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In [18], Mendes and Remmel showed how Gessel’s generating function for the distributions of the number of descents, the major index, and the number of inversions of permutations in the symmetric group could be derived by applying a ring homomorphism defined on the ring of symmetric functions to a simple symmetric function identity. We show how similar methods may be used to prove analogues of that generating function for compositions.

In this paper we present a case study of an individual student who consistently used semantic reasoning to construct proofs in calculus but infrequently used semantic reasoning to produce proofs in linear algebra. We hypothesize that the differences in these reasoning styles can be partially attributed to this student's familiarity with the content, the teaching styles of the professors who taught him, and the amount of time he was given to complete the tasks. We suggest that there are factors that have been ignored in previous research, including domain, instruction, and methodological constraints, that researchers should consider when ascribing to students a proving style.

In this qualitative study, we explored how oral interviews can inform instructors about students' understanding of abstract algebra and their ability to construct a proof in this setting. Our findings indicate that some students had a good understanding of the ideas needed for a subgroup proof, but could not write a coherent proof. On the other hand, these same students provided fairly sound verbal proofs. We discuss the implications of these results for teaching and future research.

Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics. Building on Yang and Lin's (Educational Studies in Mathematics 67:59-76, "2008") model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the proof's high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples. We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory. (Contains 5 footnotes.)

Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students' treatment of symbols will be largely non-referential.

Mammalian DNA polymerase delta (Pol delta) is believed to replicate a large portion of the genome and to synthesize DNA in DNA repair and genetic recombination pathways. The effects of mutation in the polymerase domain of this essential enzyme are unknown. Here, we generated mice harboring an L604G or L604K substitution in highly conserved motif A in the polymerase active site of Pol delta. Homozygous Pold1(L604G/L604G) and Pold1(L604K/L604K) mice died in utero. However, heterozygous animals were viable and displayed no overall increase in disease incidence, indicative of efficient compensation for the defective mutant polymerase. The life spans of wild-type and heterozygous Pold1(+/L604G) mice did not differ, while that of Pold1(+/L604K) mice was reduced by 18%. Cultured embryonic fibroblasts from the heterozygous strains exhibited comparable increases in both spontaneous mutation rate and chromosome aberrations. We observed no significant increase in cancer incidence; however, Pold1(+/L604K) mice bearing histologically diagnosed tumors died at a younger median age than wild-type mice. Our results indicate that heterozygous mutation at L604 in the polymerase active site of DNA polymerase delta reduces life span, increases genomic instability, and accelerates tumorigenesis in an allele-specific manner, novel findings that have implications for human cancer.