This morning’s water-cooler topic here at the Math Forum: threshold concepts. I’m not sure if it’s a phrase we just coined or not, it’s definitely not original, I think Erik Meyer and Ray Land coined the term, but it led to really interesting conversations and storytelling, more of which I hope will continue in the comments.

Our working definition of a threshold concept: something that changes the way you understand mathematics, which after you understand it, you can’t go back to previous ways of thinking, and you find yourself interpreting and understanding math through the lens of that concept.

We have a hunch that threshold concepts might have been those that were controversial on being discovered (e.g. irrational numbers, negative numbers, infinitesimals…) though I’m not sure that’s a necessary or sufficient condition.

Here’s one example: knowing that the ratio of the circumference to the diameter of a circle is π is not necessarily a threshold concept (it can be as boring as knowing the first 20 digits of π, if you just think of it as a fact). But understanding that there is a constant relationship between the circumference and diameter of a circle that can be described by a number that cannot be measured, represented as a fraction, or written out completely (and yet can still be represented, calculated with, etc.) is a big deal. When you encounter numbers like e and √2, you might understand them in terms of your understanding of π. You might go out looking for more numbers like π. You might begin to wonder about the concept of limits. A new door in mathematics has been opened, and after going through it, you can’t go back…

What other concepts might be threshold concepts? Do you have a personal story of a new door in mathematics opening up? And, as educators, should threshold concepts inform our teaching? How?