# Are kindergartners supposed to be steered from squares being rectangles?

Question 1: What are the literature, status, debates, references, etc regarding this matter please?

Apparently, some (woohoo weasel words!) consider that squares are rectangles too advanced a topic for kindergartners so either

1. The topic is avoided
2. Kindergartners are allowed to think squares are not rectangles while the teacher does not confirm or deny such.
3. Kindergartners are taught that squares are not rectangles.
4. Same as #2 or #3 but the same kindergartners would be taught that squares are rectangles later on. In the case of #3, this is an outright contradiction.

Question 2: Which of the above 4 cases are acceptable? Which are not? Why/why not?

Related:

In Korea, are squares considered rectangles?

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?

• I suspect that many K-6 teachers do not understand that a rectangle is a square and would not understand this fact if someone tried to explain it to them. – Ben Crowell Sep 8 '20 at 14:43
• @BenCrowell I think you meant "a square is a rectangle". – Mark Fantini Sep 8 '20 at 23:53
• @MarkFantini: yes, thanks – Ben Crowell Sep 9 '20 at 13:22

Kindergartners are generally at an early stage of geometric development, in which shapes are recognized by how well they resemble prototypical images, rather than by whether or not they conform to a definition. Thus, for example, the shape on the left below is likely to be recognized as a "triangle" (despite the fact that it has four sides), the shape in the middle is is unlikely to be recognized as a "triangle" (despite the fact that it has three sides), and the shape on the right is likely to be recognized as a "diamond" (and not as a square, despite the fact that it is a regular quadrilateral). In the Van Hiele model, children at this level would be described as being at the "visualization" level, or Level 0. In order to recognize that a square is a rectangle they would need to be at the "abstraction" level, or Level 2, at which hierarchical relationships can be understood.

I would say that one of the goals of early childhood education (roughly K-2) is to move kids from Level 0 to Level 1, with the transition to Level 2 taking place in later elementary (say grades 3-5 or 6). While certainly some kindergartners are developmentally ready for understanding some hierarchical relationships (e.g. "poodles are dogs, and dogs are mammals"), I think expecting this to be a goal of kindergarten (something they are "supposed to learn") is unrealistic.

Parenthetically, it's probably worth mentioning that the Van Hiele model is widely regarded by math ed researchers as outdated, overly simplistic, and extremely reductionist. However,

much as Euclidean geometry maintains a stable position in the secondary curriculum despite the efforts of some reformers of the early and mid-twentieth century to jettison it in favor of more modern approaches to the field, so too does the van Hiele theory continue to play a dominant role in the discourse of students’ thinking about geometry, notwithstanding several decades of critique regarding its value and validity as an empirically-grounded theory.

(Source: Herbst, Fujita, Halverscheid and Weiss, The Learning and Teaching of Geometry in Secondary Schools, pp. 92).

There are decades and decades worth of research on these matter, but an accessible point of entry might be:

Hannibal, M. A. (1999). Young children's developing understanding of geometric shapes. Teaching Children Mathematics, 5(6), 353+.

In fact, the entire issue of Teaching Children Mathematics in which that article was published may also be of use, as it was a themed issue on geometry.

• @BenjaminDickman Whoops! Fixed. – mweiss Mar 21 '18 at 0:56
• @BCLC Added a couple of references, including one to a book I co-authored. – mweiss Mar 21 '18 at 1:37
• @BCLC Changing the question and then asking the author of an excellent answer to change their answer to match it feels to me like a poor use of this platform. It also dissuades others from answering your questions in the future. – Chris Cunningham Mar 21 '18 at 15:20
• Prototype theory says that people of all ages, even all adults, primarily categorise things according to prototypical examples. So there's nothing developmentally early about children categorising shapes this way! – curiousdannii Mar 23 '18 at 8:10
• @BCLC Because I was replying to what mewiss wrote in this answer, not the question as a whole. – curiousdannii Mar 27 '18 at 7:53