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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

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

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

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

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

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

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

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

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

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

The mathematics textbooks in the sample varied enormously in quality. with the majority of textbooks earning scores low enough to label them unacceptable for use in algebra instruction. two-thirds of the courses use no textbook or a textbook that is inadequate in one or more of four critical areas of mathematics. a fact that will handicap the preparation of elementary teachers in this vital area.Executive Summary June 2008 Finding 3: Most education schools use mathematics textbooks that are inadequate. Unfortunately. algebra is shortchanged. geometry and measurement. In fact. no textbook has the strongest possible stand-alone algebra section. the algebra portions of the textbooks are the weakest. 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. Again. Predictably. algebra. with no textbook providing the strongest possible support. and data analysis and probability). Page 10 .

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

or if they do report a mathematics subscore. Under these circumstances it may be possible to answer nearly every mathematics question incorrectly and still pass the test. 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. 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.Executive Summary June 2008 Finding 5: Almost anyone can get out. Massachusetts plans to unveil in winter 2009 a stand-alone test of the mathematics an elementary teacher needs to know. 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. In addition. even if these tests require a demonstration of mathematical understanding of slightly more depth than entrance tests. Exit Tests on Mathematics Knowledge No. In almost all cases. 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 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. it is not a factor in deciding who passes. Page 12 . 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. 1 2 California’s licensing test (CSET) appears to be the most rigorous of these tests.

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

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

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

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

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

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

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

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

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

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

the solution of whose angles resolves the measurement of ACB.Executive Summary June 2008 5.) m DCB = 40º (This is an alternate interior angle to the acute angle with vertex B on line b. or perpendicular to line c through point C. An auxiliary line can also be drawn through points B and C. 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. Different approaches are possible. a 30º A D C 40º c b B Angle ACB measures 70º. its intersection with line a creates a triangle. Auxiliary lines could also be drawn perpendicular to line a through point A. creating two triangles. 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. the solution of whose angles resolves the measurement of ACB. Page 23 . creating a quadrilateral whose angles include ACB and can be solved.

Executive Summary June 2008 Page 24 .

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