conceptual understanding and problem solving

In 1980 the populations of Town A and Town B were 5,000 and 6,000, respectively. The 1990 populations of Town A and Town B were 8,000 and 9,000, respectively.

Brian claims that from 1980 to 1990 the populations of the two towns grew by the same amount. Use mathematics to explain how Brian might have justified his claim.

Darlene claims that from 1980 to 1990 the population of Town A grew more. Use mathematics to explain how Darlene might have justified her claim.

SOURCE: 1996 NAEP assessment. Cited in Wearne and Kouba, 2000, p. 186. Used by permission of National Council of Teachers of Mathematics.

toward mathematics, beliefs about one’s own ability, and beliefs about the nature of mathematics. In general, U.S. boys have more positive attitudes toward mathematics than U.S. girls do, even though differences in achievement between boys and girls are, in general, not as pronounced today as they were some decades ago. 64 Girls’ attitudes toward mathematics also decline more sharply through the grades than those of boys. 65 Differences in mathematics achievement remain larger across groups that differ in such factors as race, ethnicity, and social class, but differences in attitudes toward mathematics across these groups are not clearly associated with achievement differences. 66

The complex relationship between attitudes and achievement is well illustrated in recent international studies. Although within most countries, positive attitudes toward mathematics are associated with high achievement, eighth graders in some East Asian countries, whose average achievement in mathematics is among the highest in the world, have tended to have, on average, among the most negative attitudes toward mathematics. U.S. eighth

graders, whose achievement is around the international average, have tended to be about average in their attitudes. 67 Similarly, within a country, students who perceive themselves as good at mathematics tend to have high levels of achievement, but that relationship does not hold across countries. In Asian countries, perhaps because of cultural traditions encouraging humility or because of the challenging curriculum they face, eighth graders tend to perceive themselves as not very good at mathematics. In the United States, in contrast, eighth graders tend to believe that mathematics is not especially difficult for them and that they are good at it. 68

Data from the NAEP student questionnaire show that many U.S. students develop a variety of counterproductive beliefs about mathematics and about themselves as learners of mathematics. For example, 54% of the fourth graders and 40% of the eighth graders in the 1996 NAEP assessment thought that mathematics is mostly a set of rules and that learning mathematics means memorizing the rules. On the other hand, approximately 75% of the fourth graders and 75% of the eighth graders sampled reported that they understand most of what goes on in mathematics class. The data do not indicate, however, whether the students thought they could make sense out of the mathematics themselves or depended on others for explanations.

Despite the finding that many students associate mathematics with memorization, students at all grade levels appear to view mathematics as useful. The 1996 NAEP revealed that 69% of the fourth graders and 70% of the eighth graders agreed that mathematics is useful for solving everyday problems. Although students appear to think mathematics is useful for everyday problems or important to society in general, it is not clear that they think it is important for them as individuals to know a lot of mathematics. 69

Proficiency in Other Domains of Mathematics

Although our discussion of mathematical proficiency in this report is focused on the domain of number, the five strands apply equally well to other domains of mathematics such as geometry, measurement, probability, and statistics. Regardless of the domain of mathematics, conceptual understanding refers to an integrated and functional grasp of the mathematical ideas. These may be ideas about shape and space, measure, pattern, function, uncertainty, or change. When applied to other domains of mathematics, procedural fluency refers to skill in performing flexibly, accurately, and efficiently such procedures as constructing shapes, measuring space, computing probabilities, and describing data. It also refers to knowing when and how to use

The five strands apply equally well to other domains of mathematics such as geometry, measurement, probability, and statistics.

those procedures. Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them whether the problems arise in the context of number, algebra, geometry, measurement, probability, or statistics. Similarly, the capacity to think logically about the relationships among concepts and situations and to reason adaptively applies to every domain of mathematics, not just number, as does the notion of a productive disposition. The tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics applies equally to all domains of mathematics. We believe that proficiency in any domain of mathematics means the development of the five strands, that the strands of proficiency are interwoven, and that they develop over time. Further, the strands are interwoven across domains of mathematics in such a way that conceptual understanding in one domain, say geometry, supports conceptual understanding in another, say number.

All Students Should Be Mathematically Proficient

Becoming mathematically proficient is necessary and appropriate for all students. People sometimes assume that only the brightest students who are the most attuned to school can achieve mathematical proficiency. Those students are the ones who have traditionally tended to achieve no matter what kind of instruction they have encountered. But perhaps surprisingly, it is students who have historically been less successful in school who have the most potential to benefit from instruction designed to achieve proficiency. 70 All will benefit from a program in which mathematical proficiency is the goal.

Historically, the prevailing ethos in mathematics and mathematics education in the United States has been that mathematics is a discipline for a select group of learners. The continuing failure of some groups to master mathematics—including disproportionate numbers of minorities and poor students—has served to confirm that assumption. More recently, mathematics educators have highlighted the universal aspects of mathematics and have insisted on mathematics for all students, but with little attention to the differential access that some students have to high-quality mathematics teaching. 71

One concern has been that too few girls, relative to boys, are developing mathematical proficiency and continuing their study of mathematics. That situation appears to be improving, although perhaps not uniformly across

grades. The 1990 and 1992 NAEP assessments indicated that the few gender differences in mathematics performance that did appear favored male students at grade 12 but not before. These differences were only partly explained by the historical tendency of male students to take more high school mathematics courses than female students do, since that gap had largely closed by 1992. In the 1996 NAEP mathematics assessment, the average scores for male and female students were not significantly different at either grade 8 or grade 12, but the average score for fourth-grade boys was 2% higher than the score for fourth-grade girls. 72

With regard to differences among racial and ethnic groups, the situation is rather different. The racial/ethnic diversity of the United States is much greater now than at any previous period in history and promises to become progressively more so for some time to come. The strong connection between economic advantage, school funding, and achievement in the United States has meant that groups of students whose mathematics achievement is low have tended to be disproportionately African American, Hispanic, Native American, students acquiring English, or students located in urban or rural school districts. 73 In the NAEP assessments from 1990 to 1996, white students recorded increases in their average mathematics scores at all grades. Over the same period, African American and Hispanic students recorded increases at grades 4 and 12, but not at grade 8. 74 Scores for African American, Hispanic, and American Indian students remained below scale scores for white students. The mathematics achievement gaps between average scores for these subgroups did not decrease in 1996. 75 The gap appears to be widening for African American students, particularly among students of the best-educated parents, which suggests that the problem is not one solely of poverty and disadvantage. 76

Students identified as being of middle and high socioeconomic status (SES) enter school with higher achievement levels in mathematics than low-SES students, and students reporting higher levels of parental education tend to have higher average scores on NAEP assessments. At all three grades, in contrast, students eligible for free or reduced-price lunch programs score lower than those not eligible. 77 Such SES-based differences in mathematics achievement are greater among whites than among other racial or ethnic groups. 78 Some studies have suggested that the basis for the differences resides in the opportunities available to students, including opportunities to attend effective schools, 79 opportunities afforded by social and economic factors of the home and school community, 80 and opportunities to get encouragement to continue the study of mathematics. 81

Goals for mathematics instruction like those outlined in our discussion of mathematical proficiency need to be set in full recognition of the differential access students have to high-quality mathematics teaching and the differential performance they show. Those goals should never be set low, however, in the mistaken belief that some students do not need or cannot achieve proficiency. In this day of rapidly changing technologies, no one can anticipate all the skills that students will need over their lifetimes or the problems they will encounter. Proficiency in mathematics is therefore an important foundation for further instruction in mathematics as well as for further education in fields that require mathematical competence. Schools need to prepare students to acquire new skills and knowledge and to adapt their knowledge to solve new problems.

The currency of value in the job market today is more than computational competence. It is the ability to apply knowledge to solve problems. 82 For students to be able to compete in today’s and tomorrow’s economy, they need to be able to adapt the knowledge they are acquiring. They need to be able to learn new concepts and skills. They need to be able to apply mathematical reasoning to problems. They need to view mathematics as a useful tool that must constantly be sharpened. In short, they need to be mathematically proficient.

Students who have learned only procedural skills and have little understanding of mathematics will have limited access to advanced schooling, better jobs, and other opportunities. If any group of students is deprived of the opportunity to learn with understanding, they are condemned to second-class status in society, or worse.

A Broader, Deeper View

Many people in the United States consider procedural fluency to be the heart of the elementary school mathematics curriculum. They remember school mathematics as being devoted primarily to learning and practicing computational procedures. In this report, we present a much broader view of elementary and middle school mathematics. We also raise the standard for success in learning mathematics and being able to use it. In a significant and fortuitous twist, raising the standard by requiring development across all five strands of mathematical proficiency makes the development of any one strand more feasible. Because the strands interact and boost each other, students who have opportunities to develop all strands of proficiency are more likely to become truly competent with each.

We conclude that during the past 25 years mathematics instruction in U.S. schools has not sufficiently developed mathematical proficiency in the sense we have defined it. It has developed some procedural fluency, but it clearly has not helped students develop the other strands very far, nor has it helped them connect the strands. Consequently, all strands have suffered. In the next four chapters, we look again at students’ learning. We consider not just performance levels but also the nature of the learning process itself. We describe what students are capable of, what the big obstacles are for them, and what knowledge and intuition they have that might be helpful in designing effective learning experiences. This information, we believe, reveals how to improve current efforts to help students become mathematically proficient.

Alexander, P.A., White, C.S., & Daugherty, M. (1997). Analogical reasoning and early mathematics learning. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 117–147). Mahwah, NJ: Erlbaum.

Alibali, M.W. (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology , 35 , 127–145.

Ansell, E., & Doerr, H.M. (2000). NAEP findings regarding gender: Achievement, affect, and instructional experiences. In E.A.Silver & P.A.Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 73– 106). Reston, VA, National Council of Teachers of Mathematics.

Backer, A., & Akin, S. (Eds.). (1993). Every child can succeed: Reading for school improvement . Bloomington, IN: Agency for Instructional Television.

Baddeley, A.D. (1976). The psychology of memory . New York: Basic Books.

Ball, D.L., & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D.Phillips (Ed.), Constructivism in education: Opinions and second opinions on controversial issues (Ninety-ninth Yearbook of the National Society for the Study of Education, Part 1, pp. 193– 224). Chicago: University of Chicago Press.

Beaton, A.E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L., & Smith, T.A. (1996). Mathematics achievement in the middle school years: IEA’s Third International Mathematics and Science Study . Chestnut Hill, MA: Boston College, Center for the Study of Testing, Evaluation, and Educational Policy. Available: http://www.timss.org/timss1995i/MathB.html .

Bempechat, J. & Drago-Severson, E. (1999). Cross-national differences in academic achievement: Beyond etic conceptions of children’s understanding. Review of Educational Research , 69 (3), 287–314.

Bowman, B.T., Donovan, M.S., Burns, M.S. (Eds.). (2001). Eager to learn: Educating our preschoolers . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/9745.html . [July 10, 2001].

Bransford, J.D., Brown, A.L., & Cocking, R.R. (Eds.). (1999). How people learn: Brain, mind, experience, and school . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/6160.html . [July 10, 2001].

Brownell, W.A. (1935). Psychological considerations in the learning and the teaching of arithmetic. In W.D.Reeve (Ed.), The teaching of arithmetic (Tenth Yearbook of the National Council of Teachers of Mathematics, pp. 1–31). New York: Columbia University, Teachers College, Bureau of Publications.

Brownell, W.A. (1987). AT classic: Meaning and skill—maintaining the balance. Arithmetic Teacher , 34 (8), 18–25. (Original work published 1956).

Bruner, J.S. (1960). The process of education . New York: Vintage Books.

Campbell, J.R., Hombo, C.M., & Mazzeo, J. (2000). NAEP 1999 trends in academic progress: Three decades of student performance , NCES 2000–469. Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/2000469.shtml . [July 10, 2001].

Campbell, J.R., Voelkl, K.E., & Donahue, P.L. (2000). NAEP 1996 trends in academic progress (NCES 97–985r). Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97985r.shtml . [July 10, 2001].

Carpenter, T.P., Corbitt, M.K., Kepner, H.S., Jr., Lindquist, M.M., & Reys, R.E. (1981). Results from the second mathematics assessment of the National Assessment of Educational Progress . Reston, VA: National Council of Teachers of Mathematics.

Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E., & Empson, S.B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education 29 , 3–20.

Carpenter, T.P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E.Fennema & T.A.Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 19–32). Mahway, NJ: Erlbaum.

Carpenter, T.P., & Levi, L. (1999, April). Developing conceptions of algebraic reasoning in the primary grades. Paper presented at the annual meeting of the American Educational Research Association, Montreal.

Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical meaning: Interaction in classroom cultures . Hillsdale, NJ: Erlbaum.

Cobb, P., Yackel, E., & Wood, T. (1989). Young children’s emotional acts during mathematical problem solving. In D.B.McLeod & V.M.Adams (Eds.), Affect and mathematical problem solving: A new perspective (pp. 117–148). New York: Springer-Verlag.

Cobb, P., Yackel, E., & Wood, T. (1995). The teaching experiment classroom. In P.Cobb & H.Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 17–24). Hillsdale, NJ: Erlbaum.

Committee for Economic Development, Research and Policy Committee. (1995). Connecting students to a changing world: A technology strategy for improving mathematics and science education: A statement . New York: Author.

Davis, R.B., & Maher, C.A. (1997). How students think: The role of representations. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 93– 115). Mahwah, NJ: Erlbaum.

Donovan, M.S., Bransford, J.D., & Pellegrino, J.W. (Eds.). (1999). How people learn: Bridging research and practice . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/9457.html . [July 10, 2001].

Druckman, D., & Bjork, R.A. (Eds.). (1991). In the mind’s eye: Enhancing human performance . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/1580.html . [July 10, 2001].

Dweck, C. (1986). Motivational processes affecting learning. American Psychologist , 41 , 1040–1048.

English, L.D. (1997a). Analogies, metaphors, and images: Vehicles for mathematical reasoning. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 3–18). Mahwah, NJ: Erlbaum.

English, L.D. (Ed.). (1997b). Mathematical reasoning: Analogies, metaphors, and images . Mahwah, NJ: Erlbaum.

Fennema, E. (1995). Mathematics, gender and research. In B.Grevholm & G.Hanna (Eds.), Gender and mathematics education (pp. 21–38). Lund, Sweden: Lund University Press.

Fennema, E., & Romberg, T.A. (Eds.). (1999). Mathematics classroom that promote understanding . Mahwah, NJ: Erlbaum.

Fuson, K.C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction 7 , 343–403.

Fuson, K.C. (1992a). Research on learning and teaching addition and subtraction of whole numbers. In G.Leinhardt, R.T.Putnam, & R.A.Hattrup (Eds.), The analysis of arithmetic for mathematics teaching (pp. 53–187). Hillsdale, NJ: Erlbaum.

Fuson, K.C. (1992b). Research on whole number addition and subtraction. In D.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–275). New York: Macmillan.

Fuson, K.C., & Briars, D.J. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place-value and multidigit addition and subtraction. Journal for Research in Mathematics Education 21 , 180–206.

Fuson, K.C., & Burghardt, B.H. (1993). Group case studies of second graders inventing multidigit addition procedures for base-ten blocks and written marks. In J.R.Becker & B.J.Pence (Eds.), Proceedings of the fifteenth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 240– 246). San Jose, CA: San Jose State University. (ERIC Document Reproduction Service No. ED 372 917).

Fuson, K.C., Carroll, W.M., & Landis, J. (1996). Levels in conceptualization and solving addition and subtraction compare word problems. Cognition and Instruction , 14 , 345– 371.

Geary, D.C. (1995). Reflections of evolution and culture in children’s cognition. American Psychologist , 50 (1), 24–37.

Greeno, J.G., Pearson, P.D., & Schoenfeld, A.H. (1997). Implications for the NAEP of research on learning and cognition. In R.Linn, R.Glaser, & G.Bohrnstedt (Eds.), Assessment in transition: Monitoring the nation’s educational progress (Background Studies, pp. 151–215). Stanford, CA: National Academy of Education.

Hagarty, M., Mayer, R.E., & Monk, C.A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology , 87 , 18–32.

Hatano, G. (1988, Fall). Social and motivational bases for mathematical understanding. New Directions for Child Development , 41 , 55–70.

Hiebert, J. (Ed.). (1986). Conceptual and procedural knowledge: The case of mathematics . Hillsdale, NJ: Erlbaum.

Hiebert, J., & Carpenter, T.P. (1992). Learning and teaching with understanding. In D. A.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.

Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding . Portsmouth, NH: Heinemann.

Hiebert, J., & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J.Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199–223). Hillsdale, NJ: Erlbaum.

Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction , 14 , 251–283.

Hilgard, E.R. (1957). Introduction to psychology (2nd ed.). New York: Harcourt Brace.

Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence . New York: Basic Books.

Katona, G. (1940). Organizing and memorizing . New York: Columbia University Press.

Kilpatrick, J. (1985). Doing mathematics without understanding it: A commentary on Higbee and Kunihira. Educational Psychologist , 20 (2), 65–68.

Knapp, M.S., Shields, P.M., & Turnbull, B.J. (1995). Academic challenge in high-poverty classrooms. Phi Delta Kappan , 76 , 770–776.

Kouba, V.L., Carpenter, T.P., & Swafford, J.O. (1989). Number and operations. In M. M.Lindquist (Ed.), Results from the fourth mathematics assessment of the National Assessment of Educational Progress (pp. 64–93). Reston, VA: National Council of Teachers of Mathematics.

Kouba, V.L., & Wearne, D. (2000). Whole number properties and operations. In E.A. Silver & P.A.Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 141–161). Reston, VA: National Council of Teachers of Mathematics.

Krutetskii, V.A. (1976). The psychology of mathematical abilities in schoolchildren (J.Kilpatrick & I.Wirszup, Eds.; J.Teller, Trans.). Chicago: University of Chicago Press. (Original work published 1968).

Ladson-Billings, G. (1999). Mathematics for all? Perspectives on the mathematics achievement gap. Paper prepared for the Mathematics Learning Study Committee, National Research Council, Washington, DC.

Leder, G.C. (1992). Mathematics and gender: Changing perspectives. In D.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 597–622). New York: Macmillan.

Lemaire, P., & Siegler, R.S. (1995). Four aspects of strategic change: Contributions to children’s learning of multiplication. Journal of Experimental Psychology: General , 124 , 83–97.

Mack, N.K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education , 26 , 422–441.

Maher, C.A., & Martino, A.M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education , 27 , 194–214.

Mason, D., Schroeter, D., Combs, R., & Washington, K. (1992). Assigning average achieving eighth graders to advanced mathematics classes in an urban junior high. Elementary School Journal , 92 , 587–599.

Mayer, R.E. (1999). The promise of educational psychology . Upper Saddle River, NJ: Prentice Hall .

Mayer, R.E., & Hegarty, M. (1996). The process of understanding mathematical problems. In R.J.Sternberg & T.Ben-Zee (Eds.), The nature of mathematical thinking (Studies in Mathematical Thinking and Learning Series, pp. 29–53). Mahwah, NJ: Erlbaum.

Mayer, R.E., & Wittrock, M.C. (1996). Problem-solving transfer. In D.C.Berliner & R. C.Calfee (Eds.), Handbook of educational psychology (pp. 47–62). New York: Macmillan.

McLeod, D.B. (1992). Research on affect in mathematics education: A reconceptualization. In D.A.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575–596). New York: Macmillan.

Mullis, I.V.S., Jenkins, F., & Johnson, E.G. (1994). Effective schools in mathematics . Washington, DC: National Center for Education Statistics.

Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Gregory, K.D., Garden, R.A., O’Connor, K. M., Chrostowski, S.J., & Smith, T.A. (2000). TIMSS 1999 international mathematics report: Findings from IEA’s repeat of the Third International Mathematics and Science Study at the eighth grade . Chestnut Hill, MA: Boston College, Lynch School of Education, International Study Center. Available: http://www.timss.org/timss1999i/math_achievement_report.html . [July 10, 2001].

National Assessment Governing Board. (2000). Mathematics framework for the 1996 and 2000 National Assessment of Educational Progress . Washington, DC: Author. Available: http://www.nagb.org/pubs/96–2000math/toc.html . [July 10, 2001].

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/1199.html . [July 10, 2001].

National Research Council. (1998). High school mathematics at work: Essays and Examples for the education of all students . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/5777.html . [July 10, 2001].

Nunes, T. (1992a). Cognitive invariants and cultural variation in mathematical concepts. International Journal of Behavioral Development , 15 , 433–453.

Nunes, T. (1992b). Ethnomathematics and everyday cognition. In D.A.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 557–574). New York: Macmillan.

Oakes, J. (1990). Multiplying inequalities: The effects of race, social class, and tracking on opportunities to learn mathematics and science . Santa Monica: CA: RAND Corporation.

Pólya, G. (1945). How to solve it: A new aspect of mathematical method . Princeton, NJ: Princeton University Press.

Pesek, D.D., & Kirshner, D. (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education , 31 , 524–540.

Reese, C.M., Miller, K.E., Mazzeo, J., & Dossey, J.A. (1997). NAEP 1996 mathematics report card for the nation and the states . Washington, DC: National Center for Education Statistics. Available: http://nces.ed.gov/spider/webspider/97488.shtml . [July 10, 2001].

Resnick, L.B. (1987). Education and learning to think . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/1032.html . [July 10, 2001].

Resnick, L.B., & Omanson, S.F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (vol. 3, pp. 41–95). Hillsdale, NJ: Erlbaum.

Rittle-Johnson, B., & Alibali, M.W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology , 91 , 175– 189.

Rittle-Johnson, B., & Siegler, R.S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C.Donlan (Ed.), The development of mathematical skills (pp. 75–110). East Sussex, UK: Psychology Press.

Saxe, G. (1990). Culture and cognitive development . Hillsdale, NJ: Erlbaum.

Schifter, D. (1999). Reasoning about operations: Early algebraic thinking in grades K-6. In L.V.Stiff (Ed.), Developing mathematical reasoning in grades K-12 (1999 Yearbook of the National Council of Teachers of Mathematics, pp. 62–81). Reston, VA: NCTM.

Schoenfeld, A.H. (1988). Problem solving in context(s). In R.I.Charles & E.A.Silver (Eds.), The teaching and assessing of mathematical problem solving (Research Agenda for Mathematics Education, vol. 3, pp. 82–92). Reston, VA: National Council of Teachers of Mathematics.

Schoenfeld, A.H. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education , 20 , 338–355.

Schoenfeld, A.H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.

Secada, W.G. (1992). Race, ethnicity, social class, language, and achievement in mathematics. In D.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 623–660). New York: Macmillan.

Shannon, A. (1999). Keeping score . Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/9635.html . [July 10, 2001].

Siegler, R.S., & Jenkins, E.A. (1989). How children discover new strategies . Hillsdale, NJ: Erlbaum.

Silver, E.A., Alacaci, C., & Stylianou, D.A. (2000). Students’ performance on extended constructed-response tasks. In E.A.Silver & P.A.Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 301– 341). Reston, VA: National Council of Teachers of Mathematics.

Silver, E.A., & Kenney, P.A. (2000). Results from the seventh mathematics assessment of the National Assessment of Educational Progress . Reston, VA: National Council of Teachers of Mathematics.

Silver, E.A., Strutchens, M.E., & Zawojewski, J.S. (1997). NAEP findings regarding race/ethnicity and gender: Affective issues, mathematics performance, and instructional context. In E.A.Silver & P.A.Kenney (Eds.), Results from the sixth mathematics assessment of the National Assessment of Educational Progress (pp. 33–59). Reston, VA: National Council of Teachers of Mathematics.

Steele, C.M. (1997). A threat in the air: How stereotypes shape intellectual identity and performance. American Psychologist , 52 , 613–629.

Steele, C.M., & Aronson, J. (1995). Stereotype threat and the intellectual test performance of African-Americans. Journal of Personality and Social Psychology , 69 , 797–811.

Steinberg, R.M. (1985). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education , 16 , 337–355.

Sternberg, R.J., & Rifkin, B. (1979). The development of analogical reasoning processes. Journal of Experimental Child Psychology , 27 , 195–232.

Stevenson, H.W., & Stigler, J.W. (1992). The learning gap: Why our schools are failing and what we can learn from Japanese and Chinese education . New York: Simon & Schuster.

Strutchens, M.E., & Silver, E.A. (2000). NAEP findings regarding race/ethnicity: Students’ performance, school experiences, and attitudes and beliefs. In E.A.Silver & P.A. Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 45–72). Reston, VA: National Council of Teachers of Mathematics .

Swafford, J.O., & Brown, C.A. (1989). Attitudes. In M.M.Lindquist (Ed.), Results from the fourth mathematics assessment of the National Assessment of Educational Progress (pp. 106–116). Reston, VA: National Council of Teachers of Mathematics.

Tate, W.F. (1997). Race, ethnicity, SES, gender, and language proficiency trends in mathematics achievement: An update. Journal for Research in Mathematics Education , 28 , 652–679.

Thompson, A.G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127– 146). New York: Macmillan.

Thornton, C.A., & Toohey, M.A. (1985). Basic math facts: Guidelines for teaching and learning. Learning Disabilities Focus , 1 (1), 44–57.

U.S. Department of Labor, Secretary’s Commission on Achieving Necessary Skills. (1991). What work requires of schools: A SCANS report for America 2000 . Washington, DC: Author. (ERIC Document Reproduction Service No. ED 332 054).

Wearne, D., & Hiebert, J. (1988). A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers. Journal for Research in Mathematics Education , 19 , 371–384.

Wearne, D., & Kouba, V.L. (2000). Rational numbers. In E.A.Silver & P.A.Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 163–191). Reston, VA: National Council of Teachers of Mathematics.

Wertheimer, M. (1959). Productive thinking . New York: Harper & Row.

Wiest, L.R. (2000). Mathematics that whets the appetite: Student-posed projects problems. Mathematics Teaching in the Middle School , 5 , 286–291.

Wu, H. (1999, Fall). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator , 14–19, 50–52.

Yaffee, L. (1999). Highlights of related research. In D.Schifter, V.Bastable, & S.J. Russell with L.Yaffee, J.B.Lester, & S.Cohen, Number and operations: Making meaning for operations. Casebook (pp. 127–149). Parsippany, NJ: Dale Seymour.

Zernike, K. (2000, August 25). Gap widens again on tests given to blacks and whites: Disparity widest among the best educated. The New York Times , p. A14.

Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years.

The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning:

• Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction.
• Details on the processes by which students acquire mathematical proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics.

The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.

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Common Core’s Focus on Concepts Is Key to Improving Math Education, Report Says

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The Common Core State Standards’ emphasis on conceptual understanding in math will improve students’ problem-solving skills and ultimately help prepare them for jobs of the future, argues a new report by the Center for American Progress, a Washington-based think tank.

Math instruction in the United States has traditionally focused on algorithms and procedures, explain Catherine Brown and Max Marchitello, the authors of the report. Under the common core, students spend more time learning the underlying math concepts and how to apply them. “As a result, students become stronger critical thinkers and problem solvers and will be better prepared for the rigorousness of today’s job market,” they write.

While much of the writing about the common core remains at the the 30,000-foot, theoretical level, this report, for the most part, is focused on specifics: It drills down into individual standards and gives examples for how the skills are taught differently under the common core vs. previous state standards.

For instance, it shows an addition problem done two ways. First, with the traditional, vertical algorithm:

And then a more conceptual way of teaching addition, which focuses on understanding place value and having good number sense:

The report rightly points out that students do still learn the traditional algorithm under the common core. (I t shows up in 4th grade .) They’re just introduced to the concepts first ( beginning in 1st grade ). “Learning the conceptual approach to math is just as important as learning to add with paper and pencil before using a calculator,” the authors write.

It’s worth noting though, as I’ve written before , that the common core is not actually a very prescriptive document. In fact, Jason Zimba, one of the lead writers of the math standards, has argued that teachers can introduce the algorithm right away in 1st grade along with the addition and subtraction concepts, and still be consistent with the standards. (He’s even drawn out a table for how to do that .) But in truth, that’s not how most math teachers will interpret the standards.

The Center for American Progress report also explains that in the common core, concepts build on each other year after year—1st grade students, for instance, are exposed to foundational concepts in fractions and geometry. The long, slow rollout of important mathematical ideas helps prevent misunderstandings when students get to more complicated problems, it says.

“At a time when calculators and computers are ubiquitous, workers need to move beyond the surface to understand underlying concepts, be deep thinkers and problem solvers, and apply their skills to new situations,” the authors write.

The Center for American Progress has long been supportive of the common-core standards. The report concludes with recommendations for ensuring conceptual math is taught well, such as providing ongoing professional development, training parents in the new methods, and incorporating conceptual math into teacher-preparation programs.

Images: From the Center for American Progress report, “ Math Matters: How the Common Core Will Help the United States Bring Up Its Grade on Mathematics Education .”

Related stories:

• Approach to Fractions Seen as Key Shift in Common Standards
• Schools Teach Common Core Math to Two Generations
• Should ‘Regrouping’ Be Taught Earlier Under Common-Core Math Standards?
• Common Core Redoes the Math

A version of this news article first appeared in the Curriculum Matters blog.

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An IERI – International Educational Research Institute Journal

• Open access
• Published: 05 February 2015

Teaching for conceptual understanding: A cross-national comparison of the relationship between teachers’ instructional practices and student achievement in mathematics

• Laura M O’Dwyer 1 ,
• Yang Wang 2 &
• Katherine A Shields 3  

Large-scale Assessments in Education volume  3 , Article number:  1 ( 2015 ) Cite this article

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Educators, researchers and policy-makers worldwide continue to struggle to understand the changes needed for improving educational outcomes and attainment for students, particularly in content areas such as mathematics and science that are essential for developing a highly skilled workforce. Given the continuing emphasis on linking teacher practices with student outcomes, this study aims to explore teachers’ use of particular instructional strategies and whether use is associated with student achievement.

Using data from the Trends in International Mathematics and Science Study (TIMSS) 2007, this study examines eighth grade teachers’ use of instructional practices that support students’ conceptual understanding, and examines the relationship between those practices and mathematics test scores. Data from four countries was examined: U.S., Korea, Japan, and Singapore. Descriptive statistics and a series of multilevel regression models were used.

The results show that teachers in the U.S. were most similar to Korean teachers in the frequency and types of teaching practices reported, and that they reported using some practices at least as frequently, or more frequently than teachers in the other countries. A sub-set of teachers’ instructional practices are related to mathematics achievement in the U.S. and Singapore, but not in Korea or Japan.

Conclusions

Teacher practices explained very little of the variability in test scores, even in cases where particular instructional practices were positively and significantly related to achievement. This research highlights the challenges for examining the teacher-level correlates of student test scores using data from cross-sectional, international studies that do not include measures of students’ prior achievement, classroom observations, or estimates of students’ out-of-school instruction.

Researchers and policymakers worldwide have an ongoing interest in understanding the ways in which teachers contribute to students’ learning and academic achievement. In the U.S. for example, prompted by the most recent federal investments in education, the Race to the Top program authorized under sections 14005 and 14006 of the American Recovery and Reinvestment Act of 2009 (ARRA), which places specific emphasis on improving teacher quality, researchers, policy-makers and practitioners are increasingly interested in the interplay between teacher quality and student achievement. However, as with many constructs in education, teacher quality is a multidimensional construct that includes not only teacher qualifications, but also teachers’ pedagogical content knowledge, their professional development activities, their instructional practices and quality, as well as their attitudes toward students and teaching (Darling-Hammond and Youngs, 2002 ; Goe, 2007 ). While several studies have linked overall measures of the quality of instructional practice (such as standardized teacher evaluation ratings) to student test scores (Borman and Kimball, 2005 ; Heneman et al., 2006 ; Holtzapple, 2003 ; Kimball et al., 2004 ; Milanowski, 2004 ), other studies have found no such connections (e.g., Gallagher, 2004 ; Rowan et al., 1997 ; D’Agostino, 2000 ). Overall, there appears to be little consensus on the specific instructional strategies that are consistently associated with student achievement.

The purpose of this research is to describe patterns in teachers’ instructional practices in eighth-grade mathematics classrooms in four countries that participated in the Trends in International Mathematics and Science Study (TIMSS) 2007 (the U.S., Korea, Japan, and Singapore), and to examine the relationship between teachers’ instructional practices and students’ achievement in mathematics. This research also highlights the challenges for using data from international cross-sectional comparative studies to link student outcomes with teachers’ instructional practices.

Instructional practices and student achievement

Demonstrating relationships between specific teaching practices and student achievement has proven difficult. From a methodological perspective, cross-sectional educational research designs do not allow us to know whether a teacher is using a particular strategy because that teacher has high-achieving students, or whether that strategy is producing high-achieving students. Likewise, cross-sectional studies that lack information about students’ prior achievement make it difficult to disentangle a single teacher’s contribution to students’ achievement.

Beyond these challenges, Hiebert and Grouws ( 2007 ) attribute the lack of consensus on the relationship between instructional strategies and student achievement to an absence of well-developed theories describing how teaching practices affect learning; to the varying effectiveness of a given teaching approach that depends on the learning goals; and to the mediating effects of student and contextual variables. However, evidence has accumulated for the effectiveness of some teaching practices. For example, Hill et al. ( 2005 ) found that teachers’ pedagogical content knowledge, the specialized mathematical knowledge and skills used in teaching mathematics, were positively associated with gains in students’ mathematical achievement over time.

Likewise, Hattie’s ( 2009 ) exhaustive synthesis of meta-analyses of research on student achievement identifies a number of teaching strategies with moderate-to-large effect sizes across multiple studies. These strategies include “problem-solving teaching,” which is defined as a set of methods that help students to define the problem, then assess and select among possible solutions. In alignment with Hattie’s findings, Camburn and Han ( 2011 ) found support for a significant effect for problem-solving and related instructional strategies in a review of evidence from the past twenty years of student achievement studies based on large-scale assessments. The associated student activities included “solving novel problems, engaging in reasoning and analysis, and making connections to real world applications of knowledge” (p. 593). These practices can be understood within the framework of conceptual and procedural mathematics knowledge (Hiebert and Lefevre, 1987 ; Hiebert, 2003 ).

Conceptual knowledge is the understanding of core principles and the relationships among them (Hiebert and Lefevre, 1987 ; diSessa and Sherin, 1998 ; Star, 2005 ). Hiebert and Grouws ( 2007 ) outline two key aspects of instructional practices that help students develop conceptual understanding: giving students the opportunity to “struggle” with problems (p. 387), and discussing conceptual relationships “explicitly” (p. 383). These approaches help students develop the ability to transfer their skills and knowledge to new contexts. Prawat ( 1989a ), whose work was a resource for the TIMSS conceptual framework underlying the teacher and classroom context surveys, connects conceptual knowledge development with teacher practices that attend to students’ explanations and problem-solving strategies. By contrast, procedural knowledge is limited to recognizing the mathematical system of symbols and rules, and carrying out procedures using that system (Star, 2005 ). Conceptual knowledge allows the student to select and apply the right procedure for the situation, and to incorporate new information correctly into what he or she already knows (Prawat, 1989b ). In the U.S., the National Council of Teachers of Mathematics (NCTM) ( 2000 ) recognizes the importance of conceptual understanding, calling for students to learn “with understanding, actively building new knowledge from experience and previous knowledge” and “reflecting on their thinking” (p. 2).

Teaching practices that help students to develop conceptual knowledge have been linked to achievement in the literature. These practices include: giving students challenging assignments (Newmann et al., 2001 ; Schacter and Thum, 2004 ); providing opportunities to apply learning and solve unique problems (Wenglinsky, 2002 ); spending time on conceptual activities such as estimation, writing equations, and word problems (Lubienski, 2006 ); and having students discuss the reasons behind their answers to questions (Gales and Yan, 2001 ). The research companion to the NCTM Standards (Mewborn, 2003 ) also cites studies that link higher achievement to providing students with opportunities to solve problems and talk with each other about their approaches (Fennema, et al., 1996 ), as well as helping students see the associations between concepts and mathematical symbols (Gearhart et al., 1999 ).

Evidence from international comparative studies

With the almost 20 years of available data from international comparative studies in education such as the TIMSS, the Progress in International Reading Literacy Study (PIRLS), and the Programme for International Student Assessment (PISA), secondary analyses have been conducted in individual counties that examined the relationship between teachers’ instructional practices and student achievement. For example, using TIMSS 1995 eighth grade data in the U.S., Tomoff et al. ( 2000 ) compared the NCTM-recommended practices to those reported by the sample of TIMSS mathematics teachers and found that the use of drill and review exercises was significantly negatively associated with mathematics achievement. Their study also showed that more interactive, conceptual activities such as group and project-based work showed no relationship. In Japan, House ( 2009 ) found that higher-achieving fourth grade students in the 2007 TIMSS sample reported being asked more frequently to work on problems independently and to explain their answers, although procedural skills such as memorization of procedures and practicing computation were also positively related to achievement in this younger population.

A small number of researchers have used data from international comparative studies to describe the differences in teachers’ instructional practices across countries, and to examine the relationship between instructional practices and achievement. For example, as part of a multi-country study using 1999 TIMSS eighth grade data, Desimone et al. ( 2005 ) analyzed whether teachers’ use of conceptual versus computational instructional strategies was predictive of classroom-level mathematics achievement. The authors found weak relationships within countries, and no meaningful variation in the strength of this relationship across countries. In country-specific analyses, they found a weak positive association between conceptual instruction and achievement in the U.S., but no such relationship in Singapore or Japan.

Evidence on instructional practice in the U.S. and East Asia

In the U.S., researchers, policy-makers, practitioners, and even the general public are keenly aware that multiple international studies have shown that our students consistently lag behind students in some countries, particularly students in East Asian countries. For example, on the 2007 TIMSS and PISA assessments, U.S. middle school students performed below the OECD mean in mathematics, and lagged behind the East Asian countries; Singapore, Korea and Japan ranked among the top five countries in TIMSS and the top 10 in PISA, compared to the U.S. at 9 th in TIMSS and 31 st in PISA (Mullis et al., 2008 ; OECD, 2010 ). Moreover, evidence shows that only 6% of U.S. students scored in the advanced category of TIMSS, compared to 40% of students in Korea and Singapore, and 26% in Japan (Mullis et al., 2008 ).

Studies that have explored explanations for what distinguishes East Asian nations from lower performing countries, including the U.S., reported various differences including stronger teacher preparation (Heck, 2007 ; Kim et al., 2011 ) and a uniform, coherent national curriculum (Schmidt et al., 2005 ). Some strands of research also explore the cultural roots of these differences, such as a predominant focus on the collective, a greater emphasis on hard work and practice, and the role of the teacher as content expert in the Eastern tradition, compared to a focus on the individual and child-centered pedagogy in the Western tradition (Kaiser and Blömeke, 2013 ). Other researchers point to the amount of time spent both in and outside of school on mathematics instruction as an important distinguishing characteristic (Lewis and Seidman, 1994 , Bray, 2003 ; Bray and Kwo, 2013 ). In particular, valid comparisons across education systems may be hampered by the existence of for-profit, supplementary “shadow education” systems that operate in parallel to those sampled in international comparative studies. Despite their strong education systems, cram schools, referred to as juku in Japan and hagwons in Korea, are ubiquitous in many East Asian countries (Bray and Kwo, 2013 ). In Singapore for example, the number of juku almost tripled between 1994 and 2002 (Bray, 2003 ) and in 2008 in Korea, more than 70% of middle school students received extensive tutoring outside of the country’s formal education system sampled and assessed by international comparative studies (Bray and Kwo, 2013 ).

Related to instructional practice differences, classroom observational studies suggest that teachers in higher-performing countries tend to use strategies that support conceptual knowledge development more frequently than their peers in the U.S., where teachers spend more time nurturing basic procedural skills at the expense of critical thinking and understanding of underlying concepts (Prawat, 1989a ). This finding is supported by the 1999 TIMSS video study, which collected classroom observations from randomly selected classrooms in six high-performing countries and the U.S., and reported that teachers in U.S. classrooms spent more time completing repetitive exercises rather than applying and extending skills to new, different problems; reviewed previously taught procedures and material more frequently; covered less advanced content; covered a scattered mix of skills and concepts rather than a single coherent topic; and suffered more interruptions to the lesson (Hiebert et al., 2005 ). Researchers have also found that teachers in Japan (Whitman and Lai, 1990 ) and Singapore (VanTassel-Baska et al., 2008 ) focused more on critical thinking and lesson structure, when compared to U.S. teachers who focused more on discipline and rules. Japanese teachers also reported a more unified view of what constitutes “good” mathematics instruction than their American counterparts (Jacobs and Morita, 2002 ).

Research purpose

Recognizing the limitations of international cross-sectional studies of educational outcomes that use self-report instructional practices, the present study seeks to extend the field’s understanding of the relationship between teaching for conceptual understanding and mathematics achievement using TIMSS 2007 data. The objective of this research is to describe patterns in teaching for conceptual understanding in eighth-grade mathematics classrooms in the U.S. and three East Asian countries, Korea, Japan, and Singapore, and to examine how teaching for conceptual understanding in mathematics classrooms is associated with students’ achievement in mathematics. Korea, Japan, and Singapore were selected as comparisons to the U.S. because, over repeated international studies, they have out-performed the U.S. For example, while the mean mathematics scores for the U.S., Korea, Japan, and Singapore were each above the TIMSS international mean (i.e., 500) across the 49 participating countries/jurisdictions in 2007, the means in 2007 for Singapore (593), Japan (570), and South Korea (597) were each higher than the mean in the U.S. (508). Moreover, this pattern of lower mean achievement for the U.S. compared to these countries was observed in all past TIMSS administrations for mathematics – TIMSS 1995, 1999, 2003, and most recently in 2011.

Using TIMSS 2007 data from individual students, data from their mathematics teachers, as well as information about their schools, this research addresses the following research questions:

How does teaching for conceptual understanding in mathematics classrooms vary across the U.S., Korea, Japan, and Singapore?

Within these four countries, is teaching for conceptual understanding associated with students’ mathematics achievement? And, do the associations between teaching for conceptual understanding and students’ mathematics achievement vary across the U.S., Korea, Japan, and Singapore?

These questions were addressed using teacher data and individual student data, and analyses were conducted separately by country. Descriptive analyses and ANOVA with teacher data were used to address the first research question, and a series of two-level hierarchical linear regression models with student, teacher and school data was used to address the second research question. In addition to addressing these research questions, this study highlights the challenges for using data from cross-sectional, international comparative studies to link student outcomes with teachers’ instructional practices.

This section describes the TIMSS 2007 sample, the variables and measures derived from the TIMSS student, teacher, and school questionnaire data, and the data analyses used to address the research questions. Explicit descriptions of the model specifications used to estimate the relationship between teachers’ instructional practices and student achievement are also provided a .

TIMSS 2007 sample

The final analysis sample for each country was defined after the data files were cleaned according to specific criteria. First, students with missing cases on the student-level covariates were removed from the analysis sample (2% or 164 students in the U.S., 0.03% or 14 students in Korea, 1% or 35 students in Japan, and 0.5% or 21 students in Singapore). Studies have shown that estimates are not sensitive to the choice of treatment of missing data at such minimal levels (Little & Rubin, 2002 ), and so we do not expect the listwise deletion of these small numbers of cases to introduce bias.

Second, students and classrooms linked with more than one mathematics teacher were removed from the analysis sample. This step ensured that only students whose test scores could be linked with one teacher were included in the analyses, a necessary condition for examining the association between teachers’ instructional practices and student achievement. Through this process, 216 students were removed in U.S. (3%) and 58 students in Korea (1%). In Japan, the number of students removed at this step was large, 1212 (22%), and because in many schools all students were linked with more than one teacher, 37 schools (of the available 146) were removed from the sample at this stage. In Singapore, 171 students (4%) in two schools (of the available 164) were removed.

Third, some classrooms were randomly removed from the analysis sample to allow comparisons across countries. The TIMSS sample is selected using a two-stage stratified sampling procedure to produce a representative sample of students from participating countries; schools are randomly sampled at the first stage and intact classrooms are sampled within each sampled school at the second stage (Joncas, 2008 ). The majority of TIMSS countries randomly sampled only one intact classroom in each school, resulting in classroom effects being confounded with school effects. However, some countries (e.g., the U.S. and Singapore) routinely sampled more than one classroom per school. To allow direct comparisons among the four countries included in this study, one classroom was randomly selected within each school to be included in the analyses. This process resulted in the random removal of 192 classrooms in the U.S., 20 in Japan, and 133 in Singapore. No classrooms were removed from the Korean sample since all schools sampled only one classroom. As a consequence of including only students from one mathematics classroom and their mathematics teacher, the classroom and school levels are confounded. In describing the analyses and the results we refer to the between school effects, recognizing that they could also be considered between classroom effects.

Finally, schools fewer than five students were removed from the analysis sample (only one in Japan and none in the other three countries) and schools with missing teacher instructional practice data were removed from the analysis samples (23 schools in the U.S., no schools in Korea, one school in Japan, and two schools in Singapore).

Overall, the analyses were conducted with data from 217 schools and 3255 students in the U.S., 150 schools and 4102 students in Korea, 107 schools and 2719 students in Japan, and 160 schools and 2270 students in Singapore. Table  1 shows that mean achievement and the variability in achievement were similar in the reported samples (Mullis et al. 2008 ) and the analysis samples. Moreover, the percentages of students whose teachers asked them to engage in the six activities in about half the lessons or more were similar in the reported and analysis samples.

Variables and measures

The research questions were addressed using students’ eighth grade mathematics achievement, the instructional practices reported by their teachers, as well as information about the student body in the sampled schools. The teacher instructional practice measures were derived from teachers’ responses to the mathematics teacher survey. In addition, the analyses included student and school covariates derived from the student and school questionnaires. The student achievement and teacher practice measures, as well as the student and school covariates are discussed in turn.

• Mathematics achievement

At the eighth grade, the student mathematics assessment comprised four sub-domains, each contributing to a total mathematics score: Number (30%), Algebra (30%), Geometry (20%), and Data and Chance (20%) (Mullis et al., 2005 ). TIMSS uses item response theory (IRT) in combination with conditioning and multiple imputation to summarize students’ total mathematics and sub-domain achievement on a scale with a mean of 500 and a standard deviation of 100, and a set of five plausible values are provided to account for the fact that not all students are administered all test items (Foy et al., 2008 ). HLM multilevel regression modeling software was used so that all five plausible values representing students’ total mathematics achievement were used appropriately in the analyses.

Teaching for conceptual understanding measures

The teaching for conceptual understanding measures used to address the research questions were derived from a sub-set of the questions relating to teachers’ self-reported instructional practices in the mathematics classroom (Mullis et al., 2008 ). Specifically, six of the available items were used to represent teachers’ practices related to teaching for conceptual understanding. As discussed in the previous section, academic achievement has been shown to relate to instructional practices that help students develop conceptual understanding. TIMSS developed the set of teacher practice items in part by drawing on research on these issues (Schmidt and Cogan, 1996 ), particularly work by Prawat ( 1989a ; 1989b ) on the aspects of teaching that promote conceptual understanding, such as analyzing how students explain and solve problems, and giving students opportunities to engage with fundamental concepts. TIMSS developers used three of these six items in the 1999 administration as part of an index of Teachers’ Emphasis on Mathematics Reasoning and Problem-solving (Mullis et al., 2000 ).

Measured on a scale ranging from “Every or almost every lesson” (3) to “Never” (0), the following six items that asked teachers to report how often they asked students to do the following were used to represent teachers’ practices related to teaching for conceptual understanding:

… write equations and functions to represent relationships

… interpret data in tables, charts or graphs

… apply facts, concepts and procedures to solve routine problems

… relate what they are learning in mathematics to their daily lives

… decide on their own procedures for solving complex problems

… work on problems for which there is no immediately obvious method of solution

Teachers’ responses to the six individual items were used to represent teachers’ practices related to teaching for conceptual understanding. While the TIMSS instructional practice items do not map neatly onto the constructs of conceptual versus procedural knowledge development, the six items used in this study were chosen to represent several key elements of teaching for conceptual understanding. “Writ[ing] equations and functions to represent relationships” gives students the opportunity to explicitly examine conceptual relationships, using the symbolic language of mathematics (Hiebert, 2003 ; Prawat, 1989a ). When students “relate what they are learning to their daily lives, ” they must transfer mathematical knowledge to a real-world context and connect different pieces of information (Hiebert and Lefevre, 1987 ). To “interpret data in tables, charts or graphs, ” “apply facts, concepts and procedures to solve routine problems, ” and “decide on their own procedures for solving complex problems, ” students must select the appropriate procedure and use it correctly for the given context (Hiebert and Lefevre, 1987 ). When students select their own methods of solution, the teacher also has an opportunity to observe students’ thought processes and misconceptions (Hiebert and Grouws, 2007 ; Prawat, 1989a ). Taking these problem-solving skills a step further, when students “work on problems for which there is no immediately obvious method of solution, ” they grapple with problems in which they have incomplete information or skills, creating a productive opportunity for growth (Hiebert and Grouws, 2007 ). In contrast, TIMSS instructional practice items representing activities that focus on procedural knowledge development were not included in this study (e.g., “memorize formulas and procedures” and “practice adding, subtracting, multiplying and dividing without using a calculator” (Mullis et al., 2008 )).

The authors explored the creation of a composite variable across all six items since composites have the advantage of providing a more comprehensive measure of the construct when compared to a single variable (DeVellis, 2012 ). However, principal axis factoring (PAF) indicated that the composite was not unidimensional across countries. Specifically, the six items split across two or three factors in the four countries. The frequency with which teachers asked students to “relate what they are learning in mathematics to their daily lives,” “decide on their own procedures for solving complex problems” and “work on problems for which there is no immediately obvious method of solution” loaded on a single factor in all four countries. The reliability of the three-item scale ranged from 0.6 in Korea and Singapore to 0.7 in the U.S. and Japan. The remaining three items, the frequency with which teachers asked their students to “interpret data in tables, charts or graphs,” “write equations and functions to represent relationships” and “apply facts, concepts and procedures to solve routine problems” loaded on a single factor in the U.S., Japan, and Singapore, but across two factors in Korea. The reliability of this scale was lower than optimal, ranging from .4 in the U.S. and Korea to .5 in Singapore and .7 in Japan. Based on the dimensionality of the items across countries and their lack of alignment with a theoretical construct, a composite variable was not created and teachers’ responses to the six items were used individually in the analyses to represent teachers’ practices related to teaching for conceptual understanding.

Student and school covariates

As student and school characteristics have been found to be associated with academic achievement, covariates measured at the student and school levels were included in the analyses. The purpose of the covariates was to account for student-to-student, and school-to-school differences that were unrelated to teachers’ instructional practices. The student-level covariates included characteristics that prior research has linked with achievement: students’ gender (Else-Quest et al., 2010 ; Liu and Wilson, 2009 ; Robinson and Lubienski, 2011 ), home background (Henderson and Berla, 1994 ; Berliner, 2006 ; Schreiber, 2002 ; Sirin, 2005 ; Votruba-Drzal, 2006 ; Yeung et al., 2002 ), and attitudes toward and beliefs about mathematics (Choi and Chang, 2011 ; Ho et al., 2000 ; Ma and Kishor, 1997 ; Singh et al., 2002 ). Likewise, the socioeconomic status of the school has been found to be associated with outcomes for individual students (Rothstein, 2004 ; Rumberger and Palardy, 2005 ).

Students’ gender was included as a covariate (male = 1, female = 0) at the student level and multiple student-level measures were combined to create a composite variable to represent students’ home resources and family background. The home background composite was created from several related questions: home educational resources (possession of a calculator, study desk, and dictionary), number of books in the home, and parents’ highest education. The inter-relatedness of the items was confirmed using PAF and within each country, a standardized factor score with a mean of 0 and a standard deviation of 1 was created to represent students’ home background. The reliability of the student home background composite is shown in Table  2 for each country. Also, to account for school-to-school differences related to the economic resources available in students’ homes, the student home background composite was aggregated to the school level and included as a school-level covariate.

In an effort to control for student-to-student differences in attitudes toward and perceptions about mathematics, perceptions about their own ability in mathematics, as well differences in motivation to learn, three composite indices provided in the TIMSS student database were included as covariates in the analyses. The indices measured students’ Positive Affect Toward Mathematics (PAT-M), students’ Self-Confidence in Learning Mathematics (SCM) and Students’ Valuing Mathematics (SVM) and were created by computing the mean of the student’s scores on the constituent items (Martin and Preuschoff, 2008 ). The response categories for the items used to create the indices were agree a lot  = 1, agree a little  = 2, disagree a little  = 3, and disagree a lot  = 4. On each index, students were classified into low, medium and high based on the mean across the constituent items. Subsequently, a categorical variable was created in which students with a mean score of 2 or lower received a classification of high (+1); students with mean scores greater than 2 but lower than 3 received a classification of medium (0); and students with scores of 3 or more received a low (−1) classification (Martin and Preuschoff, 2008 ). Since the three indices were categorical, two dummy variables were created for each index in which the high (+1) group was the reference group for all comparisons. The categorical versions of the indices were used in the models because these characteristics were included as student-level covariates and were not central to addressing the research questions. The reliabilities of the three indices across countries are shown in Table  2 .

Analysis procedures

Analyses were conducted separately by country and included teacher-level analyses (research question 1) and student-level analyses (research question 2). To address the first research question, descriptive statistics were calculated using teachers’ responses to the six individual teaching for conceptual understanding items and the results were represented both graphically and numerically. In addition, one-way ANOVAs with Bonferroni post hoc tests were conducted to examine whether the frequency of teaching for conceptual understanding in the U.S. was significantly different from the frequency in the East Asian countries. The descriptive statistics and ANOVAs allowed comparisons among countries with respect to mathematics teachers’ use of instructional practices that support conceptual understanding. In each country, sampling weights were used to make adjustments for non-response and to ensure that subgroups of the sample were properly represented in the estimation of population parameters (Joncas, 2008 ). Since the descriptive analyses and ANOVAs were based on teacher data, the mathematics teacher sampling weight (MATWGT) was applied to generate accurate estimates in each country.

To address the second research question, a series of multilevel linear regression models was formulated for each country. HLM software was chosen so that all five plausible values could be analyzed appropriately and so that the sampling weights could be applied at the appropriate level. In the series of models, students’ achievement on the TIMSS mathematics assessment, represented by five plausible values, was regressed on teachers’ practices related to teaching for conceptual understanding along with the student and school covariates. Multilevel regression procedures ensured that the statistical dependence among students within schools was accounted for by the complex residual structure thereby producing correct estimates of the standard errors associated with the regression coefficients; allowed student, teacher and school characteristics to be included in a single model to predict students’ mathematics achievement; and allowed all five plausible values to be handled simultaneously (Raudenbush and Bryk, 2002 ). The multilevel regression analyses required the calculation of student population estimates within each country and so the authors followed the guidelines set forth in the TIMSS documentation and applied the student sampling weight (TOTWGT) at the student level. Since this weight includes the probability of both the school and student being selected from the population, no weights were required at the school level (Joncas, 2008 ).

At the first stage, an unconditional (or null) two-level regression model was specified for each country. This model was used to partition the total variability in mathematics achievement scores into within- and between-school variance components. These unconditional variance components were used to estimate the degree of statistical dependence (nesting) among students within schools for each country, as indicated by the intra-class correlation coefficient (ICC) and served as a comparison for subsequent models that included teachers’ instructional practices as well as the student and school covariates. For each country, the unconditional model used to calculate the ICC was as follows:

At the second stage, teachers’ practices related to teaching for conceptual understanding and the student and school covariates were included in the model to predict students’ mathematics scores. The six individual teaching for conceptual understanding items were added to the models separately, thereby allowing the association between each one and mathematics scores to be examined independently. In each country, the level-1 model included the student covariates:

In this model, the mathematics score for student i in school j ( Y ij ) was predicted by the linear combination of student covariates and a random student effect, r ij . The regression coefficients β 1 through β 8 represent the association between each covariate and mathematics achievement, holding all other predictors in the model constant. The level-1 covariates were entered into the model uncentered since they were dichotomous (i.e., gender), standardized to have a mean of 0 and a standard deviation of 1 (i.e., home background composite), or were coded as dichotomous dummy variables (i.e., PAT-M, SVM, and SCM).

At level-2, the aggregated student home background composite was included as a school covariate along with the teaching for conceptual understanding measures, each added separately. In each country, six level-2 models were formulated, one for each of the six teaching for conceptual understanding items, as follows:

In each of the six intercept-only models, the γ 02 regression coefficient represented the association between a particular teaching for conceptual understanding measure and mathematics test scores, holding all covariates in the model constant. The individual teaching for conceptual understanding measures were entered into the model uncentered. The percentage of economically disadvantaged students in the school was grand mean centered around the mean for all schools.

During the model development phase, the significance and reliability of the variability in the level-1 regression coefficients associated with the student covariates across schools was evaluated using McCoach’s guidelines ( 2010 ). With some exceptions, the level-1 regression coefficients were found to be constant across schools and so these regression coefficients were fixed. However, in the U.S. and Japan, the relationships between mathematics test scores and gender, and between test scores and the SVM index, were not constant across schools. Also, in Japan, the relationship between mathematics test scores and the SCM index was not constant across schools. Additional exploratory analyses revealed that the variability in these relationships across schools was not related to the level-2 variables included in the analyses, the teaching for conceptual understanding measures and the percent of economically disadvantaged students in the school covariate. Therefore, in the final models, the slopes associated with these level-1 variables in the U.S. and Japan were allowed to vary randomly across schools, but no level-2 predictors were included in the models to predict that variability. In all other countries and for the other level-1 variables in the U.S. and Japan models, the level-1 slopes were fixed. For the fixed P level-1 slopes, the models were:

And, for the random P level-1 slopes (e.g., for the gender and SVM slopes in the U.S. and Japan, and for the SCM slope in Japan) the models were:

Comparisons between the fixed effects across models with different varying slopes is deemed to be appropriate since allowing slopes to vary randomly has little effect on the fixed effects or the standard errors associated with these fixed effects (Raudenbush and Bryk, 2002 ).

For each country, the regression coefficients and variance components from these models were evaluated to examine whether teaching for conceptual understanding predicted students’ mathematics achievement within each country, after controlling for student characteristics, and whether these relationships varied across countries.

Patterns of teaching for conceptual understanding

Figure  1 and Table  3 summarize the weighted percentages of teachers in the four countries that reported using the six individual measures of teaching for conceptual understanding (never (0), some lessons (1), about half the lessons (3), and every or almost every lesson(4)) and the descriptive statistics (mean, standard error of the mean, and standard deviation), respectively. Comparing across the six individual measures of teaching for conceptual understanding in all four countries, the largest percentage of mathematics teachers reported that they spent time in every or almost every lesson having their students apply facts, concepts and procedures to solve routine problems (48.8% in the U.S., 42.7% in Korea, 27.1% in Japan, and 34.4% in Singapore). The lowest percentage of mathematics teachers in all four countries spent time in every, or almost every lesson having their students interpret data in tables, charts or graphs (approximately 1.9% in the U.S., 2.7% in Korea, 2.8% in Japan, and 1.3% in Singapore). In general, the results in Figure  1 and Table  3 show that there is variability among the four countries in how teachers teach for conceptual understanding.

Percentages of teachers using each instructional practice. *As TIMSS uses a complex sampling design, mathematics teachers’ sampling weights (MATWGT) were applied in the descriptive analyses to generate accurate estimates on the teacher population in each participating country.

The results of the ANOVAs in Table  3 show that for each item, there were significant differences among at least two of the country means, even with a conservative alpha of .05/6 or .008. Focusing on how the U.S. compared to the other countries on the frequency of teaching for conceptual understanding, the post hoc tests indicate patterns in the differences. First, teachers in the U.S. reported that they had their students engage in four of the six practices significantly more frequently than teachers in some of the other countries. Specifically, mathematics teachers in the U.S. (M = 1.86, SD = 0.83) reported significantly higher rates for having their students relate what they are learning in mathematics to their daily lives than teachers in Japan (M = 1.26, SD = 0.62), Korea (M = 1.65, SD = 0.67), or Singapore (M = 1.38, SD = 0.61). Likewise, the post hoc tests showed that teachers in the U.S. (M = 2.29, SD = 0.78) reported having their students apply facts, concepts and procedures to solve routine problems significantly more frequently than teachers in Japan (M = 1.85, SD = 0.88) or Singapore (M = 2.00, SD = 0.83). Teachers in the U.S. (M = 1.54, SD = 0.75) also had their students decide on their own procedures for solving complex problems significantly more often than teachers in Japan (M = 1.18, SD = 0.61) or Singapore (M = 1.13, SD = 0.66). Finally, teachers in the U.S. (M = 1.15, SD = 0.72) reported having their students work on problems for which there is no immediately obvious method of solution significantly more frequently than teachers in Singapore (M = 0.88, SD = 0.59).

This pattern was reversed for the practice of having students write equations and functions to represent relationships and interpret data in tables, charts or graphs . Teachers in the U.S. (M = 1.49, SD = 0.69) were significantly less likely to have their students write equations and functions to represent relationships than teachers in either Japan (M = 1.80, SD = 0.75) or Korea (M = 1.78, SD = 0.65). Although the frequency of teachers having their students interpret data in tables, charts or graphs was low in both countries (see Figure  1 ), teachers in the U.S. (M = 1.14, SD = 0.46) were significantly less likely than teachers in Korea (M = 1.28, SD = 0.57) to engage in this practice.

These patterns across the four countries support the findings from classroom observation studies (Hiebert et al., 2005 ; VanTassel-Baska et al., 2008 ; Whitman and Lai, 1990 ) that there is variability in teachers’ use of the instructional practices measured on the TIMSS teacher survey across countries.

Teaching for conceptual understanding and mathematics test scores

Tables  4 , 5 , 6 , and 7 present the regression coefficients associated with the six individual teaching for conceptual understanding measures (Models 1–6) for predicting students’ total mathematics test scores in the U.S., Korea, Japan, and Singapore, respectively. Since the student and school covariates are included in Models 1 through 6, the tables also present those regression coefficients. Using the standard deviation of students’ mathematics scores in the sample, the coefficients were transformed into predicted standardized differences by dividing the regression coefficient by the standard deviation of the outcome variable (see Table  1 ). For example, if an independent variable X is associated with a regression coefficient b for predicting the outcome variable Y , the standardized regression coefficient is interpreted as the predicted standard deviation change in Y for a one unit increase in X , holding all else constant. In this case, a one unit increase in X corresponds to increasing teaching for conceptual understanding practices from never to some lessons, from some lessons to about half the lessons, or from about half the lessons to every or almost every lesson. Subsequent to the interpretation of the fixed effects, the percentage of variance explained by the models is discussed.

For the U.S., Table  4 shows that the frequency with which teachers asked their students to write equations and functions to represent relationships (b = 24.78, standardized difference = 0.33, p < .001), decide on their own procedures for solving complex problems (b = 9.49, standardized difference = 0.13, p < .01), and to work on problems for which there is no immediately obvious method of solution (b = 7.05, standardized difference = 0.09, p < .05) were each associated with increased mathematics scores, after controlling for the covariates included in the models. The largest predicted increase in achievement was associated with U.S. teachers having their students write equations and functions to represent relationships ; recall that teachers in the U.S. were significantly less likely than teachers in Japan and Korea to have their students engage in this practice (Table  3 ). For every one point increase in teachers’ practice of having their students write equations and functions to represent relationships (e.g., increasing the frequency of using this practice from some lessons to about half the lessons), students’ mathematics scores were predicted to increase by 0.33 standard deviations.

Tables  5 and 6 show that in Korea and Japan, none of the six individual teaching for conceptual understanding items was significantly associated with students’ mathematics test scores; the regression coefficients and standardized difference associated with the individual practice items were small; between −0.08 and 0.01 in Korea and between −0.003 and 0.11 in Japan.

Table  7 shows that in Singapore, the frequency with which teachers asked their students to write equations and functions to represent relationships (b = 28.38, standardized difference = 0.32, p < .001), apply facts, concepts and procedures to solve routine problems (b = 20.43, standardized difference = 0.23, p < .01), and to work on problems for which there is no immediately obvious method of solution (b = 33.68, standardized difference = 0.37, p < .001) were each associated with increased mathematics test scores, after controlling for the student and school covariates in the model.

Two patterns were apparent in these results. In Korea and Japan the items used to represent teaching for conceptual understanding were not associated with students’ mathematics test scores. Conversely in the U.S. and Singapore, students in classrooms where teachers asked their students to write equations and functions to represent relationships or work on problems for which there is no immediately obvious method of solution were predicted to have higher mathematics scores, after controlling for all other variables in the model. The standardized regression coefficients for having students write equations and functions to represent relationships were almost identical in both countries; 0.33 in the U.S. compared to 0.32 in Singapore.

The percentage of variance explained by the predictors in the models corroborates these findings. Table  8 shows the percentage of variance in mathematics achievement within and between schools, and the variance explained (within and between schools, and in total) by the items representing teaching for conceptual understanding, and the student and school covariates. The variance in achievement within schools and between schools (the ICC) varied across the four countries; however, the U.S. is most similar to Singapore with respect to how the variance in mathematics scores is distributed. In both countries, more than half of the variability in students’ scores lies among schools, 57.1% in the U.S. and 73.9% in Singapore, suggesting that students in the U.S. and Singapore are likely to attend schools with students who are similar to themselves.

In Korea and Japan, the majority of variance in mathematics scores is among students within schools, 89.4% in Korea and 80.1% in Japan. The small percentages of variance in achievement between schools suggests that teacher-to-teacher differences in teaching for conceptual understanding practices, or any teacher- or classroom-level practice or characteristic for that matter, will be of limited use in predicting student-to-student differences in achievement in these countries.

Recall that only one classroom per school is included in the analysis sample. As a consequence, the variance in achievement between classroom within schools and the between school variance in achievement are confounded, so the proportion of variability between schools is larger than is observed when classroom-to-classroom differences are separated from between school differences. For example, when multiple classrooms per school are included in the analysis in the U.S., a three-level model indicates that approximately 40% of the variability in mathematics performance lies between students within classrooms, 47% between classrooms within schools, and 12% between schools. In Singapore, the only other country with large numbers of schools with more than one classroom, approximately 23% of the variability in mathematics performance lies between students within classrooms, 53% between classrooms within schools, and 24% between schools.

Table  8 also presents the variance components for the covariates-only model (fixed effects not presented), and for each of Models 1 through 6 in the four countries. The student and school covariates explained similar percentages of the total variance in students’ mathematics scores in the U.S. and Korea, 44.6% and 43.1%, respectively. In Singapore however, the student and school covariates explained approximately 62% of the total variance in achievement – 22.5% of the 26.1% available within schools, and 75.4% of the 73.9% between schools.

Despite being significantly associated with achievement in the U.S. and Singapore (Tables  4 and 7 ), the frequency with which teachers asked their students to write equations and functions to represent relationships and work on problems for which there is no immediately obvious method of solution explained only very small percentages of the total variance in students’ scores after controlling for the covariates in the model (between 0.2 and 1.6 additional percentage points). Likewise, having students decide on their own procedures for solving complex problems did not explain any additional variance in achievement in the U.S., and having students apply facts, concepts and procedures to solve routine problems did not explain any additional variance in achievement in Singapore.

The percentage of variance explained in Korea and Japan corroborate the findings in Tables  5 and 6 that these measures of teaching for conceptual understanding were not associated with mathematics test scores; Models 1 through 6 explained no additional variance over and above the percentage of variance explained by the covariates.

Student and school covariates and mathematics test scores

Though not central to the research questions posed, patterns were also evident in the regression coefficients associated with the student and school covariates across the four countries. The only significant effect for gender was observed in the U.S., with the difference favoring males. In the three East Asian countries, students’ with more positive attitudes toward mathematics (higher on the PAT-M index), students who valued mathematics (higher on the SVM index), and had greater self-confidence in learning mathematics (higher on the SCM index) were predicted to have higher mathematics scores. However, in the U.S., only students’ self-confidence in learning mathematics was associated with mathematics test scores. In all countries, the composite representing students’ home background and the school aggregated home background composite were significantly associated with mathematics test scores; students from better resourced homes and students in schools where the student body come from well-resourced homes were predicted to have significantly higher mathematics scores.

Conclusion and discussion

Educators, researchers and policy-makers worldwide continue to struggle to understand the changes that need to be made to improve educational outcomes and educational attainment for students, particularly in content areas such as mathematics and science that are essential for developing a highly skilled workforce. It is evident from the vast body of educational research that the correlates of student achievement and attainment are many and varied, particularly across countries, and that it is unlikely we will be able to isolate a single policy or practice that will alone, resolve all the concerns around improving student outcomes.

In the U.S., state and federal initiatives such as the Race to the Top program have renewed the emphasis on linking teacher quality with student outcomes. In reality, teacher quality is a complex construct that almost certainly hinges on teachers’ instructional practices, pedagogical content knowledge, attitudes toward students and teaching, as well as pre-service qualifications and professional development activities (Darling-Hammond and Youngs, 2002 ; Goe, 2007 ). However, while U.S. federal policies support the idea that how teachers teach is central to teacher quality, there appears to be little consensus on the specific instructional practices that are consistently associated with student achievement. The focus of this research was on developing a deeper understanding of the types of instructional practices, particularly practices supporting conceptual understanding, used by U.S. teachers compared to teachers in a subset of higher-performing East Asian countries, and to examine whether those instructional practices were related to students’ mathematics test scores.

In conducting this study, the authors acknowledge that one of the most significant challenges for comparing the relationships between student test scores in U.S. to the test scores of students in East Asian countries is the pervasiveness of out-of-school instruction in those countries, particularly in mathematics (Bray, 2003 ; Bray and Kwo, 2013 ). The proliferation of for-profit, supplementary instruction through juku or hagwons is likely to affect the validity of the comparisons, and unfortunately the extent of the effect is difficult to estimate using data from large-scale, international studies such as TIMSS, PIRLS, and PISA.

The descriptive results presented here support the findings from previous studies, including classroom observation studies, that there is variability across countries in teachers’ use of instructional practices for supporting conceptual understanding (Hiebert et al., 2005 ; VanTassel-Baska et al., 2008 ; Whitman and Lai, 1990 ). Focusing on how the U.S. compares to the other countries, it appeared that teachers in the U.S. were most similar to Korean teachers in the frequency of their reported practices, and that they reported using some of the practices at least as frequently, or more frequently than teachers in the other countries. This pattern is somewhat at odds with previous research findings about teaching practices in the U.S. For example, Hiebert et al. ( 2005 ) concluded from the 1999 TIMSS video study that, compared to higher performing countries, teachers in the U.S. spent more time completing repetitive exercises rather than applying and extending skills to new, different problems. In contrast, these data show that teachers in U.S. classrooms reported rates for having their students apply facts, concepts and procedures to solve routine problems, relate what they are learning in mathematics to their daily lives, decide on their own procedures for solving complex problems , and work on problems for which there is no immediately obvious method of solution that were as high, or higher than the top-performing comparison countries.

One possible explanation for the patterns observed may be differences in how the survey items are interpreted across countries, a significant challenge in any international study that uses self-report surveys. For example, when presented with a survey item that uses the term “complex problem” there may be cultural differences in what constitutes “complex”; something that is considered complex in one culture may be characterized as basic in another. Likewise, it is also possible that the different patterns observed in this study are due to the data being gathered from self-reports only, with no classroom observations or interviews to verify the results. Unfortunately, in the absence of observational or interview data, it is difficult to disconfirm the hypothesis that cross-cultural interpretation differences underlie the response patterns.

In Japan, the pattern of responses about use of practices that support conceptual understanding was at odds with the findings from previous TIMSS administrations. For example in TIMSS 1999, Japan led the other nations in teaching for conceptual understanding in eighth grade mathematics classrooms, ranking first in the frequency of using such practices (Mullis et al., 2000 ). Moreover, the TIMSS video study, also conducted in 1999, came to the same conclusion based on classroom observations (Hiebert et al., 2005 ). However, the patterns observed here for TIMSS 2007 suggest there may have been changes in Japanese teachers’ practices in the intervening years; teachers in Japan no longer reported using the six strategies as frequently as in previous TIMSS administrations. Overall, teachers in Korea, and in some cases the U.S., employed these strategies more frequently than teachers in Japan.

In looking at whether teachers’ instructional practices for supporting conceptual understanding were related to students’ mathematics achievement, two interesting patterns were observed. The relationships observed between teachers’ instructional practices and mathematics achievement in the U.S. and Singapore were similar. Of the six teaching for conceptual understanding practices examined, two were significantly associated with achievement in both countries. These were: the frequency with which teachers had their students write equations and functions to represent relationships and work on problems for which there is no immediately obvious method of solution , one of the most complex conceptual skills at the higher end of the construct of teaching for conceptual understanding (Hiebert and Grouws, 2007 ). In terms of a standardized effect size, the largest predicted change in mathematics achievement was associated with teachers having their students write equations and functions to represent relationships . Holding all else constant, increasing the frequency of using this practice from say some lessons to about half the lessons was associated with a predicted increase in achievement of 0.33 standard deviations in the U.S. and 0.33 standard deviations in Singapore. In each of these two countries, one additional instructional practice was associated with higher mathematics test scores: having students decide on their own procedures for solving complex problems in the U.S., and in Singapore, having students apply facts, concepts and procedures to solve routine problems , one of the less complex skills at the lower end of the construct (Hiebert and Lefevre, 1987 ).

Although teachers in the U.S. had their students write equations and functions to represent relationships less frequently than teachers in Korea and Japan, the significant regression coefficient in the U.S. indicates that, even after controlling for student and school covariates, this instructional strategy may be effective when used. This result in the U.S. aligns with Camburn and Han’s meta-analysis (Camburn and Han, 2011 ) that reported statistically significant support for the positive association between achievement and certain instructional practices. However, it is at odds with Tomoff et al.’s ( 2000 ) study of the earlier TIMSS 1995 data, which based on the analysis of slightly different survey items relating to teaching practices, did not find even a weak relationship between achievement and instructional practices in the U.S.

Likewise, although teachers in Singapore reported some of the lowest frequencies of having their students write equations and functions to represent relationships , apply facts, concepts and procedures to solve routine problems , and work on problems for which there is no immediately obvious method of solution among the four countries studied, these practices were each positively associated with mathematics achievement.

However, despite observing a statistically significant relationship between some of the instructional practice measures and mathematics achievement in the U.S. and Singapore, the measures explained very small percentages of the variability in mathematics achievement (between 0.2 and 1.6 additional percentage points). These percentages were similar to those reported by Desimone, Smith, Baker and Ueno’s TIMSS 1999 study (Desimone et al. 2005 ) who found that the relationship between mathematics achievement and the frequency of using strategies for teaching for conceptual understanding were weak in the U.S. The results from this study confirm that differences in instructional practices may not be the largest contributing factor to the observed differences in achievement between U.S. students and students in higher-scoring, East Asian countries.

First, teachers’ uses of instructional practices were not associated with students’ total mathematics scores in Korea or Japan. Though not presented as part of this research, this finding held when we looked at students’ sub-domain scores in Numbers, Algebra, Geometry, and Data and Chance, and when we included or excluded the student and school covariates from the model. This finding may be due to the small proportion of variability in achievement among schools in Korea and Japan that is available to be explained by differences in teachers’ instructional practices. Compared to the U.S. and Singapore where 57.1% and 73.9% of the variability in mathematics achievement existed between schools, respectively, only 10.6% of the variability in mathematics achievement exists between schools in Korea, and 19.9% in Japan. Furthermore, in Korea, more than two-thirds of the small proportion of available variance in mathematics scores between schools was accounted for by the covariates, leaving even less to be explained by differences in teachers’ uses of instructional practices.

As with any study of this nature, there are additional considerations in the interpretation of the findings presented here, some of which are intractable issues when conducting secondary analyses with large-scale, international comparative study data. First, the most obvious limitation of this research is that it cannot support causal claims about the observed relationships between teachers’ instructional practices and student test scores. Being a cross-sectional observational study that does not include information about students’ prior achievement, these data do not support claims about whether student achievement is influenced by teachers’ instructional practices, or whether teachers are using particular practices because they have high achieving students. Related to this point is the self-reported nature of the information provided by the teachers; stronger evidence for the existence of the relationship, or lack of relationship, between instructional practices and student outcomes would require studies to include a classroom observation component.

Second, the data analyzed to address the research questions were derived from the survey data collected as part of the TIMSS 2007 study. So, although the TIMSS documentation reports that the practices measured on the surveys were those deemed by TIMSS developers and its partners to be most effective for mathematics instruction (Mullis et al., 2005 ), they may not be culturally attuned to the practices that teachers use in some East Asian countries. In this sense, we cannot be certain that other, perhaps similar instructional practices are not associated with student test scores. In addition, there is the possibility that teachers from different cultures interpreted the survey items in different ways. Also, since teachers were not asked to indicate the amount of time spent per lesson on the different teaching practices, it was not possible to discern whether time spent using particular teaching strategies was associated with the observed differences between countries.

Third, the analysis sample was selected according to strict exclusion criteria and resulted in the removal of many students and schools. The comparison presented in Table  1 indicates that the reported samples (Mullis et al., 2008 ) and analysis samples were similar in terms of mean achievement, variability in achievement, and in the percentages of students whose teachers asked them to engage in the six activities in about half the lessons or more. However, it is possible that the reported and analysis samples differ on other characteristics and that bias may have been introduced with the removal of students and schools.

Finally, one possible explanation for the observed differences in mathematics achievement between countries may be the pervasiveness of supplementary, out-of-school instruction for East Asian students (Bray, 2003 ; Bray and Kwo, 2013 ). While in 1995, 1999, and 2003 TIMSS asked students how much time they spent outside of school receiving instruction in mathematics, this question was omitted from TIMSS 2007, precluding researchers and policy makers from exploring this hypothesis. Given the increase in out-of-school instruction in some countries, we would recommend that a question relating to out-of-school instruction be re-introduced in future TIMSS studies.

To conclude, while acknowledging these limitations, the authors believe that the evidence presented in this research contributes to our understanding of the complexity of teachers’ pedagogical practices in the classroom and points to the need for more nuanced research in this area. In particular, additional efforts should be undertaken in international comparative studies to estimate the effects of out-of-school instruction on students’ test scores, and to disentangle those effects from the effects of countries’ formal education systems.

a As the research conducted here is a secondary data analysis no human subjects were involved and the research is exempt under our organizations' Institutional Review Boards.

Berliner, DC. (2006). Our impoverished view of educational research. Teachers College Record, 108 (6), 949–995.

Article   Google Scholar  

Borman, GD, & Kimball, SM. (2005). Teacher quality and educational quality: Do teachers with higher standards-based evaluation ratings close student achievement gaps? The Elementary School Journal, 106 (1), 3–20.

Bray, M. (2003). Adverse Effects of Supplementary Private Tutoring: Dimensions, Implications, and Government Responses (UNESCO International Institute for Educational Planning, p. 84). Downloaded November, 2013.

Google Scholar  

Bray, M, & Kwo, O. (2013). Behind the façade of fee-free education: Shadow education and its implications for social justice. Oxford Review of Education, 39 (4), 480–497.

Camburn, EM, & Han, SW. (2011). Two decades of generalizable evidence on U.S. instruction from national surveys. Teachers College Record, 113 (3), 561–610.

Choi, N, & Chang, M. (2011). Interplay among school climate, gender, attitude toward mathematics, and mathematics performance of middle school students. Middle Grades Research Journal, 6 (1), 15–28.

D’Agostino, J. (2000). Instructional and school effects on students’ longitudinal reading and mathematics achievements. School Effectiveness and School Improvement, 11 (2), 197–235.

Darling-Hammond, L, & Youngs, P. (2002). Defining “highly qualified teachers”: What does “scientifically-based research” actually tell us? Educational Researcher, 31 (9), 13–25.

Desimone, LM, Smith, T, Baker, D, & Ueno, K. (2005). Assessing barriers to the reform of U.S. mathematics instruction from an international perspective. American Educational Research Journal, 42 (3), 501–535.

DeVellis, RF. (2012). Scale Development: Theory and Applications (3rd ed.). Washington, D.C.: SAGE Publications.

diSessa, AA, & Sherin, BL. (1998). What changes in conceptual change? International Journal of Science Education, 20 (10), 1155–1191.

Else-Quest, NM, Hyde, JS, & Linn, MC. (2010). Cross-national patterns of gender differences in mathematics: A meta-analysis. Psychological Bulletin, 136 (1), 103–127.

Fennema, E, Carpenter, TP, Franke, ML, Levi, L, Jacobs, VR, & Empson, SB. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27 (4), 403–434.

Foy, P, Galia, J, & Li, I. (2008). Scaling the data from the TIMSS 2007 mathematics and science assessments. Chapter 11. In JF Olson, MO Martin, & IVS Mullis (Eds.), TIMSS 2007 technical report (pp. 225–280). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

Gales, MJ, & Yan, W. (2001). Relationship between constructivist teacher beliefs and instructional practices to students’ mathematical achievement: Evidence from TIMSS (Paper presented at the annual meeting of the American Educational Research Association, Seattle, WA).

Gallagher, HA. (2004). Vaughn Elementary's innovative teacher evaluation system: Are teacher evaluation scores related to growth in student achievement? Peabody Journal of Education, 79 (4), 79–107.

Gearhart, M, Saxe, GB, Seltzer, M, Schlackman, J, Ching, CC, Nasir, N, & Sloan, TF. (1999). Opportunities to learn fractions in elementary mathematics classrooms. Journal for Research in Mathematics Education, 30 (3), 286–315.

Goe, L. (2007). The link between teacher quality and student outcomes: A research synthesis . Washington, DC: National Comprehensive Center for Teacher Quality.

Hattie, JAC. (2009). Visible learning: A synthesis of over 800 meta-analyses relating to achievement . New York: Routledge.

Heck, RH. (2007). Examining the relationship between teacher quality as an organizational property of schools and students' achievement and growth rates. Educational Administration Quarterly, 43 (4), 399–432.

Henderson, AT, & Berla, N. (1994). A new generation of evidence: The family is critical to student achievement . St. Louis, MO: Danforth Foundation and Flint, MI: Mott (C. S.) Foundation.

Heneman, HG, Milanowski, A, Kimball, SM, & Odden, A. (2006). Standards-based teacher evaluation as a foundation for knowledge- and skill-based pay (CPRE Policy Brief No. RB-45) . Philadelphia: Consortium for Policy Research in Education.

Hiebert, J. (2003). What research says about the NCTM Standards. In J Kilpatrick, WG Martin, & D Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 5–23). Reston, VA: The National Council of Teachers of Mathematics, Inc.

Hiebert, J, & Grouws, DA. (2007). The effects of classroom mathematics teaching on students’ learning. In FK Lester (Ed.), Second handbook for research in mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 371–404). Charlotte, NC: Information Age Publishing.

Hiebert, J, & Lefevre, P. (1987). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–28). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Hiebert, J, Stigler, JW, Jacobs, JK, Givvin, KB, Garnier, H, Smith, M, & Gallimore, R. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 video study. Educational Evaluation and Policy Analysis, 27 (2), 111–132.

Hill, HC, Rowan, B, & Ball, DL. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal., 42 , 371–406.

Ho, H, Senturk, D, Lam, AG, Zimmer, JM, Hong, S, Okamoto, Y, & Wang, C. (2000). The affective and cognitive dimensions of mathematics anxiety: A cross-national study. Journal for Research in Mathematics Education, 31 (3), 362–379.

Holtzapple, E. (2003). Criterion-related validity evidence for a standards-based teacher evaluation system. Journal of Personnel Evaluation in Education, 17 (3), 207–219.

House, JD. (2009). Elementary-school mathematics instruction and achievement of fourth-grade students in Japan: Findings from the TIMSS 2007 Assessment. Education, 130 (2), 301–307.

Jacobs, JK, & Morita, E. (2002). Japanese and American teachers' evaluations of videotaped mathematics lessons. Journal for Research in Mathematics Education, 33 , 154–175.

Joncas, M. (2008). TIMSS 2007 Sample Design. In JF Olson, MO Martin, & IVS Mullis (Eds.), TIMSS 2007 technical report (pp. 77–92). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

Kaiser, G, & Blömeke, S. (2013). Learning from the Eastern and the Western debate: The case of mathematics teacher education. ZDM Mathematics Education, 45 , 7–19.

Kim, RY, Ham, S, & Paine, LW. (2011). Knowledge expectations in mathematics teacher preparation programs in South Korea and the United States: Towards international dialogue. Journal of Teacher Education, 62 (1), 48–61.

Kimball, SM, White, B, Milanowski, AT, & Borman, G. (2004). Examining the relationship between teacher evaluation and student assessment results in Washoe County. Peabody Journal of Education, 79 (4), 54–78.

Lewis, KA, & Seidman, LS. (1994). Mathematics-time capital matters: A cross-country analysis. Economics of Education, 13 (3), 215–226.

Little, RJA, & Rubin, DB. (2002). Statistical analysis with missing data (2nd ed.). New York: Wiley.

Liu, OL, & Wilson, M. (2009). Gender differences in large-scale mathematics assessments: PISA Trend 2000 and 2003. Applied Measurement in Education, 22 (2), 164–184.

Lubienski, ST. (2006). Examining instruction, achievement, and equity with NAEP mathematics data. Education Policy Analysis Archives, 14(14) . Retrieved Sep. 23, 2010 from http://epaa.asu.edu/ojs/article/view/85 .

Ma, X, & Kishor, N. (1997). Assessing the relationship between attitude toward mathematics and achievement in mathematics: A meta-analysis. Journal for Research in Mathematics Education, 28 (1), 26–47.

Martin, MO, & Preuschoff, C. (2008). Creating the TIMSS 2007 Background Indices. In JF Olson, MO Martin, & IVS Mullis (Eds.), TIMSS 2007 technical report (pp. 281–338). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

McCoach, DB. (2010). Hierarchical linear modeling. In GR Hancock & RO Mueller (Eds.), Quantitative methods in the social and behavioral sciences: A guide for researchers and reviewers (pp. 123–140). New York: Routledge.

Mewborn, DS. (2003). Teaching, teachers’ knowledge, and their professional development. In J Kilpatrick, WG Martin, & D Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 45–52). Reston, VA: The National Council of Teachers of Mathematics, Inc.

Milanowski, A. (2004). The relationship between teacher performance evaluation scores and student achievement: Evidence from Cincinnati. Peabody Journal of Education, 79 (4), 33–53.

Mullis, IVS, Martin, MO, Gonzalez, EJ, Gregory, KD, Garden, RA, O'Connor, KM, & Smith, TA. (2000). TIMSS 1999 international mathematics report: Findings from IEA's repeat of the Third International Mathematics and Science Study at the eighth grade . Chestnut Hill, MA: The International Study Center, Boston College.

Mullis, IVS, Martin, MO, Ruddock, GJ, O'Sullivan, CY, Arora, A, & Erberber, E. (2005). TIMSS 2007 assessment frameworks . Chestnut Hill, MA: TIMSS& PIRLS International Study Center, Lynch School of Education, Boston College.

Mullis, IVS, Martin, MO, Foy, P, Olson, JF, Preuschoff, C, Erberber, E, Arora, A, & Galia, J. (2008). TIMSS 2007 international mathematics report: Findings from IEA’s trends in international mathematics and science study at the fourth and eighth grades . Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics . Reston, VA: NCTM.

Newmann, FM, Bryk, AS, & Nagaoka, JK. (2001). Authentic intellectual work and standardized tests: Conflict or coexistence? Chicago: Consortium on Chicago School Research.

OECD. (2010). PISA 2009 at a glance . Paris, France: OECD Publishing.

Prawat, RS. (1989a). Teaching for understanding: Three key attributes. Teaching and Teacher Education, 5 , 315–328.

Prawat, RS. (1989b). Promoting access to knowledge, strategy, and disposition in students: A research synthesis. Review of Educational Research, 59 , 1–41.

Raudenbush, SW, & Bryk, AS. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks, CA: Sage.

Robinson, JP, & Lubienski, ST. (2011). The development of gender achievement gaps in mathematics and reading during elementary and middle school: Examining direct cognitive assessments and teacher ratings. American Educational Research Journal, 48 (2), 268–302.

Rothstein, R. (2004). Class and schools: Using social, economic, and educational reform to close the Black-White achievement gap . Washington, D.C.: Economic Policy Institute.

Rowan, B, Chiang, FS, & Miller, RJ. (1997). Using research on employees’ performance to study the effects of teachers on students’ achievement. Sociology of Education, 70 , 256–284.

Rumberger, RW, & Palardy, GJ. (2005). Does segregation still matter? The impact of student composition on academic achievement in high school. Teachers College Record, 107 (9), 1999–2045.

Schacter, J, & Thum, YM. (2004). Paying for high- and low-quality teaching. Economics of Education Review, 23 , 411–430.

Schmidt, WH, & Cogan, LS. (1996). Development of the TIMSS context questionnaires. In MO Martin & DL Kelly (Eds.), Third international mathematics and science study technical report, volume I: design and development . Boston College: Chestnut Hill, MA.

Schmidt, WH, Wang, HC, & McKnight, CC. (2005). Curriculum coherence: An examination of U.S. mathematics and science content standards from an international perspective. Journal of Curriculum Studies, 37 (5), 525–559.

Schreiber, JB. (2002). Institutional and student factors and their influence on advanced mathematics achievement. Journal of Educational Research, 95 (5), 274–286.

Singh, K, Granville, M, & Dika, S. (2002). Mathematics and science achievement: Effects of motivation, interest, and academic engagement. Journal of Educational Research, 95 (6), 323–332.

Sirin, S. (2005). Socioeconomic status and academic achievement: A meta-analytic review of research. Review of Educational Research, 75 (3), 417–453.

Star, JR. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36 , 404–411.

Tomoff, J, Thompson, M, & Behrens, J. (2000). Measuring NCTM-recommended practices and student achievement with TIMSS (Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA).

VanTassel-Baska, J, Feng, A, MacFarlane, B, Heng, M, Teo, CT, Wong, ML, & Khong, BC. (2008). A cross-cultural study of teachers’ instructional practices in Singapore and the United States. Journal for the Education of the Gifted, 31 (3), 338–363.

Votruba-Drzal, E. (2006). Economic disparities in middle childhood development: Does income matter? Developmental Psychology, 42 (6), 1154–1167.

Wenglinsky, H. (2002). How schools matter: The link between teacher classroom practices and student academic performance. Education Policy Analysis Archives, 10 (12), 1–30.

Whitman, NC, & Lai, MK. (1990). Similarities and differences in teachers' beliefs about effective teaching of mathematics: Japan and Hawai'i. Educational Studies in Mathematics, 21 , 71–81.

Yeung, WJ, Linver, MR, & Brooks-Gunn, J. (2002). How money matters for young children’s development: Parental investment and family processes. Child Development, 73 (6), 1861–1879.

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LOD, YW and KS participated in the development of the rationale for the study. YW prepared the data for analysis. LOD and YW conducted the analyses. LOD drafted the manuscript with contributions from YW and KS. All authors read and approved the final manuscript.

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O’Dwyer, L.M., Wang, Y. & Shields, K.A. Teaching for conceptual understanding: A cross-national comparison of the relationship between teachers’ instructional practices and student achievement in mathematics. Large-scale Assess Educ 3 , 1 (2015). https://doi.org/10.1186/s40536-014-0011-6

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Mindset and Levels of Conceptual Understanding in the Problem-Solving of Preservice Mathematics Teachers in an Online Learning Environment

International Journal of Learning, Teaching and Educational Research, 21(6), 18-33 (2022)

Posted: 20 Jul 2022 Last revised: 31 Jan 2023

Ma Luisa Mariano-Dolesh

Dr. Henry A. Wise, Jr. High School

Leila M. Collantes

Central Luzon State University (CLSU)

Edwin D. Ibañez

Jupeth pentang.

Central Luzon State University

Date Written: June 22, 2022

Mindset plays a vital role in tackling the barriers to improving the preservice mathematics teachers’ (PMTs) conceptual understanding of problem-solving. As the COVID-19 pandemic has continued to pose a challenge, online learning has been adopted. This led this study to determining the PMTs’ mindset and level of conceptual understanding in problem-solving in an online learning environment utilising Google Classroom and the Khan Academy. A quantitative research design was employed specifically utilising a descriptive, comparative, and correlational design. Forty-five PMTs were chosen through simple random sampling and willingly took part in this study. The data was gathered using validated and reliable questionnaires and problem-solving tests. The data gathered was analysed using descriptive statistics, analysis of variance, and simple linear regression. The results revealed that the college admission test, specifically numerical proficiency, influences a strong mindset and a higher level of conceptual understanding in problem-solving. Additionally, this study shows that mindset predicts the levels of conceptual understanding in problem-solving in an online environment where PMTs with a growth mindset have the potential to solve math problems. The use of Google Classroom and the Khan Academy to aid online instruction is useful in the preparation of PMTs as future mathematics teachers and problem-solvers. Further studies may be conducted to validate these reports and to address the limitations of this study.

Keywords: conceptual understanding, growth mindset, mathematics education, online learning, preservice teachers

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Dr. Henry A. Wise, Jr. High School ( email )

Central luzon state university (clsu) ( email ).

Barangay Bantug Science City of Munoz Science City of Muñoz, Nueva Ecija 3119 Philippines

Jupeth Pentang (Contact Author)

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7 Strategies to Teach Conceptual Understanding in Math

These strategies will give you a head start on getting rid of math tips and tricks.

Multiplication is repeated addition.

Keep, switch, flip. 

The butterfly method.

These are all examples of math shortcuts, tips, or tricks that many students learn to rely on from an early age. I taught many students throughout my 16 years in the classroom who quickly pulled out these strategies!

But my students couldn’t explain why these tips and tricks work. Sometimes they would struggle and get upset. This happened when they faced situations where the tricks didn't work or they forgot what to do.

That’s why math education has changed recently to focus on teaching students a deep understanding of concepts instead of relying on shortcuts.  Educators know that teaching children to deeply understand math leads to the development of problem-solvers and critical thinkers. 

But how can we stop focusing on teaching shortcuts and instead help students become real mathematicians?

Don’t worry; we’ve got a few ideas for you! Check out these seven tips for getting rid of the shortcuts and teaching true conceptual understanding in math.

1. Spiral Practice Through a Well-Thought-Out Scope and Sequence

Mathematics is a body of conceptual knowledge made up of interrelated concepts. It isn’t just a list of disconnected topics to check off a list as students move from grade to grade. Plan your school year carefully to avoid math pitfalls by following a structured scope and sequence.

I used the   Carnegie Learning High School Math Solution  for Algebra 1 and Geometry in my last years of teaching. For the first time, I saw how much the scope and sequence really matter. My Algebra 1 students used what they learned in Module 1 to understand quadratic functions in Module 5. It was a lightbulb moment for all of us!

My Algebra 1 students used their prior knowledge and noticed recurring concepts. This helped them avoid relying on shortcuts or tricks.

A thoughtful scope and sequence incorporating spiral review is key to teaching deep conceptual understanding in math. If we rely on teaching the “easy” shortcuts instead of giving students the time and space to master grade-level skills and see the connections between concepts, they’ll struggle to develop a body of conceptual knowledge that will help them understand more complex ideas in the future.

2. Use High-Order Tasks to Build Critical Thinking Skills

Many students (and teachers!) love math shortcuts for quick “success.” But having a toolbox packed with critical thinking skills and problem-solving strategies is so much more valuable. These skills will serve your students not only in class, but in the real world.

One way to help students develop their critical thinking and problem-solving skills is to assign high-order math tasks in your classroom. Rich tasks help students think about what they already know and test out different methods until they identify one that works. In the process, your students gain skills and strategies that eliminate the need for tips and tricks.

Some of my favorite high-order tasks to use with my Algebra 1 students were in a lesson titled, “Do You Mean: Recursion ?” This lesson is filled with activities that encourage students to think critically about arithmetic and geometric sequences and explicit and recursive formulas. They’re even asked to compare the pros and cons of using explicit or recursive formulas, using evidence developed over the last series of lessons!

The fact that there’s no “plug and chug” in this series of high-order tasks meant that my students were constantly using and developing their critical thinking skills and problem-solving strategies. 

I was amazed by the intelligent conversations happening in the room. Students were discussing cell division tables and explaining why explicit and recursive formulas worked!

3. Visual Representations for Better Retrieval

Visual aids are powerful tools for helping students to develop a deep, conceptual understanding of mathematical concepts. I loved supplementing as many lessons as possible with diagrams, graphs, anchor charts, manipulatives, and even high-quality math videos . In doing so, every learner had an entry point into even the most upper-level mathematic concepts.

Visualizing math concepts helps students see patterns and make connections that they may not immediately understand from written or verbal explanations. And when they have a visual cue stored in their brain, it makes retrieving information much more manageable. 

For example, suppose a student can recall that a quadratic function looks like a parabola because they’ve interacted with graphs illustrating a pumpkin catapult or diving into a swimming pool. If that happens, they're more likely to understand and use the formula of a quadratic function in various situations.

4. Manipulatives and Hands-On Learning

Another way to eliminate tips and tricks (“A negative times a negative is a positive,” anyone?) is with manipulatives. I love algebra tiles, counting chips, and even interactive number lines.

And I promise those hands-on materials aren’t just for the younger kids. Your high schoolers won’t mind abandoning note-taking in favor of digging into algebra tiles! 

I’ll never forget using algebra tiles for various purposes with my high schoolers. From watching a student with complex special needs finally understand the meaning and applications of a zero pair to seeing upper-level students suddenly “get” factoring trinomials, each visual and hands-on learning experience was pure magic!

5. Connect Concepts Instead of Teaching Math Shortcuts

Teaching is all about making connections. In this case, we're talking about mathematical connections.

Teach your students to look for the interconnectedness of mathematical concepts. Show them how ideas fit together and build on one another. Watch as they develop a deeper understanding of the underlying concepts. Then, it’s time to kiss the shortcuts goodbye!

For example, the scope and sequence I used encouraged my students to apply their foundational knowledge of concrete geometric investigations and reasoning with shapes to formalize their understanding. Circles were also integrated throughout the course, rather than treating them as isolated geometric figures (as many other curriculums do). 

Watching my Geometry students make connections between circles and angle relationships and complete constructions using arcs was a game changer! They remembered more when they understood how concepts were connected and could use their knowledge in unexpected ways.

6. Help Your Students Make Real-World Connections

Another vital connection that will lead to the elimination of shortcuts, tips, and tricks is between the mathematics your students learn in the classroom and the real-world applications of the concepts.

When you help your students discover these links to the real world, math suddenly loses its abstract nature. It becomes relevant, practical, and motivating.

Your students will stay interested and learn concepts that can be applied in different situations. Here are some examples using real-world scenarios to model integer subtraction that could be used in a 7th-grade class.

7. Don’t Use Math Tips and Tricks—Collaborate!

Most kids love to work in groups, right? It enhances the social aspect of school that many students value. And when structured correctly, these collaborative learning experiences can be the perfect setting for developing deep mathematical understanding.

Collaborating to create their conceptual knowledge is a powerful experience for your students. They may productively struggle , disagree, and even argue a bit, but these experiences are where the magic happens. 

“Allow students to experience and play and notice and wonder,” writes Tina Cardone, author of Nix the Tricks: A Guide to Avoiding Shortcuts That Cut Out Math Concept Development . “They will surprise you! Being a mathematician is not limited to rote memorization…Being a mathematician is about critical thinking, justification, and using tools from past experiences to solve new problems.”

And I can think of no better opportunity to notice, wonder, think critically, and justify those thoughts than when collaborating with peers. It may be hard to give up that “sage on the stage” lecture style (I definitely struggled!), but hearing your students engage in rich, mathematical conversations and watching them abandon the shortcuts in favor of deeply understanding the math is worth it. The feeling is second to none!

Don’t Let Tips and Tricks Take Away the Beauty of Math

Math is a beautiful, creative, and thought-provoking subject that sets the perfect stage for your students to become critical thinkers, problem solvers, and leaders of tomorrow. Don’t let a reliance on math shortcuts, tips, and tricks rob them of that experience!

I hope you’re ready to ditch the tips and tricks in your classroom. I you need more convincing, check out this case study from Muleshoe Independent School District in Texas. They were able to teach their students deep conceptual understanding in math and get rid of the shortcuts—with some great results to show for it!

• Karen Sloan
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Before joining Carnegie Learning's marketing team in 2022, Karen spent 16 years teaching mathematics and social studies in Ohio classrooms. She has a passion for inclusive education and believes that all learners can be meaningfully included in academic settings from day one. As a former math and special education teacher, she is excited to provide educators with the latest in best-practices content so that they can set all students on the path to becoming confident "math people."

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• April 17, 2023

Math is a beautiful, creative, and thought-provoking subject that sets the perfect stage for your students to become critical thinkers, problem solvers, and leaders of tomorrow. Don’t let a reliance on math shortcuts, tips, and tricks rob them of that experience!

Karen Sloan, Math and Special Education Teacher of 16 Years

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A Math Word Problem Framework That Fosters Conceptual Thinking

This strategy for selecting and teaching word problems guides students to develop their understanding of math concepts.

Word problems in mathematics are a powerful tool for helping students make sense of and reason with mathematical concepts. Many students, however, struggle with word problems because of the various cognitive demands. As districtwide STEAM professional development specialists, we’ve spent a lot of time focusing on supporting our colleagues and students to ensure their success with word problems. We found that selecting the right word problems, as well as focusing on conceptual understanding rather than procedural knowledge, provides our students with real growth.

As our thinking evolved, we began to instill a routine that supports teaching students to solve with grit by putting them in the driver’s seat of the thinking. Below you’ll find the routine that we’ve found successful in helping students overcome the challenges of solving word problems.

Not all word problems are created equal

Prior to any instruction, we always consider the quality of the task for teaching and learning. In our process, we use word problems as the path to mathematics instruction. When selecting the mathematical tasks for students, we always consider the following questions:

• Does the task align with the learning goals and standards?
• Will the task engage and challenge students at an appropriate level, providing both a sense of accomplishment and further opportunities for growth?
• Is the task open or closed? Open tasks provide multiple pathways to foster a deeper understanding of mathematical concepts and skills. Closed tasks can still provide a deep understanding of mathematical concepts and skills if the task requires a high level of cognitive demand. 
• Does the task encourage critical thinking and problem-solving skills?
• Will the task allow students to see the relevance of mathematics to real-world situations?
• Does the task promote creativity and encourage students to make connections between mathematical concepts and other areas of their lives?

If we can answer yes to as many of these questions as possible, we can be assured that our tasks are rich. There are further insights for rich math tasks on NRICH and sample tasks on Illustrative Mathematics and K-5 Math Teaching Resources .

Developing conceptual understanding

Once we’ve selected the rich math tasks, developing conceptual understanding becomes our instructional focus. We present students with Numberless Word Problems and simultaneously use a word problem framework to focus on analysis of the text and to build conceptual understanding, rather than just memorization of formulas and procedures. 

• First we remove all of the numbers and have students read the problem focusing on who or what the problem is about; they visualize and connect the scenario to their lives and experiences. 
• Next we have our students rewrite the question as a statement to ensure that they understand the questions.
• Then we have our students read the problem again and have them think analytically. They ask themselves these questions: Are there parts? Is there a whole? Are things joining or separating? Is there a comparison? 
• Once that’s completed, we reveal the numbers in the problem. We have the students read the problem again to determine if they have enough information to develop a model and translate it into an equation that can be solved.
• After they’ve solved their equation, we have students compare it against their model to check their answer.  

Collaboration and workspace are key to building the thinking

To build the thinking necessary in the math classroom , we have students work in visibly random collaborative groups (random groups of three for grades 3 through 12, random groups of two for grades 1 and 2). With random groupings, we’ve found that students don’t enter their groups with predetermined roles, and all students contribute to the thinking.

For reluctant learners, we make sure these students serve as the scribe within the group documenting each member’s contribution. We also make sure to use nonpermanent vertical workspaces (whiteboards, windows [using dry-erase markers], large adhesive-backed chart paper, etc.). The vertical workspace provides accessibility for our diverse learners and promotes problem-solving because our students break down complex problems into smaller, manageable steps. The vertical workspaces also provide a visually appealing and organized way for our students to show their work.  We’ve witnessed how these workspaces help hold their attention and improve their focus on the task at hand.

Facilitate and provide feedback to move the thinking along

As students grapple with the task, the teacher floats among the collaborative groups, facilitates conversations, and gives the students feedback. Students are encouraged to look at the work of other groups or to provide a second strategy or model to support their thinking. Students take ownership and make sense of the problem, attempt solutions, and try to support their thinking with models, equations, charts, graphs, words, etc. They work through the problem collaboratively, justifying their work in their small group. In essence, they’re constructing their knowledge and preparing to share their work with the rest of the class. 

Word problems are a powerful tool for teaching math concepts to students. They offer a practical and relatable approach to problem-solving, enabling students to understand the relevance of math in real-life situations. Through word problems, students learn to apply mathematical principles and logical reasoning to solve complex problems. 

Moreover, word problems also enhance critical thinking, analytical skills, and decision-making abilities. Incorporating word problems into math lessons is an effective way to make math engaging, meaningful, and applicable to everyday life.

Understanding the Difference between Procedural vs. Conceptual Understanding

What’s the difference.

Procedural understanding is when students hoard steps and algorithms. They rely on the memorization of these formulas to answer questions, and they rarely make deep connections during instruction.

Conceptual understanding is knowing the procedural steps to solving a problem and understanding why those algorithms and approaches work, similar to a recognition that there is a man hiding behind the giant head in The Wizard of Oz. This level of understanding has students reaching higher depths of knowledge because they are making connections from one skill to another. As you plan your math lessons, ensure that you are targeting both procedural and conceptual understanding with an unbalanced approach. I use the 80-20 rule. Of the questions I pose to my students, 80% are conceptual-based. This is no easy feat when the majority of textbooks follow a 20-80 rule, where most of the work is procedural-based. We all know those textbooks and workbooks. They have 20+ rote questions for long division on a single page. As educators, we have to go after those meaningful questions.

Here are examples of questions aligned to the 4.MD.3 standard: Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

Take the time to find materials that require your students to deeply evaluate curriculum. With specific questioning you’ll be able to cross over the threshold from procedural to conceptual understanding.

My line of Power Problems are designed to target conceptual understanding of standards. They are available for grades 3rd-6th.

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Published: July 26, 2001 Subjects Levels Resource Types Education - Applied Research Education - Basic Research PER-Central Type Intended Users Ratings Want to rate this material? Primary Details Citation formats. %A William Gerace %T Problem Solving and Conceptual Understanding %S PER Conference Invited Paper %D July 25-26 2001 %C Rochester, New York %U https://www.compadre.org/Repository/document/ServeFile.cfm?ID=4375&DocID=1028 %O Physics Education Research Conference 2001 %O July 25-26 %O application/pdf %0 Conference Proceedings %A Gerace, William %D July 25-26 2001 %T Problem Solving and Conceptual Understanding %B Physics Education Research Conference 2001 %C Rochester, New York %S PER Conference Invited Paper %8 July 25-26 %U https://www.compadre.org/Repository/document/ServeFile.cfm?ID=4375&DocID=1028 The AIP Style presented is based on information from the AIP Style Manual . 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AI Magazine Privacy & TC Cookie Policy Privacy Overview 246 IMAGES (PDF) Conceptual Understanding, Procedural Knowledge and Problem-Solving Skills in Mathematics Conceptual Understanding Problem Solving for Conceptual Understanding The Four Stages Of Problem Solving Adapted From The Integrated Model The 5 Steps of Problem Solving What is Growlity's Problem Solving Approach for 100% Business Success? VIDEO Try solving this conceptual problem on Application of Derivatives from JEE ADVANCED 2022 Thermodynamics Two Graphical Problem Difference Between Factual Knowledge and Conceptual Knowledge Lecture 11 Middle School Algebra Problem 02 Conceptual Dynamics Review Problem 5-14: Particle Newtonian Mechanics COMMENTS PDF Conceptual Understanding, Procedural Knowledge and Problem- Solving C: conceptual understanding. P: problem solving scale. iii. The test will have a total of 60 questions, 20 questions for each different type of ability. Questions will be numbered as the following: D1-1K, D1-1C, D1-1P and so on. iv. Each question will be graded using a four-point Likert scale. 0: if they did not write anything. PDF What Is Conceptual Understanding? twenty years, mathematics educators have often contrasted conceptual understanding with procedural knowledge. Problem solving has also been in the mix of these two. A good starting point for us to understand conceptual understanding is to review The Learning Principle from the NCTM Principles and Standards for School Mathematics (2000). THE STRANDS OF MATHEMATICAL PROFICIENCY work has some similarities with the one used in recent mathematics assessments by the National Assessment of Educational Progress (NAEP), which features three mathematical abilities (conceptual understanding, procedural knowledge, and problem solving) and includes additional specifications for reasoning, connections, and communication. 2 The strands also echo components of mathematics learning ... PDF Problem Solving and Conceptual Understanding W. J. Gerace, "Problem Solving and Conceptual Understanding" 1 Problem Solving and Conceptual Understanding William J. Gerace Physics Department and Scientific Reasoning Research Institute University of Massachusetts, Amherst MA 01003-4525 A framework for thinking about knowledge and its organization is presented that can ac-count for known ... Common Core's Focus on Concepts Is Key to Improving Math Education The Common Core State Standards' emphasis on conceptual understanding in math will improve students' problem-solving skills and ultimately help prepare them for jobs of the future, argues a new ... Teaching for conceptual understanding: A cross-national ... Teaching for conceptual understanding and mathematics test scores. Tables 4, 5, 6, and 7 present the regression coefficients associated with the six individual teaching for conceptual understanding measures (Models 1-6) for predicting students' total mathematics test scores in the U.S., Korea, Japan, and Singapore, respectively. Since the ... Mastering Mathematics and Problem Solving Ofsted recently reported that 'the degree of emphasis on problem solving and conceptual understanding is a key discriminator between good and weaker provision,' (Ofsted, 2015). The current paucity in provision can result in classrooms where 'many pupils spend too long working on straightforward questions, with problem solving located at the ... Mindset and Levels of Conceptual Understanding in the Problem-Solving Mindset plays a vital role in tackling the barriers to improving the preservice mathematics teachers' (PMTs) conceptual understanding of problem-solving. As the COVID-19 pandemic has continued to pose a challenge, online learning has been adopted. This led this study to determining the PMTs' mindset and level of conceptual understanding in ... 7 Strategies to Teach Conceptual Understanding in Math Check out these seven tips for getting rid of the shortcuts and teaching true conceptual understanding in math. 1. Spiral Practice Through a Well-Thought-Out Scope and Sequence. Mathematics is a body of conceptual knowledge made up of interrelated concepts. PDF Improving conceptual understanding and problem-solving in ... Conceptual understanding 34.1 55.53 34.82 73.92 Problem-solving 19.64 45.35 19.68 71.25 Generally, both pre-test and the post-test scores improved the ability of the students (conceptual understanding and problem-solving). Table 1 also informs that there was no significant difference between thepre-tests in either experimental class. conceptual understanding of problem solving Best Teaching Practices Conceptual Understanding of Problem Solving. Research Findings. Research at the secondary and even post-secondary level on understanding of basic concepts that are involved in solving biology, chemistry, and physics problems (many of which require the application of algebraic or other mathematical concepts) indicates that students do not understand the concepts. Improving conceptual understanding and problem-solving in mathematics Request PDF | Improving conceptual understanding and problem-solving in mathematics through a contextual learning strategy | Mathematics is a basic fundamental of science and technology. However ... Math Manipulatives: A Student-Centered Approach to Teaching Mathematics To develop the conceptual understanding needed for future success, students need access to a wide array of tools that help them grasp, inte grate, ... In a Eureka Math 2 classroom, students gain an understanding of why and how manipulatives and other tools aid their problem-solving processes, ... Conceptual Understanding, Procedural Knowledge and Problem-Solving The results show that conceptual understanding and problem-solving skills are positively correlated. This research also endeavors to correlate students" performance in this test with their high ... A Strategy for Teaching Math Word Problems A Math Word Problem Framework That Fosters Conceptual Thinking. This strategy for selecting and teaching word problems guides students to develop their understanding of math concepts. Word problems in mathematics are a powerful tool for helping students make sense of and reason with mathematical concepts. Many students, however, struggle with ... PDF Assessment Strategies for Enhancing Students' Mathematical Problem conceptual understanding and mathematical problem-solving between students who were taught with authentic assessment and those who were taught with the conventional assessment strategies. Besides that, Cai et al. (2013) conducted a longitudinal research project employing the Connected Mathematics Program (CMP) (PDF) CONCEPTUAL UNDERSTANDING AND PROBLEM-SOLVING ... Students' conceptual understanding was measured using 13 multiple-choice questions while their problem-solving skill was measured using 3 essay questions. The normality, the linearity, and the ... Understanding the Difference between Procedural vs. Conceptual Conceptual understanding is knowing the procedural steps to solving a problem and understanding why those algorithms and approaches work, similar to a recognition that there is a man hiding behind the giant head in The Wizard of Oz. This level of understanding has students reaching higher depths of knowledge because they are making connections ... PDF Investigating Students' Mathematical Difficulties with Quadratic ... intertwined relationship between conceptual understanding, procedural fluency, and problem solving and reasoning due to the hierarchical nature of mathematics (AAS, 2015, p. 17). The cognitive load work by Kirschner, Sweller, and Clark (2006) gives an explanaton for the necessity of fluency with prerequisite knowledge. Conceptual & Procedural Math: What's the Difference? It is often contrasted with "procedural math," which teaches students to solve problems by giving them a series of steps to do. Procedural math approaches an elementary problem such as two-digit subtraction (72 − 69, say) by teaching students to "borrow."Since you can't subtract 9 from 2, strike through the 7 next to the 2, turn it ... (PDF) Assessing conceptual understanding in mathematics Given that math is an active process that encourages higher-order thinking and problem solving, an assessment focusing on the growth of conceptual understanding is required. Problem Solving and Conceptual Understanding Problem Solving and Conceptual Understanding. written by William J. Gerace. This paper, presented at the 2001 Physics Education Research Conference, presents a framework for thinking about knowledge and its organization that can account for known expert-novice differences in knowledge storage and problem solving behavior. 6 crucial soft skills every student needs to master in the 21st ... CRITICAL THINKING AND PROBLEM-SOLVING. Creating an environment that focuses on conceptual understanding and application of that knowledge to real-world skills leads to lifelong learning and ... Top Artificial Intelligence AI Books to Read in 2024 The Alignment Problem: Machine Learning and Human Values. This book talks about a concept called "The Alignment Problem," where the systems we aim to teach, don't perform as expected, and various ethical and existential risks emerge. Life 3.0: Being Human in the Age of Artificial Intelligence assignment essay homework paper phd report resume review thesis