Chapter 5




Summary of the Project


The purpose of this study was to determine the effect of instruction to promote mental computation on third grade students. A series of activities was designed to facilitate the development of different visualizations of numbers and their properties through varied number models and activities to aid in the development of number sense. These activities offered the students opportunities to develop creatively and share their strategies for computation in order to encourage mental computational development and promote number sense.


An instrument was designed, refined and administered as both a pre and post treatment measure. The treatment lasted a month and a half. Most of the activities lasted a total of 15 to 20 minutes of the mathematics class three to four times each week, after which the students received regular instruction. Four groups and two teachers participated in the study: two treatment and two control groups. One of the teachers had the two treatment groups and one control group. The other teacher had a control group. The groups were selected from a total of eight groups and were classified as two low achiever arid two high achiever groups (n = 103) according to the results of an achievement test provide by the district authorities to the investigator, whom administered the examination.


A random sample of the students in the upper half of each group (n = 16) was interviewed to determine: if they conserved, the types of strategies they used and their flexibility in the use of nonstandard strategies which could suggest the development of number sense.


It should be pointed out that, before the study, the students had never been exposed to formal instruction involving alternate ways of mentally computing. Neither of the two teachers who participated in the study, had experience in teaching mental computation strategies.


The teachers who participated in the study were interviewed before and after the treatment to determine if participation in the study had affected their knowledge or beliefs about mental computation, learning strategies and assessment. They were also given three questionnaires in the course of the study.


The questions to be answered by the study were:

1.        What is the effect of instruction upon student's achievement in mental computation as measured in a pre/posttest?

2.        What are the different strategies used by third grade students for mental computation classified as: standard, transition, nonstandard with no reformulation and nonstandard with reformulation (Markovits and Sowder, 1994)?

3.        Does the use of the instructional materials help to increase number sense in the students as related to the flexibility in the use of different computational strategies?

4.        Does the use of the instructional materials have any effect on the teachers' general pedagogical knowledge and beliefs about mental computation, learning strategies and assessment?




The pre/post treatment measures and the results of the on the clinical interview questions reflect that the treatment did not affect the high achiever groups significantly but seemed to have helped the low achiever group. This effect could also be attributed to the greater amplitude of their potential gain when compared to the high achiever group. The gains appeared greater for both treatment groups when the pre and posttest were compared by items.


The analysis of the clinical interview showed that, on the average, nonstandard procedures were the preferred strategies on the average by all groups (540). The most frequently used nonstandard procedure by the treatment groups and the high achiever control group was left to right computation. The low achiever control group used counting on as the preferred nonstandard procedure. The low achiever treatment group used nonstandard procedures 660 of the times as compared to the low achiever control group which used it 41a of the times and the high achiever control group 54%. The high achiever treatment group used nonstandard procedures with the same frequency as did the high achiever control group (54%), however, the high achiever treatment group used nonstandard strategies with reformulation twice as often as did the high achiever control group. These results suggest that the treatment could have been effective in promoting flexibility in the use of different computational strategies that could aid in the development of number sense in both groups, especially in low achiever groups.


From a constructivist view, mental computation is a higher order thinking skill, where the making of the strategy is as important as the using of the strategy. Choosing efficient strategies and inventing nonstandard ways that use number properties to a good advantage, are considered manifestations of number sense by recent studies.


The amount of change that took place after limited instruction (less than two months), may indicate that not a great deal of learning of new concepts took place. But rather that existing knowledge was used in new ways. Informal notions were recalled and new connections were formed. This is in keeping with the view of number sense as a well organized conceptual network of number information that enables one to relate numbers and operations to solve problems in flexible and creative ways. Similar results were, found by Markovits and Sowder (1994) after limited instruction with materials with a focus similar to those used in this study. They found that not only did the students use more nonstandard procedures after the treatment, but measures that were taken after a few months, revealed that the students were more likely to choose to use strategies that reflect number sense and that this was a long term change.


The teachers that participated in the study were found to have certain beliefs, two of which changed during the course of the study and one was added at the end of the study. These were, the meaning of number sense, which neither teacher understood before participating in the study, and the belief that mathematics was equivalent to basic facts and procedures. A belief that was added after the last interview was that "speed is important in mathematics."


The teachers were asked on various occasions about the meaning of number sense. While the treatment teacher received information about number sense, the control teacher did not receive any information from the investigator. It would seem that the control teacher looked for information elsewhere after having been asked about number sense in various occasions. At the time of the last interview both teachers understood clearly what number sense involved. At no time did the activities or the investigator emphasize time in the course of the study except when administering the timed mental computation instrument, for which 9 seconds proved to be sufficient. The teachers major concern seemed to be with the timed assessment of their students by external testing. They felt that their job was to prepare the students for these test; however, the treatment teacher indicated that this was not the sole purpose of teaching mathematics and that helping students to learn how to think was important. She felt that, in this way, mathematics was portrayed not only as procedures and basic facts but also as a way of thinking. This change could be attributed to the use of the treatment activities.




Several variations in the procedures could improve the study, such as a pretreatment interview with the students of the study in order to compare the preference of strategies before and after the treatment. An formal in-service training for the treatment teachers in the teaching of mental computations strategies and the development of number sense could be a useful improvement to the study because the teachers could apply what they learned to regular class instruction and not only to the 15 to 20 minute activity.


The results in this study suggests that teachers could help develop mental computation skills and number sense in low achiever groups with the use of the types of activities used for the study. It could then prove useful that more activities of this type be developed and tested for use with low achiever groups in English; however, the activities developed for the study could be used in bilingual mathematics education for low achiever groups. The treatment activities could also be used for a future investigation which analyzes the retention of the use of the nonstandard strategies over a six month period or longer.


The mental computation instrument could be used by the classroom teacher to determine mental computation skills in third grade students. The investigator suggests that the test could be incorporated into video or computerized form to facilitate its use by classroom teachers. The application of the mental computation instrument developed for the study to a larger sample, could provide benchmarks that facilitates data for curricular and instructional decision-making in the area of mental computation such as the performance level of third grade student in a certain school district. Using a larger sample of teachers, pretreatment shared beliefs could be studied in further detail to determine common views of elementary school teachers.


The study of mental computation is much in need of the testing of instructional materials that promote the use of student generated computation strategies. Viewing mental computation as efficient student invented strategies for computation, these materials could be used to contribute to the development of this higher-order thinking skill.


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