The Suuji Approach to Multi-Digit Addition: Using Length to Deepen Students’ Understanding of the Base 10 Number System

We describe the Suuji representation of numbers which aims to deepen elementary students’ understanding of the base 10 system. (“Suuji” means “number” in Japanese.) This representation takes a two pronged approach of (1) making the place value more explicit and (2) using length to represent numbers, thus allowing students to reason spatially. We taught multi-digit addition using the Suuji representation to 20 second and third grade students. The article uses lesson descriptions and student work to illustrate the Suuji approach, as well as its impact on student learning

Mathematical Habits of Mind for Teaching: Using Language in Algebra Classrooms

The notion of mathematical knowledge for teaching has been studied by many researchers, especially at the elementary grades. Our understandings of this notion parallel much of what we have read in the literature, but are based on our particular experiences over the past 20 years, as mathematicians engaged in doing mathematics with secondary teachers. As part of the work of Focus on Mathematics, Phase II MSP, we are developing, in collaboration with others in the field, a research program with the ultimate goal of understanding the connections between secondary teachers’ mathematical knowledge for teaching and secondary students’ mathematical understanding and achievement. We are in the early stages of a focused research study investigating the research question: What are the mathematical habits of mind that high school teachers use in their professional lives and how can we measure them? The main focus of this paper is the discussion of the habit of using mathematical language, and particularly how this habit plays out in a classroom setting

A PROGRAMMING ENVIRONMENT FOR NOVICES WITH VISUAL BLOCK AND TEXTUAL INTERFACES

Abstract Visual block languages such as Scratc

Spectral evolution of GRB 060904A observed with Swift and Suzaku -- Possibility of Inefficient Electron Acceleration

We observed an X-ray afterglow of GRB 060904A with the Swift and Suzaku satellites. We found rapid spectral softening during both the prompt tail phase and the decline phase of an X-ray flare in the BAT and XRT data. The observed spectra were fit by power-law photon indices which rapidly changed from $\Gamma = 1.51^{+0.04}_{-0.03}$ to $\Gamma = 5.30^{+0.69}_{-0.59}$ within a few hundred seconds in the prompt tail. This is one of the steepest X-ray spectra ever observed, making it quite difficult to explain by simple electron acceleration and synchrotron radiation. Then, we applied an alternative spectral fitting using a broken power-law with exponential cutoff (BPEC) model. It is valid to consider the situation that the cutoff energy is equivalent to the synchrotron frequency of the maximum energy electrons in their energy distribution. Since the spectral cutoff appears in the soft X-ray band, we conclude the electron acceleration has been inefficient in the internal shocks of GRB 060904A. These cutoff spectra suddenly disappeared at the transition time from the prompt tail phase to the shallow decay one. After that, typical afterglow spectra with the photon indices of 2.0 are continuously and preciously monitored by both XRT and Suzaku/XIS up to 1 day since the burst trigger time. We could successfully trace the temporal history of two characteristic break energies (peak energy and cutoff energy) and they show the time dependence of $\propto t^{-3} \sim t^{-4}$ while the following afterglow spectra are quite stable. This fact indicates that the emitting material of prompt tail is due to completely different dynamics from the shallow decay component. Therefore we conclude the emission sites of two distinct phenomena obviously differ from each other.Comment: 19 pages, 9 figures, accepted for publication in PASJ (Suzaku 2nd Special Issue

Lethal Bleeding from a Duodenal Cancerous Ulcer Communicating with the Superior Mesenteric Artery in a Patient with Pancreatic Head Cancer

Pancreatic cancer often invades the duodenum and causes obstruction, but rarely causes massive duodenal bleeding. A 68-year-old male was admitted to our hospital because of vomiting. Enhanced abdominal CT showed a hypovascular tumor with air bubbles in the uncinate process of the pancreas. The tumor invaded the duodenum and metastasized to the liver and peritoneum. The main trunk of the superior mesenteric artery (SMA) was circumferentially involved. After admission, he had hematemesis and melena. Emergency gastroduodenoscopy revealed pulsating vessels in the third portion of the duodenum and he eventually experienced hemorrhagic shock. Severe bleeding occurred from his mouth and anus like a catastrophic flood. It was difficult to sustain blood pressure even with massive blood transfusion with pumping. After insertion of an intra-aortic balloon occlusion catheter, the massive bleeding was eventually stopped. Although we attempted interventional radiography, aortography revealed direct communication between the main SMA trunk and the duodenal lumen. The tumor was considered anatomically and oncologically unresectable. Thus, we did not perform further intervention. The patient died 2 h after angiography. Herein, we report the case of pancreatic head cancer causing lethal bleeding associated with tumor-involved SMA. Duodenal bleeding associated with pancreatic cancer invasion should be considered as an oncogenic emergency

Development of unit for elective subject from fifth to ninth grade to improve cooperative creation (3)

本研究は, 「21世紀型の教科学力」の新たな観点としての「協同的創造力」の育成をめざして, 自分たちで新たな文化を創造する子どもを育てる協同的創造学習のあり方について実証的に研究を進め, 単元モデルと評価方法を開発することを目的としている。そこで, 教科学習を「協同的創造学習」としてとらえ直すとともに, 中学校での従来の選択教科の時間に加えて, 小学校第5・6学年合同の選択教科の時間を新設して「協同的創造力」を特化して育むことにし, 本年度は, 選択教科の単元モデルの充実・改善と評価方法の確立に取り組んだ。その結果, 選択教科において, これまで開発した単元モデルをより充実させたり, 新たな単元モデルを開発したりすることができた。また, 評価の観点を整理し, 子どもの意識調査やカリキュラム評価に継続して取り組むことによって, 子どもの思いを汲み取り単元を見直していくことができた。今後も必修教科と選択教科のつながりや関連性, 各学年の系統性を整理するとともに, 協同的創造力育成の手だてを整理し, 来年度に向けて, これまで培ったものを生かす新たな学習開発を模索していきたいと考えている

Problem posing in Pósa Problem thread

Problem posing is considered as a crucial element of learning and doing mathematics, and there is a constant quest for how this can be taught effectively. The Pósa method is a Hungarian instructional method of mathematical guided discovery through specifically designed problem threads, where problem posing plays an important role. The chapter presents how this method encourages students to pose problems, and hence create their own problem threads. To illustrate this pedagogical approach, to delineate various problem-posing heuristics, and to describe how a teacher guides his middle school students to engage in problem posing, we describe a sample problem thread involving geometric transformations