Structural identifiability of surface binding reactions involving heterogeneous analyte : application to surface plasmon resonance experiments

Binding affinities are useful measures of target interaction and have an important role in understanding biochemical reactions that involve binding mechanisms. Surface plasmon resonance (SPR) provides convenient real-time measurement of the reaction that enables subsequent estimation of the reaction constants necessary to determine binding affinity. Three models are considered for application to SPR experiments—the well mixed Langmuir model and two models that represent the binding reaction in the presence of transport effects. One of these models, the effective rate constant approximation, can be derived from the other by applying a quasi-steady state assumption. Uniqueness of the reaction constants with respect to SPR measurements is considered via a structural identifiability analysis. It is shown that the models are structurally unidentifiable unless the sample concentration is known. The models are also considered for analytes with heterogeneity in the binding kinetics. This heterogeneity further confounds the identifiability of key parameters necessary for reliable estimation of the binding affinit

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

K-12 Mathematics: What Should Students Learn and When Should They Learn it?

The conference was held at the National Rural Electric Cooperative Conference Center on February 5-6, 2007 and Organized by the Center for the Study of Mathematics Curriculum. The conference was also webcast and archived sessions are available at: http://cltnet.org/cltnet/misc/csmcmath07/In the fall of 2006 several influential national groups (Achieve, American Statistical Association, College Board, and the National Council of Teachers of Mathematics) developed and released documents that recommended curriculum standards or focal points for K-12 mathematics. In response, the Center for the Study of Mathematics Curriculum (CSMC) organized several meetings of representatives of each group to discuss the nature of their work and possible collaboration. As a result of those meetings, the groups agreed to cosponsor a national conference to highlight the recommendations and to engage “users” of standards (state and district curriculum specialists, textbook and assessment publishers, K-12 district and teacher leaders, and representatives from higher education and business) in discussions about implications for their work.This report is based on the work of the Center for the Study of Mathematics Curriculum, supported by the National Science Foundation under Grant No. ESI-0333879. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation

A Case Study of Culturally Responsive Teaching in Middle School Mathematics

This dissertation is a case study of culturally responsive teaching in middle school mathematics. The study contributes to the emerging definition of learning experiences in mathematics that support all middle school students through the use of culturally responsive teaching strategies. These experiences are organized to actualize the balance between accommodating the individual student’s culture, needs and interests and supporting the attainment of appropriate mathematics learning goals. The research site was chosen because of its diverse and changing student demographics, and because of the achievement and growth rates of these students. Data collected from classroom observations of three middle school mathematics teachers at this site; observations of school activities; interviews with these teachers and their school leaders; a student focus group with representative students from the three teacher’s classrooms; and, a review of school and classroom artifacts contributed to the findings for this study. All data were collected during the period of December 2008 through March 2009. Results of this study give evidence of the significance of a strong school culture that supports the development of understandings in mathematics for each learner. These three teachers provided learners with regular and comprehensive opportunities to learn by organizing strong learning communities in their classrooms; by encouraging students to help each other; by adjusting their instruction based on students’ needs; and, by establishing and acting on challenging learning goals for all students. Additionally, the teachers gave students regular opportunities to make sense of the mathematics they were studying and to share leadership and decision-making within the strong learning community that they established and maintained with each section of students that they taught. Leaders and others may believe that the components of culturally responsive teaching do not apply in a mathematics classroom. Thus, the findings from this study further educators’ reflection about these ideas. The results of culturally responsive teaching include not only deeper learning of content, but also, an opportunity for students’ to learn to value their own and each other’s differing perspectives that supports the development of stronger democratic citizenship

Workforce Preparedness

[Excerpt] Concern about slackening productivity growth and deteriorating competitiveness has resulted in a new public focus on the skills and education of frontline workers. The introduction of Lean Production and Total-Quality- Management is apparently raising the cognitive demands placed on blue collar workers (Womack, Jones and Roos, 1990). Increasingly they are working in production cells in which every member of the team is expected to learn every job and to take on responsibilities formerly the sole province of supervisors, specialized technicians and industrial engineers. Higher order thinking and problem solving skills are believed to be in particularly short supply so much attention has been given to mathematics and science education because it is thought that these subjects are particularly relevant to their development

The Impact of Number Talks on Kindergarten Math Growth in a Large Private Independent School

This mixed methods action research study examined the implementation process and impact of a 9-week Number Talk intervention to build number sense in kindergarten students. Fifty-eight kindergarten students engaged in daily Number Talk lessons for 9 weeks. Qualitative data were collected to evaluate the strengths and challenges of the implementation process with the teacher participants through a twice weekly observation tool and through bi-weekly math professional learning community discussions. The qualitative data gathered were coded for themes using Tesch’s Eight Steps for coding information. The data allowed me to evaluate the implementation process to determine if Number Talks were implemented with fidelity during the implementation process. Quantitative data were collected with the Number Sense Screener (NSS) assessment instrument. The students were given the NSS as a pretest before Number Talk implementation and as a posttest after the 9 weeks of implementation. A paired sample t test was utilized to analyze the pre and posttest results. The t test was completed using the results of the pre- and post-NSS, and analysis showed a significant gain in the mean score for the targeted group. The qualitative and quantitative data were utilized by the participating teachers and me to create an action plan for future Number Talk implementation within the school. The action plan includes four steps for implementation to support lesson planning, implementation reflection, implementation walk-throughs, and formative assessment of number sense

Interpretable Graph Networks Formulate Universal Algebra Conjectures

The rise of Artificial Intelligence (AI) recently empowered researchers to investigate hard mathematical problems which eluded traditional approaches for decades. Yet, the use of AI in Universal Algebra (UA) -- one of the fields laying the foundations of modern mathematics -- is still completely unexplored. This work proposes the first use of AI to investigate UA's conjectures with an equivalent equational and topological characterization. While topological representations would enable the analysis of such properties using graph neural networks, the limited transparency and brittle explainability of these models hinder their straightforward use to empirically validate existing conjectures or to formulate new ones. To bridge these gaps, we propose a general algorithm generating AI-ready datasets based on UA's conjectures, and introduce a novel neural layer to build fully interpretable graph networks. The results of our experiments demonstrate that interpretable graph networks: (i) enhance interpretability without sacrificing task accuracy, (ii) strongly generalize when predicting universal algebra's properties, (iii) generate simple explanations that empirically validate existing conjectures, and (iv) identify subgraphs suggesting the formulation of novel conjectures

Using a peer supervision model to implement recommendations of the NCTM standards in algebra classes in an urban school system.

The NCTM Standards have established new directions for math teaching and learning. The problem of implementation, particularly in urban school systems remains. This study focuses on an urban school system in Western Massachusetts. Of particular interest to the researcher is the lack of success of students in Algebra I. This particular course has traditionally been the pivotal course that determines if a student gets into and remains in the College Preparatory sequence. The fact that too many minority and women students are left out of these choices due to lack of mathematics preparation can be traced back to being left out of algebra in high school. What happened to these students? Why were they left out? Why is the failure rate nearly 45 percent in Algebra I in this public school system? Teachers working in the traditional classroom structure of the current school setting are isolated without opportunities to work in cooperation with other teachers. Without a process for sharing ideas and a method to support new teaching strategies, it will not be possible for the vision of the Standards to become a reality. The challenge for a supervisor is to bring the message of the Standards to the secondary mathematics teachers in an urban school system. This study develops and tests a supervision model, based on peer supervision, for the implementation of teaching strategies recommended in the Standards. The findings of this study show that peer supervision can help school systems bring new teaching strategies, like cooperative learning and hands-on activities, into its Algebra I classrooms