From Mathematical to Imaginative Thinking

What is the nature of mathematical thinking?  How does a child engage in mathematics at different stages of development?  What is the step from mathematical to imaginative thinking? 

In just 10 days, we will meet in NYC to explore these aspects of teaching math! Colleagues will describe classroom experiences with the Ten Theorems.   We will work with mathematical exercises in projective geometry, described by Rudolf Steiner in 1921 (GA76, Lecture 3), to practice moving on the path from mathematical to imaginative thinking.   You can still register on the workshops page online.  Meanwhile, here is a glimpse into the weekend’s theme.

What is the chance of rolling a “6” with a fair die? Nearly every 9th grader in a recent Counting Theory class immediately recognized that the chance is one out of six.  They were certain.  They could “see” the answer clearly.  Next, we passed out dice, and each pair of students had the task of determining whether their die was fair.  After some conversation, students decided that they would have to roll the die many times and record the result.  They embarked on this task with relish, making a mark for each point on a graph.   One group completed 400 rolls.  It was apparent from the histograms whether a die was close to being fair or not.  Only then did the discussions begin about how to quantify the fairness, and it became much more heated when students considered whether they would play a game with that die.    

In experiencing the initial insight, a student engaged in an inner activity, mathematical thinking.  The correct answer feels right, and it is right for all time.  Contrast this to the experience of tossing a single die, perceiving and describing outcomes.  This die is unique, maybe warped or chipped, certainly not here for all time.   In the process a 9th grade student brings their inner world into relationship with the outer world. 

A student can find sureness in the inner process of mathematics, whereas the sense world is an enigma.  She establishes a relationship between her inner and outer worlds by seeing the mathematical lawfulness in the outer world.  Rudolf Steiner describes how humans have evolved through previous epochs to use number, weight and measure to describe objects in increasingly abstract ways which has resulted in our current mode of intellectual thinking (GA204, Lecture 8).   In science, we now rely on mathematical representations of phenomena to give us a firm footing in the physical world.   When working in the world of these mathematical representations, we can predict, through logical extensions, how inanimate objects might behave.   However, by doing this, a person limits his experience of the sense world to fit the quantitative conceptual frameworks developed mathematically out of his inner experience.

 To really comprehend the outer sense world, the world of nature, a person must be willing to put aside the inner stability of the quantitative mathematical models, and to perceive imaginatively the qualities present in the sense world.  This is a new type of imaginative thinking, more akin to listening.  It does not “roll forward” in a logical, consequential way.

 Developmentally, a 9th grade student who is 14- or 15-years old is ready to encounter abstract, mathematical thinking in a Waldorf school, making conscious connections from their inner world to the sense world.  In 10th grade, wrestling with logic and analytical geometry further strengthens a student’s intellectual thinking.  By the end of 11th grade, the groundwork has been laid for imaginative thinking through studying projective geometry. The mathematical journey which began with physical movement and qualitative experiences in the early grades and became increasingly quantitative and abstract in high school, returns in 11th grade to become descriptive and qualitative.  There is again movement, but this time it is the inner movement of imaginative thinking!