Eighth-grade Mathematics Research Articles

Watch What You Say

Although it has been many years, I still remember the story of Ricky as if it were only yesterday. I was a teacher consultant in a district where one of my roles was to support teachers in their classroom practice. A colleague had invited me to join one of her eighth-grade mathematics classes. The students were studying quadrilaterals, and our objective for the lesson was to discover what the students knew about quadrilateral properties. The lesson was informed by the van Hiele model of geometric thinking (Fuys, Geddes, and Tischler 1988). The van Hieles hypothesized that students' thinking begins at a visual level. At this level, the learner recognizes shapes by an overall gestalt based on visual clues rather than defining properties. From this level, the model suggests, learners move to a level of analysis that uses defining properties to categorize shapes and classes of shapes.

Teaching Math and Science

Local Systemic Change projects, funded by the National Science Foundation, were designed to help teachers of mathematics and science deepen their content knowledge and improve their instructional practices. Ms. Weiss and Ms. Pasley describe how these aims were accomplished through combination of intensive professional development and follow-up support when the teachers took their new knowledge and skills back to the classroom. ********** IN THE FALL of 2001, University of Delaware Professor Jon Manon observed mathematics lesson about factoring in an eighth-grade classroom of an urban middle school, a school which is home to what is often reputed to be one of the most challenging student populations in Delaware. Jenni Green (a pseudonym), the teacher he was observing, had been participant in an intensive summer institute, and Manon wanted to find out if any of the summer's workshop lessons had had an effect on her work. decided that I would look for 'residue' from the summer professional development across grades and units, writes Manon in case study published on LSC-net, an interactive website for the project leaders of the Local Systemic Change (LSC) initiative, funded by the National Science Foundation (NSF) since 1995. In particular, I was seeking evidence that our emphasis on the instructional model, the launch-investigate-summarize lesson cycle, was finding root in actual practice. (1) A LESSON IN FACTORING Jenni Green used to teach sixth grade. This year, was an eighth-grade mathematics teacher. According to Manon, she did have special concerns about particular students, e.g., one 14-year-old had baby and often stayed home to care for it. But when these kids were in school, Jenni was prepared to help them focus on learning mathematics. Manon's visit to her classroom came at what Jenni described as a very critical moment in the Mathematics in Context curriculum algebra unit, Building Formulas. (2) Manon continues: This was the lesson in which factoring was first introduced. We agreed that it was seminal concept, gatekeeper to success in more formal algebra course. As the name of the unit suggests, building situations were used to introduce algebraic representations. Section B, Basic Patterns, exploited the context of construction with bricks to introduce representation for rows of bricks involving Standing, S, and Lying, L, bricks. For example, the length of [a simple] pattern, made up of two Standing [bricks] and one Lying brick, might be represented symbolically as 2S + L. A such as this one might be repeated to produce longer string of bricks. And that new configuration might be represented as either three groups of two Standing and one Lying--3(2S + L)--or, dropping the reference to the pattern, as 6S + 3L. Before the students arrived, Jenni had described the approach planned to take to introduce her lesson. She decided to challenge her students to identify the basic pattern in longer string and then determine how many times that was repeated. As the first students entered the room, Jenni apparently made up her mind to try something different. Seated at the overhead projector, Jenni wrote 15S + 10L and asked, How can I figure out what the is in this case? Immediately, Ellen, diminutive girl in the front row, answered, Five ... it fits into [15] ... three times. For 10, it will be 2. Initially caught off guard by both the promptness and content of Ellen's response, Jenni recovered and asked, So the is repeated how many times? She drew 15 Standing and 10 Lying bricks on the overhead and, indicating in the drawing, said, I'll try to clarify Ellen's thinking ... for every three Standing, there are two Lying bricks ... pretty cool. I like that Ellen! Then, looking quite pleased and shifting her gaze from to the rest of the class, said, Ellen stole my thunder! …

Reconsidering the Study of Mathematics Instructional Practices: The Importance of Curricular Context in Understanding Local and Global Teacher Change

This paper discusses the case of one teacher, Jackie, whose instructional practices illuminate the importance of textbooks and student/parent expectations in shaping pedagogy. Jackie teaches in the Plainview district, which offers parents and students a choice between a reform-oriented, integrated curriculum (Core Plus) and a more conventional algebra sequence (the University of Chicago series). Each day, Jackie teaches two very different sections of accelerated eighth-grade mathematics using each of these curricular materials. Drawing from students’ survey responses, classroom observations, and teacher interview data, we show ways in which Jackie’s pedagogy differs considerably between the two courses and we shed light on reasons underlying this variation.

The Thinking of Students: What's the Area?

The following problem appeared in the December 2004/January 2005 issue of this journal: Square ABCD has the centers of four circles of radius 5 cm as its vertices. Find the area of the shaded region. Ten teachers submitted the work of their students, illustrating their methods for solving this problem. Kathy Hawes, an eighth-grade mathematics teacher at Graham Middle School in Mountain View, California, wrote that she found this problem “particularly interesting, because it featured geometry and students often have little chance to work on geometry challenges in today's climate of algebra for all.”

Motivations and meanings of students' actions in six classrooms from Germany, Hong Kong and the United States

This article presents an analysis of about 100 interviews with students from eight-grade classrooms in Berlin, Hong Kong and San Diego that reconstructs student motivations and the meanings they attribute to classroom activities. The data of the six classrooms were produced in the Learner's Perspective Study (LPS). The LPS is an international collaboration of researchers investigating practices in eighthgrade mathematics classrooms in 13 countries. Although not the central focus of the research, the case study of six classrooms revealed a variety of students' beliefs and perceptions, which are the focus of this article. These correspond to the possibilities the classroom practices offer. The study also reveals some similarities among student motives and concerns across classrooms. The findings are an important reminder that basing a curriculum upon an alternative vision calls for changing mathematical content as well as the social relations that are established through teaching methods and principles of evaluation.

The Thinking of Students: Have You Lost Your Marbles? Three Creative Problem-Solving Approaches

As an eighth-grade mathematics teacher, I have come to value the process of communicating about mathematics. I have learned that it is vital for students to explain and describe their work, not just simply submit their solutions. I use several methods to encourage my students to share their problem-solving processes, such as asking them to show their work on the board for the class, write essays, and answer a simple question like “How did you arrive at your answer?” Of these tasks, writing takes the most time and effort, but the results can be extraordinary. The formal essay requires students to organize their thoughts, prepare a logical sequence, and explicitly communicate the details of their work. This “thinking about thinking” translates to an improved problem-solving ability and greater conceptual understanding.

Effects of an Online Instructional Application on Reading And Mathematics Standardized Test Scores

Standardized tests have become commonly used tools for accountability in public education in the United States. In Florida, the Florida Comprehensive Assessment Test (FCAT) is used to measure student achievement on grade-specific standards and benchmarks. Various agencies have developed computer-based and web-based software applications to improve student performance on these tests. The purpose of this study was to examine the impact of one such application, FCAT Explorer, on student FCAT scores. We used hierarchical analysis of variance and analysis of covariance to compare scores for schools that used FCAT Explorer, and schools that did not. We examined fourth, fifth, eighth, and tenth grade FCAT reading and mathematics scores for selected elementary schools and high schools. Student scores from elementary schools using FCAT Explorer were significantly higher than scores from elementary schools that did not use FCAT Explorer. At the high school level, we found no significant differences in scores between schools that used FCAT Explorer and schools that did not use the application.

Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons

Although psychological theories assign to mistakes a fundamental role in the learning process and suggest ways to deal with them, only few studies have analyzed ways teacher–student interactions surrounding mistakes occur in actual classroom settings. This study focuses on such analysis. Also examined here are cultural factors that inform teacher mistake-handling. The data include a sample of 30 Italian and 30 US videotaped eighth-grade mathematics lessons. The methodology combines a discourse analytic approach with quantification of some aspects of the interactions observed. The differences observed in Italian and US lessons are interpreted as a complex interplay of two elements: beliefs about learning and culturally specific instructional practices.

Study of a Dual-Language Test Booklet in Eighth-Grade Mathematics

The purpose of this study was to address the effectiveness of a Spanish-English dual-language test booklet in 8th-grade mathematics. This study used analyses of test data (n = 402) as well as information gathered through focus groups (n = 68) and cognitive interviews (n = 18) to assess the accuracy and utility of the dual-language format as a testing accommodation. The data indicate that the translation procedures were effective and that the 2 test booklet formats were comparable once English-language proficiency and language used to answer the test were accounted for. Students found the dual-language format to be very useful, and no undue advantage appeared to be accorded to students given the dual-language booklet over the English-only booklet. In our discussion, we address translation issues and the challenges in accurately identifying students who should receive language accommodations.

Commentary on Ballou's paper: Galois - The Myths and the Man

Ballou (2005) provides a possibility for the introduction of the cubic into the eighth grade mathematics curriculum. As seen in her paper, the solution of the cubic essentially involves the use of a clever substitution in the general cubic in order to reduce it into a cubic without the square term, which in turn is factorable as a quadratic provided one makes another substitution. A natural question to ask ourselves, which mathematicians in the colorful theory of equations undoubtedly did as well, is how far can we push this technique of clever substitutions to solve higher degree equations. These substitutions are called Tschirnhaus transformations and have a pattern of the form x = y -a/n. In Ballou’s (2005) paper the transformation used in the cubic was x = y a/3 which allowed us to get rid of the so-called pesky x term. For the general quartic the transformation is x = y –a/4 which in turn transforms the equation into a cubic solveable by Cardano’s method. The question is what happens when we try our technique of Tschirnhaus transformations into the general quintic. To find out we first need to make a sidetrack into a little history packed with drama.

Block Scheduling and Mathematics: Enhancing Standards-Based Instruction?

This study investigates differences in eighth-grade mathematics students' engagement in standards-based curriculum and instruction practices between blockand traditional-schedule schools. Survey data were gathered from 156 middle level mathematics teachers to access the use of standards-based curriculum and instruction practices in their classrooms. Results indicate there are few differences in curriculum and instruction based on the type of school schedule.

Math and Science Achievement

T he results of the Trends in International Mathematics and Science Study (TIMSS) tests for 2003 were released on 14 December 2004. They contain some important messages about what's happening globally in math and science education—messages that are taken increasingly seriously by the nearly 50 participating nations and, we hope, by the readers of Science . The tests, also given in 1995 and 1999, measure mathematics and science achievement at grades 4 and 8 (reflecting an average of 4 or 8 years of schooling, respectively). Because they are closely linked to school mathematics and science curricula in the participating countries, the tests yield data on student achievement that relate to the concepts learned in their own schools. Comparative national data for mathematics and science in both grades reveal a near-monopoly by Asia in the top-scoring group, including Singapore, Korea, Hong Kong, Taiwan, and Japan. Several European nations cluster below that, and the United States and several other nations are in the next set. There is a considerable spread of scores among nations, with the average scaled scores from eighth-grade mathematics ranging from 605 (Singapore) to 264 (South Africa). Some of the “best performer” nations were those who ranked high on the United Nations Development Programme's Human Development Index (HDI), but students in Hungary, Malaysia, and South Korea, for example, did much better than their country's HDI. Between 1995 and 2003, scores for both fourth- and eighth-graders in both disciplines increased or held constant in most nations in the TIMSS samples, with improvement being especially noteworthy in fourth-grade mathematics scores. For international science, that's good news. ![Figure][1] CREDIT: PHOTOS.COM In the United States, mathematics and science education has been a core item in the national debate over education reform at the K-12 level, and active national and local efforts have focused on curriculum improvement. So how did U.S. students do on TIMSS? At grade 4, between 1995 and 2003, U.S. student scores held constant, although their international ranking declined slightly. But the average scores of U.S. eighth-graders made statistically significant improvements between 1995 and 2003 in both mathematics and science. In other good news, as described by the U.S. Subpopulation Performance analysis, eighth-grade African-American and Hispanic students demonstrated improvement in both mathematics and sciences over that 8-year span, and their average science scores narrowed the achievement gap that previously separated their performance from that of whites. These gains should provide some comfort to U.S. scientists who worry about the health of elementary education. Can we find a cause? Some will surely give credit to the No Child Left Behind (NCLB) legislation, but that can be set aside as political optimism. NCLB was passed only in 2001, too late for much influence on the assessment results. On the other hand, national standards, including the National Council of Teachers of Mathematics (NCTM) Standards (1989), the American Association for the Advancement of Science (AAAS) Benchmarks for Science Literacy (1993), and the National Research Council (NRC) National Science Education Standards (1995), have been around for just about the right length of time. There has been a significant cumulative effect of these national standards on the educational system as they have influenced state standards, textbook selection, and assessments and, less directly, teacher education. So we think that a strong argument can be made for linking these standards to the improvement we see in the TIMSS results. Education reform takes time, but the results suggest that some progress is beginning to show. TIMSS is always a sobering test for U.S. educators, because the results force a comparison of their system—which insists on the right of 50 states and 15,000 school districts to shape their own mathematics and science programs—with centrally organized systems such as that of Singapore, the world leader. The results represented by TIMSS 2003 would suggest giving accelerated emphasis to strong voluntary national standards. That is a challenging prescription for a nation devoted to local control, and one that is made more difficult by the increasing efforts in a number of school districts to mandate courses that offer religious “alternatives” to scientific theories. Perhaps the 2003 TIMSS results will, in addition to providing a baseline for comparing international educational systems, persuade U.S. policy-makers to count the costs of leaving everything to local decision-making. [1]: pending:yes

Math and Science Achievement

T he results of the Trends in International Mathematics and Science Study (TIMSS) tests for 2003 were released on 14 December 2004. They contain some important messages about what's happening globally in math and science education—messages that are taken increasingly seriously by the nearly 50 participating nations and, we hope, by the readers of Science . The tests, also given in 1995 and 1999, measure mathematics and science achievement at grades 4 and 8 (reflecting an average of 4 or 8 years of schooling, respectively). Because they are closely linked to school mathematics and science curricula in the participating countries, the tests yield data on student achievement that relate to the concepts learned in their own schools. Comparative national data for mathematics and science in both grades reveal a near-monopoly by Asia in the top-scoring group, including Singapore, Korea, Hong Kong, Taiwan, and Japan. Several European nations cluster below that, and the United States and several other nations are in the next set. There is a considerable spread of scores among nations, with the average scaled scores from eighth-grade mathematics ranging from 605 (Singapore) to 264 (South Africa). Some of the “best performer” nations were those who ranked high on the United Nations Development Programme's Human Development Index (HDI), but students in Hungary, Malaysia, and South Korea, for example, did much better than their country's HDI. Between 1995 and 2003, scores for both fourth- and eighth-graders in both disciplines increased or held constant in most nations in the TIMSS samples, with improvement being especially noteworthy in fourth-grade mathematics scores. For international science, that's good news. ![Figure][1] CREDIT: PHOTOS.COM In the United States, mathematics and science education has been a core item in the national debate over education reform at the K-12 level, and active national and local efforts have focused on curriculum improvement. So how did U.S. students do on TIMSS? At grade 4, between 1995 and 2003, U.S. student scores held constant, although their international ranking declined slightly. But the average scores of U.S. eighth-graders made statistically significant improvements between 1995 and 2003 in both mathematics and science. In other good news, as described by the U.S. Subpopulation Performance analysis, eighth-grade African-American and Hispanic students demonstrated improvement in both mathematics and sciences over that 8-year span, and their average science scores narrowed the achievement gap that previously separated their performance from that of whites. These gains should provide some comfort to U.S. scientists who worry about the health of elementary education. Can we find a cause? Some will surely give credit to the No Child Left Behind (NCLB) legislation, but that can be set aside as political optimism. NCLB was passed only in 2001, too late for much influence on the assessment results. On the other hand, national standards, including the National Council of Teachers of Mathematics (NCTM) Standards (1989), the American Association for the Advancement of Science (AAAS) Benchmarks for Science Literacy (1993), and the National Research Council (NRC) National Science Education Standards (1995), have been around for just about the right length of time. There has been a significant cumulative effect of these national standards on the educational system as they have influenced state standards, textbook selection, and assessments and, less directly, teacher education. So we think that a strong argument can be made for linking these standards to the improvement we see in the TIMSS results. Education reform takes time, but the results suggest that some progress is beginning to show. TIMSS is always a sobering test for U.S. educators, because the results force a comparison of their system—which insists on the right of 50 states and 15,000 school districts to shape their own mathematics and science programs—with centrally organized systems such as that of Singapore, the world leader. The results represented by TIMSS 2003 would suggest giving accelerated emphasis to strong voluntary national standards. That is a challenging prescription for a nation devoted to local control, and one that is made more difficult by the increasing efforts in a number of school districts to mandate courses that offer religious “alternatives” to scientific theories. Perhaps the 2003 TIMSS results will, in addition to providing a baseline for comparing international educational systems, persuade U.S. policy-makers to count the costs of leaving everything to local decision-making. [1]: pending:yes

Mathematics Teaching in the United States Today (and Tomorrow): Results From the TIMSS 1999 Video Study

The Third International Mathematics and Science Study (TIMSS) 1999 Video Study examined eighth-grade mathematics teaching in the United States and six higher-achieving countries. A range of teaching systems were found across higher-achieving countries that balanced attention to challenging content, procedural skill, and conceptual understanding in different ways. The United States displayed a unique system of teaching, not because of any particular feature but because of a constellation of features that reinforced attention to lower-level mathematics skills. The authors argue that these results are relevant for policy (mathematics) debates in the United States because they provide a current account of what actually is happening inside U.S. classrooms and because they demonstrate that current debates often pose overly simple choices. The authors suggest ways to learn from examining teaching systems that are not alien to U.S. teachers but that balance a skill emphasis with attention to challenging mathematics and conceptual development.

Assessing Barriers to the Reform of U.S. Mathematics Instruction From an International Perspective

The authors assessed five commonly perceived barriers to increased use of conceptual teaching in mathematics in the United States related to teacher autonomy, trade-offs with computational strategies, student achievement, class size, and teacher qualifications. These barriers were examined through the use of data from nationally representative samples of eighth-grade mathematics classrooms across 38 nations that took part in the Third International Mathematics and Science Study in 1999 and follow-up analyses involving data from the high-achieving nations of Japan and Singapore. Findings suggest that most of these perceived barriers are not impediments to the use of conceptual teaching strategies in other countries, and the comparative findings hold promise for alternative paradigms for organizing better mathematics instruction in the United States.

“Are you joking or are you sleeping?” Cultural beliefs and practices in Italian and U.S. teachers’ mistake-handling strategies

This study investigates the role of cultural beliefs and practices in teaching. The focus is the process of socialization by which students learn to deal with mistakes. Analyzed are teacher–student interactions surrounding mistakes videotaped in 30 Italian and 30 U.S. eighth-grade mathematics lessons. These are complemented by interviews with teachers and students and videotapes of elementary-school classes. The study combines discourse analytic approaches with quantification of some aspects of the interactions observed. Italian and U.S. students have different experiences with mistakes that largely depend on how teachers organize instructional activities and frame classroom participant roles. Mistakes are discussed publicly in Italy twice as often as in the United States. U.S. teachers mitigate their responses; Italian teachers aggravate them. Mistake-handling activities may be understood as culturally specific interplays of beliefs and practices. The complexity of these interplays is suggested here as a significant obstacle to effective teaching innovation.

Analogy Use in Eighth-Grade Mathematics Classrooms

Analogical reasoning has long been believed to play a central role in mathematics learning and problem solving (see Genter, Holyoak, & Kokinov, 2001); however, little is known about how analogy is used in everyday instructional contexts. This article examines analogies produced in naturally occurring U.S. mathematics lessons to explore patterns between analogy generation and instructional context. One hundred and three analogies were identified in a random sample of 25 eighth-grade mathematics classrooms videotaped as part of the Third International Mathematics and Science Study (see Stigler et al., 1999). Qualitative codes were used to gather quantitative data about the set of analogies, and emergent patterns considered. Specifically, the study examines patterns of teacher-student participation, analogy source and target construction, and contexts for analogy production. These data suggest that teachers regularly use analogy as instructional mechanisms to teach concepts and procedures, differentially generating sources to match the content goal of the analogy. The source and target construction is also related to whether the analogy is in response to students' needs for help. Teachers typically maintain control of each analogy by producing the majority of the comparison, though students are regularly involved in the components of the analogy that require minimal analogical thinking.

Teachers around the World

A GREAT deal has been made of student achievement data in international comparisons. Less information has been gathered about in various countries. We have a few field studies, such as those reported in The Teaching Gap, by James Stigler and James Hiebert, and in Knowing and Teaching Elementary Mathematics, by Liping Ma. Neither depicts a particularly robust portrait of what American know about mathematics or how to teach it. With the new emphasis on highly qualified teachers from the U.S. Department of Education, teacher quality has moved to center stage. Two new studies add a bit to our knowledge of here and abroad. The first is Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study, in which videotapes of actually teaching eighth-grade mathematics in seven countries were studied. The report (NCES 2003013) was put together by Hiebert and a host of other researchers and is available at www.nces.ed.gov/timss. The second is Preparing Teachers Around the World, from Aubrey Wang, Ashaki Coleman, Richard Coley, and Richard Phelps of the ETS Policy Information Center. It is available at www.ets.org/research/pic/teachprep.pdf. In 1995, as part of the original TIMSS (Third International Mathematics and Science Study), videos of were made in three nations: Germany, Japan, and the United States. In general, U.S. and German focused on teaching procedures, while Japanese emphasized conceptual matters. One accidental finding cropped up when people coding the videos had a What the hell was that? reaction to noises on the tapes. Those noises were mostly interruptions by the intercom, visitors, etc. More than 30% of American classes suffered interruptions. Zero percent of Japanese classes suffered an intrusion. The TIMSS 1999 Video Study extends the original research to seven countries, adding Australia, the Czech Republic, Hong Kong, the Netherlands, and Switzerland and omitting Germany. (A more detailed summary of this study, by the TIMSS Video Mathematics Research Group, can be found in the June 2003 Kappan.) These countries all scored high or relatively high in TIMSS and TIMSS-R (Repeat). In TIMSS-R, only the Czech Republic did not have a score that was significantly higher than that of the U.S., which scored 502 as compared to the Czech Republic's 520. (The international average was 487.) Getting coders to agree on what they were seeing was difficult in the original study, and it was exponentially more difficult to get coders to agree about what they were seeing when viewing videos from seven countries. In addition, they might have missed something important, as overall judgments of quality did not correlate with judgments on the specific indicators of quality. (These results are from a presentation made at the annual meeting of the American Educational Research Association in April, not from the report itself.) Coders first looked at how much time in different countries spent introducing new content, practicing new content, or reviewing earlier content. American spent more time reviewing old material than anyone else except the Czechs. None of the countries spent a great deal of time practicing new material, but Japanese did allot 60% of their time to introducing new content. Only Japan's lessons were considered to be of high or moderate complexity, 39% and 45% respectively. The U.S. had the fewest lessons considered to be of high complexity (6%) and the third-highest number judged to be of low complexity (67%), behind Australia (77%) and the Netherlands (69%). Only Japanese offered many proofs. Twenty- six percent of the problems in any given lesson contained proofs, and 39% of the lessons contained at least one proof. Second-place Switzerland posted just 3% and 11% respectively. The researchers asked whether the lessons consisted of stating concepts, using procedures, or making connections between mathematical facts. …

Proportionality: A Unifying Theme for the Middle Grades

The mathematics studied in high school and beyond is organized around large themes, such as algebra, geometry, trigonometry, calculus, statistics, and so on. Even though the boundaries between these topics blur at times, the themes give connections and structure to the mathematics studied. For example, a huge body of knowledge is recognized as algebra. Students know that they are doing algebra when they formulate and solve equations, even when they perform these tasks in geometry or calculus class. In contrast, the study of mathematics in the middle grades, with the exception of prealgebra, lacks an overarching theme. Most middle-grades' courses, generally known merely as sixth-, seventh-, or eighth-grade mathematics, lack big ideas and tend to be a mass of discrete topics with little connection or continuity. The notion that prealgebra is a course to be done right before algebra, as opposed to a long-term theme to be developed over several years, is a prime example of this situation.

Assessing the Basic Math Skills of First-Year Doctor of Pharmacy Students

ABSTRACTThe objective of this study was to assess the basic math skills of two classes of doctor of pharmacy students at a private southeastern school of pharmacy. The Basic Math Skills Test (BMST) was used to assess 121 first-year doctor of pharmacy students from the classes of 2003 and 2004. All entering students present during orientation week were tested using the BMST. The BMST consists of a 50-question, timed math test that measures basic math skills in 9 different areas. The areas and level of difficulty were taken from an eighth-grade mathematics textbook. In accordance with the recommendation of the developer, students were allotted 20 minutes to complete the exam. Results revealed that the cumulative mean score on the BMST of this sample was 68.90%. Implications of the study: (1) poor basic math skills impede successful completion of the pharmacy curriculum and (2) poor basic math skills may increase the probabilities of mathematical mistakes when filling prescriptions in pharmacy practice. Two ...