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Strategies for Math

Try to identify which of these areas cause you difficulty.

• knowledge of basic math concepts (numeration, place values, etc.)
• calculation of basic math processes
• detecting errors in calculation
• knowledge of fractions, decimals, percentages, and ratios
• application of math concepts in word problems
• solving algebra problems
• geometry

Frequently, students do not understand why they are having math problems. Students may struggle with one or more areas of math including computational math, fractions, applied word problems, geometry, and/or algebra. Cognitive processes that may underlie one or more of these areas of math difficulty could include poor abstract reasoning skills, visual spatial difficulties, sequencing problems or slow processing of information which has limited the learning of prerequisite knowledge for college math.

Learning math is more challenging for some students because of

• their difficulty in understanding math vocabulary (for example, set, numeral, proportion),
• poor auditory memory and word retrieval,
• difficulties in understanding and using syntax, and/or
• problems in explaining how they reached a solution.

Planning and organizational skills, and attentional factors also can impact math skills. It is important for students to understand the nature of their math difficulties and the reasons underlying them, so they can explain their needs to instructors and tutors, and choose the most helpful learning strategies.

GENERAL STRATEGIES AND ACCOMMODATIONS

1. To improve math skills, students should begin by consulting with their learning disabilities services coordinator to select the most appropriate courses and learning strategies. Consider taking a remedial math course or using tutorial services, if you feel you need help with basic math concepts and calculation. Many students will benefit from increasing their understanding of decimals and fractions, percentages, ratios, proportions and signed numbers, with the focus on solving word problems. An introductory course which includes algebra, with an emphasis on reasoning and solving word problems rather than on calculation, may be helpful.
2. To maximize the benefits of tutorial services, work with your tutor to define the aspects of math that are creating obstacles, then develop strategies to overcome or circumvent them. Think about how to highlight your strengths and use them to compensate for areas of difficulty.
3. It may be appropriate to request extended time on tests, and to take your test in a distraction- free setting.
4. Allow time to practice solving problems. Try to figure out where you are having difficulty, so you can request help. Ask for or create extra practice tests and homework assignments.
5. Identify a sample problem to illustrate each mathematical operation or type of problem you have learned. Organize these samples into a reference guide for study and homework.
6. If you are stronger verbally, use verbal mediation to talk your way through the steps of a problem. Write the steps on a cue card and use the card to work practice problems.
7. If visual cues help you, try the following strategies:

1. Draw or diagram problems before you solve them,
2. Make flash cards to help you recall signs, math facts, or equations,
3. Create mental images of material you want to remember, and create mental pictures of problems, and/or
4. Turn ruled paper horizontally to create columns for math, if you have difficulty keeping numbers aligned correctly.
5. If reading rate or comprehension are difficult, see if there is a taped version of your math textbook that you can follow.

STRATEGIES FOR ENHANCING BASIC CALCULATION SKILLS

1. Make a concerted effort to memorize math facts for addition, subtraction and multiplication. Work to understand the logic behind the answers, by looking for patterns. Allow time for repeated drill of math facts and calculations. The rapid automaticity you may achieve will allow you to allocate time and mental resources to higher level processes.

1. A game-oriented approach to fact learning may be productive. For example, using number cards or dice, pick a sum (addition) or a product (multiplication) and see how many different cards or dice can be used to create that answer. For example, the sum can be achieved by adding 1 + 8, 2 + 7, 3 + 6 etc.
2. Record math facts on a tape recorder; then play the tape, trying to give the correct response before you hear it on the tape.
3. Use relational thinking when solving computational problems. For example, to learn the addition facts from 10 to18, think 8 + 8 = 16, so 8 + 9 must equal 17, because 9 is 1 more than 8.
4. Try to learn as many ways as possible of solving a given type of problem, so that if you forget one way, you'll have an alternative. For example, 3 x 4= 2 x 4 + 4.

2. Use a calculator, when allowed, for quick, accurate computation. A talking calculator may be helpful for students who profit from auditory input.
3. If monitoring computational and process errors is a weakness, use a checklist to monitor math work for computational and process errors. The items on your list would, of course, depend on your particular difficulties.

To prevent careless errors in computation, the following strategies may be helpful:

1. Put the number you are carrying or borrowing in a circle
2. Whisper the problem and procedures to yourself as you do it;
3. Circle signs (=, -, x);
4. Use slash lines after borrowing;
5. Use procedures for checking your answers. For example:

	512 -400 112
Checking procedure	112 +400 512

6. Recheck to be certain that you followed all the steps required, and in the correct sequence;
7. Recheck signs and calculations for addition, multiplication, subtraction and division;
8. Check to be sure columns of numerals are lined up correctly, so you won't make an error like,

342 +43 772

9. Check to see if your answer makes sense based on a rough estimate;
10. Spot check by redoing some problems to see if you get the same answer the second time; and
11. Try to put the problem into a real-life, meaningful context. For example, for 5 + x = y, you might say "If I have 5 donuts, and I want to give one to each member of my class of 15 students (y), how many more do I need to buy? (x) Also, make up a real-life problem, and write an equation which describes it.

STRATEGIES FOR ENHANCING APPLIED MATH, GEOMETRY AND ALGEBRA SKILLS

1. In doing applied or word problems, determine the essential information, and make a list of the steps to follow in sequence in finding the solution.
2. Before solving word problems, think through the following steps:

1. Understand the problem. What do you know? What do you want to know?
2. Make a plan.
3. Use the plan.
4. Check to be sure your answer makes sense.

3. Have someone make up problems with extraneous information. Identify the irrelevant information. For example, there are 10 people working the evening shift at a restaurant. Six women have ordered steak, and three men have ordered salads. How many people have placed orders? (extraneous information includes "10 people," "men," "women," "steak," and "salad").
4. Have someone make up problems with insufficient information. Identify the missing essential information. For example,"Bill has six more CDS than Charla. How many CDS does Bill have?" (You must have the number of CDS owned by Charla before you can solve the problem).
5. Concepts of money, time and measurement may need to be strengthened. You may be able to find workbooks that provide practice, but often self-developed problems are even more effective. You will increase your understanding of math concepts both when you create a problem and when you solve it.
6. With respect to practical money skills:

1. If making change is difficult, .practice with real money, writing down the problems and responses as you do them.
2. Create and solve problems using a sample checkbook. You may want to work with someone initially who can check your accuracy.
3. Figure out how much money you need to put in an envelope each day in order to have enough to buy a particular item in ten days. Did you have enough? Did you remember to include tax?
4. Purchase or make a variety of denominations of postage stamps; then determine what combination of stamps you need to have sufficient postage for letters of varying weights.
5. Use catalog purchases and menus, for example, as opportunities to tone your skills.

7. To enhance time concepts, consider using the following strategies:

1. Use a TV guide to develop problems. For example, how long a video tape will you need to tape three particular shows? If it is now 1:30, at what time will a 2 hour and 30 minute show end.
2. Use a watch with a second hand to help you achieve a more concrete understanding of time concepts.
3. Estimate how long it will take you to complete a task; record the starting and ending time, and calculate the interval. Were you accurate?
4. Use bus, train and plane schedules for practice as well as real opportunities.

8. To develop measurement concepts try the following exercises:

1. Measure and chart the height of family members or household objects.
2. Compare light and heavy objects around your house, and create a graph from lightest to heaviest.

9. When studying geometry, make flash cards and over learn the formuli for areas, circumference, perimeter, etc. Then create and solve problems, initially using real objects such as the area to be covered by wallpaper or the amount of lawn to be fertilized.

REFERENCES

Johnson, D. Myklebust, H.R. (1967). Learning disabilities: Educational Principles and Practices. NY: Grune & Stratton.

Spiers, Paul A. (1987). Acalculia revisited: Current issues. In G. Deloche & X. Seron (eds..), Mathematical disabilities: A

cognitive neuropsychological perspective. (pp. 1 - 25) Hillsdale, NJ: Lawrence Erlbaum Associates.

Starke, M.C. (1993). Strategies for college success, 2nd edition. Englewood Cliffs, NJ: Prentice Hall.

Tobias, S. (1978). Overcoming math anxiety. Boston: Houghton Mifflin.

Wren, Carol T. (1983). Language learning disabilities: Diagnosis and remediation. Rockville, MD: Aspen Systems Corporation.

Vogel, S. A. & Adelman, P. B. (1993) Success for college students with learning disabilities. NY: 4. Springer-Verlaag.

ERROR ANALYSIS*

Most math errors can be described using the following categories. It will be helpful to know what types of errors you most commonly make.

• Basic Fact Errors
• Table value
• Retrieval of an incorrect table value
• Zero/Identity

Basic fact errors that appear only when a 0 or 1 is present in the problem being computed.

• Symbol Errors
• Loss of symbols
• Omits the symbol. May still compute correctly.
• Substitution
• Writes an incorrect sign. May still compute correctly.
• Rotation
• Rotates the "+" so it becomes "x" or vice versa. May still compute correctly, or may perform incorrect procedure.
• Algorithm Errors
• Incomplete
• Initiates correct operation, but fails to carry out all necessary steps.
• Incorrect alignment

Does not accurately line up the elements spatially in order to solve the problem correctly. For example, misaligns the columns in the intermediate steps of multi digit multiplication or division.

• Incorrect sequence

May complete computations correctly, but the sequence in which they are carried out leads to incorrect carries, borrows, and intermediate products. For example, may work from left to right, rather than right to left.

• Subtraction inversion

The minuend and the subtrahend are reversed in the act of computation. For example, 15 - 6 = 11.

• Inappropriate

The problem is written in the manner of another operation, typically multiplication substituted for division.

• Substitution

Executes a different operation than the one requested by the sign, for example, a multiplication problem is added.

• Confounded

Substitutes parts of different operations within the same problem.

• Inconsistent

Uses incorrect, inconsistent, or idiosyncratic procedures, or fails to access any correct computational strategy.

• Place-holding Errors

Number value

Unable to distinguish the larger of two numbers due to a lack of understanding of units, tens, hundreds,etc., for example 835 vs. 384.

• Number expansion

Incorrectly expands tens, hundreds, etc., for example 3,529 is written as 3000500209.

• Mirror reversal

Digit sequence is written, copied, or repeated in reverse, for example 835 is written as 538.

• Partial reversal

Only one set of digits is involved in reversal, for example, 3529 is written as 5329.

• Digit errors

A digit is written incorrectly, or a digit is omitted from the computation. A digit may be taken from the problem and incorrectly inserted in the solution.

• Borrow and Carry Errors
• Neglect of Carry

Fails to use a verbalized or clearly indicated carry in the process of computation.

• Defective Carry

All the digits of an intermediates solution are written and the higher place-holding digits are not carried, though they still may be added to the next column in computation.

265 +149 711

• Incorrect Placement

The carry is made, but is added to the wrong column.

• Wrong Carry

The smaller rather than larger place-holding digit from the sum of the previous column is carried and added to the next column.

265 +149 711

• Zero carry/borrow

Error is made due to confusion of the carry/borrow process when a zero is involved.

• Neglect of Borrow

The leftmost or higher place-holding digit is not reduced after a clearly verbalized or indicated borrow.

265 -149 126

• Defective Borrow

The borrowed amount is added to the lesser place-holding value

265 -149

Borrowing from a lesser to a greater place-holding digit

265 -149

*Spiers, Paul A. (1987). Acalculia revisited: Current issues. In G.Deloche & X. Seron (eds.), Mathematical disabilities: Acognitive neuropsychological perspective. (pp.1-25) Hillsdale, NJ: Lawrence Erlbaum Associates.