# Rich Math Tasks

Thursday evening I asked Frank Noschese (@fnoschese) and Elizabeth (@cheesemonkeysf) a question about whether they thought there was any value in students using a reductive, drill-and-kill math practice exercise platform, provided that it was accompanied by rich tasks and assessments in a classroom that demands synthesis and critical thinking, and provides students with opportunities to develop mathematical habits of mind.

I received a reply, but I’m more interested in a tangential question @NatBanting asked in the middle of the exchange:

I’ve been throwing that phrase (“rich tasks”) around a lot more lately, in tweets, blog posts, workshops, and conversations at school and at home (my wife is either amazingly longsuffering or genuinely interested—maybe both?—when it comes to talking about math education). So what exactly is a “rich task”? What do other people have in mind when they use the phrase? What do I mean when I use it?

I told Nat that I’m still working that out in my own mind, but I think it’s time to clarify what I mean, if for no other reason than to have a better sense of what I’m looking for when I try to find, adapt, and/or create rich tasks for my own students.

#### Here’s What I Do Mean

A rich mathematical task is one that…

1. Has a low floor and a high ceiling. The first of many ideas I’ve stolen from others, this one from so many sources I don’t even know who to credit anymore. Most recently, Dan Meyer has me thinking about this in his Makeover Monday series. The bottom line: everyone can start, no one can truly say they’ve exhausted the problem’s potential (at least not in a 50-minute period).
2. Has multiple entry points, invites use of multiple representations. Student A starts by exploring numerically, Student B begins by investigating graphically, Student C jumps in by reasoning algebraically, and I don’t have to tell two of them that their approach is a dead end because—even if they don’t always make it—there is fruitful territory a little further down the path in any of their approaches.
3. Has multiple solution paths, provides opportunity for rich discussion. If there’s only one way to solve the task students lose out on the rich discussion of making connections between various approaches and teachers lose the opportunity to build a mathematically coherent, concrete-to-abstract storyline as they orchestrate these discussions.
4. Integrates multiple topics. I owe a lot of what I’m thinking here to a single word Daniel Schneider used in a post about assessment. After my initial foray into standards based grading left me dissatisfied with an overly fractured curriculum, I’m now placing a high priority (philosophically, at least) on tasks and assessments that bring multiple topics together. A rich task, in my estimation, should demand that students wrestle with multiple topics from multiple domains (if I can use the term in the CCSSM sense).
5. Engages student interest, is mathematically/cognitively challenging. I’m a little mixed up here, because I believe engaging students’ interest is massively important, but I’m not ready to throw away tasks that fail to generate buzz among students if I know they nevertheless provide great opportunities for exploration and discussion.

#### Here’s What I Don’t Mean

To clarify what I think a rich task is, I’ll share a few thoughts on what I think a rich task is not:

1. A well-crafted, but constrained guided-discovery activity. I value this kind of activity, but when students are guided along a specific path to a specific goal, it pushes the lesson into a different category for me.
2. A thoughtfully constructed lecture or an engaging presentation. If the instructor is doing the heavy lifting during class, I would say the students are not engaged in a rich mathematical task. I’m not opposed to heavy lifting, especially in preparation outside of class, but students need to play an active, central role in exploring/solving/reporting if I’m going to use the “rich task” label for an activity.
3. A challenging problem for which students already have a tried-and-true method. I have a large stack of started-but-not-finished books, some related to math and education, others not. George Polya’s How To Solve It is on the list, though I’ve read enough of it to be provoked and inspired, particularly by the distinction Polya provides between a problem (solution/method not known) and an exercise (solution/method already known).

#### I’ll Close with a Link…

Incidentally, after typing out the post, I Googled “what is a rich mathematical task” and found this.

#### …And an Invitation

I would love to hear what you think are the key characteristics of a rich mathematical task. Drop a line in the comments or send me a line on Twitter.

## Comments 4

1. What a terrific and timely post. I am preparing my thoughts for opening department meetings in a little over a week and I now have some important ideas to chew on. Two things here that I love. (1) Synthesis – This is CRUCIAL whether you are teaching 3rd grade or Algebra I or AP Calculus or anything in between. I feel that it is SO important to ask people to draw upon past knowledge, not just the present ideas with which they are engaged. Ideally (and this is more possible at the higher levels) you can ask students to reach out to ideas and skills from other subjects as well. (2) Any mention of Polya makes me happy. Trying to instill either through modeling problem-solving behavior or by explicitly outlining an appropriate strategy such as the one Polya presents can give the students a powerful base on which to grow. I need to organize these thoughts rattling around my head to share with my department. Thanks for setting off sparks!

2. Quick follow up thought – I agree with your original supposition that there is still a place for some skills-based practice for our kids. The discussion I’ll be having with my department about this will go like this: (1) We need to find time/space for rich, challenging tasks for our students. That space should (almost) always be when we are together in class and can share ideas. (2) When we ask students to work at home on our own we should be careful that these HW assignments concentrate on skills practice and that there is time for checking in on answers for these. I am trying to set up a distinction between exercises and problems.

3. Welp, generally when the teacher doesn’t like a task, it doesn’t work.
When I taught at The New Community School, I gained a real respect for practicing the stuff that the students had trouble with. Now, these were students with specific language-based learning disorders, so stuff that most folks developed fluency in easily were challenging to them… mmoste of ’em.
I had the job of crunching the evaluative data at the end of the year. I knew all the teachers and some of ’em were very. methodical. about. the. drill. [Extremely important detail: the ‘drill’ was in 1:1 or at most 1:5 settings, so it was the drill *for that student,* not ‘here’s the drill for today from the book.”]
It really annoyed me that… year after year… even the smartest, quickest students who I wouldn’t have thought would benefit from the drill– I’d have thought they’d disengage — did better if they had one of the teachers that always. did. the. drill.
Even before I was crunching data, I was trying it, on the grounds that these folks really were utterly invested in the students… and I was surprised to find that … no, the students didn’t mind the daily drill. It wasn’t always their favorite thing (though for some it was), and some didn’t like it for a while… but… appropriate drill works, and they started seeing the benefits. Danged if it wasn’t pretty much a life lesson about delayed gratification.
OUt here last year we had a pilot of a new “Math Literacy” course that’s much more conceptual and rich and all that good stuff you talk about. HOwever, students also have to grind out a mess of stuff in ALEKS. That *isn’t* as good at figuring out their individual needs (no matter what the claims of tech fanboys are — there’s a guy on a discussion list now claiming that “most instructional strategies are too difficult to implement by human teachers,” and it’s the ‘diagnostic/intervention’ stuff that computers do better…. um, yea.
Still, at the end of the semester, after grumbling and grousing along the way, at least half a dozen students acknowledged that it had *really* helped in the long run to have ground out the drill, because knowing how to just grab that tool and use it when a problem asked for it let them get things done.