Education vs. Training

The dumbing down of America

There is substantial evidence of efforts to dumb down education in order to train "workers, not thinkers." See for example various accounts of Rockefeller's philanthropic activities while funding the first Boards of Education. Whether or not this is the cause, many of us may have been trained to dislike mathematics because we did not have sufficient opportunity for two essential experiences:

• The chance to invent, discover, explore, and test our own mathematical ideas in an open, experimental, creative, and judgment-free environment.
• The chance to discover the power, beauty and mystery of the ocean of universal patterns that surround and shape us. 

While teaching, I try to clearly distinguish between the memorization of skills (training) and the development of mathematical thinking (education). Each has its place, and a complete education requires both. I hope students will also appreciate the difference as awareness of this distinction will serve them well as they enter the propaganda drenched adult world.

"While training narrows a person's perspective, education broadens it.
While training is about learning the right answers, education is about
learning how to ask the right question so that one can be
innovative, creative and responsive to change." 

-- Horace Jeffery Hodges

Waldorf-inspired education

Some believe that a true Waldorf-inspired education is incompatible with a state-funded school, in which national propaganda and narrow training for workingclass jobs takes prioirity over the development of free thinking, couragious and creative individuals. Others believe we must do what we can given the current political situation, and even when it is less than the ideal, it is still better than nothing.

Emil Molt's primary goal for founding the first Waldorf school in 1919 was to educate students to be free and independent thinkers who would not be easily manipulated into marching unthinkingly into battle as so much of Europe did in WW I. This is a challenging goal in the US, where state and corporate propaganda intentionally confuses nationalism, patriotism and individual identity, i.e., “Be all you can be. Join the Army.”

Mathematics in Waldorf-inspired schools

Quality mathematics education is central to a Waldorf program as it builds directly on the insights and discoveries of the Pythagoreans, Plato and others whose ideas are central to the Waldorf curriculum, and who used mathematics to explore questions of reality, identity, place, meaning and purpose. Current mainstream teaching intentionally isolates mathematics from the universal search for meaning. Although there are some advantages to this differentiation, a primary purpose of Waldorf schools is to act as an antidote to the extremes of this reductionist mindset.

Mainstream (state-sponsored) schools tend to focus on training, with the stated goal of providing society with appropriately skilled employees and appropriately enthusiastic citizens. Waldorf schools focus on education, in which the goal is to help students discover and strengthen their own inner capacities. Despite such differing goals, Waldorf graduates should measure well on mainstream standardized math tests. This is because the Waldorf goal of understanding deep mathematical principles makes the mainstream goal of applying specific facts and skills easier, not harder.

Although Waldorf and mainstream graduates will not always have equal amounts of practice in specific skills, Waldorf graduates should have developed richer schemas for understanding how a particular skill or method relates to mathematics as a whole. This enables continued and rapid understanding of new ideas and skills.

Due to this emphasis on schema development, Waldorf graduates should be better able to see underlying patterns and principles — the lawfulness of mathematics. This more complex level of thinking is a prerequisite for success in higher levels of math, such as calculus and beyond.

Emil Molt and Rudolf Steiner recognized that their school could not be completely divorced from outside society, and so they made many compromises with government, religious, and economic authorities. In particular, they recognized that their working class students would need employable skills immediately upon graduation from eighth grade. In their time, these skills included woodworking, sewing, machine making, and stenography. We should also be teaching the employable skills of our time. In our time these skills include a strong understanding of mathematics. Therefore Waldorf schools should be leaders in the teaching of mathematics.

This is not an abandonment of Waldorf ideals; it is an appropriate adaptation to current economic and social realities. Some areas of future opportunity are likely to include ecology, biology, medicine, systems analysis, economics, politics, computer sciences, and of course the arts. Waldorf schools should be consciously preparing the next generation of thinkers and leaders for these subjects. Given the ever increasing relevance of mathematics in all fields, graduates with strong mathematical ability will have clear employment advantages. This is very likely to be true whether or not we can predict the exact nature of future jobs.

Despite the above, we should not confuse training with education as so often happens in mainstream schools. If all we do is train students for employment, we will have missed the primary reason for Waldorf schools. Given the central importance of mathematics for exploring questions of reality, meaning, justice and identity, mathematics should remain a core part of a Waldorf curriculum whether or not it furthers future employment prospects.

"How can it be that mathematics, being after all a product of human thought
which is independent of experience, is so admirably appropriate to the objects of reality?
Is human reason, then, without experience, merely by taking thought,
able to fathom the properties of real things?"

-- Albert Einstein

Elements of a quality Waldorf-inspired mathematics program

• Mathematics should be taught as an exploration, i.e., from mystery and questions to theories and conclusions.
• Mathematics should be taught from the whole to the parts, from large patterns to specific facts and formulas.
• Mathematics should be taught phenomenologically, i.e., by observing outer reality (nature observations) and inner reality (insight into universal patterns).
• Mathematics should be integrated into all subjects.
• Starting at about sixth grade and no later than seventh grade, subject math classes should be taught by specialists. This frees the main lesson teacher to focus on other critical main lesson topics and allows students to share in the enthusiasm and expertise that someone with time to focus on the subject can demonstrate.
• Mathematics should be taught historically, i.e., linking why a particular concept was developed at a particular time and place. Questions to consider might include: 
  •  - Who were the important innovators/inventors/discoverers of new mathematical ideas?
  •  - What new questions and challenges did they face for which earlier math was inadequate?
  •  - What were the typical responses of various segments of society to the new mathematics and the ideas it modeled?
  •  - Was there resistance to the new mathematical ideas, i.e., perhaps from religion, scientific, or political perspectives? If so, why and how was this conflict eventually resolved?
• Every class should include daily mental math practice for 10 to 15 minutes. This should promote the discovery of new mathematical patterns, and not simply reenforce the memorization of previously learned skills.
• Algebra, geometry and trigonometry should be fully linked by the end of eighth grade.
• The facts and skills measured on mainstream tests, such as CA STAR, should be mastered by eighth grade, but this should not become the primary goal of education.
• Starting with symbolic reasoning in Algebra I, mathematics classes may need to be split into developmentally appropriate groups. For example, in seventh grade there could be one group (Concrete Operational) focusing on Geometry and Physics, while another group focuses more on Algebra and symbolic reasoning. With careful coordination, these groups can work together at times, each contributing their unique perspective to the whole.
• The need for upper grades math remediation can be reduced if more quality education occurs in earlier grades. It is too late to re-teach basic arithmetic skills in an Algebra class. It embarrasses the unprepared, bores the prepared, and causes severe delays and time restrictions for the class as a whole.
• Mathematics classes can---and usually should--include textbooks, but to enable a deep exploration of each topic, the course should be taught thematically, such as through Waldorf blocks.
• Avoid main lesson books that become busy work.
• Include projects that call for deep exploration. Such projects might include research papers in which students work out mathematical principles and methods, and practical projects that make use of these principles. Project examples include, navigation and map making, flight and kite design, cooking and convection, the physics of simple machines, game theory, electronic relays and simple circuit boards, ecological restoration and statistics, raising fish for local streams, weather and climate tracking, color and the EMF spectrum, illusion and perspective drawing, music and ratios, temperature and weather, color, rainbows and refraction, and much more. The exact choice of
• Multiple forms of assessment should be used, including frequent informal assessments such as daily mental math, and formal summative assessments at the beginning and end of each block, and at  the beginning and end of each school year.
• Grading is often not helpful below seventh grade, but becomes essential around eighth grade. Numerical grades should be balanced with thoughtful, qualitative assessments that focus on aspects of student development that are difficult to measure with a grading system.