The thinking is what matters approach is invalidated if the students don’t get to the correct answer.
How many students reinvent the Peano Axioms? Or see the necessity of the Dedekind recursion theorem? Or invent the Grassmann induction proof that addition of whole numbers is associative and commutative?
How many of these classes get to correct versions? How many leave out details?
In a subject like math or even more so in physics, there is confusion and also intentional false statements as pitfalls.
Working with the correct versions is hard enough.
What benefit is it to stop at a wrong understanding?
Devlin doesn’t understand that addition and multiplication of natural numbers are just a way to keep one’s place in the count.
In the process of spelling it out, the author, teacher, tutor, etc. learns something and often doesn’t know the correct version unless they spell out the proof or method in every detail.
Computer programmers, users, etc. are always complaining the specs, documentation, etc. has left things out. Do we want specifications for VCRs, computers, etc. that leave things out?
If you look at videos for home repair, they spell it out. Unplug the refrigerator before replacing the automatic ice maker. Do we really want this left out? Which wires to connect? Leave it out?
Should anything involving safety leave out steps and entire concepts? The answer is no. This tells us people can’t invent those steps and safety precautions in all cases.
How many people continue with math courses taught this way? How many people search out Schaum’s outlines to see detailed step by step examples?
How many people learn something in math or applied math from an article that spells out in detail every step and obvious concept and connection?
The Dedekind book on natural numbers spells out each little lemma and its proof. How many have rediscovered it?
Math needs to be spelled out more not less.
Furthermore, important insights into math have been lost in current textbooks. Look at the book Lennes Veblen
This book is 100 years old and contains an approach to calculus reflecting the Lagrange and Cauchy way of doing calculus in some points. Most of this is completely lost to teachers of calculus today in high school and college.
In a series called Vanishing Calculus, I will demonstrate a superior way of teaching calculus incorporating these prior approaches. Base Point Test Point Calculus is a series already available in part that emphasizes the logic of limits closer to how they are taught currently.
Base Point Test Point Calculus spells out in complete words the logic of limits in calculus. Most students don’t understand this at all. This disproves the self discovery approach.
Base Point Test Point Calculus at e-vendor.
The don’t explain the details approach lets the textbook writer and thus the teacher off the hook to explain. In practice, such gaps are never made up. Tests are not more detailed and nit picky than are the text materials in practice. I would like to see a case where self discovery was used without a text that spells things out and the test asked for detailed distinctions of the students and marked off for not doing them correctly.
How could a test have the words and terminology to make distinctions that are common to the students taking the test if those words were not taught by the text or class materials?
If each student invents the restrictions and limitations of a method or theorem with their own words, how does the test have common words to use to ask them for those distinctions or mark them off for the wrong words to explain the distinctions?
How about a function is a set of ordered pairs in which no two ordered pairs have a common first element and different second element? Is this self discovered? Including the words?
Or the epsilon delta definition? Including the requirement of a test point in the domain of the function in each interval around the base point? This question illustrates the value of having terms for the parts of the limit definition. Why didn’t these teachers invent these words already? A few have, but the combination of base point and test point in the limit definition is new as far as I know.