Research on Teaching and Learning Probability

This book summarizes the vast amount of research related to teaching and learning probability that has been conducted for more than 50 years in a variety of disciplines. It begins with a synthesis of the most important probability interpretations throughout history: intuitive, classical, frequentist, subjective, logical propensity and axiomatic views. It discusses their possible applications, philosophical problems, as well as their potential and the level of interest they enjoy at different educational levels. Next, the book describes the main features of probabilistic thinking and reasoning, including the contrast to classical logic, probability language features, the role of intuitions, as well as paradoxes and the relevance of modeling. It presents an analysis of the differences between conditioning and causation, the variability expression in data as a sum of random and causal variations, as well as those of probabilistic versus statistical thinking. This is followed by an analysis of probability’s role and main presence in school curricula and an outline of the central expectations in recent curricular guidelines at the primary, secondary and high school level in several countries. This book classifies and discusses in detail the three different research periods on students’ and people’s intuitions and difficulties concerning probability: early research focused on cognitive development, a period of heuristics and biases programs, and the current period marked by a multitude of foci, approaches and theoretical frameworks

Stacked porphyrin—quinone triads as models for the primary charge-separation in photosynthesis

We present time-resolved fluorescence decay measurements on a cylophane-bridged donor-acceptor triad containing a porphyrin and two different quinones in the order porphyrin—quinone1—quinone2. The time scale of photoinduced electron transfer, as well as its energetics, suggests a strong analogy with the primary charge-separation in photosynthetic reaction centers

Energy Gap and Temperature Dependence of Photoinduced Electron Transfer in Porphyrin-Quinone Cyclophanes

We have investigated intramolecular photoinduced charge separation and recombination in a series of cyclophanebridged porphyrinquinone systems by means of time−resolved fluorescence decay measurements. Rates of charge separation have been determined as a function of the free energy change of the reaction, of the polarity of the solvent, and of the temperature. In some systems a long−lived fluorescence is observed which is attributed to a thermal repopulation of the initially excited state from the charge transfer state. This delayed fluorescence allows the calculation of the rate of recombination in these cases. The observation of delayed fluorescence for a particular donor−acceptor compound in some solvent serves as a reference for the reaction free energy of the respective charge separation (AG, N 0 eV). The free energy change in other systems is estimated by correcting for differences in the redox potentials of the respective porphyrins and quinones. Electronic couplings and reorganization energies are determined by globally fitting standard rate expressions as a function of the free energy change to the experimental rate data. Three different kinds of fits are performed by (a) using both charge separation and recombination within the nonadiabatic approximation, (b) allowing for Landau−Zener adiabaticity corrections, and (c) fitting rates of charge separation (in the normal region) only. A particular focus lies in the specific effects imposed by the compact structure of the porphyrin−uinone cyclophanes. It is shown that electron transfer in these systems is nonadiabatic and dominated by intramolecular reorganization whereas the influence of the surrounding solvent is minimized by the close packing of electron donor and accepto

Absence of dielectric relaxation effects on intramolecular electron transfer in porphyrin-quinone cyclophanes

We have measured photoinduced intramolecular electron transfer rates ket in two porphyrin-dichloroquinone cyclophanes varying the polarity, the temperature, and the dielectric relaxation time of the solvent. The reaction is most likely activationless or in the inverted region in all solvents. The adiabaticity parameter k = 4πV2eff/khλsτℓ for dynamic dielectric solvent control is estimated to cover the range 0.2 ⪕k ⪕ 400. Nevertheless the rate ket was found to be almost independent of the solvent properties (ket > 5 × 1011 s−1) casting doubt on current ideas of dielectric relaxation effects on electron transfer

Electron transfer in porphyrin-quinone cyclophanes

We have measured the fluorescence properties of five cyclophanes containing, in a vertical face-to-face arrangement, a porphyrin as electron donor and one of several substituted quinones as electron acceptor with varying acceptor strength in a series of solvents covering a wide range of polarity. The results of time-resolved fluorescence decay measurements are interpreted in terms of a very fast intramolecular charge separation within 1–10 ps with driving forces decreasing from −ΔGcs = 0.7 to about 0 eV and a charge recombination in the order of 10–200 ps