urff.app

Written arithmetic

An app for the understanding-oriented exploration of written addition and subtraction.

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Written arithmetic
Developer: Christian Urff
Price: 0,99 €

The app displays each invoice simultaneously in multiple representations:

  • notation: the classic place value chart with numbers, carry-overs and underwrites, as it is written in the notebook.
  • Plates: 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.
  • Dienes 3DAn isometric view made up of cubes, rods, plates, and thousand cubes. It makes the spatial relationship between the place values immediately visible—a ten is ten times as large as a one, a hundred ten times as large as a ten.
  • Language supportEvery step is accompanied by spoken guidance, both in writing and via speech output.

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.

Subtraction method

The app supports both subtraction methods commonly used in Germany:

  • Unbundling (Subtraction method): The next highest digit of the minuend is reduced by one, and ten units are added to the current column.
  • Expand (Fill-in method): The subtrahend is increased by one in the next higher column, the minuend remains unchanged.

Both methods can be switched between within the same task context. Even the difficult one. cascaded unbundling (for example, at) 900 − 191, where a borrowed unit has to travel across several locations) is fully animated and narrated.

Prediction tasks

Optionally, the app can ask a multiple-choice question before each step or require direct action on the representations: „"What happens next?"“ The child predicts the next step. The app only executes the step afterward. The distractors deliberately depict typical misconceptions, such as "the bigger minus the smaller" or the forgotten carry-over.

Didactic background

Educational policy framework

Since the adoption of the revised KMK (Standing Conference of the Ministers of Education and Cultural Affairs) mathematics standards for 2022, the didactic emphasis on written calculation methods has changed significantly. While the 2004 standards identified written algorithms as a central competency area in primary school, the new standards place greater emphasis on... semi-written arithmetic and assign a function building upon the written procedures. The essential changes are explained in an accompanying article by the Society for Didactics of Mathematics (Barzel et al. 2023). Three points are particularly relevant to our question:

  1. Semi-written arithmetic is becoming more prominent. 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).
  2. Understanding becomes the guiding principle. „Understanding before automating“ is included programmatically in the implementation brochure.

The federal states have implemented this inconsistently. The Bavarian CurriculumPLUS For elementary school, for example, the following is explicitly written for subtraction: Deduction method (= 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).

This means that a learning software that can be used nationwide both procedures It must at least be possible to represent it in order to allow connection to the respective textbook. Moreover, this even allows for a comparative, understanding-oriented examination of the two methods.

What does "understanding-oriented" mean?

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 & Selter 2014; Selter et al. 2014). Understanding-oriented mathematics teaching is characterized by three interconnected features:

  • Priority of ideas over calculation. 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 & Selter 2014).
  • Consistent change of representation. 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.
  • Structural, intra-mathematical concepts. 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).

The app is therefore designed as a tool that, in the sense of Building up ideas and Demand a change of representation which helps, and brings with it linguistic support that serves as a model for communicative processing.

Written procedures as bundling procedures

The written algorithms are taken directly from the decimal place value system 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 ≥ 10, group into a unit of the next higher place value. Copying is simply writing down this group (Padberg & Benz 2021, Chapter 8).

Written subtraction is a two-step process: comparing the units in the upper place value and, if the upper place value is insufficient, making one unit from the next higher place value available as ten units from the current column. Whether this is done by reducing the minuend ("unbundling") or by increasing the subtrahend plus adding ten to the minuend ("expanding") is a matter of procedure; mathematically, both are equivalent (Padberg & Benz 2021).

In order for children to truly grasp this logic and not just mindlessly calculate with numbers in columns, they need a Understanding of bundling on the enactive and iconic level, before they work (purely) symbolically with notation.

Typical comprehension barriers

addition

The subject-specific didactics literature primarily documents the following error strategies for written addition (PIKAS (2024) and Padberg & Benz (2021)):

  • Separation or complete transcription errorsThe 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.
  • Forgot to transferColumns are added correctly, but the carry-over from the previous column is not included.
  • Incorrect handling of zero: Columns with 0 are misinterpreted (e.g. 0 + n = 0), or a carry to zero is not performed.
  • Place value reversal: When operands have a different number of digits, the digits are written one above the other, left-aligned instead of right-aligned.

subtraction

The range of obstacles is significantly broader in subtraction. The most common errors are (KIRA 2024):

  1. Difficulties with the transfer (no carry-over entered, no carry-over to empty spaces, one carry-over too many).
  2. Error with zero („0 − x = 0″, „x − 0 = 0″, no carry to or over zero).
  3. Calculation direction error (counted from left to right).
  4. Operation reversal (Addition instead of subtraction).
  5. Column-wise differentiation "larger minus smaller"„ as the most common systematic error.
  6. Incorrect understanding of importance.
  7. Mixing several processes — for example, the simultaneous application of expansion and unbundling in the same task.
  8. One plus one error (Numbering error in the column itself).

The frequency of the „"Larger minus smaller" error. 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 & Benz 2021). Wartha & Schulz (2014) show that this error reliably correlates with a weak understanding of place value.

Based on these common errors, the app specifically adapted the impulses and predictions for each step and additionally visualized them in order to address such errors, misconceptions and offer learning opportunities to overcome these misconceptions through representational networking.

Procedural mix-ups

A separate hurdle arises when children expand and unbundle. mix. 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 & Benz 2021).

Expanding vs. Unbundling: A subject-specific didactics debate

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:

Unbundling (pulling method)

Logic of justification: The next highest digit of the minuend is reduced by one; the borrowed unit is added to the current column as ten units.

Strengthen (Padberg & Benz 2021; Krauthausen 2018; Bavarian State Ministry 2024):

  • Direct connection to the enactive material activity: A ten-rod is divided into ten one-cubes.
  • Consistent logic: only the minuend changes, the subtrahend remains the same.
  • Cascaded borrowing can be represented as a spatial movement.

Weaken:

  • The notation ("cross out the tens, write −1 next to them") becomes confusing, especially when borrowing multiple times.
  • For tasks such 1000 − 1 A unit must cascade across three places.

Expansion (refilling process)

Logic of justification: The constancy of the difference: If you add the same value to both numbers (10 ones = 1 tens), the difference remains the same.

Strengthen (Schipper 2009; Padberg & Benz 2021):

  • Clean notation, no deletions.
  • Connection to the Add to ("6 plus how much is 13?"), which is cognitively more accessible to many children than subtraction.

Weaken:

  • The rationale ("we add 10 at the top and have to compensate for that at the bottom") is complex and more difficult to demonstrate using the actual material. Selter (1995) describes this discrepancy between the material operation and the recorded operation as typical. Fiction trap.

The app offers visualizations for both methods to make them easier to understand. In the unbundling method, a tile from the next higher place value is crossed out, becoming 10 tiles from the next lower place value. In the extension method, a "minus tile" (disappearing tile/hole) is added, and to compensate, 10 tiles from lower places are added.

Implementation in the app: Didactic design decisions

The following design decisions are based on the subject-specific didactic principles outlined above.

Multiple representations synchronously

The central didactic decision is the synchronous coupling Multiple representation levels: every action (changing a digit, performing a step) simultaneously affects notation, 2D tiles or 3D dienes, and linguistic description. This is precisely what Bruner (1966) describes as representational shifting and what Krauthausen (2018) identifies as the central task of digital learning software: not animation as an end in itself, but structurally identical Animating the mathematical relationships.

The roller shutters above each panel allow teachers to selectively one 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 & Müller 1990 ff.) to actively demand and increasingly anticipate shifts in representation. This allows for a wide variety of given-required tasks.

Explicit choice of procedure

Since the federal state curricula prefer different procedures, the app both procedures as equally valid options. This also addresses the mixing hurdle mentioned in 4.3: Teachers can set the procedure of their own textbook and simultaneously make the alternative procedure visible for comparison purposes as a possibility, because children often cannot clearly assign their own mental strategy to a specific procedure.

Speech output as a model language

The speech output uses consistently Technically correct termsThe words "bundle," "unbundle," "carry over," "minuend," and "subtrahend" are used. This transforms the algorithm into an explanatory sequence that children can use as a linguistic guide. They can use this linguistic model as a template to translate into their own words. This corresponds to the QuaMath principle. Promoting communication (QuaMath 2024) and the one in Can definitely do mathpropagated practice, Verbalization to treat as a central part of mathematical understanding (Selter et al. 2014).

Prediction tasks

The optional forecast Before each step, the Predict-Observe-Explain-format of science education (White & Gunstone 1992). Instead of a mere animation that one watches, each step is framed by appropriate questions: What is happening here? What do I need to do here? Which distractor is plausible but false?

The app's distractors deliberately reflect the documented error strategies, for example „"the larger minus the smaller"“, forgotten carry, reversed place values, procedural confusion, based on error analysis literature (Schipper 1983, 2009; Padberg & Benz 2021; Wartha & Schulz 2014; KIRA 2024). This results in a Cognitive activation (QuaMath 2024) takes place.

Task generator and adaptive difficulty

The task generator offers several difficulty levels and, in "adaptive" mode, adjusts the difficulty of the next offered ("rolled") task based on the prediction success rate.

Virtual Action: Challenging the Relationship Between Representations

During certain phases (placing the carry-over, entering the expansion correction), the multiple-choice question is replaced by a Tip task The material or the notation is replaced. The child points directly to the correct column and thus works on the iconic-symbolic level.

Limitations, research needs and practical recommendations

As diverse as the app's possibilities are, it does not replace the tangible material (Dienes cube you can touch) and that Talking about mathematics in the social space of the class. Krauthausen (2018, p. 215 ff.) points out that digital animations enhance the haptic experience add to, It cannot replace oral instruction. The same applies to oral work: communication about mathematics is essential and cannot be delegated (QuaMath 2024). However, the app can be used as a supplement, especially for independent practice and within the framework of research tasks (some ideas are included in the app).

literature

  • Barzel, B., Gasteiger, H., Greefrath, G., Maritzen, N., Nührenbörger, M. & Stanat, P. (2023). Further development of educational standards in mathematics for primary and lower secondary education. Communications of the Society for Didactics of Mathematics. https://ojs.didaktik-der-mathematik.de/index.php/mgdm/article/view/1113
  • Bruner, JS. (1966). Toward a Theory of Instruction. Cambridge, MA: Harvard University Press.
  • Krauthausen, G. (2018). Introduction to Mathematics Didactics — Primary School (4th ed.). Berlin/Heidelberg: Springer Spektrum.
  • Padberg, F. & Benz, C. (2021). Didactics of Arithmetic — well-founded, versatile, practical (5th ed.). Berlin/Heidelberg: Springer Spektrum.
  • Preacher, S. (2024). „Mathematics doesn’t have to hurt.“ Article on the German School Portal. https://deutsches-schulportal.de/expertenstimmen/susanne-prediger-didaktik-mathematik-muss-nicht-wehtun/
  • Prediger, S. & Selter, C. (2014). Mathematics Education Update — Current Perspectives for Teaching and Learning Research and Teacher Education. https://wwwold.mathematik.tu-dortmund.de/prediger/veroeff/14-FuCo-Prediger-Selter.pdf
  • Schipper, W. (1983). Typical student errors in written subtraction of natural numbers. Bielefeld University. https://pub.uni-bielefeld.de/record/1775198
  • Schipper, W. (2009). Handbook for mathematics teaching in primary schools. Braunschweig: Schroedel.
  • Selter, C. (1995). On the fictitiousness of the 'zero hour' in early arithmetic instruction. Mathematical teaching practice, 16(2), 11–19.
  • Selter, C., Prediger, S., Nührenbörger, M. & Hußmann, S. (eds.) (2014). Mastering Mathematics with Confidence — A Diagnostic and Support Concept for Securing Basic Mathematical Skills. Thematic module „Natural Numbers“. Berlin: Cornelsen.
  • Wartha, S. & Schulz, A. (2014). Preventing calculation problems (2nd ed.). Berlin: Cornelsen.
  • White, R. & Gunstone, R. (1992). Probing Understanding. London: Falmer Press.
  • Wittmann, E. Ch. & Müller, GN. (1990 ff.). Handbook of Productive Arithmetic Exercises, Volumes 1 and 2. Stuttgart: Klett.

Educational policy documents

Online platforms and funding programs