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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We also propose initiatives that would build on the 3/1 framework in order to achieve a truly rigorous integration of content and methods instruction. states.nctq. 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.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. higher education institutions. that is. These are the schools that now have the basic “3/1” framework already in place for adequate preparation. Exit with Expertise: Do Ed Schools Prepare Elementary Teachers to Pass This Test? (an excerpt is on page 5 and the full test is available on our website: www. States States must set thresholds for acceptable scores for admission to education schools on standardized achievement tests. college placement tests. Our recommendations here are addressed to professionals responsible for elementary teacher preparation: professional organizations. program completion. and textbook publishers. Page 16 .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. We offer a sample test. course exemption. three mathematics courses that teach the elementary mathematics content that a teacher needs to know and one well-aligned mathematics methods course. and licensure. 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. They should be clearly differentiated from those assessments one might find in an elementary or middle school classroom. and high school exit tests. The guiding principle in setting these scores should be to ensure that every teacher candidate possesses a competent grasp of high school geometry and second-year high school algebra. education schools.

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

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

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

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. must be the new “common denominator” of our preparation programs for elementary teachers within education schools. Washington D. Selecting the best of current textbooks. National Academic Press. n n A deeper understanding of elementary mathematics. 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 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.C. Unfortunately. The reforms that will make classroom teachers more mathematically competent could improve mathematics specialists as well.Executive Summary June 2008 CONCLUSION American elementary teachers as a group are caring people who want to do what is best for children. Nonetheless.1 Until such time as an improved instructional model is developed that combines mathematics content and mathematics methods instruction. the standards against which these education schools were judged only lay a solid foundation. But we are only at the beginning of the process of seeing how that new measure might be calculated.. Further improvement is still necessary. 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. their mathematics preparation leaves far too many of them ill-equipped to do so. Page 20 . 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. with more attention given to the foundations of algebra.

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. The number 3 is the least odd number that satisfies this condition: 32 = 9. with the remainder less than k (and greater than or equal to 0).2…). b. it can be represented as 2w +1. Neither is cheaper since both approaches yield the same total purchase price.) 1.1. Thus the remainder when dividing by 8 is 1. Helpful reminder for (b) and (c): In division with a remainder. The expression w 2+w = w (w +1). and when this is divided by 16 the remainder is 9. 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. a. Many odd numbers when their square is divided by 16 leave a remainder that is not 1.nctq. n 2 = (2w+1) 2 = 4w 2+4w +1 = 8[(w 2+w )/2] + 1. 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 let p represent any purchase price: a.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. and either w or w+1 is even. If n is an odd number. where w represents a whole number (0. c. 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. so (w 2+w )/2 is a whole number. To determine this. the result is a whole number and a remainder. Page 21 . the remainder when dividing by 4 is 1.

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

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

Executive Summary June 2008 Page 24 .


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