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No Common Denominator

The Preparation of Elementary Teachers in Mathematics by America’s Education Schools

Executive Summary June 2008

National council on teacher quality

The full report of No Common Denominator: The Preparation of Elementary Teachers in Mathematics by America’s Education Schools is available online from www.nctq.org. authors: Julie Greenberg and Kate Walsh our thanks to: Research analysts: Emmanuel Caudillo, Aileen Corso, Elizabeth McCorry, Stephanie Parry, Felicity Messner Ross, Michael Savoy, Nate Sheely, and Kevin Walsh Database design and technical support: Jeff Hale Graphic design: Colleen Hale Mathematics Advisory Group: Richard Askey, Andrew Chen, Mikhail Goldenberg, Roger Howe, Jason Kamras, James Milgram, Robin Ramos, and Yoram Sagher Consultation: Francis (Skip) Fennell and Mark Thames with principal funding from: The Brookhill Foundation, The Louis Calder Foundation, Ewing and Marion Kauffman Foundation, Exxon Mobil Foundation, and Searle Freedom Trust nctq board of directors: Clara M. Lovett, Chair, Stacey Boyd, Chester E. Finn, Jr., Ira Fishman, Marti Garlett, Jason Kamras, Donald N. Langenberg, Carol G. Peck, Andrew J. Rotherham, Kirk Schroder, Danielle Wilcox, and Kate Walsh, President nctq advisory board: Steven J. Adamowski, Roy E. Barnes, Alan D. Bersin, Lawrence S. Braden, Cynthia G. Brown, Cheryl Ellis, Michael Feinberg, Ronald F. Ferguson, Eleanor Gaines, Michael Goldstein, Eric A. Hanushek, Frederick M. Hess, Paul T. Hill, E.D. Hirsch, Frank Keating, Paul Kimmelman, Martin J. Koldyke, Wendy Kopp, Hailly Korman, Amy Jo Leonard, Deborah McGriff, Ellen Moir, Robert H. Pasternack, Michael Podgursky, Michelle Rhee, Stefanie Sanford, Laura Schwedes, Thomas Toch, and Daniel Willingham

Unfortunately. mathematics educators. consisting of mathematicians and distinguished teachers with a long history of involvement in K-12 education. in particular Singapore.” we refer to them as education schools because the phrase is commonly understood. The link from there to the capability of elementary teachers to provide effective instruction in mathematics is immediate. as well as the best state standards. many American elementary teachers are weak in mathematics and are too often described. whose students lead the world in mathematics performance.Executive Summary June 2008 Executive summary In this second study of education schools. in particular the National Council on Teachers of Mathematics (NCTM).1 the National Council on Teacher Quality (NCTQ) examines the mathematics preparation of America’s elementary teachers.pdf Page 1 . Though improving American students’ relative performance depends on a variety of factors. and economists. The recommendations of professional associations. we devoted two years of study to develop a set of five standards that would be the mark of a high quality program of teacher training. by a variety of measures. 2 In May 2006 we issued What Education Schools Aren’t Teaching about Reading and What Elementary Teachers Aren’t Learning http://www. Numerous mathematicians.nctq. cognitive psychologists. Education ministries of other nations with higher performance in mathematics than our own. n n n 1 While teacher preparation programs do not always reside in “education schools. a particularly critical consideration must be the foundations laid in elementary school because mathematics relies so heavily on cumulative knowledge. To ensure that these standards were well-founded and comprehensive. and other key national studies. we consulted: n Our own Mathematics Advisory Group.2 The impetus for this study is the mediocre performance of American students in mathematics compared to their counterparts around the world. as “math phobic.” Absent a conclusive body of research on how best to prepare elementary teacher candidates. both by themselves and those who prepare them.org/p/publications/docs/nctq_reading_study_app_20071202065019. social scientists.

3. focusing on four critical areas: 1. with emphasis placed on the delivery of mathematics content. no current assessment is up to this task. 2. aspiring elementary teachers should demonstrate that their knowledge of mathematics is at the high school level (geometry and coursework equivalent to second-year algebra). Page 2 . numbers and operations. college placement tests. Appropriate tests include standardized achievement tests. Careful attention must be paid to the selection of instructors with adequate professional qualifications in mathematics who appreciate the tremendous responsibility inherent in training the next generation of teachers and who understand the need to connect the mathematics topics to elementary classroom instruction. moving well beyond mere procedural understanding. Unfortunately. This course should provide numerous opportunities for students to practice-teach before elementary students. and — to a lesser degree — 4. Standard 3: As conditions for completing their teacher preparation and earning a license.Executive Summary June 2008 Five Standards for the Mathematics Preparation of Elementary Teachers Standard 1: Aspiring elementary teachers must begin to acquire a deep conceptual knowledge of the mathematics that they will one day need to teach. Standard 5: The job of teaching aspiring elementary teachers mathematics content should be within the purview of mathematics departments. Standard 4: Elementary content courses should be taught in close coordination with an elementary mathematics methods course that emphasizes numbers and operations. Required mathematics coursework should be tailored to the unique needs of the elementary teacher both in design and delivery. algebra. Standard 2: Education schools should insist upon higher entry standards for admittance into their programs. data analysis and probability. As a condition for admission. geometry and measurement. elementary teacher candidates should demonstrate a deeper understanding of mathematics content than is expected of children. and sufficiently rigorous high school exit tests.

Executive Summary June 2008 This study evaluates the elementary education programs at a sample of 77 education schools located in every state except Alaska. in reality. we always gave the school the benefit of the doubt. We analyzed their mathematics programs. In all. such as final exams and study guides. The syllabus represents a professor’s goal for what he or she wishes to accomplish in a course. we understand that a course’s intended goals and topics as reflected in syllabi and texts may differ from what actually happens in the classroom. Also. DEPTH: Is enough time available to devote sufficient attention to the essential topics? Page 3 . We recognize that less than what the syllabi and certainly the texts contain. that professors develop their syllabi and choose texts not for some empty purpose. Our sample represents elementary education schools at higher education institutions of all types and constitutes more than 5 percent of the institutions that offer undergraduate elementary teacher certification. however. considering every course that they require of their elementary teacher candidates. but for quite an important one: to serve as an outline for the intended progression of a course and to articulate instructional objectives. twice invited the selected schools to submit additional materials. however. Our analysis provides a reasonable assessment and the most comprehensive picture to date of how education schools are preparing — or failing to prepare — elementary teachers in mathematics. when we encountered any sort of ambiguity. is apt to be covered in class. we looked at 257 course syllabi and required textbooks as the source of information. The schools did not volunteer to participate in this study. In selecting this methodology. How were schools rated? We considered three factors: 1. Given the extremely low threshold that we set for schools to earn a good rating. but were notified early on that they had been selected. we expected many more schools to pass than ultimately did. BREADTH: Does the coursework cover essential mathematics topics? 3. We assert. We acknowledge the inherent limitations of this methodology and for this reason. there are the inevitable interruptions and distractions that almost always leave that goal to some degree unmet. as opposed to coursework requirements intended for any student on the campus? 2. RELEVANCE: Does the education school require coursework that is relevant to the job of the elementary teacher. in order to enhance our understanding. not more.

it is indeed university mathematicians who lead the charge against these general-audience mathematics courses. it did not really matter what those courses were. we presumed. we created a tear-out test containing the kinds of mathematics problems that should be taught to teacher candidates and which they should be able to solve.nctq. as opposed to offering a remedial program. but which is by no means remedial.org. While perhaps counterintuitive. The logic behind this approach is that if a teacher candidate can pass a college-level. every expert we consulted told us we were wrong. Materials we obtained from schools did not allow us to do a comprehensive evaluation of whether they delivered a college-level program in elementary mathematics content. At the outset of this study. With remarkable consensus. Nevertheless. Page 4 . To better illustrate what the learning objectives would be for such courses. mathematicians and mathematics educators believe that the “anything goes” practice of educating aspiring elementary teachers is both inefficient and ineffective. general-audience mathematics course. The full test is available at www. Though the full report contains an extensive discussion of all three criteria. that while elementary teachers should be required to take some mathematics at the college level. we could not evaluate schools on the basis of a fourth factor: rigor. A few sample problems follow. as we know many people do. then he or she should not have much difficulty wrestling with mathematics as an instructor in an elementary classroom. arguing instead that elementary teacher candidates need a rigorous program of study that returns them to the topics they encountered in elementary and middle school grades. Any instructional strategies that a teacher needs to know could be taught in a mathematics methods course.Executive Summary June 2008 Unfortunately. some attention here is needed to explain the first: relevance.

The measurement of the acute angle with its vertex at point B created by CB is 40º. John’s shop sells bicycles and tricycles. 3. Connect points A and C. Figure the tax on the full price and get the discount on that amount. A store has a sale with a d % discount and must add a t % sales tax on any item purchased. 2. How many bicycles are available for sale in John’s shop that day? Solve arithmetically and algebraically. 2: A trapezoid with vertices EFJG Polygon No. Which would be cheaper for any purchase: a. Let n be an odd number. Polygon No. Prove that when n 2 is divided by 4. Find an odd n such that n 2 divided by 16 leaves a remainder that is not 1. Find the measurement of ACB. c. A B C D E F Let b represent the base of the rectangle and h represent its height. the measurement of the acute angle created by CA with its vertex at point A is 30º. a. 4. d. 3: A triangle with vertices ALH How do the areas of the three polygons compare? Justify your answer. The complete test is available at www. a A C b B Page 5 . Prove that n 2 is odd. Get the discount first and pay the tax on the reduced amount. Lines a and b are parallel.Executive Summary June 2008 Sample Problems Exit with Expertise: Do Ed Schools Prepare Elementary Teachers to Pass This Test? (Answer Key can be found on page 21.nctq. G H I J K L A different polygon is drawn within each of three rectangles with vertices AFLG. b. the remainder is 1.) 1. Justify your answer.org. 1: A parallelogram with vertices DFIG Polygon No. Prove that when n 2 is divided by 8. the remainder is 1. One day there are a total of 176 wheels and 152 pedals in the shop. and points B and C with line segments. 5. b.

and we suspect reflects the variation found across all American education schools. There is one unfortunate area of agreement: a widespread inattention to algebra. Page 6 . is unacceptable. all preparing individuals to do the same job. breadth. the education schools in our sample are remarkable for having achieved little consensus about what teachers need.” Within this variation. With the exception of the University of Georgia. the listings are in alphabetical order within the group rankings. The table on page 7 lists the institutions by rankings. few education schools stand out for the quality of their mathematics preparation. The content of this coursework ranges from “Integrated Mathematics Concepts” (described as a survey course in contemporary mathematics that presents mathematics as a human endeavor in a historical context) to “Calculus. which we single out as an exemplary program. elementary teacher candidates are required to take anywhere from zero to six mathematics courses in their undergraduate careers. and depth. These schools met all three of our criteria: relevance. The variation in requirements across the sample 77 education schools. Only ten schools in our sample (13 percent) rose to the top in our evaluation of the overall quality of preparation in mathematics. Depending upon the institution. In fact.Executive Summary June 2008 FINDINGS Finding 1: Few education schools cover the mathematics content that elementary teachers need.

PA* University of Alabama at Birmingham* University of Arizona University of Memphis. Bloomington Lourdes College. TN MacMurray College. ND Viterbo University. MO Concordia University. WA West Virginia University at Parkersburg * Programs requiring no elementary content coursework at all. NC* Gustavus Adolphus College. College Park University of Michigan University of Montana† University of New Mexico† Western Oregon University† † Although these schools pass for providing the right content. CA* University of Rhode Island* University of Richmond. Reno University of Portland. HI Columbia College. VA* University of Texas at Dallas Utah State University Valley City State University. VA Saint Joseph’s College of Maine Saint Mary’s College. KY University of Mississippi University of Nevada. MO Seattle Pacific University. VT** Greensboro College. OK The College of New Jersey Towson University. PA Chaminade University of Honolulu. KS Norfolk University. DE Education Schools that Fail on All Measures Albion College. Stanislaus* Colorado College* Florida International University Green Mountain College. MI American University. OH† University of Louisiana at Monroe University of Maryland. TN University of Nebraska at Omaha University of New Hampshire. MD Western Connecticut State University Wilmington University. Education Schools that Would Pass if They Required More Coursework Arizona State University Boston University Calumet College of St. Durham University of Redlands. WA Southern Adventist University. IN Cedar Crest College. OR Georgia College and State University King’s College. IN Southern New Hampshire University State University of New York (SUNY) College at Oneonta University of Central Arkansas University of Louisville. Page 7 . San Marcos* California State University. OR University of South Carolina University of South Dakota University of Texas at El Paso University of Wyoming West Texas A&M University Education Schools that Would Pass with Better Focus and Textbooks Benedictine University. IL Metropolitan State College of Denver. IL Northeastern State University. they still fall short on mathematics methods coursework. PA Lewis-Clark State College. DC California State University. John’s University. MN* Hampton University. VA* Iowa State University Lee University. MA† Indiana University. ID Minnesota State University Moorhead Radford University. ** New coursework requirements are not publicly available. Joseph. VA Park University. WI Walla Walla College. They do not require a full course dedicated solely to elementary mathematics methods. CO Newman University. TN* St. NY* Saint Joseph’s University.Executive Summary June 2008 ARE EDUCATION SCHOOLS PREPARING ELEMENTARY TEACHERS TO TEACH MATHEMATICS? Education Schools with the Right Stuff An exemplary teacher preparation program University of Georgia Boston College.

They should learn that a large variety of word problems can be solved with either arithmetic or algebra and should understand the relationship between the two approaches. algebra instruction is most anemic: over half of all schools (52 percent) devote less than 15 percent of class time to algebra. they need to understand algebra as the generalization of the arithmetic they address while studying numbers and operations. 3) geometry and measurement. and models that will occupy their elementary students. and 4) data analysis and probability. 2) algebra. The table below shows how much programs deviate from the recommended time allocation. as well as algebra’s connection to many of the patterns. While elementary teachers do not deal explicitly with algebra in their instruction. We considered the time spent on the four critical areas of mathematics that an elementary teacher needs to understand: 1) numbers and operations. relationships. rules.) Numbers and operations Algebra Geometry and measurement Data analysis and probability 40 30 35 10 13 24 14 1 Page 8 . Deficiencies in Mathematics Instruction for Teachers Critical areas Recommended distribution (hours) Average hours shortchanged (Estimated for the sample. with another third effectively ignoring it entirely. devoting less than 5 percent of class time to that area. By a number of measures. including the recommendation of our Mathematics Advisory Group. roughly 25 percent of the preparation in mathematics that elementary teachers receive. Of the four areas. algebra should comprise a large part of an entire elementary content course.Executive Summary June 2008 Improving the Heft and Focus of Mathematics Preparation for Elementary Teachers A fundamental problem observed in most of the programs is that there is a large deficit in the amount of time devoted to elementary mathematic topics. properties.

Since all aspects of public K-12 education in the United States are regulated by the states. Even without national oversight states could be more consistent in their requirements regarding coursework. whether at private or public colleges. Ohio. Arkansas. California. regulation of the preparation of K-12 teachers. and Oregon pertaining only to foundations of mathematics and geometry: Alaska. Wisconsin. Massachusetts. New Mexico. Nebraska. Connecticut. is also within the purview of states. Missouri. states do not appear to know what is needed. and geometry: Source: NCTQ’s State Teacher Policy Yearbook 2007. New Jersey. Tennessee. South Carolina. Illinois. and Wyoming 1 3 state has requirements pertaining only to geometry: Minnesota states have requirements Colorado. District of Columbia. Arizona. Louisiana. Iowa. or no requirements pertaining Maine. Michigan. Vermont. Kansas. Delaware. Oklahoma. Utah. Florida. to specific areas of math: Virginia. standards. South Dakota. Hawaii. algebra. Kentucky. Mississippi. Washington. www. and/or preparation for assessments in specific areas of mathematics. Montana. North Carolina. Maryland.org/stpy Page 9 . Pennsylvania.Executive Summary June 2008 Finding 2: States contribute to the chaos. Rhode Island. New York.nctq. Indiana. Texas. Nevada. Idaho. Georgia. North Dakota. While most state education agencies issue guidelines for the mathematics preparation of elementary teachers. and West Virginia 29 states have requirements pertaining to foundations of mathematics. States’ Guidance is Confusing 18 states have no requirements Alabama. New Hampshire.

two-thirds of the courses use no textbook or a textbook that is inadequate in one or more of four critical areas of mathematics. Most Courses Use Inadequate Textbooks use texts rated inadequate in three critical areas 20% Courses Using Adequate Texts Use texts that adequately cover all four critical areas 34% Courses Using Inadequate Texts Use texts rated inadequate in two critical areas 30% Use texts rated inadequate in one critical area 10% do not use a text 6% Only one-third of the elementary content courses in our sample use a textbook that was rated as adequate in four critical areas of mathematics (numbers and operations. and data analysis and probability). a fact that will handicap the preparation of elementary teachers in this vital area. algebra is shortchanged. the algebra portions of the textbooks are the weakest. The mathematics textbooks in the sample varied enormously in quality. algebra. Unfortunately. Page 10 . with the majority of textbooks earning scores low enough to label them unacceptable for use in algebra instruction.Executive Summary June 2008 Finding 3: Most education schools use mathematics textbooks that are inadequate. Again. Predictably. In fact. no textbook has the strongest possible stand-alone algebra section. geometry and measurement. with no textbook providing the strongest possible support.

2 The total number of schools noted in the table is more than 77 because some schools have multiple options for 3 entrance tests. Page 11 . the Praxis I. The majority of the 77 schools require applicants to take a form of basic skills test for admission. including the most popular choice.Executive Summary June 2008 Finding 4: Almost anyone can get in. as they address only those mathematics topics taught in elementary and middle school grades. None of these tests. measures the proficiency one should expect from a high school graduate. 1 We classify algebra as a middle school course because it is such in most developed countries. American education schools set exceedingly low expectations for the mathematics knowledge that aspiring teachers must demonstrate. and mathematics. typically a three-part assessment of skills in reading. Colorado College requires applicants to score at least 600 on the SAT math. of schools Do they have tests? 11 No test at all 14 Test requirements or test expectations not clear 54 Basic skills test 3 1 Test for high school proficiency Only one schoool in our sample of 77 clearly has adequate entry requirements.1 Entrance Tests on Mathematics Knowledge2 No. Compared to the admissions standards found in other countries. writing. Sixteen percent of the education schools do not require applicants to pass any sort of mathematics test to get into their programs.

Massachusetts plans to unveil in winter 2009 a stand-alone test of the mathematics an elementary teacher needs to know. even if these tests require a demonstration of mathematical understanding of slightly more depth than entrance tests. of schools Do they have tests? 17 No test at all or test requirement not clear Only assess elementary and middle school proficiency and do not use a stand-alone test 601 0 2 Stand-alone test for what an elementary teacher needs to know Not a single state requires an adequate exit test to ensure that the teacher candidate knows the mathematics he or she will need. Exit Tests on Mathematics Knowledge No. The fact that education schools are relying on tests that allow prospective teachers to pass without demonstrating proficiency in all subject areas with “stand-alone” tests makes it impossible for either the institution or the state in which they are going to teach to know how much mathematics elementary teachers know at the conclusion of their teacher preparation program. but the mathematics portion is not stand-alone. There are two major failings of these tests: they either do not report a subscore for the mathematics portion of the test. 1 2 California’s licensing test (CSET) appears to be the most rigorous of these tests. In almost all cases. In addition.Executive Summary June 2008 Finding 5: Almost anyone can get out. The standards used to determine successful completion of education schools’ elementary teacher preparation programs are essentially no different than the low standards used to enter those programs. it is insufficient to establish whether elementary teacher candidates are truly prepared for the challenges of teaching mathematics. Most education schools told us that they require an exit test in mathematics. Page 12 . Under these circumstances it may be possible to answer nearly every mathematics question incorrectly and still pass the test. or if they do report a mathematics subscore. these exit tests are the same tests that teachers need to take for state licensure (Praxis II or a test specific to a state). it is not a factor in deciding who passes.

Executive Summary June 2008 The Other Dimension of Mathematics Preparation: Mathematics Methods Coursework Our study focused primarily on the content preparation of elementary teachers in mathematics. University of Georgia 5. University of Texas at El Paso In the methods syllabi found in these six programs we saw instructor expectations for practice teaching such as this: The student has demonstrated an appreciation of what it means to teach mathematics for conceptual understanding. but courses in which aspiring teachers learn the methods of mathematics instruction are essential in their overall preparation for the classroom. University of Michigan 2. in particular whether it was generally adequate and how instructors designed practice teaching experiences to ensure that teacher candidates focused on conveying mathematics to their child audiences. yet a large share of the education schools we studied (42 percent) do not have even one methods course dedicated to elementary mathematics methods and 5 percent have only a two credit course. Many mathematics educators report that it is difficult to adequately cover all elementary topics in even one methods course. Finding 6: The elementary mathematics in mathematics methods coursework is too often relegated to the sidelines. we found only six education schools that appeared to emphasize the need for aspiring teachers to consider how to communicate mathematical content and how to determine if children understood what they had been taught: Education schools which put mathematics at the center of practice teaching 1. an aspiring teacher might be asked to answer a question such as: What part of your teaching philosophy did you demonstrate in your experience? Page 13 . syllabi from other courses requiring practice teaching tended to make the mathematics instruction almost beside the point. In contrast. Therefore we also examined mathematics methods coursework. University of Louisville 6. University of Nevada. Reno 3. Looking at programs that had a course devoted solely to elementary mathematics methods and required practice teaching. In particular. For example. Greensboro College 4. any practice teaching that may occur fails to emphasize the need to capably convey mathematics content to children.

the person assigned to teach mathematics to elementary teacher candidates is not professionally equipped to do so. and not perceived as the assignment of the instructor who drew the short straw. Page 14 . No matter which department prepares teachers in mathematics. Elementary mathematics courses are neither demanding in their content nor their expectations of students.Executive Summary June 2008 Finding 7: Too often. however. make use of assessments that some education schools provided us. although the issue of just who is best qualified and motivated to impart the content of elementary mathematics to teachers remains a conundrum. both on a related topic. Commendably. We could not evaluate the rigor in mathematics content courses taught in our sample education schools using syllabi review because too few syllabi specified student assignments. It pairs three problems that would be appropriate for an elementary classroom with three problems appropriate for a college classroom. their general level of rigor is dismaying. and exams used in courses in programs in our sample. The table on page 15 demonstrates the contrast between two types of questions taken from actual quizzes. About a third of the questions in assessments we obtained from mainstream education schools were completely inappropriate for a college-level test. Finding 8: Almost anyone can do the work. tests. elementary content mathematics courses must be taught with integrity and rigor. most elementary content courses are taught within mathematics departments. We did. The fact that prospective teachers may have weaker foundations in mathematics and are perceived to be more math phobic than average should not lead to a conclusion that the mathematics presented must be watered down. With a cautionary note that these assessments may not be representative of all the schools in our sample.

The number 0. 32 c. The big dog weighs 5 times as much as the little dog. giving a teacher-style solution. 18 b. answer. 5 3a. Simplify the fraction (1/2 + 1/3) ÷ (5/12) (1 – ½) (1 – 1/3) (1 – ¼) 3b. 1a. How much does the big dog weigh? Solve the problem and explain your solution process. 1 ½ d. Do not simplify your final answer. one hundredth and three ten-thousandths 2a. 42 Mathematics questions that are closer to hitting the mark for what teachers should be able to answer – taken from actual college course assessments. The little dog weighs 2/3 as much as the medium sized dog. 2b. 1b.00561616161… as a quotient of two integers (that is. Which of the following is (2 ½) ÷ (1/2)? a. how many students are in the course? a. If 24 of them are passing. thirteen thousandths b. 1 ¼ b.0013 is equal to the following: a. 36 d. Show step-by-step arithmetic leading to your final. in fractional-rational form). Solve the problem and explain your solution process. Write the number 1. Page 15 . The medium dog weighs 9 pounds more than the little dog.Executive Summary June 2008 contrasting Problems: The mathematics that teachers need to know – and children do not Mathematics questions children should be able to answer – taken from actual college course assessments. zero point one three d. Exactly three-fourths of the students in a certain class are passing. 2 ¼ c. thirteen ten-thousandths c.

They should be clearly differentiated from those assessments one might find in an elementary or middle school classroom. that is.org) as a jumping-off point for the development of a new generation of tests that will drive more rigorous instruction and ensure that teachers entering the elementary classroom are well prepared mathematically. Page 16 . and licensure. These are the schools that now have the basic “3/1” framework already in place for adequate preparation. We also propose initiatives that would build on the 3/1 framework in order to achieve a truly rigorous integration of content and methods instruction. States States must set thresholds for acceptable scores for admission to education schools on standardized achievement tests. course exemption. college placement tests. We offer a sample test. Our recommendations here are addressed to professionals responsible for elementary teacher preparation: professional organizations. and textbook publishers. The guiding principle in setting these scores should be to ensure that every teacher candidate possesses a competent grasp of high school geometry and second-year high school algebra. education schools. higher education institutions. program completion. and high school exit tests.nctq. three mathematics courses that teach the elementary mathematics content that a teacher needs to know and one well-aligned mathematics methods course. The Association of Mathematics Teacher Educators (AMTE) The Association of Mathematics Teacher Educators (AMTE) should organize mathematicians and mathematics educators in a professional initiative and charge them with development of prototype assessments that can be used for course completion. These assessments need to evaluate whether the elementary teacher’s understanding of concepts such as place value or number theory is deep enough for the mathematical demands of the classroom. Exit with Expertise: Do Ed Schools Prepare Elementary Teachers to Pass This Test? (an excerpt is on page 5 and the full test is available on our website: www. states.Executive Summary June 2008 RECOMMENDATIONS We suspect that in several decades we will look back on the current landscape of the mathematics preparation of elementary teachers and have the benefit of hindsight to realize that some education schools were poised for significant and salutary change.

A unique stand-alone test of elementary mathematics content that a teacher needs to know is the only practical way to ensure that a state’s expectations are met.Executive Summary June 2008 While these proposed thresholds are significantly higher than current ones. Currently. not currently available from any testing company. In fact. we recommend a 3/1 framework: three mathematics courses designed for teachers addressing elementary and middle school topics and one mathematics methods course focused on elementary topics and numbers and operations in particular. in the process requiring too few courses specific to teaching any grade span. and data analysis and probability. to test for these standards. The test could also be used as a vehicle to allow teacher candidates to test out of required coursework. they still may be lower than what is required of elementary teachers in nations reporting higher levels of student achievement in mathematics. These certifications encourage education schools to attempt to broadly prepare teachers. geometry and measurement. not just relegate the task to a few courageous volunteers. With the exception of the most selective institutions. Page 17 . Teacher preparation programs should make it possible for an aspiring teacher to test out of mathematics content course requirements using a new generation of standardized tests that evaluate mathematical understanding at the requisite depth. 23 states offer some form of PreK-8 certification. States need to adopt wholly new assessments. they are reasonable. algebra. For most programs. States need to develop strong coursework standards in all four critical areas: numbers and operations. there is a quite plausible perception that an education school cannot raise its admission standards without putting itself at a disadvantage in the competition for students. moving well beyond mere procedural understanding. Education Schools Education schools should require coursework that builds towards a deep conceptual knowledge of the mathematics that elementary teachers will one day need to convey to children. States need to eliminate their PreK-8 certifications. The pressure these institutions face to accept a sufficient number of students makes it incumbent upon states to raise the bar for all education schools.

Education schools should eliminate any of the following: mathematics programs designed for too many grades. Teacher preparation programs do a disservice to the material that future elementary teachers need to learn by trying to accomplish too many instructional goals at the same time. the practice of teaching methods for science or other subjects as companion topics in mathematics methods coursework. fractions. as well as algebra topics typically covered in an introductory algebra course. Page 18 . provided they are willing to redirect their general education requirements to more relevant coursework for the elementary teacher. only slightly below our recommendation of three elementary content mathematics courses. The five-year model for teacher preparation. although much of that coursework bears little relation to the mathematics that elementary teachers need.5 courses in mathematics.” adequate preparation of elementary students for algebra requires that their teachers have a strong mathematics background in those critical foundations. and particular aspects of geometry and measurement are the “critical foundation of algebra. Algebra must be given higher priority in elementary content instruction. whereby prospective teachers complete coursework for an undergraduate major taking the same courses as would any other major in that subject and than devote a fifth year to courses about teaching and learning. can quickly move towards meeting this standard by substituting requirements for elementary content mathematics courses. these programs as currently structured are inadvisable for the appropriate preparation of elementary teachers for teaching mathematics. Five-year programs. does not accommodate coursework designed for teachers in elementary mathematics topics. and the practice of combining content and methods instruction if only one or two combined courses are required. Institutions. As the National Mathematics Advisory Panel made clear in its 2008 report. For that reason.Executive Summary June 2008 The higher education institutions in our sample require an average of 2. such as those found in California. such as PreK-8. need to be restructured if they are going to meet the mathematics content needs of elementary teachers. while proficiency with whole numbers.

. Professionals dedicated to improvements in elementary teacher preparation should collaborate to develop a textbook that can serve as a resource both in content and methods coursework..g. and innovation that would make the mathematics preparation of elementary teachers coherent.Executive Summary June 2008 Higher Education Institutions On too many campuses. Higher education institutions housing education schools must take the lead in orchestrating the communication. and demands made that they be more systematic — neither of which is an expensive proposition — change could be dramatic. This ideal “combo-text” would augment a core of solid mathematics content with discussion of a process for continuous improvement of instruction focused on student learning. teacher preparation is regarded by university professors and administrators as a program that is beneath them and best ignored. coordination with content courses. Page 19 . coordination. Many changes cannot be made in isolation and most will not be undertaken without explicit encouragement by institutional leadership. The connection of our national security to the quality of the teachers educating new generations of Americans goes unrecognized. establishing more rigorous standards) and mathematics methods courses (e. Much of what has to be changed about the preparation of teachers connects to decisions regarding instruction in mathematics courses (e. possibly through concurrent registration. but content textbooks that are more consistently good across all topics are still needed. especially in practice teaching). Textbook Publishers Several elementary content textbooks (particularly those by Thomas Parker and Scott Baldridge. textbook selection. and Sybilla Beckmann) are excellent and we recommend their use.g. emphasizing the mathematics in mathematics methods. Were education schools to receive more university scrutiny. the priority attached to algebra.

1 The prospect that mathematics specialists will become increasingly common in elementary classrooms due to initiatives promoted by groups including the National Academies (Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future.Executive Summary June 2008 CONCLUSION American elementary teachers as a group are caring people who want to do what is best for children. Selecting the best of current textbooks. the standards against which these education schools were judged only lay a solid foundation. Further improvement is still necessary. must be the new “common denominator” of our preparation programs for elementary teachers within education schools. The reforms that will make classroom teachers more mathematically competent could improve mathematics specialists as well. Washington D. But we are only at the beginning of the process of seeing how that new measure might be calculated.C. 2007) does not change this imperative for improvement since those specialists can emerge from the same courses and programs as regular elementary classroom teachers.. We are confident that the education schools that rose to the top in our evaluation process are preparing teachers relatively well compared to the majority of education schools in this study which rated so poorly. Page 20 . with more attention given to the foundations of algebra. Unfortunately. Their teachers stand readier than most to forestall the frustrations of youngsters leaving the familiar world of the counting numbers or dealing with the debut of division with fractions.1 Until such time as an improved instructional model is developed that combines mathematics content and mathematics methods instruction. National Academic Press. their mathematics preparation leaves far too many of them ill-equipped to do so. teacher preparation programs should increase the efficacy of existing content courses: n Intensifying teacher preparation on essential topics with the same “laserlike focus” endorsed by the National Mathematics Advisory Panel for K-12 mathematics instruction. Nonetheless. Setting high standards for student performance in courses and in exit tests. n n A deeper understanding of elementary mathematics.

) 1. the result is a whole number and a remainder. Discounted price: p – p *(d /100) = p (1– d /100) Tax on discounted price: p (1– d /100) (t /100) Adding the two and simplifying: p (1– d/100) + p (1– d /100)(t /100) = p (1– d /100)(1 + t /100) b. Thus the remainder when dividing by 8 is 1. and either w or w+1 is even. with the remainder less than k (and greater than or equal to 0). where w represents a whole number (0. Helpful reminder for (b) and (c): In division with a remainder.Executive Summary June 2008 Answer Key for Sample Problems on page 5 Exit with Expertise: Do Ed Schools Prepare Elementary Teachers to Pass This Test? (The complete test is available at www. Neither is cheaper since both approaches yield the same total purchase price. The expression w 2+w = w (w +1). n 2 = (2w+1) 2 = 4w 2+4w +1 = 8[(w 2+w )/2] + 1. so (w 2+w )/2 is a whole number. b. a. Many odd numbers when their square is divided by 16 leave a remainder that is not 1. d.2…). If n is an odd number. n 2 = (2w+1) 2 = 4w 2+4w +1 = 4(w 2+w ) + 1 Since w 2+w is a whole number and 1 is less than 4.nctq. To determine this. The number 3 is the least odd number that satisfies this condition: 32 = 9. n 2 = (2w +1)2 = 4w 2+4w +1 = 2(2w 2+2w ) + 1 so n 2 is odd. c. when dividing by a number k. and when this is divided by 16 the remainder is 9. the remainder when dividing by 4 is 1. let p represent any purchase price: a. Full price with tax: p + p * (t /100) = p (1+ t /100) Discount on full price with tax: [p + p * (t /100)]*d /100 = p (1+ t /100)(d /100) Subtracting the discount from the full price and simplifying: p (1+ t /100) – p (1+ t /100)(d /100) = p (1+t /100)(1-d /100) These are the same since a *b = b *a 2. it can be represented as 2w +1.1. Page 21 .org.

so the number of wheels on bicycles is 176–72=104. For each tricycle. The number of bicycles is half the number of wheels. developed using number of wheels: 2b +3t = 176 Equation B. Equation A. developed using number of pedals: 2b +2t = 152 Subtracting equation B from A: 1t = 24 Substituting this value for t into equation B and solving for b. Solved arithmetically: Each bicycle has two wheels and each tricycle has three wheels. and both have two pedals. Solved algebraically: Let b represent the number of bicycles in the store and t the number of tricycles. All the polygons have the same area: A 1 = A 2 = A 3 Area of parallelogram: A 1 = 2/5b *h h b Area of trapezoid: A 2 = 1/2h ( 3/5b+ 1/5b) = 1/2h * 4/5b = 2/5b *h h b Area of triangle: A 3 = h 1/2 ( 4/5b) *h = 2/5b *h b Page 22 . There are 176–152=24 extra wheels. so there are 24 tricycles. there is one more wheel than pedals. 104/2=52. These have 24*3=72 wheels.Executive Summary June 2008 3. There are 52 bicycles in the shop. b = 52 4.

a 30º A D C 40º c b B Angle ACB measures 70º. often ones which have already been solved. Different approaches are possible. creating a quadrilateral whose angles include ACB and can be solved. its intersection with line a creates a triangle. but one approach is to draw an auxiliary line1 parallel to lines a and b through point C and add point D to line c : m ACD = 30º (This is an alternate interior angle to the acute angle with vertex A on line a. An auxiliary line can also be drawn through points B and C. the solution of whose angles resolves the measurement of ACB.) m ACD + m DCB = m ACB = 30º+ 40º = 70º 1 The function of auxiliary lines is to change difficult probelms to simpler ones.Executive Summary June 2008 5. creating two triangles. Auxiliary lines could also be drawn perpendicular to line a through point A.) m DCB = 40º (This is an alternate interior angle to the acute angle with vertex B on line b. Page 23 . the solution of whose angles resolves the measurement of ACB. or perpendicular to line c through point C.

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nctq. We must not have the mathematically blind leading the blind. President Emeritus of the University of Texas “This report should help counter the common belief that the only skill needed to teach second-grade arithmetic is a good grasp of third-grade arithmetic.org/p/tab/subscribe. University of Maryland To download the full report. and local teacher policies and the events that help to shape them. Langenberg Chancellor Emeritus.” — Larry R.C. Suite 720 Washington. contact: National Council on Teacher Quality 1341 G Street NW.nctq. go to www.jsp). Subscribe to NCTQ’s free monthly electronic newsletter. to stay abreast of trends in federal.” — Donald N. Our education schools urgently need to ensure that our elementary teachers do not represent in the classroom the substantial portion of our citizenry that is mathematically disabled.org The National Council on Teacher Quality advocates for reforms in a broad range of teacher policies at the federal.“ I commend this valuable report from the National Council on Teacher Quality for addressing a critical need in improving teacher capacity: more effective assessments of mathematical knowledge as part of the process by which candidates qualify for entry into elementary teacher preparatory programs. .org. Houston Endowment Inc. Teacher Quality Bulletin (www. and local levels in order to increase the number of effective teachers. Faulkner President. state. D. For additional copies of the executive summary.nctq. 20005 Tel 202 393-0020 Fax 202 393-0095 www. state.