HOW TO FACILITATE LEARNING IN MATH 1. Make materials meaningful. The number one question students ask is, Why do I have to learn this? It is important to relate math to real-life events. Balance rote learning with conceptual understanding. What is the difference between 3 x 2 and 2 x 3? Think about elephants and their trunks. In that context, the first equation depicts 3 elephants with 2 trunk each, and the second equation is 1 elephant with 3 trunks. What does it mean? and Why is that the answer? must be balanced with What is the answer? Teach first with concrete materials and words (concrete stage), then with pictures and words (semi-abstract stage), and finally only with words (abstract stage). Base-ten blocks are indispensable. 2. Use small sequential steps. Teach in patterns. Chunk information to build skills. 3. Elicit the correct response. Use verbal rehearsal (self-verbalization) strategies. What do I ask myself? Use mnemonics (memory aids). Give cues to prompt rather than giving the answer. 4. Consequate correct responses. Feedback should be immediate. 5. Teach in several contexts to promote generalization. The same self-verbalization cues should be used in the classroom as in the learning centre. As math skills develop, more than one skill is required to complete a question. Students don t always generalize the application of a skill; they have difficulty extending the skill beyond isolated practice. Conceptual understanding of a skill is prerequisite to its application in word problems. When a skill becomes part a complex question, the question must be task analyzed and its parts connected to previously mastered skills. 34 SEA Magazine, Fall 2001
Real-life examples of a concept should be varied. Apply math skills in other subjects, e.g., time lines in humanities and measuring in science. 6. Pro vide distributed practice. More is learned if one studies for less time each day than for more time one day. Some students require up to two hundred repetitions bef ore information is automatically recalled. Constantly review. If students don t consistently practice previously learned material throughout the year, many skills are lost by the year-end. 7. Pro vide overlearning Practice until the skill is automatic. 8. Have high expectations. Self-fulfilling prophecy: If less is expected of a student, the student will do and achieve less; if more is expected of a student, the student will do and achieve more. Once a student achieves success, he/she blossoms. This accomplishment positively affects other subjects. This article was inspired by John Salvia, Penn State University, and his class assignment on the seven steps to facilitate learning. Julia Fraser, 2000 VERBAL REHEARSAL Verbal rehearsal is a very powerful strategy that allows a student to become more actively involved in his/her learning. It activates the retrieval component of memory. It is very effective as a questioning technique. When steps are phrased into questions or statements, they answer students questions, What do I know?, What question should I ask myself? or What should I say? Verbal rehearsal strategies should follow conceptual understanding of the skill. Example: Adding & Subtracting Integers Signs Same: + Signs Different: - SEA Magazine, Fall 2001 35
1. If 2 signs are together without a number between them, first change them to 1 sign. a. Are the signs the same or different? If the signs are the same, change to + sign. If the signs are different, change to - sign. b. Are the signs the same of different? If the signs are the same, add (+) the numbers. If the signs are different, subtract (-) the numbers. 2. Take the sign of the largest number 1. Doubles, e.g., 3 + 3 Addition 2. Doubles + 1, e.g., 3 + 4 X X X X X X X The larger numeral is just 1 more than the smaller numeral. Do the numbers go together? (Are they in sequence?) If so, Double the smallest and add 1 more. 3. 1 Number Missing (Sharing), e.g., 3 + 5 X X X X X X X X If 1 from the larger group is given to the smaller group, then both groups will have the same amount. Is 1 number missing? If so, Double the missing number. 4. + 0 Have the child generate a rule for adding 0, e.g., The answer is always the other number because nothing is added to it. 5. + 1 Have the child generate a rule for adding 1, e.g., The answer is always the other 1 larger than the other number because it s just 1 more. 36 SEA Magazine, Fall 2001
6. + 10 Use base ten blocks to demonstrate the ones and stick of ten. 7. + 9 Use base ten blocks as for adding 10. Ten is too many. How can I get rid of 1 block? Put down the 10 (instead of the 9). Take 1 away from the other number. OR As 9 is just 1 less than 10, think of adding 10 when adding 9. If (for example) 10 + 6 = 16, then 9 + 6 = 15; it s just 1 less. 8. Numbers That Add to 10 8 + 2 and 7 + 3 are the only new facts. Tens (House of Games Corp. Ltd., Bramalea, Ont.) is a good game. 9. 1 More and Less Than 10, e.g., 7 + 4 and 7 + 2 If 7 + 3 = 10, the 7 + 4 is just 1 more and 7 + 2 is just 1 less. 10. Sharing Numbers 8 becomes 0, e.g., 8 + 4 = 9 + 3 X X X X X X X X X X X X the 8 gets larger, so the 4 gets smaller. 2 becomes 1, e.g., 2 + 5 = 1 + 6 X X X X X X X The 2 gets smaller, so the 5 gets larger. 7 becomes 6 (only 7 + 4) & + 4 = 6 + 5 X X X X X X X X X X X The 7 gets smaller, so the 4 gets larger. SEA Magazine, Fall 2001 37
1. Addition/Subtraction Relationship a + b = c Subtraction b + a = c (Although the answer is the same, the story is not.) c - a = b c - b = a Think of addition facts when subtracting. Check subtraction with addition. Have students make above four equations with three given numbers, e.g., 3, 8, 5. This is a good check for concept development. 2. - 0 Have the child generate a rule for subtracting 0, e.g., The answer is always the other number because nothing is ever taken away. 3. c - c = 0 Have the child generate a rule for subtracting a number from itself, e.g., The answer is always 0 because nothing is left. 4. - 1 Have the child generate a rule for subtracting 1, e.g., the answer is always 1 smaller than the other number because it s just 1 less. 5. c - a = 1, e.g., 6-5 Have the child generate a rule for subtracting a subtrahend which is 1 less than the minuend, e.g., The answer is always 1 because the starting number (or minuend) is just 1 more than the number being taken away (or subtrahend). 6. - 10, e.g., 15-10 Use base ten blocks to depict the 1 ten and the (5) ones. Just take away the ten. The (5) ones are left. 7. - Ones, e.g., 15-5 Use base ten blocks to depict the 1 ten and the (5) ones. 38 SEA Magazine, Fall 2001
Just take away the (5) ones. The ten is left 8. - 9 Use base ten blocks to depict the 1 ten and the (5) ones. Take away the ten (1 too many) and give 1 back. Take away the ten (1 too many) and give 1 back. 9. Doubles, e.g., 6-3 10. Doubles + 1, e.g., 11-5 11-5 = n 5 + 5 = 10 5 + 6 = 11 5 + 5 = 10. As 11 is 1 larger than 10, the answer is 1 larger than 5. 11. Doubles - 1, e.g., 11-6 11-6 = n 6 + 6 = 12 6 + 5 = 11 6 + 6 = 12. As 11 is 1 smaller than 12, the answer is 1 smaller than 6. 1. Addition/Multiplication Relationship Multiplication Multiplication is just adding groups of numbers (x means groups of). 2 x 3 = 3 + 3 = 6 3 X 2 = 2 + 2 + 2 = 6 Commutative Property - Although the answer is the same, the story or picture is different. 2. x 0 n x 0 = 0 SEA Magazine, Fall 2001 39
The answer is 0 because n 0 s added together equal 0. Have the child generate a rule. 0 X n = 0 The answer is impossible. Have the child generate a rule and an explanation. 3. x 1 n x 1 - n The answer is n because n 1 s equal n. Have the child generate a rule. 1 x n = n The answer is n because 1 n equals n. Have the child generate a rule. 4. 2 x These are doubles when added. 5. x 5 One just counts by 5. Relate to nickels and minutes. 6. x 9 Use finger trick The ten s digit in the answer is always 2 less than the number multiplied by 9. The digits in the answer equal 9 when added. (to 9 x 10) 7. x 10 Use base ten blocks to depict n 10 s. 8. Arrays 40 SEA Magazine, Fall 2001
9. Distributive Property of Multiplication Over Addition Rename multiplier 6 x 8 = (5 + 1) x 8 Apply distributive property of multiplication over addition. 6 x 8 = (5 x 8) + (1 x 8) 8 8 8 8 8 40 8 + 8 8 48 Division 1. Subtraction/Division Relationship Division is subtracting groups of a number until 0 is reached. 2. Multiplication/Division Relationship a x b = c b x a = c c a = b c b = a Think of multiplication facts when solving division facts. Check division with multiplication. Have students make four equations with three given numbers, e.g., 3, 24, 8. This is a good check for concept development. SEA Magazine, Fall 2001 41
3. Sharing 4. = How Many Groups Of? 6 3 = n How many groups of 3 can be made with 6? X X X X X X 5. 0 0 n = 0 The answer is impossible; if you haven t got anything, you can t share it or group it. Have the child generate a rule and an explanation. Division is not defined for the case in which zero is the divisor. 6. 1 n 1 = n The answer is n because n groups can be formed. Have the child generate a rule. 7. n n n n = 1 The answer is 1 because all are put into the 1 group. If n children shared n items, each would receive 1. Have the child generate a rule. 8. 2 Ref er to halves. 42 SEA Magazine, Fall 2001
STORY PROBLEMS 1. Classify/Categorize key words. 2. Construct vocabulary maps. 3. Cre ate oral problems. Relate to activities experienced by your students. 4. Have your students write their own problems and then exchange them. 5. Progress from simple to complex problems. Show that given information is often irrelevant by creating problems which include very unnecessary information. 6. Enact/Construct the problem. 7. Diagram the problem. PROFILE Julia Fraser, M.Ed. (learning disabilities) has taught for 23 years. For 19 years she has taught students with learning disabilities in district elementary, multi-aged programs, for students with severe learning disabilities, and in elementary, junior secondary, and middle school integrated learning assistance and resource programs. She is a past executive member of LATA. Currently she is on leave from Coquitlam School District and operates a private math consulting and tutoring business, Math Advantage, for students in K through grade 10. Students severely disabled in math when taught basic arithmetic facts by Julia, following the information presented, have mastered them with long-term memory retention. Many of her students, formerly on modified math programs in school, have been able to complete regular math classes, some with A and B marks, with little or no learning assistance. SEA Magazine, Fall 2001 43