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New understandings of the functioning of human brains engaged in mathematics raise interesting questions for mathematics educators. Novel lines of research are suggested by neuroscientific findings, and new light is shed on some longstanding issues in mathematics education., Comment: 21 pages, 11 figures

Data science is a discipline that provides principles, methodology and guidelines for the analysis of data for tools, values, or insights. Driven by a huge workforce demand, many academic institutions have started to offer degrees in data science, with many at the graduate, and a few at the undergraduate level. Curricula may differ at different institutions, because of varying levels of faculty expertise, and different disciplines (such as Math, computer science, and business etc) in developing the curriculum. The University of Massachusetts Dartmouth started offering degree programs in data science from Fall 2015, at both the undergraduate and the graduate level. Quite a few articles have been published that deal with graduate data science courses, much less so dealing with undergraduate ones. Our discussion will focus on undergraduate course structure and function, and specifically, a first course in data science. Our design of this course centers around a concept called the data science life cycle. That is, we view tasks or steps in the practice of data science as forming a process, consisting of states that indicate how it comes into life, how different tasks in data science depend on or interact with others until the birth of a data product or the reach of a conclusion. Naturally, different pieces of the data science life cycle then form individual parts of the course. Details of each piece are filled up by concepts, techniques, or skills that are popular in industry. Consequently, the design of our course is both "principled" and practical. A significant feature of our course philosophy is that, in line with activity theory, the course is based on the use of tools to transform real data in order to answer strongly motivated questions related to the data., Comment: 25 pages, 5 figures

We elaborate on an alternative representation of conditional probability to the usual tree diagram. We term the representation `turtleback diagram' for its resemblance to the pattern on turtle shells. Adopting the set theoretic view of events and the sample space, the turtleback diagram uses elements from Venn diagrams---set intersection, complement and partition---for conditioning, with the additional notion that the area of a set indicates probability whereas the ratio of areas for conditional probability. Once parts of the diagram are drawn and properly labeled, the calculation of conditional probability involves only simple arithmetic on the area of relevant sets. We discuss turtleback diagrams in relation to other visual representations of conditional probability, and detail several scenarios in which turtleback diagrams prove useful. By the equivalence of recursive space partition and the tree, the turtleback diagram is seen to be equally expressive as the tree diagram for representing abstract concepts. We also provide empirical data on the use of turtleback diagrams with undergraduate students in elementary statistics or probability courses., Comment: 23 pages, 14 figures

We examine a measure of individual student gain by preservice elementary teachers, related to Richard Hakes use of mean gain in the study of reform classes in undergraduate physics. The gain statistic assesses the amount individual students increase their test scores from initial test to final test, as a proportion of the possible increase for each student. We examine the written work in mathematics classes of preservice elementary teachers with very high gain and those with very low gain and show that these groups exhibit distinct psychological attitudes and dispositions to learning mathematics. We show a statistically significant, small, increase in average gain when course goals focus on patterns, connections, and meaning making in mathematics. A common belief is that students with low initial test scores will have higher gains, and students with high initial-test scores will have lower gains. We show that this is not correct for a cohort of preservice elementary teachers., Comment: 14 pages, 5 figures, 3 tables

The mathematical growth of a pre-service elementary teacher over the period of a semester is detailed. This student had an unusual combination of very poor basic computational skills and a capacity for higher level problem solving involving algebraic notation. The extent to which one or two semesters of mathematics content, however well-structured, is sufficient to develop deep understanding, much less pedagogical content knowledge, particularly when growth of mathematical reasoning and understanding are coupled with incorrect or inappropriate algorithms which have become automated over several years of use is detailed. The paper provides evidence that this student acquired new insights and understandings about mathematics and experienced changes in her attitudes and beliefs. However, it is not known whether the changes in attitudes and beliefs are short- or long-term nor whether what she learned is truly "owned" by her. (Contains 13 references.) (Author/MM)

This paper explores how college students understand ideas of functions, and which representations are productive for them in promoting their ability to work flexibly across representations. The study used pre- and post-test scores, and triangulations via student self evaluations, to generate a hypothesis related to flexible thinking and success in algebra. It used confidence intervals to provide evidence for a highly significant change in student flexibility in algebraic thinking, and to assist in generating a plausible model of how the use of function machines in a developmental algebra course is instrumental in stimulating that flexibility. (Contains 19 references.) (Author/MM)

This article addresses the question: "What are the implications for the preparation of prospective elementary teachers of 'early algebra' in the elementary grades curriculum?" Part of the answer involves language aspects of algebra: in particular, how a change in pre-service teachers' attitudes to algebra, from instrumental to relational, is correlated with learning the conventional meaning of algebraic and other mathematical terms. (Contains 13 references.) (Author/MM)

This study addresses the question of how pre-service elementary teachers can be induced to change their attitudes to mathematics. Pre-service elementary teachers want to get mathematical answers right. They want to know which formulas to use, and how to get the correct answer. The paper presents transcriptions of student writing that illustrate and support this point. Changing what students value in mathematics is a much harder challenge than teaching them mathematical procedures and applications of formulas. An antidote is needed to a severely procedural orientation to mathematics focused on "correct answers" that prospective teachers have learned to value above all. How can we explicitly emphasize connections, and assist students to construct relationships between parts of mathematics that they see as different? (Contains 21 references.) (Author/MM)

This paper details one young woman's journey in a 16-week mathematics content course for pre-service teachers. It analyzes her initial efforts at problem solving in groups, and her ability to see and recall explicitly connections that were inherent in the instructor's conceptions of the problems. Using her writing during the course, the paper compares her memories with those of other students of differing achievement. She made remarkable strides in coming to terms with the language and ideas of mathematics, and understanding how engaging with these can help her be a better teacher. (Author/MM)