In one of my LinkedIn group, someone started a discussion with this question *What is the bedrock principle of mathematics?* May I share some of the answers.

1. “The bedrock principle of mathematics is the axiomatic system. The realization that there are propositions that must be taken for granted in order to have something to build upon.”

2. “Speaking about foundations, in my opinion the bedrock should be enlarged at least as follows:

- discerning that two things are different;
- identifying two things which share same property;
- discovering relations among properties.”

3. “I would say the bedrock principles of mathematics are:

- The ability to differentiate two things
- The ability to rank two things (as to most value, shortest route, least danger, etc.)
- The ability to expand the above to more than two things”

Bedrock principles of any discipline of course can’t tell us how one can know if he or she understands a piece of that discipline. So I asked

*How can one tell if he/she understands a piece of mathematics?*According to Peter Alfeld you

*understand*a piece of mathematics if you can do all of the following:- Explain mathematical concepts and facts in terms of simpler concepts and facts.
- Easily make logical connections between different facts and concepts.
- Recognize the connection when you encounter something new (inside or outside of mathematics) that’s close to the mathematics you understand.
- Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

You may also want to read my post To understand is to Make Connection.

The challenge of course will still be this question: *What does all these imply about teaching mathematics?*