Comparison of IMP Students with Students Enrolled in Traditional
Courses on Probability, Statistics, Problem Solving, and Reasoning April 1997 Norman
Webb Wisconsin Center for Education Research The Interactive Mathematics Program Evaluation Project operates under the auspices of the Wisconsin Center for Education Research and is funded under a contract with the San Francisco State University Foundation, Inc., with resources that are provided by the National Science Foundation (award number ESI9255262). The opinions, findings, and conclusions that are expressed in this paper do not necessarily reflect those of the supporting agencies. Table of Contents
A series of three studies compared the performance of students enrolled
in the Interactive Mathematics Program (IMP) with students enrolled in the
traditional algebra 1, geometry, and algebra 2 sequence. The studies,
conducted at the end of the 199596 school year, compared the performance of
IMP students with students enrolled in the traditional collegepreparatory
course sequence on activities using probability, statistics, quantitative
reasoning, and problem solving. The studies were done independently of each
other at three different schools, with ethnically diverse populations,
different outcome measures, and different grade levels: grade 9, grade 10, and
grade 11. The grade 9 test consisted of all four statistics items released by the Second International Mathematics Study (SIMS). The multiplechoice items used by SIMS were modified to a constructedresponse format. The results on these four items were analyzed for 115 students who were enrolled in either IMP Year 1 or algebra 1. The grade 10 test consisted of two performance assessment activities prepared for the Wisconsin Student Assessment System. Each activity required students to produce a response by solving a multistep mathematics problem, making a generalization, and writing an explanation of the procedures used. One activity required some conceptual knowledge of probability. The other activity required some knowledge of combinatorics and pattern recognition. The test was administered to 184 students who were enrolled either in IMP Year 2 or geometry. The grade 11 test consisted of 10 multiplechoice items taken from a practice version of a quantitative reasoning test developed by a prestigious university to test the knowledge of entering students. This test required data and graph interpretation, and basic understanding of probability and statistics concepts such as the standard deviation and the mean of a distribution. The test was administered to 133 students enrolled in either IMP Year 3 or algebra 2. The results obtained across the three schools and three grade levels support the assertion that students who took IMP gained knowledge and reasoning skills beyond what students learned in the traditional mathematics sequence. Comparison of IMP Students with Students Enrolled in Traditional
Courses Norman L. Webb and Maritza Dowling Wisconsin Center for Education Research, School of Education, An increasing number of studies have compared the mathematical knowledge of students enrolled in the Interactive Mathematics Program (IMP) with that of students enrolled in the traditional algebra 1, geometry, algebra 2 sequence. Accumulating evidence shows that IMP students perform as well as, if not better than, students taking the traditional curriculum, as measured by standardized normreferenced tests such as the Scholastic Assessment Test (SAT), PreScholastic Assessment Test (PSAT), and the Comprehensive Test of Basic Skills (Interactive Mathematics Program, 1995; Schoen, 1993; S. Chew, personal communication, November, 1996; Webb & Dowling, 1995a, 1995b, 1995c, 1996). These traditional instruments measure students' knowledge of very general mathematics skills and reasoning in mathematics. Probability, statistics, quantitative reasoning, and problem solving are given little or no attention on these instruments however. It is increasingly important for students to have knowledge in these areas. This has been recognized by the NCTM Curriculum and Evaluation Standards for School Mathematics (1989) and by various state standards such as South Carolina Mathematics Framework (1993), Illinois Academic Standards Project (1996), and New Jersey Mathematics Curriculum Framework (Rosenstein, Caldwell, & Crown, 1996). Therefore, we set out to examine the performance of IMP students on these critical criteria, which have not been focused upon by past research. The IMP curriculum is a fouryear collegepreparatory sequence of courses designed for grades 9 through 12. The IMP curriculum integrates traditional areas of mathematics such as algebra, geometry, and trigonometry with probability, statistics, discrete mathematics, and matrix algebra. Students are challenged to actively explore openended situations, in a way that closely resembles the inquiry methods used by mathematicians and scientists. IMP calls on students to experiment with examples, look for and articulate patterns, and make, test and prove conjectures. The problembased curriculum is organized into fivetoeight week units, each centered on a central problem or theme. Students engage in solving both routine and nonroutine problems, use graphing calculators, and are encouraged to work cooperatively.
IMP, as a comprehensive collegepreparatory fouryear course sequence,
incorporates mathematical ideas and concepts from the traditional high school
curriculum of algebra, geometry, and trigonometry. Other subject areas,
including probability and statistics, are incorporated throughout the program.
Students study topics such as how to calculate simple probabilities, the
distinction between theoretical and experimental probabilities, the meaning of
a bestfitting line for a set of data, properties of normal and other
distributions, the calculation and use of standard deviation, and the notion
of testing a null hypothesis and methods for doing so.
Teaching techniques used in IMP are designed to help students gain deep
understanding of mathematical ideas, reason mathematically, and apply
mathematics to solve problems. For example, one night's homework involves four
sets of data about which students are to answer a number of questions,
including about the spread from the mean, the standard deviation, and the
similarity among the data sets (Fendel, Resek, Alper, & Fraser, 1997, p.
343). Students also are assigned Problems of the Week (POWS) to work on for
five or more days in addition to daily homework. Students' ability to reason
is enhanced in many ways. For instance, students are asked to design
experiments, to state their conclusions based on evidence and analyses, to
compare mathematical ideas, and to explain general cases of specific
situations. Classroom experiences such as presentations, written explanations, and small group activities are structured for students to verbalize their thinking. This verbalization is designed to increase students' understanding and to improve their communications of mathematics. Students are to become more independent learners by using multiple sources of information including their teachers, the textbook, classmates, and other references. To make valid comparisons between IMP and the traditional mathematics course sequence, specific controls were imposed on the design of these three studies. Each participating school had to provide grade 8 standardized normreferenced mathematics test scores on a significant number of IMP and traditional mathematics students who were tested. An adequate number of students, as close to 50 as possible or more in each program, had to be available to be tested. This meant that there had to be more than one IMP class at the target grade level. One or more teachers of the traditional mathematics course taken by students at the target grade level had to agree to participate in the study. The tests had to come from a source independent of IMP and had to be easily administered under the same conditions to classes of students in both course sequences. The inconvenience to teachers and students had to be kept to a minimum. Students' knowledge on as wide a range of content as possible was sought. Regional and school factors had to be reduced as much as possible. Effects had to be attributed to the curriculum rather than other factors such as teacher, school, or region.
Three studies were conducted. Each study was done in a different school
at a different grade level with a different outcome measure. This design was
chosen to meet the criteria above while maximizing the comparative information
and minimizing the interference within any one school (Figure 1, p. 3). Three
different instruments were used. The grade 9 test (Appendix A) was composed of
all statistics items (a total of four) released from the Second International
Mathematics Study (SIMS) (Crosswhite et al., 1986). These items were
administered to grade 12 students in the 198182 school year. The items were
modified from the multiplechoice format used in SIMS to an openresponse
format. The grade 10 test consisted of two performance assessment activities
prepared for the Wisconsin Student Assessment System under the auspices of the
Wisconsin Department of Public Instruction. Each activity requires students to
construct a response by solving a multistep mathematics problem, generalizing
the results obtained, and writing an explanation of the reasoning and
procedures used. One activity required some knowledge of probability and one
activity required some knowledge of combinatorics. These activities are only
available for field testing and cannot be released. The grade 11 test was 10
multiplechoice items taken from a practice version of a quantitative
reasoning test (QRT) developed by a prestigious university for its firstyear
students. Students at this university are required to pass the QRT or a
specified course as a graduation requirement. Permission was not received from
the university to disclose its name or the test items.
Three different schools were selected for each of the studies. Each school came from a different region of the country, was considered to be effectively implementing the IMP curriculum, and had an IMP coordinator who would attend to all of the controls on the study. All testing was done within the last six weeks of the 199596 school year. Some classes tested in these studies included students not in the target grade. Those students were not included in the analyses.
An ethnically diverse group of students participated in the study
(Table 1, p. 16). A total of 240 students who were taking IMP were included in
the analysis—47 percent white, 24 percent black, 11 percent Hispanic, 9
percent Asian/Pacific Islander, 1 percent American Indian, and 8 percent other
or unknown. A total of 192 students who were taking the traditional curriculum
were included in the analysis—33 percent black, 24 percent white, 18 percent
Asian/Pacific Islander, 11 percent Hispanic, 1 percent American Indian, and 13
percent other or unknown.
A higher proportion of female students than male students participated
in this study in both the IMP group and the traditional group—55 percent
female for IMP and 61 percent female for the traditional curriculum (Table 2,
p. 17). Because IMP and traditional groups varied some on their composition by
ethnicity and sex at each of the three schools, analyses were performed to
determined if differences associated with group characteristics were
significant. To test for statistical significance, the student was used as the unit of analysis rather than the teacher or the class. Except for grade 9, the tests measured knowledge and skills that were not specific to the current course and teacher but reflected the results of two or more years in the given program. In all three studies, analyses were performed to determine if findings were upheld for classes as well as students.
At each of the three schools, IMP students obtained statistically
significant higher test scores compared to students in the traditional
mathematics course (Table 3, p. 18). These results were true even when
accounting for any differences between groups in prior achievement as measured
by grade 8 standardized normreferenced test scores. An analysis of covariance
was used to account for prior achievement for each of the studies.
Matchedgroup analyses also were performed for each school to control for any
differences due to prior achievement.
Grade 9 Year 1 IMP students at school 1 outperformed those taking
algebra 1 on the statistics items from the SIMS (Table 3, p. 18). On the
average, IMP students attained three of the possible 5 points, compared to the
algebra 1 students who attained one of the possible 5 points. The
internalconsistency reliability estimate was .73, high for a test consisting
of only five items. Eleven of the 60 IMP students (18 percent) attained a
perfect score, compared to none of the algebra 1 students. A total of 24
students did not receive any points, 4 IMP students (7 percent) and 20 algebra
1 students (36 percent). No significant differences were found in the
performance by sex or ethnicity.
SIMS items were originally administered in a multiplechoice format to
precalculus students, as compared to the openresponse format used in the
current study. The openresponse format is more difficult. The same percentage
of correct answers on an openresponse item, compared to the corresponding
multiplechoice item, represents a greater degree of understanding. In
analyzing results of the current study, performance of IMP students was
compared to that of the SIMS precalculus students as well as to that of
current grade 9 algebra 1 students.
On item 1 (requiring students to determine the approximate average
weekly rainfall from a bar graph), IMP students did better than algebra 1
students (78 percent correct compared to 55 percent). The IMP students'
performance matched that of the SIMS precalculus students, of whom 78 percent
correctly answered this item, even though the IMP students had to produce the
correct answer as an open response rather than selecting the correct response
among five options.
Item 2 required the computation of a weighed average. On this item, 48
percent of IMP students found the correct answer, compared to 11 percent of
the algebra 1 students. Only 22 percent of the SIMS precalculus students
selected the right answer, a percentage that is only slightly better than
randomly selecting from five choices.
Item 3 required students to analyze a linear transformation on the mean
and standard deviation of a distribution. Concerning the effect on the mean,
87 percent of the IMP students got the right answer, compared with 33 percent
of the algebra 1 students. Both groups had more difficulty with the item
determining the effect on the standard deviation. The IMP students again did
much better than the algebra 1 students (57 percent correct compared to 2
percent). In the 198283 SIMS, the two questions were combined as a single
multiplechoice item, and only 24 percent of the precalculus students selected
the correct choice.
Item 4 required the application of properties of the normal curve
(identifying the proportion of the area under the curve related to +/ one
standard deviation). On this item, 37 percent of the IMP students were able to
produce the correct answer, while none of the algebra 1 group were able to do
so. Only 22 percent of the precalculus students identified the correct answer
from five choices.
The IMP curriculum exposes students to the properties of the normal
curve of mean and standard deviation among other mathematical concepts and
ideas. Not all IMP students are expected to master these ideas by the end of
grade 9. More instruction is provided on these ideas later in the curriculum.
These ideas are not included in most traditional algebra curricula so algebra
1 students would not be expected to know these ideas. The findings from this
study identify at least some differences between what students learn from
taking IMP compared to the traditional algebra course. The findings support
that students who take IMP Year 1 gain knowledge about statistics and the
normal distribution. The IMP students scored as well as, or better than, the
SIMS precalculus students, who had taken the items 14 years before. Again,
this finding denotes difference in curriculum coverage.
Although students were in three different IMP classes, they all had the
same teacher. This teacher was considered to be using IMP as intended. The
algebra 1 students were taught by two different teachers, two classes each. We
have no special information about these teachers. There was a variation in
class means both within IMP classes (2.25, 2.89, and 4.13) and within algebra
1 classes (.47, .90, 1.29, 1.31). The variation in class means indicates some
differences in achievement by class. Of the possible 12 pairwise comparisons
of an IMP class mean with an algebra 1 class mean, for 10 of the comparisons
the IMP class mean was significantly higher than the algebra class mean. One
IMP class (mean of 2.25) had a significantly higher mean than two of the
algebra 1 classes (means of .47 and 1.29), but not significantly higher mean
than the other two algebra 1 classes (means .90 and 1.31). An analysis of
covariance, using the grade 8 CTBS score as the covariate, was used to make
the pairwise comparison of class means.
Grade 10 IMP Year 2 students performed significantly better than
geometry students on an instrument measuring students' mathematical reasoning
and problem solving (Table 3, p. 18). The instrument was composed of two
performance assessment activities, each designed to require students about 20
minutes to complete. Students' work on each activity was scored using a
sixlevel rubric ("advanced response" (5), "proficient"
(4), "nearly proficient" (3), "minimal" (2),
"attempted" (1), and "not scorable" (0)). A
"proficient" or "advanced" score on one activity, Connecting
Nodes, indicates that the student demonstrated skills in solving problems,
reasoning, developing and testing conjectures, writing clear and correct
explanations, computing, and forming a generalization. A
"proficient" or "advanced" score on the second activity, New
Cubes, indicates that the student performed a number of skills. These
included comparing probabilities between two or more sums when rolling two
dice; computing all possibilities for an invented set of dice meeting
specified conditions; determining the expected value for a large number of
trials; and describing their reasoning in writing. Each student response was
scored by at least two raters. A third rater scored the response if there was
an unacceptable range in scores by the first two raters. Exact agreement by
raters on scoring was 78 percent for Connecting Nodes and 80 percent
for New Cubes. Acceptable interrater agreement—exact agreement or
only varying by one level between "attempted" and
"minimal" or "proficient" and "advanced"—on
both activities was 87 percent.
IMP Year 2 students, compared to geometry students, had a significantly
higher (p < .01) mean score on both activities—2.53 compared to 2.04 on Connecting
Nodes and 3.91 compared to 2.52 on New Cubes. On Connecting
Nodes, 21 percent of the IMP Year 2 students, compared to 5 percent of the
geometry students, performed at a "proficient" or
"advanced" level. On New Cubes, 64 percent of the IMP Year 2
students, compared to 16 percent of the geometry students, performed at a
"proficient" or "advanced" level. These results were true
even though the geometry students had a higher mean score on the grade 8 ITBS.
A total of 15 IMP students (17 percent) achieved a "proficient"
score or higher on both of the activities. Fiftynine of the IMP students (69
percent) achieved a "proficient" score or higher on one of the two
activities. This is compared to one geometry student (1 percent) who achieved
at least a "proficient" score on both activities and 46 students (47
percent) who achieved at least a "proficient" score on one activity.
The differences in the respective percentages are statistically significant (p
< .01). Nearly all of the students participating in the study, 87 percent,
were identified as black, Asian, or Hispanic. There were no significant
differences by sex or ethnicity.
Four classes of IMP Year 2 students—two classes for each of two
teachers—and six classes of geometry students—three classes for each of
two teachers—were tested. Two of the geometry classes were honors classes.
Mean scores among the four IMP classes and between the two teachers varied
little (6.0 and 6.79 for one teacher and 6.16 and 6.75 for the other teacher).
The mean scores among the six geometry classes differed significantly between
honors and regular classes. The mean scores for the geometry honors classes,
both taught by the same teacher, were 5.10 and 6.00. The mean scores for the
regular geometry classes were 4.45 (taught by the same teacher as the honors
courses) and 3.50, 4.07 and 4.63 (taught by the other geometry teacher).
Of the 24 possible pairwise comparisons of an IMP class with a geometry
class, IMP classes had significantly higher means for 20 of the comparisons.
One of the four IMP Year 2 classes (mean of 6.79) had a significantly higher
mean than all six of the geometry classes. Two of the IMP Year 2 classes
(means of 6.75 and 6.16) had significantly higher means than five of the six
geometry classes. Only the honors geometry class, with a mean of 6.00, was not
significantly different from the means of these two IMP classes. Only one IMP
Year 2 class (mean of 6.00) was not significantly different from either of the
two honors geometry classes (means of 6.00 and 5.10), but this class was
significantly higher than all four of the regular geometry classes. An
analysis of covariance, using grade 8 ITBS score as the covariate, was used to
make the pairwise comparisons.
We did two analyses to compare grade 10 IMP Year 2 students with grade
10 geometry honors students. One analysis involved only those students for
whom we had grade 8 ITBS‑math scores (71 out of 87 IMP Year 2 students,
20 out of 23 geometry honors students). The IMP students had a significantly
lower mean score on grade 8 ITBS‑math (77.34 compared to 92.50; p <
.01), but performed higher on the problemsolving test (mean of 6.35 compared
to mean of 5.65). Using analysis of covariance to control for grade 8 scores,
this difference on the problemsolving test is statistically significant (p
< .01). In the second analysis, we compared scores on the problemsolving
test, using all grade 10 IMP Year 2 students and all grade 10 geometry honors
students (including those without grade 8 ITBS scores). Using analysis of
variance, the IMP students performed significantly higher than the geometry
honors students (p < .05). These two analyses suggest that the experiences
students had in IMP raised their level of performance on these two activities
above the highest achieving students from the traditional curriculum at this
school.
Performance assessment is designed to measure students' performance on
complex openresponse situations with multiple parts. An important part of the
IMP curriculum is for students to work complex problems, some over an extended
period of time. This attention to complex problems solved over extended time
periods differs greatly from the approach of most traditional curricula, which
are designed for students to become accomplished in doing more limited tasks.
The two performance activities used in this study required some knowledge of
probability and computing combinations, two topics included in the IMP
curriculum. Again, as in the first study, grade 10 IMP students outperformed
students taking the traditional curriculum on activities aligned with the IMP
curriculum. A higher percentage of IMP students were able to apply reasoning
to analyze a situation using ideas from probability, and to make a
generalization, compared to geometry students. The fact that the honors
geometry classes outperformed the regular geometry classes suggests that
general mathematics achievement is related to doing well on these performance
assessment activities. All IMP classes did as well as, or better than, both of
the two honors geometry classes. This was the case even though IMP students'
prior achievement in grade 8 was even lower than geometry students in general.
This suggests that the experiences students had in IMP had raised them to a
level of performance on these two activities above the highest achieving
students from the traditional curriculum at this school.
Grade 11 IMP Year 3 students performed significantly better than grade
11 algebra 2 students on 10 items from a practice version of a quantitative
reasoning test developed for a prestigious university to administer to
entering first year students (Table 3, p. 18). In order to attain second year
status, students are required to obtain a specific score on this test or to
take a designated course. All of the items are multiple choice. University
students are given 90 minutes to work 25 items, or 3.6 minutes per item. Ten
items aligned with the IMP curriculum were selected from the practice test to
be administered in one class period of nearly 50 minutes. Knowledge measured
by these items include data and graph interpretation, probability of
independent events, and statistical concepts of mean and standard deviation
for a distribution. The measure of internal consistency for the test of 10
items was .65.
Students in all four IMP Year 3 classes and students in three algebra 2
classes were tested. IMP Year 3 students had a significantly higher mean score
than the algebra 2 students, 5.04 compared to 2.40 (Table 3, p. 18). The
difference in means was found to be statistically significant using an
analysis of covariance to account for the difference in mean score on the
grade 8 achievement measure. For the total group, IMP and algebra students
combined, male students had a significantly higher (p < .05) mean than
females. But no significant interactions by sex or ethnicity were found
between the two curricula. Of the 12 possible pairwise comparisons of IMP
class mean with algebra 2 class mean, on 11 comparisons IMP classes had
significantly higher means. Only the IMP class with the lowest mean (4.44) was
not significantly higher than the algebra 2 class with the highest mean
(3.13). Means were compared using an analysis of variance.
The percentage of IMP students who correctly answered any one item
ranged from 18 percent to 75 percent. The percentage of algebra 2 students who
correctly answered any one item ranged from 3 percent to 53 percent. Two IMP
students correctly answered all 10 times. The highest score obtained by an
algebra 2 student (one student) was 8. Three students obtained a score of 0,
one IMP student and two algebra 2 students. IMP students outperformed algebra
2 students on all ten items. The highest percentage of both groups answered
correctly a probability question requiring students to determine the number of
balls in a bag given the probability for drawing each of three of four colors
of balls and the number of one color of balls. Of the IMP students, 75 percent
correctly answered this question, compared to 53 percent of the algebra 2
students. On a second question based on the same situation, where students
were to identify the number of balls of one color to be replaced by a second
so that the probability to pick either color was the same, 67 percent of the
IMP students answered correctly, compared to 38 percent of the algebra 2
students.
On another item, given graphs of growths in revenue for four divisions
of a company, students were asked to select a true written description about
the comparison of the divisions' rates of growth. Seventyone percent of the
IMP students answered this correctly, compared to 45 percent of the algebra 2
students. The most difficult item for all of the students was a statistical
question. Students were asked to identify the number of people who were within
a specific range of reading speeds on a reading test. The test item gave the
total number of students who took the reading test, the fact that reading
speeds were normally distributed, the mean, and the standard deviation. On
this item, only 18 percent of the IMP students and 3 percent of the algebra 2
students gave a correct answer. Students in both groups performed better on
the next item. Students were asked to use the mean and standard deviation of
the distribution to approximate the reading speed for a student at the 80th
percentile. Over half of the IMP students answered this correctly (54
percent), compared to 28 percent of the algebra 2 students.
None of the 10 items asked a question in the same form as it would
appear in the IMP curriculum. Half of the 10 items required some understanding
of probability and statistics, two areas given some emphasis in IMP. The other
five items required knowledge of rate of change and the interpretation of
slope of a graph, topics given as much attention in IMP as in the traditional
curriculum. The inclusion of these items suggests that the university where
the test was developed felt that it was important for entering college
students to know what is measured by these items. The findings from this study
indicated that IMP students performed significantly better on these items than
students in the traditional algebra 2 course. Part of the differences in
performance could be explained by the increased emphasis by IMP on probability
and statistics. The other part related more to differences in general
quantitative reasoning skills demonstrated by students in the two different
curricula.
For each of the three studies, the IMP students and students in the
traditional college preparatory mathematics courses had a different mean score
on the grade 8 mathematics standardized normreferenced test. For two schools,
the mean for IMP students was significantly higher at the .05 level. For the
third school, the mean for IMP students was lower than the mean for the
traditional students, but not significantly so (Table 3, p. 18). An analysis
of covariance was performed, using the grade 8 score as the covariate, to
adjust for differences in prior achievement by group. Other factors, such as
ethnicity and sex, were not controlled. We performed a matchedgroup analysis
to consider the results in a second way and to control for prior achievement,
ethnicity (when possible), and sex (Tables 4 and 5, pp. 1920). For schools 1
and 2, the students within each school had scores from the same grade 8 test.
School 3 students were from a large number of middle schools, both public and
private. Thus, not all of the students had taken the same test in grade 8. In
the matchedgroup analyses, only students with scores on the same grade 8 test
(California Achievement Test) were included in the analysis.
For the matchedgroup analysis, we tried to generate two comparison
groups for each school so that the IMP group and the traditional group had
comparable statistical qualities on the independent variables. Student scores
were not included in the analysis if a close match could not be found. As a
result, the numbers of students included in the matchedgroup analyses are
lower than the numbers included in the analyses of covariance. For all three
schools, the matching process successfully produced very similar groups on
grade 8 mathematics score and distribution of sex, but for school 1 the groups
varied some in ethnic composition (Tables 4, 5 and 6, pp. 1921).
As expected, the matchedgroup analysis produced the same results as
the analysis of covariance. IMP students in each of the three schools,
compared to those in the same grade in the traditional mathematics course,
performed significantly higher on the given measure of mathematical knowledge
(Table 6, p. 21; Figure 2, p. 23). Grade 9 IMP students had a significantly
higher score on the SIMS statistics items than the algebra students. Grade 10
IMP students had a significantly higher score on the performance assessment
instrument than the geometry students. Grade 11 IMP students had a
significantly higher score on the 10item quantitative reasoning test designed
for entering university students. All results were statistically significant
(p < .01).
This series of three studies provides evidence that IMP students in
mathematics are learning beyond what students are learning in the traditional
algebra 1, geometry, and algebra 2 course sequence. IMP students performed
better than students from the same grades in the traditional mathematics
courses on activities requiring knowledge of probability and statistics and
skills in reasoning, problem solving, and forming generalizations. The
measures used in these studies were developed independently of the IMP
curriculum. One instrument was items from an international study, one
instrument was activities developed for a state assessment, and one instrument
was items developed by university faculty. Grade 8 mathematics test scores
were obtained to control for differences in prior achievement upon entering
high school. In all three studies, IMP achieved a mean score that was
significantly higher using common statistical tests. The number of students in this series of three studies is small. Schools were selected to participate in these studies because they were implementing IMP as designed. Two of the schools had enrollment criteria. The findings support that students taking IMP are learning mathematics beyond what students learn in the traditional curriculum. There is mounting evidence that IMP students do as well as students enrolled in the traditional collegepreparatory mathematics course sequence on standardized normreferenced tests such as the SAT. Combined, these studies indicate that IMP not only prepares students to perform well on traditional measures, but that the curriculum provides students knowledge and skills that are becoming increasingly important. Crosswhite,
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N. L. & Dowling, M. (1995c). Impact of the Interactive Mathematics
Program on the retention of underrepresented students: Class of 1993
transcript report for school 3: Valley High School. Project Report 955
from the Interactive Mathematics Program Evaluation Project. Madison:
University of WisconsinMadison, Wisconsin Center for Education Research. Webb, N. L. & Dowling, M. (1996). Impact of the Interactive Mathematics Program on the retention of underrepresented students: Crossschool analysis of transcripts for the class of 1993 for three high schools. Project Report 962. Madison: University of WisconsinMadison, Wisconsin Center for Education Research. Table 1
Students by Ethnicity by School and Curriculum
Table 2
Students by Sex by School and Curriculum
Pre and PostTest Scores for Students by School and Curriculum
^{a} Significantly higher mean score from traditional curriculum group using an analysis of covariance to account for any differences on grade 8 achievement, p < .01. ^{b}
Significantly higher mean score from other group within school, p <
.05. ^{c} Mean national percentiles scores.
Students by Ethnicity by School and Curriculum
Students by Sex by School and Curriculum
Pre and PostTest Scores for Students by School and Curriculum
^{a}
The zstatistic approximation for the largesample Wilcoxon test was
found statistically significant, Z = 5.38, p < .01. ^{b}
The zstatistic approximation for the largesample Wilcoxon test was
found statistically significant, Z = 5.07, p < .01. ^{c} The zstatistic approximation for the largesample Wilcoxon test was found statistically significant, Z = 4.58, p < .01. * Mean scores for both groups in school 1 were doubled to have these scores on the same scale as the measures for the other two schools. Figure 2. Test means across schools comparing IMP to traditional math students. Grade
9 Instrument
SAMPLE COVER SHEET ONLY Student's Name: Class Period: Grade:
Teacher: Mathematics Gender:
(circle one) F
M Optional
This
test consists of four questions. Write
your name at the top of each page. For
each question, find the answer and then write your answer on the line provided.
Each question will be scored as right or wrong.
The total score for the four problems will be the total number of
questions you answered correctly. You
are free to show your work and write on the pages even though only your answer
will be scored. You can use a
calculator. You
may answer the questions in any order. If
you have difficulty answering one question, go to the next question.
Go back to any unanswered question and do your best to determine an
answer. Please check all of your answers before returning the test
booklet to your teacher. Please
complete the information above before beginning.
Name 1.
Answer 1: 2. The same test was given in two classes. The first class, with 20 pupils, obtained an average score of 12.3 points. The second class, which had 30 pupils, obtained an average score of 14.8 points. What was the average score for the whole group of 50 pupils? 3. The mean of a
population is 5 and its standard deviation is 1.
If 10 is added to each element of the population, what will be the new
mean and standard deviation? Answer 3: New mean: New standard deviation: 4. A test is taken by
all first year university students in a country.
The mean is 50 and the standard deviation 20.
Assuming the scores are normally distributed, approximately what
percentage of students score more than 30? Answer 4: Scoring Key
Five Modified SIMS Statistics Items 1.
Average weekly rainfall during the period: 1.9 to 2.3
cm 2.
Average score for the whole group of 50 ss:
13.8 points 3a.
New Mean:
15 3b.
New Standard Deviation: 1 4. Percentage of students scoring more than 30: 83 to 85%

