Middle school mathematics classroom practices and achievement: a TIMSS-R analysis.

Abstract

Recent debate in mathematics education has focused attention on the value of teaching mathematics reform-oriented strategies and/or to take a more "back-to-basics" approach. Data from the TIMSS-R Population 2 (13 year-olds) International Database were analyzed using a multiple regression model. The model was composed of variables related to classroom practices from the student questionnaire. Separate analyses were conducted for the overall achievement score, and achievement scores for each of the five content areas, and differentiated by gender. The results were mixed. The more often a student worked on projects the lower the achievement score. There was a negative relationship between asking to explain their thinking in front of the class at the board and at the overhead and achievement. The more frequently a calculator was used the greater the achievement levels. The use of the calculator correlates strongly with achievement in the Geometry content area. Female students had a stronger negative correlation than males when the teacher asks what they know related to a new mathematics topic in Geometry.

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Introduction

The way mathematics is taught has recently gained the attention of policy makers, parents, and other stake-holders as the result of recent reports of low performance in an international comparison of United States students to students in other nations. The National Council of Teachers of Mathematics [NCTM] (2000) has taken the position that, "students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge" (p. 11). This position implies that students should receive mathematics instruction that builds on their prior knowledge and should be related to what they know and to real-life situations. Moreover, teachers are encouraged to establish and nurture a classroom climate where students collaborate and are comfortable in discussing their ideas, strategies, and solutions (NCTM, 2000).

Former Secretary of Education Riley (1998) recognized the debate between reform-oriented instruction and the back-to-basic movement. The debate centers on teaching strategies, where the back-to-basic traditionalists argue that mathematics should be taught by encouraging students to memorize and practice basic facts and skills (Starr, 1998), whereas, reform oriented instruction focuses on developing mathematical understanding through communicative strategies, dialogue among students and between the students and the teacher. The debate continues.

The Third International Mathematics and Science Study (TIMSS)-R database offers a wealth of information concerning the mathematics teaching practices, achievement, and curriculum of several countries and is the largest and most comprehensive international comparison to date of mathematics and science achievement (Martin, 1996). However, this paper focuses on United States eighth-grade students. As part of the study, data were collected regarding student achievement and factors related to mathematics performance (Schmidt & Logan, 1996). This database offers the opportunity to examine relationships between selected student characteristics and mathematics achievement. The research question was "How do mathematics classroom strategies that involve communication relate to United States eighth grade students' mathematics achievement?"

Teaching for understanding in mathematics involves incorporating various reform-oriented strategies. The strategies include building on students' prior understanding, building from informal to formal knowledge, the use of calculators, having students work cooperatively, involving students with projects, relating mathematics to real-life, and communication of ideas (NCTM, 2000). In a reform-oriented classroom, communication plays a crucial role. Communication in a classroom can occur in various ways. Students may be called upon to share ideas at the board, the overhead, or within a group. A traditional activity that occurs in many mathematics classes is discussing or going over homework. All of these various communication avenues can assist in developing mathematical understanding (Fennema & Romberg, 1999).

Moreover, social constructivism as a theory of how people arrive at "knowing" describes knowledge as being in flux, where an individual internally constructs knowledge through social and cultural mediation where communication has an important role (Ernest, 1998; Steffe & Gale, 1995; Fosnot, 1996). The construction of knowledge occurs through one's perceptions and experiences, which are judged, or assessed by previous experiences and currently held briefs (Ernest, 1991). Social activity and discourse play important roles for understanding to occur. Thus, the classroom is viewed as a mini society--a community of learners engaged in activity, discourse, and reflection. The teacher provides contextually meaningful experiences where students are permitted to raise their own questions, construct models, concepts, and strategies (Fosnot, 1996).

Steffe and D'Ambrosio (1995) argue for reflection from a constructivist perspective. It is through reflection that one critiques and/or compares and contrasts one's own thoughts with those of others in a dialogic format. In order to encourage reflection, there are three prevailing actions that mathematics teachers can use to assist in the development of a learning space as discussed above: (1) pose situations that students regard as genuine problems, (2) model reflective actions, and (3) facilitate interactive, mathematical communication to activate schemes such as prior knowledge. It is thought that activating prior knowledge leads to generalizing assimilation (or, transfer of learning). This implies that social interaction, "underlies all teaching actions" (p. 156). In this regard, verbal and nonverbal communication becomes a major component of a constructivist environment.

Method

Sample

The TIMSS sample design was a two-stage cluster design where schools were selected in the first stage followed by classrooms in the second stage. Various types of information were collected as part of the TIMSS-R student questionnaire, including student characteristics, instructional activities, family characteristics, out-of-school activities, learning resources, and achievement. Students included in this analysis were from the TIMSS Population 2 International Sample (13 year-olds) from the United States. The reader is reminded that the survey was administered to students who self-reported their perception of the frequency that the selected activities occurred in their classrooms. As with any survey data, the data may be questioned as to whether the respondents are responding truthfully to the statements.

Measures

The TIMSS-R measures achievement in five mathematics categories and provides an overall mathematics achievement score. The five reporting categories were: (1) Fractions and Number Sense, (2) Measurement, (3) Data Representation, Analysis and Probability, (4) Geometry, and (5) Algebra. There was a variety of item types that included, multiple-choice, short answer, and extended response, with a total of 162 items, but students do not respond of all of the items (Gonzalez & Miles, 2001a). Items have been released to the public, which may be accessed at the following website: http://isc.bc.edu.

Six variables were chosen from the instructional activities section of the TIMSS-R student survey in response to the question, "How often does this happen in your mathematics lesson?" The six selected variables were the following: (1) "We work on mathematics projects," (2) "We use calculators," (3) "We work together in pairs or small groups," (4) "We discuss our completed homework," (5) "Students use the board," and (6) "Students use the overhead projector." The author presumes that students would be required to explain their thinking while at the board or the overhead; the survey does not make this clear. Two other variables from the survey were selected in response to the question: "When we begin a new topic in mathematics we begin by ...?" (1) "Discussing a practical or story related to everyday life, and (2) "Having the teacher ask us what we know related to the new topic." On each of these instructional activities, students indicated the frequency using a Likert-scale where a one indicated "almost always," a two, "pretty often," a three, "once in a while," and a four, "never." The dependent measures for this study were the TIMSS-R Overall Mathematics score, and the content area scores for Algebra, Data and Probability, Fractions/Number Sense, Geometry, and Measurement.

Procedure

Simple random sampling involves assumptions that are inappropriate for data collected using two-stage stratified cluster designs. Jackknife variance estimation procedures using replicate weights were used to compute appropriate standard errors of each variable included in this study, as a result of the complex sampling design (National Center for Education Statistics, 1998). These procedures provide unbiased variance estimates to enable appropriate statistical tests of significance to be made with a complex sample design. The data were analyzed using the software program, WesVar v4.1 (Westat, 2000).

A least-squares multiple regression analysis was conducted. The eight variables from the TIMSS-R student questionnaire main survey were used to simultaneously assess the relative contribution of each instructional activity toward the explanation of mathematics achievement (Gonzales & Miles, 2001a). Because students in the TIMSS 1999 assessment were given relatively few test items in each specific content area, statistical procedures have been developed to generate plausible values for each student based on responses given in order to estimate student proficiency (Yamamoto & Kulick, 2000). Each plausible value provides an estimate of the performance of each student had they actually been given all possible items on the assessment. Because there is error in the generation of these imputed proficiency values, five plausible score values were computed for each student (Gonzalez & Miles, 2001a). The dependent measures used were one of the five plausible values generated for each student on their overall mathematics performance and the score in each of the five content areas. Due to complex sampling, replicate weights were used to ensure an accurate analysis. The number of replicate weights used was 53. The total number of observations read was 9,072, which was equivalent to 3,336,295 observations. Gender comparisons were also conducted on the data. The data were analyzed in accordance with the procedures outlined in Gonzalez and Miles (2001b).

Results

Tables 1 and 2 present results, for females and males respectively, of the multiple regression analyses. Each table presents parameter estimates for the eight classroom activity variables, overall score, and scores for each content area. The direction of the relationships is reversed since the survey used a Likert scale where the greater the frequency the lower the number. Table 3 presents a summary of the relationships between the achievement variables and instructional strategy as a visual guide.

Tables 1 and 2 provide an [R.sup.2] for each analysis. The values for females were generally lower than for the male sample. The highest value, for females, 0.12, occurred in the content area of geometry. The next largest value, 0.09, was in algebra. The model for males produced [R.sup.2] values in the range of 0.06 to 0.15, with the content area Data Analysis and Proportionality for both genders having the lowest value, indicating a weak model for that content area.

With Overall Achievement as the dependent variable, the model shows similar results for male and female students. The classroom activities, discussing homework and using calculators, had the greatest positive impact on the model for Overall Achievement for both genders, while the frequent use of projects was associated with lower achievement scores as measured by TIMSS-R 1999 test.

Gender Differences

Table 3 presents a summary of the direction of each relationship by gender. There was a differential result between male and female students regarding the demonstration of one's knowledge at the chalkboard or the overhead. There was a strong negative association between Overall Achievement and using the overhead for male students, while it makes little difference in achievement for female students. Also, the use of small groups had no association with overall achievement; however, there was a slight positive trend in achievement for female students. Hence, there was a negative relationship between using the overhead and overall achievement for male students, while working in small groups had little or no impact on the achievement score of students.

Moreover, when the teacher introduced a new mathematics lesson by relating the topic to what they know, there was a fairly strong negative association with achievement, as indicated by strength of the parameter estimate. Similarly, when students reported a greater frequency in discussing a practical or story problem related to real life, there was a negative relationship with the overall mathematics achievement score.

The analysis also included the association of the five content areas tested by TIMSS-R and the eight variables. In the algebra content area, when males reported a greater frequency of using the overhead, their performance level was lower. For female students the introduction of a new topic in algebra, or the relating of a new idea to a practical or everyday situation, correlated negatively with achievement levels, while for male students, these activities had little or no relationship with their achievement. The strength of the relationship was greater for female students than for male students even though it had a small impact on the model. The use of calculators had a high positive relationship with achievement in the model for algebra. The strength of the relationship was nearly equal for both genders as indicated by the parameter estimates. Another positive relationship was found between the variables, discussing homework, and algebra achievement for both female and male students, with it being slightly stronger for male students. There was a strong negative magnitude for the use of projects in algebra classes for both genders, with it being stronger negatively for male students. Also, for female students, there was a fairly strong negative relationship between algebra achievement and relating a new topic to what students know.

In the content area of measurement, working in pairs or small groups had the smallest magnitude regarding measurement achievement. The magnitude of the relationships were fairly strong negatively for the variables related to the use of projects for both genders, with the parameter estimate stronger for males than females. A negative association, for both genders, was also found for the teaching strategies of relating new information to practical and real life situations, and relating new information to what students know. The parameter estimates where nearly equal for both genders.

However, in geometry, a positive association was found between working in small groups or pairs when introducing a new topic with achievement; the magnitude of the relationship was greater for females than male students. Geometry was the one content area where each of the variables in the model had significant parameter estimates with the exception of male students going to the chalkboard; there was a negative association for female students. In addition, the more often male students "go to the overhead" the lower their achievement levels are in other content areas such as fractions and number sense, and algebra. The relationship between going to the overhead and achievement was strongly negative for male students, while for female students, there is a slightly negative impact, or no relationship.

Other results regarding reform-oriented practices include the use of calculators and projects. Generally, the model revealed that calculator use had a significant positive impact on achievement in all content areas and in overall achievement. Projects in the mathematics classroom on the other hand had a significant negative impact on achievement. The use of small groups had a slight positive impact, while indicated no significance in any area except for geometry. It appears that small group learning may be an effective instructional strategy for teaching geometry content.

In summary, achievement levels for both male and female students in the content areas correlate negatively with asking students to relate what they know to a new topic. The more often a teacher discussed a practical or story problem related to real life, when introducing a new topic, the lower the achievement of their students. Frequent discussion of homework correlates positively with the achievement in each of the five content areas for both genders. The greater the occurrence of working in small groups or in pairs has a slight positive relationship to achievement levels with female students tending to have higher achievement when working in small groups or pairs, although this is not a statistically significant finding. Using the chalkboard correlates negatively with achievement for female students in Geometry; while for male students, going to the overhead negatively correlates with their overall achievement, and their achievement in Algebra, Fractions/Number Sense, Data and Probability, and in Geometry. Frequent use of a calculator correlates positively with performance in each of the content areas, with the strongest relationship being in Geometry for both genders. The use of projects is negatively correlated with performance levels for both genders.

Discussion and Conclusion

Some aspects of reform-oriented instructional practices such as the use of the calculator and communication that deals with going over homework are supported by this study. However, it appears that communication activities that focus on students being the center of attention, such as going to the chalkboard, negatively relates to student achievement. Social cultural theory suggests that the use of communicative practices and projects improve concept attainment. These findings contradict that notion. It appears that spotlighting students at the board or the overhead may not be a productive strategy in an eighth grade mathematics classroom. A surprising result was related to the use of small groups, which did not have much of an impact on achievement except in geometry content. Middle school students may be discussing issues other than the mathematics topic at hand. House (in press) had a similar result regarding cooperative learning strategies (working together in pairs or small groups on a problem or project) was negatively related to mathematics achievement when used more frequently for introducing a new mathematics topic. Not surprisingly, discussing homework had a positive relationship with achievement. The discussion of homework problems often involves the students in selecting problems that were difficult or did not understand. This coupled with the fact that going over homework involves further practice and finding errors may improve learning of skills, with students describing how they completed a problem or exercise. Tripp (1998) reported that discussions of completed homework problems during class were effective for motivating students to do their homework.

Frequent use of the calculator apparently improves student achievement. The calculator is a tool that students may rely on when solving problems or exercises. Combined with homework, this tool can aide in raising achievement. The calculator may be used simply for calculating or checking answers, or it may be used to directly teach concepts. How the calculator was used in the classroom is not evident from the data presented in the database, and is a topic for further investigation.

This study found that using some reform-oriented mathematics teaching strategies has a negative relationship with achievement. Moreover, some of the strategies adversely impacted male students' achievement. Regardless of the mathematics content, the use of frequent projects was negatively associated with achievement. Depending how students understood the word "project" in the survey, the findings suggest that the use of projects should not be employed in middle school mathematics classrooms. It appears that there is some credence to the notion that reform-oriented instruction is not necessarily the most appropriate. For eighth grade students, a balance must be struck between practicing skills and developing conceptual understanding. The results of this study suggest further research in order to understand specific instructional conditions where cooperative learning may be less effective for student performance, such as when learning new mathematics topics.

Further study is also needed to assess the relationships between these instructional strategies and other types of outcome measures, such as conducting actual observations of classrooms to verify the students' self-reports in the survey. Qualitative analysis of student communication, as they present their work at the board or the overhead, is necessary to understand the negative relationship with achievement. Other questions arise from this study such as, Why is it, that when a new topic is introduced by asking eighth graders what they know about the topic, it negatively relates to achievement? Why does relating a new topic to a practical problem have an adverse relationship with achievement? Are cooperative learning groups a productive instructional strategy in middle school?

Some caution should be taken when considering the results for two reasons. One, given the moderate [R.sup.2] values, the classroom has many influences and students' achievement may be affected by other variables other than those chosen for the analysis. Two, the data are a result of students self-reporting and so should be interpreted as perhaps not being as accurate as might be expected. However, the findings do suggest areas for further study related to the teaching of mathematics at the middle grades.

TABLE 1 Multiple Regression Parameter Estimates for Female Students

Variable                           Overall Score     Algebra
                                   [beta]     SE     [beta]     SE

Intercept                          -470.35**  11.35  -464.83**  10.53
How often does the teacher ask
what you know about the topic
when beginning new math topics?     -11.43**   1.74   -11.26**   1.98
How often do you discuss
completed homework in your
mathematics lesson?                  15.44**   2.80    11.12**   2.15
How often do you work in pairs
or small groups in your
mathematics lesson?                   3.08     2.85     4.39     2.75
How often do students use the
board in your mathematics
lesson?                              -0.74     2.67    -3.47     2.55
How often do students use the
over head projector in your
mathematics lesson?                  -0.61     2.52    -2.04     2.65
How often do you discuss a
practical problem when beginning
new math topics?                     -9.35**   2.29    -4.20*    1.84
How often do you use calculators
in your mathematics lesson?          14.58**   2.04    13.18**   1.98
How often do you work on
mathematics projects in your
mathematics lesson?                 -13.46**   2.23   -15.83**   2.00
Model [R.sup.2]                                0.12              0.09

Variable                           Data ...            Fractions/
                                   Proportionality     Number Sense
                                   [beta]       SE     [beta]    SE

Intercept                          -499.29**    12.42  494.74**  11.73
How often does the teacher ask
what you know about the topic
when beginning new math topics?      -5.45*      2.29  -15.71**   1.56
How often do you discuss
completed homework in your
mathematics lesson?                  10.22**     3.43   10.84**   2.95
How often do you work in pairs
or small groups in your
mathematics lesson?                   4.47       3.83    4.03     3.02
How often do students use the
board in your mathematics
lesson?                               2.05       2.84    2.74     2.64
How often do students use the
over head projector in your
mathematics lesson?                  -0.89       2.92   -1.98     2.46
How often do you discuss a
practical problem when beginning
new math topics?                     -8.01**     2.25   -4.67     2.48
How often do you use calculators
in your mathematics lesson?          -8.56**     2.83   13.88**   2.35
How often do you work on
mathematics projects in your
mathematics lesson?                  -7.56**     2.79   -7.86**   2.46
Model [R.sup.2]                                  0.03             0.08

Variable                           Geometry          Measurement
                                   [beta]      SE    [beta]     SE

Intercept                          -429.90**  9.51   -431.00**  11.92
How often does the teacher ask
what you know about the topic
when beginning new math topics?     -10.60**  1.83     -8.54**   1.85
How often do you discuss
completed homework in your
mathematics lesson?                  11.07**  2.15     12.93**   2.48
How often do you work in pairs
or small groups in your
mathematics lesson?                   8.27**  2.71      1.84     3.04
How often do students use the
board in your mathematics
lesson?                              -6.13*   2.39     -1.74     2.96
How often do students use the
over head projector in your
mathematics lesson?                  -7.45**  2.12     -5.66     2.99
How often do you discuss a
practical problem when beginning
new math topics?                     -7.89**  1.86    -14.72**   1.95
How often do you use calculators
in your mathematics lesson?          21.13**  1.73      5.76*    2.15
How often do you work on
mathematics projects in your
mathematics lesson?                 -11.07**  2.35    -13.60**   2.36
Model [R.sup.2]                               0.12               0.08

Note: [beta] = parameter estimate; SE = Standard Error of estimate.
*p < 0.05
**p < 0.001

TABLE 2 Multiple Regression Parameter Estimates for Male Students

Variable                          Overall Score     Algebra
                                  [beta]     SE     [beta]     SE

Intercept                         -454.45**  11.11  -443.58**  11.06
How often does the teacher ask
what you know about the topic
when beginning new math topics?     -9.77**   1.71    -9.92**   1.68
How often do you discuss
completed homework in your
mathematics lesson?                 18.18**   2.52    15.48**   2.51
How often do you work in pairs
or small groups in your
mathematics lesson?                  0.87     3.22     2.79     3.48
How often do students use the
board in your mathematics
lesson?                              2.20     3.19    -0.72     2.96
How often do students use the
overhead projector in your
mathematics lesson?                 -9.40**   2.93   -10.48**   3.14
How often do you discuss a
practical problem when beginning
new math topics?                    -7.66**   2.52    -3.07     2.42
How often do you use calculators
in your mathematics lesson?         14.06**   2.40    13.80**   2.30
How often do you work on
mathematics projects in your
mathematics lesson?                -13.46**   2.23   -20.69**   2.57
[R.sup.2]                                     0.15              0.12

Variable                           Data ...          Fractions/
                                   Proportionality   Number Sense
                                   [beta]     SE     [beta]    SE

Intercept                          -501.19**  11.36  481.82**  10.30
How often does the teacher ask
what you know about the topic
when beginning new math topics?      -5.75*    1.94  -14.26**   1.79
How often do you discuss
completed homework in your
mathematics lesson?                  17.13*    2.67   11.57**   2.78
How often do you work in pairs
or small groups in your
mathematics lesson?                   5.79     3.13    1.48     3.41
How often do students use the
board in your mathematics
lesson?                               3.27     3.53    3.47     3.54
How often do students use the
overhead projector in your
mathematics lesson?                  -7.30*    3.09   -7.51**   2.88
How often do you discuss a
practical problem when beginning
new math topics?                     -8.09**   2.61   -1.68     2.11
How often do you use calculators
in your mathematics lesson?          -8.74**   2.85   12.85**   2.69
How often do you work on
mathematics projects in your
mathematics lesson?                 -11.18*    3.42  -11.88**   2.72
[R.sup.2]                                      0.06             0.09

Variable                           Geometry          Measurement
                                   [beta]     SE     [beta]    SE

Intercept                          -443.08**  10.14  434.35**  10.31
How often does the teacher ask
what you know about the topic
when beginning new math topics?      -7.67**   1.35   -8.91**   1.93
How often do you discuss
completed homework in your
mathematics lesson?                  13.02**   2.60   16.51**   2.50
How often do you work in pairs
or small groups in your
mathematics lesson?                   5.94**   2.32    0.59     2.80
How often do students use the
board in your mathematics
lesson?                              -4.71**   2.56    0.22     2.91
How often do students use the
overhead projector in your
mathematics lesson?                 -11.06**   2.25   -1.15     2.23
How often do you discuss a
practical problem when beginning
new math topics?                     -6.09**   1.75  -12.96**   2.51
How often do you use calculators
in your mathematics lesson?          21.56**   2.10    6.74*    2.12
How often do you work on
mathematics projects in your
mathematics lesson?                 -13.51**   2.17  -17.44**   2.59
[R.sup.2]                                      0.13             0.11

Note: [beta] = parameter estimate; SE = Standard Error of estimate.
*p < 0.05
**p < 0.001

TABLE 3 Summary of Results: Relating Instructional Strategy to
Achievement by Gender

Instructional Strategy           Content Area
                           Algebra  Data/Proportionality
                           M     F      M         F

Asking what students know
about the new topic.       -     -      -         -
Discussing completed
homework.                  +     +      +         +
Working in pairs or small
groups.                    nr    nr     nr        nr
Students using the board.  nr    nr     nr        nr
Students using the
overhead projector.        nr    +      nr        -
Discussing a practical
problem for a new topic.   -     nr     -         -
Using calculators.         +     +      +         +
Working on projects.       -     -      -         -

Instructional Strategy           Content Area
                           Fractions/Number Sense  Geometry  Measurement
                                 M          F      M     F    M     F

Asking what students know
about the new topic.             -          -      -     -    -     -
Discussing completed
homework.                        +          +      +     +    +     +
Working in pairs or small
groups.                          nr         nr     +     +    nr    nr
Students using the board.        nr         nr     -     nr   nr    nr
Students using the
overhead projector.              nr         -      -     -    nr    nr
Discussing a practical
problem for a new topic.         nr         nr     -     -    -     -
Using calculators.               +          +      +     +    +     +
Working on projects.             -          -      -     -    -     -

Note: (+) refers to "higher achievement;" (-) refers to "lower
achievement;" (nr) refers to "no relationship."

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James A. Telese

University of Texas, Brownsville