Rotated square, “not a square”? Recognising shapes in grades 1 and 2
Why children no longer see a rotated square as a square, what idea lies behind this and how teachers can show shapes in many different ways.
In short: Children often say a square standing on one corner is “not a square any more”, because they tie the concept to a typical picture, usually a square resting on one side. Position, size and colour are not part of the concept. What counts are four sides of equal length and four right angles, and these stay the same when the square is turned. Turning, laying out and sorting with many examples and non-examples help. Mazio Math organises its content by the five content areas, including space and shape, and gives the teacher a flag when a common misconception shows up in the class.
This article is for teachers in grades 1 and 2. It describes typical errors in recognising plane shapes, their causes, support ideas using simple materials, and questions you can use in lessons to find out what a child really thinks.
Typical errors in recognising shapes
Children first learn geometric concepts from a few very typical examples. In picture books and on worksheets the square almost always rests on one side, the triangle has its point at the top and three sides of equal length, and the rectangle lies horizontally. Mathematics education calls this model picture a prototype. A child who only knows the prototype compares every new shape with this picture and pays attention to the overall appearance instead of the properties.
| What the child says or does | Possible cause | What can help |
|---|---|---|
| A square standing on one corner is “not a square any more”. | Position is seen as a feature. | Turn a paper square in front of the child and ask what has changed. |
| A narrow triangle, or one with its point at the bottom, is “not a triangle”. | The equilateral triangle with its point at the top counts as the only triangle. | Make many different triangles on the geoboard and collect them together. |
| A shape with rounded corners or a gap is still called a triangle. | The overall impression counts for more than corners and straight sides. | Sort examples and non-examples into “triangle” and “not a triangle”, with reasons. |
| A large and a small square are “different shapes”. | Size is seen as a feature. | Place shapes of different sizes on top of each other and compare what is the same. |
| A square is “not a rectangle”. | In everyday language the names exclude each other. | Not yet a goal in grades 1 and 2; let children compare the properties of both shapes. |
Why this happens
Mathematics education distinguishes levels of conceptual understanding. At the beginning there is an intuitive understanding in which children recognise a shape as a whole. At the next level they connect the concept with properties and therefore recognise a shape regardless of its size or position. Only after that do children see relationships between concepts, for example that a square is also a rectangle. This is how it is described in a handout on the content area space and shape from QUA-LiS NRW (in German), which comes from the didactic materials for VERA-3, the comparative assessment in grade 3 in Germany.
A child who rejects a rotated square is therefore thinking consistently at the first level and needs a wider range of experiences.
Support ideas with simple materials
Turning and observing. Each child gets a paper square, places it on the table and turns it slowly. The children describe what stays the same: four corners, four sides of equal length. The sides can be compared by folding, and the corners checked with a folded paper right angle.
Sorting with reasons. Lay out shapes in different positions, sizes and colours, together with non-examples that have rounded corners or unequal sides. The children sort them into two groups and say why each time.
Making and drawing. On the geoboard or on dot paper, the children make a square and then “a square that looks different from yours”. Tilted squares often appear by themselves.
Asking diagnostic questions
A wrong word does not yet reveal what a child thinks. These questions help you follow up:
- “How can you tell that this is a square?” Notice whether the child names properties or says “it looks like one”.
- “I’m turning it now. Is it still the same? What has changed?”
- “Can you make a square that looks different from mine?”
- “Is this still a triangle? Why, or why not?”
If a child names corners and sides of equal length but remains unsure about the rotated square, a short phase of turning and checking is often enough. If the child reasons only from appearance, they need more time laying out and sorting shapes.
How Mazio Math supports space and shape in lessons
Mazio Math is a learning platform for maths lessons in primary school. The content is organised by the five content areas of the German national education standards, including space and shape, and designed for the two-year bands of grades 1/2 and 3/4. There are units for grade 1 too.
Finding suitable content. Under “Lessons”, the shelf shows the same content as the children’s learning area. Choose the “Units” section, set the grade under “Grade” and use the search to filter by title, topic or competence. Tapping a card opens the details with the worksheet and teaching notes, where available. With “Open & play” you see the unit as a child sees it. The “Curriculum” page also helps: with the filter “Grade 1/2” and a focus area expanded, “Matching content in Mazio Math” shows which material fits.
An example from grade 2. The lesson starts with paper squares at the tables, as described above. Then you use the plus to put a unit from space and shape on the class board. New content appears on the board as compulsory, and with a note you can add a sentence such as “Look closely at what stays the same when you turn it”. The row shows you how many children have finished. If a child makes a mistake, the learning assistant Mazio helps directly in the task, with a question, with markings in the picture and, if needed, with a given partial step. He does not give the solution.
After the lesson. Under “Results” there is an evaluation for each level of every unit worked on. Once a week Mazio Math checks whether a common misconception shows up in the class’s wrong answers. The support card explains the error in thinking with an example, lists steps for the class under “In lessons” and suggests units and Knobelbox tasks under “Suitable content for the board”. For reference, the library collects common misconceptions, organised by content area. Talking with the child remains important. With shapes in particular, much becomes visible in the reasons a child gives aloud.
A teacher in the scientific study writes: “The children are motivated. The tasks are very inviting. Even the grade 2 classes can already work with them.” The page on the study of the effectiveness of Mazio Math describes what was investigated.
Next step. The page Maths in grades 1 and 2 (in German) describes further typical hurdles. Register for free to create an account, then set up a class and print the access cards. Using Mazio Math with a class, trying it out and licensing it is currently available to schools in Germany. If your school is in another country, registration records your interest and we will let you know as soon as Mazio Math is available there. How Mazio Math evaluates recurring errors is explained in the article How Mazio Math analyses the class’s errors in thinking.
Frequently asked questions
Is a rotated square a rhombus?
Yes. Every square is also a rhombus, because all four sides are the same length. A child who says “rhombus” is therefore right. What is missing is the insight that the same shape also remains a square, because its angles are right angles.
When should children know that a square is a rectangle?
That is set by the curriculum where you teach. In mathematics education, this insight belongs to a later level of conceptual understanding. In grades 1 and 2 the focus is on children recognising shapes by their corners, sides and angles, regardless of position and size.
Is Mazio Math available for grade 1?
Yes. There are units for grade 1 too. With “Open & play” you can check beforehand, as a child sees it, which ones are suitable.