A great math activity shouldn't be too easy. If a problem is too easy it holds no interest to us. This is a claim that's all over the comments of Dan's recent post.

But, why not? I mean, we do easy things all the time. Why would a math problem become less interesting if it's too easy.

"Because people have an intrinsic love of challenges."Really? Is our perception that something would be challenging (but doable) sufficient to get us interested in an activity? Aren't most of our worksheets doable but challenging? And why would some sort of activities be more interesting than others? Do they appear more challenging? More doable? What's the theory?

Drop a comment, and we'll hash this out.

A math problem that's too easy feels like busy-work to me, and as a student I always resented that. I suppose if I recognize a need for practice (like with times tables), I might be willing to do a limited amount, especially if you put it in a game.

ReplyDeleteWhat's wrong with busy-work?

DeleteBusy-work is an insult. It's the stuff adults give kids just to keep the kids out of their hair. Busy-work is like when an adult says, "Do what I say, not what I do." Or answers an honest question with, "Because I said so."

DeleteDenise

DeleteCan you replace 'busy-work' with 'practice' and feel better about it? For example, my 10 yr old is starting to play basketball. He needs to take plenty of free throws to get good eventually. There is no inherent challenge (he's not good at it, but it is not 'challenging' in a meaningful way) or inherent interest just in standing there and repeating a shooting motion.

If your son wants to improve at basketball, then that gives him reason to practice. The practice is not "too easy," even if it's not mentally challenging: training muscles to do what we want is hard. It's different in math, because we aren't training physical muscles, so repeating the same motions over and over doesn't bring improvement.

DeleteI guess part of the problem is that "too easy" wasn't clearly defined. I took it as the student having to crank through a set of tedious calculation problems of a type he/she has already mastered. If the problems are interesting, then they aren't "too" easy, just like an adult can still enjoy reading a story written for kids, if it's a well-told story.

I don't think ease or difficulty is necessarily linked to interestingness, except that if a problem is just at the right level of challenge, it can be like a dare, drawing me in to see if I can "beat" it. It's hard for a teacher or textbook to hit that sweet spot regularly, though I think the Art of Problem Solving texts do a better job than any others I've seen.

Like Chris just below here, I let my kids skip problems that are too easy so that we can focus on learning to think through the harder ones. For instance, we will fairly often skip calculating the actual answer of a word problem, after we've worked out the logic of how to solve it. Or we will jump straight to the end of a problem set and just try the hardest ones, to see if we've really mastered the material. This gives us more time to focus on mathematical reasoning and justification/proof without getting distracted by minor details.

Why do we ask students to do problems in the first place?

ReplyDeleteFor me, I ask students to do work that will expand their understanding and knowledge. If a problem is "too easy" that means that students inherently already know how to solve it with relative ease. Based on that, how would doing easy problems expand their knowledge or ability. They won't, so I advocate not doing them.

When I assign homework, I tell my students to skip any that they know they can do (actually I tell them to try a few easy ones just to be sure they are not overconfident, but that distracts from the comment). When I assign homework, I ask my students to only do the ones that they don't instantly know how to do. These can help them grow and develop mathematical power. Easy problems should really be limited to topics that we expect quick factual recall for.

In Chris' comment we have

DeleteTheory #1. Chris' explanation is a version of theIt's Good For Me Hypothesis."Students are motivated to do things that will help them advance. A task that is too easy is unmotivating, because it won't lead to much learning."

Agree? Disagree? Why do you think that kids don't like tasks that are too easy?

Unfortunately, I do not always agree that kids don't like tasks that are too easy. By the time many students reach me in high school, many have learned to solely seek the grade in a course, not understanding and knowledge. As a result, if a problem is graded or students think that the problem they are given is representative of what they will be assessed with, they love an easy problem.

DeleteAs for the hypothesis you reference, I am not sure that student enjoyment of a challenging problem is a result of the fact that it is good for them: that is my reason for assigning such problems. Honestly, I have seen my students interacting with challenging problems and when they make progress and/or find a solution, they feel empowered (as do I when I tackle a problem). I think it is similar to many other things. Sports teams feel far better when winning a game against a difficult opponent compared to beating an opponent that they expected to win against. I wonder if this is a testament to the fact that somewhere, deep inside of us as human beings, we are naturally growth mindset individuals.

Michael

DeleteIs it reasonable to think that doing a handful of easy problems as practice falls into the category of helping me to advance? If I 'warm up' with some easy ones I might remind my self of those skills and be better able to make progress later in the problem set.

I'm not sure, but I'll posit that students' reactions to math work that is easy for them is bound up with their "rank" or "status" as math students.

ReplyDeleteUnless you've already won or lost the math status game--unless you're in the remedial class once again, or you've just won the school math competition once again--then easy work means you're treading water. And worse, that someone else thinks you shouldn't be going faster. That you're a dummy, and that your peers in a different section or group are zipping right by you.

If you've lost the math status game, then you can be relieved when you're handed easy work that you can actually do. If you've won, then you can just shine while others struggle. But if you're still racing, then easy work means you're falling behind.

This is not a complete theory, but I think it's a frequent contributing factor.

Justin's given us

DeleteTheory #2, which we might call theStatus Hypothesis:"The real problem with easy work isn't that it's less engaging. The real problem is that kids perceive easy work to be an affront to their ability to keep up with their peers. Easy work kills any existing interest."

Predictions:

* The interest of our top students won't be harmed by easy work. Ditto to our remedial kids.

* Remove the status issue, and kids will find easy tasks just as interesting as challenging tasks.

Interesting, in Justin's response, I hear a lot of notes of "growth vs. fixed mindset." Students that have "won' or "lost" the status race already either feel they are good or bad at math. Isn't this another way of stating that they are operating with a fixed mindset/

DeleteQuestion: How would you suggest testing your predictions Michael given our current educational system and without harming the educational goals we have for our youth?

Good eye Chris. I think you got to the heart of Justin's response. Justin - is that where your mind was?

Delete

DeleteHow would you suggest testing your predictions?I don't think that it'd be that hard. We can use polling to figure out which students don't have status issues in math, and then give those kids easy tasks. (This isn't a direct test of the theory, but I'd love to see if there's a difference between what kids find interesting when alone and when in a classroom.)

Mindset is something of a buzzword, and a tool where everything starts looking like a nail. I'm talking about something else--namely, social dynamics. Regardless if a kid has a fixed or a growth mindset, it can feel bad to be given easy work if that's interpreted as a signal that they're "behind" their peers--either in a fixed way or in a dynamic way.

DeleteEveryone wants to feel smart and capable. If students are given tasks that are too easy, they may believe that their teacher does not think them capable of more difficult tasks. They might even think that the teacher's assessment is correct.

ReplyDeleteThis sounds a bunch like the

DeleteStatus Hypothesis.Does anybody think that easy tasks are inherently less interesting?

Ok, I'll bite -- I'm not sure kids recognize easy tasks as easy rather than as proof of their *amazing* intellect (emphasis and mild sarcasm mine).

DeleteInteresting!

DeleteDo you disagree with the premise? Do you find that kids are equally motivated by easy and challenging tasks?

I need to think more on it to commit to a side. Speaking for my own engagement, the mathematical challenge isn't what makes a task interesting. Isn't it about the question itself being interesting? Cause when we set out on a task, we don't know how hard it'll be to solve.

DeleteI think the interest of a task can come from multiple sources: the challenge of the task, the content of the task, how useful I perceive the task to me personally, ...

DeleteIn addition to the

DeleteIt's Good For Me Hypothesis(see above), Chris is positing another source of motivation and interest.Theory #3: Students are intrinsically interested by challenges.Question: Is a challenge sufficient to create interest? Are there challenging tasks that don't create interest? If so, why don't they?

You are full of great questions. Personally, I know sometimes it depends on how the challenge is presented to me or who it is presented by. At other times, if I am in the right mood, it doesn't matter at all. I wonder if my students would agree. This sounds like a worthwhile short opener question for a slight down day.

DeleteWay back in HS I read an interesting series of math essays in a book called "Bridges to Infinity." One of those essays was called "Between Checkers and Chess." The author makes the following point:

ReplyDelete"We could get a very specific idea of where we stand along our theoretical scale of reasoning ability by taking a particularly thorough inventory of which games challenge us and which ones do not. But on the basis of the few observations made, it can be estimated that the limits of our capacity to reason as a species places us somewhere between the limits of checkers and chess."

It is the only thing I've read that even begins to address the question you are asking. I haven't re-read the entire book, so all I can really say about it is that I found it to be really interesting when I was a kid.

I'd also add that an easy math activity is not often so easy to spot. The Collatz Conjecture, for example, involves either dividing by 2, or multiplying by 3 and adding 1. Not exactly difficult math on the surface, but the problem is unsolved.

The Monty Hall problem is another example. There are only 9 cases for the "switching" and "non-switching" games - 3 spots for the prize, and 3 initial guesses - and so it could (and probably should) be thought of as being a not so challenging problem. Yet it has caused endless arguments.

So, I guess I'd say that easy is in the eye of the beholder, but that just because something is viewed as easy, doesn't mean that it isn't also interesting, and certainly doesn't mean that it isn't a great gateway to something even more interesting.

I guess the last example I'd give is something that I did with my kids last weekend - the Chaos Game. Pretty easy, and, frankly pretty dull . . . until it suddenly isn't dull at all:

https://www.youtube.com/watch?v=FIsUsnfLVXQ

Good lord, I don't know who you people are teaching, that all the kids are saying "that's too easy"

ReplyDeleteTheory #4: The "Projection" Theory

Math teachers think they have anything in common with the average or lower math student.

I have never seen a kid insulted by problems that were too easy, and most math teachers dramatically overestimate the content their kids can handle and remember.

Oddly enough, I just wrote about a related aspect of this:

http://educationrealist.wordpress.com/2013/12/07/the-release-and-dumbing-it-down/

Your "Projection" idea is irrelevant. First, because the issue at hand isn't the prevalence of kids saying that work is too easy for them. It's whether easy tasks are interesting for kids. Second, because you know full well that there are people who teach high-performing kids. That you teach low-performing kids is pretty uninteresting, as far as this conversation is concerned.

DeleteBut the core idea of your comment is interesting.

Ed offers Theory #4, an example of what philosophers of morals call an Error Theory. Ed thinks that easy tasks are just as interesting to kids as challenging tasks. (Or, at least that's what I gather from his/her comment. Let me know if that sounds right/wrong, Ed.)

Ed's claim ought to be easily testable, and I'm sure that the test has been done with students. All that I was able to find now, though, was a (good) UK <a href = "http://www.kent.ac.uk/careers/Choosing/career-satisfaction.htm> poll of employees</a> that found that, of those that are bored, 61% attributed their boredom to a lack of challenge at work.

Of course, being a pencil pusher is different than being a student, but Ed's claim is that students are equally motivated by easy and challenging work. I don't see why <i>that</i> would only be true for students, and not also for employees.

Don't know who you are teaching, I guess. I have routinely had his who feel that something is too easy. They don't exactly complain about it, but I've seen kids lose motivation due to it.

ReplyDeleteI'm with you, Dardy, and I can say with confidence that I've taught low-performing students.

DeleteMaybe Ed just teaches at a higher level of difficulty than I do, but when I taught low-performing students, I would aim to challenge the majority of the class. I'd always have a few kids, though, who told me that the lesson was too easy, or that the class was too easy. Ed's comment doesn't resonate with my experience working with low-performing kids.

I have, literally, never been told by a low ability kid that my math class is too easy. Not ever. Five years.

ReplyDeleteI have, on occasion, been told that Ms. X or Mr. Y taught much better than I did, that they made it "easy". I would then call up their test scores from the previous year and tell them that, alas, it may have been easy, but it wasn't effective.

So, seriously, if you always have genuinely low ability kids who think math is easy, who fail math or get low test scores but tell you it's too easy, then that's interesting. I not only don't have kids like that, I have never run into teachers (in person) who have kids like that.

So to be clear: we're talking about teaching advanced math (algebra plus, hs level math) to kids who have "below basic" scores, kids who would be pleased to hit 400 on the SAT in math? And these kids are always telling you they are bored, that work is too easy?

Because the comments I see here are not addressing the kind of work you would give to kids of that ability level, whether or not they thought it was easy.

No, they're not "always" doing anything. But, yes, last year, sometimes, when they thought that work was too easy, they said so.

DeleteWell, now we're getting somewhere. That literally never happens to me, and I don't think it happens to any teacher of my acquaintance.

DeleteAnd so, I bring up the next point: why would you give kids with far below basic proficiency the kind of tasks listed here in the comments as a way to engage them, when they will not be able to learn them? Why not instead just give them work that shows them (in a good way) that they still aren't prepared, even if they think they are?

I'm not making accusations, I'm trying to figure out the whole line of questioning (that is clearly not just you) in which teachers are trying to engage kids who say math is too easy when their performance shows clearly that it is not.

Here are too scenarios that, in the past, lead me to give tasks that were "easy" for some students:

Delete1. Because classes have a range of abilities. If a student is at the "top" of a low ability class, then even when I find tasks that are challenging for the majority of the class, it will sometimes be too easy for individual members of the class.

2. Sometimes there are very basic skills that some students of a class struggle with, though the majority don't. In such a case, I might still include a lesson on the easy stuff, since it's so important. (This happened last year in my Algebra class when I decided to spend a very frustrating week going over some fraction basics with four kids.)

Also, for the record, I'm not interested in figuring out a way to avoid these situations. I'm interested in the theoretical point. Many people think that easy tasks are uninteresting. Are they right? Is this true in general? If so, why?

Delete"Why would you give kids with far below basic proficiency the kind of tasks listed here in the comments as a way to engage them, when they will not be able to learn them?"

DeleteI think some assumptions about what classifies as an "easy" and "not easy" task are being made. When I am thinking of tasks to give my students, I don't have predefined challenging tasks for all students. Instead, I think about their ability both mathematically and in regards to general problem solving (there are discrepancies between the two regularly in students). Next, I find a problem that would push and challenge them.

A few years ago, when I was doing a lot of research on cognitively challenging questions, I read that teachers are more prone to avoid such questions when working with low performing students. In such cases, the researchers found that we attempt to make things easy for students by making everything procedural. The researchers also suggested that this has a negative impact on students mathematical learning. I would tend to agree.

Think about problems from which you have learned a lot. Think about problems from which your students have learned a lot. Were these problem easy for you and/or for the individual student or did they require some significant thought from the learner?

So Michael, I'm confused. I specifically discussed low ability kids saying that the math is too easy, and you respond with an example of your high ability kids.

DeleteSo let me back up, because you appear to be misunderstanding me:

1) Math should never be *easy*. It should, occasionally, be familiar, which is not the same thing. That is, I should be able to give my kids a group of linear/quadratic/exponential problems in Algebra 2, and in a perfect world, the problems would be familiar. It would serve to remind kids how to recognize the problems, but shouldn't be a complete snap.

2) When introducing new material, the problems should definitely not be too easy. That is, the material should assuredly be new to all students; if not, then give something that is either new or challenging to the new students. Because all students, regardless of skill, must sruggle with something challenging. This is neither the vegetables nor the status hypothesis, but the Reality of Learning Axiom.

3) Do teachers avoid cognitively challenging questions with low ability students? Probably, but only because it is very hard, without practice, to find the appropriate level of cognitive challenge for low ability students. I spend a lot of time working on this, and I do not do this. Also, any teacher who consistently works with low ability kids realizes that the rote procedures don't help, either. (I thought they would at first, but apparently it takes brains to remember procedures.)

4) Under no circumstances should teachers give students work they are very familiar with. They should come up with another task. So if 90% of your students don't know fractions, but 10% know them very well, you come up with some other fraction-related challenge (say, percentages) for that 10% and then review with the other 90.

So going back to your original post, you set up a dichotomy: like vs. not like. I am uninterested in what my students like or don't like to do. I am interested in what they *can* do, and in increasing what they can do, however incrementally. This I refer to as the Don't Waste Time in Class Axiom.

Thanks for the thoughtful comment, Ed.

DeleteFirst, what I thought I was describing was giving low-achieving students something that was easy for them. The students at the top of my low-achieving class were still low-achieving, even though they knew what a fraction was.

Also, I agree with almost all of the substance of your comment. I'm even OK with your dismissal of the original questions, whether kids find easy tasks interesting. That's cool, and I get why you prefer thinking about what they can and can't do.

But I'm interested in the question of like/not like. Not even necessarily for practical, it'll-change-what-I-do-in-the-classroom reasons. Maybe it will. But right now, I'm just eager to understand what our students find interesting, and whether easy things are interesting for kids.

What is analogous to “easy” and “difficult” as far as being a teacher goes? Are most teachers more interested in difficult than easy?

ReplyDelete