Mathematical Intimidation: Driven by the Data
applied mathematics in doing so). More recently, groups of mathematicians tried to organize a boycott of the Star Wars project on the grounds that it was an abuse of mathematics. And even more recently some fretted about the role of mathematics in the financial meltdown. But the most common misuse of mathematics is simpler, more pervasive, and (alas) more insidious: mathematics employed as a rhetorical weapon--an intellectual credential to convince the public that an idea or a process is "objective" and hence better than other competing ideas or processes. This is mathematical intimidation. It is especially persuasive because so many people are awed by mathematics and yet do not understand it--a dangerous combination. The latest instance of the phenomenon is valued-added modeling (VAM), used to interpret test data. Value-added modeling pops up everywhere today, from newspapers to television to political campaigns. VAM is heavily promoted with unbridled and uncritical enthusiasm by the press, by politicians, and even by (some) educational experts, and it is touted as the modern, "scientific" way to measure educational success in everything from charter schools to individual teachers. Yet most of those promoting value-added modeling are ill-equipped to judge either its effectiveness or its limitations. Some of those who are equipped make extravagant claims without much detail, reassuring us that someone has checked into our concerns and we shouldn't worry. Value-added modeling is promoted because it has the right pedigree--because it is based on "sophisticated mathematics". As a consequence, mathematics that ought to be used to illuminate ends up being used to intimidate. When that happens, mathematicians have a responsibility to speak out. Background
Value-added models are all about tests--standardized tests that have become ubiquitous in K-12 education in the past few decades. These tests have been around for many years, but their scale, scope, and potential utility have changed dramatically. Fifty years ago, at a few key points in their education, schoolchildren would bring home a piece of paper that showed academic achievement, usually with a percentile score showing where they landed among a large group. Parents could take pride in their child's progress (or fret over its lack); teachers could sort students into those who excelled and those who needed remediation; students could make plans for higher education. Today, tests have more consequences. "No Child Left Behind" mandated that tests in reading and mathematics be administered in grades 3-8. Often more tests are given in high school, including high-stakes tests for graduation. With all that accumulating data, it was inevitable that people would want to use tests to evaluate everything educational--not merely teachers, schools, and entire states but also new curricula, teacher training programs, or teacher selection criteria. Are the new standards better than the old? Are experienced teachers better than novice? Do teachers need to know the content they teach? Using data from tests to answer such questions is part of the current "student achievement" ethos--the belief that the goal of education is to produce high test scores. But it is also part of a broader trend in modern society to place a higher value on numerical (objective) measurements than verbal (subjective) evidence. But using tests to evaluate teachers, schools, or programs has many problems. (For a readable and comprehensive account, see [Koretz 2008].) Here are four of the most important problems, taken from a much longer list. 1. Influences. Test scores are affected by many factors,
including the incoming levels of achievement, the influence of previous teachers, the attitudes of peers, and parental support. One cannot immediately separate the influence of a particular teacher or program among all those variables. 2. Polls. Like polls, tests are only samples. They
cover only a small selection of material from a larger domain. A student's score is meant to represent how much has been learned on all material, but tests (like polls) can be misleading. 3. Intangibles. Tests (especially multiple-choice
tests) measure the learning of facts and procedures rather than the many other goals of teaching. Attitude, engagement, and the ability to learn further on one's own are difficult to measure with tests. In some cases, these "intangible" goals may be more important than those measured by tests. (The father of modern standardized testing, E. F. Lindquist, wrote eloquently about this [Lindquist 1951]; a synopsis of his comments can be found in [Koretz 2008, 37].) 4. Inflation. Test scores can be increased without
increasing student learning. This assertion has been convincingly demonstrated, but it is widely ignored by many in the education establishment [Koretz 2008, chap. 10]. In fact, the assertion should not be surprising. Every teacher knows that providing strategies for test-taking can improve student performance and that narrowing the curriculum to conform precisely to the test ("teaching to the test") can have an even greater effect. The evidence shows that these effects can be substantial: One can dramatically increase test scores while at the same time actually decreasing student learning. "Test scores" are not the same as "student achievement". This last problem plays a larger role as the stakes increase. This is often referred to as Campbell's Law: "The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to measure" [Campbell 1976]. In its simplest form, this can mean that high-stakes tests are likely to induce some people (students, teachers, or administrators) to cheat. . . and they do [Gabriel 2010]. But the more common consequence of Campbell's Law is a distortion of the education experience, ignoring things that are not tested (for example, student engagement and attitude) and concentrating on precisely those things that are. Value-Added Models
In the past two decades, a group of statisticians has focused on addressing the first of these four problems. This was natural. Mathematicians routinely create models for complicated systems that are similar to a large collection of students and teachers with many factors affecting individual outcomes over time. Here's a typical, although simplified, example, called the "split-plot design". You want to test fertilizer on a number of different varieties of some crop. You have many plots, each divided into subplots. After assigning particular varieties to each subplot and randomly assigning levels of fertilizer to each whole plot, you can then sit back and watch how the plants grow as you apply the fertilizer. The task is to determine the effect of the fertilizer on growth, distinguishing it from the effects from the different varieties. Statisticians have developed standard mathematical tools (mixed models) to do this. Does this situation sound familiar? Varieties, plots, fertilizer. . .students, classrooms, teachers? Dozens of similar situations arise in many areas, from agriculture to MRI analysis, always with the same basic ingredients--a mixture of fixed and random effects--and it is therefore not surprising that statisticians suggested using mixed models to analyze test data and determine "teacher effects". This is often explained to the public by analogy. One cannot accurately measure the quality of a teacher merely by looking at the scores on a single test at the end of a school year. If one teacher starts with all poorly prepared students, while another starts with all excellent, we would be misled by scores from a single test given to each class. To account for such differences, we might use two tests, comparing scores from the end of one year to the next. The focus is on how much the scores increase rather than the scores themselves. That's the basic idea behind "value-added". But value-added models (VAMs) are much more than merely comparing successive test scores. Given many scores (say, grades 3-8) for many students with many teachers at many schools, one creates a mixed model for this complicated situation. The model is supposed to take into account all the factors that might influence test results-- past history of the student, socioeconomic status, and so forth. The aim is to predict, based on all these past factors, the growth in test scores for students taught by a particular teacher. The actual change represents this more sophisticated "valueadded"-- good when it's larger than expected; bad when it's smaller The best-known VAM, devised by William Sanders, is a mixed model (actually, several models), which is based on Henderson's mixed-model equations, although mixed models originate much earlier [Sanders 1997]. One calculates (a huge computational effort!) the best linear unbiased predictors for the effects of teachers on scores. The precise details are unimportant here, but the process is similar to all mathematical modeling, with underlying assumptions and a number of choices in the model's construction. History
When value-added models were first conceived, even their most ardent supporters cautioned about their use [Sanders 1995, abstract]. They were a new tool that allowed us to make sense of mountains of data, using mathematics in the same way it was used to understand the growth of crops or the effects of a drug. But that tool was based on a statistical model, and inferences about individual teachers might not be valid, either because of faulty assumptions or because of normal (and expected) variation. Such cautions were qualified, however, and one can see the roots of the modern embrace of VAMs in two juxtaposed quotes from William Sanders, the father of the value-added movement, which appeared in an article in Teacher Magazine in the year 2000. The article's author reiterates the familiar cautions about VAMs, yet in the next paragraph seems to forget them:
Over the past decade, such cautions about VAM slowly evaporated, especially in the popular press. A 2004 article in The School Administrator complains
that there have not been ways to evaluate teachers in the past but excitedly touts valueadded as a solution:
And newspapers such as The Los Angeles Times
get their hands on seven years of test scores for students in the L.A. schools and then publish a series of exposÃƒƒÃ‚Â©s about teachers, based on a valueadded analysis of test data, which was performed under contract [Felch 2010]. The article explains its methodology: The Times used a statistical approach
known as value-added analysis, which rates teachers based on their students' progress on standardized tests from year to year. Each student's performance is compared with his or her own in past years, which largely controls for outside influences often blamed for academic failure: poverty, prior learning and other factors. Though controversial among teachers and others, the method has been increasingly embraced by education leaders and policymakers across the country, including the Obama administration. It goes on to draw many conclusions, including:
The writer adds the now-common dismissal of any concerns:
The article goes on to do exactly what it says "no one suggests"--it measures teachers solely on the basis of their value-added scores. What Might Be Wrong with VAM?
As the popular press promoted value-added models with ever-increasing zeal, there was a parallel, much less visible scholarly conversation about the limitations of value-added models. In 2003 a book with the title Evaluating Value-Added Models for Teacher Accountability laid out some of the problems and concluded:
In the next few years, a number of scholarly papers and reports raising concerns were published, including papers with such titles as "The Promise and Peril of Using Valued-Added Modeling to Measure Teacher Effectiveness" [RAND, 2004], "Re-Examining the Role of Teacher Quality in the Educational Production Function" [Koedel 2007], and "Methodological Concerns about the Education Value-Added Assessment System" [Amrein- Beardsley 2008]. What were the concerns in these papers? Here is a sample that hints at the complexity of issues.
These last two points were raised in a research paper [Lockwood 2007] and a recent policy brief from the Economic Policy Institute, "Problems with the Use of Student Test Scores to Evaluate Teachers", which summarizes many of the open questions about VAM.
In addition to checking robustness and stability of a mathematical model, one needs to check validity. Are those teachers identified as superior (or inferior) by value-added models actually superior (or inferior)? This is perhaps the shakiest part of VAM. There has been surprisingly little effort to compare valued-added rankings to other measures of teacher quality, and to the extent that informal comparisons are made (as in the LA Times article),
they sometimes don't agree with common sense. None of this means that value-added models are worthless--they are not. But like all mathematical models, they need to be used with care and a full understanding of their limitations. How Is VAM Used?
Many studies by reputable scholarly groups call for caution in using VAMs for high-stakes decisions about teachers.
And yet here is the LA Times, publishing valueadded
scores for individual teachers by name and bragging that even teachers who were considered first-rate turn out to be "at the bottom". In an episode reminiscent of the Cultural Revolution, the LA Times reporters confront a teacher who
"was surprised and disappointed by her [valueadded] results, adding that her students did well on periodic assessments and that parents seemed well-satisfied" [Felch 2010]. The teacher is made to think about why she did poorly and eventually, with the reporter's help, she understands that she fails to challenge her students sufficiently. In spite of parents describing her as "amazing" and the principal calling her one of the "most effective" teachers in the school, she will have to change. She recants: "If my student test scores show I'm an ineffective teacher, I'd like to know what contributes to it. What do I need to do to bring my average up?" Making policy decisions on the basis of valueadded models has the potential to do even more harm than browbeating teachers. If we decide whether alternative certification is better than regular certification, whether nationally board certified teachers are better than randomly selected ones, whether small schools are better than large, or whether a new curriculum is better than an old by using a flawed measure of success, we almost surely will end up making bad decisions that affect education for decades to come. This is insidious because, while people debate the use of value-added scores to judge teachers, almost no one questions the use of test scores and value-added models to judge policy. Even people who point out the limitations of VAM appear to be willing to use "student achievement" in the form of value-added scores to make such judgments. People recognize that tests are an imperfect
[emphasis added]
measure of educational success, but when sophisticated mathematics is applied, they believe the imperfections go away by some mathematical magic. But this is not magic. What really happens is that the mathematics is used to disguise the problems and intimidate people into ignoring them--a modern, mathematical version of the Emperor's New Clothes. What Should Mathematicians Do?
The concerns raised about value-added models ought to give everyone pause, and ordinarily they would lead to a thoughtful conversation about the proper use of VAM. Unfortunately, VAM proponents and politicians have framed the discussion as a battle between teacher unions and the public. Shouldn't teachers be accountable? Shouldn't we rid ourselves of those who are incompetent? Shouldn't we put our students first and stop worrying about teacher sensibilities? And most importantly, shouldn't we be driven by the data? This line of reasoning is illustrated by a recent fatuous report from the Brookings Institute, "Evaluating Teachers: The Important Role of Value- Added" [Glazerman 2010], which dismisses the many cautions found in all the papers mentioned above, not by refuting them but by asserting their unimportance. The authors of the Brookings paper agree that value-added scores of teachers are unstable (that is, not highly correlated year to year) but go on to assert: The use of imprecise measures to make high-stakes decisions that place societal or institutional interests above those of individuals is widespread and accepted in fields outside of teaching [Glazerman 2010, 7]. To illustrate this point, they use examples such as the correlation of SAT scores with college success or the year-by-year correlation of leaders in real estate sales. They conclude that "a performance measure needs to be good, not perfect". (And as usual, on page 11 they caution not to use valueadded measures alone when making decisions, while on page 9 they advocate doing precisely that.) Why must we use value-added even with its imperfections? Aside from making the unsupported claim (in the very last sentence) that "it predicts more about what students will learn--than any other source of information", the only apparent reason for its superiority is that value-added is based on data. Here is mathematical intimidation in its purest form--in this case, in the hands of economists, sociologists, and education policy experts. Of course we should hold teachers accountable, but this does not mean we have to pretend that mathematical models can do something they cannot. Of course we should rid our schools of incompetent teachers, but value-added models are an exceedingly blunt tool for this purpose. In any case, we ought to expect more from our teachers than what value-added attempts to measure. A number of people and organizations are seeking better ways to evaluate teacher performance in new ways that focus on measuring much more than test scores. (See, for example, the Measures of Effective Teaching project run by the Gates Foundation.) Shouldn't we try to measure longterm student achievement, not merely short-term gains? Shouldn't we focus on how well students are prepared to learn in the future, not merely what they learned in the past year? Shouldn't we try to distinguish teachers who inspire their students, not merely the ones who are competent? When we accept value-added as an "imperfect" substitute for all these things because it is conveniently at hand, we are not raising our expectations of teachers, we are lowering them. And if we drive away the best teachers by using a flawed process, are we really putting our students first? Whether naÃƒƒÃ‚Â¯fs or experts, mathematicians need to confront people who misuse their subject to intimidate others into accepting conclusions simply because they are based on some mathematics. Unlike many policy makers, mathematicians are not bamboozled by the theory behind VAM, and they need to speak out forcefully. Mathematical models have limitations. They do not by themselves convey authority for their conclusions. They are tools, not magic. And using the mathematics to intimidate-- to preempt debate about the goals of education and measures of success--is harmful not only to education but to mathematics itself. References
Audrey Amrein-Beardsley, Methodological concerns about the education value-added assessment system, Educational Researcher 37 (2008), 65-75. http:// dx.doi.org/10.3102/0013189X08316420 Eva L. Baker, Paul E. Barton, Linda Darling- Hammond, Edward Haertel, Hellen F. Ladd, Robert L. Linn, Diane Ravitch, Richard Rothstein, Richard J. Shavelson, and Lorrie A. Shepard, Problems with the Use of Student Test Scores to Evaluate Teachers, Economic Policy Institute Briefing Paper #278, August 29, 2010, Washington, DC. http://www. epi.org/publications/entry/bp278 Henry Braun, Using Student Progress to Evaluate Teachers: A Primer on Value-Added Models, Educational Testing Service Policy Perspective, Princeton, NJ, 2005. http://www.ets.org/Media/Research/pdf/ PICVAM.pdf Henry Braun, Naomi Chudowsky, and Judith Koenig, eds., Getting Value Out of Value-Added: Report of a Workshop, Committee on Value-Added Methodology for Instructional Improvement, Program Evaluation, and Accountability; National Research Council, Washington, DC, 2010. http://www.nap.edu/ catalog/12820.html Donald T. Campbell, Assessing the Impact of Planned Social Change, Dartmouth College, Occasional Paper Series, #8, 1976. http://www.eric.ed.gov/PDFS/ ED303512.pdf Jason Felch, Jason Song, and Doug Smith, Who's teaching L.A.'s kids?, Los Angeles Times, August 14, 2010. http://www.latimes.com/news/local/ la-me-teachers-value-20100815,0,2695044. story Trip Gabriel, Under pressure, teachers tamper with tests, New York Times, June 11, 2010. http://www. nytimes.com/2010/06/11/education/11cheat. html Steven Glazerman, Susanna Loeb, Dan Goldhaber, Douglas Staiger, Stephen Raudenbush, Grover Whitehurst, Evaluating Teachers: The Important Role of Value-Added, Brown Center on Education Policy at Brookings, 2010. http://www.brookings.edu/ reports/2010/1117_evaluating_teachers.aspx Ted Hershberg, Virginia Adams Simon and Barbara Lea Kruger, The revelations of value-added: An assessment model that measures student growth in ways that NCLB fails to do, The School Administrator, December 2004. http://www.aasa.org/ SchoolAdministratorArticle.aspx?id=9466 David Hill, He's got your number, Teacher Magazine, May 2000 11(8), 42-47. http://www.edweek.org/ tm/articles/2000/05/01/08sanders.h11.html Cory Koedel and Julian R. Betts, Re-Examining the Role of Teacher Quality in the Educational Production Function, Working Paper #2007-03, National Center on Performance Initiatives, Nashville, TN, 2007. http:// economics.missouri.edu/working-papers/2007/ wp0708_koedel.pdf Daniel Koretz, Measuring Up: What Educational Testing Really Tells Us, Harvard University Press, Cambridge, Massachusetts, 2008. E. F. Lindquist, Preliminary considerations in objective test construction, in Educational Measurement (E. F. Lindquist, ed.), American Council on Education, Washington DC, 1951. J. R. Lockwood, Daniel McCaffrey, Laura S. Hamilton, Brian Stetcher, Vi-Nhuan Le, and Felipe Martinez, The sensitivity of value-added teacher effect estimates to different mathematics achievement measures, Journal of Educational Measurement 44(1) (2007), 47-67. http://dx.doi.org/10.1111/ j.1745-3984.2007.00026.x Daniel F. McCaffrey, Daniel Koretz, J. R. Lockwood, and Laura S. Hamilton, Evaluating Value-Added Models for Teacher Accountability, RAND Corporation, Santa Monica, CA, 2003. http://www.rand.org/ pubs/monographs/2004/RAND_MG158.pdf Daniel F. McCaffrey, J. R. Lockwood, Daniel Koretz, Thomas A. Louis, and Laura Hamilton, Models for value-added modeling of teacher effects, Journal of Educational and Behavioral Statistics 29(1), Spring 2004, 67-101. http://www.rand.org/pubs/ reprints/2005/RAND_RP1165.pdf RAND Research Brief, The Promise and Peril of Using Value-Added Modeling to Measure Teacher Effectiveness, Santa Monica, CA, 2004. http://www.rand.org/ pubs/research_briefs/RB9050/RAND_RB9050.pdf William L. Sanders and Sandra P. Horn, Educational Assessment Reassessed: The Usefulness of Standardized and Alternative Measures of Student Achievement as Indicators of the Assessment of Educational Outcomes, Education Policy Analysis Archives, March 3(6) (1995). http://epaa.asu.edu/ojs/article/ view/649 W. Sanders, A. Saxton, and B. Horn, The Tennessee value-added assessment system: A quantitative outcomes-based approach to educational assessment, in Grading Teachers, Grading Schools: Is Student Achievement a Valid Evaluational Measure? (J. Millman, ed.), Corwin Press, Inc., Thousand Oaks, CA, 1997, pp 137--162. John Ewing is president of Math for America. His email address is jewing@mathforamerica.org
— John Ewing |

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