and Andrzej Ehrenfeucht, University of Colorado, andrzej.ehrenfeucht@colorado.edu
Abstract
Students taking math classes for future teachers at a university in the southwestern United States learned in one semester a novel multiplication algorithm for multi-digit numbers. To execute this algorithm students do not need to know the products of one-digit numbers, such as 7 times 8 equals 56. But the actual written computation is approximately twice as long as for the standard multiplication algorithm. The new algorithm is flexible. It gives the user options about how to proceed, and therefore it never becomes automatic, i.e., it cannot be executed without thinking. The students' progress and the amount they practiced were monitored. We present the novel algorithm, the level of mastery students attained, the types of errors they made, and their evaluation of the difficulty and usefulness of the new algorithm. The results challenge the common belief that a large amount of practice is needed in order to learn arithmetic algorithms. 1. Introduction The standard algorithm for written multiplication is difficult for students because it requires memorizing multiplication “facts” up through 9 times 9, and executing it puts considerable stress on the executive part of students’ working memory. They need to recall and process the next fact while remembering the result that was carried out from the previous step of the computation. The amount of school time spent on the algorithm, compared to the unsatisfactory results, suggests that teaching a different algorithm could produce a better outcome.
346 * 157 =
Standard Algorithm 2. A Different Multiplication Algorithm The algorithm described here doesn’t require one to memorize multiplication facts, and executing it puts a very small load on a student’s working memory. But it requires more writing, so it is significantly slower than the standard algorithm.
An Example of the New Algorithm: Multiply 346 by 157.
Step 1 The user may choose which number is the multiplicand (here we choose 157) and which is the multiplier (here 346). Step 2 Multiples of the multiplicand, 157, by 2, 3, and 6 are computed by doubling and adding. (Students are shown an efficient algorithm in which no multiplying is required.) 1 * 157: 157 2 * 157: 314 (double 157) 3 * 157: 471 (157 + 314) 6 * 157: 942 (double 417) (Any multiple of 157 by digits 1 through 9 can be computed singly or by adding at most 2 of them. For example, 8*157 = 2*157 + 6*157.) Step 3 The multiplier, 346, is viewed as the sum of two numbers that are written with the digits 0, 1, 2, 3, and 6. We choose 346 = 336 + 10 (other choices such as 323 + 23 are possible), and it is written in two lines. The pre-computed multiples of 157 from step 2 are aligned below the multiplier, and added:
The right-most zeroes may be omitted or included as the student wishes. Students also learn a “dot addition” algorithm for adding the numbers to yield the final product. This addition algorithm lightens the load on working memory (Baggett, Ehrenfeucht, Jeffrey, & Robles, 2008). The main differences between this algorithm and the standard one:
3. Subjects and Procedure Students in two elementary math classes for future and practicing K-8 teachers were taught this new algorithm during one semester. Enrollment in the two classes totaled 62 students. They practiced the multiplication algorithm approximately 15 minutes per session over 15 sessions (once or twice a week) spanning 10 weeks. In total, they practiced the new algorithm on 24 problems. During an additional session occurring in week six, they were given the first quiz; and during a session in the tenth week, they were given a second quiz and a questionnaire. Both quizzes and the questionnaire were anonymous and did not count toward a student’s grade. Problems on the Quizzes:
The Questionnaire:
1. Do you think that the dot multiplication algorithm is
useful? Yes No interesting? Yes No 2. Is dot multiplication easier or more difficult than the algorithm taught in schools? Easier More Difficult 3. Could you explain the dot multiplication algorithm to someone else? Yes I'm Not Sure No 4. Should children in school learn the dot multiplication algorithm? Yes No Why? or Why not? (open question) 5. What is good about the dot multiplication algorithm and what is bad about it? (open question) As mentioned above, at the beginning of the course students were also shown a “dot addition” algorithm that makes adding several multi-digit numbers easier, and they were shown an efficient algorithm for doubling a number. (Not all students took advantage of these techniques.) 4. Results Thirty-four students did both skill quizzes. Distribution of the sum of the number of correct answers on both quizzes:
Looking at the types of errors (conceptual or arithmetic) students made allows us to partition these students into three categories. (1) Students with scores 0 through 2 (3 students, 9%) These students did not understand the algorithm. (2) Students with scores 3 through 7 (23 students, 68%) These students understood the algorithm (21 of them already understood it at the time of the first quiz), but they were making a large variety of errors during adding and doubling. (3) Students with a perfect score of 8 (8 students, 24%) These students showed both understanding and skills in handling this algorithm. The average scores for the first and second quiz were 2.705 and 2.735 respectively (out of 4 possible; about 68%), showing that students’ skills in executing the algorithm did not improve from the sixth to the tenth week. Fifty students took the second quiz (34 took both quizzes, and 16 others took only the second). Five of the 50 did not understand the algorithm (they did not follow the instructions). The distribution of errors made by the remaining 45 is shown here:
The distribution of the number of correct answers (out of 4) on the final quiz:
Answers (yes or no) given on the first four questions on the questionnaire from 50 students:
1. Is the new algorithm useful?
Is it interesting?
2. Is it easier than the algorithm currently taught?
3. Could you explain it to someone else?
4. Should it be taught to school children?
The statement that the new algorithm is more difficult was positively correlated with all “no” answers. But its relation to the results of the second quiz is more complex. Distribution of the number of correct answers given by the 28 students who thought that the new algorithm was easier:
and by the 22 who thought that the new algorithm was more difficult or who gave no answer:
All five students who did not understand the algorithm, and who got all problems wrong, thought that the new algorithm was difficult, but when we exclude them from the distribution, the average scores of the two groups are the same, 2.96 and 2.97. Critical Comments The main criticisms of the new algorithm (from question number 5 on the questionnaire above) were, “It takes too much writing” and “It can be confusing”. (Both are true.) 5. Final Comments During one semester, with a minimal amount of practice, one third of the students became proficient in a new algorithm (they did 4 out of 4 problems on a final quiz), and another third made considerable progress (they did 3 out of 4 problems). Only the remaining third had considerable difficulties, either with understanding what to do or with the execution of a plan of action. This calls into question the wisdom of putting all students through years of repetitive practice, and demanding that they learn algorithms which require them to memorize a large number of independent and disconnected facts. Note: More details about this study are in the handout. Baggett, P., Ehrenfeucht, A., Jeffery, M., & Robles, L. (2008). An addition algorithm with lower cognitive load. Poster. 49th Annual Meeting, Psychonomic Society, Chicago, IL, Nov. 15. Sloane, T. O’Conor (1922). Rapid arithmetic. New York: D. Van Norstrand Company. lesson index |