A Critique of the School Mathematics Curriculum
Written in 2002, “A Mathematician’s Lament” by a mathematician by the name of Paul Lockhart was published online in 2008 at the Devlin’s Angle on the website of the Mathematical Association of America. It immediately caused a widespread stir as it gave a scathing attack on the traditional mathematics curriculum at school. The purpose of this Blog is to give a brief overview and comment on some of his criticisms. Readers are invited to download and read his full paper directly from (click on link):
In addition, it is hoped that readers, whether they agree or disagree with Lockhart, will give feedback by commenting and discussing some of the important issues raised by Lockhart in his provocative paper.
Lockhart’s paper starts off by describing a nightmarish, educational situation experienced by a musician where suddenly children are not allowed:
‘… to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.’ (p. 1)
After describing a similar ludicrous situation for painting, he draws an analogy between these nightmarish situations and the traditional mathematics curriculum as follows (p. 2):
‘Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul- crushing ideas that constitute contemporary mathematics education.’
Lockhart then goes on to describe mathematics as an art and that it is about ‘wondering, playing, amusing yourself with your imagination’ and that ‘inspiration, experience, trial and error, dumb luck’ all have their role to play in mathematics’ (p. 4). He criticizes the mathematics curriculum (and associated textbooks) for focusing largely only on the learning of facts and manipulative procedures and calculations instead of the creative spirit underlying mathematical thinking as follows:
‘By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject …
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.’ (p. 5)
It is interesting that in the above quotation (and also on p. 8), Lockhart is making the ‘process-product’ distinction that was made popular in mathematics education in the late 1970’s and 1980’s by Hans Freudenthal, Alan Bell, and many others. This distinction tried to draw attention to the processes of mathematical creativity that lead to the eventual, finished products. In other words, to focus more in our teaching on the processes of discovery and creativity that lead to theorems and their proofs, as well as those of algorithmitizing, classifying, defining and axiomatizing that respectively lead to algorithms, definitions and axioms. Noteworthy in the above quotation is also that he assigns an ‘explanatory’ function to proof (argument) and not just traditional one of the ‘verification’ of results.
On p. 9, Lockhart pleads for the connection of the school mathematics curriculum with the historical (and philosophical) development of the subject, and refers to the work of Eudoxus and Archimedes. He strongly argues that problems (and problem-solving) should be made central to mathematics teaching as follows (also compare with p. 16):
‘Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process— that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique.’ (p. 9)
‘So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.” (p. 10)
In this regard, he argues very similarly to George Polya and others about the importance and centrality of problems and problem-solving not only historically, but also in current mathematical research. Though Lockhart doesn’t say a mathematics teacher has to be a professional mathematician, but that they should not be mere ‘data transmitters’. In essence, they should at least be mathematically active and good problem solvers themselves to be able to engage their learners in genuine mathematical activity (p. 11).
On p. 15, he strongly criticizes the early introduction of ‘a priori’ definitions of terminology and concepts if the need for those definitions have not yet been developed in some relevant problem setting. In particular, on p. 22, he gives a lengthy criticism of the way definitions are handled in the curriculum:
“Even the traditional way in which definitions are presented is a lie. In an effort to create an illusion of “clarity” before embarking on the typical cascade of propositions and theorems, a set of definitions are provided so that statements and their proofs can be made as succinct as possible. On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it.
The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.
This is yet another example of the way that students are shielded and excluded from the mathematical process. Students need to be able to make their own definitions as the need arises— to frame the debate themselves. I don’t want students saying, “the definition, the theorem, the proof,” I want them saying, “my definition, my theorem, my proof.” (p. 22)
Clearly, his comments about the poor teaching of definitions remind one strongly of similar criticisms already given by Freudenthal, Krygowska, and others in the 1970’s. However, despite these, hardly any changes in the mathematics curricula around the world have been affected.
In line with the above, Lockhart describes the typical High School Geometry course at school as an ‘instrument of the devil’ if it is introduced in a highly formalized manner as follows (p. 18):
‘The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.”’
His viewpoint in this regard is in correspondence with a learning theory perspective such as the Van Hiele theory, which argues that learners ought to develop through at least two levels of Recognition and Properties of Shapes before the Deductive Ordering of Shapes would become meaningful to learners.
Lockhart further argues on p. 20 that ‘formal, rigorous proof’ should not be a student’s first introduction to logical argument, and that it only becomes important when there are crises or paradoxes that have developed naturally. He opposes the two—column proof method often used by teachers in the classroom saying that it is artificial, ugly and unnatural. Specifically, on p. 21 he writes: “No mathematician works this way. No mathematician has ever worked this way. This is a complete and utter misunderstanding of the mathematical enterprise. Mathematics is not about erecting barriers between ourselves and our intuition, and making simple things complicated. Mathematics is about removing obstacles to our intuition, and keeping simple things simple.”
Continuing in the same mode (p. 22), Lockhart concludes: “The problem with the standard geometry curriculum is that the private, personal experience of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a rigid step-by step pattern of uninspired formal deductions. The textbook presents a set of definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the “good” students.”
Lockhart finishes of his critique by giving a short global critique of each phase of the whole school mathematics curriculum starting at Lower and Middle School, and proceeding through Algebra, Geometry through to Pre-Calculus and Calculus.
- Michael de Villiers, Professor Extraordinaire: Mathematics Education, University of Stellenbosch.