** **

This chapter presents the results of the analysis of
the data obtained in the study. The purpose of the study was to determine the
effect of instruction to promote mental computation in third grade students.
The questions to be answered 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, inefficient, nonstandard with no
reformulation and nonstandard with reformulation?

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?

In order to answer the first question, the students
that participated in the study were administered an examination that was
constructed and refined by the investigator. This examination served as a pre
and post treatment measure. Appendix Q presents the scores obtained by each of
the four groups, the gains obtained by calculating the difference between the
pre and post examination score, and average score for each group. The table
also presents the number of students that took the test in each of the four
groups, which was an total of 103 students. The examination had a total of 57
items.

When the pre and posttest scores of each of the four
groups was compared using a t test for equality of means, each group obtained a
significant gain when compared to itself as can be observed in Table 2.

An analysis of variance (ANOVA) was used to compare
the mean scores of the low achiever treatment and control groups and the high
achiever control and treatment groups as can be seen in Table 3.

__Analysis of Variance (ANOVA) Between Group Types on__

It can be observed from Table 3 that there was no
significant difference between the two low achiever groups or between the two
high achiever groups on the pretest. This suggests that the two low achiever
groups and the two high achiever groups started out the same. However, on the
posttest when comparing the two the low achiever groups there was a significant
difference, while there was no significant difference between the two high
achiever groups. This could be due to fact that the high achiever groups had
less to gain because they were closer to the ceiling level; on the other hand
the lower achiever groups had more to gain because they were further from the
top.

Analyzing the gain in each item for treatment versus control group for low achievers and high achievers, on pretest compared to posttest (Table 4), the "greater gains" referred to were determined using the criterion of a difference between the posttest and the pretest of 8 students or more, answering the item correctly which represents 25% or more of students from any group.

It can be observed that on 65% of the items the 57
items, the low achiever treatment group (t_{1}) obtained a "greater
gain" (25% or more of the students) than the low achiever control group (c_{1}). On 9% of the 57 items
both groups (t_{1} and c_{1}) obtained equal gains. In
12 items (21%) the treatment group (t_{1}) did not have gains when
compared to the posttest and the control group (c,,) did not have gains also in
12 items (not necessarily the same). In particular in Part A, the low achiever
treatment group (t_{1}) obtained a "greater gain" (25p or more
of the students) than the low achiever control group (c_{1}) in the item (#6) which is
a missing addend word problem; in Part B on items #6, 9 and 19, which deal with
one digit combinations of ten, the addition of more than two addends, and mixed
operations using multiples of 10; and Part C on items # 7, 8, and 11, which
deal with addition of more than two addends, skip counting and one digit
subtraction involving 9. The low achiever control group (c_{1}) did not obtain a
"greater gain" than the low achiever treatment group (t_{1}) on any item.

In 49% of the items the high achiever treatment
group (t_{2}) obtained "greater gains" (25% or more of the students) than
the control high achiever group (c_{2}). Both groups (t_{2} and c_{2}) obtained equal gains in 7%
of the items. In 18 items (320) gains were not obtained by the treatment high
achiever group (t_{2}) while for the control high achiever group (c_{2}) no gains were observed in
22 items (390). In particular the treatment high achiever group (t_{2}) obtained a much greater
gain in item #12 of Part B which deals with one digit subtraction. The high
achiever control group (c_{2}) did not obtain a "greater gain" in any
item.

The second question to be answered by this study pertains
to the types of strategies the students use. To answer this question the
results of a clinical interview after the treatment with a sample of students
from each group was used.

The types of strategies used by the students were
classified as standard, nonstandard with and without reformulation according to
the Markovitz and Sowder (1994) study defined in Chapter 1. The standard
strategy classification was redefined in Chapter 3 to include recall of basic
facts. The transition strategy classification was not found and the inefficient
invented strategy was added. This category is similar to one used by Stevens
(1992) as referred to in Chapter 2.

The
questions used in the interview were constructed parallel to those items in the
mental computation examination that seemed to be more difficult. This can be
observed in the criteria table in Appendix 15.

__Frequency and Average Percents of Strategies Used by
Students in Mental Computation in Posttreatment Clinical Interview__

__Note. __Percent of use by group was calculated dividing the
frequency of use of the strategy by 56 (4 students x 14 items). Percent of
average use was calculated dividing total use of the strategy by total of
possibilities of use by all the students (56 x 4 groups).

Table 5 shows the average percentage of strategies used by students on the clinical interview. The kinds of inefficient strategies used by students, on the clinical interview questions, were of two types. One of the types of inefficient strategies was left to right computation in which units were combined with tens and tens with units in addition or subtraction depending on the problem These results were then added or subtracted. For example for item #8, some students did the following: 23 + 37 -> 3 + 3 + 2 + 7. A second type of inefficient strategy used by some students was right to left computation which was used in the same manner. For example, for item #5, some students did the following: 39 - 24 -> 9 - 2 = 7, 3 - 4 = 1 so 7 + 1 = 8. This could be attributed to a lack of work with mental computation, not understanding what two digit addition is about or very little exposure to addition placed horizontally or in word problems.

According to the way "standard strategies"
was defined for the study these include the use of algorithms and recall (a
lower level thinking skill) of basic facts. The total frequency of the use of
standard strategies was according to Table 6, 31% of all strategies used by all
groups. This finding is similar to that of Carraher and Schiemann (1985) who
found that the preference of school taught algorithms was limited to 34% in
addition and 24% in subtraction. They also noted that when school taught
algorithms were used there were frequent errors. This can also be observed in
the types of inefficient procedures used.

The use of recall of basic facts can be observed in
Table 6. It can be noted from Table 6 students used algorithms over recall of
basic facts in all groups in the use of standard strategies.

__Note.
__Percents
were calculated by dividing total use of the strategy by the total possibilities
of use by all the students (14 items x 4 students x 4 groups). C_{1} and T_{1} are low achiever groups and
C_{2} and T_{2 }are low achiever groups.

__Frequency of Use of Nonstandard Strategies Without
Reformulation by Students in Mental Commutation on Post Treatment Clinical
Interview__

__ __

__Note.
__C_{l} and T_{1} are low achiever groups and
C_{2} and T_{2} are low achiever groups.

It can be observed from Table 7 that across the groups the nonstandard strategy without reformulation most used by the students on the post treatment clinical interview of the thirteen strategies found was left to right computation. Left to right computation accounted for 46% of all the nonstandard strategies without reformulation used and 19% of all the strategies used by all the groups. This can be compared to the use of standard strategies (Table 5) which accounted for 300 of all the strategies used by all groups, inefficient strategies which accounted for 26% and the use of all the nonstandard strategies were 540. It can be noted through Table 7 that left to right computation was used on the average more by the treatment groups than by the control groups (lea to 6a). The low achiever treatment group used left to right computation more than any of the other groups. Left to right computation was the nonstandard strategy most used by the two treatment groups (low and high achievers) and the high achiever control group. The low achiever control used the "counting on" strategy more frequently. The more frequent use of left to right computation by the treatment groups could be attributed to the use of base ten blocks in the treatment activities.

__Frequent of Use of Nonstandard Strategies With
Reformulation by Students in Mental Computation on Post Treatment Clinical
Interview__

__Note.
__C_{1} and T_{1} are low achiever groups and
C_{2} and T_{2} are low achiever groups.

Observing Table 8, it can be noted the left to right computation with reformulation was on the average the most frequently used strategy. The high achiever treatment group used it more frequently than any other group as can be noted. This finding is similar to a those in a study by Hope and Sherill (1987) which suggested that a common characteristic of proficient students was their tendency to perform a calculation in left to right fashion and the tendency to incorporate progressively the interim calculations into a single result. This can be observed in some examples of the nonstandard with reformulation procedures used by the students for addition. On item #9, 28 + 26 + 2, a student did the following: 2 + 2 = 4, 8 + 6 = 14 + 2 = 16, 40 + 16 = 56. For item #13, 29 + 32, a student used: 30 + 30 = 60 + 1 = 61. Examples of reformulation procedures in subtraction are for item #14, 54 - 30, was calculated as 50 - 30 = 20 + 4 = 24 and item #4, 23 - 12, was calculated as 12 + 12 = 24 - 1 = 23 so 12 - 1 = 11.

The third question to be answered by this study is:
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?

Returning to Tables 5, 7 and 8 it
can be noted that both treatment groups used on the average nonstandard
procedures more frequently than the control groups. The treatment groups also
used on the average inefficient procedures fewer times than the control groups.

__Frequency and Percents of Nonstandard Strategies
Used in Mental Computation bar Students in Post Treatment Clinical Interview__

__Note.__ Percents = Total use of
strategy / 56 (4 students x 14 items) . C_{1} and T_{1} are low achiever groups and
C_{2} and T_{2} are low achiever groups.

Table 9 shows the frequency in which samples of
students from each group used nonstandard strategies. It can be noted that on
the average students in the treatment groups (t_{1} and t_{2})
used nonstandard procedures without reformulation more frequently than the
control groups. The low achiever treatment group (t_{1}) used nonstandard procedures of both type on the average
more frequently than the high achiever control group (c_{2}) and the low achiever control group,
as can be noted on the Tables 9. Even though the treatment high achiever
control (c_{2}) group used
nonstandard procedures with the same frequency as the high achiever treatment
group (t_{2}), the high achiever
treatment group (t_{2}) used
nonstandard procedures with reformulation almost twice as much as the high
achiever control group (c_{2}).
These results may be indicative of greater flexibility in the use of strategies
that reflect number sense in both treatment groups.

Some examples of the nonstandard with reformulation
procedures used by the students for addition were for item #9, 28 + 26 + 2, a
student used 2 + 2 = 4, 8 + 6 = 14 + 2 = 16, 40 + 16 =56. Another examples is
for item #13, 29 + 32. A student used: 30 + 30 = 60 + 1 = 61. Examples of
reformulation procedures in subtraction are: for item #14, 54 - 30, a student
used 50 - 30 = 20 + 4 = 24; for item #4, 23 - 12, a student used 12 + 12 = 24 -
1 = 23 so 12 - 1 = 11. In the problem where the students had to calculate how
much money there was in an illustration of eight 5 cent coins, most students
used skip counting while other made arrangements to facilitate the calculation.
One arrangement was 15 + 15 + 10. Another arrangement was 25 + 15.

The fourth question to be answered by the study was:
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
preliminary conclusions were drawn from the analysis of the first interview and
the questionnaires applied to the two teachers who participated in the study.

It
was found that the following were shared beliefs that were concluded from the
answers of both teachers.

- The teachers did not
understand what number sense was.
- They considered that
they had to teach for the standardized examinations the students had to
take.
- The teachers considered
that mathematics was equivalent to basic facts and procedures.
- They believed that a
student's social behavior can reflect his not understanding.
- The teachers considered
games were good teaching strategies.
- They did not understand
how mental computation skills could be developed.

Questions pertaining to all the shared beliefs on
the preliminary list were asked on various occasions during the study in the
first interview and in the questionnaires used in the course of the study.
After the last clinical interview two of these beliefs seemed to have changed
in both teachers. These were the first and the third of these shared beliefs.
Both teachers at the time of the interview seem to have a good grasp of what
number sense involves and what were characteristics of students who had a developed
number sense. They used phrases that conveyed that they believed it was
"the ability to solve problems in different ways." Both teachers also
changed how they felt about mathematics consisting solely of facts and
procedures. Both teachers mentioned the importance of mathematics for daily use
and for professional use. The treatment teacher pointed out that mathematics
involves ways of thinking about numbers, that it was more than just facts and
procedures. The control group teacher stressed the importance of conveying that
mathematics is a tool for daily living.

The following shared belief surfaced in the post
treatment interview was added to the preliminary list of shared beliefs:

7. Speed is
important in mathematics.

This change could be attributed to the teachers
perceiving that because the mental computation test was timed, they could
should consider time as an important factor in mathematics teaching. However,
the treatment teacher pointed out that in today's society it is necessary to
make quick decisions so speed should be cultivated in mathematics calculation.
Both teachers also argued that the standardized tests the students take for
group assignment for Federal government funding purposes are timed, as are
other district tests. They felt that student should be taught how to respond
quickly for these examinations. The control group teacher said that the
participation in the project motivated her to do better work and dedicate more
of her time to designing new activities for her students. She also said that
she tried to teach in a more sense-making,way. She indicated however, that she
would not accept left to right computation, that the objective of teaching
mathematics was so that students perform well on standardized examinations;
furthermore, she has no idea of what mental computation strategies are.

The treatment group teacher indicated that she had
learned about: the importance of using patterns in mathematics teaching, mental
computation strategies for counting and addition, the utility of the use of
base ten blocks before teaching regrouping in addition, how to use different
visual number models, and how to use card games and dominoes for teaching. The
teacher indicated: that she could now accept left to right computation if the
students demonstrated that they understood what they were doing and that the
purpose of teaching mathematics was not solely for preparing students for
examinations but that it was also about helping them learn how to think.