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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 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

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

Appropriate tests include standardized achievement tests. Required mathematics coursework should be tailored to the unique needs of the elementary teacher both in design and delivery. Standard 4: Elementary content courses should be taught in close coordination with an elementary mathematics methods course that emphasizes numbers and operations.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. elementary teacher candidates should demonstrate a deeper understanding of mathematics content than is expected of children. and sufficiently rigorous high school exit tests. 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 . focusing on four critical areas: 1. algebra. 3. Standard 3: As conditions for completing their teacher preparation and earning a license. Standard 2: Education schools should insist upon higher entry standards for admittance into their programs. This course should provide numerous opportunities for students to practice-teach before elementary students. 2. Unfortunately. data analysis and probability. 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. college placement tests. moving well beyond mere procedural understanding. As a condition for admission. Standard 5: The job of teaching aspiring elementary teachers mathematics content should be within the purview of mathematics departments. and — to a lesser degree — 4. numbers and operations. with emphasis placed on the delivery of mathematics content. geometry and measurement. no current assessment is up to this task.

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

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

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

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

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

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

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

Page 10 . the algebra portions of the textbooks are the weakest. a fact that will handicap the preparation of elementary teachers in this vital area. Predictably. algebra. no textbook has the strongest possible stand-alone algebra section. In fact. Unfortunately. 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. with no textbook providing the strongest possible support. and data analysis and probability). geometry and measurement. 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. The mathematics textbooks in the sample varied enormously in quality. two-thirds of the courses use no textbook or a textbook that is inadequate in one or more of four critical areas of mathematics. Again. algebra is shortchanged.

including the most popular choice. The majority of the 77 schools require applicants to take a form of basic skills test for admission.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.1 Entrance Tests on Mathematics Knowledge2 No. the Praxis I. Colorado College requires applicants to score at least 600 on the SAT math. 2 The total number of schools noted in the table is more than 77 because some schools have multiple options for 3 entrance tests. American education schools set exceedingly low expectations for the mathematics knowledge that aspiring teachers must demonstrate. Page 11 . measures the proficiency one should expect from a high school graduate. Compared to the admissions standards found in other countries. writing. 1 We classify algebra as a middle school course because it is such in most developed countries. None of these tests. and mathematics. Sixteen percent of the education schools do not require applicants to pass any sort of mathematics test to get into their programs. 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. typically a three-part assessment of skills in reading.

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.Executive Summary June 2008 Finding 5: Almost anyone can get out. Under these circumstances it may be possible to answer nearly every mathematics question incorrectly and still pass the test. Exit Tests on Mathematics Knowledge No. or if they do report a mathematics subscore. Massachusetts plans to unveil in winter 2009 a stand-alone test of the mathematics an elementary teacher needs to know. these exit tests are the same tests that teachers need to take for state licensure (Praxis II or a test specific to a state). 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. There are two major failings of these tests: they either do not report a subscore for the mathematics portion of the test. Most education schools told us that they require an exit test in mathematics. it is not a factor in deciding who passes. even if these tests require a demonstration of mathematical understanding of slightly more depth than entrance tests. In addition. 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. 1 2 California’s licensing test (CSET) appears to be the most rigorous of these tests. it is insufficient to establish whether elementary teacher candidates are truly prepared for the challenges of teaching mathematics. In almost all cases. Page 12 . but the mathematics portion is not stand-alone.

University of Louisville 6. 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 . 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. 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.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. Looking at programs that had a course devoted solely to elementary mathematics methods and required practice teaching. Many mathematics educators report that it is difficult to adequately cover all elementary topics in even one methods course. University of Georgia 5. University of Michigan 2. Reno 3. any practice teaching that may occur fails to emphasize the need to capably convey mathematics content to children. In particular. Greensboro College 4. Finding 6: The elementary mathematics in mathematics methods coursework is too often relegated to the sidelines. 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. In contrast. For example. syllabi from other courses requiring practice teaching tended to make the mathematics instruction almost beside the point. University of Nevada. 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. Therefore we also examined mathematics methods coursework. but courses in which aspiring teachers learn the methods of mathematics instruction are essential in their overall preparation for the classroom.

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

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

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. Page 16 . They should be clearly differentiated from those assessments one might find in an elementary or middle school classroom. and licensure. 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. Our recommendations here are addressed to professionals responsible for elementary teacher preparation: professional organizations.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. that is. States States must set thresholds for acceptable scores for admission to education schools on standardized achievement tests. and high school exit tests.nctq. 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. program 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. higher education institutions. course exemption. states. These are the schools that now have the basic “3/1” framework already in place for adequate preparation. 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. 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. education schools. college placement tests. We offer a sample test. and textbook publishers. three mathematics courses that teach the elementary mathematics content that a teacher needs to know and one well-aligned mathematics methods course.

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. not just relegate the task to a few courageous volunteers. algebra. 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. geometry and measurement. States need to adopt wholly new assessments. they are reasonable. and data analysis and probability. 23 states offer some form of PreK-8 certification. Currently. to test for these standards. 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. 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. not currently available from any testing company. moving well beyond mere procedural understanding. With the exception of the most selective institutions. In fact. For most programs.Executive Summary June 2008 While these proposed thresholds are significantly higher than current ones. States need to eliminate their PreK-8 certifications. These certifications encourage education schools to attempt to broadly prepare teachers. 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. Page 17 . they still may be lower than what is required of elementary teachers in nations reporting higher levels of student achievement in mathematics. in the process requiring too few courses specific to teaching any grade span. The test could also be used as a vehicle to allow teacher candidates to test out of required coursework. 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.

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

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

2007) does not change this imperative for improvement since those specialists can emerge from the same courses and programs as regular elementary classroom teachers. must be the new “common denominator” of our preparation programs for elementary teachers within education schools. 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. The reforms that will make classroom teachers more mathematically competent could improve mathematics specialists as well. Selecting the best of current textbooks. Setting high standards for student performance in courses and in exit tests.C. Further improvement is still necessary. 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.1 Until such time as an improved instructional model is developed that combines mathematics content and mathematics methods instruction.Executive Summary June 2008 CONCLUSION American elementary teachers as a group are caring people who want to do what is best for children. n n A deeper understanding of elementary mathematics. Nonetheless. with more attention given to the foundations of algebra. Page 20 . their mathematics preparation leaves far too many of them ill-equipped to do so. National Academic Press. the standards against which these education schools were judged only lay a solid foundation. 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. Unfortunately. But we are only at the beginning of the process of seeing how that new measure might be calculated.. Washington D. 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.

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. the result is a whole number and a remainder. b. The number 3 is the least odd number that satisfies this condition: 32 = 9. and when this is divided by 16 the remainder is 9. c. 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. n 2 = (2w +1)2 = 4w 2+4w +1 = 2(2w 2+2w ) + 1 so n 2 is odd.1. 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. when dividing by a number k. the remainder when dividing by 4 is 1. so (w 2+w )/2 is a whole number. The expression w 2+w = w (w +1). and either w or w+1 is let p represent any purchase price: a. with the remainder less than k (and greater than or equal to 0). n 2 = (2w+1) 2 = 4w 2+4w +1 = 8[(w 2+w )/2] + 1. Helpful reminder for (b) and (c): In division with a remainder. 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. d.2…). it can be represented as 2w +1. Thus the remainder when dividing by 8 is 1.) 1. If n is an odd number. a. Many odd numbers when their square is divided by 16 leave a remainder that is not 1.nctq. To determine this. Neither is cheaper since both approaches yield the same total purchase price. where w represents a whole number (0. Page 21 .

104/2=52. These have 24*3=72 wheels. There are 176–152=24 extra wheels. The number of bicycles is half the number of wheels. 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 is one more wheel than pedals. Solved arithmetically: Each bicycle has two wheels and each tricycle has three wheels. and both have two pedals. b = 52 4. so the number of wheels on bicycles is 176–72=104. so there are 24 tricycles. There are 52 bicycles in the shop. Equation A. developed using number of wheels: 2b +3t = 176 Equation B.Executive Summary June 2008 3. For each tricycle. 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.

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

Executive Summary June 2008 Page 24 .


Langenberg Chancellor Emeritus.“ 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. Faulkner President. We must not have the mathematically blind leading the blind. Subscribe to NCTQ’s free monthly electronic newsletter. For additional copies of the executive summary. University of Maryland To download the full report. D. state. 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.nctq.jsp). and local teacher policies and the events that help to shape them.” — Donald and local levels in order to increase the number of effective teachers.” — Larry The National Council on Teacher Quality advocates for reforms in a broad range of teacher policies at the federal. to stay abreast of trends in federal. go to www. 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. 20005 Tel 202 393-0020 Fax 202 393-0095 www. state. Houston Endowment Inc. .nctq. contact: National Council on Teacher Quality 1341 G Street NW. Suite 720 Washington.C.nctq. Teacher Quality Bulletin (www.