Teacher and coach

Another Mathematician’s Lament

There’s a fascinating little book called “A Mathematician’s Lament,” written by research-mathematician-turned-math-teacher Paul Lockhart, in which the author bemoans the sorry state of mathematics teaching today and offers his thoughts on how it could be done better.  In some ways, it’s similar to a lot of other books and articles that complain about how math is taught (the poor curriculum, ineffective teaching methods, etc.) so I was prepared to largely ignore it.  I don’t generally have much use for these kinds of screeds, as I think most math teachers already know what the problems are and are doing the best they can under difficult circumstances; having someone shouting “You’re not doing it right!” from the sidelines isn’t particularly helpful.

I was pleasantly surprised, however, as Lockhart does in fact offer some useful insights into things that can and should be done differently for many math teachers, and the fact that he’s a math teacher himself, and one who used to be an actual mathematician, gives him a bit more street cred than most others.  So I wholeheartedly recommend this book, especially to math teachers, as there’s a lot of good stuff here.

That said, I was somewhat troubled by the book.  Lockhart’s main thesis is that math is a naturally fascinating and creative and joyful activity, and the way schools and math teachers present it turns it into something dull and dreary (and he uses some pretty strong language to make his points).  He’s a big proponent of allowing students to “play” with math so they can experience this joy themselves, and he’s a big critic of treating math as a utilitarian tool that students need to master in order to be able to function well in society.  He doesn’t like worksheets or textbooks or memorizing formulas or learning particular topics just because “that’s how we’ve always done it.”

That was all fine; I don’t have any big problem with any of that, in theory at least, and I actually agree with Lockhart’s basic point of view.  I try to use “discovery learning” as often as I can and do my best to allow my students to experience the joy of math.  But there seemed to be something that Lockhart wasn’t addressing, and it took me a while before I could put my finger on it.  I finally happened across an anonymous comment from another discussion of Lockhart’s book that summed up the dissonance I was experiencing.

I have an interesting situation with regards to this. The students I teach are quite difficult at most times, and are generally quite behind in maths. They can often remember ways of doing things by rote, but have troubles generalising and get completely stuck if they find themselves in an unexpected situation. Probably for this reason, they really like worksheets and textbooks. They can get the answers (one way or another) – in fact there _is_ an answer. They have no idea of what this answer means, or, for that matter what the question means or why the hell they’re doing it. But that doesn’t really matter if they can get the answer. On the other hand, whenever I have tried more open, exploratory or play based activities, the students tend to revolt (in a manner of speaking). Not always, but often. There’s no clear end in sight, and they have difficulties working independently, so things start going awry. A lot of this, no doubt, comes from their lives outside of school, and earlier schooling.

It’s a conundrum – I can work through the state’s curriculum via a textbook and tick all the boxes, and the administrators will be happy and so will the students (though they won’t actually have learnt anything useful), or I can tear my hair out trying something different where the students get confused, I don’t cover the expected material, and the outcome might not be much better. I’m tending towards the latter option, in the hope that over time the students will become more responsive…

I think Anonymous nailed it, and I suspect that most math teachers struggle with this type of student to one degree or another.  Discovery- or play-based activities are not some magic bullet, and in fact require some degree of effort to implement, not just on the part of the teacher but on the part of the student.  The students have to be willing to cut the triangles out of the paper, or graph different equations on the graphing calculator, or change the different parameters on the computer program.  They have to be willing to work (or play) with a math problem that is open-ended and may not have a specific solution.  Many of us (including, I suspect, Mr. Lockhart) might not consider this much “effort” at all, and would certainly balk at calling it “work,” but the fact is that many of my students struggle with exactly this issue.  If it’s anything more complicated or creative than a standard worksheet, they’re completely befuddled and will often, as Anonymous says, revolt.  “This is too hard!”  “You never showed us how to do this, how are we supposed to know what to do?”  “Why don’t you give us some real math problems?”

Mr. Lockhart’s book seems to assume that if math teachers would just stop presenting the material in such a dreary and boring way, the students would immediately respond with shouts of joy and begin exploring and playing with math concepts and begin really “learning” and not just memorizing a bunch of facts and formulas.  But before students can experience the true joy of math (or, I would argue, any field of endeavor), they must be willing to do at least a little bit of work.  Instilling this work ethic in some students is often quite a challenge.

Image by Combined Media on flickr.

12 Responses to “Another Mathematician’s Lament”

  1. bledsoe says:

    Hi Jonny,

    You may want to take a look at Dan Meyer’s blog ( http://blog.mrmeyer.com/ ). He has a lot to say about the value of choosing good problems (and presenting them in the right way) as a means of enticing students into problem-solving.

  2. jonny says:

    I’m a maths student at university and my mum is a maths teacher at high school. I have always liked maths.

    I have ADHD and while I often drifted off in English lessons of in French at school, I drifted off much less in maths. I think it’s simply because I love to solve problems and I think (although with no evidence to support it) that many children naturally love to solve problems too.

    I’m sure that many of the kids that rebel when given a maths problem would enjoy other types of puzzles/problems…perhaps crosswords, sudoku, lateral thinking puzzles or even those physical puzzles where you have to get two strangely shaped pieces of metal apart or a bead off a piece of string.

    For me anyway, it’s all the same kind of enjoyement…the challenge and curiousity of not knowing how to solve a problem and then the satisfaction and perhaps pride of figuring it out.

    I’m wondering if a kind of weaning onto maths problems may work. By giving kids other puzzles to do and getting them in that problem solving frame of mind they may then feel the same enjoyement with maths problems?

    Perhaps a weekly riddle with a prize for the ones who get it right? I don’t know.

    But I feel that this enjoyement for problem solving is natural and is in all kids, it jsut needs to be transfered to maths problems too.

  3. Sue says:

    Thanks for the recommendation, Pearl!

  4. Pearl says:

    You asked about books that offered instructions/examples a la the Lockhart philosophy. I am currently reading “About Teaching Mathematics, 2nd Edition” by Marilyn Burns. I love some of the activities and exercises that Burns has come up with. They all emphasize “playing” with numbers and getting kids to see patterns. I tried out one with my kids ages 7-12, and despite their age differences, the activity (Palindrome Numbers) worked for each one of them. The youngest one could do some of the easier addition, the middle child worked on addition w/regrouping and started to see some patterns, and the oldest was really noticing the patterns. And in the end, all three said “That was fun!”
    Here’s a link to the book:
    Hope you find it helpful!

  5. bledsoe says:

    I agree, it sounds like your daughter may have some real interest/ability in math. If she’s interested in binary numbers, the whole world of computers and computer programming comes immediately to mind. There are plenty of math-related activities and projects related to binary numbers (and their cousins, the hexadecimal system), from simple (or not-so-simple) conversion of numbers from one base to another, to the use of truth tables and logic to design computer circuits, to the many different levels of computer programming, from higher level languages like Python and Java down to assembler language and machine code. All three of my boys have dabbled in computer programming in one way or another, even as young as 8 years old, and I can tell you from personal experience (I’m a former engineer and computer programmer) that there’s a ton of math used in all those areas.

    Regarding taking an unconventional approach to education, I understand your uneasiness; I felt it when I was homeschooling my boys. I suppose some homeschooling parents feel completely comfortable with less traditional approaches but I was never one of them (I was something of a “great rule-following straight A math star” myself). I remember being particular fascinated by a book by Grace Llewellyn (author of “The Teenage Liberation Handbook,” something of a classic in the unschooling movement). Llewellyn also wrote a book called “Real Lives: Eleven Teenagers Who Don’t Go To School Tell Their Own Stories” which is just what it sounds like. It was the first time I ever considered the fact that one might grow into a healthy young adult without going the traditional school route.

    Good luck and let me know how things go. Sounds like you’re doing a great job.

  6. Sue says:

    Thanks so much for responding. Yes, what you recommended has been my approach so far and we have been doing a lot of hand-on problem solving together. I think my daughter saying she hates math is purely a function of the way it has been taught. I actually think she would love it, if it were taught right. In fact, we were watching The Story of One (a PBS video on the history of numbers) and from that, she became really interested in converting numbers into binary. She picked up the concept so easily, so I sense she has some real math talent that has been buried beneath all of the focus on arithmetic and mind-numbing worksheets of the past few years.

    So while I can just follow her lead and use my own creativity to teach her math, I do feel a little insecure about my haphazard approach to teaching. I’d love to embrace Lockhart’s advice to just forget about the computational component of math for now, and leave it for when she wants to learn it — but I am also afraid of taking such an unconventional approach. I’d like to at least read about someone else who tried it and see how it worked out.

    As an aside, regarding your remark on history: I always hated history when I was a student. But now I love it. And again, my dislike of history was because it was taught in the most mind-numbing way: memorizing random dates and names. Through homeschooling, i discovered a series of history books by Joy Hakim that really make history come alive. My daughter can’t put them down. So I think people’s dislikes of many subjects if often more about how they were taught than the subject itself.

  7. bledsoe says:

    Hi Sue,

    I’d actually like to know more about Lockhart’s methods myself. I suspect much depends on the fact that he is able to work with kids at very young ages at his school and presumably form much of their mathematical thinking early on.

    One of the great things about homeschooling is that you don’t have to approach it any particular way, so if your daughter hates worksheets then by all means stay away from them as much as possible. There are plenty of things to do with manipulatives or just “real world” math, those places where math touches on actual life (measuring, estimating, drawing geometrical designs, data analysis, number games, etc.). If your daughter expresses an interest in something math-related, help her to explore it and let that be your curriculum.

    That said, I will note that I’ve never agreed with many parents who think that if their child doesn’t love math then they need to “do something” to make them love math. Some kids just don’t find math all that interesting (I never found history all that interesting), and that’s okay. You may (or may not) still want to make sure your daughter covers some basic math in her schooling, but if there are other things she’s more interested in, I’d say let her explore those.

  8. Sue says:

    I loved reading Lockhart’s lament. I was one of those great rule-following straight A math stars who thought I was good in math — because I really had no idea what math was all about.

    Flash forward 30 years, and I am now (out of necessity, not choice) homeschooling my 4th grader. She says she hates math….because she equates it with mindnumbing worksheets and arithmetic drills. So it is not too late for her and I would love to adopt Lockhart’s approach into my homeschooling. Can you by chance recommend a book/curriculum/website that would offer some practical guidance on this?

  9. bledsoe says:

    Hi Stephanie, and thanks for the kudos. Glad you enjoyed the post.

  10. I’m just ready to name a Huffington Blog Post entry: Teachers Are My Rock Stars, and my Director of Development casually said “Oprah said that,” and in horror I went to google it with a “say it ain’t so” feeling fomenting.

    Instead I found your lovely blog, and I much enjoyed this entry and could not agree more with the sentiments of the author. I will buy his book today. Happy writing!

  11. bledsoe says:

    I think you may be talking about something that’s related but slightly different from what I was talking about. Yes, there’s something inherently troubling about questions that don’t have a single right answer (and something inherently comforting about those that do), but I think that’s different from the issue of students who are simply uncomfortable with the idea of DOING WORK.

    Though now you’ve got me wondering about whether there might be a relationship between the two.

  12. browse says:

    I definitely think you’re onto something. I’m one of those students who enjoys the validation of finding “the right answer”. To that extent, math classes like algebra and calc, which are usually presented in a more formulaic, tool-based approach, are really rewarding and enjoyable to me. Classes like Discrete Math and Number Theory end up being rather more challenging and occasionally frustrating. I am (slowly) learning to be more comfortable with uncertainty, and learning how to flail about with a topic for quite some time before some sense of pattern emerges from it. But whew, none of my previous schooling ever prepared me for that kind of math. It’s a stretch.

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