Pre-Service Primary School Teachers’ Knowledge and Their Interpretation of Students’ Answers to a Measurement Division Problem with Fractions

During the last decades, research in teacher noticing has increased since its development is considered important in teacher training programs. An issue that needs more research is the relationship between teachers’ mathematical knowledge for teaching in a specific mathematical domain and their ability to notice. This study focuses on how pre-service primary school teachers (PPTs) solve a measurement division problem with fractions and interpret (score and justify) students’ answers to this problem. The participants were 84 PPTs who answered two tasks. Task 1 consisted of solving a measurement division problem with fractions. Task 2 involved interpreting (scoring and justifying) the answers of four primary school students to the problem. Responses to Task 1 were classified based on their accuracy and the procedure used. For Task 2, the scores given along with their justifications were analyzed. The results show that PPTs’ knowledge of division with fractions is limited and that they had difficulties in identifying conceptual errors in students’ answers. This study provides information on the relationships between PPTs’ knowledge of these types of problems and how PPTs interpret students’ answers. This information could aid in adjusting mathematical teaching knowledge in training programs.

Dynamic Fair Resource Division

A single homogeneous resource needs to be fairly shared between users that dynamically arrive and depart over time. Although good allocations exist for any fixed number of users, implementing these allocations dynamically is impractical: it typically entails adjustments in the allocation of every user in the system whenever a new user arrives. We introduce a dynamic fair resource division problem in which there is a limit on the number of users that can be disrupted when a new user arrives and study the trade-off between fairness and the number of allowed disruptions, using a fairness metric: the fairness ratio. We almost completely characterize this trade-off and give an algorithm for obtaining the optimal fairness for any number of allowed disruptions.

The Parameterized Complexity of Connected Fair Division

We study the Connected Fair Division problem (CFD), which generalizes the fundamental problem of fairly allocating resources to agents by requiring that the items allocated to each agent form a connected subgraph in a provided item graph G. We expand on previous results by providing a comprehensive complexity-theoretic understanding of CFD based on several new algorithms and lower bounds while taking into account several well-established notions of fairness: proportionality, envy-freeness, EF1 and EFX. In particular, we show that to achieve tractability, one needs to restrict both the agents and the item graph in a meaningful way. We design (XP)-algorithms for the problem parameterized by (1) clique-width of G plus the number of agents and (2) treewidth of G plus the number of agent types, along with corresponding lower bounds. Finally, we show that to achieve fixed-parameter tractability, one needs to not only use a more restrictive parameterization of G, but also include the maximum item valuation as an additional parameter.

PROSPECTIVE TEACHERS UNDERSTANDING FRACTION DIVISION USING RECTANGLE REPRESENTATION

Fraction division is one of the most difficult subjects in elementary school. Not only elementary students but many prospective teachers don’t understand the fraction division concept yet—most of them using a keep-change-flip algorithm to solve fraction division problems. A study using rectangle representation was conducted by us to prospective teachers. This study aims to see whether this rectangle representation will make prospective teachers understand or not. To do so, we made a mixed-method study with 80 prospective teachers as participants. The results show that 53,75% of prospective teachers use the keep-change-flip algorithm without understanding the concept of fraction division, and just 15% of prospective teachers understand fraction division. We assume that most prospective teachers still can’t imagine how fraction division works in a real-life context. They remember what they used to do to finish the fraction division problem that their teacher has introduced in primary school. Based on the results, we conclude that the study with rectangle representation still needs an improvement, whether the teacher’s explanation or the rectangle media.

An Analysis on Stylistic Features of Donald Trump’s Speech

For government or leaders, public speaking is an important way to show the statesmanship and eloquence. It is a means of attracting groups of people who come from different classes. As the president of the United States, Donald Trump’s speaking talent plays an important role in the general election. Stylistics, which uses theories of modern linguistics to solve problems, aims at studying linguistic features and revealing the effect and function of pragmatic expression. This article selected Donald Trump’s three typical speeches, which studies from the perspective of stylistics on three major aspects—language description, textual analysis and contextual analysis. The analysis yielded the following results, 1) Language description consists of lexical analysis and syntactic analysis. On lexical level, Trump tends to use more abstract nouns and first person plural pronoun to make the addresses persuasive and more acceptable. Syntactically, for the sake of expressing information effectively and attracting more support, simple sentences and declarative sentences are prevailing in the speeches; 2) On the aspect of textual analysis, Trump employs topical division, problem-solution division and chronological division in an overlapping way in main body of speeches and creates crescendo in closure; 3) Contextual analysis shows that language varies from situations and they are formal and highly-structured. In a word, to analyze Donald Trump’s speech on stylistic features is significant for us on observing the features of his speeches and word-using habits.

Conclusion

Can risk assessment be made fair? The conclusion of Calculating Race returns to actuarial science’s foundations in probability. The roots of probability rest in a pair of problems posed to Blaise Pascal and Pierre de Fermat in the summer of 1654: “the Dice Problem” and “the Division Problem.” From their very foundation, the mathematics of probability offered the potential not only to be used to gain an advantage (as in the case of the Dice Problem), but also to divide material fairly (as in the case of the Division Problem). As the United States and the world enter an age driven by Big Data, algorithms, artificial intelligence, and machine learning and characterized by an actuarialization of everything, we must remember that risk assessment need not be put to use for individual, corporate, or government advantage but, rather, that it has always been capable of guiding how to distribute risk equitably instead.

Parallelized Additive Manufacturing of Variably Partitioned Volumes for Large Scale 3D Printing With Localized Quality

Despite the growing application of additive manufacturing (AM) in fabricating complex designs, most machines suffer from small working envelopes and slow processing speeds. One workaround to the problem of small throughput in AM is to partition the volume of a desired object and fabricate sub-volumes in parallel. Prior related work has focused on two problems. One is the geometric division problem, disregarding AM benefits and challenges in determining partitions. Others attempt to install multiple AM processing heads on the same machine, ensuring seamless bonding between deposited material from different heads while avoiding interference among them. A missed opportunity lies in deploying many independent machines simultaneously while considering benefits and limitations of AM. To that end, objects too large to be fabricated on one machine, are divided primarily into cubes that exploit benefits of AM. Specifically, the cubes are hollowed out in the direction of printing to reduce weight while avoiding the need for support structure, and depending on load conditions, packed in different orientations to mitigate material anisotropy.