Means & Ends

Problem Solving
I have changed much of my teaching in the last year. One big change has been the way I approach teaching problem solving, and it has to do with a larger shift in my thinking about the difference between means and ends in education.

Here are some fun problems:

#1: You have two strings that take an hour to burn from end to end, although they do not burn at constant rates (maybe the first half of a string burns in 5 minutes, and the second half burns in 55).You also have a lighter and a pair of scissors. How can you time 15 minutes?

#2: 100 prisoners are slated for execution. They will be lined up, and each prisoner given either a blue or a red hat. He can’t see his own hat, but he can see the hat of every man in front of him. Starting at the back of the line, the executioner will ask each prisoner what color his hat is. If he is right, he is spared. The men can make a plan together before the execution. How many men can be saved?

I’ve enjoyed working on both of these problems. I would argue that someone who possesses strong problem solving skills is likely to do well with these problems, and that those skills are an important goal of math education.

But these problems are the ends of a math education. In the past, I have “taught problem solving” by giving students problems like these and letting them struggle. But I think I was confusing means and ends — while these problems, and problems like them, are useful to gauge problem solving skills, I don’t think giving them to students is a particularly effective way to teach problem solving. I was confusing the ends of a quality math education with the means of getting there.

Lesson Planning
One approach to lesson planning floated at my school this year that has caught on with several teachers is the idea that, in each lesson, students should read, write, speak, reflect, and move. This strikes me as a useful goal, but is not an end in and of itself. The end is student learning and engagement. This approach is one means of getting there — but it isn’t the only one, and it is far from perfect. When I teach Des-Man, one of my favorite lessons, students are typically sitting, working, largely silently, for a long period of time. And it is one of the most engaging and, I think, one of the most successful lessons I teach. To read, write, speak, reflect, and move in a lesson is a useful means to move more consistently toward high student engagement, but it is not an end in and of itself.

Number Talks
I’ve had a similar thought about number talks recently. I don’t use number talks as often as I did when I taught middle school, but in looking back on those lessons, I think my number talks may have focused more on trying to get kids to produce expert mathematical thinking — focusing on the ends — rather than thinking about the means of building that thinking component by component. Putting kids in a situation where they might excel if they have the knowledge, no matter how clear and valuable my vision, is not a substitute for thinking carefully about the means of getting there.

Vertical Non-Permanent Surfaces
Vertical non-permanent surfaces are one of my favorite tools I’ve learned about in the last few years, but putting students in front of a VNPS with an Expo marker does not create learning. The purpose of a vertical non-permanent surface is to increase knowledge mobility throughout the room and decrease barriers to trying new ideas and making mistakes. If they aren’t serving those functions, then I’m focusing on the means at the expense of the ends, and shorting my students.

Means & Ends
I could give many more examples. I’m writing this post as a reminder to myself that the goals of my class are for students to learn math, to be able to transfer some of that knowledge outside of the math classroom, and to enjoy the whole process. Those are the ends, and there are lots of little pieces that bring my students in that direction. It’s easy to get caught up in cool-sounding trends and new clever ways of doing things that distract from my larger goals. It’s also easy to just focus on where I want students to end up without thinking critically about the little steps along the way that they need to get there. I hope this perspective questioning my view of means and ends can help me look at my pedagogical decisions more critically, and teach a little better for my students.

10 thoughts on “Means & Ends

  1. Michael Paul Goldenberg

    Too much in this rich post to comment on it all. So let me ask a question: how would you teach problem-solving differently? You haven’t said enough for me to feel like I know what you have been doing exactly, but clearly “giving problems” is part of the process. And I think that giving/trying to solve problems is a major piece of teaching/learning problem-solving. Rather than say more, I will wait for your response. This is a topic that would be easy to write about at length. 🙂

    1. dkane47 Post author

      My experience has been that throwing problem solving-type tasks at students tends to result in strong students being successful and reinforcing positive beliefs, and weaker students getting frustrated and giving up or copying an answer from another student. I think it’s possible but very difficult to facilitate this in a productive way for all students.

      My current approach with problem solving is to do it within the context of a unit of study. I teach exponential functions, say, and after developing those ideas and giving students a base to build off of, challenge students with problem solving scenarios within exponential functions.

      My idea is to level the playing field, in that problem solving happens within the context of content that we are learning. Not that this is always successful, but makes feedback around problem solving more actionable for students, and show them a potential path to success. The end is a positive experience solving non-routine problems; the means is giving students specific, concrete tools to do so.

      1. Michael Paul Goldenberg

        I understand (I think) the concern with throwing problems at kids (seemingly) at random and with no context. At the same time, consider that so much school math consists of a structure in which students see problems for which they’ve been incrementally prepared: teacher introduces new concept; a few examples are worked in class on the board; then seatwork; then homework; next day – review of homework (which is likely overly long and incremental). Maybe a deeper level of difficulty is introduced in the new lesson or a topic that follows from that of the previous day. There are virtually no surprises.

        Even if the teacher exposes students to more challenging problems as you mention, the key idea has likely already been presented in class and students may only need to nudge that a few angstroms to solve the challenge problem. (Granted, that might be the low end of things. I’ve had classes where we needed to sort through a number of ideas, focus on exactly the one that applied to the challenge, and then figure out how to make it work, and then come up with a working solution. Solving something like that can be pretty gratifying).

        I also want to put on the table the nature of various tests, including the ACT & SAT, but also math contests, where problems get incrementally harder but which draw on all sorts of topics, not just what’s on the table in math class this week, and can call for students to think outside the box or draw on insights gleaned from struggling with something just a little beyond their comfort zone (Vygotsky definitely would have something to say about that, I imagine).

        So I’m not rejecting your ideas or your reasons for employing the approach you describe. But I think that has to be blended with problems that are mathematically within the knowledge base of one’s students but aren’t simplistic, almost knee-jerk responses to completely familiar questions. And that means drawing upon an enormous body of intriguing recreational mathematics, puzzles, games (where analyzing strategies might become the problem), etc. How do we prepare students for the unexpected and non-routine if they never face anything of the kind?

        But it’s not simply a matter of throwing kids into deep water and telling them to swim. We know that there are good problem-solving strategies that students can learn and apply to other problems. There are habits of mind that make for effective mathematical problem-solving. And if nothing else, students can (and, in my view, must) learn particular practices (some of which can be gleaned from the Common Core Standards of Mathematical Practice), including keeping organized records of their attempts to solve problems, then reflecting with peers and the instructor on what they’ve tried and why seem to be stuck. Building up habits of reflection or metacognition, if you will, are valuable beyond a given problem or the entire field of mathematics.

        My concern is that if we do not give students legitimate challenges (along with reasonable tools and support, of course) at all, they will not build up any tolerance at all for frustration. And that seems to me to be pretty much what we find in so many math classrooms. Given even the smallest wrinkle, kids throw up their hands and say, “I can’t do this. I don’t get it.” And I’m talking about kids putting in no time at all to think about the problem or try even an obvious guess. Or if they try something and it’s not right, they fold up their mental tent.

        I think there’s a significant middle ground between throwing nothing but 100 mph heaters at students that only one or two can even take a stab at and serving up nothing but softballs (even if that isn’t what we mean to be doing). What do you think?

    1. dkane47 Post author

      Really interesting. Not sure if I agree with everything he’s seeing, but I think he’s hitting a similar point about confusing our goals in education with the means of getting there.

  2. hpicciotto

    I definitely agree with the thrust of the post. Context is everything, and the teacher who thinks that any one technique is the universal answer is in for disappointment. Nothing works in every situation. Not raising means to the status of ends is super-helpful in avoiding that trap.

    I also agree with Michael’s response: if you only work within one unit at a time, you are being too helpful, making the problems too predictable. It’s important to mix it up a little. Easiest step in that direction is lagging homework, so that there are a couple of units at play at any one time. If you teach in long periods like I did, you can and should pursue two units at a time, so that combined with lagged homework, kids have to develop some agility and think outside the one-unit box. (Working on two units at a time may work in shorter periods, I have no idea.)

    This article is not exactly on this, but is not unrelated:

    Yet another angle on this is the “anchor problem”, which can serve as an intro to a course or a unit, and by definition is about something that hasn’t been studied yet, but helps set up for that.

    Anyway, those sorts of questions are very hard to discuss in general terms. You have to look at the specific situation. Lacking that, it’s possible to think you have a disagreement, when in fact what you have is different context (e.g. student with different levels of’ experience with problem solving, or different attitudes towards it, or teachers with different levels of experience / expertise in guiding student discovery and managing heterogeneous classes, etc.)

    1. dkane47 Post author

      In response to both Henri and Michael:

      I definitely agree in the importance of mixing problem types and units when possible. Tracy Zager wrote a great post a little while back that helped to capture much of what I want to do:

      That said, my writing was in response to my own mistakes: I focused too much on non-routine problems, and I reacted in the opposite direction. What each of you talked about is undoubtedly important, and there are lots of ways to get at the big ideas (I would label it “conditionalized understanding”, based on my reading of How People Learn (here: Students need to know under what conditions different strategies or concepts are useful through varied, interleaved practice. That said, they still need a concrete idea of what those strategies or concepts consist of before we go around interleaving everything. That first step is easy to ignore in the name of non-routine problems, but I think it’s critically important, and it’s a big focus of mine in my teaching at the moment.

      1. Michael Paul Goldenberg

        Dylan, I’m glad you said, “That said, my writing was in response to my own mistakes: I focused too much on non-routine problems, and I reacted in the opposite direction,” because I was pretty sure that there was an element of reacting to (perhaps) being towards one extreme and heading to the other. That is an understandable response; I hope, however, that you carve out some space for yourself and your students in which to build strength across a range of problem-solving challenges and tasks.

        No one has a perfect formula for how much of each “nutrient” should go into the diet of a mathematics student, but I’m pretty sure that if we look around the country, the tendency is to give extremely short-shrift to the non-routine. And it’s important, too, to consider that what comprises a challenging problem for any given student depends a great deal on the individual. So it’s not a case of throwing out major mind-benders all the time, but perhaps just making sure that students get to try to hit against slightly faster or trickier pitching on a regular basis. If they never see a split-finger fastball, it’s asking a lot to expect them to hit against one on an “outside” exam of some sort.

        And I think it’s possible to draw on our student’s natural (if dormant) puzzle-solving, recreational tastes when we present some of these problems. Maybe I’m way off base there, but I hope that we haven’t irrevocably killed off such things in young people by the time they get to your class.

  3. Pingback: Maths teaching rhetoric that doesn’t add-up | Filling the pail

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