We don’t often connect the words mathematics and trust together, but the essence of mathematics is in trust. Today, as usual, I was pondering the issues that people have in life when it comes to learning math and in learning problem solving. It occurred to me that it really is an issue of trust:
- Why do I have to learn this stuff?
- How does it make sense in my life?
- Who says this stuff actually works?
- Why use methodical problem solving when it takes away from creativity?
- I don’t get how memorizing rules to solve stuff actually works. I would rather figure things out for myself.
Well, here is the interesting twist. If you don’t trust math and other people’s reasoning then you are already a born mathematician!
Mathematicians have a long history of not trusting anything. Consider Euclid’s writing inElements, where the first three definitions that open up this quintessential math textbook break down the most basic elements of physical reality.
- A point is that which has no part.
- A line is breadthless length.
- The ends of a line are points.
These three initial definitions of mathematics are the basis of western mathematical thought. Consider Euclid’s process. All mathematics start in the most basic and mutually agreeable definitions. This initial list is 23 definitions of simple space. Later Euclid introduces the idea of common notions. The first common notion is the most famous:
Common Notion 1.
- Things which equal the same thing also equal one another.
This common notion needed to be tested and reproved so that we could depend on that rule in mathematics.
Browse through Elements to see how basic the definitions are and how they all depend on each other as a house depends on each brick in a foundation. Each definition, postulate, common notion, and proposition needed to be tested and retested.
Historically, mathematicians demonstrated very little trust in thought in creating thirteen books in Euclid’s Elements to explain elementary mathematics. I liken mathematicians to a child afraid of the water, fearful of letting go of the side of a pool till they can reach out to another fully trusted object like a parent or a ladder. Mathematicians do not accept a new rule until it is fully rock solid.
Traditionally, mathematicians have struggled with each new discovery, argued it, tested it, and proved it through collaborative agreement. This is the same angst that has given us rigorous scientific investigation and the threshold of overwhelming proof.
So, how is it that in a tradition of such process driven freaks have we ended up with mathematics education that has turned into taking rules and instructing children to just accept them as fact and to make haste using these rules? It works because we said so!
In successful mathematics education, the process starts with creative play and tinkering, where math learners begin to test what they have discovered. Math learners develop their own construction of these basic rules so that they can intrinsically build trust in their feel and understanding of mathematics. It is through the development of these sound principals of elementary math that math learners can begin to trust their own reasoning and to then become creative in solving problems using these rules.
There is an exciting moment when math learners begin to trust a process that they create and to flow through a multistep problem to find a solution. Having the ability to track back and forth through the process and to see that there is no gap in the logic and that there is no fault in their reasoning is freeing. This moment is personal empowerment. Not just that the math learner becomes a successful doer, but because the learners become successful problem solvers with a template for success. Our most precious tool, our functions of reasoning, become powerful.
It is this process of problem solving where we share a common need to be developed. Consider a life where we could all share common rules for understanding our reality, engaging in a shared process, solving problems both big and small, only to question the process and start over to let the next generation make it better.
I could trust a world like that!