Chapter 3




The setting for the study


The setting of the study is a rural school in Ponce which is the second largest city of the island of Puerto Rico. Ponce is on the southern coast. The school, which is situated on the lower part of a very hilly area, is part of a system of two schools: a primary level (K-3) and a middle school (4-6) level. The school population consists of 514 students and 80 teachers with two principals.


The students who initially began in the study were 103 third grade students only two of whom did not take the posttest. The age range was from 7 to 9 year old and there were 57 females and 46 males.

In the community where the study took place ninety-nine percent of the families receive government financial and medical aid.




Four out of eight third grade classes were selected based on results of an achievement examination that was provided to the investigator by the school district supervisor. The investigator administered the test to the eight third grades and the selection of the sample was made using the highest and the lowest mean scores on the achievement examination. Two high and two low achiever groups were selected that best accommodated to the school's program. The treatment (t1 and t2) and control groups (c1 and c2) each had a high (t2 and c2) and a low (t1 and c1) achiever group. Two teachers participated in the study. One of which had three of the four groups (two treatment and one control group) and the second teacher taught one control group (see Table 1) .


Table 1

Design of the study


Low achievers

High achievers

Teacher A




Group t1

Group t2






Group c1

Teacher B




Group c2



The subjects for the clinical interviews from each of the four groups were selected at random from two envelopes for each group. Each envelope contained the names 5 students from the group, that attained scores close to the mean and 5

who attained scores of above one standard deviation from the mean. Two students were selected from each of the two

envelopes for each of the four groups for a total of 4 students per group. The total number of students interviewed

was 16, 8 high achievers and 8 low achievers.


Instruments and Procedures


Pre and Posttest


The pre and posttest was designed using the criteria of mental computation skills for addition and subtraction most commonly cited in the literature. The identical test was used both as a pre-treatment and a post-treatment measure. The examination consisted of three parts: one oral and visual (using transparencies) consisting of nine short word problems, the second part was completely oral consisting of 25 exercises and a third part which was oral and visual consisting 23 exercises similar to those of the second part. As can be seen in the criteria table in Appendix A, each item of the second part was matched to one of the third part to compare the two modes of presentation. A first pilot examination was given in June 1995 at a private school in Ponce, Puerto Rico. The group consisted of 18 second grade students. The second grade was selected for this pilot because it was the end of the year and the examination was intended for third graders at the beginning of the year.

The materials used for the examination were the following: student answer sheets, a portable overhead projector, 4 large white poster boards for a screen and a chronometer. The answer sheets were specially designed so that there was no room for written computation and to discourage students from copying. The examination was given by the investigator. An external evaluator did the timing for the test and did the test correction.


Before the second pilot testing, the instrument was given to a retired elementary school principal for Spanish grammatical revision and test evaluation commentary. The main change recommended was to not present the missing addends problems with a blank but instead with a picture in the place of the blank, because she considered that it would be confusing for the students. She also suggested that numbers be placed next to the sets of objects presented on the transparency, so that the students would not necessarily have to count them.


The second pilot examination was given to 140 third grade students at a local public school in the Ponce, Puerto Rico. It was administered at the beginning of the school year.


An item analysis test (Iteman) was applied to measure difficulty and discrimination of examination items. According to these indexes particular items were eliminated, modified or left on the test (see Appendix B and C for Iteman test interpretation scale and table sample).


During each stage of the examination revision an external evaluator who is a mathematics professor who is a working statistician in a university was consulted. The final version was used as the pre and post treatment measure (Appendix D).

After administration the pre and posttest results were analyzed to obtain the following data:

  1. An analysis for each student of gain or loss with respect to the pretest and the posttest.

  1. A t-test for independent samples to see if there was a significant difference among the means of the pretest and posttest for each group. (low achievers and high achievers).
  2. An analysis of variance (ANOVA) to determine statistically significant difference between group types.


The Interviews and Questionnaires


The interview questions for the teachers were selected as conversation openers so that the teachers would feel at ease with the investigator but at the same time begin the process of discovering their beliefs and understanding of mental computation, number sense, and other areas of pedagogy (Appendix E). The first questionnaire was constructed using the same questions that were included in the interview as confirmation (Appendix F). The second questionnaire included questions about the teachers background, in-service training, and questions about pedagogical strategies that they were using during this year (Appendix G). They were also asked about their participation in the study. The third questionnaire was constructed using statements about assessment that included some common myths or beliefs about assessment as pointed out by NCTM (1991) (Appendix H). Teachers were asked to write what they thought of each statement. The questions for the last interview were similar to those of the first interview but also included selected questions from the three questionnaire for purpose of comparison (Appendix I). They also were asked to evaluate the process of their participation in the study. A summary of the teachers' shared beliefs obtained through the analysis of the interviews and the questionnaires was prepared.


The questions for the clinical interview for the students were constructed in the following manner:

1.     Eight addition, five subtraction and one mixed operations problem types were selected from the criterion referenced table used to construct the pre/posttest (Appendix J),

  1. Problems parallel to those on the examination for the corresponding criteria were constructed,
  2. A conservation problem was designed to check if the students were in the concrete operational stage according to Piaget's theory of cognitive development. Each student was shown eight colored markers and eight rubber bands placed in two parallel rows so that each marker had a rubber band next to it. After the student had enough time to observe both sets, the chips and the rubber bands were removed and placed again on the table. This time the rubber bands were placed closer together and the markers were placed the same way as before. The student was asked if there were more rubber bands or markers, or was there the same amount. The purpose of the exercise is to determine if the student understands the law of conservation which is indicative that the child is in the concrete operational stage (which is usual for the third grade students).
  3. Recording materials were prepared and a table was designed to analyze the types of strategies the students used (Appendix K),
  4. Visual aids of each problem were prepared (see Appendix L).


To analyze the results, of the clinical interviews of the students, the percent of use of inefficient, standard and nonstandard procedures were compared. This clasification is defined in the following manner:

  1. Standard: The student uses a mental version of a pencil and paper algorithm or uses recall of basic facts.
  2. Nonstandard with no reformulation: The left to right process is used. (38+45=30+40+13)

  1. Nonstandard with reformulation: the numbers are reformulated to make the computation easier (38+45= 40+45-2).
  2. Inefficient invented strategies: the students use either a left to right or a right to left process but combine the place values incorrectly. (23 + 36 = 3+3+2+6).

Percents of students using standard and non-standard computational strategies were also computed. A clasification of the types of nonstandard strategies with and without reformulation was computed.


The Instructional Material


Theoretical Basis and Description


Throughout much of the literature reviewed in Chapter 2 on mathematics learning, a common theme kept revealing itself. This theme was the need for children to "invent" their own ways of "doing" mathematics. The activities for the treatment in this study were developed with a constructivist framework. Mental computation strategies were not taught. Throughout the activities, students were motivated to share their strategies, evaluate their own work, visualize and write about their thinking, in order to promote the development of number sense and mental computation skills.


The identification of patterns was a theme that was included in most of the activities for the treatment. There was also emphasis on connecting ideas presented in the different activities. Various visual models were used to help students develop number and number properties visualization skills. The models used were dominoes, playing cards, ten frames and painted wooden sticks (a variation of bean sticks). Game-like activities were frequently used. Place value concepts and addition of two digit numbers were developed using base-ten blocks. The hundreds table was used for pattern identification, addition and subtraction.


Materials Development


In order to begin the development of the materials for this study a database of games and instructional activities from diverse resources was collected. The database and the studies on addition, subtraction and counting were the starting points for the materials development.


The central themes to be included were patterns, computation using concrete and pictorial models, place value and computation using the 0 to 99 table (see Appendix M for Curriculum Plan). Using a list of mental computation skills developed through the review of the literature (see Appendix N) and the central themes, the activities were developed.


For the evaluation of the activities, the investigator developed an instrument to evaluate the material through classroom observation or by reading the activity (Appendix O). The instrument was developed originally in the form of a checklist based on one by Caine and Caine (1994) for use in the evaluation in the activities for the treatment. It was redesigned and given to a panel of four professors and a retired principal with an open questionnaire for evaluation purposes. After the copies of the instrument were returned, the suggestions were incorporated and the instrument was given a trail run in the evaluation the first five activities. This was done by a retired elementary school principal. These were evaluated and the instrument was also evaluated for the purpose it was used. Again the suggestions were incorporated and the instrument was used by two external evaluators, to evaluate the activities developed for the treatment in the study. Their suggestions were incorporated into the final design of the activities (Appendix P). The instrument was also used by the investigator and the treatment teacher for observation and evaluation of activities in class. These evaluations were useful in the refining of the activities in the classroom setting.


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