Discover the relation symbols interactively

A learning and experimentation environment for primary school focusing on relational symbols =, < and >.

Direct link: https://waage.urff.app

The web app Is the same simulates a virtual balance scale, allowing children to compare quantities, numbers, and expressions through hands-on activities. Counters are generated by tapping or clicking (multiple counters simultaneously via multitouch), placed in trays on both sides of the scale, stacked, automatically bundled into rods of ten, and displayed in real time as equations or inequalities. The app alternates between hands-on activity, iconic visual representation (points on the scale), and symbolic notation (=, <, > (plus numbers and expressions) takes place synchronously. Each element can also be covered, so that any given-sought problem can be formulated.

The app is intentionally designed to be user-friendly: no login, no data collection, no advertising. Configurations can be easily shared via link/QR code.

What children can do with the scales:

• Compare quantities. Place the plates in the left and right bowls. The scale will immediately show which side contains more.
• Restore balance. By adding, removing, or redistributing, consciously experience and experiment with the equals sign. How much do I need to add to balance the scales?
• Work with two bowls on each side. Term structures such as 5 + 3 build it up directly and compare it with other terms.
• Use negative shells. Subtractive relationships such as 9 − 4 Represent materially and symbolically in parallel. Removed quantities are collected in negative bowls.
• Stacking and bundling. In stacking mode, tiles are lined up; every 10 (or 5) tiles are automatically animated to form a rod of ten, if desired (can be changed in the settings).
• Cover up. Slide blinds over individual bowls or numbers to make assumptions and check them later.
• Split. Each configuration can be shared with learners as a compact link or QR code.

Sample tasks / task ideas:

• Find five examples of numbers greater than 8.
• Place 3 dots on the left and 5 dots on the right. How many do you need to add or remove to balance the scale?
• Stelle die Aufgabe „3 + 4 = 7“ auf der Waage dar — und dann „7 = 3 + 4“. Vergleiche beide. Warum kann man die Seiten vertauschen?
• Double the number of points on both sides. When does the scale remain balanced, and when does it not?
• There are 12 points on one side and 15 on the other. Can you balance the scale by yourself? Redistribute Bring into balance? Explain.
• On one side are 8 dots. On the other side are two bowls: one contains 5 dots, the other is empty. How many dots must go into the empty bowl to create a balance?
• Richtig oder falsch: „Die Waage bleibt im Gleichgewicht, wenn ich auf jeder Seite dieselbe Anzahl dazufüge.“ Begründe.

Eine umfangreichere Sammlung von Aufgabenbeispielen ist in der App selbst hinterlegt (Info-Bereich → „Aufgabenbeispiele“).

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Didactic background

Operational interpretation of the equals sign as a problem

Since the clinical interviews by Behr, Erlwanger and Nichols (1980), it has been empirically well established that primary school children predominantly perceive the equals sign as a symbol. Call to action to interpret, that is, as a request to perform an operation and write down the result. Equations of the form 3 + 4 = ☐ are processed effortlessly, while tasks such as ☐ = 3 + 4, 3 + 4 = 5 + ☐ or 5 = 5 häufig als „falsch“ oder „nicht erlaubt“ abgelehnt werden (Behr et al., 1980; Falkner, Levi & Carpenter, 1999; Knuth, Stephens, McNeil & Alibali, 2006). Falkner et al. (1999) berichten, dass nur etwa ein Viertel der Sechstklässler die Aufgabe 8 + 4 = ☐ + 5 correctly solves. A significant portion is either 12 or 17 One. Both solutions are typical indicators of a purely operational interpretation.

Dieses Phänomen ist sprach- und schulsystemübergreifend dokumentiert: für deutschsprachige Lernende durch Borromeo Ferri und Blum (2012), durch Hagemeister (2013) sowie durch Unteregge (2017); im englischsprachigen Raum durch Carpenter, Franke und Levi (2003); in weiteren Kontexten durch Oksuz (2008), Wahyuni und Herman (2019) sowie Farfan und Schoen (2021).

Knuth et al. (2006) were able to show that the understanding of the equals sign as Relationship or equivalence sign significantly correlated with success in solving simple equations: learners who = interpreting relationally, solving tasks such as 4m + 10 = 70 In secondary school, they are significantly more successful than their peers with operational interpretations. The question is how = Understanding this is therefore a key lever for the transition from arithmetic to algebra (Kieran, 1981; Sfard, 1991).

Operational and relational interpretation of the equals sign

Winter (1982) already pointed out that both perspectives – the operational interpretation („= heißt: rechne aus“) und die relationale Deutung („= heißt: beide Seiten sind gleichwertig“) – bereits in der Grundschule angelegt werden sollten. Borromeo Ferri und Blum (2012) sowie Unteregge (2017) betonen, dass das one-sided Focusing on a task-outcome interpretation reinforces misconceptions and creates obstacles in secondary school algebra instruction. Sfard (1991) explored this connection in her theory of Process-Object Duality Mathematical concepts are theoretically underpinned: Learners must understand a term such as 3 + 4 both as process (an invoice) as well as object (a number that is related to other objects). The equals sign is the interface between the two interpretations.

The key consequence for lesson planning is therefore: task formats should systematically address both interpretations, starting in elementary school, in order to counteract misconceptions early on. This includes equations with the variable on the left (☐ = a + b), equations without operation signs (5 = 5), Term comparisons (3 + 4 = 5 + 2) and especially that Justifying equivalence regardless of the calculation.

The scales as a visual aid

Die Balkenwaage ist im deutschsprachigen wie im englischsprachigen Raum das verbreitetste enaktiv-ikonische Material zur Erarbeitung der relationalen Bedeutung des Gleichheitszeichens (Wittmann, 1981; Mann, 2004; Dooley & Kirwan, 2014). Sie hat zwei wesentliche didaktische Vorzüge:

1. Symmetry is visible. A balanced scale is immediately recognizable as symmetrical; an unbalanced one shows the difference through its tilt. Furthermore, the relational symbols <, >, and = can be directly derived from the position of the beam, since the beam always tilts in the appropriate direction or is horizontal when the balance is equal.
2. Operational changes are demonstrably comprehensible. Adding or removing equal amounts on both sides maintains the balance. This experience is the action-oriented precursor to... Equivalent transformation, which is later formalized in secondary education (Carpenter, Franke & Levi, 2003; Prediger, 2009).

Mann (2004) bezeichnet das Gleichheitszeichen in diesem Kontext als „a balancing act“, also als eine Metapher, die in der digitalen Umsetzung der vorliegenden App buchstäblich umgesetzt wird: Der Balken neigt sich, sobald die Mengen nicht übereinstimmen, und richtet sich beim Herstellen des Gleichgewichts in Echtzeit waagerecht aus. Waagen sind deshalb ideal als Experimentierumgebung geeignet, weil Operationen und ihre Auswirkungen direkt erfahrbar werden im Sinne des operativen Prinzips.

Understanding the importance and bundling

The optional automatic grouping of ten (or alternatively five) tiles into a ten-rod adopts the classic proposal of Dienes' multi-system blocks (Dienes, 1960) without forcing a change of material. Two things are pedagogically crucial here:

• Continuity of representation. The rod remains visibly composed of ten tiles. Learners do not lose sight of the one-to-one ratio. This corresponds to the recommendation of Krauthausen and Scherer (2007) to design bundling materials in such a way that the bundling relationship remains reversible at any time.
• Animated connection and resolution. Upon reaching the tenth unit, the tiles visibly slide into a column and are connected by a subtle border. When a unit is removed, the connection breaks. This reversibility supports the understanding of bundling and unbundling as complementary operations, as Padberg and Benz (2021) emphasize as a prerequisite for place value understanding.

Die Wahlfreiheit zwischen 5er- und 10er-Bündeln ermöglicht Anschluss an Konzepte des „Five-Frame“- und „Ten-Frame“-Materials (Van de Walle, Karp & Bay-Williams, 2019), das als Brücke zwischen subitisierender Mengenwahrnehmung und Stellenwertdarstellung genutzt wird.

Multiple representations and the switching between them

The app simultaneously displays each state of the scale in three representational forms:

• Acting-iconic: as a plate on a physically reacting scale,
• symbolic: as an equation or inequality over the image,
• linguistic-verbal: als ausgesprochener Satz auf Wunsch („Die Drei ist kleiner als die Vier“), wahlweise auch per Sprachausgabe.

Bruner's (1966) classic EIS principle (enactive-iconic-symbolic) and Ainsworth's (2006) DeFT framework (Design, Functions, Tasks of multiple external representations) provide the theoretical justification for why the simultaneous and Consistent The availability of different representations facilitates concept development: learners can actively explore transitions between representations instead of passively observing them. The ability to hide individual representations using a blind makes the app a valuable tool for... Given-Sought Tasks in the sense of Selter and Spiegel (1997).

Conceptual design decisions

element	Didactic rationale
One or two bowls per side	Allows term structures such as (a + b) = c or (a + b) = (c + d) and thus relational comparisons
Optional negative shells	Erfahrbarmachung der Subtraktion als „Wegnehmen“ mit demselben Operationsverständnis wie Addition, die Gesamtmenge bleibt sichtbar, es „zählt“ aber nur die Restmenge auf der Waage.
Roller shutters for bowls and numbers	Structured conjecture learning, reduction of display to promote mental operations
Stack mode with comparison lines	Iconic precursor of the order-preserving bijection between two sets, derivation of the symbols from the visual representation of the two bars above and below.
Limiting the number of tiles	Differentiation according to number range (Grade 1: 10–20; Grade 2: 100; Grade 3+: up to 500)
Share via link/QR code	Low-threshold task transfer between teacher and learner, also in distance learning
No registration, no tracking data, only local data	GDPR-compliant use in the school context without additional organizational effort

Recommended use

The app is particularly suitable for primary school and for supporting students at the beginning of lower secondary school, especially those experiencing difficulties with the relational interpretation of the equals sign (Prediger, 2009). It does not replace physical learning materials but complements them: In line with the spiral learning principle (Bruner, 1966; Wittmann, 1981), this exploration opportunity can be repeatedly accessed and built upon. The web app offers the possibility of using the same structures. between to repeat, vary, and divide the material phases in a way that is not possible with physical material.

literature

Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198.

Behr, M., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92, 13–15.

Borromeo Ferri, R., & Blum, W. (2012). Learners' conceptions of the use of the equals sign at the interface between primary and secondary education. Contributions to mathematics education 2012, 127–130.

Bruner, JS (1966). Toward a theory of instruction. Belknap Press of Harvard University Press.

Carpenter, T.P., Franke, M.L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann.

Dienes, ZP (1960). Building up mathematics. Hutchinson Educational.

Dooley, T., & Kirwan, A. (2014). Facilitating young children’s understanding of the ‚equal‘ sign. Vortrag auf der NCCA Mathematics Conference, Dublin, 24. November 2014. https://www.ncca.ie/media/2006/ncca-maths-conf_equal-sign_24-11-14-2.pdf

Falkner, KP, Levi, L., & Carpenter, T.P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232–236.

Farfan, G., & Schoen, R.C. (2021). Elementary students' understanding of the equals symbol: Do Florida students outperform their peers? Dimensions in Mathematics, 41(1), 27–38.

Hagemeister, V. (2013). Primary school problems with the equals sign. MNU, 66(7), 393–398.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326.

Knuth, EJ, Stephens, AC, McNeil, NM, & Alibali, MW (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.

Krauthausen, G., & Scherer, P. (2007). Introduction to Mathematics Didactics (3rd ed.). Spektrum Akademischer Verlag.

Mann, R.L. (2004). Balancing Act: The Truth behind the Equals Sign. Teaching Children Mathematics, 11(3), 65–69.

Oksuz, C. (2008). Children's understanding of equality and the equal symbol. International Journal for Mathematics Teaching and Learning, August.

Padberg, F., & Benz, C. (2021). Didactics of Arithmetic (5th ed.). Springer Spektrum.

Prediger, S. (2009). „… nee, so darf man das Gleich doch nicht denken!“ — Lehramtsstudierende auf dem Weg zur fachdidaktisch fundierten diagnostischen Kompetenz. In B. Barzel et al. (Hrsg.), Algebraic Thinking. Festschrift for Lisa Hefendehl-Hebeker (pp. 89–99). Franzbecker.

Selter, C., & Spiegel, H. (1997). How children count. Velcro.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.

Unteregge, S. (2017). Algebraic equality relations in the context of primary school arithmetic instruction. Contributions to mathematics education 2017, 1009–1012.

Van de Walle, JA, Karp, KS, & Bay-Williams, JM (2019). Elementary and middle school mathematics: Teaching developmentally (10th ed.). Pearson.

Wahyuni, R., & Herman, T. (2019). Students' understanding of the equal sign: A case in suburban school. Proceedings of the 1st International Conference on Educational Sciences, 351–355.

Winter, H. (1982). The equals sign in primary school mathematics instruction. mathematica didactica, 5(4), 185–211.

Wittmann, EC (1981). Fundamental questions of mathematics teaching (6th ed.). Vieweg.