Pluto and the Mini-Planets.

“We recommend Edric Cane’s book, Teaching to Intuition, as an excellent source for how mathematics should be learned.”  NASA mathematician Fred Krogh (Principal Mathematician, Ret., Jet Propulsion Laboratory) was kind enough to recommend my book on the home page of the official site that made available to Universities and Industry the math and algorithms developed at JPL over 30 years of planetary exploration. I was taken by surprise and greatly appreciative of seeing a book that uses only examples from Elementary and Middle School math in such proximity with sophisticated,  pioneering mathematics. Krogh added: “This (Teaching to Intuition) is mainly for elementary mathematics, but the principles hold at all levels.”

Now I learn that one of the junior siblings of Pluto, one of those mini-planets or main-belt asteroids that, along with Pluto, do not qualify as official planets, has been named after Fred Krogh. The NASA page that gives the known characteristics of 5927 Krogh says of its eponym: “American mathematician Fred T. Krogh developed the accurate, flexible and fast numerical integration algorithm used to track and navigate NASA interplanetary spacecraft since the 1970s. His work is at the core of JPL planetary, asteroid and comet orbit solutions and ephemerides.”  

Most of us earthlings have little connection with Mercury, Mars, Jupiter, Saturn, Uranus or Neptune.  As for Venus, we can only wish. But Krogh was kind enough to look down on a book I wrote. Let me congratulate Fred Krogh on 5927 Krogh.

Intuition and critical thinking

I got an e-mail from a friend that said: “I can’t find your book on Google!” Well, it takes a few days. But as I checked for myself, I came across an article: Making Critical Thinking Intuitive.  It’s by the Foundation for Critical Thinking about which I know nothing. It makes many points that echo what I try to do in Teaching to Intuition, and speaks of “teaching for intuitive understanding.” It makes no reference to mathematics but expands the general approach to knowledge and education in general and to critical thinking in particular.  Let me give a few quotations from the article that can be found at (Articles Tab, K-12 instruction strategies.)

  • Some people erroneously believe that critical thinking and intuitive thinking are incompatible opposites. If one means by intuitive thinking a form of inexplicable, non-rational thought, the claim is correct. But if one means by intuition the process by which one translates the abstract into the concrete…then, not only are critical thinking and intuitive thinking not incompatible, they are necessarily conjoined. Solid critical thinking always requires fundamental insights, fundamental intuitions, to guide it.
  • Whatever we are teaching, we should therefore continually ask ourselves, “What are the intuitions and insights essential to this mode of knowledge and thought?” and “How can I most effectively foster them with these students?”
  • Helping students to develop critical thinking intuitions is, then, helping them gain the practical insights necessary for a quick and ready application of concepts to cases in a vast array of circumstances. We want critical thinking principles to be “intuitive” to our students in the sense that we want those principles ready and available in their minds for immediate translation into their everyday thoughts and experience.
  • A powerful way to make the abstract more intuitive, is, therefore, to use stories and dramatized characters for that purpose.
  • This, then, is probably the fundamental reason why so much school learning is not effectively transferred to real life. It lacks the intuitive basis, the insights, for the translation.

I couldn’t have said it better myself, and I am grateful for the extra dimension it gives to my own thinking.