Why is it that students find it easier to calculate the area of triangle ABC but will have difficulty calculating the area of triangle DEF? Middle school students even believe that it’s impossible to find the area of DEF because the triangle has no base and height!

That knowing the invariant properties that makes a triangle a triangle (or any geometrical shape for that matter), is not an easy concept to learn is illustrated by this conversation I had with my 4-year old niece who proudly announced she can name any shape. The teacher in me has to assess.

Thinking about how a four-year old could possibly think of these meaning of the shapes made me ask:** If four-year olds are capable of thinking this way then why do we think that there are students who can’t do math or doubt the idea that algebra is for all? **

This is an exciting topic! Something I’ve written about for college and university faculty: pre-school children developmentally don’t perceive that a shape is a triangle unless it is an equilateral triangle with its base on the horizontal. And how do we help them see otherwise when they get to school?

But…that middle grade students couldn’t mentally reorient a right triangle and see that it had a base and and altitude? That someone needs to tell them to rotate the page until they can “see it”? That floored me!

Second is the pervasiveness of technology in the world of preschoolers with access to computers. Those little ones are our REAL “digital natives”: “Play.” “Download.”

PS: This triangle topic hits a chord with me, b/c it reminds me of being 5 years old. My introduction to the word “triangle” was musical, not mathematical. I thought they named the shape called a triangle after the little musical instrument I hit with a metal stick in my Kindergarden “rhythm band” of percussion instruments.

Bottom line: Young children desperately try to make sense of the world given the contexts they encounter.

Thank you for this wonderful article!

Isn’t the reason for this not that kids don’t get it, even a four year old, but that when we introduce a triangle, we always introduce it in the standard position (your first picture for your four year old niece). And continue to use that illustration throughout the learning, so that we basically force the learning that a triangle looks a certain way. Here’s where I think introducing concepts before attaching formal words and definitions and algorithms is important and where technology can be very useful. A static picture cannot truly demonstrate all the possible triangle (or any shape) positions, but dynamic math software such as Sketchpad can introduce a triangle, and then move that triangle to hundreds of different positions and orientations in a matter of seconds, providing a much broader understanding of a triangle, which then leads to a broader understanding of the algorithms we eventually attach to the representations.