Notes on an unpublished article by Paul Lockhart titled, “A
Mathematician’s Lament”. It was later published on the Devlin’s Angle
website under Lockhart’s
Lament and followed by a post two months later that focused on
critic of the article and a response from the author (see Lockhart’s
Lament - The Sequel). In 2009 the 25-page article was expanded into
a 140-page book.
- Mathematicians are working artists
- It is more abstract and therefore allows for more freedom than
poetry, art or music
- While poetry is built upon words and music upon sound, mathematics
is solely built upon imaginary ideas
- A student lost in thought better represents mathematics than one
focused on finding the correct answer to a series of problems
- A rectangle containing a triangle with two vertices at the same
point as two neighboring corners of the rectangle. The problem of
finding the triangles size can be solved by simplifying the problem by
dividing the triangle using its third vertex.
- The key to discovering the solution is based upon inspiration,
experience, trial and error or simply dumb luck
- “The art is not in the”truth” but in the explanation, the argument.”
It is like looking at a painting and only thinking literally about the
image and not trying to understand or develop an explanation or
argument
- “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 just very good at following
directions.”
- Mathematics are subject to critical appraisal
- Practical math is actually hardly ever applied
- The fantasy that mathematics can provide is far more interesting
than how mathematics applies to reality (e.g. compound interest)
- The history of mathematics is incredibly important because it
provides a narrative
- Allow students to struggle with a problem before providing help and
even then only hints and steps, not the entire solution
- “Have an honest intellectual relationship with our students and our
subject.”
- Young children in math class should play games like Chess, Go, Hex,
Backgammon, Sprouts, Nim. Also do puzzles and be encouraged to make up
games.
- “You learn things by doing them and you remember what matters to
you.”
- “As Gauss once remarked,”What we need are notions, not
notations.””
- “Mental activity of any kind comes from solving problems yourself,
not from being told how to solve them.”
- The history of mathematics provides a narrative and context for
learning.
- Redundant nomenclature like mixed number and improper function,
quadrilateral and four-sided shape, secant and the reciprocal of
cosine
- Being engaged in the process means having ideas, not having ideas,
discovering patterns, making conjectures, constructing examples and
counterexamples, devising arguments and critiquing each other’s
work
- Students designing their own problems based on what they are
learning
- What about writing mathematics using prose? For example, the use of
formal rules as reasons for statements that lead to a proof, rather than
a simple explanation
- It is not the concept that is difficult, but the language used to
formalize it
- It’s not the amount of material covered, but rather the fundamental
concepts that are important
- Definitions are developed historically by working on problems, not
as a prelude to it