see Figure 2), and Hutchings' low-stress algorithm (see Figure 3). Subtraction for students with mathematics disabilities is made easier through the use of Hutchings' low-stress subtraction method (McCoy & Prehm, 1987) (see Figure 4) where all renaming is done first. Multiplication and division (McCoy & Prehm, 1987) can be illustrated through the use of partial products (see Figure 5) . Further, arrays that use graph paper to allow students to plot numbers visually on the graph and then count the squares included within the rectangle they produce. Arrays can be used in combination with partial products to modify the multiplication process, thereby enabling students with math disabilities to gain further insight into the multiplication process.

Providing adaptations is often very effective for helping students with mathematics disabilities successfully use facts to solve computational problems. Salend (1994) lists suggestions for modifying mathematics assignments in computation. These suggestions are shown in Table 1.

Figure 1. Expanded Notation
29 = 2 tens and 9 ones +43 = 4 tens and 3 ones
Step one: Add the ones and tens. 6 tens and 12 ones
Step two: Regroup the ones, if neccessary 6 tens and (1 ten 2 ones)
Step three: Put the tens together. (6 tens and 1 ten) and 2 ones
Step four: Write the tens in a simpler way. 7 tens and 2 ones
Step five: Write the answer in number form. 72

Figure 2. Partial Sums

39
+65
--------

(sum of the ones) 14
(sum of the tens) 90
104

Figure 3. Hitchings' Low-Stress Algorithm

Problem: 45 + 77 + 56 + 83 + 27 + 39 = 45
77 1) Add 5 + 7 and record 12, put the "1" above the tens.
56 2) Add 2 + 6 and record 8, no tens to carry.
83 3) Add 8 +3 and record 11, put the "1" above the tens.
27 4) Add 1 +7 and record 8, no tens to carry.
39 5) Add 8 + 9 and record 17, put the "1" above the tens.
6) Add 3 + 4 and record 7, no tens to carry.
7) Add 7 + 7 and record 14, put the "1" in the hundreds
8) Add 4 + 5 and record 9, no hundreds to carry.
9) Add 9 + 8 and record 17, put the "1" in the hundreds.
10) Add 7 + 2 and record 9, no hundreds to carry.
11) Add 9 + 3 and record 12, put the "1" in the hundreds.
12) Add the hundreds place.

Figure 4. Hutchings' Low-Stress Subtraction Algorithm

3247 3247 3247 3 247 3 247
-1736 47 1247 21247 21247
-1736 -1 736 -1 736 -1 736

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1 511
1) Rewrite the tens and ones places.

2) Determine if renaming is necessary.
3) Rewrite the hundreds, tens and ones places.
4) Determine if renaming is necessary.
5) Renaming is necessary to complete subtraction in the hundreds place. Rewrite
the number in the hundreds place.
6) Complete subtraction with renaming already accomplished.

Figure 5. Partial Products

1) 2*3=6 23

2) 2*20=40*12

3) 10*3=30

4) 10*20=200/276
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Table 1. Tips for Modifying Mathematics Computational Assignments.

1. Reduce the number of problems on worksheets for independent practice.

2. Increase the amount for time students have time to complete the assignment.

3. Provide adequate space for students to write out solutions.

4. Follow a standard format for developing worksheets.

5. Cut the worksheet in halves or fourths requiring students to complete one section at a time.

6. Assign only odd or even problems.

7. Highlight the operation to be performed.

8. Move gradually to increasing the number of problems (not more than 20 problems) and decreasing the amount of time to complete the assignment.

Further adaptations and modifications in computational instruction include color coding of the desired function for the computation problem (Ariel, 1992), either ahead of time by the teacher or during independent practice by the student. This process serves as a reminder to the student to complete the desired function and also may be used as an evaluation device by the teacher to determine the student's knowledge of the mathematical symbols and processes they represent.

Matrix paper allows students a physical guide for keeping the numbers in alignment (Ariel, 1992), thus decreasing the complexity of the task and allowing the teacher and student to concentrate on the mathematical process. In simplifying the task, the teacher then can identify problems in the student's understanding of the process rather than in the performance of the task.

Finally, modeling is another effective strategy for helping students solve computational problems. For example, Rivera and Deutsch-Smith (cited in Salend, 1994) recommend the use of the demonstration plus permanent model strategy, which includes the following three steps designed to increase skill in comprehending the computation process: (a) the teacher demonstrates how to solve a problem while verbalizing the key words associated with each step in solving the computation problem; (b) the student performs the steps while verbalizing the key words and looking at the teacher's model; and (c) the student completes additional problems with the teacher's model still available. Other modeling examples provided by Salend (1994) include the use of charts that provide definitions, correct examples, and step-by-step instructions for each computational process.

Problem-Solving:

Problem-solving can be adapted and modified for students with mathematics disabilities in several different ways (see Kelly & Carnine in this series for additional word problem-solving instructional strategies). Polloway and Patton (1993) note that students with math disabilities improve their problem-solving skills through teacher-directed activities that include (a) having students read or listen to the problem carefully; (b) engaging students in focusing on relevant information and/or significant words needed to obtain the correct answer while discarding the irrelevant by writing a few words about the answer needed (e.g., number of apples), by identifying aloud or circling the significant words in the problem, and by highlighting the relevant numbers; (c) involving students in verbalizing a solution for the problem using a diagram or sketch when appropriate; (d) developing strategies for working through the story problem by writing an appropriate mathematical sentence; and (e) performing the necessary calculations, evaluating the answer for reasonableness, and writing the answer in
appropriate terms.

Lack of critical thinking skills compounds problem-solving difficulties. Several cognitive and meta-cognitive strategies can be used effectively. For example, (1992) recommends the use of six problem-solving strategies that students can monitor on an implementation sheet. Students verbalize the steps while completing the problem and note their completion of the steps on the monitoring sheet. The six steps are:

1. Read and understand the problem.
2. Look for the key questions and recognize important words.
3. Select the appropriate operation.
4. Write the number sentence (equation) and solve it.
5. Check your answer.
6. Correct your errors.

Further, Mercer (1992) identifies the components necessary for students to engage in successful problem-solving. According to Merger, the problem-solving process involves 10 steps, which can be expanded into learning strategies to enable students with math disabilities to be more effective in solving word problem. The 10 steps are:

1. Recognize the problem.
2. Plan a procedural strategy (i.e., identify the specific steps to follow).
3. Examine the math relationships in the problem.
4. Determine the math knowledge needed to solve the problem.
5. Represent the problem graphically.
6. Generate the equation.
7. Sequence the computation steps.
8. Check the answer for reasonableness.
9. Self-monitor the entire process.
10. Explore alternative ways to solve the problem.

Hammill and Bartel (in Polloway & Patton,1993) offer many suggestions for modifying mathematics instruction for students with LD. They encourage teachers to think about how to alter instruction while maintaining the primary purpose of mathematics instruction: Competence in manipulating numbers in the real world. Their suggestions include:

1. Altering the type or amount of information presented to a student such as giving the student the answers to a story problem and allowing the student to explain how the answers were obtained.

2. Using a variety of teacher-input and modeling strategies such as using manipulatives during the instructional phase with oral presentations.

TECHNIQUES TO ENHANCE MATHEMATICS INSTRUCTION

For students with math disabilities, effective mathematics instruction is the difference between mathematics as a paper-and-pencil/right-answer type of task and an important real-life skill that continues to be used throughout their lifetime. This section examines effective instructional techniques that the general educator can incorporate into the classroom for all learners, and especially for students with math disabilities.

Increasing Instructional Time

Providing enough time for instruction is crucial. Too often, "math time" according to Usnick and McCoy (cited in McCoy & Prehm, 1987) includes a long stretch of independent practice where students complete large numbers of math problems without feedback from the teacher prior to completion. Instructional time is brief, often consisting of a short modeling of the skill without a period of guided practice. By contrast, small-group practice where students with math disabilities complete problems and then check within the group for the correct answer, use
self-checking computer software programs, and receive intermittent teacher interaction are positive modifications for increasing time for mathematics instruction. Additionally, time must be provided for students to engage in problem-solving and other math "thinking" activities beyond the simple practice of computation, even before students have shown mastery of the computational skills. Hammil and Bartel (cited in Polloway & Patton, 1993) suggest slowing down the rate of instruction by using split mathematics instructional periods and reducing the number of problems required in independent practice.

Using Effective Instruction

Polloway and Patton (1993) suggest that the components of effective instruction play an important role in the success of students with disabilities in general education mathematics instruction. One suggested schedule for the class period includes a period of review of previously covered materials, teacher-directed instruction on the concept for the day, guided practice with direct teacher interaction, and independent practice with corrective feedback. During the guided and independent practice periods, teachers should ensure that students are allowed opportunities to manipulate concrete objects to aid in their conceptual understanding of the mathematical process, identify the overall process involved in the lesson (i.e., have students talk about "addition is combining sets" when practicing addition problems rather than silent practice with numerals on a worksheet), and write down numerical symbols or mathematical phrases such as addition or subtraction signs.

Teaching key math terms as a specific skill rather than an outcome of basic math practice is essential for students with LD (Salend,1994) . The math terms might include words such as "sum," "difference," "quotient," and "proper fraction," and should be listed and displayed in the classroom to help jog students' memories during independent assignments.

Varying Group Size

Varying the size of the group for instruction is another type of modification that can be used to create an effective environment for students with math disabilities. Large-group instruction, according to McCoy and Prehm (1987), may be useful for brainstorming and problem-solving activities. Small-group instruction, on the other hand, is beneficial for students by allowing for personal attention from the teacher and collaboration with peers who are working at comparable levels and skills. This arrangement allows students of similar levels to be grouped and progress through skills at a comfortable rate. When using grouping as a modification, however, the teacher must allow for flexibility in the groups so that students with math disabilities have the opportunity to interact and learn with all members of the class (see Rivera in this series for cooperative learning information).

Using Real-Life Examples

Salend (1994) recommended that new math concepts be introduced through everyday situations as opposed to worksheets. With everyday situations as motivators, students are more likely to recognize the importance and relevance of a concept. Real-life demonstration enables students to understand more readily the mathematical process being demonstrated (see Scott & Raborn in this series for additional ideas). Further, everyday examples involve students personally in the instruction and encourage them to learn mathematics for use in their lives.
Changing the instructional delivery system by using peer tutors (see Miller et al. in this series for ideas about peer tutoring); computer-based instruction; or more reality-based assignments such as "store" for practice with money recognition and making change also provide real life math experiences (Hammill & Bartel cited in Polloway & Patton,1993).

Varying Reinforcement Styles

Adaptations and modifications of reinforcement styles or acknowledgment of student progress begin with teachers being aware of different reinforcement patterns. Beyond the "traditional" mathematical reinforcement style, which concentrates on obtaining the "right answer," students with mathematics disabilities may benefit from alternative reinforcement patterns that provide positive recognition for completing the correct steps in a problem regardless of the outcome (McCoy & Prehm, 1987). By concentrating on the process of mathematics rather than on the product, students may begin to feel some control over the activity. In addition, teachers can isolate the source of difficulty and provide for specific accommodations in that area. For example, if the student has developed the ability to replicate the steps in a long division problem but has difficulty remembering the correct multiplication facts, the teacher should reward the appropriate steps and provide a calculator or multiplication chart to increase the student's ability to obtain the solution to the problem.

SUMMARY

The mathematical ability of many students with LD can be developed successfully in the general education classroom with proper accommodations and special education instructional support. To this end, teachers should be aware of the necessity for adapting and modifying the environment to facilitate appropriate, engaging instruction for these students. Use of manipulatives is encouraged to provide realistic and obvious illustrations of the underlying mathematical concepts being introduced. Reliance on problem-solving strategies to improve students' memories and provide a more structured environment for retention of information also is appropriate. Finally, teachers must evaluate the amount of time spent in instruction, the use of effective instructional practices, student progress (see Bryant in this series), and the use of Real-life activities that encourage active, purposeful learning in the mathematics classroom.

References

Ariel, A. (1992). Education of children and adolescents with learning disabilities. NY: Merrill.

McCoy, E. M., & Prehm, H. J. (1987). Teaching mainstreamed students. Methods and techniques. Denver, CO: Love Publishing Company.

Mercer, C. D. (1992). Students with learning disabilities (4th ed.). NY: Merrill.

Polloway, E. A., & Patton, J. R. (1993). Strategies for teaching learners with special needs (5th ed.). NY: Merrill.

Salend, S. J. (1994). Effective mainstreaming. Creating inclusive classrooms (2nd ed.). NY: MacMillian.

Wood, J. W. (1992). Adapting instruction for mainstreamed and at risk student (2nd ed.). NY: Merrill.
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Learning Disabilities in Mathematics
C. Christina Wright, Ph.D.
October 1996

Article Outline:

I. What constitutes a learning disability in mathematics?
II. How is mathematics learning related to mathematics learning disabilities?
III. How do you assess a mathematics disability?
IV. How do you help a child who is having difficulty?
V. Summary and References

What constitutes a learning disability in mathematics?

There is no single mathematics disability. In fact, mathematics disabilities are as varied and complex as those associated with reading. Furthermore, there are some arithmetic disabilities which can exist independent of a reading disability and others which do not. One type of learning disability affecting mathematics can stem from an individual's difficulty processing language, another might be related
to visual spatial confusion, while yet another could include trouble retaining math facts and keeping procedures in the proper order. While extremely rare, there are some learners who cannot successfully compare the lengths of two sticks and others who have almost no ability to estimate. Finally, some people experience emotional blocks so overwhelming as to preclude their ability to think responsibly and clearly when attempting math, and these students are disabled, as well.

How is mathematics learning related to mathematics learning disabilities?

Ginsburg (1977) and Baroody (1987) have identified the initial, intuitive stages of mathematics learning as the "informal" stage. A young child learns the language of magnitude (more, less; bigger, smaller) and equivalence (same) at home, long before schooling begins. In much the same way a child learns to chant the alphabet before knowing how to use it, children learn the counting sequence. This sequence is a kind of song, they discover, and it must go in a particular order.

Informal mathematics includes the ability to match one item with another item, as in setting the table. Later, sometime during the first years of formal school, the child comes to realize that five objects, no matter what size, no matter how spread out, no matter what the configuration, are still counted as five. This gradual realization, called "conservation" of number is an exciting transition and cognitive metamorphosis. It heralds the child's growing ability to use numerals symbolically with real meaning.

A learning disability at this age may revolve around using language, manipulating objects, or judging size at a glance. Those who are visually impaired require experiences touching and judging more/less, bigger/smaller. There is a very small group of children who seem unable to visually compare length and amount.

When children enter school, they will gradually learn the format aspects of number ,i.e., adding with exchanging and trading. In the best circumstances, children begin with informal mathematics, usually with manipulatives, and gradually build to the more abstract, less inherently meaningful formal procedures.

Many children do not make this connection and characterize math as a collection of unconnected facts which must be memorized. They don't look for patterns or meaning and can feel puzzled by classmates who seem to learn with so much less effort. In other cases, adults move in prematurely with children who are eager and excited to memorize, teaching them procedures which they can imitate but not understand. While this informal/formal gap is not, strictly speaking, a learning disability, it probably is a factor in a majority of math learning difficulties.

The pace at which children move from informal to formal arithmetic is far more gradual than most educators or parents realize. Even as adult learners we need a considerable chunk of time with the concrete, "real" aspect of a new piece of learning before we move on to making generalizations and other abstractions.

There are some children who have a language impairment, who do not easily process and understand the words and sentences they hear. Sometimes these children also have difficulty grasping the connection and the organizing hierarchy of "little" ideas and "big" ones. These children are also likely to view math as an ocean full of meaningless facts and procedures to be memorized.

Visual processing difficulties play a different sort of role in reading than they do in mathematics. In math there are fewer symbols to recognize, produce, and decode, and children can "read" math successfully even when they cannot yet read words. Children with visual/spatial perceptual difficulties may exhibit two kinds of problems. In the less severe instance, some will understand math quite clearly but be unable to express this using paper and pencil. More severe is the case where children cannot translate what they see into ideas which make sense to them.

How do you assess a mathematics disability?

One need not be a mathematics expert to evaluate a child's ability and style of doing math. A one-to-one mathematics interview is the best format for noting details. In the interview one focuses as intently on how the child does mathematics as on what or how correct they do it. It is essential to keep in mind that you are searching for what does work at the same time as you are probing to find out what doesn't work.

A mathematics interview should include the use of manipulatives, i.e. coins, base ten blocks, geoboards, cuisenaire rods, and tangrams. A calculator is an important tool and can be used to uncover the difference between comprehension and computation difficulties.

The interviewer needs to remember to look at the full range of mathematical areas. In addition to computation, one should explore the child's ability to make predictions based on understanding patterns, to sort collections of blocks or objects in a logical way, to organize space with flexibility, and to measure.

To aid in making a diagnosis which will result in useful recommendations, look carefully at strengths and weaknesses. Note whether the child talks to herself, whether she draws a picture to help her understand a situation, or whether he asks you to repeat. See if the child has a mathematics "proofreading" capacity by asking him to estimate before he computes. This is an important strength.

How do you help a child who is having difficulty?

The fundamental principle in helping a child with a disability in mathematics is to work with the child to define his or her strengths. As these strengths are acknowledged, one uses them to reconfigure what is difficult.

When learners have lost (or never had) the connection between mathematics and meaning, it is helpful to encourage them to estimate their answers before they begin computing. When children work together in small groups to solve problems, they often ask more questions, get more answers, and do more quality thinking than when they work quietly, alone.

When children have difficulty organizing their written work on a page, they often do better with graph paper. A less expensive solution is to turn lined paper sideways so that the lines serve as vertical columns. This is especially helpful for long division.

The task of learning the facts can be transformed into one requiring Verbal reasoning. Instead of being asked to memorize 7 + 8, one boy was asked, "How do you remember that 7 + 8 = 15?" His strategies, in this case, that 7 + 7 = 14, so 7 + 8 = 15, were practiced and reinforced and he became able to retain his facts. A general principle is that through drill and practice children will get faster at whatever they're already doing. This technique of focusing on strategies is one which fosters a healthy sense of self reliance and diminishes the need for meaningless memorization.

When children do not have a strong language base, it is even more important for the language of explanations to be absolutely accurate (concrete) and parsimonious. In other words, elaborations confuse rather than help this type of child. Give the instructions or explanation once and give the child time and the materials to think about what has been said so that he or she can formulate a meaningful question, if necessary. Asking these children to process quickly is unrealistic and not helpful.

By contrast, the group of children who use language as a tool to keep themselves on track and to organize their thinking are often extremely quick to respond. Language is their preferred medium, after all. These children often respond well to the use of metaphor in explanations. These children are often impatient and do not understand that good thinking is not instantaneous. They need reassurance and a relaxed structure so that they go beyond the superficial quickness and do some real thinking.

Finally, those who are afraid to even attempt math are often unaware of their very real strengths. This group believes that math = computation, when in fact computation is but a small slice of mathematics. The increasing acceptance of calculators refocuses teachers and students on the real issue at hand: problem solving. Math anxious students often will take risks if their fears are acknowledged and support is provided. Students will gradually feel more powerful as they experience themselves as successful thinkers.

Summary

Mathematics learning disabilities do not often occur with clarity and simplicity. Rather, they can be combinations of difficulties which may include language processing problems, visual spatial confusion, memory and sequence difficulties, and/or unusually high anxiety. With the awareness that math understanding is actively constructed by each learner, we can intervene in this process to advocate for or provide experience with manipulatives, time for exploration, discussion where the "right" answer is irrelevant, careful and accurate language, access to helpful technologies, and understanding and support.
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Mathematics and Dyslexia
Perspectives, Fall 1998
International Dyslexia Association

Not all individuals with dyslexia have problems with mathematics, but many do. There are those who have a good memory for sequences and can execute procedures in a "recipe" style, i.e., step-by-step. They are able to remember formulas, but may not understand why the formula makes sense. They prefer to do paper and pencil tasks and are attentive to the details, but do not see the big picture. Then, there are those who see the big picture and have insight into the patterns of mathematics, but are poor at computation and have problems with remembering step-by-step procedures. They also understand mathematical concepts and like to solve problems mentally and quickly, yet their answers may be inaccurate. These individuals may have difficulty in verbalizing and explaining their answers.

Too frequently and too readily, individuals with dyslexia who have difficulty with mathematics are misdiagnosed as having dyscalculia - literally trouble with calculating, a neurologically based disability. True dyscalculia is rare (Steeves, 1983).1 We know that for individuals with dyslexia, learning mathematical concepts and vocabulary and the ability to use mathematical symbols can be impeded by problems similar to those that interfered with their acquisition of the written language (Ansara, 1973).2 Additionally, we know that the learning of mathematical concepts, more than any other content area, is tied closely to the teacher's or academic therapist's knowledge of mathematics and to the manner in which these concepts are taught (Lyon, 1996).3 Therefore, there are individuals with dyslexia who will exhibit problems in mathematics, not because of their dyslexia or dyscalculia, but because their instructors are inadequately prepared in mathematical principles and/or in how to teach them.

In addition, we know that individuals with dyslexia may have problems with the language of mathematics and the concepts associated with it. These include spatial and quantitative references such as before, after, between, one more than, and one less than. Mathematical terms such as numerator and denominator, prime numbers and prime factors, and carrying and borrowing may also be problematic. These individuals may be confused by implicit, multiple meanings of words, e.g., two as the name of a unit in a series and also as the name of a set of two objects. Difficulties may also occur around the concept of place value and the function of zero. Solving word problems may be especially challenging because of difficulties with decoding, comprehension, sequencing, and understanding mathematical concepts. In understanding the complex nature of dyslexia, Ansara (1973)4 made three general assumptions about learning, in particular, for individuals with dyslexia. These assumptions affect the way one needs to provide instruction. They are:

learning involves the recognition of patterns which become bits of knowledge that are then organized into larger and more meaningful units;
learning for some children is more difficult than for others because of...deficits that interfere with the ready recognition of patterns; (and)
some children have difficulty with the organization of parts into wholes, due to ... a disability in the handling of spatial and temporal relationships or to unique problems with integration , on, sequencing ng or memory.

Therefore, teachers and academic therapists who provide remedial instruction in mathematics to these individuals must have an understanding of the nature of dyslexia and how it affects learning, not only in written language, but also in mathematics. Additionally, the instructor needs to have an understanding of the mathematics curriculum; the ability to use a variety of instructional techniques that are simultaneously multisensory and which provide for explicit instruction that is systematic, cumulative, diagnostic, and both synthetic and analytical- as well as a knowledge of current research in mathematical instruction.

Simply just being good at mathematics is not enough. The teacher and academic therapist need to understand that mathematics is problem-solving which involves reasoning and the ability to read, write, discuss and convey ideas using mathematical signs, symbols and terms. This requires an understanding of mathematical knowledge, both conceptual (relationships constructed internally and connected to already existing ideas) and procedural (knowledge of symbols used to represent mathematics, and the rules and procedures that are used to carry out mathematical tasks). Both are important and need to be understood. For procedural knowledge, the most important connection is to the conceptual knowledge that supports it; otherwise, procedural knowledge will be learned rigidly and used narrowly. Usually, when there is a connection to a conceptual basis, the procedure is not only understood, but the learner will have access to other ideas associated with the concept (Van de Walle, 1994).5 For individuals with dyslexia, this linkage is critical and language plays an important role.

To assist individuals with dyslexia in making this linkage, it is essential that teachers and academic therapists provide instruction that allows the learner to work through the following cognitive developmental stages when teaching mathematical concepts at all grade levels: concrete, pictorial, symbolic, and abstract. Individuals with dyslexia will learn best when provided with concrete manipulatives with which they can work or experiment. These help build memory as well as allowing for revisualization when memory fails. The next stage, pictorial, is one which may be brief, but is essential for beginning the transition away from the concrete. This is where individuals recognize or draw pictures to represent concrete materials without the materials themselves. Symbols, i.e., numerals, plus signs, etc., are introduced when individuals understand the basic concept, thereby making the connection to procedural knowledge. Finally, the abstract stage is where individuals are able to think about concepts and solve problems without the presence of manipulatives, Pictures, and symbols. (Steeves & Tomey, 1998a).6

According to Steeves and Tomey (1998a),7 it is important that the four developmental stages are linked through language for these individuals. There are three kinds of language which allow one to fully integrate mathematical learning. First, is the individual's own language. No matter how imperfect this language is, it is important that the individual discusses, questions, and states what she/he has learned. Second, is the language of the instructor, or standard English, which clarifies the learner's own language, and links to the third language, the language of mathematics. The language of mathematics is not just the vocabulary but the use of sign, symbols, and terms to express mathematical ideas, such as 2 + 4 = 6. Also, language allows the instructor to determine if the learner understands the concept and is not just following steps demonstrated by the instructor to complete a process, even at the concrete stage.

For these reasons, teachers and academic therapists who, in mathematics, work with individuals with dyslexia, must be well-trained in multi-sensory structured techniques both in language and mathematics instruction and remediation. They must not only demonstrate competencies in knowledge and skills in teaching language to these individuals, but also demonstrate the following competencies in mathematics (Steeves and Tomey, 1998b)8:

Understanding of the mathematics and the use of appropriate methodology, technology, and manipulatives within the following content:

Number systems, their structure, basic operations and properties;

Elementary number theory, ratio, proportion and percent;

Algebra;

Measurement systems - U.S. and metric;

Geometry: geometric figures, their properties and relationships;

Probability;

Discrete mathematics: symbolic logic, sets, permutations and combinations; and Computer science: terminology, simple programming, and software applications;

Understanding of the sequential nature of mathematics, and the mathematical structures inherent in the content strands;

Understanding of the connections among mathematical concepts and procedures and their practical applications;

Understanding of and the ability to use the four processes - becoming mathematical problem- solvers, reasoning mathematically, communicating mathematically, and making mathematical connections at different levels of complexity;

Understanding the role of technology, and the ability to use graphing utilities and computers to teach mathematics;

Understanding of and ability to select, adapt, evaluate, and use instructional materials and resources, including technology;

Understanding of and the ability to use strategies for managing, assessing, and monitoring student learning, including diagnosing student errors; and

Understanding of and the ability to use strategies to teach mathematics to diverse learners.

The editors thank Harley A. Tomey, III (VA) and Joyce Steeves, Ed.D. (MD) for their suggestions for and review of this article, and especially Mr Tomey for his help in its preparation.

References and Endnotes

1Steeves, K.J. (1983). Memory as a factor in the computational efficiency of dyslexic children with high abstract reasoning ability. Annals of Dyslexia, 33,141-152. Baltimore: International Dyslexia Association.

2, 4 Ansara A. (1973). The language therapist as a basic mathematics tutor for adolescents. Bulletin of the Orton Society, 23, 119-138.

3 Lyon, G.R. (1996). State of Research. In Cramer, S. & Ellis, W. (Eds.), Learning disabilities: Lifelong issues (pp. 3-61). Baltimore: Brooks Publishing.

5 Van de Walle, J. A. (1994). Elementary school mathematics: Teachi ng developmentally (2nd ed.). White Plains, NY. Longman.

6,7 Steeves, K. J., & Tomey, H.A. (1998a). Mathematics and dyslexia: The individual who learns differently may still be successful in math. Manuscript in preparation.

8 Steeves, K. J., & Tomey, H.A. (1998b). Personal written communications to the editors.
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Math Study Skills for Elementary Through High School

Attendance
Attend class regularly and take notes.

Ask Questions

Don't be afraid to ask questions in class. There are always other students in class who have the same questions but are too shy too ask.

Read Your Text

Many students depend solely on the teacher's lecture and explanations in class to do their assignments, without reading the explanation or studying the examples in their text. If you get stuck, one of the best resources you have is your text! Many exercises are keyed to specific examples or objectives that will explain the procedures for working them. Math texts should be read slowly and carefully with pencil and paper at hand. Try to identify the main ideas that are in each lesson. Often they are clearly highlighted or boxed in the text.

Homework

Review your class notes and try reworking the problems the teacher did in class. Many students say "I understand it perfectly when you do it, but I get stuck when I try to work the problem myself." Reworking the problems the instructor did in class will help reinforce what you have learned.

Try to do your math homework as soon after class as possible, and reserve some time later in the day to attack the problems that you get stuck on the first time around.

Check your work with the answers in the back of the book. If you get a problem wrong, check to see if there is an example in the book similar to the problem you are working on. Study the example and then try to do the example problem with your book closed. Then give the homework problem another try. If the answer is not correct, check your computations (arithmetic and algebra). If you still have difficulty solving the problem, mark that problem so that you can get help with it.

Keep your homework in a notebook. The more organized it is, the more it will serve you as a guide for seeking help in resolving difficulties and studying for tests.

Getting Help

Visit your teacher during office hours.Take advantage of the tutoring services on campus.Visit the Tutoring Center.Form a study group with your classmates.

Preparing for a Math Test

Many students work hard preparing for a test and yet they don't do as well as they expected. Here are some suggestions that might help:

Work regularly on your math homework. If you have difficulties get them cleared up right away.

A couple of days before the test, start to work through the sections to be covered on the test. Work sample problems. Make notes on what each section is about. Get help with material you don't understand.

Make a summary of the different kinds of problems that will be on the test. Describe each problem type, describe the steps in working the problem, do an example of each. Your summary should include key definitions and/or theorems.

The last studying you do before the test should be in reviewing your summary. For each problem type it includes, make sure that you know the steps needed for its solution. Also, make sure that you know and understand all key definitions and/or theorems.

Taking a Math Test

Many students feel that they knew the material for a test and yet they did not do as well as they should have.

A few tips: Prepare as well as possible. Start preparing yourself mentally and, as the instructor passes out the test materials, think about your test taking strategies. The following should be part of your plan:

Keep cool

Do a "memory dump". While your head is still clear and before looking at the test, write yourself some notes. Include those things you think you might forget or "cautions" about typical errors you have made before.

Make a reasonable attempt at each problem. If you get it, fine. Go on. If you don't get it, put a check by it and then forget it until you have finished the rest of the problems.

When you have tried all of the problems on the test, go back and do the best you can on the checked problems.

If you can do only part of a problem, do it. Partial credit is better than no credit.Above all, remain cool. Don't think about how you are doing or what happens if you don't do well. (You can do that after the test.) Don't worry if you find something you don't know.
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Math Study Skills High School and College

Active Study vs. Passive Study

Be actively involved in managing the learning process, the mathematics and your study time:

Take responsibility for studying, recognizing what you do and don't know, and knowing how to get your Instructor to help you with what you don't know.

Attend class every day and take complete notes. Instructors formulate test questions based on material and examples covered in class as well as on those in the text.

Be an active participant in the classroom. Get ahead in the book; try to work some of the problems before they are covered in class. Anticipate what the Instructor's next step will be.

Ask questions in class! There are usually other students wanting to know the answers to the same questions you have.

Ask the teacher questions. The teacher will be pleased to see that you are interested, and you will be actively helping yourself.
Good study habits throughout the semester make it easier to study for tests.

Studying Math is Different from Studying Other Subjects

Math is learned by doing problems. Do the homework. The problems help you learn the formulas and techniques you do need to know, as well as improve your problem-solving prowess.

A word of warning: Each class builds on the previous ones, all year long. You must keep up with the teacher: attend class, read the text and do homework every day. Falling a day behind puts you at a disadvantage. Falling a week behind puts you in deep trouble.

A word of encouragement: Each class builds on the previous ones, all year long. You're always reviewing previous material as you do new material. Many of the ideas hang together. Identifying and learning the key concepts means you don't have to memorize as much.

College Math is Different from High School Math

A College math class meets less often and covers material at about twice the pace that a High School course does. You are expected to absorb new material much more quickly. Tests are probably spaced farther apart and so cover more material than before. The Instructor may not even check your homework. Take responsibility for keeping up with the homework. Make sure you find out how to do it. You probably need to spend more time studying per week - you do more of the learning outside of class than in High School. Tests may seem harder just because they cover more material.

Study Time

You may know a rule of thumb about math (and other) classes: at least 2 hours of study time per class hour. But this may not be enough!
Take as much time as you need to do all the homework and to get complete understanding of the material. Form a study group. Meet once or twice a week (also use the phone). Go over problems you've had trouble with. Either someone else in the group will help you, or you will discover you're all stuck on the same problems. Then it's time to get help from your Instructor. The more challenging the material, the more time you should spend on it.

Problem Solving

Problem Solving (Homework and Tests) The higher the math class, the more types of problems: in earlier classes, problems often required just one step to find a solution. Increasingly, you will tackle problems which require several steps to solve them. Break these problems down into smaller pieces and solve each piece - divide and conquer!

Problem types:

Problems testing memorization ("drill"),
Problems testing skills ("drill"),
Problems requiring application of skills to familiar situations ("template" problems),
Problems requiring application of skills to unfamiliar situations (you develop a strategy for a new problem type),
Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation.
In early courses, you solved problems of types 1, 2 and 3. By College Algebra you expect to do mostly problems of types 2 and 3 and sometimes of type 4. Later courses expect you to tackle more and more problems of types 3 and 4, and (eventually) of type 5. Each problem of types 4 or 5 usually requires you to use a multi-step approach, and may involve several different math skills and techniques.

When you work problems on homework, write out complete solutions, as if you were taking a test. Don't just scratch out a few lines and check the answer in the back of the book. If your answer is not right, rework the problem; don't just do some mental gymnastics to convince yourself that you could get the correct answer. If you can't get the answer, get help. The practice you get doing homework and reviewing will make test problems easier to tackle.

Tips on Problem Solving

Apply Pólya's four-step process:

The first and most important step in solving a problem is to understand the problem, that is, identify exactly which quantity the problem is asking you to find or solve for (make sure you read the whole problem). Next you need to devise a plan, that is, identify which skills and techniques you have learned can be applied to solve the problem at hand. Carry out the plan. Look back: Does the answer you found seem reasonable? Also review the problem and method of solution so that you will be able to more easily recognize and solve a similar problem. Some problem-solving strategies: use one or more variables, complete a table, consider a special case, look for a pattern, guess and test, draw a picture or diagram, make a list, solve a simpler related problem, use reasoning, work backward, solve an equation, look for a formula, use coordinates.

"Word" Problems are Really "Applied" Problems

The term "word problem" has only negative connotations. It's better to think of them as "applied problems". These problems should be the most interesting ones to solve. Sometimes the "applied" problems don't appear very realistic, but that's usually because the corresponding real applied problems are too hard or complicated to solve at your current level. But at least you get an idea of how the math you are learning can help solve actual real-world problems.

Solving an Applied Problem
First convert the problem into mathematics. This step is (usually) the most challenging part of an applied problem. If possible, start by drawing a picture. Label it with all the quantities mentioned in the problem. If a quantity in the problem is not a fixed number, name it by a variable. Identify the goal of the problem. Then complete the conversion of the problem into math, i.e., find equations which describe relationships among the variables, and describe the goal of the problem mathematically. Solve the math problem you have generated, using whatever skills and techniques you need (refer to the four-step process above). As a final step, you should convert the answer of your math problem back into
words, so that you have now solved the original applied problem. For Further Reading: George Pólya, How to Solve It, Princeton University Press, Princeton (1945)

Studying for a Math Test

Everyday Study is a Big Part of Test Preparation. Good study habits throughout the semester make it easier to study for tests. On tests you have to solve problems; homework problems are the only way to get practice. As you do homework, make lists of formulas and techniques to use later when you study for tests. Ask your teacher questions as they arise; don't wait until the day or two before a test. The questions you ask right before a test should be to clear up minor details.

Studying for a Test

Start by going over each section, reviewing your notes and checking that you can still do the homework problems (actually work the problems again). Use the worked examples in the text and notes - cover up the solutions and work the problems yourself. Check your work against the solutions given.

You're not ready yet! In the book each problem appears at the end of the section in which you learned how do to that problem; on a test the problems from different sections are all together.

Step back and ask yourself what kind of problems you have learned how to solve, what techniques of solution you have learned, and how to tell which techniques go with which problems.

Try to explain out loud, in your own words, how each solution strategy is used (e.g. how to solve a quadratic equation). If you get confused during a test, you can mentally return to your verbal "capsule instructions". Check your verbal explanations with a friend during a study session (it's more fun than talking to yourself!).

Put yourself in a test-like situation: work problems from review sections at the end of chapters, and work old tests if you can find some. It's important to keep working problems the whole time you're studying.

Also:
Start studying early. Several days to a week before the test (longer for the final), begin to allot time in your schedule to reviewing for the test.Get lots of sleep the night before the test. Math tests are easier when you are mentally sharp.

Taking a Math Test

Test-Taking Strategy Matters. Just as it is important to think about how you spend your study time (in addition to actually doing the studying), it is important to think about what strategies you will use when you take a test (in addition to actually doing the problems on the test). Good test-taking strategy can make a big difference to your grade!

Taking a Test

First look over the entire test. You'll get a sense of its length. Try to identify those problems you definitely know how to do right away, and those you expect to have to think about. Do the problems in the order that suits you! Start with the problems that you know for sure you can do. This builds confidence and means you don't miss any sure points just because you run out of time. Then try the problems you think you can figure out; then finally try the ones you are least sure about. Time is of the essence - work as quickly and continuously as you can while still writing legibly and showing all your work. If you get stuck on a problem, move on to another one - you can come back later. Work by the clock. On a 50 minute, 100 point test, you have about 5 minutes for a 10 point question. Starting with the easy questions will probably put you ahead of the clock. When you work on a harder problem, spend the allotted time (e.g., 5 minutes) on that question, and if you have not almost finished it, go on to another problem. Do not spend 20 minutes on a problem which will yield few or no points when there are other problems still to try. Show all your work: make it as easy as possible for the Instructor to see how much you do know. Try to write a well-reasoned solution. If your answer is incorrect, the Instructor will assign partial credit based on the work you show. Never waste time erasing! Just draw a line through the work you want ignored and move on. Not only does erasing waste precious time, but you may discover later that you erased something useful (and/or maybe worth partial credit if you cannot complete the problem). You are (usually) not required to fit your answer in the space provided - you can put your answer on another sheet to avoid needing to erase. In a multiple-step problem outline the steps before actually working the problem. Don't give up on a several-part problem just because you can't do the first part. Attempt the other part(s) - if the actual solution depends on the first part, at least explain how you would do it. Make sure you read the questions carefully, and do all parts of each problem. Verify your answers - does each answer make sense given the context of the problem? If you finish early, check every problem (that means rework everything from scratch).

Getting Assistance

When

Get help as soon as you need it. Don't wait until a test is near. The new material builds on the previous sections, so anything you don't understand now will make future material difficult to understand.

Use the Resources You Have AvailableAsk questions in class. You get help and stay actively involved in the class. Ask the teacher when she/he has time to help you. . Teachers like to see students who want to help themselves.Ask friends, members of your study group, or anyone else who can help. The classmate who explains something to you learns just as much as you do, for he/she must think carefully about how to explain the particular concept or solution in a clear way. So don't be reluctant to ask a classmate.Go to the Math Help Sessions or other tutoring sessions on campus.Find a private tutor if you can't get enough help from other sources.All students need help at some point, so be sure to get the help you need.

Asking Questions

Don't be afraid to ask questions. Any question is better than no question at all (at least your Instructor/tutor will know you are confused). But a good question will allow your helper to quickly identify exactly what you don't understand. Not too helpful comment: "I don't understand this section." The best you can expect in reply to such a remark is a brief review of the section, and this will likely overlook the particular thing(s) which you don't understand. Good comment: "I don't understand why f(x + h) doesn't equal f(x) + f(h)." This is a very specific remark that will get a very specific response and hopefully clear up your difficulty.

Good question: "How can you tell the difference between the equation of a circle and the equation of a line?"

Okay question: "How do you do #17?"

Better question: "Can you show me how to set up #17?" (the Instructor can let you try to finish the problem on your own), or "This is how I tried to do #17. What went wrong?" The focus of attention is on your thought process.Right after you get help with a problem, work another similar problem by yourself.
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Text:
Precalculus with Limits, a Graphing Approach, 3rd edition. Authors: Larson, Hostetler & Edwards

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