Top Posts
 Notes on Representation  The Number Line
 Operators, Functions, and Properties  part 42
 Running the Math Gauntlet
 Making Change  A Sam Lloyd puzzle
 Notes on Deferred Computation  the Pythagorean Spiral
 Notes on Fractions
 Operators, Functions, and Properties  part 35
 Operators, Functions, and Properties  part 38

Recent Posts
 Mathematical Notations and Schools – 15
 Mathematical Notation and Schools – 14
 Mathematical Notation and Schools – 13
 Mathematical Notation and Schools – 12
 Mathematical Notation and Schools – 11
 Mathematical Notation and Schools – 10
 Mathematical Notation and Schools – 9
 Mathematical Notation and Schools – 8
 Mathematical Notations and Schools – 7
 Mathematical Notations and Schools – 6
 Mathematical Notation and Schools – 5
 Mathematical Notation and Schools – 4
 Mathematical Notation and Schools – 3
 Mathematical Notation and Schools – 2
 Mathematical Notation and Schools: The Series
Blogroll
 attribution black box deferred computation denominations discoverable education embedding equivalence extending patterns gauntlet graded hurdles invariants lookup math class math in the comics matrix multiplication models naming notation puzzle recurrence relationship representational proof representations reverse engineering selection series on math learning sieve sorting standing on nothing state unlearning whole vs. parts
Archives
 July 2011
 June 2011
 May 2011
 April 2011
 February 2011
 January 2011
 December 2010
 November 2010
 October 2010
 September 2010
 August 2010
 July 2010
 May 2010
 April 2010
 March 2010
 February 2010
 January 2010
 December 2009
 September 2009
 August 2009
 July 2009
 June 2009
 May 2009
 April 2009
 March 2009
 February 2009
 January 2009
 December 2008
Tag Archives: embedding
Operators, Functions, and Properties – part 13
In this series, we have been looking at little machines that do things like “add 2”, and we’ve been chaining such machines together by connecting the output of one to the input of the next. We’ve been exploring the behavior … Continue reading →
Posted in Uncategorized

Tagged black box, denominations, embedding, extending patterns, models, representations, reverse engineering

1 Comment
Representations – Black Boxes – Equivalence
I introduced the notion of a black box in an earlier post as some thing that has an internal organization that drives its behavior, and though we can see the behavior, we don’t have perfect knowledge of that internal organization. … Continue reading →
Posted in Uncategorized

Tagged black box, embedding, extending patterns, models, representations, reverse engineering, unlearning

1 Comment
Representations – Formulas and Some Alternatives
There are systems of notation for mathematical expressions that are in wide use. One of them is so widely used and so wellknown that we often think of it as the only one, the real one, the true one and … Continue reading →
Posted in Uncategorized

Tagged embedding, extending patterns, math class, models, representations, unlearning, whole vs. parts

Leave a comment
Quantity – Different Kinds of Numbers: Classes
Let me recap what this series of blog posts has been about. I’ve been looking at different kinds of quantities, different kinds of numbers. I started with observations like: it’s meaningless to add 1951 AD to 2009 AD but completely … Continue reading →
Posted in Uncategorized

Tagged denominations, embedding, invariants, models, naming, representations, unlearning

3 Comments
Quantity – Different Kinds of Numbers: Vectors
Sometimes, when you assign a number to a quantity, a single number really isn’t enough. Take blue jeans sizes, as a simple example. Jeans sizes aren’t expressed with a single number, from small to large, like shoes. Since at least … Continue reading →
Posted in Uncategorized

Tagged denominations, embedding, extending patterns, models, representations

2 Comments
Quantity – Different Kinds of Numbers: Keys
I introduced this series by noting that there are different kinds of numbers – more precisely, different kinds of quantities represented by numbers – and that each allows you to draw different kinds of conclusions. The most basic kind of … Continue reading →
Posted in Uncategorized

Tagged embedding, extending patterns, invariants, models, representations, reverse engineering

3 Comments
Quantity – Different Kinds of Numbers
We’re very used to representing quantities by numbers. The cash register will tell us how much we have to pay, and the pump at the gas station will tell us how much gasoline we poured into our tank, and the … Continue reading →