“A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made – all without the advice or participation of a single working musician or composer.” (Paul Lockhart, 2002)
It is with this that Paul Lockhart begins his criticism of the current (read 1800′s to present) mathematics instruction. Lockhart uses this analogy, along with a secondary story about painting instruction, to illustrate how mathematics has lost its art. He suggests that teachers, who were indoctrinated themselves, are doing a disservice to students by withholding the beauty and artistic nature of mathematics. The problem, according to Lockhart, is that the curriculum of mathematics reduces the art and artistic critique of doing mathematics to rote memorization of formulae and the regurgitation of mathematical ‘facts’.
Lockhart suggests that mathematics is seen by society (and hence taught) as a functional tool to create technology and not as an artistic endeavor. “The art is not in the ‘truth’ but in the explanation, the argument…..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.”
I would suggest that in many mathematics classrooms, particularly elementary, there are many students struggling with problems that are authentic to their experiences and provide the platform to learn the mathematics set out as the provincial curriculum standards. These students are not learning through traditional problems such as “How old is Maria if you know that she is two years older than twice her age seven years ago?”. Lockhart states ” as if anyone would ever have access to that ridiculous kind of information, and not her age.” Questions like ” suppose I am given the sum and difference of two numbers. How can I figure out what the numbers are themselves?” opens the same conceptual outcomes with a more open-ended and multiple entry points for all students.
With differentiated instruction, various assessment strategies (like Assessment for Learning and formative assessments), and great questionning techniques like those espoused by Mariam Small, Peter Sullivan and Pat Lilburn, are all parts of a larger development in mathematics instruction that Lockhart ignores. Although the Lament is meant as a thought-provoking piece, those teachers who already include authentic, personalized learning in their classrooms probably do not recognize the mathematics classroom he presents.
Mathematics is art, as Lockhart states, and should be presented as such. But it is also beauty and simple. When presented with an elegant solution to any problem, the simplicity and careful nuances should be celebrated in the classroom. Discoveries in mathematics should be celebrated and displayed, allowing for more problems to present themselves.
Getting back to the music classes, Lockhart shares some insight to how mathematics is traditionally taught and how ridiculous it seems if music education followed the same recipe:
“Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding music notation and theory. Playing and listening to music, let along composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.”
To read the entire essay.