In an attempt to reduce the growth of its population, China has instituted a policy that limits a family to one child. This policy has been particularly unpopular among rural Chinese, who have suggested revising the policy to limit families to one son. Suppose you were among those in the government considering the implications of adopting this proposal.
Probability is a notoriously difficult concept. Even after much instruction, many students remain confused both about methods used to calculate a probability and about its meaning (Konold 1991). In this article, I shall describe a modified version of the tree diagram that many of my students at both the high school and college levels have found helpful in making probabilities more meaningful. I refer to these representations as pipe diagrams. Although most readers are probably familiar with tree diagrams, I review a few of their basic features before introducing pipe diagrams and discussing their educational advantages.
Clifford Konold and Alexander Pollatsek
The idea of data as a mixture of signal and noise is perhaps the most fundamental concept in statistics. Research suggests, however, that current instruction is not helping students to develop this idea, and that though many students know, for example, how to compute means or medians, they do not know how to apply or interpret them.
Clifford Konold, Alexander Pollatsek, Arnold Well, Jill Lohmeier and Abigail Lipson
Subjects were asked to select from among four possible sequences the “most likely” to result from flipping a coin five times. Contrary to the results of Kahneman and Tversky (1972), the majority of subjects (72%) correctly answered that the sequences are equally likely to occur. This result suggests, as does performance on similar NAEP items, that most secondary school and college-age students view successive outcomes of a random process as independent. However, in a follow-up question, subjects were also asked to select the “least likely” result. Only half the subjects who had answered correctly responded again that the sequences were equally likely; the others selected one of the sequences as least likely. This result was replicated in a second study in which 20 subjects were interviewed as they solved the same problems. One account of these logically inconsistent responses is that subjects reason about the two questions from different perspectives. When asked to select the most likely outcome, some believe they are being asked to predict what actually will happen, and give the answer “equally likely” to indicate that all of the sequences are possible. This reasoning has been described by Konold (1989) as an “outcome approach” to uncertainty. This prediction scheme does not fit questions worded in terms of the least likely result, and thus some subjects select an incompatible answer based on “representativeness” (Kahneman & Tversky, 1972). These results suggest that the percentage of secondary school students who understand the concept of independence is much lower than the latest NAEP results would lead us to believe and, more generally, point to the difficulty of assessing conceptual understanding with multiple-choice items.
Richard Lesh, Kathryn B. Chval, Karen Hollebrands, Clifford Konold, Michelle Stephan, Erica N. Walker and Jeffrey J. Wanko
For roughly 35 years, the NCTM Research Presession has been held 1 or 2 days prior to the NCTM Annual Conference—hence the word presession. Beginning with the 2014 meeting in New Orleans, the NCTM Research Presession will be rebranded as the NCTM Research Conference. This change of name is intended to emphasize the critical role that research should play in our efforts to improve mathematics education. The NCTM Research Committee thought this an appropriate occasion to invite Richard Lesh, who was instrumental in the founding of the Research Presession, to join the members of the current Research Committee in reflecting on its formation, the hopes he and other kindred spirits had in mind when they started it, and the current state and future of research in our field.
Michelle L. Stephan, Kathryn B. Chval, Jeffrey J. Wanko, Marta Civil, Michael C. Fish, Beth Herbel-Eisenmann, Clifford Konold and Trena L. Wilkerson
Mathematics education researchers seek answers to important questions that will ultimately result in the enhancement of mathematics teaching, learning, curriculum, and assessment, working toward “ensuring that all students attain mathematics proficiency and increasing the numbers of students from all racial, ethnic, gender, and socioeconomic groups who attain the highest levels of mathematics achievement” (National Council of Teachers of Mathematics [NCTM], 2014, p. 61). Although mathematics education is a relatively young field, researchers have made significant progress in advancing the discipline. As Ellerton (2014) explained in her JRME editorial, our field is like a growing tree, stable and strong in its roots yet becoming more vast and diverse because of a number of factors. Such growth begs these questions: Is our research solving significant problems? How do we create a system and infrastructure that will provide an opportunity to accumulate professional knowledge that is storable and shareable as we work together to address significant problems (Hiebert, Gallimore, & Stigler, 2002)? How do we “facilitate research and development that is coordinated, integrated, and accumulated” (Lesh et al., 2014, p. 167)?