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, allowing for arbitrary given-sought-problem statements.
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 + 3build it up directly and compare it with other terms. - Use negative shells. Subtractive relationships such as
9 − 4Represent materially and symbolically in parallel. Removed quantities are compensated for by minus signs. - 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. Pull blinds over individual bowls or numbers to make guesses 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?
- Represent the equation "3 + 4 = 7" on the scale — and then "7 = 3 + 4". Compare both. Why can the sides be switched?
- 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?
- True or false: "The scale remains balanced if I add the same number to each side." Explain.
A more extensive collection of sample tasks is available in the app itself (Info section → „Sample Tasks“).
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 are frequently rejected as "wrong" or "not allowed" (Behr et al., 1980; Falkner, Levi & Carpenter, 1999; Knuth, Stephens, McNeil & Alibali, 2006). Falkner et al. (1999) report that only about a quarter of sixth graders complete the task. 8 + 4 = ☐ + 5 correctly solves. A significant portion is either 12 or 17 One. Both solutions are typical indicators of a purely operational interpretation.
This phenomenon has been documented across language and school systems: for German-speaking learners by Borromeo Ferri and Blum, by Hagemeister (2013) and by Unteregge (2017); in English-speaking countries by Carpenter, Franke and Levi (2003) and Prestwood (2017); in other contexts by Oksuz (2008), Wahyuni and Herman (2019) and Farfan and 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 („= means: calculate“) and the relational interpretation („= This means: both sides are of equal value“ – and should be established as early as elementary school. Borromeo Ferri and Blum, as well as Unteregge, emphasize that the 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
The balance scale is the most widespread enactive-iconic material for developing the relational meaning of the equals sign in both German-speaking and English-speaking countries (Wittmann, 1981; Mann, 2014; Dooley & Kirwan, 2018). It has two key didactic advantages:
- 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 balance beam, as they always tilt in the corresponding direction or are horizontal when equal.
- 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 (2014) describes the equals sign in this context as "a balancing act," a metaphor that is literally implemented in the digital realization of the present app: The bar tilts as soon as the quantities do not match and realigns itself horizontally in real time when equilibrium is established. They are therefore ideally suited as experimental environments because operations and their effects can be directly experienced in accordance with the operational principle.
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.
The freedom to choose between bundles of 5 and 10 allows connection to concepts of the „Five-Frame“ and „Ten-Frame“ material (Van de Walle, Karp & Bay-Williams, 2019), which is used as a bridge between substitutive quantity perception and place value representation.
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: as a spoken sentence on request ("Three is smaller than four"), optionally also via speech output.
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 | Experiencing subtraction as "taking away" with the same understanding of the operation as addition; the total amount remains visible, but only the remaining amount on the scale "counts". |
| 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 because it depends on physical conditions.
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. (2018). Facilitating young children's understanding of the 'equal' sign. Mathematics Education in the Early Years, Special Issue, IPC.
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.
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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.
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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). „…no, you can’t think of it like that!“ — Teacher trainees on the path to subject-didactically sound diagnostic competence. In B. Barzel et al. (Eds.), Algebraic Thinking. Festschrift for Lisa Hefendehl-Hebeker (pp. 89–99). Franzbecker.
Prestwood, SP (2017). Children's understanding of the equal sign [Unpublished manuscript]. Georgia College and State University.
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.