Using a Technology-Enriched Environment to Improve Higher-Order Thinking Skills Michael H. Hopson Stephen F. Austin State University
Richard L. Simms and Gerald A. Knezek The University of North Texas
Abstract This study examined the effect of a technology-enriched classroom on student development of higher-order thinking skills and student attitudes toward computers. A sample of 80 sixthgrade and ad fifth-grade students was tested using the Ross Test of Higher Cognitive Processes and surveyed using the Computer Attitude Questionnaire. The creation of a technologyenriched classroom environment appears to have had a positive effect on student acquisition of higher-order thinking skills. This study identified several implications related to classroom design to enhance the development of higher-order thinking skills. Teachers reported that the technology-enriched classroom differed from the traditional classroom in several significant ways. (Keywords: classroom environment, higher-order thinking skills, instructional change, instructional technology.)
The need to prepare students for adulthood is a recurring theme throughout educational reform. The advent ofthe Information Age has made the development of problem solving, critical thinking, and higher-order thinking skills crucial to future success (Fontana, Dede, White, & Cates, 1993; Morgan, 1996; Norris & Poirot, 1990; Ramirez & Bell, 1994). Hence, experiences that engage students at higher levels of Bloom's Taxonomy (analysis, synthesis, evaluation) need to become common practice (Morgan). According to Harris (1996), "Information Age citizens must learn not only how to access information, but more importantly how to manage, analyze, critique, cross-reference, and transform it into usable knowledge" (p. 15). Kelman (1989) identifies higher-order thinking skills as one ofthe instructional areas that could be improved by using the computer. Salomon (1990) concludes that for the computer to be an effective classroom tool, "most everything in the classroom needs to change in a way that makes curriculum, learning activities, teacher's behavior, social interactions, learning goals, and evaluation interwoven into a whole newly orchestrated learning environment" (p. 51). In light of what is known about learning, using the computer and other technology as tools for meaningful projects seems reasonable as a method for engaging students in problem solving and critical thinking (Muir, 1994; Peck & Dorricot, 1994). Ragsdale (1989) challenged educators to teach with the computer because "tool" applications are independent of subject matter and can be used for curriculum integration across grade levels and subject areas.
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In his review of earlier studies, Atkins (1993) noted that the richer and more comprehensive the interactions between the learner and the material, the more the learner learns. Levine (1990) poinrs out that with increased placement of computers in the classroom has come an increased interest in assessing their potentially diverse efFects and that "investigators typically find themselves searching for new study designs and data collection strategies" (p. 461). Recent studies suggest that research in the field has moved beyond a focus on the computer to an interest in designing an environment that fosters within students the disposition for critical thinking (Facione, Facione, & Sanchez, 1994; Taube, 1995; Wiburg, 1995-1996). This restructuring ofthe classroom includes the use of computers to provide active learning, authentic tasks, challenging work, complex problem solving, and higher-order thinking skills (Dalton & Goodrum, 1991; David, 1993). Dede (1990) suggests that higher-order thinking skills for structured inquiry are best acquired where: 1. learners construct knowledge rather than passively ingest information; 2. sophisticated information-gathering tools are used to stimulate the learner to focus on testing hypotheses rather than on plotting data; 3. there is collaborative interaction with peers, similar to team-based approaches underlying today's science; and 4. evaluation systems measure complex, higher-order skills rather than simple recall of facts. According to Ryan (1991), the perceived need for improving instruction and student achievement through the use of computer technologies has challenged educational administrators to fmd optimal ways of integrating computers into learning environments. Research has not clearly delineated the relationship between implementation characteristics and increased academic achievement (Chen, 1985; Hoot, 1986; Stennett, 1985). In their comparative study ofthe use ofthe computer for improving higherorder thinking skills. Cousins and Ross (1993) conclude "there is little research which would inform practice as to the use ofthe computer as a tool to accomplish pre-specified tasks" (p. 94). An additional conclusion is that studies designed to measure change in student performance are needed, specifically in higher-order thinking skills. METHODOLOGY Purpose of the Study We investigated the effect of a technology-enriched classroom on student development of higher-order thinking skills and student attitudes toward computers. We defined higher-order thinking skills as those cognitive skills that allow students to function at the analysis, synthesis, and evaluation levels of Bloom's Taxonomy. We addressed the following research questions: 1. Do students in a technology-enriched classroom demonstrate better use of higher-order thinking skills than students in a traditional classroom? 2. Do attitudes toward computers differ between students in a technologyenriched classroom and students in a traditional classroom? 110
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Sample Participants included fifth- and sixth-grade students in a suburban North Central Tcxa.s school district. The treatment group was comprised of students who were enrolled in the technology-enriched classroom magnet program in the 1996-1997 and 1997-1998 school years. These students came from each of the district's six elementary campuses selected at random from among the students who applied for the program. Comparison groups included students not accepted into the magnet program and students from comparable campuses without a technology-enriched curriculum. Instrumentation The Ross Test of Higher Cognitive Processes was selected because of its stated purpose to judge the effectiveness of curricula or instructional methodology designed to teach the higher-order thinking skills of analysis, synthesis, and evaluation. The reliability coefficients for the 105-item test were obtained using testretest and split-half procedures. The reported test-retest reliability coefficient was .94, while the coefficient derived from the split-half procedure is reported as .92. The test validity was determined by correlation with chronological age and was found to be r = .674. The Computer Attitude Questionnaire (CAQ, Knezek, Christensen, & Miyashita, 1998) was used to determine student attitudes toward the computer. The questionnaire used 65 Likert-type questions for eight psychological dispositions. The reliabiliry for the eight attitude measures ranges from 0.80 to 0.86 (Knezek & Christensen, 1996). The reliability coefficients were calculated using data from 1995 (A' = 588). The values reflect the internal consistency ofthe instrument and are all within the very good range, according to guidelines for research scales provided by DeVilli.s (1991). Research Design This study used a po.sttest and a quasi-experimental design (Campbell & Stanley, 1981). The treatment and comparison groups were given the Ross Test of Higher Cognitive Processes and the Computer Attitude Questionnaire. Four distinct groups were identified for the study. Sixth-grade srudents (20 male, 16 female) who had been in the program for one year and five months comprised Treatment Group 1, and fifth-grade students (20 male, 23 female) who had been in the program for five months constituted Treatment Group 2. Sixthgrade students (21 male, 22 female) enrolled in social studies classes at the middle school were selected for Comparison Group 1. Students in Comparison Group 1 were selected from preexisting middle school classes to which they had been randomly assigned by a computer-scheduling program. Students for Comparison Group 2 (23 male, 21 female) were identified at an elementary school with comparable demographics and selected at random from all fifthgrade students. The treatment groups were instructed using the district's fifth-grade curriculum in a technology-rich environment, and they were provided access to the computer as a tool for learning. Treatment classrooms were equipped with one computer for every two students. Treatment teachers were trained in the use Journal of Research on Technology in Education
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of- and equipped with a multimedia teaching station that was used for direct instruction. Students were taught to use spreadsheet, database, and word processing software. Students were required to use these tools to take notes, produce assignments, and construct projects. In addition, the classroom was equipped with Internet access and "electronic resource materials," such as a thesaurus, encyclopedia, and adas. Additionally, students were taught to use a scanner, QuickTake Camera, and the multimedia presentation software HyperStudio (1989-2000). The comparison groups were instructed in a traditional classroom setting using the district's prescribed curriculum for the fifth grade. The teachers for the comparison groups were not trained in the use of technology, and no computerbased teaching stations were available to them. The comparison group classrooms had no computers. The only exposure to technology for students in the comparison groups was through the campus computer labs that were used for computer literacy and remediation. Data Analysis A univariate analysis of variance was used to establish initial equivalence for the comparison and treatment groups on the Ross Test (Table 1). The results of the ANOVA indicated no significant difference bervveen the fifth-grade treatment and comparison groups. Thus, a one-way ANOVA conducted on posttest data was used. The differences between the sixth-grade groups was significant, so an analysis of covariance was required for these students on the Ross Test results. The Computer Attitude Questionnaire data for both grades were analyzed using Analysis of Variance. Table 1. Comparison of Ross Test Scores for Group One and Group Two Sum ol Squares
Variable Sixth-gradt' TAAS-" I'ifth-grade 'lAAS
Between Within Berween Within
' TAAS = TexiU Aansmmtof is significant fp < 0.0.1).
Significance of/"
df
F-value
1 1 242.957 9797.517 77 1 .107 5576.786 82
9.764
0.003**
0.002
0.968
AcademicSkilU. '* I heV-siiore for the hctuvcn-groiips roivparison
RESULTS Research Question 1: Do students in a technology-enriched classroom demonstrate better use of higher-order thinking skills than do students in a traditional classroom? Statistics for all groups on the Ross Test are reported in Table 1. Mean scores are shown in Table 2. Maximum scores for each subtest are: analysis, 36; synthesis, 39; and evaluation, 30. 112
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Table 2, Mean Scores for the Ross Test of Higlier Cognitive Ability
Sixth grade
Group
Subtext
Comparison
Analysis Synthesis Evaluation Analysis .Synthesis Evaluation .A.naly.sis Synthesis Evaluation Analysis SynthesJ.s Evaluation
Ireaiment
Fikh grade
Comparison
Treatment
M
SD
22.53 23.00 18.93 23.89 25.92 22.19 19.05 23.05 15.21 19.36 21.47 20.36
15.07 6.43 4.06 4.28 5.70 3.82 5.23 5.17 4.72 4.22 4.67 3.49
Analysis ol variance results for Grade 6 (Table 3) indicate tluit the difference between the .scores for the treatment group and the comparison group on the evaluation subtestwas significant at tlie/> < .01 level. Treatment group students exhibited a higher level of evaluation skill as measured by the Ro.ss Test. There wa.s no significant difference in the performance of the rwo groups on the analysis and synthesis subtexts. Table 3. Comparison of Ross Test Results for Sixtb-Grade Treatment and Comparison Groups
Variable
Sum of Squares
df
F-valuc
Significance of/'
Analysis Synthesis Evaluation
3.047 28.697 122.110
1 1 1
0.024 0.885 8.111
0.878 0.350 0.006*^'
' ' Sig/iiftc/mt at p < 0.01.
The results tor Grade 5 (lable 4) indicate that the difference between the scores lot the treatment group atid the comparison group on the evaluarion subtest was significant at (he p < .01 level. As with the sixth-grade students, treatment group scores were higher than comparison group scores. ] here was no significant difference in the performance of the two groups on the analysis and sytithesis subtests. Research Question 2: Do attitudes toward computers difiFer between students in a technology-enricbed classroom and students in a traditional classroom? Group mean scores are presented in Table 5. The results of the ANOVA (onetailed) for the Computer Attitude Questionnaire for grade (Table 6) indicated no significant difference between the treatment and comparison groups on any Journal of Research on Technology in Education
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Table 4. Comparison of Ross Test Results for Fifth-Grade Treatment and Comparison Groups
Analysis Synthesis Evaluation
Variable
Sum of Scjuates
Between Within Between Within Between Within
2.060 1904.216 54.298 2053.105 574.227 1451.383
df
F-valiie
Sigriificarux; of F
1
0.092
0.762
1
2.248
0.137
85 1 85
33.629
85
0.000*'
'* Significiint at p < 0.01.
of the eight attributes measured. As previously noted in the analysis of the Ross lest data, the extensive variation within the groups for sixth-grade scores made the results difficult to interpret. The analysis of fifth-grade student scores (Table 7) indicated that the treatment group scores were significantly higher {p = .05) on subtests measuring iniporcance, motivation, and creativity. No significant difference."; were fotind on enjoyment, study habits, empathy, anxiety (reported as le.ss anxiety), or seclusion (reported as less seclusion). DISCUSSION This study added to the limited research on the use of computers to enhance the student development of higher-order thinking skills. It provides data that may be used to create a new paradigm for classroom organization and structure. The results will also be useful for educators who are formulating long-range technology plans. This study was limited by the characteristics of the population. The suburban district's profile was not comparable to that of the state or narion; therefore, generalizations will require additional research. A second concern was the inability to control for the effect of personal and home computers on the compari.son group. In addition, the higher-order thinking skills studied were limited to analysis, synthesis, and evaluation, as identified by Bloom and measured by the instrument. The creation of a technology-enriched cla.ssroom environment appears to bave had a minimal but positive effect on student acquisition of higher-order thinking skills. Although the difference in .scores was not significant for every level of Bloom's Taxonomy, the scores were generally higher for analysis and .synthesis and significantly higher for evaluation. The argument can be made that the minimal effect was less related to an ineffective treatment and more a result of the short duration of the treatment (20 weeks) and the inability of the study to control for home use of the computer.
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Table 5. Computer Attitude Questionnaire Mean Scores Sixth Grade Variable Computer importance Computer enjoyment Motivation Study habits Empathy Creativity Computer anxiety Computer seclusion
M
Treatment (Comparison Treatmenc Comparison Treatinent Comparison Trfainient Comparison Treatment Comparison freatment Comparison Treatment Comparison Treatment Comparison
2.8750 2.7562 2.7708 2.7069 2.2118 2.1743 2.4844 2.4534 2.8531 2,8828 3.1707 3.2042 1.9453 2.0905 2.4447 2.4814
Fifth Grade
SD
M
SD
.3513 .4032 .2257 .2200 .2417 .2778 .2371 .2556 .3910 .5086 .4166 .3618 .3283 .4373 .2367 .2636
2.7352 2.5679 2.6721 2.6531 2.33.33 2.2222 2.6488 2.5878 2.9366 2.8537 3.3002 3.0882 1.9453 2.0122 2.5253 2.4916
.4245 .4754 .2249 .2572 ..3033 .3012 .2873 .2629 .3878 .4032 .3680 .5187 .3500 .3706 .2383 .2407
Table 6. Comparison of Computer Attitude Questionnaire Results for Sixth-Grade Treatment and Comparison Groups
Variable Computer importance C'ompiiter enjoyment Motivation Study habits Empathy (Creativity Computer anxiety Computer seclusion
Between Within Between Within Between Within Between Within Between Within Between Within Between Within Between Within
Sum of Squares 0.291 13.092 .0843 4.337 .02896 6.209 .01972 5.466 .01811 19.482 .02324 2.844 0.435 14.242 .02781 5.696
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df
/"-value
Significance ofF
1 88 1 88 1 88 1 88 1 88 1 88 1 88 1 88
1.958
0.083
1.711
0.097
0.410
0.262
0.318
0.288
0.082
0.388
0.159
0.346
2.687
0.053
0.430
0.257
115
Table 7. Comparison of the Results of the Computer Attitude Questionnaire for Fifth-Grade Treatment and Comparison Croups
Variable Computer importance Computer enjoyment Motivation Study habits Etiipathy Creativicy Computer anxiety (Computer seclusion
Between Within Between Within Between Within Between Within Between Wivhin Between Within Between Within Between Within
Sum of Squares 0.573 16.247 7.377E-03 4.670 0.253 7.309 7.622E-02 6.066 0.141 12.517 0.92 16.182 6.879E-02 10.393 2.338E-02 4.589
df
F-valite
Significance off
1 80 1 80 1 80 1 80 1 80
2.824
0.049'
0.126
0.362
2.770
0.050*
1.005
0.160
0.901
0.173
1
4.555
0.018'
80 1 80
0.5.30
0.235
1
0.408
0.263
80
' Significiuit lit p < 0.05 (aiie-taileii test).
The teachers reported that the technology-enriched classroom differed from the traditional classroom in several significant ways. The learning was more student centered and less teacher/textbook driven. The environment facilitated the use of cooperative groups and student participation focused on application rather than knowledge acquisition. Current research in the Fields of cognition and brain theory are reflected in this shift to a more learner-centered instrtictional paradigm in which students actively manipulate information in a variety of contexts from a number of different re.sources in order to solve meaningful and relevant problems (Ramirez & Bell, 1994). This problem-solving environment resulted from the introduction of information management and collaboration technology into the classroom (Schwen, Goodrum, & Dorsey, 1993). The almost exponential increase in available sources of information in the technology-enriched classroom created the need for student learning to be assessed using non-traditional methods. The use of individual student products and group projects replaced tests and homework as the primary assessment tools. There are several implications for this study related to the design of classrooms to enhance the development of higher-order thinking skills. We have identified technology as the catalyst for restructuring and redesigning the classroom to create an environment that promotes and encourages the development of the higher-order skill evaluation. In this study, technology was the tool tbat allowed the students to move beyond knowledge acquisition to knowledge ap-
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plication. In earlier studies. Peck and Dorricot (1994) and Van Dusen and Worthen (1995) reported that the use of technology applications allowed students to organize, analyze, interpret, and evaluate their work. As the students began to use the technological resources to manage their learning, the role of the teacher was transformed from lecturer to guide. The availability of vast amounts of easily accessible information freed the teacher from the role of purveyor of facts and allowed the teacher to encourage the students to use the computer as a tool for problem solving and decision making. The technology-enriched classroom environment had a significant effect on the fifth-grade student attitudes oi computer importance, the perceived value or significance of knowing how to use computers,/) = .05; motivation, including attitudes such as unceasing effort, perseverance, and never giving up, p = .05; and self-reported creative tendencies, the inclination toward exploring the unknown, taking individual initiative, and finding unique solutions, p = .02, as measured by the Computer Attitude Questionnaire. Exposure to technology and training in its use results in a more positive attitude relative to computer importance. Such a positive attitude indicates that once students are successful using technology and recognize the associated benefits, they will choose to continue using it as a learning tool. More positive attitudes toward motivation and creativity indicate that, when provided with technology, students are more likely to take control of their learning, stay focused until the task is complete, and pursue more obscure and hypothetical solutions to problems. • Contributors Michael H. Hopson is an assistant professor of secondar)' education and educational leadership at Stephen F. Austin State University in Nacogdoches, Texas. He received his PhD in curriculum and instruction in 1998 from the University of North Texas. Dr. Hopson has 26 years of public school experience as teacher, middle .school principal, high school principal, and assistant superintendent. His research interests are educational technology and administrative organization. Richard L. Simms is a professor of teacher education and administration at the University of North Texas (UNT) in Denton. He has been a professor. Teacher Corps Director, and administrator at UNT since 1970. Prior to earning his doctorate at the University of Missouri, he was a public school teacher and administrator. His writings have appeared in many ofthe major education journals. Gerald A. Knezek is a professor of technology and cognition and coordinator ofthe doctoral program in educational computing at UNT. He is principal investigator ofthe U.S. Department of Education Technology Innovation Challenge Crant R303A99030, external evaluation for 1999-2004, and lead principal investigator for the U.S. Department of Education Preparing Tomorrow's Teachers to Use Technology (PT') Millennium Project Capacity and Implementation Grants (P342A990474 & P342A000123A) for 1999-2003. He held the Matthews Chair for Research in Education at the University of North Texas from 1995-1997. He was a Fulbright Scholar at the Tokyo Institute ofTechnol-
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ogy and Japan's National Center for University Entrance Examinations during 1993-1994. He received his BA in mathematics and the Social Sciences from Dartmouth College and his MEd and PhD degrees in educational psychology from the University of Hawaii. (Address: Dr. Michael H. Hopson, Stephen F. Austin State University, College of Education, Department of Secondary Education, Nacogdoches, TX 75962-3055;
[email protected].) References Atkins, M. J. (1993). Evaluating interactive technologies for learning./o«r«(z/ of Curriculum Studies, 26{4), 333-342. Campbell, D. T., & Stanley, J. (1981). Experimental and quasi-experimental designs for research. Chicago: Rand McNally. Chen, M. (1985). A macro-focus on microcomputers: Eight utilization and effects issues. In M. Chen & W. Paisley (Eds.), Children and microcomputers: Research on the newest medium (pp. 37—58). Beverly Hills, CA: Sage. Cousins, J. B., & Ross, J. A. (1993). Improving higher order thinking skill by teaching "with" the computer: A comparative study. Journal ofResearch on Computing in Education, 26{\), 94-115. Dalton, E. W., & Coodrum, D. A. (1991). The effects of computer programming on problem-solving skills and attitudes. Journal of Educational Computing Research, 7(4), 483-506. David, J. L. (1993, April). Realizing the promise of technology: The needfor systemic education reform. Paper presented at the meeting of the American Educational Research Association, Atlanta, GA. Dede, C. (1990). Imaging technology's role in restructuring for learning. In K. Sheingold & M. S. Tucker (Eds.), Restructuringfor learning with technology (pp. 49-72). New York: Center for Technology in Education, Bank Street College of Education, and National Center on Education and the Economy. DeVillis, R. E (1991). Scale development: Theory and applications. Newbury Park, CA: Sage. Facione, N. C , Facione, P. A., & Sancez, C. A. (1994). Critical thinking as a measure of competent clinical judgment: The development of the California critical thinking dispositions itwentory. Journal ofNursing Education, 33(8), 345-350. Fontana, L. A., Dede C , White, C. S., & Cates, W. M. (1993;. Multimedia: A gateway to higher-order thinking skills. Fairfax, VA: George Mason University, Center for Interactive Educational Technology. Harris, J. (1996). Information is forever in formation, knowledge is the knower: Global connectivity in K—12 classrooms. Computers in the Schools, 72(1-2), 11-22. Hoot, J. D. (1986). Computers in early childhood education: Issues and practices. Englewood Cliffs, NJ: Prencice-Hall. HyperStudio [Computer sofiware]. (1989-2000). Torrance, CA: Knowledge Adventure, Inc. Kelman, P. (1989, June). Alternatives to integrated instructional systems. Paper presented at the National Educational Computing Conference, Nashville, TN. 1 18
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Knezek, G., & Christensen, R. (1996, January). Validating the computer attitude questionnaire (CAQ). Paper presented at the annual meeting of che Southwest Educational Research Association, New Orleans, LA. Knezek, G., Christensen, R., & Miyashita, K. (1998). Instruments for assessing attitudes toward information technology. Denton: Texas Center for Educational Technology. Levine, H. (1990). Models of qualitative data use in the assessment of classroom-based microcomputer education progTums. Joumal of Educational Computing Research, 6, 461—477. Morgan, T. (1996, February). Using technology to enhance learning: Changing the chunks. Learning & Leading with Technology, 23(5), 49-51. Muir, M. (1994). Putting computer projects at the heart ofthe curriculum. Educational Leadership, 5(7), 30-33. Norris, C , & Poirot, J. L. (1990). Problem solving and critical thinking for computer science educators [Monograph]. Eugene, OR: International Society for Technology in Education. Peck, K. L., & Dorricot, D. (1994). Why use technology? Educational Leadership, 5l{7), 11-14. Ragsdale, R. G. (1989). Teacher development: The implications of using computers in education. Canadian Journal ofEducation, 14{A), 444-456. Ramirez, R., & Bell, R. (1994). Byting back: Policies to support the use oftechnology in education. Oak Brook, IL: North Central Regional Educational Laborator}'. Ryan, A.W. (1991). Meta-analysis of achievement effects of microcomputer applications in elementary schools. Educational Administration Quarterly, 27(2),161-184. Salomon, G. (1990). The computer lab: A bad idea now sanctified. Educational Technology, 30(10), 50-52. Schwen, T, Goodrum, D., & Dorsey, T. (1993). On the design of an enriched learning and information environment. Educational Technology, 33, 5-9. Stennet, R. G. (1985/1. Computer assisted iftstruction: A revieiv ofthe reviews. London: The Board of Education for the City of London. (ERIC No. ED 260 687) Taube, K. (1995, April). Critical thinking ability and disposition as factors of performance on a written critical thinking test. Paper presented at the annual meetingof the American Educational Research Association, San Francisco. Van Dusen, L. M., & Worthen, B. (1995). Can integrated in.structional technology transform the classroom? Educational Leadership, 53(2), 28-33. Wiburg, K. (1995-1996, December/January). Changing teaching with technology. Learning & Leading with Technology, 23{4), 46-48.
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354
A. Tamura: Coordinatewise domain scaling algorithm for M-convex function minimization
21. Lovasz, L.: "Submodular functions and convexity." In Mathematical Programming — The State of the Art A. Bachem, M. Grotschel and B. Korte, (eds.), pp. 235-257, Springer-Verlag, Berlin, 1983 22. Moriguchi, S., Murota, K., Shioura, A.: Scaling algorithms for M-convex function minimization. IEICE Trans. Fundamentals E85-A, 922-929 (2002) 23. Murota, K.: Convexity and Steinitz's exchange property. Adv. Math. 124, 272-311 (1996) 24. Murota, K.: Discrete convex analysis. Math. Programming 83, 313-371 (1998) 25. Murota, K.: Submodular flow problem with a nonseparable cost function. Combinatorica 19, 87-109 (1999) 26. Murota, K.: Matrices and Matroids for Systems Analysis, Springer-Verlag, Berlin, 2000 27. Murota, K.: Discrete Convex Analysis, Society for Industrial and Applied Mathematics, Philadelphia, 2003 28. Murota, K., Shioura, A.: Quasi M-convex and L-convex functions — quasiconvexity in discrete optimization. Discrete Appl. Math. 131,467-494 (2003) 29. Murota, K., Tamura, A.: New characterizations of M-convex functions and their applications to economic equilibrium models with indivisibilities. Discrete Appl. Math. 131,495-512 (2003) 30. Murota, K., Tamura, A.: Application of M-convex submodular flow problem to mathematical economics. Japan J. Indust. Appl. Math. 20, 257-277 (2003) 31. Rockafellar, R. T.: Convex Analysis, Princeton University Press, Princeton, 1970 32. Saito, K., Oshiro, T.: On M-convex function minimization algorithms (in Japanese). Bachelor Thesis, Dept. of Management Science, Tokyo University of Science, (2003) 33. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Sen B 80, 346-355 (2000) 34. Shioura, A.: Minimization of an M-convex function. Discrete Appl. Math. 84, 215-220 (1998) 35. Shioura, A.: Fast scaling algorithms for M-convex function minimization with application to the resource allocation problem. Discrete Appl. Math. 134, 303-316 (2004)