Exploring the Paradox: Adolescent Reasoning in Mathematical and Non-mathematical Domains

January 15, 2012

A perennial concern in mathematics education is that students often fail to understand the nature of evidence and justification in mathematics, that is, the nature of proof. As a consequence, significant attention has been directed toward enhancing the role of proof in school mathematics.

Mathematical proofs play a critical role in promoting deep learning. Proofs constitute the basis of mathematical understanding and are essential in developing, establishing, and communicating mathematical knowledge. Yet, despite its importance to learning as well as the attention being placed on proof in school mathematics, research has often painted a bleak picture of adolescent students’ abilities to reason mathematically.

If true, that lack is especially unfortunate because skill in reasoning goes beyond mathematics to help students become citizens who can think logically and critically.

In contrast, cognitive science research has revealed surprising strengths in children’s abilities to reason inferentially in non-mathematical domains. Traditional views have posited children as limited to understanding obvious relations among observable properties. Yet there is growing evidence that children are capable of developing sophisticated causal theories, and of using powerful strategies of inductive inference when reasoning about the natural world.

This raises a paradox: Why do children appear so capable when reasoning in non-mathematical contexts, yet seemingly appear much less capable when reasoning in mathematical domains?

UW Madison education professors Eric Knuth, Charles Kalish, Amy Ellis, and colleagues explored that  paradox. Their research bridges the research on adolescents’ reasoning capabilities within mathematics education and within cognitive science.

A primary challenge students face in developing an understanding of deductive proof is overcoming their reliance on empirical evidence. In fact, the wealth of studies investigating students’ proving competencies demonstrates that students overwhelmingly rely on examples to justify the truth of statements.

Knuth and colleagues categorize as empirical those justifications in which people use examples to support the truth of a statement. General justifications (or proofs) are those in which one demonstrates that the statement is true for all members of the set.

Adolescents are limited in their understanding of what constitutes evidence and justification in mathematics. Moreover, they demonstrate a tendency toward empirical-based, inductive reasoning, rather than more general, deductive reasoning. In prior work, Knuth found that middle school students attempt to produce more general, deductive justifications as they progress though middle school, yet still less than half the students produce such justifications even by the end of their middle school mathematics education.

Adolescents’ difficulty with mathematical reasoning raises the question of whether there is some developmental constraint that limits their mathematical reasoning. There is probably nothing special about adolescence in terms of acquiring deduction. Younger children have been shown to appreciate that deductive inference leads to certain conclusions, and is stronger than inductive inference. At the same time, however, even adults struggle to reason formally and deductively. Thus, deductive inference seems neither impossible before adolescence nor guaranteed after.

Deductive and inductive arguments have very different qualities. In making a deductive argument, one tries to show that the hypothesized conjecture must be true as a logical consequence of the premises. One common form of deductive reasoning is the syllogism: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.

On the other hand, in making an inductive argument, one seeks supporting evidence as the means for justifying that the conjecture is likely to be true. When repeated observations reveal patterns that persist over time, one can infer things; one can generalize. For example, every raven in a random sample of 3200 ravens is black. This strongly supports the hypothesis that all ravens are black.

Deductive arguments prove their conclusions through logic, while inductive arguments provide evidence that conclusions are likely.

By the time they reach middle school, children can see that many examples are more convincing than are fewer, that a diverse set of examples is better than a set of very similar examples, and that an argument based on a typical example is stronger than an argument based on an atypical example.

Knuth and colleagues want to know more about how adolescents use such abilities to evaluate both mathematical and non-mathematical conjectures, and how they think about the nature of evidence used to support conjectures. Insight into this process may suggest a means for leveraging their inductive reasoning skills to foster the development of more sophisticated (deductive) ways of reasoning in mathematical domains.

An important first step toward developing a deeper understanding of students’ inductive reasoning is to explore their representations of similarity relations among mathematical objects. Successful inductive reasoning depends on seeing objects as similar to the degree they really do share important features or characteristics.

How do students’ similarity judgments compare with experts’ similarity judgments? Answers to such questions may provide insight into students’ choices for the empirical evidence they use to justify mathematical conjectures. These, in turn, may provide insight into means to foster their transition to more deductive ways of reasoning.

In one study, Knuth and colleagues interviewed 14 middle school students, 14 undergraduates, and 14 doctoral students in the fields of science, technology, engineering, and mathematics (STEM experts). Participants examined various numbers and shapes on individual cards and then sorted and re-sorted them into groups, according to whatever principles they chose.

In a number-sorting task the researchers noted a number of similarities between the middle school students and the STEM experts, in terms of which features they noticed. These similarities between the students and the experts led the researchers to wonder, were there any features that one group attended to but the other did not?

Findings from another exercise showed that, in general, there were not many differences between the middle school students, the undergraduates, and the STEM experts in terms of the characteristics of number and shape that they attended to. Further, some of the most salient features of number included multiples, parity, primes, and intervals (i.e., the relative size of numbers). Several of the more salient features of shape included the number of sides, the shape’s size, recognizable features of a shape, and the size of its angles. Knuth says these findings are important:  They reveal particular characteristics that participants find noticeable, such as a number’s relative size or a shape’s size, that matter to students, but in many cases are not mathematically important.

The results from this initial study suggest that adolescents and experts use similar representations of similarity among (some) mathematical objects. In particular, adolescents did notice mathematically significant relations among the objects.  Part of what makes two numbers or shapes similar is that they share relevant mathematical properties.

The pressing question for future research is how adolescents they use such relations to evaluate conjectures.

Knuth and colleagues have taken a relatively new perspective (in mathematics education research) on adolescents’ use of empirical strategies for evaluating mathematical conjectures, and in particular, how students make decisions regarding their selection of evidence. Rather than seeing such strategies as limited, or as failures to adopt deductive strategies, Knuth suggests that there may be value in such inductive strategies.

His research team proposes considering inductive inference about mathematical conjectures as an object of study in and of itself. To that end, they seek to better understand the adolescents’ inductive reasoning in the domain of mathematics.

This research is a first step in a larger project of exploring how adolescents use empirical examples and inductive methods to reason about mathematical objects. Knuth and colleagues believe inductive inference strategies should play an important role in mathematics. Understanding adolescents’ inductive reasoning may provide important insights into helping them transition to more sophisticated, deductive ways of reasoning in mathematics.