Book: A Mathematician's Lament
Overview
Title: A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form
Author: Paul Lockhart
Published: 2009 (as a book; the essay circulated from 2002)
Genre: Essay / Education / Mathematics
Pages: ~140
Stars: 10
Description
A Mathematician’s Lament began as a 25-page essay by Paul Lockhart, a research mathematician who left academia to teach K-12 mathematics in New York City. The essay was passed around informally for years before being expanded into a short book published by Bellevue Literary Press.
Lockhart argues passionately that the way mathematics is taught in schools is fundamentally broken — that by reducing mathematics to a set of rote procedures and formulas to be memorised, schools strip away the very thing that makes mathematics beautiful: it is a creative art form. He draws an analogy to music education: imagine if music class consisted only of reading and writing sheet music notation, never actually playing or hearing music. That, he argues, is what mathematics education does.
The essay is provocative, funny, and at times infuriating (deliberately so). It is widely read and debated in mathematics education circles.
“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”
— G.H. Hardy (quoted approvingly by Lockhart in A Mathematician’s Lament)
| Source: Bellevue Literary Press – A Mathematician’s Lament | Original essay (PDF) via the Mathematical Association of America |
My Notes
This is clearly a favorite of the year so far, and a recommendation for parents, kids (middle school and up) and math nerds everywhere. For the first to categories it is especially recommended for those that hated math class and think they are ‘bad at math’. The book is super short and a really entertaining read so it is likely to be completed in a short hour or two. This is clearly a favorite of the year so far, and a recommendation for parents, kids (middle school and up) and math nerds everywhere. For the first two categories it is especially recommended for those who hated math class and think they are ‘bad at math’. The book is super short and a really entertaining read so it is likely to be completed in a short hour or two. Lockhart’s joy in the art of math leaps off the page, as does his joy of instilling this passion in the seventh graders of his math class. I only wish I’d had more teachers that could bring this level of mentoring a self-learning attitude when I was younger. From the very first page, he reframes what you think math classes purport to be, and enrage in you a passionate desire to go change the way your kids are being taught this art. Lockhart’s joy in the art of math leaps off the page, as does his joy of instilling this passion in the seventh graders of his math class. I only wish I’d had more teachers that could bring this level of mentoring a self-learning attitude when I was younger. From the very first page, he reframes what you think math classes purport to be, and awakens in you a passionate desire to go change the way your kids are being taught this art. The mental shift of math as a science, to math as an art, is exemplified in a series of great metaphors from page one. The allegory of a musician having a nightmare schools teach music class purely by teaching kids how to read and write sheet music, things like the theory of fifths and how to accurately and neatly draw musical notes. All this is done prior to ever hearing music, playing an instrument, singing any songs, or even hearing any musical pieces. The only time a student would be able to enjoy musical art is once they were in graduate schools. The best students in high school and college would be able to adhere to evaluation of peer notation of music. They may go to graduate school in music and discover that they actually have no talent in the art of music and then either fail as musicians or drop out.
This is how we are teaching math currently in grade school middle school high school and even colleges. We teach the basic alphabet of math, we teach the formulas that will get an answer to an obscure question that no one has defined why we want to know. All a student needs to do is memorize these formulas and plug in the new values and run through the algorithm. This requires no deep understanding of the art of, of the fundamental foundation of mathematics, and how numbers and associated geometric objects interact and ‘play’ together. The student is left bewildered with how any concept intertwines with another and only the rote steps to answer standardized test questions. It is only until upper level graduate schools when a student would be doing research in math that they would suddenly discover the art of math and be able to apply previous concepts they didn’t even understand to this new fuzzy logic of finding answers to puzzles in mathematics. This is how we are teaching math currently in grade school, middle school, high school, and even colleges. We teach the basic alphabet of math, we teach the formulas that will get an answer to an obscure question that no one has defined why we want to know. All a student needs to do is memorize these formulas and plug in the new values and run through the algorithm. This requires no deep understanding of the art of the fundamental foundation of mathematics, and how numbers and associated geometric objects interact and ‘play’ together. The student is left bewildered with how any concept intertwines with another and only the rote steps to answer standardized test questions. It is not until upper-level graduate school, when a student is doing research in math, that they suddenly discover the art of math and can apply previous concepts they didn’t even understand to this new fuzzy logic of finding answers to puzzles in mathematics. Lockhart’s answer to this is to have students explore math as if no concepts in math have ever been discovered until the student discovers them. In this way they are explorers and an exciting adventure in solving puzzles with numbers. He allows his student to explore a problem come up with assumptions, conjectures, and theorems with proofs as diverse as the class size allows. Each student ‘proves’ a conjecture to a problem in their own unique way. Each proof retains the artistic skill of the student theorist. This means each student can have opinions of the ‘beauty’ of other’s works of art, and have a conversation about how to improve on each others work, similar to their art classes.
Lockhart’s answer to this is to have students explore math as if no concepts in math have ever been discovered until the student discovers them. In this way they are explorers and an exciting adventure in solving puzzles with numbers. He allows his students to explore a problem, come up with assumptions, conjectures, and theorems with proofs as diverse as the class size allows. Each student ‘proves’ a conjecture to a problem in their own unique way. Each proof retains the artistic skill of the student theorist. This means each student can have opinions of the ‘beauty’ of others’ works of art, and have a conversation about how to improve on each other’s work, similar to their art classes.
They also gain a core ownership of knowledge in their bones of how math works, because they themselves ‘discovered’ the concept learned. They did not get it spoon fed in one end and out the other not to be remembered; it was generated solely by them and is now an intrinsic part of their DNA. They can use this deep knowledge of that to further study new problems that the first ‘puzzle’ uncovered, unveiling new and exciting assumptions, conjectures, and ultimately proofs in their own math journey. They also gain a core ownership of knowledge in their bones of how math works, because they themselves ‘discovered’ the concept learned. They did not get it spoon-fed in one end and out the other not to be remembered; it was generated solely by them and is now an intrinsic part of their DNA. They can use this deep knowledge to further study new problems that the first ‘puzzle’ uncovered, unveiling new and exciting assumptions, conjectures, and ultimately proofs in their own math journey.
Truly this is the way math should be taught in school, not as a mind-numbing exercise in standardized test subjugation, but as a journey in artistic adventures in mathematics.
Questions:
- Did Lockhart’s critique of mathematics education resonate with your own school experience?
Especially so, I was always a lover of math, but was never great in math class. This was partly because I was easily distracted (I may have had ADD, but who knows — it was diagnosed so frequently in kids who naturally had high energy like I did). I tended to ‘tune out’ when the lesson was boring, and miss most of the important bits when the teacher had moved to something I hadn’t learned yet.
A few exceptions emerged while reading this book however. One was my experience with a math class (easily forgetting which) where I had a calculator in hand and was just playing with numbers. I had taken the conversion calculation for Fahrenheit to Celsius as $ F° = \frac{9}{5} C° + 32 $ which is not super difficult on a calculator, but try that in your head please 😒. So I started ‘playing’ with the $9/5$ part… and ‘discovered’ that this is so close to 2 aka $10/5$ as to be a good path for mental math. How could I use this to my advantage? Could I just multiply the C° by 2? what would I need to do to ‘fix’ it afterwards? How close is the answer if I don’t? These are the problems I started mulling over. I ‘discovered’ by playing with this that $9/5$ is 1.8, close to 2 but off by .2 or 2 tenths. If I took two tenths of the original C° and reduced the 2 x C° by that, I’d get what would have been $9/5 C°$ . Easy to get a tenth of something, easy to multiply by 2… and easy to do in your head. All you’d then need to do is add 32! So my mental math method became, $F° = 2C° - 2(C°/10) + 32$ which anyone can do easily. To reverse it is the opposite (only use 1/2 of a 10th): $C° = (F° - 32)/2 + (F°/20)$ These give you very close to the correct answers and don’t require a calculator when you are simply approximating a conversion.
The second experience with this was a favorite physics teacher in my high school (New Jersey). He would start our class with a goal to teach that day’s physics lesson… but it was known to all the students that we could easily distract him from this by asking science-like questions about various other subjects, such as a Shark Week documentary we’d seen the night before. This would get him going on the tangent of answering these questions, and proposing questions back to us… the entire class would then be engaged in an interesting conversation on science that had nothing to do with his daily lesson plan. Though he’d end class by saying “Oh sorry, we didn’t cover enough for you to do the homework assignment… why don’t we take the last 20 min of class to sit here and do your homework, and if anyone has any questions I’ll come by to assist you.” We would believe that we’d gotten away with something, but really in retrospect I think he was teaching physics as an art form, with all of us as collaborating artists.
- As someone with an engineering background, did his argument about maths as “creative art” surprise you, or did you already feel that?
I have always believed this… many of us who are mainly self-taught or guided self-learners primarily play with ideas to learn new concepts. In order to learn any new skill, be it DIY upholstery, technical concepts, or a new language, one must get immersed in the subject and have their hands dirty trying → failing → retrying before succeeding.
- Do you agree that the way maths is taught kills curiosity? Or is there a counter-argument he misses?
I think anyone who has been subjected to the last 50yrs or more of math education can’t argue with this. The question is how to change the status quo. In his book he does not advise ‘revising’ the plan, since this has been attempted with disastrous effect in the past. The current structure of math education kills the very thing it claims to teach. His argument is that reform always comes in the incremental form, add a bit of this here and a bit of that there… He thinks incremental reform just makes the cage prettier. Mathematics is an art, and you can’t reform a system that denies that.
- Is there a topic in mathematics you wish you’d been taught the “beautiful” way?
I recently did a refresher on polynomials in Khan Academy and was blown away at how it tied all the concepts beautifully to show how the polynomials can represent all different sides of a rectangular area (only one way to use them!) but this was WAY more insightful from the ‘just memorize it’ way I was taught in class. Class was completely boring to me because it was so oblivious how the concepts could be used in life. “When are we ever going to use this?!” is said all the time, precisely because Math is not taught in an artful way. You never hear anyone say “When will I use this in life?” in an art class, they are just having fun creating and playing with the medium. It would have been way more effective to have us play with geometry and solve problems in any way we choose throughout all levels of math.
Quotes
Location: 199
Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions.
Location: 756
School has never been about thinking and creating. School is about training children to perform so that they can be sorted.
Location: 757
It’s no shock to learn that math is ruined in school; everything is ruined in school!