This is the other book I mentioned finding at the Book Barn. It is a collection of eight lectures delivered by the author in 1912 and 1913, and a fascinating historical document. Some features that I found particularly interesting follow.
The first essay sets out the theoretical foundations of mathematics (and therefore of mathematics education), and boy is it a trip. Mathematics is divided into four areas: arithmetic, geometry, analysis, and mechanics. There is a long discussion of the importance of understanding the distinction between postulates and axioms, and another about the distinction between deductions and proofs. We get a list of The Three Functions of Mathematics: apparently, starting with a collection of “postulates, the truth of which is no concern of mathematics”, they are to perform deductions via the use of logical axioms, “to ascertain whether the set of postulates is consistent”, and finally to demonstrate that the chosen postulates are irredundant. (This description sounds comical to a modern mathematician, but Gödel’s work was two decades in the future when Carson was writing.)
The absurd high-level description is followed by quite reasonable practical discussions: for example, Carson argues that using technical jargon can be challenging for students and that demanding extreme rigor while proving obvious things is alienating, and he suggests that instructors should instead use rigorous methods in the exploration of new ideas. The second essay contains a good discussion of the value of relatability in word problems. The fifth essay includes makes a good case for the importance of introducing fractions via concrete examples before abstracting, and for developing their properties conceptually before developing mechanical rules for manipulating them.
Some other things that caught my eye: the fifth essay recommends teaching (e.g.) base 8 to help students appreciate the significance of base 10 (fifty years before Lehrer), and he advocates introducing students to elementary combinatorics (though he does not call it by that name) as part of arithmetic. There’s a discussion of whether students should learn to use ready reckoners (he’s for it: “Those who deprecate this suggestion might as well deprecate the use of sewing-machines, and their introduction into girls’ schools”) and of infinite series (he takes the view that they are not useful, and thinks people who want to keep them in the curriculum should not pretend otherwise). The discussion of arithmetic wraps up with the suggestion that “the meaning of the value of money and its variations in time and place as matters which should be considered by teachers of arithmetic; the fallacy of the thirty-shilling wage would find no wide acceptance if education were all that it should be” and I would love it if someone could explain to me what exactly this is a reference to.
Anyhow, in summary: as a curiosity, certainly worth the $4 I paid for it.