Transition to college math, development age and Harvard Math 55

The question of how age links to ability to understand a proof, e.g. not leave steps out, is obviously important to transition to college math.

At Harvard they have the course Harvard Math 55.

http://en.wikipedia.org/wiki/Math_55

Harvard Math 55

http://www.math.harvard.edu/pamphlets/freshmenguide.html

If students in high school have difficulty identifying left out steps in proofs because of age, then Harvard Math 55 for intro freshman would be a mistake.

This then raises the issue of what is going on in the mind at age levels that makes it hard to learn proofs or complicated proofs.

Does this mean at younger ages, drill and procedural math should be the focus and only at later age levels should critical thinking be stressed?  Or does it mean that critical thinking has to be introduced at younger ages to create the development in critical thinking needed at later age levels?

This hypothesis or way of thinking would also suggest the use of procedural techniques as much as possible to teach concepts as opposed to the other way around.

Successor prime notation can be used to teach natural number concepts.  1960s New Math did not use prime notation for successor.  Dedekind 1888 did.  So Dedekind may have been ahead pedagogically.

n+0 = n

n+m’ = (n+m)’

This notation helps teach that addition as a concept comes from successor as a concept.  Here the operator nature is stressed.

When we use ordered pairs, we stress the function.

(n,0) -> n

(n,m’)-> (n+m)’

((n,0),n)

((n,m’),(n+m)’)

If one looks at the Devlin blog article on Mathematical Thinking, one will see that it is not focused on age level development issues and whether we can teach not leaving out proof steps at younger ages, or we should try anyhow, or anything of the kind.

http://devlinsangle.blogspot.com/2012/08/what-is-mathematical-thinking.html

Truth tables and propositional calculus with their symbols are a way for younger people to learn proofs if that is something they are particularly challenged on because of lower critical thinking skills at lower age levels.  If this is correct, New Math in the 1960s was right to try to introduce such methods.  They are suited to teaching concepts in a procedural and mechanical way suited to lower age levels.  If we wait until people are in their 20s to teach proofs, that may not work either.  We may have missed pushing and stimulating the needed development for critical math thinking in the math focus.

 

 

 

Advertisements

About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s