Tuesday, January 17, 2012

Failing by Solving

The upcoming February 2012 issue of the Notices of the American Mathematical Society contains an interesting essay titled "A Modest Proposal" by Alan H. Schoenfeld, a Professor of Education and Mathematics at UC Berkeley. He writes of some "horror stories" of how kids being taught mathematics respond to questions that don't match the usual pattern:

Reusser [2] asked ninety-seven first- and second-grade students the following question:  
There are 26 sheep and 10 goats on a ship. How old is the captain?  
More than 3/4 of the students “solved” the problem, obtaining their answers by combining the integers 26 and 10.
( Note: The cited paper is K. Reusser, Problem solving beyond the logic of things, Instructional Science 17 (1988), 309–338. )

Schonfeld concludes that kids are not being taught how to think correctly in classrooms and advocates teaching in such a way that they come to view mathematics as providing  "a set of sensible answers to a set of reasonable questions".

This is certainly commendable, and I like the example he gives of how to make the arc length formula simple and intuitive. But, as I've argued repeatedly on this blog, the problem runs deeper.

Over-schooling, by its very nature, so constrains young children's thinking that they are unable to see outside the patterns that they have been drilled on and made to repeat. They need to learn in a more carefree environment, ideally one where they have plenty of time to think of many "reasonable" questions and possible solutions on their own through their natural habits of play and curiosity-driven exploration, free of the overwhelming pressures of adult expectations.

Here is a relevant piece from John Holt's foreword to his classic "How Children Fail"; the italicized words in brackets are mine, connecting the quote to the above setting:

...there is a more important sense in which almost all children fail: Except for a handful, who may or may not be good students, they fail to develop more than a tiny part of the tremendous capacity for learning, understanding, and creating with which they were born and of which they made full use during the first two or three years of their lives.  
Why do they fail?  
They fail because they are afraid, bored, and confused.  
They are afraid, above all else, of failing, of disappointing or displeasing the many anxious adults around them, whose limitless hopes and expectations for them hang over their heads like a cloud. [Every question posed to them at school has a clear and definite answer; and failure to answer disappoints the teacher. So they know once they are asked a question, they *must* set out to solve it.
They are bored because the things they are given and told to do in school are so trivial, so dull, and make such limited and narrow demands on the wide spectrum of their intelligence, capabilities, and talents. [Face it. At the end of the day, solving word problems about addition are simply not as interesting as any of the dozen other things related to playing and exploring and being with friends that little kids would rather be doing.]
They are confused because most of the torrent of words that pours over them in school makes little or no sense. It often flatly contradicts other things they have been told, and hardly ever has any relation to what they really know — to the rough model of reality that they carry around in their minds. [If much of what is presented to them in their workbooks or by the teacher doesn't make sense anyway, most of the time,  why should this puzzle be any different? What all this confusion sadly does is to breed a special form of intellectual laziness --- it's easier just to play along and mindlessly apply some known formula, rather than say over and over again, "I don't understand".]
I asked my eight-year old the same question. The amazing school he goes to leaves him to his resources almost entirely. I do teach him a bit myself from time to time, mostly because of my own interest in mathematics, but I try hard not to over-do it. And when we do talk about math, our discussions are very much in the spirit of Schonfeld's "reasonable questions" approach. So I was quite sure how he would respond.

Nevertheless, it was with some relief that I heard him say, "Wait, that doesn't make sense! What does that have to do with anything?"


This post seems as good place a place as any to put in a pointer to one of my favorite songs:


David Kempe said...

This article about the sheep and cows has made the news in a bunch of places. I've also seen plenty of articles and papers advocating a joyful, exploring approach to math, papers advocating that children be given the freedom to explore their natural curiosity. I forget the name of the author, but there is the paper that likens the current math education to teaching students music without ever letting them hear music or touch instruments, rather learning music theory entirely in the abstract.

What these papers often share is a list of a few beautiful results, often combinatorial or geometric, and an explanation of how these could be taught in a way that students explore and discover them, and experience the joy from doing so. I have to admit that I have not read the Berkeley paper yet, but the articles I've seen in the past were remarkably short on how a concrete math curriculum would evolve or look like for the 10+ years that students usually study math. It's easy to be shiny and beautiful if you get to pick out the gems. But to return to the musical analogy, to become a musician who can play beautifully, you must have gone through a lot of drilling, in addition to working on seeing the beauty.

While I don't use my fast multiplication/addition skills or various other things that were drilled into me at some point very regularly, I think that these drills often breed a notion of familiarity. You are more comfortable building on concepts that you have good skills in. Yes, it would be very useful to truly understand these topics, and I like the fact that now I do. But it would be difficult if every time you need to apply a previous skill, you kind of had to rederive it from scratch. You wouldn't get very far.

Frankly, I have become convinced that the problem is not just in the schools; the problem seems to lie in whether children have role models who themselves enjoy exploring, learning, and understanding. Your children are extremely blessed to have parents who like to think like scientists, who enjoy the process of discovery, and who are able to admit when they don't know something, and work at finding solutions. Many/most children are not lucky. I think that children can sense very easily if their parents are scared of math, or just find curious questions annoying. If parents model a lack of curiosity, and a desire to have things follow old patterns and routines, children will often pick this up early.

Once children arrive at school with a mindset of "I want you to tell me what I need to do," it will become much harder to implement this idea of learning by exploring. I have talked with quite a few people who genuinely felt that with regards to math, they want to be told what formula solves their problem, and how to apply it. Many people like patterns and routines. And few people genuinely see the beauty of math and mathematical discovery. You and I both try to awaken it in as many students as we can, but I think that in a lot of people, it's not dormant, it's absent.

So I think that if implemented in practice, even from a young age, creative and exploration-driven ways of teaching math would meet with a lot more resistance from students and parents than implied by many of these writers. Schools like Shriram's probably have a lot of selection bias, in that the parents believe in exploration learning, and thus will have set very different role models to begin with. Those students may be much more inclined towards this kind of approach.

Anyway, I felt like sharing, since I'd also thought about this question before, and am not as idealistic any more as I used to be.

Bhaskar Krishnamachari said...

Thanks for the thoughtful comments, David. I think the other paper you are referring to was perhaps Lockhart's "A mathematician's lament" (

I don't disagree that practice is necessary to build skill, but I do feel strongly that it should not be made the centerpiece of the educational process, particularly at younger ages.

Your point about the impact of early exposure to role models is fair. Parents and family do have a huge influence on a child's aspirations and ways of thinking. What can be done about this? One nice effort I've seen to address this issue is Iridescent ( , founded by a USC Viterbi Alumna) which organizes joint family science classes in LA for kids and their parents, many of whom have had limited experience of a good science education themselves.

As to whether the ability to appreciate and do mathematics is utterly absent in some portion of the population, my gut feeling is that most kids have the raw potential to enjoy and achieve a significant level of proficiency in most endeavors. The key is that they have to perceive it themselves as "cool" and/or necessary, and have sufficient time and a great deal of freedom from too much close adult supervision for immersion in it. The near-universal appreciation of video-games among kids exposed to them, and their ability (in most cases) to speak fluently in their mother-tongue are evidence enough of this. I do believe that this potential can be "schooled-out" of kids through the combination of fear, boredom and confusion that Holt refers to.

At the same time, I am not arguing that everyone will appreciate math to the same extent. The catch with an autonomous, exploration-driven approach to education is that there is no fundamental guarantee that a particular kid will enjoy or feel inclined to consider as "cool" or important any particular activity, be it math, writing, music, or sports. Their individual interest in any particular direction is shaped by a complex blend of chance and interaction. An education system which is organized on the principle that everyone has to learn the same amount of "material" at the same pace, is fundamentally misguided.

Anonymous said...


I'd like to share some personal experiences that seem marginally relevant to this post. I'm a volunteer at-home tutor for a 6th-grade boy. He and his family are refugees resettled in the US. He had insufficient schooling before coming here, and is quite behind. I spend a few hours with him once a week, helping him catch up.

The two most important things he can be taught right now are English and math. Since his math seemed okay, I had been completely focused on English until the day he pulled out his math homework, which he had "solved."

The page was full of fraction-comparing problems -- the goal of each of them was to compare a given pair of fractions and write down the sign representing the relationship between them (< and >, no equalities). I started going over them, and was immediately quite pleased to see that he had inserted the correct sign between 2/3 and 3/4. I asked him how he had arrived at the answer. He said 2 (in 2/3) is less than 3 (in 3/4) and 3 (in 2/3) is less than 4 (in 3/4), so 2/3 must be less than 3/4. It's quite unfortunate that his flawed reasoning led to the right answer in this particular case, because I had to struggle to convince him his method was incorrect despite producing the correct answer. I thought that the counterexample of 99/100 and 100/200 would help, that he would clearly see that 99 was almost all of 100, but 100 was only half of 200, but it didn't help -- he complained that the numbers were too large for him to work with. I had a twin struggle at this point -- to help him understand the magnitudes represented by fractions, and to disentangle the concepts of method and answer. With my limited time running out, I gave up after a while, and instead worked on drilling some mechanical rules for comparing fractions. His problem set included negative numbers as well, and there was no way I was going to be able to give him an intuitive understanding of it all. He just doesn't have enough time with me. I just hope that he learns and remembers his mechanical rules well enough to make progress in school, and over time is able to understand the meaning of what he is doing, hopefully with my continued help.

Another incidental problem that came up was that he associated the word "answer" only with numerical results. I had to take some to explain to him that "<" can also be an answer to a problem. But he may not remember that next week.

Oh, and as for the problems that could not be solved using his "method", he had just eyeballed them and arrived intuitively at the answers (~ 50% wrong, as expected), which makes me wonder what he's being taught in school.

In my case at least, I know teaching by mechanics is a necessity. And I fear I will not have enough time with him to plug this stop-gap by eventually catching him up with the real meaning of what he's doing.

Bhaskar Krishnamachari said...

Dear Anonymous,

Thank you for sharing your experience. I admire you for volunteering your time to work with this child in need.

If you feel that a drill-based approach may be the only way to go at least for now, may I recommend taking a look at the JUMP math curriculum ( if possible? It's a particularly well-thought out curriculum and you can find workbooks titled "JUMP at Home Grade x" on Amazon, for 1st to 6th grade. I recommend starting with a much lower grade to see what he gets and what he is missing from earlier grades. The core philosophy of JUMP math (which I do agree with, despite my strong advocacy of free, inquiry-driven learning in general), is that in mathematics you do have to master certain basic skills with solid understanding before going to the next steps; any gaps result in a lot of confusion. The JUMP program has a very systematic, thorough approach. In many cases, because the teachers are pressed for time to cover the curriculum at a standard pace, some students don't quite master these critical skills and the class moves on leaving them further and further behind. The JUMP philosophy tries to avoid this from ever happening.

I have worked with inner city school children in new york and LA in the past, and it affected me deeply to see how many kids from disadvantaged backgrounds lack positive role models and exposure to richer learning experiences in their life. What you are doing for this child already and what you could do for him to expand his horizons is priceless.

Besides working with him on his math and english, I hope you also talk to him about other aspects of your life, your interests, the kind of work you do, about people you know and what they do. Beyond helping him catch up with his classes, do tell him about and show him things that will evoke a sense of wonder about the world outside his home and school. You may also find it useful to hear him tell you about his world, and spend some time understanding what matters and is interesting to him. You may find this way some keys to help him learn and grow.

Anonymous said...


Thank you, those are some very useful recommendations and suggestions. I've ordered the JUMP books and will be contacting them for their Tutor Guides.

I do make sure I spend time on expanding his horizons and getting him interested in the larger world (astronomy nights with my telescope, field trip to see ballet, and more in the future -- I'd like to do a hike and a museum soon). But your suggestions have made me realize that I don't ask much about the goings-on in his world, and say almost nothing about life in my world. I'll make sure to incorporate those.

Good stuff!