In the thread on Jonathan Bartlett and priorities for Math education, I raised two comments that I think it would be profitable to further reflect on. First, from 33 on how the US National Academy of Sciences tried to classify Mathematics as a “science”:
The Nature of Mathematics
(These paragraphs are reprinted with permission from Everybody Counts: A Report to the Nation on the Future of Mathematics Education. ©1989 by the National Academy of Sciences. Courtesy of the National Academy Press, Washington, D.C.)
Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
The special role of mathematics in education is a consequence of its universal applicability [–> hint, hint!]. The results of mathematics–theorems and theories–are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty.
In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power–a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world in which we live. [–> bursting with implications]
Notice, the implicit scientism, which tries to shoe-horn Mathematics into the sciences even though it is patently different? That’s a reflection of the notion that the sciences effectively monopolise credible approaches to and the content of knowledge.
Mathematics is the case that blows up that notion, and as a bonus, the sciences cannot get along without it. If you needed smoking gun evidence that something is very wrong with evolutionary materialistic scientism [= naturalism] this is it. That’s why I commented: “[t]ake particular note on how the US NAS speaks of Mathematics as a “Science” even as it is forced to admit that it does not work in the way that Natural Sciences work, and as it is forced to admit the pivotal role of logic in Mathematics. “
Clip No. 2 in 86, is about radical constructivism (not the Richard Skemp type):
Thompson Patrick W. (2013).: Constructivism in Mathematics Education. In Lerman, S. (Ed.)
Encyclopedia of Mathematics Education: Springer
[. . . ]
There are two principle schools of thought within constructivism: radical constructivism (some people say individual or psychological), and social constructivism. Within each there is also a range of positions. While radical and social constructivism will be discussed in a later section, it should be noted that both schools are grounded in a strong
Skeptical stance regarding reality and truth: Knowledge cannot be thought of as a copy of an external reality, and claims of truth cannot be grounded in claims about reality.
The justification of this stance toward knowledge, truth, and reality, first voiced by the Skeptics of ancient Greece, is that to verify that one’s knowledge is correct, or that what one knows is true, one would need access to reality by means other than one’s knowledge
of it . . . .
Constructivism did not begin within mathematics education. Its allure to mathematics educators is rooted in their long evolving rejection of Thorndike’s associationism (Thorndike, 1922; Thorndike, Cobb, Orleans, Symonds, Wald, & Woodyard, 1923) and Skinner’s behaviorism (Skinner, 1972). Thorndike’s stance was that learning happens by forming associations between stimuli and appropriate responses. To design instruction from Thorndike’s perspective meant to arrange proper stimuli in a proper order and have students respond appropriately to those stimuli repeatedly. The behaviorist stance that mathematics educators found most objectionable evolved from Skinner’s claim that all
human behavior is due to environmental forces. From a behaviorist perspective, to say that children participate in their own learning, aside from being the recipient of instructional actions, is nonsense . . . .
The gradual release of mathematics education from the clutches of behaviorism, and infusions of insights from Polya’s writings on problem solving (Polya, 1945, 1954, 1962), opened mathematics education to new ways of thinking about student learning and the importance of student thinking
Notice, the self-referential incoherence implicit in the skepticism advocated? That’s why I commented:
>>Fail spectacular, self-referential fail.
Notice, how the hyperskeptical stance implies what it denies, knowledge of the truth about reality?
Similarly, take Josiah Royce’s proposition E, error exists. This is undeniably, self evidently true not just a figment of the individual or socially constructed model of “reality.” Try to deny it, ~E. Immediately, this means — and ability to understand is key to self-evidence — E is an . . . ah, um, err, . . . ERROR. So, the attempt to deny E itself implies E and refutes itself. E is and must undeniably be so on pain of instant, patent absurdity. (Never mind that over the years we have seen cases of those willing to cling to absurdity on this.)
So, truth exists and in some cases it is warranted to utter certainty. Thus too, knowledge up to and including undeniably certain knowledge, exists. Any scheme of thought or praxis that tends to undermine such, fails and is unworthy of acceptance. It is error, and our plain duty is to move beyond error to truth and knowledge wherever we can.
Now, too, F H Bradley long since decisively addressed the underlying kantian claim of an unbridgeable ugly gulch between our world of appearances and the realities of things in themselves:
We may agree, perhaps, to understand by metaphysics an attempt to know reality as against mere appearance, or the study of first principles or ultimate truths, or again the effort to comprehend the universe, not simply piecemeal or by fragments, but somehow as a whole [–> i.e. the focus of Metaphysics is critical studies of worldviews] . . . .
The man who is ready to prove that metaphysical knowledge is wholly impossible . . . himself has, perhaps unknowingly, entered the arena . . . To say the reality is such that our knowledge cannot reach it, is a claim to know reality ; to urge that our knowledge is of a kind which must fail to transcend appearance, itself implies that transcendence. For, if we had no idea of a beyond, we should assuredly not know how to talk about failure or success. And the test, by which we distinguish them, must obviously be some acquaintance with the nature of the goal. Nay, the would-be sceptic, who presses on us the contradictions of our thoughts, himself asserts dogmatically. For these contradictions might be ultimate and absolute truth, if the nature of the reality were not known to be otherwise . . . [such] objections . . . are themselves, however unwillingly, metaphysical views, and . . . a little acquaintance with the subject commonly serves to dispel [them]. [Appearance and Reality, 2nd Edn, 1897 (1916 printing), pp. 1 – 2; INTRODUCTION. At Web Archive.]
Going further, knowledge is shot through with ethical import, as “warrant” implies. That is, our intellectual lives are morally governed through inescapable duties of care to truth, right reason, prudence, justice etc. This then applies with particular force to Mathematics and to education in Mathematics, as this is the field that studies, builds the habits and skills of and applies right reason to things that may be quantified and/or symbolised and analysed or modelled structurally. So, it is particularly important to get Math and Math Education ethically right.>>
That is our challenge, and as ever, it has an inescapable ethical component. One that is compounded by the need and duty to soundly educate our children. END