Don’t I Know You? A Misstep in Teaching Mathematics with and for Social Justice in a Rural Context

In this paper, I document my own struggles and insights in moving toward a pedagogy of teaching mathematics with and for social justice within a rural high school. Teaching mathematics for social justice has been presented as a way to address the inequities present in the classroom, and the world at large, by having students work with mathematics to question and analyze inequities in their world (Gutstein, 2006). Inclusive education has been presented as a means for providing all students, regardless of their needs, abilities and interests, access to engaging content in the classroom (Villa & Thousand, 2005). These approaches to education can be summarized as teaching with and for social justice (Wager, 2008). I offer teaching mathematics with and for social justice as a way to make mathematics meaningful within a rural setting

A model for fire‐induced sediment yield by dry ravel in steep landscapes

Sediment flux from hillslopes to channels commonly increases following wildfires, with implications for the carbon cycle, river habitats, and debris-flow hazards. Although much of this material is transported via dry ravel, existing ravel models are not applicable to hillslopes with gradients greater than the angle of repose, which can constitute the majority of mountainous terrain. To fill this knowledge gap, we develop a continuity model for sediment storage by vegetation dams on steep hillslopes to predict sediment yields following wildfire. The maximum volume of sediment stored prior to wildfire is set to be a function of vegetation density, the capacity of plants to impound sediment, and the contributing hillslope area. Time is required after fire to establish vegetation and replenish hillslope sediment storage, which introduces vegetation regrowth rate, soil production rate, and fire recurrence interval as important variables that affect ravel yield. Model results for the San Gabriel Mountains, California, predict that sediment yield can increase by several orders of magnitude following fire. These results are consistent with field data of ravel yield (~30 mm per contributing area of hillslope in 5 months) we collected following the 2009 Station Fire, as well as postfire sediment flux recorded by 93 debris basins. In contrast to previous work, our model shows that heightened postfire sediment yields can be explained by a change in hillslope sediment storage independent of major changes in the soil production rate and landscape form over geomorphic timescales

Commercializing Biorefinery Technology: A Case for the Multi-Product Pathway to a Viable Biorefinery

While there may be many reasons why very interesting science ideas never reach commercial practice, one of the more prevalent is that the reaction or process, which is scientifically possible, cannot be made efficient enough to achieve economic viability. One pathway to economic viability for many business sectors is the multi-product portfolio. Research, development, and deployment of viable biorefinery technology must meld sound science with engineering and business economics. It is virtually axiomatic that increased value can be generated by isolating relatively pure substances from heterogeneous raw materials. Woody biomass is a heterogeneous raw material consisting of the major structural components, cellulose, lignin, and hemicelluloses, as well as minor components, such as extractives and ash. Cellulose is a linear homopolymer of D-glucopyrano-units with β-D(1®4) connections and is the wood component most resistant to chemical and biological degradation. Lignin is a macromolecule of phenylpropanoid units, second to cellulose in bio-resistance, and is the key component that is sought for removal from woody biomass in chemical pulping. Hemicelluloses are a collection of heteropolysaccharides, comprised mainly of 5- and 6-carbon sugars. Extractives, some of which have high commercial value, are a collection of low molecular weight organic and inorganic woody materials that can be removed, to some extent, under mild conditions. Applied Biorefinery Sciences, LLC (a private, New York, USA based company) is commercializing a value-optimization pathway (the ABS Process™) for generating a multi-product portfolio by isolating and recovering homogeneous substances from each of the above mentioned major and minor woody biomass components. The ABS Process™ incorporates the patent pending, core biorefinery technology, “hot water extraction”, as developed at the State University of New York College of Environmental Science and Forestry (SUNY-ESF). Hot water extraction in the absence of mineral acids and bases is preferred because of its ability to generate multiple high value output products without chemical input, recovery, or disposal costs. Instead of added chemicals in the cooking phase, the ABS Process™ relies upon an autocatalytic reaction in which acetyl groups, bound through an ester linkage to hemicellulose chains, are hydrolyzed at high temperature in water. The resulting acidic conditions (final pH ~3.5) and temperatures of 160–170 °C permit further solubilization and diffusion of oligomeric 5- and 6-carbon sugars, acetic acid, aromatic substances, monomeric sugars, and other trace compounds into the extract solution. These conditions also avoid extensive degradation of monosaccharides, enabling membrane fractionation and other chemical separation techniques to be used in the following separations. A range of separation techniques are applied on the extract solution to isolate and purify fermentable sugars, acetic acid, lignin, furfural, formic acid, other hemicellulose related compounds, lignin, lignin degradation products, and phenolic extractives for commercial sale. The extracted lignocellulosic biomass, with reduced hemicellulose content and is thus less heterogeneous, carries the value-added advantages of: (1) enhanced product characteristics, and (2) reduced energy and chemical manufacturing costs. Thus, by fractionating woody biomass into more homogeneous substances, the ABS Process™ holds potential as an economically viable pathway for capturing sustainable, renewable value not currently realized from lignocellulosic biomass

An Exponential Growth Learning Trajectory: Students' Emerging Understanding of Exponential Growth Through Covariation

This article presents an Exponential Growth Learning Trajectory (EGLT), a trajectory identifying and characterizing middle grade students' initial and developing understanding of exponential growth as a result of an instructional emphasis on covariation. The EGLT explicates students' thinking and learning over time in relation to a set of tasks and activities developed to engender a view of exponential growth as a relation between two continuously covarying quantities. Developed out of two teaching experiments with early adolescents, the EGLT identifies three major stages of students' conceptual development: prefunctional reasoning, the covariation view, and the correspondence view. The learning trajectory is presented along with three individual students' progressions through the trajectory as a way to illustrate the variation present in how the participants made sense of ideas about exponential growth

Quantifying exponential growth: Three conceptual shifts in coordinating multiplicative and additive growth

This article presents the results of a teaching experiment with middle school students who explored exponential growth by reasoning with the quantities height (y) and time (x) as they explored the growth of a plant. Three major conceptual shifts occurred during the course of the teaching experiment: (1) from repeated multiplication to initial coordination of multiplicative growth in y with additive growth in x; (2) from coordinating growth in y with growth in x to coordinated constant ratios (determining the ratio of f(x(2)) to f(x(1)) for corresponding intervals of time for (x(2)) > 1), and (3) from coordinated constant ratios to within-units coordination for corresponding intervals of time for (x(2)) < 1. Each of the three shifts is explored along with a discussion of the ways in which students' mathematical activity supported movement from one stage of understanding to the next. These findings suggest that emphasizing a coordination of multiplicative and additive growth for exponentiation may support students' abilities to flexibly move between the covariation and correspondence views of function. (C) 2015 Elsevier Inc. All rights reserved