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To receive a degree in a foreign language from an accredited US university, students must demonstrate a high level of language mastery, which includes their ability to communicate effectively in the target language. For majors who are working in a second language, reaching professionally required levels of proficiency is a daunting task. For example, the state of Michigan requires that language teachers score a minimum of Advanced Low on a Simulated Oral Proficiency Examination (SOPI) in order to teach in public high schools. It is important to address the needs of these second language learners by assessing their outcomes upon degree completion and adjusting our programs to optimize this outcome (Carroll, 1967).Each language confronts the learner with unique learning challenges. Learners of Chinese must master a large number of characters to read a newspaper. Conversely, speakers of some foreign languages find the correct usage of English articles to be an ongoing challenge. For L2 German, one of the greatest hurdles is learning to understand, process, and produce the differing gender, number, and case markings in determiner phrases (DPs) (DPs in German always show marking on their non-root parts, i.e., articles and adjectives. In some cases, an indefinite article may not receive overt marking, but the lack of marking does, in fact, provide case and gender information as it is understood within the morphological paradigm. The following example shows how German DPs are marked within sentences. For clarity, DPs are underlined and overt marking is bolded: Deralte Mann kaufte ein neues Auto. The (masc-nom) old (masc-nom) man bought a (neuter-acc) new (neuter-acc) car. 'The old man bought a new car.'The marking of case, gender, and number is important, because it provides essential information about the role of the DP in the sentence. Although the above sentence uses standard SVO (Subject-Verb-Object) word order, speakers of German can also use OVS order to emphasize the object, because it is the morphological marking on the DP that provides the role information, not the order of the words: Einneues Auto kaufte der alte Mann. A (neuter-acc) new (neuter-acc) car bought the (masc-nom) old (masc-nom) man. 'The old man bought a new car.' 26 UP 48.1 (Spring 2015)

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In recent decades, the instructional theory of Realistic Mathematics Education has exerted a powerful influence on mathematics education around the world. The idea of progressive mathematisation has gained international acceptance. In this chapter, we will illustrate the way in which we benefited from the idea of organising the teaching and learning of mathematics in keeping with this guiding principle. After some personal memories of the first author, we start by describing what we consider to be the central elements of the principle of progressive mathematisation. This is followed by a description of two methods, the mathematics conferences and mathematics language tools, for rendering the learning and teaching concepts entailed by the principle of progressive mathematisation-especially its vertical component-even more expedient and fruitful. The contribution concludes with an explanation of how we understand the term 'realistic' in Realistic Mathematics Education. KeywordsProgressive schematisation • Mathematics conferences • Language and mathematics • Individual learning processes • Co-operative learning • Mathematics language tools The Santa Claus ProblemIt must have been at the end of 1983 that the first author-at the time studying to become a primary school teacher-became aware of Adri Treffers' paper "Fortschreitende Schematisierung -ein natürlicher Weg zur schriftlichen Multiplikation und Division im 3. und 4. Schuljahr" (Treffers, 1983). Taking the multiplication of large