Connecting proportionality and slope: middle school students' reasoning about steepness

Thesis (Ed.D.)--Boston UniversityThis study investigates the relationship between middle school students' proportional reasoning abilities and their understanding of steepness, since steepness may be a key developmental understanding that students need in order to understand slope in algebra. This study uses mixed methods, with a large-scale survey and one-on-one interviews with students. The large-scale survey involves two tests: an adapted version of the Ratio and Proportion Test administered in England for the Concepts in Secondary Mathematics & Science (CSMS) and Increasing Competence & Confidence in Algebra and Multiplicative Structures (ICCAMS) projects, and a steepness test which I created and for which I established content validity and test-retest reliability. The problems on the steepness test involve a comparison of two roofs, two staircases, or two lines. The problems vary by structural difficulty based upon the values of the slopes involved. Analysis of 413 research participants' survey data indicate that approximately 25% of the variability in students' scores on the Steepness test is explained by their performance on the Ratio and Proportion Test. Linear regression has shown that there is a positive correlation between participants' proportional reasoning abilities and their abilities to solve steepness problems. Using paired t-tests, there is evidence of significant difference in students' performance on the three contexts. The difficulties of contexts in the order of difficulty from hardest to easiest are: stairs, roofs, lines. Analyses of interviews with middle school students solving steepness problems indicate that they do use different strategies to solve problems based upon their proportional reasoning levels. Participants who attained higher levels of proportional reasoning used quantitative considerations of two measurements, norming, and rates and ratios to determine relative steepness more frequently than participants who attained lower levels of proportional reasoning. It was concluded that participants' abilities to solve steepness problems are related to their abilities to reason proportionally. The findings of this research contribute to literature on early algebraic reasoning that explores ways in which algebraic topics such as slope can be made accessible to students prior to their formal studies of algebra.2031-01-0

Making Algebra More Accessible: How Steep Can it be for Teachers?

Teacher educators need to support middle grades teachers in developing mathematical knowledge for teaching algebraic concepts. In particular, teachers should become familiar with possible introductions and sequencing to the concept of slope, and common middle school students’ limited conceptions about measuring the steepness of an incline. Steepness can be expressed directly in terms of an angle or indirectly as a slope. Encouraging middle school students to find a measure of steepness using a ratio may help support students’ transition to multiplicative thinking. This mixed – methods study explores middle school students’ responses in solving a comparison problem involving the steepness of two inclines, in order to gain insight into common student strategies. The quantitative portion of the study involved written surveys distributed to 256 Grade 7 participants in the United States. We examined the frequency and types of solutions offered by these participants. We found that 27% of the participants provided an incorrect solution which was consistent with additive reasoning. The qualitative portion of this study consisted of small group interviews of 19 Grade 7 participants, who were conflicted in the different solutions they produced from using additive reasoning and their geometric knowledge

Syntenin-1 is a promoter and prognostic marker of head and neck squamous cell carcinoma invasion and metastasis.

Metastasis represents a key factor associated with poor prognosis of head and neck squamous cell carcinoma (HNSC). However, the underlying molecular mechanisms remain largely unknown. In this study, our liquid chromatography with tandem mass spectrometry analysis revealed a number of significantly differentially expressed membrane/membrane-associated proteins between high invasive UM1 and low invasive UM2 cells. One of the identified membrane proteins, Syntenin-1, was remarkably up-regulated in HNSC tissues and cell lines when compared to the controls, and also over-expressed in recurrent HNSC and high invasive UM1 cells. Syntenin-1 over-expression was found to be significantly associated with lymph node metastasis and disease recurrence. HNSC patients with higher syntenin-1 expression had significantly poorer long term overall survival and similar results were found in many other types of cancers based on analysis of The Cancer Genome Atlas data. Finally, knockdown of syntenin-1 inhibited the proliferation, migration and invasion of HNSC cells, and opposite findings were observed when syntenin-1 was over-expressed. Collectively, our studies indicate that syntenin-1 promotes invasion and progression of HNSC. It may serve as a valuable biomarker for lymph node metastasis or a potential target for therapeutic intervention in HNSC

Endangered Species: A Population Simulation

This article describes an activity which incorporates biology and mathematics at the secondary level. A hands-on simulation of a decreasing animal population is provided. Students collect their data, record it in a table, and analyze it using technological tools. The mathematics used in this activity includes rates; fitting functions to data; creating and interpreting linear, quadratic, and exponential functions; and using functions to solve real-world problems

DanceSport and Power Values

DanceSport is a competitive form of ballroom dancing. At a DanceSport event, couples perform multiple dances in front of judges. This paper shows how a goal for a couple and the judges\u27 evaluations of the couple\u27s dance performances can be used to formulate a weighted simple game. We explain why couples and their coaches may consider a variety of goals. We also show how prominent power values can be used to measure the contributions of dance performances to achieving certain goals. As part of our analysis, we develop novel visual representations of the Banzhaf and Shapley-Shubik index profiles for different thresholds. In addition, we show that the quota paradox is relevant for DanceSport events

Higher education and political transition : the University of Macau in comparative perspective

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Metabolomic analysis of human oral cancer cells with adenylate kinase 2 or phosphorylate glycerol kinase 1 inhibition.

The purpose of this study was to use liquid chromatography-mass spectrometry (LC-MS) with XCMS for a quantitative metabolomic analysis of UM1 and UM2 oral cancer cells after knockdown of metabolic enzyme adenylate kinase 2 (AK2) or phosphorylate glycerol kinase 1 (PGK1). UM1 and UM2 cells were initially transfected with AK2 siRNA, PGK1 siRNA or scrambled control siRNA, and then analyzed with LC-MS for metabolic profiles. XCMS analysis of the untargeted metabolomics data revealed a total of 3200-4700 metabolite features from the transfected UM1 or UM2 cancer cells and 369-585 significantly changed metabolites due to AK2 or PGK1 suppression. In addition, cluster analysis showed that a common group of metabolites were altered by AK2 knockdown or by PGK1 knockdown between the UM1 and UM2 cells. However, the set of significantly changed metabolites due to AK2 knockdown was found to be distinct from those significantly changed by PGK1 knockdown. Our study has demonstrated that LC-MS with XCMS is an efficient tool for metabolomic analysis of oral cancer cells, and knockdown of different genes results in distinct changes in metabolic phenotypes in oral cancer cells