Grant Wiggins threw down a challenge, and it has taken me so long in thinking about it that there have been responses and followups already. I hope I am not too late to the discussion.

I have a different idea of “Big Ideas” than Mr. Honner or Mr. Wiggins, and I think it fits Mr. Wiggin’s idea of what a big idea should look like.

#### My first big idea is Inverses & Identities.

I want to start with a statement Mr. Wiggins makes in his response, “I am looking for those ideas that are big – powerful and fecund – for both novice and expert.”

Does the circular idea of Inverses and Identities meet this criteria? I think it does.

In elementary school, learners focus on the most basic inverses, addition and subtraction, before moving on to the still basic, multiplication and division. In Algebra 1, these inverses are used to solve equations through the application of identities.

Every equation solving is about the application of inverses to get the identity, whether it is just adding and multiplying or using roots (exponent of 1/2 times exponent of 2 = 1: how many learners never see this because of the radical sign!) or trig functions, all the way up to derivatives and integrals. All of solving can be viewed through the big idea of inverses and identities.

We can extend the idea of inverse and identity to functions as well. This explain WHY two functions are inverse if f(g(x)) and g(f(x)) both equal 1x . It isn’t the “x” that is important in the composition statements, it is the ONE!

Viewing algebra through the lens of Identity and Inverses allows us to see why we do mathematics the way we do, from arithmetic all the way through trigonometry and calculus and beyond. Abstract algebra makes use of these ideas as well in the structure and definition of operations even.

Novices and experts alike must use and understand just how powerful the concept of inverse and identity are in doing mathematics. I think this is a “big idea” in the spirit of Mr. Wiggins definition.

#### My second big idea is Transformations

I have been trying to make a lot of connections in Algebra class, and I think the idea of function transformations fits this definition as well. What do I mean by transformations?

These transformation rules are shocking to a beginner when they realize that all of the functions have the same properties for all transformations.

And even the expert makes use of the fact that there really is only one set of math “rules” through all of mathematics. The connection between them is demonstrated through the use of the transformation properties and can be taught through the properties. The underlying structure of mathematics that the expert uses to is still evident as well.

I am not sure if this completely fits the big ideas that Mr. Wiggins was asking for, but I know these are ideas that I have been working towards as big ideas for my classroom and my own thinking.

I know my learners have responded to the use of Identity and Inverse in solving and graphing, and the transformation rules concept has definitely had an impact on the learning of mathematics in my classroom.