I'm reading a certain set of kindergarten/lower primary maths textbooks that is written by American authors for a non-American 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:





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

Could this happen in French? Or a French 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 France? 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.


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?

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    It's not clear from your question where the translation error could have occurred. Are you saying that the answer booklets were translated while the textbooks were not? Who translated what and why? Mar 23, 2018 at 13:03
  • @guillaume31 I'm not sure myself. It could be 1. textbooks english originally then read by non-fluent english 2. textbooks non-english originally then translated then read by non-fluent english 3. non-english textbooks read (either by fluent or non-fluent of that language) though that's not the one i came across 4. in any case, it could be that the ones who wrote the answer booklets were not the ones who wrote the textbooks 5. any valid combination of the previous
    – BCLC
    Mar 23, 2018 at 14:58
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    Now I'm confused. Is it a hypothetical question? It sounded like you'd had these booklets in your hands before. You must surely know which language they are in? Mar 23, 2018 at 15:13
  • @guillaume31 I do know the state the company belongs to, but I don't want to say it out of fear of identification. Thus, I'm posting to various language SE sites. Here, we assume American authors wrote this for a French company. I've seen only English versions. I am not aware that these texts have non-English version. So I don't know where exactly the translation error may have taken place. In Spanish SE, a user suggests the people who wrote the answer booklets are not the same as the main (American) authors.
    – BCLC
    Mar 24, 2018 at 10:22

2 Answers 2


Yes, the confusion is also possible in French, as is arguably the case with every language that has corresponding terms for square and rectangle as we understand them in English.

The mistake comes from the fact that when a human sees

Square ___

Rectangle ___

they will tend to interpret "Rectangle" as "Rectangles that are not squares, since Squares are accounted for just above".

I'm not sure you'll find Western languages that can make this unambiguous.


To clarify: Rectangle in French mathematics means "inclusive" rectangle, i.e. squares are a subset of rectangles. That's what you learn in school. See https://fr.wikipedia.org/wiki/Carr%C3%A9.

But the natural tendency to think that Rectangle presented alongside Square means "non-square rectangles", remains. So a French translation of these instructions would remain as ambiguous as the original in English, because on the one hand you've got the formal definition and on the other hand, what your mind will spontaneously interpret.

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    It seems you are right about Spanish. I didn't know that. However, I can confirm that rectangle in French does mean "inclusive" rectangle (see my edit). I know it's the same in Dutch (vierkant / rechthoek). Mar 23, 2018 at 9:30
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    To answer your question - no, translation errors from English to French are not possible in that regard. The ambiguous English instructions you gave as an example will carry over as equally ambiguous in French. Mar 23, 2018 at 9:37
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    1. "No" to what? My "yes" was a "yes" to : same confusion possible in English and French. 2. and 3. French doesn't have a word for exclusive rectangle, unlike Spanish. French and Spanish are different in that regard. 4. Yes, I disagree with @jiliagre and so does Wikipedia Mar 23, 2018 at 11:26
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    Spanish Wikipedia states En algunas traducciones, no aceptadas en general, el cuadrado es un rectángulo de cuatro lados iguales, and there is no mention of a cuadrado being a specialized form of rectangulo, which speaks in favor of an "exclusive" definition of rectangle in that language. Mar 23, 2018 at 11:29
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    1. No, the confusion exists in the original text ; translation into French doesn't add or remove any of it. 5. I don't know enough about Spanish ; guess you'll have to ask on spanish.stackexchange.com ;) Mar 23, 2018 at 11:38

While technically a square is indeed a rectangle too, French kindergarten students (and more generally people outside mathematicians) are very unlikely to consider it as such.

Here is an excerpt of a FAQ for teachers of students from 6 to 9 years old (CP - CE2 / 1st grade to 3rd grade)

Par exemple, il n’est pas d’usage de dire qu’une table carrée est rectangulaire, pourtant, du point de vue mathématique, c’est vrai. En mathématiques on peut dire qu’un carré est un rectangle. Ainsi dès le cycle 1 le nombre de côtés des polygones peut être utilisé pour différencier les formes. On peut également travailler, même si cela peut paraître ambitieux, le fait que les rectangles et les carrés appartiennent à une même famille, celle des quadrilatères (figures à 4 côtés) ayant quatre angles droits.

We do not use rectangle allongé (oblong rectangle) to describe a geometrical form in French. I'd even expect carré allongé (oblong square) to be used in that case.

For example, there is a monument in Nîmes called la maison carrée, which is not squared but rectangular. Until the 16th century, carré was used for all rectangular forms so an oblong rectangle was a carré/quarré long while the modern square was called carré/quarré parfait (perfect square).

A word that encompass oblong rectangles and squares is quadrilatère (quadrilateral), but some quadrilaterals like losanges (rhombus) are not rectangles.

Including carrés in the rectangle class might only be teached at the end of primary (CM2 - grade 5) or possibly later, e.g.:

Nous avons commencé par du calcul mental, ensuite des problèmes de proportionnalités. Puis nous avons travaillé sur la carré, le rectangle et le losange. J’ai appris que le carré était une sorte de rectangle et une sorte de losange. Nous sommes ensuite sortis de la classe.

Here is a also as document stating the fact carrés and rectangles are exclusives concepts in French.

Apport pour les apprentissages de l’explicitation des relations d’inclusion de classes, Laurence Dupuch* et Emmanuel Sander, Université Paris 8

Par exemple, par l’analyse de la phrase «Est-ce un carré ou un rectangle ? » , Politzer (1991) montre que l’usage commun du français place les carrés et les rectangles en relation contrastive : ils forment alors deux classes disjointes si bien qu’un locuteur n’a aucune raison de croire qu’une forme rectangulaire puisse être carrée. Ces deux classes n’ont pas lieu d’être réunies et sont également disjointes de la classe des parallélogrammes et des losanges. L’organisation conceptuelle est alors formée de relations exclusives lexicalisées : quadrilatère – parallélogramme – rectangle – losange – carré. De ce fait, la relation inclusive n’est pas lexicalisée, alors que du point de vue de leur définition mathématique, qui est parfois, mais pas systématiquement, énoncée de façon explicite dans les manuels scolaires, les carrés sont des rectangles : «Un carré est un rectangle qui a des côtés égaux » (Math élèm. manuel CM2, 2000, p. 18).

  • jlliagre, thanks! So your answer is that you think that it's possible (not necessarily probable) that this could be a French language issue? Cf: In Mandarin, the term for oblong (exclusive rectangle) is 长方形(的) but the term for rectangle (I guess inclusive rectangle) is 长方形. Thus, for Mandarin, it's possible the answer booklets were mistaken due to language issues rather than geometric misconception issues. Cf: For Spanish, it maybe that it's unlikely, language would be an issue because Spanish doesn't have a word for inclusive rectangle. Now: What about French?
    – BCLC
    Mar 23, 2018 at 9:17
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    I guess many French people and even more young students do not consider a square to be a rectangle at all. Exceptions would likely be due to a recent lecture stating that mathematical fact.
    – jlliagre
    Mar 23, 2018 at 10:19
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    @jiliagre the last excerpt you added highlights the fact that language definition and mathematical definition are indeed at odds. But maybe we could agree about the answer to BCLC's original question - that a French translation wouldn't change a thing about the comprehension of the exercise by students compared to the English version? Mar 23, 2018 at 13:08
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    There are far more chances that booklets don't consider squares to be rectangles because they follow the same intuition that students have than because of language specifics, if you ask me. Mar 23, 2018 at 15:22
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    FWIW, I just made a small poll with five kids playing in front of my house, four 10 years old in CM2/last primary grade, and one 8 years old and they were all definitive about the fact squares are not rectangles. One told me a rectangle might be made by two squares joined by one side. None of them accepted rectangle carré as a correct description of anything.
    – jlliagre
    Mar 23, 2018 at 17:42

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