brief history of mathematics educationperformance by American students after fourth grade:       the TIMSS and the NAEPsummaryMathematics Education since 19601	There have been three major shifts in what and how mathematics has been taught since the 1950s.  Prior to the early 1960s behaviorism was the steering mechanism for describing and understanding how students learn.  Curricula were developed with the aim in mind of drilling and practicing problems in order that a learning bond would form between an environmental stimulus and the correct learner response, or, behavior.  For example, when a student adds two numbers correctly and receives a reward, such as teacher approval or a good grade, the bond grows.  Through repetition of this stimulus-response cycle, then, students would learn and remember math.  While most teachers did not completely embrace the precepts of behaviorism, this repetition of procedure became the main form of teaching.	Oddly enough, the launching of the Soviet rocket, Sputnik, in 1957 brought critical attention to the state of mathematics and science instruction in the United States.  The general feeling of the American public was that we were behind at least the Soviets in these areas, and that the curricula were to blame.  Prominent mathematicians and scientists assembled to create a ônew mathö curriculum which made an attempt to teach the concepts underlying the procedures.  However, only those individuals who were career-bound in such fields as engineering and the sciences selected this curriculum.  Thus, it had little impact on improving mathematics achievement for most American students, and was very short-lived.  A back-to-basics movement won out, and curricula shifted back to the traditional drill and practice method of instruction until the late 1970s.	At this time, cognitive science, supported by research into how the mind learns, supplanted behaviorism as the best approach to teaching.  Research studies in mathematical problem-solving examined how an expert solved a problem, how a novice approached the same problem, and how a novice became an expert, as well as the mental processes involved in the latter.  Over the 1980s, much of cognitive research was focused on the student, as well as the material or the instruction.  The studentsÆ prior knowledge, extra-curricular experiences with math, informal mathematics knowledge, and the teacherÆs influence became major areas of study.  Therefore, a holistic view of the student emerged, including childrenÆs development.  The idea of the student as a ôblank slateö began to retreat, and changes in curricula reflected this idea.  The following table illustrates these contrasting methods:    	Important studies were undertaken during the 1990s, such as A Nation At Risk 2 by the National Commission on Excellence in Education (1983), which said, ôsecondary school curricula have been homogenized, diluted, and diffused to the point that they no longer have a central purpose,ö and Reshaping School Mathematics: A Philosophy and Framework for Curriculum 4 (1990) by the National Research Council, which called for a redesign in curricula in response to the findings of cognitive science and to the changes brought about by an increasingly technological world.  Lastly, in 1989 the National Council of Teachers of Mathematics (NCTM) released Curriculum and Evaluation Standards for School Mathematics3 (1989), which built upon the ideas generated by research in cognitive science, and called for less emphasis on rote learning and more on real-world problem solving as the basis for learning.  In this approach, the teacherÆs role is to guide the students to bring what they know into new situations and discover many possible strategies to solve a given problem.  As new curricula were introduced into schools, opposition to this approach arose.  Some teachers, mathematicians, scientists, and, especially, parents objected to the ônew new mathö as ambiguous and flawed.  This sparked the ômath warö which continues to the present.  The opposing ideas of traditionalist views VS reform views in mathematics education will be discussed in depth later in this paper.	Notes:	1.  Knuth R.A. and Jones, B.F.  ôWhat Does Research Say About Mathematics?ö.  North 	 	     Central Regional Educational Laboratory. 1991. 1-14.	     www.ncrel.org/sdrs/areas/stw_esys/2math.htm	2.  National Commission on Excellence in Education.  A Nation At Risk. 1983.	     www.ed.gov/pubs/NatAtRisk	3.  National Council of Teachers of Mathematics.  Principles and Standards for 	     	  	     School Mathematics. 2000.  preface ix-xiii and 3-24.	     electronic version at:  www.standards.nctm.org	4.  National Research Council.  Reshaping School Mathematics: A Philosophy and 	 	     Framework for Curriculum.  National Acedemy Press.  Washington, D.C. (1990)The U.S. and International Mathematics Achievement:  the TIMSS?2the TIMSS research design U.S. fourth graders exceed the international average in five of six areasU.S. eighth graders fall to a seventh grade level by international standardsU.S. twelfth graders perform better than only two countries of twenty-onesummaryconclusion 	The release of the results of the Third International Mathematics and Science Study (TIMSS) in 1996 caused another stirring of the math education pot.  More than one-half million students from 41 countries participated in this study of mathematics and science in the 1994-1995 school year. The following information is taken directly from the TIMSS website at http://nces.ed.gov/timss/timss95/more.asp#topHow did the United States perform on TIMSS?U.S. students scored above the international average in both mathematics and science at the fourth-grade level. At the eighth-grade level, U.S. students performed above the international average in science and below the international average in mathematics. In the final year of secondary school (twelfth grade in the U.S.), U.S. performance was among the lowest in both science and mathematics, including among our most advanced students. Which countries participated in TIMSS?To be a part of TIMSS, countries were required to participate at the eighth-grade level. Forty-one countries took advantage of this opportunity. Participation at the other grade levels was a country option. Twenty-six nations participated at the fourth-grade level, 21 nations at the end of secondary school level, and 16 nations participated in the advanced mathematics and physics assessments in the final year of secondary school level. The U.S. participated at all grade levels assessed in TIMSS. A listing of the participating countries at each grade level is provided .What were the components of TIMSS?TIMSS was the result of a collaborative effort on the part of the 41 nations participating in the study. This process included incorporating the experience of experts in mathematics and science education, as well as authorities in educational measurement. The fundamental goal of this group was to contribute to the enhancement of the scientific knowledge about education and about the influence of a number of important variables on educational processes. In the end, assessments in both mathematics and science were developed that provide the most comprehensive database on mathematics and science education ever produced.The creation of the TIMSS assessment items was a long and arduous process. The TIMSS assessments were the result of a major consensus building effort through which the concerns of each nation were balanced against the goal of producing a reliable and valid assessment instrument for use across the participating countries. In the end, an agreed upon set of multiple choice and extended free-response items were included in TIMSS for each grade level.Each student was asked to complete a subset of the assessment items. No student at any grade level took all of the items. The assessment was designed to last approximately 90 minutes. These assessments were then scored using a complex and stringent set of criteria that was required to be implemented equally across all nations to ensure accuracy and comparability.Questionnaires (teacher, student, school)In an effort to provide additional information on student achievement and to provide a context in which the achievement data can be viewed, teachers, students and schools were asked to complete a questionnaire.The teacher questionnaire was given to the mathematics and science teachers of the students assessed in the study. These questionnaires asked about topics such as basic demographics, attitudes and beliefs about teaching and learning, teaching assignments, class size and organization, topics covered, the use of various teaching tools, and instructional practices.Each student who took the TIMSS assessments was asked to complete a questionnaire on issues including basic demographics, daily activities, family attributes, educational resources in their homes, attitudes and beliefs about learning, instructional processes in the classroom, and study habits and homework.The principal was also asked to complete a questionnaire for the school on questions concerning community attributes, staffing, teaching assignments, policy and budget responsibilities, curriculum, enrollment, behavioral problems, instructional organization, and mathematics and science courses offered.Performance AssessmentA sub-sample of countries (11 at fourth grade, and 21 at the eighth-grade level), participated in the TIMSS Performance Assessment. This TIMSS component was designed to use practical tasks to assess student performance in mathematics and science. The Performance Assessment consists of a set of 13 tasks, 12 of which were used at the fourth grade, and 12 at the eighth grade, with 11 of these tasks being common to both grade levels. For a more detailed discussion of the Performance Assessment, please explore the  report.Videotape StudyThe TIMSS Videotape Classroom Study focused on eighth-grade mathematics instruction in three countries-Germany, Japan, and the United States-videotaping one complete lesson in each sample classroom at some point during the 1994-95 school year. This study is the first to collect videotaped records of classroom instruction, in any subject, from national probability samples. In all, 231 eighth-grade mathematics classrooms were videotaped in an effort to obtain richer contextual information about what goes on inside eighth-grade mathematics classes and to compare actual mathematics teaching methods in the United States and the two other countries. For a further discussion of the video study, please visit the  homepage.Curriculum StudyTIMSS included a curriculum analysis component in an effort to develop the TIMSS curricular frameworks and assess some of the curricular factors that influence student achievement. The TIMSS Curriculum Study gathered documents that laid out official curricular intentions and plans, collected and analyzed representative mathematics and science textbooks, and looked at several topics in-depth from some 50 nations. In both mathematics and science, an international curricular framework was developed providing a context for subsequent document analysis. These analyses provide researchers with a view of how content topics enter and exit the mathematics and science curriculum across years of schooling, and provide details on specific mathematics and science curricular intentions in the participating countries. For an in-depth discussion of the curriculum analysis, please visit the homepage.Case StudiesIn an effort to gather descriptive information on the organizational and cultural factors that influence student achievement, TIMSS also included a Case Study Project in Germany, Japan, and the United States. Through interviews with administrators, teachers, parents, and students, the Case Study Project investigated four topics of interest to the education community: education standards, dealing with differences in ability, the place of school in adolescents' lives, and the training and working conditions of teachers. For more information on the Case Study Project, please consult the  homepage.Benchmarking StudyOver the last decade, many states and districts have created content and performance standards targeted at boosting student achievement in mathematics and science. States and districts saw participation in TIMSS as an opportunity to measure the effectiveness of these new programs in an international context. Between 1995 and 1997, several states and one consortia of school districts participated in the TIMSS Benchmarking Study. These states and consortia of districts took the TIMSS assessments following the same guidelines developed for the main TIMSS assessment. For an in-depth discussion of the Benchmarking Study, please visit the  homepage. What was the sample design?In the United States, a national probability sample was drawn that resulted in over 500 schools and approximately 33,000 students participating in the study. This sample design ensures the appropriate number of schools and students are participating to provide a representative sample of fourth-, eighth-, and twelfth-grade students in the United States as a whole. How can we be sure the data is comparable?The TIMSS International Study Center at Boston College diligently worked to ensure that data collection procedures across countries were comparable. To this end, the TIMSS International Study Center instituted the following procedures for quality assurance:Coordinated by the TIMSS Sampling Referee, national school and student samples were rigorously reviewed for bias and international comparability. Utilizing two independent translations within each country, the TIMSS materials were translated into the national languages of the participating countries. Once these translations were reconciled, the TIMSS International Study Center verified these results through the use of a professional translation agency.   National Research Coordinators (NRCs) and their staff were thoroughly trained in data collection and scoring procedures at international conferences designed specifically for this purpose. The TIMSS International Study Center then continued to monitor the work of the NRCs and their staff for scoring reliability. Site visits by quality control staff were conducted during the testing period to further ensure the international data collection procedures were being followed at the national level. Finally, an extensive review of data was conducted for internal and cross-country consistency. The following content categories are general areas covered at the fourth-, eighth- and final year of secondary school levels in TIMSS.What areas were assessed?  [only mathematics is examined here]MathematicsFractions and number senseAlgebraGeometryData representation, analysis, and probabilityMeasurementProportionalityThe following are the content areas included in the advanced mathematics...assessments.Advanced MathematicsNumbers and equationsCalculusGeometryThe Repeat Study:  TIMSS-RThe Third International Mathematics and Science Study-Repeat (TIMSS-R) builds on the success of TIMSS, conducted in 1994-95. One of the motivations for United States participation in TIMSS was a desire to gauge how close the U.S. was to its goal of being first in the world in mathematics and science achievement.TIMSS-R, conducted in 1999, will provide information on U.S. progress toward this goal in the four years since TIMSS. Eighth-grade students in 38 countries are participating in TIMSS-R. When will results be released?Both the international achievement reports and the U.S. national report for the main assessments from TIMSS-R will be released in December 2000.The TIMSS-R Benchmarking Study report will be released around May 2001.The report on mathematics teaching practices from the TIMSS-R Videotape Study will be available in August 2001, with a similar report for science expected to be released in August 2002. What results will be reported?Using the reports from the original TIMSS as a guide, the TIMSS-R reports will most likely include data on topics such as overall achievement results for the U.S. and the other participating countries, these achievement results broken down by specific content areas, and results from thequestionnaire data to provide a context in which to view the achievement results. What areas are being assessed?MathematicsFractions and number senseAlgebraGeometryData representation, analysis, and probabilityMeasurementProportionality SummaryThe following summary is taken directly from the TIMSS HomePage:  						http://timss.bc.edu/TIMSS/Highlights.htmlAchievement in Mathematics 3		  Eighth Grade			   Seventh Grade		Singapore 643 			Singapore 601 		Korea 607 			Korea 577 		Japan 605 			Japan 571 		Hong Kong 588			Hong Kong 564 		Belgium (Fl) 565 			Belgium (Fl) 558 		Czech Republic 564 		Czech Republic 523 		Slovak Republic 547 		Netherlands 516 		Switzerland 545 			Bulgaria 514 		Netherlands 541 			Austria 509 		Slovenia 541 			Slovak Republic 508 		Bulgaria 540 			Belgium (Fr) 507 		Austria 539 			Switzerland 506 		France 538 			Hungary 502 		Hungary 537 			Russian Federation 501 		Russian Federation 535 		Ireland 500 		Australia 530 			Slovenia 498 		Ireland 527 			Australia 498 		Canada 527 			Thailand 495 		Belgium (Fr) 526 			Canada 494 		Thailand 522 			France 492 		Israel 522 			Germany 484 		Sweden 519 			Sweden 477 		Germany 509 			England 476 		New Zealand 508 			United States 476 		England 506 			New Zealand 472 		Norway 503 			Denmark 465 		Denmark 502 			Scotland 463 		United States 500 			Latvia (LSS) 462 		Scotland 498 			Norway 461 		Latvia (LSS) 493 			Iceland 459 		Spain 487 			Romania 454 		Iceland 487 			Spain 448 		Greece 484 			Cyprus 446 		Romania 482 			Greece 440 		Lithuania 477 			Lithuania 428 		Cyprus 474 			Portugal 423 		Portugal 454 			Iran, Islamic Rep. 401 		Iran, Islamic Rep. 428 		Colombia 369 		Kuwait 392 			South Africa 348 		Colombia 385 		South Africa 354 *Eighth and seventh grades in most countries. Latvia is annotated LSS for Latvian Speaking Schools only. Countries shown in italics did not satisfy one or more guidelines for sample participation rates, age/grade specifications, or classroom sampling procedures. SOURCE: IEA Third International Mathematics and Science Study (TIMSS), 1994-95.Key Findings 3Singapore was the top-performing country in mathematics and science at both the eighth and seventh grades. Korea, Japan, and the Czech Republic also performed very well in both subjects. Hong Kong and the Flemish-speaking part of Belgium also were among the top countries in mathematics. (See pages 2 and 3 for a full listing of the results.)For most countries, even though gender differences were minimal in mathematics [authorÆs italics], they were pervasive in science. Boys outperformed girls, particularly in physics, chemistry, and earth science.Home factors were strongly related to mathematics and science achievement in every TIMSS country (i.e., educational resources, books in the home, and parents' education).In almost every country, the majority of students agreed that they did well in mathematics and science. Interestingly, some of the countries with the highest achievement also were those whose students had the most negative perceptions of success -- Japan, Korea, and Hong Kong. Achievment in Mathematics for the Final Year of Secondary School:When the results were looked at separately for mathematics and science, the top-performers in mathematics literacy were the Netherlands, Sweden, Denmark, and Switzerland. The top-performers in science literacy were Sweden, the Netherlands, Iceland, and Norway.Countries that had higher achievement in mathematics literacy than in science literacy were Denmark, France, Hungary, Lithuania, and Switzerland. Those with higher achievement in science literacy were Canada, the Czech Republic, Iceland, Norway, the Russian Federation, Sweden, and the United States.Conclusions	One fact that stands out from this research is that U.S. fourth graders scored above the international average, but eighth graders fell below this mark.  Further, American twelfth graders scored higher than only two of the twenty one nations--only Cyprus and South Africa performed worse.  Thus, the TIMSS indicates that the mathematical abilities of Untied States students declines steadily between fourth and twelfth grades.  In her curriculum article Math Wars (1998), Linda Starr from Education World 1 reports that ôsuggested reasons for this decline have run the gamut from too much television to too little homework, from too much poverty to too few classroom resources, from a lack of teacher training to a lack of student motivation.ö  However, when it comes to assessing possible reasons for this outcome, the U.S. Department of Education writes in ôMathematics Equals Opportunityö 4 (1997): 	The U.S expects less of its middle school students compared to high 	performing nations:	TIMSS data suggest that one reason U.S. students do less well at 8th grade is that the middle school mathematics curriculum in the U.S. is significantly less challenging than curricula in other countries. In Germany and Japan, virtually all students in grades 5 through 8 move beyond arithmetic to the foundations of algebra and geometry. By 8th grade, mathematics courses in virtually all other countries participating in TIMSS include significant algebra and geometry, while in the U.S., only students in college-preparatory classes receive significant exposure to algebra, and very few students study geometry. As a result, the content taught in U.S. 8th grade mathematics classrooms is usually at a 7th-grade level compared to the 40 other nations in the TIMSS study.	TIMSS also found that U.S. mathematics classes require students to 	engage in less high-level mathematical thought and to solve fewer 	multi-step problems than classes in Germany or JapanA U.S. mathematics teacher's typical goal is to teach students the mechanics of solving a problem versus understanding the concepts behind it, while a Japanese teacher's goal is to help them learn the basics as well as understand the relevant mathematical concepts. In a typical U.S. classroom, students follow the teacher as he or she leads them through solutions to mathematics problems. In Japan, students are asked to solve problems, present them to the class, and describe how they approached the problem to increase their own understanding. How Does Our Curriculum Compare Internationally?	The 8th grade mathematics curricula in both Germany and Japan are 	more advanced than in the United StatesThe TIMSS analysis of U.S. curricula examined both the content of textbooks and how it is implemented in classrooms. An analysis of curricula in the U.S. and other countries found that  algebra and geometry  occupy more space in German and Japanese textbooks than they do in the textbooks used by a majority of U.S. 8th graders         Analyses of curriculum implementation make clear that in the middle school years, the U.S. still focuses on arithmetic.  For example, 40 percent of U.S. 8th grade mathematics lessons included arithmetic topics, whereas only 13 percent of GermanyÆs and none of JapanÆs lessons at the 8th grade level included these topics.  The major focus of curricula taught in these countries is on algebra and geometry.	Thus, these results from the TIMSS strongly indicate that U.S middle school curricula must contain more algebra and geometry, and less arithmetic in order for the U.S to help its students achieve world-class results in an international arena.  Further, teaching methods must allow for less teacher-lecture and more on student reasoning, as in the case study in Japan.	Notes:	1.  Starr, L.  ôMath Wars.ö  Education World.  1998.  1-5.	     www.education-world.com/a_curr/curr071.shtml	2.  Third International Math and Science Study. 1996.	     http://nces.ed.gov/timss/timss95/more.asp#top	3.  Third International Math and Science Study. 1996.	      (Accessed from the HomePage)	      http://timss.bc.edu/TIMSS1/Highlights.html	4.  United States Department of Education.  Mathematics Equals Opportunity: White Paper 	     prepared for U.S. Secreatary of Education Richard W. Riley.  1997. 	  	  	  	     www.ed.gov/pubs/math/part5/htmlNational Mathematics Achievement:  The NAEPA Brief Outline The Main NAEP The Long Term Trend NAEPNational Trends in Mathematics: 1973-1999Highlights from the 1996 NAEP	Achievement Levels, Area Math Scores. Calculator Use, Types of Questions Asked, 	Ten Sample Questionssummaryconclusion The NAEP:  from the National Center for Education Statistics website:   http://nces.ed.gov/nationsreportcard/math/math/aspWhat Is NAEP?The Nation's Report Card, the National Assessment of Educational Progress (NAEP), is the only nationally representative and continuing assessment of what America's students know and can do in various subject areas. Since 1969, assessments have been conducted periodically in reading, mathematics, science, writing, history, geography, the arts, and other fields. By making information on student performance -- and instructional factors related to that performance -- available to policymakers at the national, state, and local levels, NAEP is an integral part of our nation's evaluation of the condition and progress of education. Main NAEPAs the content and nature of the NAEP instruments evolve to match instructional practices, however, the ability of the assessment to measure change over time is greatly reduced. Recent NAEP main assessment instruments have typically been kept stable for relatively short periods of time, allowing trend results to be reported for, at most, three time points...Because of the flexibility of the main assessment instruments, the long-term trend NAEP must be used to reliably measure change over longer periods of time.	Long-Term Trend NAEPNAEP long-term assessments are designed to give information on the changes in the basic achievement of America's youth. They measure students' performance in math, science, reading and writing and monitor trend lines first established 30 years ago. Results are reported for students at ages 9, 13, and 17 in math, reading, and science and grades 4, 8, and 11 in writing. Measuring trends of student achievement, or change over time, requires the precise replication of past procedures. Therefore, the long-term trend instrument does not evolve based on changes in curricula or in educational practices. The long-term trend assessment uses instruments that were developed in the mid-1980s and are administered every two years in a form identical to the original one. In fact, the assessments allow NAEP to measure trends from 1969 to the present. Beginning in 1999, the long-term trend assessment will be administered on a four-year schedule and in different years from the main and state assessments in mathematics, science, reading, and writing. The NAEP long-term trend assessment in mathematics that was first administered around the beginning of the 1970s is different from more recently developed assessments in the same subject area, referred to as NAEP's main assessments. The existence of the two national assessment programs--long-term trend and main--makes it possible for NAEP to meet two important objectives: measuring educational progress over time, and developing new assessment instruments that can periodically reflect current educational content and assessment methodology. The long-term trend assessment has remained substantially the same since its first administration, and thus makes it possible to meet the first objective of measuring progress over time. Because the long-term trend assessment uses different instruments from those used in the main assessments, and because students are sampled by age for the long-term trend assessment rather than by grade as in the main assessments, it is not possible to compare results from the two assessment programs.The long-term trend mathematics assessment measures students' knowledge of basic facts, ability to carry out numerical algorithms using paper and pencil, knowledge of basic measurement formulas as they are applied in geometric settings, and the ability to apply mathematics to daily-living skills (such as those related to time and money). The computational focus of the long-term trend assessment provides a unique opportunity to determine how our students are measuring up to traditional procedural skills, even as the calculator plays an increasingly greater role in today's mathematics curriculum. Calculators are permitted for a few questions on the long-term assessment, but most questions are multiple-choice and are completed without the use of a calculator.National Trends in Mathematics17-year-olds. After declining between 1973 and 1982, average scores increased during the 1980s, and more modestly in the 1990s. The average score in 1999 was higher than that in 1973. 13-year-olds. An increase in scores between 1978 and 1982, followed by additional increases in the 1990s, resulted in an average score in 1999 that was higher than that in 1973. 9-year-olds. After a period of stable performance in the 1970s, average scores increased in the 1980s. Additional modest gains were evident in the 1990s, and the 1999 average score was higher than that in 1973. Trends in Average Scale Scores for the Nation in Mathematics	* Significantly different from 1999.  	NOTE: Dashed lines represent extrapolated data.Highlights from the 1996 NAEP include:For fourth-graders, the percentage performing at or above the "basic" level was 64 percent in 1996, as compared to only 50 percent in 1990; for eighth-graders, 62 percent as compared to 52 percent; and for 12-graders, 69 percent compared to 58 percent.Scores Up on All LevelsScores were higher in 1996 than in 1992 for all three grades, and higher in 1992 than in 1990. The national average scale score for fourth graders in 1996 was 224, an increase of 11 points over the national average for 1990. The average for eighth graders in 1996 was 272, an increase of 9 points, and the average score for twelfth graders was 304, an increase of 10 points.Increased Access to CalculatorsAt grades four and twelve, students' access to calculators to do schoolwork increased from 1992 to 1996. The percentage of fourth-grade students that had access to school-owned calculators increased from 59 percent in 1992 to 84 percent in 1996. Eighty percent of eighth-grade students had access to school-owned calculators in 1996 and 95 percent of twelfth-grade students had access to calculators, an increase of 3 percent from 1992.What Types of Questions are on the NAEP?Ten Sample Questions(Sample Questions are taken directly from the National Center for Education Statistics website)http://nces.ed.gov/nationsreportcard/math/math_about_sampq.asp(Q1)	A triangle has a 120_ angle. Indicate whether each of the following MUST be true, MAY be 	true, or CANNOT be true about the triangle.	The sum of the other two angles is 60_. 	A) Must be true. B) May be true. C) Cannot be true. D) I don't know.(Q2)	Henry is older than Bill, and Bill is older than Peter. Then 	A) Henry is older than Peter. 	B) Henry is younger than Peter.	C) Henry is the same age as Peter. 	D) There is not enough information given to tell which is true. 	E) I don't know.(Q3)	What is the measure of angle X? 	A) 34_		B) 56_ 		C) 68_ 		D) 124_ 	E) 146_ 	F) Not enough information given 	G) I don't know.(Q4)	A straight line segment could NOT be drawn on the surface of which of the 	following solids?  A)        B)         C)          D)        E)     (Q5)	The nine chips shown above are placed in a sack and then mixed up. Madeline 	draws one chip from this sack. What is the probability that Madeline draws a chip 	with an even number? A) 1/9		B) 2/9		C) 4/9		D) 1/2		E) 4/5(Q6)	What is the greatest number of squares with 9-inch sides that can be cut from a 	rectangular piece of cloth 18 inches by 36 inches?	Answer: ____________________(Q7)	Semicircles are constructed on the sides of an equilateral triangle, as shown in the 	figure above. Of the following, which best approximates the sum of the lengths of 	the three darkened arcs? A) 4.4258	 B) 4.7124	 C) 6.0000 	D) 6.7124	 E) 9.4258 	Did you use the calculator on this question?	(Q8)	Which of the following figures best illustrates the statement  [for A-E]		5 _ (6 + 2) = (5 _ 6) + (5 _ 2)?	A)               	B)                C)	D)		E)(Q9)	If this pattern of dot-figures is continued, how many dots will be in the 100th figure? 	A) 100 	B) 101 	C) 199 	D) 200 	E) 201(Q10)	The total distances covered by two runners during the first 28 minutes of a race are 	shown in the graph above. How long after the start of the race did one runner pass 	the other?	A) 3 minutes 	B) 8 minutes 	C) 12 minutes 	D) 14 minutes 	E) 28 minutesSummary American students have made significant gains in mathematics achievement since the 1970s with younger age groups of nine- and thirteen-year-olds making the most progress.  However, progress seems to decelerate after the age of nine.  The 13 points that 9-year-olds gain between 1973 and 1999 in the long term trend NAEP becomes 10 points for 13-year-olds, and then 4 points for 17-year-olds.  When reporting on state testing in the Brown Center Report on American Education: 2000 2, the authors writeFrom public data available on websites maintained by the fifty states and the District of Columbia, we found thirty-six states that posted reading and math scores in 1998 and 1999, allowing us to compare achievement in these two years.                   Conclusion	The first thing one must remember when comparing results from different NAEP tests is that the long term trend assesses more arithmetic and procedural mathematical abilities, while the state and main NAEPs assess more reasoning and multi-step problem-solving abilities.  The long term trend NAEP is also the most reliable to measure mathematics achievement over time.  Thus, when looking for a way to scale student learning of mathematics across time, one must look at the long term trend NAEP and not the main or state NAEPs. There appears to be a ôslumpö after nine years of age in which students are not learning at the previous pace.  This is not to say learning has stopped or fallen off, but that students are learning less mathematics in larger amounts of time.  This suggests that a new approach is needed to facilitate greater learning of mathematics for the middle and high school grades.  	Notes:	1.  National Center for Education Statistics.  The NationÆs Report Card.  2000.	     http://nces.ed.gov/nationsreportcard/math/math/asp	2.  The Brown Center.  The Brown Center Report on American Education:  2000--How 	   	      Well Are American Students Learning? 2000.	     http://brookings.org/gs/brown/bc_report/2000/achieve2.htmSummary of Part 1	Philosophy of mathematics education has taken a meandering journey from its view of students as empty vessels waiting to be filled with knowledge imparted by the teacher and textbooks to the realization that children are active beings, both physically and mentally, and that learning comes about through the interaction of the student with the new material.  Prior experiences and learning styles play an important role in the successful transfer of knowledge.  Thus, not only curricula, teaching methods, and assessment must change to reflect this student-centered philosophy, but also expectations.  No longer is it good practice to promote a learning environment in which the teacher lectures and the students parrot the material with little or no understanding of the mathematics involved.  Rigorous standards must be in place to describe explicit expectations of student and teacher roles so that mathematics achievement continues to grow.  An excellent source for such standards is found in Principles and Standards for School Mathematics 2 (2000) by the National Council of Teachers of Mathematics.  	The Third International Mathematics and Science Study 3 (1996) provides a myriad of insights into the state of Mathematics education in the United States.  The results of the TIMSS-R released in December, 2000, may provide additional information about the impact of the gradual implementation of reform efforts in many areas of the U.S. in the four years since the TIMSS.  The TIMSS highlights a middle grade slump that seems to worsen as time progresses.  U.S. fourth graders scored above the international average in five of the six areas, but fell below this average by eighth grade (the only nation to do so) and performed toward the bottom at the last year of secondary school.  These findings emphasize the imperative need for schools to improve on the current levels of mathematical learning in middle and high school.  Other factors, such as level of parental education and involvement in their childrenÆs schooling, contribute to the TIMSS findings, and the U.S. must do better in these areas also to facilitate the success of our students.  However, the case studies, as in Japan, Germany, and the United States, offer greater insights into why the U.S. may not be achieving in parallel with these high-performing nations.  Math texts for the 1995-86 school year in the U.S. contained only 10 percent of algebraic content as compared to 40 percent and 25 percent in Japanese and German textbooks, respectively 4.  Conversely, as reported in Mathematics Equals Opportunity 4, ô40 percent of U.S. 8th grade mathematics lessons included arithmetic topics, whereas only 13 percent of GermanyÆs and none of JapanÆs lessons at the 8th grade level included these topics.  The major focus of curricula taught in these countries is on algebra and geometry.ö  It seems clear that middle school curricula must bring more algebra and geometry to its content and less on repeated algorithmic practice and arithmetic in order to compete in an international arena.  Further, there must be teaching methods similar to those in Japan and Germany to go had-in-hand with this effort which will promote understanding of this content and not simply procedural knowledge.	The NAEP 1 provides a measure of national mathematics achievement on many levels.  The long term trend NAEP shows a steady growth of mathematical achievement over a period from 1973 to 1999, but which seems to slow after the 9-year-old level.  Bearing in mind that the long term trend assessment instruments have been kept virtually unchanged since the beginning and that this test measures basic knowledge, arithmetic, and algorithmic procedures, it seems that this is a positive outcome.  Reporting averages, the National Center for Education Statistics 1saysScores were higher in 1996 than in 1992 for all three grades, and higher in 1992 than in 1990. The national average scale score for fourth graders in 1996 was 224, an increase of 11 points over the national average for 1990. The average for eighth graders in 1996 was 272, an increase of 9 points, and the average score for twelfth graders was 304, an increase of 10 points.While noting that progress is being made according to these measures, the attention stills falls on the middle grades as the crucial time period in which to reach as many students as possible and facilitate learning greater content in order that higher achievement will follow in the high school years.	Notes:	1.  National Center for Education Statistics.  The NationÆs Report Card.  2000.	     http://nces.ed.gov/nationsreportcard/math/math/asp	2.  National Council of Teachers of Mathematics.  Principles and Standards for 	     	  	     School Mathematics. 2000.  preface ix-xiii and 3-24.	     electronic version at:  www.standards.nctm.org	3.  Third International Math and Science Study. 1996.	      (Accessed from the HomePage)	      http://timss.bc.edu/TIMSS1/Highlights.html	4.  United States Department of Education.  Mathematics Equals Opportunity: White Paper 	     prepared for U.S. Secretary of Education Richard W. Riley.  1997. 	  	  	  	     www.ed.gov/pubs/math/part5/html       a back to basics approachreform mathematics:  continued growth summaryA Back to Basics ApproachScans of brains doing exact sums (left) and approximations Precision is linked to word association functionshttp://news.bbc.co.uk/hi/english/education/newsid 337000/337328.stm