Project Math Access

It is very important that students see mathematics, and the calculations they perform, as part of their daily life. Providing opportunities to apply basic concepts and operations in daily activities will reinforce students' skills and motivate them to progress in mathematics. They can use addition to figure total amounts of toys or snacks, and to keep track of their bank accounts or team equipment. Students can use subtraction to make comparisons between what they have and what they need for a game or other activity, to budget, and to calculate remaining items as they are used, or to calculate change when a purchase is made. They can multiply to figure larger totals, and to transform units from one measure into another. They can divide to determine equal portions of items, or to figure daily averages for sports scores or percent scores for quizzes or games.

In order for students to calculate using these four basic operations, they must first have developed basic concepts (including more, less, many, etc.), one to one correspondence, the concept of sets, and basic number sense. As students begin to learn to calculate, the following teaching considerations should help:

- Emphasize concept development rather than process or rote memorization.
- Apply operations to real life situations which are of interest to the student (e.g., provide opportunities for students to determine quantities of materials needed to play a game or complete a project and to estimate the price to purchase these materials). At first, provide examples for the student, then ask the student to provide his or her own examples which he or she sees as relevant uses of different operations.
- When students are using manipulatives, encourage them to search the entire "field" to make sure they are aware of all the objects with which they must work. Using trays or mats can help to identify this field and the area they must search.
- Word problems are very effective since they involve practical application of skills. To assist students in developing the skills necessary to solve word problems, it may be helpful to provide a problem solving model. First, identify the specific kinds of information needed in a particular problem; then provide two or three choices of operation statements to solve the problem. Eventually, students will be able to identify appropriate operations independently.
- Teach the concept of complements or partners for addition, subtraction, multiplication and division. For example, the number 5 is made up of 2 and 3, 1 and 4; 24 is made up of the factors 8 and 3, or 2 and 12, etc. This concept not only increases the student's ease with number facts; it also facilitates mental mathematics.
- When teaching facts, focus on 2 or 3 related facts at a time. Emphasize accuracy first, then speed. Maintain a chart of mastered facts to help the student recognize progress.
- Small flip charts can be provided at a student's desk, with cues for steps in particular types of problems as a reference or reminder.

- The number line (APH) can be used for working on operations, relationships, fractions, and decimals. Number lines are especially useful when they are stretched across the top of the student's desk, less helpful when confined to the dimensions of a braille page. The number line attached to the student's desk is particularly helpful when the student is using the braillewriter. Students can find the larger number, count forward for addition and count backward for subtraction. Students can use number lines for working on positive and negative numbers as well. Thermometers can also be useful in teaching positive and negative numbers.
- Flashcards can be used along with manipulatives for working on the basic operation facts and the Nemeth Code; later, students can braille out their own mathematical sentences.
- Students having difficulty learning addition, subtraction, multiplication or division facts can make cards of those facts with which they are having the most trouble. Write the problem on one side of the card and the answer on the reverse side. Teachers can use creativity and relevant examples, scratch and sniff stickers, etc. to make these teaching aids interesting. Many models of these types of aids can be found in teacher stores.
- To practice mathematics facts, students can roll dice, then add, subtract, multiply or divide the numbers thrown. A variety of rules can make this into a game. For example, the first student whose cumulative numbers add up to 100 wins, or students start from 100 and subtract numbers thrown and the first one reaching zero wins, or students multiply the two numbers thrown and then add these numbers cumulatively. Students could also rename fractions formed by combining the two numbers thrown.
- "Start Here. ..End There!" (Petreshene, 1985): give the student a paper with a pair of numbers brailled on the first line. The student then determines at least 3 ways to get from the first number to the second, brailling each series of computations below the number pair. He or she may use any combination of addition, subtraction, multiplication, and division. For example, if the designated numbers are 3 and 24, some possibilities would be:
3x8 = 24

3x9-3 = 24

3+10+10+1 = 24

- Mad Minute, available on disk, can be a motivating way for students to practice basic operations.

While the above tips relate to teaching any of the basic arithmetic facts, the following suggestions could be especially appropriate for working on addition and subtraction:

- For students who have great difficulty remembering number facts, an additive principle may be helpful. For example, teach the "doubles" (2+2, 3+3) for all facts up to 10. These can serve as main facts from which other facts may be derived (as in the problem "2+2+1" or "one less than 4+4").
- Magnets can be used on a small cookie sheet or magnet board to form groups for addition or subtraction problems. Unlike the manipulatives usually found in primary level classrooms, individual magnets can be moved easily without falling off the surface. At a more advanced level, a cookie sheet or magnet board could be used as a personal "chalkboard" where individual tiles labeled with Nemeth Code numbers and signs of operation (and affixed to magnetic tape) could be arranged to form a variety of problems. Wikki Stix could be used for separation lines. The student could work on a variety of problems at his or her desk while the teacher works the problems at the board. Classroom teachers would have to cooperate by verbalizing the problems clearly!
- The work tray from the American Printing House for the Blind can serve as a good organizer for making simple addition and subtraction statements. For example, a collection of small manipulatives can be placed in the larger section on the left of the tray. Four of these objects could be placed in the first of the three smaller sections and 2 more placed in the second section. The student could then take the objects from both sections and place them in the third section, counting them for the total of 6. The same procedure could be used for subtraction statements. Problems involving the addition or subtraction of zeros could also be worked out in a very concrete manner using this approach.
- To teach addition to young children, cubes that attach to each other (e.g., Unifix cubes) can be an effective aid. The student can be given a specific number of cubes, and asked to count them; then the student can be given a second group of cubes which are counted and attached to the first group. The total number of cubes can then be counted. Beads on a string can also be used by giving the child a string with first 2 beads of one shape (circles), then 3 beads of a different shape (squares). The student reads the problem from left to right (2+3=5). Subtraction could also be practiced by presenting the combined amounts first, and then having the student remove the particular number of cubes or beads (5-3=2).
- When setting up spatial arrangements involving regrouping (i.e., carrying and borrowing) or cancellation, a frame with 9 cubes arranged in a vertical line could be used as an aid. As the student fills the frame with counters, he or she can see that if another place is needed for a sum larger than the 9 available, he or she must place it above the frame. A separated section above the frame could be added to hold the carryover number.
- Students can work on the concepts of "greater than" and "less than" while they work on addition and subtraction. For example, two numbers (8,13) could be given to the student in a particular sequence, and the student would have to state the relationship of the first number to the second (8 is less than 13). Next, the student could use the operations of addition or subtraction to solve problems (e.g., how much less is 8 than 13? What number must we add to 8 to make it equal to 13? What number must we add to 8 to make it greater than 13?).
- The game "Mystery Coins" (Petreshene, 1985) provides practice for basic mathematics facts and money skills. Place a variety of coins in a paper bag, and announce the total value and number of coins (e.g., 35 cents; 8 coins). The student guesses which exact coins are in the bag. When he or she states the correct answer (in this example, it is 3 dimes and 5 pennies), he or she counts the actual coins for verification.

Following are several suggestions that might be especially helpful for teaching multiplication and division:

- Remind students of these "tricks" that can help them to remember multiplication facts:
Zero times any number is zero.

One times any number is the number itself.

Nines are "magical": the answer to the nines facts always add up to

nine (e.g., 9x8=72; 7+2=9).

- Construct a list with the numbers 0-9 in the left column, and the numbers 9-0 in the right column. The resulting number combinations will be the answers to the facts from 9x1 to 9x1:
x1: 09

x2: 18

x3: 27

x4: 36

x5: 45

x6: 54

x7: 63

x8: 72

x9: 81

x10: 90

- Start with an additive approach, using groups of manipulatives (e.g., 2+2+2= 3x2, 4+4+4+4=4x4, etc.).
- Stress the associative principle, having students make and remake statements regarding different arrangements of objects (e.g., using an egg carton, students can see that 6x2=12 and that 2x6=12).
- Students can use card games with factors (3x8) on one card and products (24) on another to make a matching pair.
- When multiplying different numbers by the factor 9, students can use the "finger trick". First, the student places both hands out in front of himself or herself, palms down, and counts the fingers as 1-10 starting with the little finger of the left hand. Then the student folds under the finger corresponding to the multiplier of nine, and reads the product by reading the number of fingers before the finger folded under, followed by the number of fingers following the folded finger. For example, in the problem 4x9, the student would fold under the index finger of the left hand because that is the 4th finger in the sequence. The student would then read the answer as 36 because there are 3 fingers before the folded finger, and 6 fingers after the folded finger.
- Students may find "multi-blocks" useful. All the relevant multipliers, multiplicands and products are labeled in Nemeth Code on the different sides of the blocks. Students can make appropriate matches by arranging the blocks so that combinations of factors and products face up.
- To practice both multiplication facts and the associative concept, students can play a tic-tac-toe game. Each square of the tic-tac-toe board contains a multiplication problem without the answer; factors should be presented in both orders but in different squares (6x8, 8x6). Students must find 3 combinations in a line which yield the same product. This activity could also be used for practicing addition facts, or other combinations of basic facts.

Petreshene, S. S. (1985). *Mind joggers! 5 to 15 minute activities that make kids think*. West Nyack, NY: The Center for Applied Research in Education, Inc.