Annotations were taken from the abstracts of the articles, the ERIC database, or the texts of the articles, except as otherwise noted.
Behr, Merlyn J., Harel, Guershon, Post, Thomas, & Lesh, Richard (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), NCTM Handbook of Research on Mathematics Teaching and Learning (pp. 296-333), New York: Macmillan Publishing Company. This handbook is somewhat dated, but for folks who have never heard of any research in math ed on the teaching/learning of rational numbers, this provides a good start.
Behr, Merlyn J., Khoury, Helen A., Harel, Guershon, Post, Thomas, & Lesh, Richard (1997). Conceptual units analysis of preservice elementary teachers' strategies on a rational-number-as operator task. Journal for Research in Mathematics Education, 28(1), 48-69. This study explores preservice teachers' understanding of the operator construct of rational number. Three related problems, given in 1-on-1 clinical interviews, consisted of finding 3/4 of a pile of 8 bundles of 4 counting sticks. Problem conditions were suggestive of showing 3/4 of the number of bundles (duplicator/partition-reducer [DPR] subconstruct) and 3/4 of the size of each bundle (stretcher/shrinker [SS] subconstruct). This study provides confirming instances that students use these 2 rational number operator subconstructs. The SS strategies are identified when the rational number, as an operator, is distributed over a uniting operation. With these SS strategies, rational number is conceptualized as a rate. However, the SS strategies were used less often than the DPR strategies. Detailed cognitive models of these strategies in terms of the underlying conceptual units, their structures, and their modifications, were produced, and a "mathematics of quantity" notational system was used as an analytical tool to describe and model the embedded abstractions.
Behr, M. J., Wachsmuth, I., & Post, T. Rational number leaning aids: Transfer from continuous models to discrete models. Focus on Learning Problems in Mathematics, 10(4), 64-82. The present study was conducted by the Rational Number Project during 1980-81 (Behr, Post, Silver, & Mierkiewicz, 1980). The Rational Number Project is a multi-site effort which began in 1979 with funding from NSF. One focus of the project continues to assess the impact of manipulative materials on the development of rational number concepts.
Boulet, G. (1999). Large halves, small halves: Accounting for children's ordering of fractions. Focus on Learning Problems in Mathematics, 21(3), 48-66. . . . just as the ordering of natural numbers depends on the cardinality of the set of things and not on the size of the things they count, the ordering of fractions must depend entirely on part-whole relationships involved and not on the size of the parts. . . . The teaching experiment was conducted over a period of four consecutive days with thirteen fourth-graders who had already completed the unit on fractions.
Cramer, Kathleen A., Post, Thomas R., & delMas, Robert C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the Rational Number Project curriculum. Journal for Research in Mathematics Education, 33(2), 111-144. This study contrasted the achievement of students using either commercial curricula (CC) for initial fraction learning with the achievement of students using the Rational Number Project (RNP) fraction curriculum. The RNP curriculum placed particular emphasis on the use of multiple physical models and translations within and between modes of representation—pictorial, manipulative, verbal, real-world, and symbolic. The instructional program lasted 28–30 days and involved over 1600 fourth and fifth graders in 66 classrooms that were randomly assigned to treatment groups. Students using RNP project materials had statistically higher mean scores on the posttest and retention test and on four (of six) subscales: concepts, order, transfer, and estimation. Interview data showed differences in the quality of students’ thinking as they solved order and estimation tasks involving fractions. RNP students approached such tasks conceptually by building on their constructed mental images of fractions, whereas CC students relied more often on standard, often rote, procedures when solving identical fraction tasks. These results are consistent with earlier RNP work with smaller numbers of students in several teaching experiment settings.
Davis, E. J. & Thipkong, S. (1991). Preservice elementary teachers' misconceptions in interpreting and applying decimals. School Science and Mathematics, 91(3), 93-99. A study on the impact of misconceptions on preservice elementary teachers' processes in solving problems involving multiplication and division of decimals is described. Included are the purpose of the study, a description of the subjects, instruments, and interviews, the findings, and discussion. (KR)
Developing proficiency with other numbers (2002). In Jeremy Kilpatrick, Jane Swafford, and Bradford Findell (Eds.), Adding It Up: Helping Children Learn Mathematics (Chapter 7, pp. 231-254). Washington, DC: National Academy Press. This chapter deals with the difficulty of learning rational numbers, representations of rational numbers, students' errors, and proportional reasoning. It is part of a larger report to the National Academy of Sciences that reviewed and synthesized relevant research on mathematics learning from pre-kindergarten through grade 8.
Graeber, A. O. & Tirosh, D. Insights Fourth and Fifth Graders Bring to Multiplication and Division with Decimals. Educational Studies in Mathematics, 21(6), 565-588. Described are the conceptions held by United States and Israeli students about multiplication and division that may impede their work with decimals. Included are the introduction, method, division and multiplication task results, and the implications for education and textbook development. (KR)
Glasgow, R., Ragan, G., Fields, W. M. (2000). The decimal dilemma. Teaching Children Mathematics, 7(2), 89-??.
Investigates why decimal understanding lags behind fractional understanding as demonstrated by students' performance on the TIMSS.
Hiebert, J. & Wearne, D. (1983). Students' conceptions of decimal numbers. Paper presented at the Annual Meeting of the American Educational Research Association (Montreal, Quebec, Canada, April 11-15, 1983). Decimal numbers have become an increasingly important topic of the elementary and junior high school mathematics curriculum. However, national and state education assessments indicate that students have incomplete and distorted conceptions of decimal numbers. This paper reports initial data from a two-year project designed to elicit and describe students' understanding of decimals. Students in grades 3, 5, 7, and 9 were given written tests and interviewed individually on a variety of decimal tasks. Of primary interest here are tasks that considered decimals as (1) quantities that have value; (2) extensions of whole numbers; and (3) equivalents of common fractions. Results indicate that students perceive decimals primarily as symbols upon which to perform syntactic maneuvers. Although many students have significant hidden understandings, they rarely connect these with the procedural rules they have memorized. (JN)
Irwin, Kathryn C. (2001). Using everyday knowledge of decimals to enhance understanding. Journal for Research in Mathematics Education, 32(4), 399-420. The study investigated the role of students’ everyday knowledge of decimals in supporting the development of their knowledge of decimals. Sixteen students, ages 11 and 12, from a lower economic area, were asked to work in pairs (one member of each pair a more able student and one a less able student) to solve problems that tapped common misconceptions about decimal fractions. Half the pairs worked on problems presented in familiar contexts and half worked on problems presented without context. A comparison of pretest and posttest results revealed that students who worked on contextual problems made significantly more progress in their knowledge of decimals than did those who worked on noncontextual problems. Dialogues between pairs of students during problem solving were analyzed with respect to the arguments used. Results from this analysis suggested that greater reciprocity existed in the pairs working on the contextualized problems, partly because, for those problems, the less able students more commonly took advantage of their everyday knowledge of decimals. It is postulated that the students who solved contextualized problems were able to build scientific understanding of decimals by reflecting on their everyday knowledge as it pertained to the meaning of decimal numbers and the results of decimal calculations.
Litwiller, B. (Ed.) (2002). Making Sense of Fractions, Ratios and Proportions, 2002 NCTM Yearbook. NCTM: Reston, VA. Excellent resource for mathematics educators! For the first time ever, an NCTM yearbook with a supplementary booklet that provides activities for teachers to use in their classrooms. . . . Emphasizes that fractions, ratios, and proportions are key concepts in the middle school, but their development and understandings begin in the elementary school. Provides insights into student's thinking and shows the importance of proportional reasoning as a foundation for many applications of mathematics.
Lo, Jane-Jane & Watanabe, Tad (1997). Developing ratio and proportion schemes: A story of a fifth grader. Journal for Research in Mathematics Education,, 28(2), 216-236. There is a growing theoretical consensus that the concepts of ratio and proportion do not develop in isolation. Rather, they are part of the individual's multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. The current study attempted to clarify the beginning of this developmental process. One fifth-grade student, Bruce, was encouraged to schematize his trial-and-error-based method, which was effective in solving so-called missing-value tasks. This study describes several advancements Bruce made during the teaching experiment and analyzes the challenges Bruce faced in attempting to schematize his method. Finally, the mathematical knowledge Bruce needed to further develop his ratio and proportion concepts is identified. The findings provide additional support for the view that the development of ratio and proportion concepts is embedded within the development of the multiplicative conceptual field.
Mack, Nancy K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422-441.
Mack, N. K. (2000). Long-term effects of building on informal knowledge in a complex content domain: The case of multiplication of fractions. The Journal of Mathematical Behavior, 19(3), 307-332. Four students participated in a 2-year study (fifth and sixth grades) that examined the development of their understanding of multiplication of fractions. During both years, students received individualized instruction that encouraged them to build on their informal knowledge of partitioning to understand and solve problems involving the multiplication of fractions. Students also received classroom instruction on algorithmic procedures for multiplication of fractions during the second year. In the long term, students consistently drew on their informal knowledge of partitioning to reconceptualize and partition units to solve problems involving multiplications of fractions in meaningful ways. At times, students' thinking was also dominate by their knowledge of algorithmic procedures of multiplication of fractions.
Mack, Nancy K. (2001). Building on informal knowledge through instruction in a complex content domain: partition, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267-295. Six fifth-grade students came to instruction with informal knowledge related to partitioning. This knowledge initially focused on partitioning "units of measure one" into a specific number of parts. Students were able to build on their informal knowledge to reconceptualize and partition units to solve problems involving multiplication of fractions in ways that were meaningful to them. Students built their knowledge by developing mental processes related to focusing on fractional amounts and to partitioning units in different ways. Students also frequently returned to their initial focus on the number of parts and to ideas embedded in equal-sharing situations.
Moloney, K. & Stacey, K. (1997). Changes with age in students' conceptions of decimal notation. Mathematics Education Research Journal, 9(1), 25-38. Examines Australian students' conceptions of ordering decimals. Fifty secondary students studied over 12 months showed little change in their misconceptions. Whole number misconceptions are important in earlier years but disappear with time. The fraction misconception persists however, being displayed by approximately 20% of year 10 students. The zero-rule misconception was uncommon. Includes a diagnostic test useful to teachers. (AIM)
Moloney, K. & Stacey, K. (1996). Understanding decimals. Australian Mathematics Teacher, 52(1), 4-8. Demonstrates ways students think about decimals and how this changes as students get older. Presents a test teachers can use to help diagnose the mistakes students make. (MKR)
Moss, Joan & Case, Robbie (1999). Developing children's understanding of the rational numbers: A new model for an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122-147. A new curriculum to introduce rational numbers was devised, using developmental theory as a guide. The 1st topic in the curriculum was percent in a linear-measurement context, in which halving as a computational strategy was emphasized. Two-place decimals were introduced next, followed by 3- and 1-place decimals. Fractional notation was introduced last, as an alternative form for representing decimals. Sixteen 4th-grade students received the experimental curriculum. Thirteen carefully matched control students received a traditional curriculum. After instruction, students in the treatment group showed a deeper understanding of rational numbers than those in the control group, showed less reliance on whole number strategies when solving novel problems, and made more frequent reference to proportional concepts in justifying their answers. No differences were found in conventional computation between the 2 groups.
Parker, M. & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421-481.
If you teach preservice teachers or have children in middle school, you will want to read this article. In a 1994 study, preservice elementary teachers were able to calculate 23% of 55, but not what percent 16 is of 55. When asked to explain the meaning of 25%, all indicated it was 1 part out of 4, but over half did not see the relevance of 25 parts out of 100. Such difficulties with percent are not new. For seventy years, studies have documented that many students, from seventh graders through preservice elementary teachers, tend to ignore the percent sign, making no distinction between 1/2 and 1/2%. In addition, many commonly use a faulty algorithm, removing the percent sign and placing a decimal point to the left of the numeral, thereby obtaining 0.55 for 55% and 0.110 for 110%. For many years, students were asked to find one of A, B, or C, given the other two, in A% of B = C, and were taught a separate rule for each case. Then, beginning in the late 1950s, proportionality was emphasized and students were taught to convert "25 is 10% of what?" into 25/x = 10/100. Unfortunately, this often turned into the mnemonic: is over of = percent over 100, with the is being 25 and the of being the unknown.
Why is percent so hard to learn? It has a long history, variously emphasizing its additive, part-whole, multiplicative, and ratio interpretations. In third century B.C. India, interest was additive -- from "12 coins on every 100 coins" one could easily figure 24 coins on 200 or 36 on 300, a method favored by students today. In 15th century Europe, the currency unit was explicitly stated with the interest, as in 6 fl. per 100. By about 1650, this had become 6 fl. percent. With the emergence of statistics for collecting data in the 18th century, percent became abstracted for use in making standardized comparisons. From 1830 to 1860, arithmetic texts underwent massive content rearrangement, including more topics under percent with specific rules for each problem type, but with little indication how the topics were related. Today's texts treat percent in much the same way.
Percent has many interpretations. It can serve to indicate the size of a subset, such as 50% of the class. It can describe various increase/decrease comparisons; for example, the new price is 125% of the original price, the price was increased by 25%, the number of girls is 25% of the number of boys, or the number of boys is 300% more than the number of girls. In addition, the conciseness of the language of percent can confuse students. Often referents must be inferred, for example, in statements like "The unemployment rate is currently 8%," neither the number of unemployed nor the total workforce is given. While percent of evokes a multiplicative schema, percent more than evokes an additive schema, so "A is 20% more than B" may need to be interpreted as "A is 120% of B." Addition and subtraction are inverse operations, yet when a price is increased by 5%, and then decreased by 5%, one does not get back where one started. [This seeming paradox manifested itself recently when two university bookstore clerks asked us to explain it.] A&JS
Piel, J. A. & Green, M. (1994). De-mystifying division of fractions: The Convergence of quantitative and referential meaning. Focus on Learning Problems in Mathematics, 16(1), 44-50. The distinction raised here is that intuitive knowledge, on the one hand, is personalized through cognitive references corresponding to objects and experiences, and is qualitatively different from algorithmic knowledge, on the other hand, which is formally tied to symbolic quantitative procedures and often the object of classroom instruction. In this paper, we argue that intuitive knowledge and computational knowledge can be combined by focusing more explicitly on the two kinds of information implied in division of fractions problems -- referential meaning and quantitative meaning.
Putt, I. J. (1995). Preservice teachers ordering of decimal numbers: When more is smaller and less is larger! Focus on Learning Problems in Mathematics, 17(3), 1-15. . . . the literature reveals glaring weakness in both preservice and practicing teachers' knowledge of a number of mathematical topics which they teach. In this paper, knowledge about rational numbers with particular reference to decimals is examined for both undergraduates, and preservice and practicing teachers.
Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S, & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8-27. This study examines children's efforts to make sense of new mathematics instruction. The study documents major categories of errors that appear consistently as children learn decimal fractions. It then establishes the conceptual sources of these errors. Whole number errors derive from children's applying rules for interpreting multidigit integers. Fraction errors derive from children's efforts to interpret decimals as fractions. Different curriulum sequenes influence th probablilty that these classes of errors will appear. It is suggested that errors are a natural concomitant of students' attemtps to integrate new materail that they are taught with already established knowledge. Since errorful rules cannot be avoided in instruction, educators are encouraged to use them as useful diagnostics tools to detect the nature of children's understanding of a mathematical topic.
Taber, S. B. (1999). Understanding multiplication with fractions: An analysis of problem features and student strategies. Focus on Learning Problems in Mathematics, 21(2), 1-27. Comparing the strategies used on a set of 30 one-step multiplication problems by 141 fourth grade students who had not studied multiplication of fractions with those of 194 sixth grade students who had studied multiplication of fractions during two school years shows that both groups of students considered multiplication a more appropriate operation for some problems than for others expressing the same mathematical relationships. These results suggest that a different instructional focus is needed in order to address directly students' beliefs about multiplication and division and to help students extend their understanding of multiplication of whole numbers to include multiplication by fractional operators less than one.
Tirosh, Dina (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25. In this article I present and discuss an attempt to promote development of prospective elementary teachers' own subject-matter knowledge of division of fractions as well as their awareness of the nature and the likely sources of related common misconceptions held by children. My data indicate that before the mathematics methods course described here most participants knew how to divide fractions but could not explain the procedure. The prospective teachers were unaware of major sources of students' incorrect responses in this domain. One conclusion is that teacher education programs should attempt to familiarize prospective teachers with common, sometimes erroneous, cognitive processes used by students in dividing fractions and the effects of use of such processes.
Zazkis, R. & Khoury, H. A. (1993). Place value and rational number representations: Problem solving in the unfamiliar domain of non-decimals. Focus on Learning Problems in Mathematics, 15(1), 38-51. The present study investigates how pre-service elementary school teachers specifically, solve problems related to place value concepts and decimal representations of rational numbers. The problems were posed to students in the unfamiliar domain of non-integer rational numbers represented in numeration systems with bases other than ten. The study examines students' problem solving strategies and the occurrence of systematic errors.