As a point of interest, as I googled to find a picture of “geometric shapes” to place here, 99% of all the shapes were PERFECT TEXTBOOK shapes. So, it is easy to understand why children have a misunderstanding about “what is a triangle?” when many of the examples are all equilateral triangles arranged with one side “sitting” horizontal and its point at the top. Wow!

**What are my definitions for these geometric terms? **Prior to reading the articles, I believe that a triangle is a closed figure with three straight lines and three angles that all add up to 180-degrees; a rectangle is a closed figure with 2 pairs of sides that are congruent and parallel and 4 90-degree angles; and, a square is a special rectangle that has four straight sides that are congruent and parallel.

**What is the difference between a definition and a list of properties/attributes? **Before reading the articles, I would think that a difference between a definition and a list of properties or attributes is not much. Maybe one might be more technical that the other, but both should contain the same type of information. When the students were asked to write the name of a shape and the properties, they rephrased it as “the rules of each different shape.” Interesting that children would refer to these as rules to be followed, just like rules of the classroom and rules of the game. Interesting …..

**What is the purpose of a definition? **Definitions provide us consistent agreed upon descriptions so that everyone understands what is being discussed.

What specific issues do they (students/children) need to consider in order to make sense of definitions for triangle, square, rectangle, and parallelogram?

When students are considering how to categorize a shape and define its properties, they must include their “scheme” meaning that they can only apply these RULES as much as their experiences provide. So, if some students have more experiences with more and different triangles, they will probably understand the overall concept of a triangle better than others.

What is the process they (children) go through as they learn to apply their definitions?

Looking beyond the specific geometric content of this set of definitions, how do children develop a sense of the purpose of a definition?

**Natalie’s Case 20** – In this case with 1st and 2nd graders, the students are realizing that as they discuss the definition for a square, it is the same definition that they had designated for a rectangle “four sides, four corners, four angles, and it’s a square.” But then “it’s a square” becomes part of the definition to help in understanding. But, then, one of the children points out that “Actually, you don’t need to say four corners and four angles; they’re the same thing.” This is a true statement, but can the students see it this way? When the square is turned so that it now looks like a diamond, some of the students say it is different, not a square anymore. And, then Charlie draws a random figure on the board that fits the “definition” but is not a square as they all agree. So, the children understand that they must refine their definition. Through their continued discussions, they decide that a square must have “angles that are the same.”

**Delores’s Case 18 and Andrea’s Case 19** – In the 3rd grade class (Delores) Mei-ying did not like that the triangle was “upside down” and that it was “skiny” but did agree that these nontypical figures were triangles. She did not let the sideways orientation of the rectangle bother her understanding or identification. Francie would not identify shape 1 as a triangle because “I know my shapes” but she would concede that shape 2 was a rectangle. Louis did identify shape 1 as a triangle but did make sure to note that is was positioned “upside down. It kind of doesn’t look like a triangal.” But he was not worried about the orientation of the rectangle either. Finally, Shannon wouldn’t identify shape 1 as a triangle but rather insisted that it “looks like a carrot. It has two long sides and 1 short side. I think it’s a real shape but I don’t know the name of it.” Again, the orientation of the rectangle did not bother this student in identification. Interestingly enough, the 2nd grade class (Andrea) seemed to have a more “defined” discussion of the shape dilemma despite their younger ages. “Well, I think *tri* means three, so triangle means three.” When asked three what, the response is “angles.” It was further discussed that an angle is when two “straight” lines meet. So the discussion about STRAIGHT lines again becomes a focus. “And triangles have to have points. They’re pointy, not like curves on the ends.” “Three points.” And then the children begin to discuss the overall LOOK of the triangles. One child disagrees that a shape is not a triangle because “It’s too stretched out.” But, then another child (Evan) makes a profound statement that “If I’m all stretched out and turned upside down, I’m still Evan.” WOW! Now that seems like true understanding to me. The students go on to discuss “slanty” sides and the ability to use these triangles as ramps, and that if you turn them around, they look different and more triangle like. So they are grappling with the idea of right triangles and how it looks “different” when they are joined. Finally, one child describes them as “It has 3 side, 3 corners, 2 slants, 1 strate side. If you turn it all around it will still be a triangle.”

I loved reading these articles because, while I don’t think my idea of what a triangle or square is has changed, I loved being able to see what younger kids thought when they heard the word “triangle” and rectangle”.

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