Back in community college I tutored calculus. I usually included one challenge question on each worksheet for my students. But most of the time they ignored them.
They were there to get extra practice to pass their classes, and they knew the challenge problems were harder than anything that would show up on their exams. So they would rather focus their effort on what they knew they needed.
I kept including challenge problems mostly because I enjoyed coming up with them, but it never worked out like I hoped.
Personally, I love the idea of Boss Questions. But it seems like it doesn't mesh very well with the typical class incentive structure.
Agreed. I think the difference here is that Boss Questions (the way my wife and I used them) came before the lessons. And the Mini-Boss questions were natural incentives: if you did them, you could skip listening to the lesson, and just start on your homework. (I actually feel like there may have been times that the Mini-Boss Questions were SO comprehensive that we rewarded them by excusing the homework entirely, but my memory might be playing tricks on me.)
Yeah, I think the way you used them sounds a lot more effective. And the fact that the strongest students can avoid being bored for an hour is a nice bonus.
I also agree there's something magical about hanging the questions on the wall. But what exactly do you think that is?
I was thinking part of it could be that it gives students tangible evidence that they learned something. It's amazing how many times people can finish a class and not remember what it was even about the next day - even if they did learn something, it's somehow not accessible without the right cue.
"...it gives students tangible evidence that they learned something. It's amazing how many times people can finish a class and not remember what it was even about the next day - even if they did learn something, it's somehow not accessible without the right cue."
I sometimes conceptualize schooling as making students pack boxes with books, then, every night, setting the boxes on fire. This isn't just because human brains suck at remembering — it's because curriculum designers don't connect things between years (or often even months). You learn something, you forget it, you're onto the next thing. Bang. As tanagrabeast writes, "everybody poops".(https://www.lesswrong.com/posts/F6ZTtBXn2cFLmWPdM/seven-years-of-spaced-repetition-software-in-the-classroom-1)
Which is all to say, yeah, maybe part of what's cool about displaying problems on walls is that, for even a short amount of time, evidence of your engagement is UP there, for EVERYONE to see. It's proof you DID something.
I wonder if students are more likely to get enthusiastic about boss questions if the questions start earlier on in education. I imagine a lot of people become switched off to math well before they get to college.
When I was in grade 6, there was a short time when my teacher gave us weekly math riddles kinda like this, and I LOVED it.
One more thought - what makes this pattern work for math and not for other subjects?
Is it that math is primarily about solving problems? Is it just that math has right answers? Or is it perhaps that math builds on itself, so solving a single hard question proves that you have a multitude of prerequisite skills?
I can imagine the same strategy working for other subjects like physics that mostly involve math. I don't see how it would work for an English class, but maybe that's just my own lack of imagination. What do you think?
My hunch right now is that this works in any context where students are problem-solving (rather than, say, story-hearing) — the parts of the curriculum where Constructivists ("kids build understanding") make stronger points than Traditionalists ("kids receiving information").
Interestingly, I don't think having a universally-agreed-upon right answer is necessary for this: I've done similar things with philosophical or political questions. (Mind, for this to work, there's still some sense that participants need to have that SOME answer is correct.)
In English — thanks for the spur — I can imagine this working in some situations. An easy one would be grammar, when the kids have already learned the relevant rules: "where should the comma go in this sentence?" (where it's a REALLY difficult sentence). A harder one would be closer to philosophy: "why did Ophelia kill herself?" But that would require you already had a group of kids who knew Hamlet, and had some ideas percolating.
Yes! I don’t know about the other subjects, but I’ve been doing student teaching in high school physics and I’ve been thinking about how useful this boss problem approach is for the students. For the last unit, I took the roughly 200 problems on Newtons laws that the teacher wanted them to do as homework and divided them into 5 or 6 problem types. What was really helpful was getting them to recognize that every problem they encountered was similar to one of these handful of problems. Like, it doesn’t matter if it is a fireman sliding down a pole like in the boss problem, or a person with a parachute, or someone standing on a scale in an elevator, or a crane lifting a piano…in all these problems you have weight pulling you down and some other force directed upward, so the solution method is the exact same. I feel like boss problems are not just motivating, but also so useful in getting kids to organize their thinking, make connections, and see the big picture.
You can't keep saying "this is an amazing book and you must own it" every time you post. It's my kryptonite I have no defense against that sort of recommendation...
This reminds me of something I encountered as a high school student in a summer math camp (not THE Math Camp, a different one).
Every morning (outside of class), one of the TAs would put a math or logic puzzle on the white board. (Usually of the flavor of: "There are 100 logicians standing in a line facing forward, each wearing either a white or black hat, and they'll be executed unless they can say which color hat they're wearing..." you get the idea.) The students would talk amongst each other about the problem during the day (mostly during lunch and the break, for fun), and then in the evening the TA would direct anyone who was interested through a formal answer. Everyone who participated had a great time, and even those who didn't participate at least didn't mind the rest of us talking about it.
Off the top of my head, this worked because, not in order: (1) the problems were voluntary, (2) they were a refreshing change of pace from what we were learning in class itself, (3) they had satisfying answers like most riddles do, (4) the social aspect -- we would discuss our ideas with each other, (5) usually no one knew what the answer was or often how to even start thinking about it at first, so it felt like an even playing field, and of course (6) I can't ignore the selection of students in the first place.
I look back on these quite fondly, even if it's a little different than what you're discussing since it's not directly about the class itself. Plus, one time instead of a normal logic puzzle, the TA walked us through something like "Build the Real numbers from the Rational numbers." Mind you, I didn't even have a concept of what the reals or rationals really were (aside from "the kind of thing 1.0123..." is or "the kind of thing 5/2 is"). But he still managed to walk us through a toy version of Dedekind cuts. Would my understanding then have passed in an analysis class? No, absolutely not. But did it help me understand what the heck the professor was talking about when I encountered the full rigorous version of the proof 5+ years later? Absolutely. (So I am quite sympathetic and optimistic about the ideas you are discussing in your post.)
And now, if nothing else, I have a huge supply of "There are 100 logicians living on an island..." problems I can deploy at my leisure.
"Would my understanding then have passed in an analysis class? No, absolutely not. But did it help me understand what the heck the professor was talking about when I encountered the full rigorous version of the proof 5+ years later? Absolutely."
Yeah! One of the cool things that "Final Boss Questions" might do: merely introduce some future stuff (e.g. weird math symbols like integrals, and even just the knowledge that math CAN solve some high-level problems) and make it interesting by wrapping it in mystery.
The mood of this reminds me of a story from the introduction of one of my Far Side collections. It was a science teacher who had put a lot of Far Side cartoons up on a wall... they weren't _assigned_ or anything.
But as the kids learned, they got into the habit of checking the wall to see if one of the obscurities had become _funny_ as they learned the science the joke relied on.
Very fun! As you know, Brandon, I'm super excited about the Boss Problems.
Here's one I had in mind for an algebra lesson on lines, something like:
"Find the intersection point of these lines: y = x - 1; y = (49/50)x+2". I tried to picked values that would make it hard to calculate from a table or graphing, so if the lesson is on solving these simple systems, they definitely have to understand the algebraic approach. This could be a superboss problem at the beginning of a unit, where kids don't even yet know common forms of line equations, if instead I just described each line, perhaps in some story form (Jim travels northeast, going 49 yards north for every 50 yards east...)
A favorite activity from high school math was when we had to make pictures on our TI-83 using various simple functions (I think we had basically just learned about lines and parabolas). "Draw Bart Simpson's face on Desmos" could be a fun Superboss Problem.
I was thinking Art of Problem Solving might make for a good source of Boss Problems. But now I'm wondering if it would miss the point. Most of the challenging AoPS problems require some additional cleverness or insight. A good Boss Problem by your definition should be the sort of thing that every student *should* be able to solve at the end of the lesson. As you point out, they might make for good Final Boss problems.
This is a bit of a tangent, but I have very strong and fond memories of my high school physics teacher using Spam -- the canned meat -- in nearly all of his examples and practice problems. It was ridiculous and silly, but made a very memorable impact. A clever problem-maker could think of some theme or character with which to journey through the boss problems. My personal favorite asthetic would probably be an 8-bit video game theme (get AI to generate images of our hero for each problem), and this would tie very well with the theme behind the nomenclature of "Boss Problem". But I think video game interest levels are too bimodal. Something goofily neutral such as Spam might be a better approach.
Matt Enlow is someone I follow on Twitter who shares interesting/challenging prompts he gives Alg II Honors students: https://twitter.com/CmonMattTHINK
I don't know if he is doing it exactly the way you describe, but some of what you are describing is how I was picturing it when he talks about problems he shares as "prompts" for his students.
Back in community college I tutored calculus. I usually included one challenge question on each worksheet for my students. But most of the time they ignored them.
They were there to get extra practice to pass their classes, and they knew the challenge problems were harder than anything that would show up on their exams. So they would rather focus their effort on what they knew they needed.
I kept including challenge problems mostly because I enjoyed coming up with them, but it never worked out like I hoped.
Personally, I love the idea of Boss Questions. But it seems like it doesn't mesh very well with the typical class incentive structure.
Agreed. I think the difference here is that Boss Questions (the way my wife and I used them) came before the lessons. And the Mini-Boss questions were natural incentives: if you did them, you could skip listening to the lesson, and just start on your homework. (I actually feel like there may have been times that the Mini-Boss Questions were SO comprehensive that we rewarded them by excusing the homework entirely, but my memory might be playing tricks on me.)
Thoughts?
Yeah, I think the way you used them sounds a lot more effective. And the fact that the strongest students can avoid being bored for an hour is a nice bonus.
I also agree there's something magical about hanging the questions on the wall. But what exactly do you think that is?
I was thinking part of it could be that it gives students tangible evidence that they learned something. It's amazing how many times people can finish a class and not remember what it was even about the next day - even if they did learn something, it's somehow not accessible without the right cue.
"...it gives students tangible evidence that they learned something. It's amazing how many times people can finish a class and not remember what it was even about the next day - even if they did learn something, it's somehow not accessible without the right cue."
I sometimes conceptualize schooling as making students pack boxes with books, then, every night, setting the boxes on fire. This isn't just because human brains suck at remembering — it's because curriculum designers don't connect things between years (or often even months). You learn something, you forget it, you're onto the next thing. Bang. As tanagrabeast writes, "everybody poops".(https://www.lesswrong.com/posts/F6ZTtBXn2cFLmWPdM/seven-years-of-spaced-repetition-software-in-the-classroom-1)
Which is all to say, yeah, maybe part of what's cool about displaying problems on walls is that, for even a short amount of time, evidence of your engagement is UP there, for EVERYONE to see. It's proof you DID something.
I wonder if students are more likely to get enthusiastic about boss questions if the questions start earlier on in education. I imagine a lot of people become switched off to math well before they get to college.
When I was in grade 6, there was a short time when my teacher gave us weekly math riddles kinda like this, and I LOVED it.
One more thought - what makes this pattern work for math and not for other subjects?
Is it that math is primarily about solving problems? Is it just that math has right answers? Or is it perhaps that math builds on itself, so solving a single hard question proves that you have a multitude of prerequisite skills?
I can imagine the same strategy working for other subjects like physics that mostly involve math. I don't see how it would work for an English class, but maybe that's just my own lack of imagination. What do you think?
Huh!
My hunch right now is that this works in any context where students are problem-solving (rather than, say, story-hearing) — the parts of the curriculum where Constructivists ("kids build understanding") make stronger points than Traditionalists ("kids receiving information").
Interestingly, I don't think having a universally-agreed-upon right answer is necessary for this: I've done similar things with philosophical or political questions. (Mind, for this to work, there's still some sense that participants need to have that SOME answer is correct.)
In English — thanks for the spur — I can imagine this working in some situations. An easy one would be grammar, when the kids have already learned the relevant rules: "where should the comma go in this sentence?" (where it's a REALLY difficult sentence). A harder one would be closer to philosophy: "why did Ophelia kill herself?" But that would require you already had a group of kids who knew Hamlet, and had some ideas percolating.
Yes! I don’t know about the other subjects, but I’ve been doing student teaching in high school physics and I’ve been thinking about how useful this boss problem approach is for the students. For the last unit, I took the roughly 200 problems on Newtons laws that the teacher wanted them to do as homework and divided them into 5 or 6 problem types. What was really helpful was getting them to recognize that every problem they encountered was similar to one of these handful of problems. Like, it doesn’t matter if it is a fireman sliding down a pole like in the boss problem, or a person with a parachute, or someone standing on a scale in an elevator, or a crane lifting a piano…in all these problems you have weight pulling you down and some other force directed upward, so the solution method is the exact same. I feel like boss problems are not just motivating, but also so useful in getting kids to organize their thinking, make connections, and see the big picture.
You can't keep saying "this is an amazing book and you must own it" every time you post. It's my kryptonite I have no defense against that sort of recommendation...
Ross, that reminds me — after your recent (wonderful) post about egregores (https://www.wearenotsaved.com/p/egregores-group-minds-and-white-magic), I wanted to tell you I think you'll love "The Journey of the Mind (https://www.amzn.com/dp/1324050578)...
;)
I'm reminded of the emotion I was experiencing at hour 2.5 of recording your book review...
Also that book is already in my library, which I'm going to count as a victory, albeit a Pyrrhic one.
This reminds me of something I encountered as a high school student in a summer math camp (not THE Math Camp, a different one).
Every morning (outside of class), one of the TAs would put a math or logic puzzle on the white board. (Usually of the flavor of: "There are 100 logicians standing in a line facing forward, each wearing either a white or black hat, and they'll be executed unless they can say which color hat they're wearing..." you get the idea.) The students would talk amongst each other about the problem during the day (mostly during lunch and the break, for fun), and then in the evening the TA would direct anyone who was interested through a formal answer. Everyone who participated had a great time, and even those who didn't participate at least didn't mind the rest of us talking about it.
Off the top of my head, this worked because, not in order: (1) the problems were voluntary, (2) they were a refreshing change of pace from what we were learning in class itself, (3) they had satisfying answers like most riddles do, (4) the social aspect -- we would discuss our ideas with each other, (5) usually no one knew what the answer was or often how to even start thinking about it at first, so it felt like an even playing field, and of course (6) I can't ignore the selection of students in the first place.
I look back on these quite fondly, even if it's a little different than what you're discussing since it's not directly about the class itself. Plus, one time instead of a normal logic puzzle, the TA walked us through something like "Build the Real numbers from the Rational numbers." Mind you, I didn't even have a concept of what the reals or rationals really were (aside from "the kind of thing 1.0123..." is or "the kind of thing 5/2 is"). But he still managed to walk us through a toy version of Dedekind cuts. Would my understanding then have passed in an analysis class? No, absolutely not. But did it help me understand what the heck the professor was talking about when I encountered the full rigorous version of the proof 5+ years later? Absolutely. (So I am quite sympathetic and optimistic about the ideas you are discussing in your post.)
And now, if nothing else, I have a huge supply of "There are 100 logicians living on an island..." problems I can deploy at my leisure.
"Would my understanding then have passed in an analysis class? No, absolutely not. But did it help me understand what the heck the professor was talking about when I encountered the full rigorous version of the proof 5+ years later? Absolutely."
Yeah! One of the cool things that "Final Boss Questions" might do: merely introduce some future stuff (e.g. weird math symbols like integrals, and even just the knowledge that math CAN solve some high-level problems) and make it interesting by wrapping it in mystery.
The mood of this reminds me of a story from the introduction of one of my Far Side collections. It was a science teacher who had put a lot of Far Side cartoons up on a wall... they weren't _assigned_ or anything.
But as the kids learned, they got into the habit of checking the wall to see if one of the obscurities had become _funny_ as they learned the science the joke relied on.
Man, "Wall of Far Side and xkcd Comics" should be a future pattern. The biggest problem might be that it'd become a fire hazard...
Some of my professors also liked to put "Proofs without words" on their office doors. There's some nice examples here: https://en.wikipedia.org/wiki/Proof_without_words.
Very fun! As you know, Brandon, I'm super excited about the Boss Problems.
Here's one I had in mind for an algebra lesson on lines, something like:
"Find the intersection point of these lines: y = x - 1; y = (49/50)x+2". I tried to picked values that would make it hard to calculate from a table or graphing, so if the lesson is on solving these simple systems, they definitely have to understand the algebraic approach. This could be a superboss problem at the beginning of a unit, where kids don't even yet know common forms of line equations, if instead I just described each line, perhaps in some story form (Jim travels northeast, going 49 yards north for every 50 yards east...)
A favorite activity from high school math was when we had to make pictures on our TI-83 using various simple functions (I think we had basically just learned about lines and parabolas). "Draw Bart Simpson's face on Desmos" could be a fun Superboss Problem.
I was thinking Art of Problem Solving might make for a good source of Boss Problems. But now I'm wondering if it would miss the point. Most of the challenging AoPS problems require some additional cleverness or insight. A good Boss Problem by your definition should be the sort of thing that every student *should* be able to solve at the end of the lesson. As you point out, they might make for good Final Boss problems.
This is a bit of a tangent, but I have very strong and fond memories of my high school physics teacher using Spam -- the canned meat -- in nearly all of his examples and practice problems. It was ridiculous and silly, but made a very memorable impact. A clever problem-maker could think of some theme or character with which to journey through the boss problems. My personal favorite asthetic would probably be an 8-bit video game theme (get AI to generate images of our hero for each problem), and this would tie very well with the theme behind the nomenclature of "Boss Problem". But I think video game interest levels are too bimodal. Something goofily neutral such as Spam might be a better approach.
Matt Enlow is someone I follow on Twitter who shares interesting/challenging prompts he gives Alg II Honors students: https://twitter.com/CmonMattTHINK
I don't know if he is doing it exactly the way you describe, but some of what you are describing is how I was picturing it when he talks about problems he shares as "prompts" for his students.