# Could a translation error lead to squares to not be considered as rectangles?

I'm reading a certain set of kindergarten/lower primary maths textbooks that is written by North American authors for a European company.

Whenever students are asked to identify the number of rectangles in a given picture, the answer booklet gives the number of oblongs instead of the number of rectangles.

While the topic may be too advanced for kindergarten students, the maths textbooks indeed explicitly say at the bottom of the first page of a textbook at the very first level to tell students that squares are special types of rectangles, where levels 1-4 are for kindergarten students.

Additionally, the accompany guide for teachers devotes a whole page of discussion as to how to teach that squares are special types of rectangles. There's even a paragraph about teaching to kindergarten students. The authors/some of the co-authors of the teacher guides are also authors/co-authors of the textbooks. They have also said that if students are taught that squares are not rectangles, then they will have misconceptions later.

Perhaps, the ones who wrote the answer booklets were not fluent in English while the ones who wrote the textbooks were.

For example

[picture with 4 circles, 2 triangles, 3 square rectangles, 2 oblong rectangles for a total of 5 rectangles]

Circle ___

Triangle ___

Square ___

Rectangle ___

The answer key would give only the numbers:

4

2

3

2

So, the last line is wrong since it should be 5.

Could this happen in German? Or a German dialect? I mean, is there something specific about the translations of any of the following words 'rectangle, square, oblong, quadrilateral, quadrangle, parallelogram, trapezoid/trapezium, rhombus' that would cause such confusion? I guess the translator/s thought that when English speakers say 'rectangle', it means 'oblong in their language/dialect, but I don't see that as specifically a problem for this particular language.

By the way, are squares considered rectangles in Germany? Apparently, these things can vary by state, curricula, culture, time, etc. Please provide a document from the education department of your government or something.

P.S. I'm a monolinguist.

Related:

Are kindergartners supposed to be steered from squares being rectangles?

In what curricula are “rectangles” defined so as to exclude squares?

Why do we have circles for ellipses, squares for rectangles but nothing for triangles?

What are/should kids (be) taught about the colour of the sun?

• Is oblong a common term in English? Not all dictionaries even have a definition for it, for instance Cambridge Dictionary (dictionary.cambridge.org/dictionary/english/oblong). Nov 4, 2022 at 11:11
• @JonathanScholbach 'Is oblong a common term in English?' --> Hell no. at least not in 2022. I did hear it as a kid iirc it meant the same thing as oval.
– BCLC
Nov 5, 2022 at 12:33
• en.wikipedia.org/wiki/Oblong Nov 5, 2022 at 12:37
• This question is really not about the German language. Nov 5, 2022 at 19:23
• @BCLC, you were also able to ask it here, but you explicitly ask for curricula, not language. The people who answered just ignored that. Nov 6, 2022 at 12:21

# "Squares are not rectangles, if both are mentioned"

I purposefully formulated it provocative, so take it with a grain of salt. Anyway:
If someone were to show me1 a picture of two rectangles and three squares, and ask me how many squared and rectangles I see, I'd answer two rectangles and three squares, because I assume the question is focusing on the difference between squares and rectangles.
If you only ask for "Wie viele Rechtecke?" (how many rectangles), I'd say 5, because now there's no difference between squares and rectangles implied.

1I have a master's degree in physics and am about to finnish my PhD, so I argue I have a reasonably sophisticated background in math

# But technically...

Technically speaking, you have the following subgroups:

The whole family of geometric shapes with 4 straight lines and angles (quadrilateral/quadrangle) is called a Viereck (tetragon). The contain the trapezoids (das Trapez), which is a quadrilateral where at least 2 opposite sides are parallel (excluding crossed trapezoids). Each (non-crossed) Trapez is a Viereck.
A parallelogram (das Parallelogramm) is more restrictive, because if requires all opposite sites to be parallel. As such, each Parallelogramm is a Trapez but not vice versa. A rhombus (die Raute or older der Rhombus) requires the opposite sites to be parallel and of equal length, making it a subgroup of the parallelogram. On the other hand, a Rechteck (rectangle/oblong) requires its opposite sides to be parallel but also each angle to be 90 degrees (das Rechteck literally means right corner). As such its a different subgroup of parallelograms. The square (das Quadrat) requires all sides to be of equal length, opposite sides to be parallel and all angles to be 90 degrees. As such, it is a special case of both the Rhombus and the Rechteck.

# But beware of...

However, as stated above, people will typically refer to the more specific version when asked. If you were to say "Das ist ein Parallelogramm" when presented with a rectangle, you'd come across as overly correct or even obnoxious. The exception to this is a Viereck, which does not have the connotation of being a smart-ass when answering the question what shape you are presented with, though it will likely prompt you to be more specific.

German language does not differentiate between oblong and rectangle - both are called "Rechteck".

(There is actually a wikipedia entry for German Oblong, but having gone to school in Germany, and having studied mathematics for a couple of years, I never heard the term before. I think, it is not commonly used.)

One might say "echtes Rechteck" ("true rectangle") in order to refer to a rectangle that is not a square. By strict mathematical definition, a square is a rectangle, and this is the same in German: Every Quadrat is a Rechteck.

But in some contexts, including the one you mention, it is common to say "rectangle" when referring to "true rectangle". This happens in contexts when a difference between squares and rectangles is the topic of the conversation. By the way, the same happens with rhombs too: Every square is a rhomb. But being shown a picture of squares and non-squared rhombs, and being instructed "point at all rhombs", most people will leave out the squares.

I don't think, this is specificic to the German language. It is rather an issue of implicature. The implicature here is that the speaker will use the most specific term that is present in the conversation and hence refer to a square as "square" (and not as "rectangle").

Yes, this happens in German. German does not have a separate word for a non-square rectangle (i.e. for an oblong). So what does the German word Rechteck [= rectangle] mean?

We must distinguish between formal mathematical definitions and colloquial speech. Of course mathematically each square is a rectangle, but I think most people without a mathematical background interpret the German word "Rechteck" as "oblong". If there is a picture with squares and oblongs and people are asked to count the number of rectangles, quite a number will only count the number of oblongs.

However, primary maths textbooks written in German should be mathematically correct and not adopt a colloquial (mis-) interpretation. I think that everybody, including childs, will understand that a square is just a special type of rectangle if it is properly explained (as everybody understands that a beagle is a dog).

As a little digression let me add two more examples where mathematical concepts differ from the popular use of words. I could imagine that it similar in English.

1. A Kreis [= circle] is mathematically a special case of an Ellipse [= ellipse]. Many people will not even know the word Ellipse, they would probably prefer to say it is an eiförmige Figur or an ovale Figur. Literally eiförmig means egg-shaped - and thus no circle will be an ellipse in colloquial speech.

2. A Gerade [= straight line] is mathematically a special case of a Kurve [= curve]. I had endless and fruitless discussions with some people to explain the mathematical point of view; my dialog partners insisted that a curve can never be a straight line simply because the word Kurve includes that it is bent. But I admit that linguistically this is a more extreme case than square and rectangle or circle and ellipse.

Also note that the German word Quadrat and its English equivalent square have a unique feature among plane polygons. We have equilateral triangles, regular pentagons, regular hexagons, etc., but their special role among triangles, pentagons, hexagons, etc., is expressed by an additional adjective and not by an individual word sounding completely different.

Here is a final non-mathematical example which works in German and in English. Imagine a picture with one woman, one cat, one dog and one chimpanzee. If people are asked to count the number of animals, I bet that almost everybody will say that there are three animals. However, there are good reasons to argue that a human being is an animal in the biological sense - in other words, that a human being is just a special type of an animal.

Update:

For those who understand German and are interested in the development of the geometric conceptual understanding in children I recommend to have a look at the article Understanding of Geometrical Concepts in Elementary School Using the Example of Quadrangle, Rectangle and Square:

Bruns, J., Unterhauser, E. & Gasteiger, H. Geometrisches Begriffsverständnis in der Grundschule am Beispiel der Begriffe Viereck, Rechteck und Quadrat. J Math Didakt 42, 581–623 (2021). https://doi.org/10.1007/s13138-021-00185-4

Quote from the article:

The theories presented for the development of (geometric) conceptual understanding assume that learners develop individual conceptual concepts about plane figure concepts that (initially) do not correspond to the formal definitions of these concepts.

That is, the formal-logical definition of the terms quadrilateral, rectangle and square may differ from childrens's understanding of these concepts.

• Even a bigger problem is "Viereck" for rectangle. But the mathematical meaning of "Viereck" is simple "has 4 corners" and could be of any shape. But in common speech a rectangle is called "Viereck"... Nov 10, 2022 at 19:32