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Describe taking a logarithm as repeated division to allow students to build on what they know about division, instead of relying solely on the exponential definition.

Eine angemessene Diagnose scheint zentral fur erfolgreichen Unterricht zu sein. Wie aber lasst sich charakterisieren, dass eine Lernstandeinschatzung und die Weiterarbeit aufeinander bezogen sind? Fur unsere explorative Studie setzten sich Personen mit und ohne mathematikdidaktische Expertise mit Text-Bild-Vignetten von Grundschulkindern auseinander: Sie schatzten in ausgewahlten arithmetischen Bereichen die Lernstande der Kinder ein und machten darauf basierend Vorschlage fur die Weiterarbeit. Wie die inhaltsanalytische Auswertung dieser Dokumente zeigt, sind fur die Expertinnen und Experten so genannte „mathematikdidaktische Konzepte“ handlungsleitend. Diese ermoglichen ihnen, ihre Einschatzungen der Lernstande fur die Planung der unterrichtlichen Weiterarbeit oder der Einzelforderung fruchtbar zu machen. Dagegen sind die mathematikdidaktischen Konzepte der Novizinnen und Novizen kaum belastbar, ihre Konzepte ubernehmen keine Vermittlungsfunktion zwischen Diagnose und Weiterarbeit.

Logarithms have a reputation for being difficult and inaccessible. As an analysis of their historical, mathematical and educational background suggests, this problem might be due to the way in which logarithms are interpreted and explained in textbooks: as the inverse of exponents. If this conclusion is right, additional interpretations of logarithms are required. By combining the theoretical construct of ‘Grundvorstellungen’ (translated as ‘basic models’) and the distinction between operational and structural conceptions, I identify and elaborate four interpretations of logarithms: (i) the basic model of ‘multiplicative measuring’, (ii) the basic model of ‘counting the number of digits’, (iii) the basic model of ‘decreasing the hierarchy level’, and (iv) the basic model of ‘inverse exponent’. Three models (i–iii) reflect operational conceptions and interpret logarithms in contexts familiar to students. In combination with (iv), a structural basic model, this paper argues on a theoretical level that they could help to make logarithms accessible and understandable to students. Following the tradition of ‘Stoffdidaktik’ (‘subject-matter didactics’), the study thus aims to unpack some of the content knowledge required for the teaching of logarithms.

International audience; We present a study of plasma conditions in the atmospheres of the Hot Jupiters HD 209458b and HD 189733b and for an HD 209458b like planet at orbit locations between 0.2 and 1 au around a Sun-like star. We discuss how these conditions influence the radio emission we expect from their magnetospheres. We find that the environmental conditions are such that the cyclotron maser instability (CMI), the process responsible for the generation of radio waves at magnetic planets in the Solar system, most likely will not operate at Hot Jupiters. Hydrodynamically expanding atmospheres possess extended ionospheres whose plasma densities within the magnetosphere are so large that the plasma frequency is much higher than the cyclotron frequency, which contradicts the condition for the production of radio emission and prevents the escape of radio waves from close-in exoplanets at distances

Mathematiklehrerinnen und -lehrer lesen Schulerprodukte je nach Verwendungszweck aus unterschiedlichen Perspektiven. In dieser Studie wurde mit Repertory Grids erhoben, wie zwolf Lehramtsstudierende und Lehrpersonen Schulerprotokolle lesen und miteinander vergleichen. Aus den Grids wurden ein Kategoriensystem entwickelt und sechs unterschiedliche Perspektiven rekonstruiert. Sie erfassen, dass die Schulerprotokolle entweder beschrieben oder gewertet wurden und sich die Lekture auf formale, inhaltliche oder personale Aspekte der Texte richtete. Damit sind Lehrpersonen in der Lage, Schulertexte aus vielfaltigen Perspektiven zu lesen, besonders wenn sie dazu angehalten werden. Dies ist bedeutsam, weil ein schulerorientierter, differenzierender Unterricht eine Vielfalt von Perspektiven bei den Lehrenden voraussetzt.

Certain three-dimensional convex bodies have a counterintuitive property; they are of constant width. In this particular respect they resemble a sphere without being one. Discovered a century ago, Meissner’s bodies have often been conjectured to minimize volume among bodies of given constant width. However, this conjecture is still open. We draw attention to this challenging and beautiful open problem by presenting some of its history and its recent development.

This retrospective study aimed to evaluate the feasibility and effectiveness of radiofrequency ablation (RFA) in patients with solitary kidney for the treatment of renal cell carcinoma (RCC). Within 2 years 10 patients (seven males, three females; age 65 ± 8 years) were treated. All patients had a history of nephrectomy of the contralateral kidney. The indications for RFA were inoperability or high probability of complete renal failure after surgical enucleation of the tumor. 13 tumors with a size between 1.9 and 4.2 cm (average 2.7 cm) were treated. In patients with a tumor diameter larger than 2.5 cm a transarterial embolization was performed prior to RFA to reduce heat sink effect and risk of bleeding. Therapeutical success was defined as a lack of contrast enhancement in follow up examinations and shrinking of the treated area. Furthermore all patients’ renal function was monitored. RFA of renal tumors under CT-fluoroscopy was feasible in all patients. Within the follow up (3 and 24 months) no tumor recurrence or major complication was detected. One patient developed another RCC and was successfully treated with a second RF-ablation. None of the patients developed renal failure with the need of hemodialysis. In one of the patients a hemorrhage into the surrounding tissue was noticed, which stopped spontaneously. RFA is a valuable and effective therapeutical option in patients with solitary kidney suffering from inoperable renal cell carcinoma. The complication rate is small and an excellent tumor control can be achieved without deterioration of the renal function.

Ischemia -modified albumin was regarded as an early marker of cardiac ischemia. On the other hand, it has been reported that increased ischemia-modified albumin levels are associated with unstable plaque processes like percutaneous coronary intervention, acute coronary syndrome or myocardial infarction. This prospective study aimed to investigate the role of ischemia-modified albumin in patients with peripheral vascular disease undergoing peripheral vascular intervention, a plaque-altering procedure without evidence of tissue ischemia. Peripheral vascular intervention was performed in 21 consecutive patients (68.2+/-13.3 years) with typical leg claudication and documented peripheral vascular disease. Additionally, 96 consecutive patients (66+/-12.0 years) undergoing routine exercise stress test for the exclusion of functionally relevant coronary artery disease were defined as controls. It was assumed that in the latter patients no unstable plaque-altering processes were present. Blood samples were drawn before, and 30 min and 3 h after, revascularization in the peripheral vascular intervention group, as well as before, and 30 min and 3 h after, maximum stress testing in the control group, respectively. Ischemia-modified albumin levels were analyzed using the albumin cobalt-binding test. In patients undergoing peripheral vascular intervention, ischemia-modified albumin increased from 116.6+/-19.1 U/ml at baseline to 132.0+/-19.3 U/ml 30 min after intervention (+14.4+/-15.7%, P

During the past two decades, digital radiography has supplanted screen-film radiography in many radiology departments. Today, manufacturers provide a variety of digital imaging solutions based on various detector and readout technologies. Digital detectors allow implementation of a fully digital picture archiving and communication system, in which images are stored digitally and are available anytime. Image distribution in hospitals can now be achieved electronically by means of web-based technology with no risk of losing images. Other advantages of digital radiography include higher patient throughput, increased dose efficiency, and the greater dynamic range of digital detectors with possible reduction of radiation exposure to the patient. The future of radiography will be digital, and it behooves radiologists to be familiar with the technical principles, image quality criteria, and radiation exposure issues associated with the various digital radiography systems that are currently available.

We study the origin and escape of catastrophically outgassed volatiles (H$_2$O, CO$_2$) from exomoons with Earth-like densities and masses of $0.1M_{\oplus}$, $0.5M_{\oplus}$ and $1M_{\oplus}$ orbiting an extra-solar gas giant inside the habitable zone of a young active solar-like star. We apply a radiation absorption and hydrodynamic upper atmosphere model to the three studied exomoon cases. We model the escape of hydrogen and dragged dissociation products O and C during the activity saturation phase of the young host star. Because the soft X-ray and EUV radiation of the young host star may be up to $\sim$100 times higher compared to today's solar value during the first 100 Myr after the system's origin, an exomoon with a mass $ < 0.25M_{\oplus}$ located in the HZ may not be able to keep an atmosphere because of its low gravity. Depending on the spectral type and XUV activity evolution of the host star, exomoons with masses between $\sim0.25-0.5M_{\oplus}$ may evolve to Mars-like habitats. More massive bodies with masses $ > 0.5M_{\oplus}$, however, may evolve to habitats that are a mixture of Mars-like and Earth-analogue habitats, so that life may originate and evolve at the exomoon's surface., 27 pages, 3 figures

Ab dem ersten postoperativen Jahr ist die Transplantatvaskulopathie die Hauptursache fur Morbiditat und Mortalitat nach orthotoper Herztransplantation. Die Wahl der adaquaten Revaskularisationsstrategie – kathetergestutzte Intervention, aortokoronare Bypassoperation oder erneute Herztransplantation – ist von vielfaltigen Faktoren, unter anderem Koronarmorphologie, linksventrikulare Funktion, Komorbiditat, Verfugbarkeit von Graftmaterial und Spenderorganen, abhangig zu machen. Katheterinterventionen an Stenosen des Hauptstammes der linken Koronararterie sind moglich, jedoch mit einer niedrigeren primaren Erfolgsrate und einer hoheren Komplikationsrate mit signifikant erhohter Mortalitat verbunden. Wir berichten uber einen vor 9 Jahren herztransplantierten Patienten, der eine schwere Transplantatvaskulopathie mit hohergradiger distaler Hauptstammstenose entwickelte. Aufgrund des Alters, der Koronarmorphologie und signifikanter Komorbiditat war der Patient fur eine aortokoronare Bypassoperation oder eine erneute Herztransplantation nicht geeignet. Wir fuhrten eine erfolgreiche, mit intravaskularem Ultraschall gefuhrte direktionale koronare Atherektomie und Stentimplantation einer distalen Stenose des ungeschutzten Hauptstammes durch. Die Kasuistik soll zeigen, dass auch bei Patienten, die fur eine operative Revaskularisationsstrategie nicht in Frage kommen, komplexe Angioplastieprozeduren mit kalkulierbarem periprozeduralen Risiko und sehr gutem primaren Resultat vorgenommen werden konnen.

Der Logarithmus birgt fur die meisten Gymnasiastinnen und Gymnasiasten grose Verstandnisprobleme. Trotz vieler Schulstunden verdichtet sich der Logarithmus nur bei wenigen SuS zu einer gedanklichen Entitat, mit der verstandig argumentiert und gearbeitet werden kann. Meistens bleibt er bis zum Schluss eine black box, mit der ein Gefuhl des Sperrigen oder gar der Bedeutungsarmut einhergeht. Im ersten Teil meines Beitrags mochte ich in einer Problemanalyse den moglichen Grunden dafur nachgehen. In der Literatur wird immer wieder auf die anspruchsvolle Schreibweise oder auf den wenig selbsterklarenden oder gar sinnstiftenden Begriff „Logarithmus“ hingewiesen. Verschiedene Autoren gehen damit unterschiedlich um. Fur die Schulerschwierigkeiten mindestens so sehr verantwortlich durfte aber auch die inverse Begriffsfassung des Logarithmus sein. Im zweiten Teil meines Beitrags mochte ich den vorlaufigen Ansatz eines verstandigen Logarithmuslernens vorstellen, der sich das Konzept der Grundvorstellungen zu Nutze macht und stellenweise in Analogie zur Behandlung des „Wurzel“-Themas verlauft: (1) Logarithmus als Rechenoperation („Werkzeug zur Berechnung einer Zahl“): → Grundvorstellung der verallgemeinerten Stellenanzahl (2) Logarithmus als Termumformung („Werkzeug zum Losen von Gleichungen“): → Grundvorstellung des Reduzierens der Operationsstufe (3) Logarithmus als Funktion („Werkzeug zum Modellieren von Wachstumsprozessen“): → Grundvorstellung einer extrem langsam wachsenden, nichtlinearen Funktion. Zur Verdeutlichung werden im dritten Teil eine mogliche unterrichtliche Umsetzung vorgestellt. Aspekte wie Schulerschwierigkeiten oder die Tragfahigkeit von (1) und (3) kann ich durch Ausschnitte aus Schulerdokumenten illustrieren. Andere Aspekte mussen hypothetisch bleiben.

Unter dem Begriff der interventionellen Onkologie werden i. Allg. nicht oder wenig invasive Eingriffe verstanden, die sowohl der Abklarung und/oder Sicherung der Diagnose als auch der Therapie dienen. Der Einsatz interventioneller Masnahmen ist in hohem Mase von der verfugbaren Technik und der Expertise der durchfuhrenden Personen abhangig und ist derzeit einer raschen Weiterentwicklung und Expansion unterworfen. Die interventionelle Onkologie wird wesentlich gepragt von der interventionellen Radiologie. Endoluminale Interventionen im Gastrointestinaltrakt sind hingegen eine Domane der gastrointestinalen Endoskopie.

Purpose: This study aimed to determine the success and complication rates of radiofrequency ablation (RFA) in treatment of osteoid osteoma (OO) and duration of pain relief. Furthermore value of bone biopsy prior to the RFA was evaluated. Materials and methods: Within 61 months 39 patients (23 male, 16 female, 7–53 years, mean 18.7 years, median 17 years) suffering from osteoid osteoma were treated. Lesions were located in femur (n = 20), tibia (n = 10), spine (n = 5), humerus (n = 1), radius (n = 1), talus (n = 1) and pelvis (n = 1). In children, RFA was performed under general anaesthesia, in adults conscious sedation was preferred. In 29 of 39 (74%) lesion biopsies were obtained. Cooling of skin was performed in OOs located in bones with minor soft tissue covering (tibia, radius) and saline flushing via an additional needle was performed if the OO was adjacent to nerval structures. Primary success rate, complications, symptom-free interval, follow-up and biopsy results were evaluated. Results: Within observation period (1–61 months; median: 32 months) 38 of 39 patients were successfully treated and had no more complaints. In 3 of 38 patients relapse occurred after 1, 14 and 32 months and RFA was repeated. Two major complications (broken drill, infection) and 2 minor complications (hematoma, prolonged pain) were observed. Biopsy was able to prove diagnosis in 14 of 29 (48%) cases. Conclusions: Biopsy prior to treatment is not mandatory due to a remarkable amount of false negative findings in clinically and morphologically unambiguous cases of OO. RFA is a highly effective, efficient, minimally invasive and safe method for the treatment of OO. © 2008 Elsevier Ireland Ltd. All rights reserved.

To evaluate the feasibility and effectiveness of combining radiofrequency (RF) ablation and osteoplasty for pain reduction in the treatment of painful osteolytic metastases.Within 5 years, 22 patients (15 men and seven women; median age, 64 years) with 28 lesions located in the thoracic and lumbar spine, sacrum, pelvis, acetabulum, femur, and tibia were treated. Underlying tumors were breast, lung, renal cell, thyroid, cancer of unknown primary, and multiple myeloma. RF ablation was performed with the patient under moderate sedation and computed tomographic fluoroscopy guidance and was immediately followed by cement injection. Pain relief was evaluated with the visual analogue scale (VAS) score and the extent to which analgesics could be reduced. Clinical success was defined as a substantial reduction in pain and/or a reduced demand for analgesics, and technical success was defined as distribution of cement between both endplates of a vertebral body or at least 75% filling of osteolyses in other bones.Technical success and pain relief was achieved in all patients. Pain ratings with the VAS decreased from a mean of 8.5 to a mean of 5.5 after 24 hours (P.01), and a further decrease was detected after 3 months to 3.5 (P.01). The amount or strength of analgesics was reduced in 15 patients and remained unchanged in five. In two patients, the amount of analgesics increased due to tumor progression elsewhere. No major complication, no clinically obvious fracture of a formerly treated bone or treatment-related death, occurred.RF ablation and osteoplasty can be combined within one session and is both feasible and useful for the treatment of osteolytic bone metastases with regard to pain relief.

To compare a hybrid magnetic resonance (MR) angiography protocol with selective digital subtraction angiography (DSA) in patients with critical limb ischemia.The study was approved by the institutional review board, and written consent was obtained from all patients. Pretreatment DSA and hybrid MR angiography were performed in 19 consecutive patients (15 men, four women; mean age, 69.8 years; range, 44-86 years). Hybrid MR angiography included submillimeter dual-phase three-dimensional gadolinium-enhanced MR angiography in lower calf and foot, and four-station bolus-chase MR angiography in pelvis, thigh, and upper calf. Three readers identified the target lesion and inflow and outflow segments and determined treatment (bypass graft placement, percutaneous transluminal angioplasty, conservative management, amputation). Results of interobserver and intermethod comparisons were expressed as percentage of agreement and 95% confidence interval (CI).On hybrid MR angiograms, no substantial venous overlay was present and image quality was excellent or adequate in 18 (95%) of 19 limbs. Readers 1, 2, and 3 selected the identical target lesion on the DSA image and the MR angiogram in 18, 17, and 18 of 18 comparable limbs, respectively. Mean percentage of agreement for readers 1 and 3 was 100% (95% CI: 81%, 100%) and for reader 2 was 94% (95% CI: 73%, 100%). Agreement of all three readers was superior with use of MR angiography for determination of inflow segments (13 [72%] of 18 limbs) and outflow segments (17 [94%] of 18 limbs), compared with agreement with use of DSA (13 [68%] of 19 inflow segments, 10 [53%] of 19 outflow segments). Agreement in therapy decisions was higher with DSA (15 [79%] of 19) than with MR angiography (11 [61%] of 18).Preliminary data strongly support the combination of submillimeter dual-phase MR angiography in lower calf and foot with four-station bolus-chase MR angiography to extend the utility of MR angiography to patients with critical limb ischemia.

A septal course of the left main coronary artery originating from the right sinus of Valsalva is a rare coronary anomaly of unknown clinical significance. We report on the morphological and functional assessment of this coronary anomaly detected incidentally in a 53-year-old asymptomatic man after orthotopic heart transplantation.

A 53-year-old man with ischemic dilated cardiomyopathy underwent successful orthotopic heart transplantation. The postoperative course was uneventful. Routine control angiography shortly after transplantation revealed an anomalous origin of the left main coronary artery (LMCA) from the right sinus of Valsalva (Figure 1). An intraseptal course between the aorta and the right pulmonary outflow tract with retrograde filling of the left anterior descending artery was demonstrated by simultaneous coronary angiography …

Dilatation of the colon and the rectum, which is not attributable to aganglionosis, is a rare finding and can be the result of intractable chronic constipation. We report a rare case of a 29-year-old male patient with impressive megacolon, in whom Hirschsprung's or Chagas disease was ruled out. In the present case, dilatation of the colon was most likely due to a behavioral disorder with habitual failure of defecation. Chronic stool retention led to a bizarre bulging of the large bowel with displacement of the other abdominal organs and severe occult blood loss. Because of two episodes of life-threatening gastrointestinal bleeding despite conventional treatment of constipation, a surgical approach for bowel restoration was necessary. Temporary loop ileostomy had to be performed for de-pressurization of the large bowel and the subsequent possibility for effective antegrade colonic lavage to remove impacted stools. Shortly after the operation, the patient was healthy and could easily manage the handling of the ileostomy. However, the course of the megacolon in this young adult cannot be predicted and the follow-up will have to reveal if regression of this extreme colonic distension with reestablishment of regular rectal perception will occur.