So Grant Wiggins threw down the gauntlet. And Patrick Honner, as Patrick Honner is wont to do, picked it up. And then Grant Wiggins -- I'm not totally sure what traditionally happens to a gauntlet at this point -- did some other gauntlet-related thing in reply. It was fast and furious. Actually, it was incredibly civil and well considered, which spirit I will try to preserve here.

For the tl;dr crowd, Wiggins posted a celebratory 100th-blog-post rant against Algebra (the course, not the content). In that post he challenged Algebra teachers to name Four Big Ideas contained in the curriculum. And Honner responded with some pretty solid candidates:

- Algebraic Structure
- Binary Relations
- The Cartesian Plane
- Function

Wiggins quasi-stipulates to a couple (binary relation, Cartesian Plane) -- with much qualification -- and more or less dismisses the others (quite politely). It seems that Wiggins agrees all those ideas are *important*, but he has a very particular notion of what makes for a Big Idea:

So, I wish to up the ante. To me a big idea is big for both: I am looking for those ideas that are big – powerful and fecund – for both novice and expert.

I always return to this simple example from soccer:

Create dangerous space on offense; collapse dangerous space on defenseis a big idea at every level of the game, from kid to pro. And it is transferrable to all space-conquest sports like lacrosse, hockey, and basketball. Truly big. [emphasis in original]

I'm not interested in debating the general lifelessness of high school Algebra, which strikes me as largely uncontroversial at this point in the conversation. In fact, the same rant could just as easily be applied to mathematical instruction at almost any level (Wiggins even includes the obligatory quotation from *A Mathematician's Lament*). I'm also not interested in trying to produce more convincing examples. Nope, I'm interested in talking about soccer.

I think the soccer analogy is on the verge of making it impossible to have a meaningful discussion about math education. Not just Algebra, but *mathematics.* And here's why: the creating and collapsing of dangerous space might be the **only** Big Idea in soccer. I submit that, if *that's* the standard for bigness, then there just aren't four Big Ideas to be had.

If soccer were taught like math, then you might take a course called Moving Without the Ball I as a freshman. And in that class there would be a unit about Overlapping Runs. And you would probably hate it, because it would be an awful lot of running, and you wouldn't ever be sure why you were *doing* all this goddamn running, because maybe your coach isn't overly concerned about unveiling to you the beautiful truth that a good, long overlapping run pulls a defender way down into the corner and stretches the whole defense and creates some dangerous space for a midfielder to run into. And you would moan and check the calendar for when you were starting the Passing Into Space unit, because you heard it was totally easy -- mostly just standing around and pushing nice, easy passes toward cones. But that really says more about your coach's ability/willingness to keep your eye on the Truly Big Idea (because hey, he's pretty much coaching the way *he* was coached in the first place, back when kids just shut up and did their Backwards Jogging homework without complaint) than it does about the bigness of the Pretty Big Ideas that you're working your way through, because the Truly Big Idea is just too ungodly huge to be useful in making you a better soccer player. After all, there's just the one.

In an attempt at analogic involution, I'm going to try to come up with a mathematical analogy for the soccer analogy for math:

I always return to this simple example from mathematics:

Create structure when you're building; look for structure when you're exploringis a big idea at every level of the subject, from kid to professional mathematician. And it is transferrable to all structure-having systems like language, chemistry, and logic. Truly big.

Maybe structure is the only big idea in math. At least the only one Wiggins might agree to. And that's just too big. What we really need in Algebra are some Pretty Big Ideas. So here's *my *gauntlet, which is admittedly significantly lighter than the one that's currently being kicked around: what are four Pretty Big Ideas in Algebra? Honner got you started.

Grant WWhat a great post! I just love the smartness, humor, and usefulness of this.

I like your line of argument - TOO BIG idea – and i agree we need some pretty big ideas in algebra (and any subject) to give the learner lighthouses and connective tissue for learning and meaning-making, but I want to stick with soccer since you threw down the yellow card, so to speak....

Here are three more ideas that are central to well-played soccer:

(Legal) deception is key to overcoming the equal number of players opposing your every move

Games are lost more than won

Luck is perhaps more important than skill - so shoot often

The more that we go back and forth in this happy exchange, the more I begin to see something important. I am more on the side of Polya than those who would identify big ideas as concepts of the math proper only. The 'game' is problem solving - in algebra or calculus. So, for me any pretty big idea is some combination of problem solving and the elements of the system that permit more of it or more success in doing it.

That's why 'using equivalences' (via the core properties) is a big idea to me; and that's why 'algebraic structure' stated just so is not.

That said, i think a virtual communal discussion of pretty big ideas is sorely needed to help teachers just marching aimlessly through the textbook. That was always the cause: identify intellectual priorities in the course to which the work keeps returning, spiral-like.

Patrick GrewI really connect with the Pretty Big Idea approach. And, yes, there are times in classrooms where it seems only the teacher knows where the class is headed and that direction has an end goal governed by the curriculum.

I am not sure that many high school teachers can relate to the Big or even Pretty Big Idea idea. As has been stated, this is likely due to the way they 'were coached soccer' as kids.

If we dicker too much about what a Big Idea is then we risk scaring off the timid teacher only to find them teaching Math with their door closed. The fact is that many teachers are willing to try this approach but afraid that their Pretty Big Idea will not be viewed positively.

Judging the bigness of a teacher's idea may certainly promote this behaviour and even worse shut down the upbeat historical discussions of best practices when teaching traditional math concepts. All because no teacher wants to be seen as inadaquate to his/her peers.

I am not sure who would be the referee of Big (or Pretty Big) ideas. I am not sure whose definition of such is correct. But I am sure that the person who loves and has a passion for the whole idea of mathematics will engage students in their classrooms. These students will become excited about the math they are learning. They will begin to ask creative questions and then maybe from these inquiring minds we will discover the Pretty Big Ideas that are important to them.

Dan MeyerTom Sallee is an emeritus math professor at UC Davis. He wrote the pretty-good curriculum CPM and is generally about as thoughtful on these issues as they come. This happened in a session of his a few years back:

Grant seems to put a fairly high bar on "Big Idea," which is probably better than a low bar. I'd be thrilled, though, if math teachers could just be persuaded to look for these thick through-lines in whatever course they taught. We could work on increasing the thickness of those lines later, but to agree that coin problems and mixture problems represent the

same kind of problemwould be a huge advance.Max Ray (@maxmathforum)As a 100% soccer novice one thing that I notice is that at no point do y'all list as a big idea in soccer "you don't touch the ball with your hands" nor "you try to get the ball into the goal." I'm hypothesizing that that's because those are the content of soccer, not the big ideas of the doing of soccer. And once you know those content things you're like, okay, fine, but how?!? How would I ever actually get good at *doing* that?

I have definitely seen students approach Algebra that way -- okay, I kinda see what this is, but how would anyone ever think of that? How would I get good at it? What does it even mean to be good at it?

[Side note: watching soccer is really boring when you can't even see that a space is dangerous, let alone see how people are collapsing & growing it]

So here are 3.5 more (pretty?) big ideas in Math then that might correspond to Grant's big ideas in soccer, and that flow from "Create structure when you're building; look for structure when you're exploring" (and that I didn't steal from Kate, or Glenn, so if they overlap that's a good sign):

Write the same thing in different, equivalent ways, and look for ways to determine if different-looking things are equivalent.

Be as concise and organized and precise in your notation as is humanly possible. Corollary: be precise in your definitions of things especially.

Lemma following 1 and 2: extend things like definitions and notations to the breaking point to find new equivalences and faults in equivalences.

Focus on and describe *relationships* among quantities [see above for tips on how to do that]

If students can do that with the content of algebra (Letters that are like numbers! Graphs! Equations! Functions! Number systems! Algebraic structure!) then they're probably doing awesome.

What's missing, I guess, are the big ideas that make Algebra different than Topology or Geometry. Like the big ideas that make soccer different from lacrosse or field hockey... Although those levels of ideas may be more useful to the expert (for example, be on the lookout for the existence of inverses, identities, closure, and the number and type of solutions whenever you meet a new operation or family thereof is sort of like do all those big ideas of space-contest sport only without using your hands, with a head-sized ball, and a garage-door sized goal. Go.).

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