{"id":998,"date":"2026-05-06T18:52:35","date_gmt":"2026-05-06T18:52:35","guid":{"rendered":"https:\/\/urff.app\/?page_id=998"},"modified":"2026-09-05T17:01:08","modified_gmt":"2026-09-05T17:01:08","slug":"schriftlich-rechnen","status":"publish","type":"page","link":"https:\/\/urff.app\/en\/schriftlich-rechnen\/","title":{"rendered":"Written arithmetic"},"content":{"rendered":"<p class=\"wp-block-paragraph\"><strong>Eine App f\u00fcr die verst\u00e4ndnisorientierte Erkundung der schriftlichen Addition und Subtraktion.<\/strong><\/p>\n\n\n\n<div class=\"urff-appbox\" style=\"display:flex;flex-wrap:wrap;align-items:center;gap:1rem;padding:1rem 1.25rem;margin:1.5rem 0;border:1px solid rgba(0,0,0,.12);border-radius:18px;background:rgba(0,0,0,.035)\">\n<a href=\"https:\/\/apps.apple.com\/de\/app\/schriftlich-rechnen\/id6765940539\" target=\"_blank\" rel=\"noopener\" style=\"flex:0 0 auto;line-height:0\"><img decoding=\"async\" src=\"https:\/\/urff.app\/wp-content\/uploads\/2026\/09\/appicon-schriftlich-rechnen-6765940539.jpg\" alt=\"App-Icon Schriftlich Rechnen\" width=\"96\" height=\"96\" loading=\"lazy\" style=\"width:96px;height:96px;border-radius:22%;box-shadow:0 1px 4px rgba(0,0,0,.15)\"><\/a>\n<div style=\"flex:1 1 12rem;min-width:0\">\n<div style=\"font-weight:700;font-size:1.15em;line-height:1.3\"><a href=\"https:\/\/apps.apple.com\/de\/app\/schriftlich-rechnen\/id6765940539\" target=\"_blank\" rel=\"noopener\" style=\"text-decoration:none\">Written arithmetic<\/a><\/div>\n<div style=\"font-size:.9em;opacity:.75;margin-top:.15em\">Christian Urff \u00b7 iPad<\/div>\n<\/div>\n<a href=\"https:\/\/apps.apple.com\/de\/app\/schriftlich-rechnen\/id6765940539\" target=\"_blank\" rel=\"noopener\" style=\"flex:0 0 auto;display:inline-block;padding:.55em 1.1em;border-radius:999px;background:#0071e3;color:#fff;font-weight:600;font-size:.95em;text-decoration:none;white-space:nowrap\">Download from the App Store<\/a>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The app displays each invoice simultaneously in multiple representations:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>notation<\/strong>: the classic place value chart with numbers, carry-overs and underwrites, as it is written in the notebook.<\/li>\n\n\n\n<li><strong>Plates<\/strong>: a two-dimensional place value chart in which each place value is represented as its own column and tiles visibly move when bundled or unbundled.<\/li>\n\n\n\n<li><strong>Dienes 3D<\/strong>An isometric view made up of cubes, rods, plates, and thousand cubes. It makes the spatial relationship between the place values immediately visible\u2014a ten is ten times as large as a one, a hundred ten times as large as a ten.<\/li>\n\n\n\n<li><strong>Language support<\/strong>Every step is accompanied by spoken guidance, both in writing and via speech output.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Changing a digit in the notation immediately shows the corresponding change in the other representations. The bundling and unbundling process is not only described but also animated. Ten ones bundle into a single ten, and a hundred flat is unbundled into ten ten rods.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"subtraktionsverfahren\">Subtraction method<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The app supports both subtraction methods commonly used in Germany:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Unbundling<\/strong>&nbsp;(Subtraction method): The next highest digit of the minuend is reduced by one, and ten units are added to the current column.<\/li>\n\n\n\n<li><strong>Expand<\/strong>&nbsp;(Fill-in method): The subtrahend is increased by one in the next higher column, the minuend remains unchanged.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Both methods can be switched between within the same task context. Even the difficult one.&nbsp;<strong>cascaded unbundling<\/strong>&nbsp;(for example, at)&nbsp;<code>900 \u2212 191<\/code>, where a borrowed unit has to travel across several locations) is fully animated and narrated.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"vorhersage-aufgaben\">Prediction tasks<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Optional kann die App vor jedem Schritt eine Multiple-Choice-Frage stellen, oder das Kind muss direkt an den Repr\u00e4sentationen handeln:&nbsp;<em>\u201eWas passiert als N\u00e4chstes?\u201c<\/em>&nbsp;Das Kind sagt den n\u00e4chsten Schritt vorher. Die App f\u00fchrt den Schritt erst danach aus. Die Distraktoren bilden bewusst typische Fehlvorstellungen ab, etwa \u201edas Gr\u00f6\u00dfere minus das Kleinere\u201c oder den vergessenen \u00dcbertrag.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"didaktischer-hintergrund\">Didactic background<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"bildungspolitischer-rahmen\">Educational policy framework<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Die schriftlichen Rechenverfahren sind seit der Verabschiedung der weiterentwickelten KMK-Bildungsstandards Mathematik 2022 deutlich anders didaktisch akzentuiert. W\u00e4hrend die Bildungsstandards von 2004 die schriftlichen Algorithmen als zentralen Kompetenzbereich der Grundschule auswiesen, betonen die neuen Standards st\u00e4rker das\u00a0<strong>semi-written arithmetic<\/strong>\u00a0und ordnen den schriftlichen Verfahren eine darauf aufbauende Funktion zu. Die wesentlichen \u00c4nderungen werden in einem Begleitartikel der Gesellschaft f\u00fcr Didaktik der Mathematik (Barzel et al. 2023) ausgef\u00fchrt. Zwei Punkte sind f\u00fcr unsere Frage besonders relevant:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Semi-written arithmetic is becoming more prominent.<\/strong>&nbsp;Semi-written strategies are understood as a prerequisite for children to learn to structure their own calculations, visualize intermediate steps, and understand their own thought processes. Written algorithms are only introduced once this foundation is solid (KMK 2022; Implementation Brochure KMK 2023).<\/li>\n\n\n\n<li><strong>Understanding becomes the guiding principle.<\/strong> \u201eVerstehen vor Automatisieren\u201c wird in der Implementierungsbrosch\u00fcre programmatisch aufgenommen.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Die Bundesl\u00e4nder haben dies uneinheitlich umgesetzt. Der bayerische&nbsp;<strong>CurriculumPLUS<\/strong>&nbsp;For elementary school, for example, the following is explicitly written for subtraction:&nbsp;<strong>Deduction method<\/strong>&nbsp;(= unbundling) and names hundreds plates, tens rods and ones as key teaching materials (Bavarian State Ministry of Education and Cultural Affairs 2024). The North Rhine-Westphalia curriculum, on the other hand, leaves the choice of procedure open and delegates the decision to the schools (PIKAS 2024).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This means that a learning software that can be used nationwide&nbsp;<strong>both procedures<\/strong>&nbsp;zumindest darstellen k\u00f6nnen muss, um den Anschluss an das jeweilige Schulbuch zu erlauben. Dar\u00fcber hinaus lassen sich dadurch sogar beide Verfahren vergleichend und verstehensorientiert untersuchen.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"was-heisst-verstehensorientierung\">Was hei\u00dft \u201eVerstehensorientierung\u201c?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The concept has been central to German mathematics education since the second half of the 2010s and was developed in particular by Susanne Prediger and Christoph Selter (TU Dortmund) (Prediger &amp; Selter 2014; Selter et al. 2014). Understanding-oriented mathematics teaching is characterized by three interconnected features:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Priority of ideas over calculation.<\/strong>&nbsp;A sound understanding of numbers and operations must be established before formal arithmetic rules are introduced. Only on this basis can procedures be condensed into schemata (Prediger &amp; Selter 2014).<\/li>\n\n\n\n<li><strong>Consistent change of representation.<\/strong>&nbsp;Concepts and operations are interconnected across multiple levels of representation (concrete, iconic, symbolic, linguistic), allowing children to test and refine their own ideas when switching between the levels.<\/li>\n\n\n\n<li><strong>Structural, intra-mathematical concepts.<\/strong> The promotion aims not only at motivating application contexts, but also at the recognition of mathematical structures, for example the bundling logic of the place value system (Prediger 2024).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Die App ist deshalb als ein Werkzeug konzipiert, das in diesem Sinne hilft, <strong>Vorstellungen aufzubauen<\/strong> and <strong>Repr\u00e4sentations\u00adwechsel einzufordern<\/strong>, und das eine sprachliche Begleitung mitbringt, die als Modell f\u00fcr die kommunikative Bearbeitung dient.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"schriftliche-verfahren-als-buendelungs-prozeduren\">Written procedures as bundling procedures<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The written algorithms are taken directly from the&nbsp;<strong>decimal place value system<\/strong>&nbsp;Derived. Each column of a place value chart represents a grouping level; written addition is then a two-step process per column: add together, and if the column yields \u2265 10, group into a unit of the next higher place value. Copying is simply writing down this group (Padberg &amp; Benz 2021, Chapter 8).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Das schriftliche Subtrahieren ist zwei\u00adschrittig: vergleichen, und falls die obere Stelle nicht ausreicht, eine Einheit der n\u00e4chsth\u00f6heren Stelle als zehn Einheiten der aktuellen Spalte verf\u00fcgbar machen. Ob dies durch Reduktion des Minuenden (\u201eEntb\u00fcndeln\u201c) oder durch Erh\u00f6hung des Subtrahenden plus Aufstockung des Minuenden um zehn (\u201eErweitern\u201c) geschieht, ist eine Verfahrens\u00adfrage; mathematisch sind beide \u00e4quivalent (Padberg &amp; Benz 2021).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Damit Kinder diese Logik tragf\u00e4hig erfassen und nicht nur verst\u00e4ndnislos mit Ziffern in Spalten rechnen, brauchen sie ein&nbsp;<strong>Understanding of bundling<\/strong>&nbsp;on the enactive and iconic level, before they work (purely) symbolically with notation. <\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"typische-verstehenshuerden\">Typical comprehension barriers<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"addition\">addition<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">The subject-specific didactics literature primarily documents the following error strategies for written addition (<em>PIKAS<\/em> 2024; Padberg &amp; Benz 2021):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Separation or complete transcription errors<\/strong>The child adds the column (e.g., 8 + 7 = 15) and writes the entire two-digit number in the ones column, instead of writing 5 and carrying over 1.<\/li>\n\n\n\n<li><strong>Forgot to transfer<\/strong>Columns are added correctly, but the carry-over from the previous column is not included.<\/li>\n\n\n\n<li><strong>Incorrect handling of zero<\/strong>: Columns with&nbsp;<code>0<\/code> are misinterpreted (e.g.&nbsp;<code>0 + n = 0<\/code>), or a carry to zero is not performed.<\/li>\n\n\n\n<li><strong>Place value reversal<\/strong>: When operands have a different number of digits, the digits are written one above the other, left-aligned instead of right-aligned.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"subtraktion\">subtraction<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">The range of obstacles is significantly broader in subtraction. The most common errors are (KIRA 2024):<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Difficulties with the transfer<\/strong>&nbsp;(no carry-over entered, no carry-over to empty spaces, one carry-over too many).<\/li>\n\n\n\n<li><strong>Error with zero<\/strong>&nbsp;(\u201e0 \u2212 x = 0\u201c, \u201ex \u2212 0 = 0\u201c, kein \u00dcbertrag zur oder \u00fcber die Null).<\/li>\n\n\n\n<li><strong>Calculation direction error<\/strong>&nbsp;(counted from left to right).<\/li>\n\n\n\n<li><strong>Operation reversal<\/strong>&nbsp;(Addition instead of subtraction).<\/li>\n\n\n\n<li><strong>Spaltenweise Unterschiedsbildung \u201egr\u00f6\u00dfere minus kleinere\u201c<\/strong>&nbsp;as the most common systematic error.<\/li>\n\n\n\n<li><strong>Incorrect understanding of importance<\/strong>.<\/li>\n\n\n\n<li><strong>Mixing several processes<\/strong>&nbsp;\u2014 for example, the simultaneous application of expansion and unbundling in the same task.<\/li>\n\n\n\n<li><strong>One plus one error<\/strong>&nbsp;(Numbering error in the column itself).<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">The frequency of the&nbsp;<strong>\u201eGr\u00f6\u00dfere minus Kleinere\u201c-Fehlers<\/strong>. In a large-scale study of 31 fourth-grade classes in the Bielefeld region, Schipper documented as early as 1983 that this error accounts for approximately 30% of all subtraction errors and is thus the most frequent systematic error (Schipper 1983; cited in Padberg &amp; Benz 2021). Wartha &amp; Schulz (2014) show that this error reliably correlates with a weak understanding of place value. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In der App wurden basierend auf diesen h\u00e4ufigen Fehlern die Impulse und Vorhersagen f\u00fcr jeden Schritt gezielt angepasst und zus\u00e4tzlich visuell verdeutlicht, um genau solche Fehler aufzugreifen, Fehlvorstellungen zu adressieren und durch die Repr\u00e4sentationsvernetzung Lerngelegenheiten zu ihrer \u00dcberwindung anzubieten.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"verfahrens-verwechslungen\">Procedural mix-ups<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">A separate hurdle arises when children expand and unbundle.&nbsp;<strong>mix<\/strong>. This error occurs empirically when textbooks and home-based support use different methods (Schipper 2009). This is partly because many parents still remember the extension method, while their children are now predominantly taught the unbundling method (Padberg &amp; Benz 2021).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"erweitern-vs-entbuendeln-eine-fachdidaktische-debatte\">Erweitern vs. Entb\u00fcndeln: Eine fachdidaktische Debatte<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The two subtraction methods commonly used in German primary schools have been the subject of controversial debate for decades. The current state of affairs can be summarized as follows:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"entbuendeln-abziehverfahren\">Unbundling (pulling method)<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Logic of justification:<\/strong>&nbsp;The next highest digit of the minuend is reduced by one; the borrowed unit is added to the current column as ten units.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Strengthen<\/strong>&nbsp;(Padberg &amp; Benz 2021; Krauthausen 2018; Bavarian State Ministry 2024):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Direct connection to the enactive material activity: A ten-rod is divided into ten one-cubes.<\/li>\n\n\n\n<li>Consistent logic: only the minuend changes, the subtrahend remains the same.<\/li>\n\n\n\n<li>Cascaded borrowing can be represented as a spatial movement.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Weaken:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Die Notation (\u201eZehner durchstreichen, daneben \u22121 schreiben\u201c) wird un\u00fcbersichtlich, vor allem bei mehrfachem Borgen.<\/li>\n\n\n\n<li>For tasks such&nbsp;<code>1000 \u2212 1<\/code>&nbsp;A unit must cascade across three places.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"erweitern-auffuellverfahren\">Expansion (refilling process)<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Logic of justification:<\/strong>&nbsp;The constancy of the difference: If you add the same value to both numbers (10 ones = 1 tens), the difference remains the same.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Strengthen<\/strong>&nbsp;(Schipper 2009; Padberg &amp; Benz 2021):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Clean notation, no deletions.<\/li>\n\n\n\n<li>Connection to the&nbsp;<strong>Add to<\/strong>&nbsp;(\u201e6 plus wieviel sind 13?\u201c), das f\u00fcr viele Kinder kognitiv zug\u00e4nglicher ist als das Abziehen.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Weaken:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Die Begr\u00fcndung (\u201ewir geben oben 10 dazu und m\u00fcssen das unten ausgleichen\u201c) ist anspruchsvoll und am gegenst\u00e4ndlichen Material schwerer zu zeigen. Selter (1995) bezeichnet diese Diskrepanz zwischen Material-Operation und notierter Operation als typische\u00a0<em>Fiction trap<\/em>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">In der App werden f\u00fcr beide Verfahren Visualisierungen angeboten, um sie verst\u00e4ndlicher zu machen. Beim Entb\u00fcndeln wird ein Pl\u00e4ttchen der n\u00e4chsth\u00f6heren Stelle durchgestrichen und wird zu 10 Pl\u00e4ttchen der n\u00e4chstniedrigeren Stelle. Beim Erweiterungsverfahren wird ein \u201eMinuspl\u00e4ttchen\u201c (Verschwindepl\u00e4ttchen\/Loch) hinzugef\u00fcgt und als Ausgleich daraus 10 Pl\u00e4ttchen der niedrigeren Stellen hinzugef\u00fcgt.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"umsetzung-in-der-app-didaktische-designentscheidungen\">Implementation in the app: Didactic design decisions<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Die folgenden Designentscheidungen orientieren sich an den oben skizzierten fachdidaktischen Prinzipien.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"mehrere-repraesentationen-synchron\">Multiple representations synchronously<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">The central didactic decision is the&nbsp;<strong>synchronous coupling<\/strong>&nbsp;mehrerer Darstellungs\u00adebenen: jede Aktion (eine Ziffer \u00e4ndern, einen Schritt ausf\u00fchren) wirkt simultan auf Notation, 2D-Pl\u00e4ttchen oder 3D-Dienes und sprachliche Beschreibung. Das ist genau das, was Bruner (1966) als Repr\u00e4sentations\u00adwechsel beschreibt und was Krauthausen (2018) als zentrale Aufgabe digitaler Lern\u00adsoftware nennt: nicht Animation als Selbstzweck, sondern&nbsp;<em>structurally identical<\/em>&nbsp;Animating the mathematical relationships.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The roller shutters above each panel allow teachers to selectively&nbsp;<strong>one<\/strong>&nbsp;The aim is to conceal one level and force the children to translate from the others. This corresponds to the requirement of operative didactics (Wittmann &amp; M\u00fcller 1990 ff.) to actively demand and increasingly anticipate shifts in representation. This allows for a wide variety of given-required tasks.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"verfahrenswahl-explizit\">Explicit choice of procedure<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Since the federal state curricula prefer different procedures, the app&nbsp;<strong>both procedures<\/strong>&nbsp;als gleichberechtigte Optionen bereit. Das adressiert auch die oben genannte Vermischungs-H\u00fcrde: Lehrkr\u00e4fte k\u00f6nnen das Verfahren des eigenen Schulbuchs einstellen und gleichzeitig das alternative Verfahren zu Vergleichs\u00adzwecken sichtbar machen als eine M\u00f6glichkeit, weil Kinder ihre eigene mentale Strategie oft nicht klar einem Verfahren zuordnen k\u00f6nnen.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"sprachausgabe-als-modell-sprache\">Speech output as a model language<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">The speech output uses consistently&nbsp;<strong>Technically correct terms<\/strong>: \u201eb\u00fcndeln\u201c, \u201eentb\u00fcndeln\u201c, \u201e\u00dcbertrag\u201c, \u201eMinuend\u201c, \u201eSubtrahend\u201c. Sie macht den Algorithmus zu einer Erkl\u00e4r-Sequenz, an der sich die Kinder sprachlich orientieren k\u00f6nnen. Sie k\u00f6nnen das Sprachvorbild als Modell nutzen, das sie in eigene Worte \u00fcbersetzen. Dies entspricht dem QuaMath-Prinzip&nbsp;<em>Promoting communication<\/em>&nbsp;(QuaMath 2024) and the one in&nbsp;<em>Can definitely do math<\/em> propagated practice,&nbsp;<strong>Verbalization<\/strong>&nbsp;to treat as a central part of mathematical understanding (Selter et al. 2014). <\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"vorhersage-aufgaben-2\">Prediction tasks<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">The optional\u00a0<strong>forecast<\/strong>\u00a0Before each step, the\u00a0<em>Predict-Observe-Explain<\/em>-format of science education (White &amp; Gunstone 1992). Instead of a mere animation that one watches, each step is framed by appropriate questions:\u00a0<em>What is happening here? What do I need to do here? Which distractor is plausible but false?<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The app&#039;s distractors deliberately reflect the documented error strategies, for example&nbsp;<em>\u201edas Gr\u00f6\u00dfere minus das Kleinere\u201c<\/em>, forgotten carry, reversed place values, procedural confusion, based on error analysis literature (Schipper 1983, 2009; Padberg &amp; Benz 2021; Wartha &amp; Schulz 2014; KIRA 2024). This results in a&nbsp;<em>kognitive Aktivierung<\/em> (QuaMath 2024) takes place.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"aufgabengenerator-und-adaptive-schwierigkeit\">Task generator and adaptive difficulty<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Der Aufgabengenerator bietet mehrere Schwierigkeits\u00adstufen und passt im Modus \u201eadaptiv\u201c die Schwierigkeit der als n\u00e4chstes angebotenen (\u201egew\u00fcrfelten\u201c) Aufgabe anhand der Vorhersage-Trefferquote an. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"virtuelles-handeln-die-bezugnahme-zwischen-repraesentationen-herausfordern\">Virtual Action: Challenging the Relationship Between Representations<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">During certain phases (placing the carry-over, entering the expansion correction), the multiple-choice question is replaced by a&nbsp;<strong>Tip task<\/strong>&nbsp;The material or the notation is replaced. The child points directly to the correct column and thus works on the iconic-symbolic level.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"grenzen-forschungsbedarf-und-praxisempfehlungen\">Limitations, research needs and practical recommendations<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">So vielf\u00e4ltig die M\u00f6glichkeiten der App sind, sie ersetzt weder das\u00a0<strong>tangible material<\/strong>\u00a0(Dienes cube you can touch) and that\u00a0<strong>Talking about mathematics<\/strong>\u00a0in the social space of the class. Krauthausen (2018, p. 215 ff.) points out that digital animations enhance the haptic experience\u00a0<em>add to<\/em>, nicht ersetzen. Dasselbe gilt f\u00fcr die m\u00fcndliche Bearbeitung: Kommunikation \u00fcber Mathematik ist essenziell und nicht delegierbar (QuaMath 2024). Die App kann aber erg\u00e4nzend und vor allem zum selbstst\u00e4ndigen \u00dcben und im Rahmen von Forscheraufgaben (in der App sind einige Ideen hinterlegt) genutzt werden.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"literaturverzeichnis\">Literature and further materials<\/h3>\n\n\n\n<ul class=\"wp-block-list literatur\">\n<li><strong>Barzel, B., Gasteiger, H., Greefrath, G., Maritzen, N., N\u00fchrenb\u00f6rger, M. &amp; Stanat, P.<\/strong>&nbsp;(2023). Further development of educational standards in mathematics for primary and lower secondary education.&nbsp;<em>Communications of the Society for Didactics of Mathematics<\/em>.&nbsp;<a href=\"https:\/\/ojs.didaktik-der-mathematik.de\/index.php\/mgdm\/article\/view\/1113\">https:\/\/ojs.didaktik-der-mathematik.de\/index.php\/mgdm\/article\/view\/1113<\/a><\/li>\n\n\n\n<li><strong>Bruner, JS.<\/strong>&nbsp;(1966).&nbsp;<em>Toward a Theory of Instruction<\/em>. Cambridge, MA: Harvard University Press.<\/li>\n\n\n\n<li><strong>Krauthausen, G.<\/strong>&nbsp;(2018).&nbsp;<em>Introduction to Mathematics Didactics \u2014 Primary School<\/em>&nbsp;(4th ed.). Berlin\/Heidelberg: Springer Spektrum.<\/li>\n\n\n\n<li><strong>Padberg, F. &amp; Benz, C.<\/strong>&nbsp;(2021).&nbsp;<em>Didactics of Arithmetic \u2014 well-founded, versatile, practical<\/em>&nbsp;(5th ed.). Berlin\/Heidelberg: Springer Spektrum.<\/li>\n\n\n\n<li><strong>Preacher, S.<\/strong>&nbsp;(2024). \u201eMathematik muss nicht wehtun.\u201c Beitrag im Deutschen Schulportal.&nbsp;<a href=\"https:\/\/deutsches-schulportal.de\/expertenstimmen\/susanne-prediger-didaktik-mathematik-muss-nicht-wehtun\/\">https:\/\/deutsches-schulportal.de\/expertenstimmen\/susanne-prediger-didaktik-mathematik-muss-nicht-wehtun\/<\/a><\/li>\n\n\n\n<li><strong>Prediger, S. &amp; Selter, C.<\/strong>&nbsp;(2014).&nbsp;<em>Mathematics Education Update \u2014 Current Perspectives for Teaching and Learning Research and Teacher Education<\/em>.&nbsp;<a href=\"https:\/\/wwwold.mathematik.tu-dortmund.de\/%3Csub%3Eprediger\/veroeff\/14-FuCo-Prediger-Selter.pdf\">https:\/\/wwwold.mathematik.tu-dortmund.de\/prediger\/veroeff\/14-FuCo-Prediger-Selter.pdf<\/a><\/li>\n\n\n\n<li><strong>Schipper, W.<\/strong>&nbsp;(1983).&nbsp;<em>Typical student errors in written subtraction of natural numbers<\/em>. Bielefeld University.&nbsp;<a href=\"https:\/\/pub.uni-bielefeld.de\/record\/1775198\">https:\/\/pub.uni-bielefeld.de\/record\/1775198<\/a><\/li>\n\n\n\n<li><strong>Schipper, W.<\/strong>&nbsp;(2009).&nbsp;<em>Handbook for mathematics teaching in primary schools<\/em>. Braunschweig: Schroedel.<\/li>\n\n\n\n<li><strong>Selter, C.<\/strong>&nbsp;(1995). On the fictitiousness of the &#039;zero hour&#039; in early arithmetic instruction.&nbsp;<em>Mathematical teaching practice<\/em>, 16(2), 11\u201319.<\/li>\n\n\n\n<li><strong>Selter, C., Prediger, S., N\u00fchrenb\u00f6rger, M. &amp; Hu\u00dfmann, S. (eds.)<\/strong>&nbsp;(2014).&nbsp;<em>Mastering Mathematics with Confidence \u2014 A Diagnostic and Support Concept for Securing Basic Mathematical Skills.<\/em>&nbsp;Themenmodul \u201eNat\u00fcrliche Zahlen\u201c. Berlin: Cornelsen.<\/li>\n\n\n\n<li><strong>Wartha, S. &amp; Schulz, A.<\/strong>&nbsp;(2014).&nbsp;<em>Preventing calculation problems<\/em>&nbsp;(2nd ed.). Berlin: Cornelsen.<\/li>\n\n\n\n<li><strong>White, R. &amp; Gunstone, R.<\/strong>&nbsp;(1992).&nbsp;<em>Probing Understanding<\/em>. London: Falmer Press.<\/li>\n\n\n\n<li><strong>Wittmann, E. Ch. &amp; M\u00fcller, GN.<\/strong>&nbsp;(1990 ff.).&nbsp;<em>Handbook of Productive Arithmetic Exercises<\/em>, Volumes 1 and 2. Stuttgart: Klett.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"bildungspolitische-dokumente\">Educational policy documents<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"literatur\"><strong>KMK \u2014 Standing Conference of the Ministers of Education and Cultural Affairs<\/strong>&nbsp;(2022).&nbsp;<em>Educational standards for the subject of mathematics. Primary level.<\/em>&nbsp;Decision of June 23, 2022. Berlin.&nbsp;<a href=\"https:\/\/www.kmk.org\/fileadmin\/Dateien\/veroeffentlichungen_beschluesse\/2022\/2022_06_23-Bista-Primarbereich-Mathe.pdf\">https:\/\/www.kmk.org\/fileadmin\/Dateien\/veroeffentlichungen_beschluesse\/2022\/2022_06_23-Bista-Primarbereich-Mathe.pdf<\/a><\/li>\n\n\n\n<li class=\"literatur\"><strong>KMK \u2014 Standing Conference of the Ministers of Education and Cultural Affairs<\/strong>&nbsp;(2023).&nbsp;<em>Educational standards for mathematics. Implementation brochure for primary and lower secondary education.<\/em>&nbsp;Berlin.&nbsp;<a href=\"https:\/\/www.kmk.org\/fileadmin\/Dateien\/pdf\/Bildung\/Qualitaet\/ImplBroschu__re_BiSta_MATHEMATIK_2023-03-23.pdf\">https:\/\/www.kmk.org\/fileadmin\/Dateien\/pdf\/Bildung\/Qualitaet\/ImplBroschu__re_BiSta_MATHEMATIK_2023-03-23.pdf<\/a><\/li>\n\n\n\n<li class=\"literatur\"><strong>KMK &amp; DZLM<\/strong>&nbsp;(2023).&nbsp;<em>Agreement on the ten-year QuaMath program.<\/em>&nbsp;Press release.&nbsp;<a href=\"https:\/\/dzlm.de\/aktuelles\/kmk_dzlm_quamath\">https:\/\/dzlm.de\/aktuelles\/kmk_dzlm_quamath<\/a><\/li>\n\n\n\n<li class=\"literatur\"><strong>Bavarian State Ministry for Education and Cultural Affairs<\/strong>&nbsp;(2024).&nbsp;<em>CurriculumPLUS Primary School \u2014 Mathematics.<\/em> <a href=\"https:\/\/www.lehrplanplus.bayern.de\/fachlehrplan\/grundschule\/3\/mathematik\">https:\/\/www.lehrplanplus.bayern.de\/fachlehrplan\/grundschule\/3\/mathematik<\/a><\/li>\n\n\n\n<li class=\"literatur\"><strong>Lower Saxony Ministry of Education<\/strong>&nbsp;(2024).&nbsp;<em>Kerncurriculum f\u00fcr die Grundschule \u2014 Mathematik<\/em> (Audio version).&nbsp;<a href=\"https:\/\/www.mk.niedersachsen.de\/download\/211766\/KC_Mathematik_fuer_die_Grundschule.pdf\">https:\/\/www.mk.niedersachsen.de\/download\/211766\/KC_Mathematik_fuer_die_Grundschule.pdf<\/a><\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"online-plattformen-und-foerderprogramme\">Online platforms and funding programs<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"literatur\"><strong>divomath<\/strong>&nbsp;\u2014 Digital teaching and learning environment for grades 3\u20136, TU Dortmund.&nbsp;<a href=\"https:\/\/www.schulministerium.nrw\/divomath-mathematische-basiskompetenzen-digital-und-verstehensorientiert-foerdern\">https:\/\/www.schulministerium.nrw\/divomath-mathematische-basiskompetenzen-digital-und-verstehensorientiert-foerdern<\/a><\/li>\n\n\n\n<li class=\"literatur\"><strong>KIRA<\/strong>&nbsp;\u2014&nbsp;<em>Children calculate differently.<\/em>&nbsp;TU Dortmund \/ DZLM.&nbsp;<a href=\"https:\/\/kira.dzlm.de\/\">https:\/\/kira.dzlm.de<\/a>&nbsp;(especially).&nbsp;<a href=\"https:\/\/kira.dzlm.de\/zahlen-und-operationen\/schriftliches-rechnen\/typische-fehler-bei-der-schriftlichen-subtraktion\">https:\/\/kira.dzlm.de\/zahlen-und-operationen\/schriftliches-rechnen\/typische-fehler-bei-der-schriftlichen-subtraktion<\/a>)<\/li>\n\n\n\n<li class=\"literatur\"><strong>Can definitely do math<\/strong>&nbsp;\u2014 Online materials. TU Dortmund \/ DZLM.&nbsp;<a href=\"https:\/\/mathe-sicher-koennen.dzlm.de\/\">https:\/\/mathe-sicher-koennen.dzlm.de<\/a><\/li>\n\n\n\n<li class=\"literatur\"><strong>PIKAS<\/strong>&nbsp;\u2014&nbsp;<em>Process- and content-related competencies.<\/em> TU Dortmund \/ DZLM.&nbsp;<a href=\"https:\/\/pikas.dzlm.de\/\">https:\/\/pikas.dzlm.de<\/a>&nbsp;(especially).&nbsp;<a href=\"https:\/\/pikas.dzlm.de\/unterricht\/zahlen-und-operationen\/zahlenraum-bis-1000\/schriftliche-addition\">https:\/\/pikas.dzlm.de\/unterricht\/zahlen-und-operationen\/zahlenraum-bis-1000\/schriftliche-addition<\/a> and&nbsp;<a href=\"https:\/\/pikas.dzlm.de\/unterricht\/zahlen-und-operationen\/zahlenraum-bis-1000\/schriftliche-subtraktion\">https:\/\/pikas.dzlm.de\/unterricht\/zahlen-und-operationen\/zahlenraum-bis-1000\/schriftliche-subtraktion<\/a>)<\/li>\n\n\n\n<li class=\"literatur\"><strong>primakom<\/strong>&nbsp;\u2014&nbsp;<em>Primary school mathematics in a nutshell.<\/em>&nbsp;TU Dortmund \/ DZLM.&nbsp;<a href=\"https:\/\/primakom.dzlm.de\/\">https:\/\/primakom.dzlm.de<\/a><\/li>\n\n\n\n<li class=\"literatur\"><strong>QuaMath<\/strong>&nbsp;\u2014&nbsp;<em>Quality of teaching and professional development in mathematics.<\/em>&nbsp;DZLM \/ IPN.&nbsp;<a href=\"https:\/\/quamath.de\/\">https:\/\/quamath.de\/<\/a><\/li>\n<\/ul>","protected":false},"excerpt":{"rendered":"<p>Eine App f\u00fcr die verst\u00e4ndnisorientierte Erkundung der schriftlichen Addition und Subtraktion. Schriftlich Rechnen Christian Urff \u00b7 iPad Im App Store laden Die App zeigt jede Rechnung gleichzeitig in mehreren Repr\u00e4sentationen: Wer in der Notation eine Ziffer \u00e4ndert, sieht in den anderen Darstellungen sofort, was sich ver\u00e4ndert. Der B\u00fcndelungs- und Entb\u00fcndelungs\u00adprozess wird nicht nur beschrieben, sondern [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-998","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/pages\/998","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/comments?post=998"}],"version-history":[{"count":12,"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/pages\/998\/revisions"}],"predecessor-version":[{"id":1358,"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/pages\/998\/revisions\/1358"}],"wp:attachment":[{"href":"https:\/\/urff.app\/en\/wp-json\/wp\/v2\/media?parent=998"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}