Lessons Learned in a Math Excel Workshop: The Importance of Maintaining High Cognitive Demands

Uri Treisman\u27s Emerging Scholars Workshop model has been implemented on many college campuses over the last twenty years. The Treisman model is based on groups of students meeting regularly in a social atmosphere to work collaboratively in solving challenging mathematics problems related to their introductory coursework. Emerging Scholars Programs (or Math Excel as it is called in many settings, including ours) have been particularly successful in increasing the academic success and participation of underrepresented groups in mathematics. The primary responsibilities of a workshop leader include the design of a session’s worksheet, as well as the facilitation of students\u27 problem solving efforts during the workshop session itself. In this paper, we discuss a mathematical tasks framework proposed by researchers in the Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project that may be especially helpful to workshop leaders in making a successful implementation of Math Excel. This framework emphasizes the notion of the cognitive demand of a mathematical task. The level of cognitive demand is not a static attribute and may well change as students undertake a task in a classroom setting. QUASAR researchers noted how the initially high demands of a task may not be maintained in the classroom, and how teachers\u27 actions may lower the demands and consequently limit learning opportunities for students. Although the QUASAR project involved middle school mathematics instruction, we believe that this mathematical tasks framework can provide valuable lessons for Math Excel workshop leaders, and it suggests how critically important both the choice of problem tasks and the workshop leaders’ facilitation of student work can be. In this paper, we review the mathematical tasks framework and illustrate its application to scenarios actually encountered in our Math Excel workshops

Oregon Mathematics Leadership Institute Project: Evaluation Results on Teacher Content Knowledge, Implementation Fidelity, and Student Achievement

The Oregon Mathematics Leadership Institute (OMLl) National Science Foundation Mathematics and Science Partnership project partners are Oregon State University, Portland State University, Teachers Development Group, and ten Oregon school districts. The primary activities of the project were a sequence of three intensive three-week residential institutes emphasizing mathematics content knowledge for teaching, collegial leadership, and the building of Professional Learning Communities. Teachers at all levels of grades K-12 participated together in the mathematics content courses. By the conclusion of the. third Summer Institute, teachers had shown significant improvements in mathematical content knowledge for teaching. Analysis of student achievement data in participating schools was initially inconclusive. However, once implementation fidelity traits were taken into account, a positive relationship between project participation and student achievement emerged. The degree to which schools implement the practices promoted by the OMLI project is a significant positive predictor of student perfomiance above and beyond what can be explained by the socioeconomic factor as indicated by the percentage of students who qualify for the free and reduced lunch program. This relationship is particularly acute at secondary levels, but additional factors appear to be at play at elementary grade levels

Fast autotuning of a hydrogen maser by cavity Q modulation

A new fast autotuner for the hydrogen maser was implemented. By modulating the cavity, a phase shift in the maser output signal is induced which is proportional to the cavity tuning error. The phase shift is detected and fed back to a varactor tuner to stabilize the cavity against long-term drifts. Also, a PIN-diode cavity modulator which gives no incidental frequency shift over a very wide range of operation was developed. Modulated at over 200 Hz, it allows variations in maser cavity frequency to be compensated with a loop gain greater than 1000. Compensation of incidental amplitude modulation of the output was demonstrated

Differential Entropy on Statistical Spaces

We show that the previously introduced concept of distance on statistical spaces leads to a straightforward definition of differential entropy on these statistical spaces. These spaces are characterized by the fact that their points can only be localized within a certain volume and exhibit thus a feature of fuzziness. This implies that Riemann integrability of relevant integrals is no longer secured. Some discussion on the specialization of this formalism to quantum states concludes the paper.Comment: 4 pages, to appear in the proceedings of the joint meeting of the 2nd International Conference on Cybernetics and Information Technologies, Systems and Applications (CITSA 2005) and the 11th International Conference on Information Systems Analysis and Synthesis (ISAS 2005), to be held in Orlando, USA, on July 14-17, 200

Persistence in Mathematics by Underrepresented Students: Experiences of a Math Excel Program

Success in mathematics by underrepresented and nontraditional college students is measured not only by academic performance (grades), but also by the continued participation and persistence of these students in mathematics coursework. The Math Excel program at Oregon State University attempts to build learning communities with a sharp academic focus in support of students concurrently taking introductory level mathematics courses. The Math Excel program is based heavily on Uri Treisman\u27s Emerging Scholars Workshop model of collaborative problem solving. In this article, we examine the experience of minority students in the Educational Opportunities Program participating in the Math Excel program. While the program had appeared successful in terms of improving academic performance in the concurrent mathematics course, the continued participation and persistence of these students in mathematics was disappointing. On a trial basis, structural changes were made to build a much stronger identification of the Math Excel learning community with a section of College Algebra. In the next term, there was a much higher incidence of participation in the subsequent Precalculus using the same Math Excel structure. While the collaborative problem solving activity provided in Math Excel was crucial to students\u27 successful academic performance, these results suggest that subtle issues related to students\u27 recognition of and identification with a learning community may be critically important to underrepresented and nontraditional students\u27 continued persistence in mathematics

Fast QMC matrix-vector multiplication

Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications from statistics. In this paper we design QMC quadrature points $\boldsymbol{y}_0, ..., \boldsymbol{y}_{N-1} \in [0,1]^s$ such that for the matrix $Y = (\boldsymbol{y}_{0}^\top, ..., \boldsymbol{y}_{N-1}^\top)^\top$ whose rows are the quadrature points, one can use the fast Fourier transform to compute the matrix-vector product $Y \boldsymbol{a}^\top$, $\boldsymbol{a} \in \mathbb{R}^s$, in $\mathcal{O}(N \log N)$ operations and at most $s-1$ extra additions. The proposed method can be applied to lattice rules, polynomial lattice rules and a certain type of Korobov $p$-set. The approach is illustrated computationally by three numerical experiments. The first test considers the generation of points with normal distribution and general covariance matrix, the second test applies QMC to high-dimensional, affine-parametric, elliptic partial differential equations with uniformly distributed random coefficients, and the third test addresses Finite-Element discretizations of elliptic partial differential equations with high-dimensional, log-normal random input data. All numerical tests show a significant speed-up of the computation times of the fast QMC matrix method compared to a conventional implementation as the dimension becomes large

Analysis and test of superplastically formed titanium hat-stiffened panels under compression

Four hat-stiffened titanium panels with two different stiffener configurations were fabricated by superplastic forming/weld brazing and tested under a moderately heavy compressive load. The panels had the same overall dimensions but differed in the shape of the hat-stiffener webs; three panels had stiffeners with flat webs and the other panel had stiffeners with beaded webs. Analysis indicated that the local buckling strain of the flat stiffener web was considerably lower than the general panel buckling strain or cap buckling strain. The analysis also showed that beading the webs of the hat stiffeners removed them as the critical element for local buckling and improved the buckling strain of the panels. The analytical extensional stiffness and failure loads compared very well with experimental results

Operational parameters for the superconducting cavity maser

Tests of the superconducting cavity maser (SCM) ultra-stable frequency source have been made for the first time using a hydrogen maser for a frequency reference. In addition to characterizing the frequency stability, the sensitivity of the output frequency to several crucial parameters was determined for various operating conditions. Based on this determination, the refrigeration and thermal control systems of the SCM were modified. Subsequent tests showed substantially improved performance, especially at the longest averaging times