CONTENTS INTRODUCTION I. ENUMERATING LEGAL INNUMERACY A. Innumeraey Through Miscalculation B. Innumeracy Through Oversimplification C. Innumeracy Through Misunderstanding II. THE SIGNIFICANCE OF LEGAL INNUMERACY A. Overvaluing Numerical Information B. Failing to Speak the Language of Numbers C. Legitimacy, Meaning, and the Practice of Law III. DEFINING THE PROBLEM: COMPETENCE OR CONFIDENCE? A. Objective Innumeracy 1. Deficits in Cognitive Functioning 2. Deficits in Mathematics Education B. Subjective Innumeracy 1. Math Anxiety 2. Relative Competence CONCLUSION: MOVING PAST INNUMERACY APPENDIX: SURVEY OF NUMERICALLY FOCUSED LAW CLASSES INTRODUCTION
It is an open secret that lawyers don't like math. Tales of lawyers who chose the profession over business or medicine at least in part because of discomfort with math are legion, as are reports of math avoidance by lawyers once in the profession. (1) Many lawyers treat explicitly math-centric fields, such as tax law and bankruptcy, as impenetrable specialties to be avoided at all costs, segregated even on the judicial level with their own dedicated courts. With the exception of empirical articles and those employing an explicitly economic approach, most legal scholars avoid even the whiff of quantitative analysis in their writings, in part to avoid discouraging use of their ideas by lawyers, legislators, and judges uncomfortable with numerical notations and formulas. (2)
To be clear, it is not only lawyers who struggle with numbers and their calculation. A 2003 study found that only 13% of American adults were "proficient" at quantitative tasks and that only 78% could perform even simple, single-step arithmetic. (3) One study, even though it focused on college-educated individuals, found that nearly half of the subjects could not solve basic problems involving probabilities or convert percentages to proportions. (4) Most of us fare little better in the real world; for example, we avoid financial calculations such as the amount needed for retirement, and we fail to assess and rebalance retirement portfolios. (5) Innumeracy is widespread, even among the most educated and successful Americans.
The profession of law, though, has embraced innumeracy in curious and significant ways that other professions have not. The Law School Admissions Test is the only major post-secondary admissions examination without a math component. (6) Law students are assumed to lack mathematical backgrounds, and it is well accepted that they are not interested in understanding even basic mathematical concepts. (7) Moreover, many law professors share the math aversion of their students so that the numerical aspects of cases are often left unexplored in class or even edited out of casebooks. As a result, little math is found in the typical law school classroom.
Not surprisingly, law students who are uncomfortable with math become lawyers who self-identify as "bad at math." Indeed, innumeracy is at times almost celebrated within the legal profession. Lawyers bond openly over their distaste for math and accept the same in others. Those who are competent at or even enjoy math are seen as an oddity. Only occasionally is the profession's math paralysis criticized or even questioned, (8) That lawyers are bad at math has become a truism, so that whether we are actually bad at math is subsumed by our image of ourselves and others' image of us as such." Yet lawyers in all types of practices must grapple with mathematical issues.I[degrees] Are we competent to do so? (11) If at least a sizable portion of the bar is innumerate, why is this the case is it objective math competence or subjective math confidence that we lack? And, either way, are our mathematical failings corrected by the checks and balances of legal practice and the legal system, or should we, as members of the legal profession, change our approach to math?
Innumeracy is an issue we must confront: numerical information is pervasive and calculations are central to the practice of many areas of law. (12) Yet many math mistakes in the law and legal practice remain unacknowledged and uncorrected because in our discomfort with numbers we assign undue weight to them, lack the language to engage with numerical ideas, and limit our ability to represent our clients with respect to some of the more interesting and novel legal issues arising in our technological world, (13) Ultimately, innumeracy prevents us from thinking critically about the information and assumptions underlying numbers and compromises transparency and comprehensibility in the law, undermining legal authority.
To date, academic attention has focused primarily on the innumeracy of jurors (14) and of the American public generally. (15) But the spotlight has rarely been focused where it belongs: on practicing lawyers (16) and lawmakers, (17) as well as law students and the law professors who prepare them for their future roles in the legal community. In this Article, I distinguish between three types of mathematical errors commonly made by lawyers: miscalculations, oversimplifications, and misapplications of mathematical principles. I argue that these errors matter because of the centrality of numerical information to the practice of many areas of law. In order to better understand the origins of this innumeracy and begin to move towards a more numerate approach to the law, I distinguish objective innumeracy a lack of math competence from subjective innumeracy a lack of math confidence. Finally, I conclude by offering practical suggestions for beginning to overcome innumeracy in the legal profession. Dealing head on with these fundamental challenges holds the promise of greatly improving how we think about and practice law.
ENUMERATING LEGAL INNUMERACY
Lawyers' discomfort with numbers and their calculation can be seen in the ways we approach mathematical issues in the law. As an initial matter, in many instances lawyers simply avoid math.
At times, this avoidance reflects our wariness about the potential for laypersons to assign undue weight to numerical evidence. So, for example, in People v. Collins, (18) mathematics professor Daniel Martinez provided expert testimony concerning probability theory in an effort to establish the likelihood that the defendants in question had committed the crime with which they were charged. Upon appeal, the California Supreme Court expressed concern with the power of probabilistic evidence and warned that "[m]athematics, a veritable sorcerer in our computerized society, while assisting the trier of fact in the search for truth, must not cast a spell over him." (19) In particular, the court worried that the jury lacked the competence to properly contextualize the probabilistic evidence with which it had been presented and, as a result, overvalued it in deciding the defendants' guilt. (20)
Professor Laurence Tribe has since expanded on this concern, (21) criticizing the use of "explicitly statistical evidence or overtly probabilistic arguments" at trial. (22) A mathematician himself, (23) Tribe believes that the risk that jurors might overvalue numerical data due to the precision and "overbearing impressiveness of numbers" is too great. (24) Thus, he prefers that jurors be allowed to make inductive inferences rather than be presented with explicitly quantified information. (25) Later commentators have echoed this idea. (26) At other times, math avoidance may reflect a conscious legal strategy, born of a belief that "[a]necdotal evidence is vivid and reaches us in a way that ... statistical information cannot." (27) In this view, storytelling is a preferred advocacy tactic, providing context and color, and thus a saliency to the jury, in a way numbers cannot. For example, Michael Saks and Robert Kidd have argued that:
Research demonstrates ... that people do not process probabilistic information well, that in the face of particularistic information, they cannot integrate the statistical and anecdotal evidence and consequently tend to ignore the statistical information. Intuitive, heuristic, human decision makers must dispense with certain information, and that tends strongly to be the quantitative information. While commentators' arguments have been that the data are inordinately persuasive, the evidence says the reverse is true. (28) This reasoning resonates with the actual experience of jurors in Collins, who later reported that they had disregarded Professor Martinez's testimony in reaching the verdict, focusing instead on the evidence provided by eyewitnesses to the crime. (29)
This research illustrates that math avoidance can be a conscious technique used by lawyers to develop a persuasive narrative that avoids confusion or the misinterpretation of numerical evidence by laypersons charged with legal decision making. But that is just the tip of the iceberg with regard to math avoidance in the law: as other commentators have noted, lawyers often avoid math simply because they are uncomfortable with it. (30)
Avoidance, though, is not the primary problem. Instead, the central issue is that when lawyers do math, (31) they often do it badly. Indeed, the ways innumeracy manifests itself are so varied and overlapping that any taxonomy of the problem is necessarily incomplete. But it is worth parsing the types of numerical errors lawyers most commonly make in order to better understand the reasons for them and identify those that are most in need of systemic correction. Three problems are of particular importance due to their pervasiveness and the ways in which they compromise transparency in the law and, thus, undermine the legitimacy of our legal system: (1) persistent computational errors, (2)the reduction of complex calculations to overly simplistic formulas that obscure their failure to accomplish their intended goals, and (3)the production and use of meaningless data through fundamental misunderstandings of the principles underlying...