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# JB: what is important in math . . . ?

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Our contributor JB asked to have a conversation on what is important in Mathematics, especially Math education.

I shared some thoughts.

>>Why not, let’s just do that, follow it up and see where it goes?

For instance, I think a key insight is to find a useful, powerful definition of what mathematics is. If we understand what it is we are exploring, it may give us a deeper, richer view on how we may understand and apply it. And for this, I have come to the view that an adaptation of a view I was taught by my very first uni prof is key:

Math is [the study of] the logic of structure and quantity.

That, is, there are two aspects, first, the substance of a certain field of reality: it forms a coherent framework of largely abstract structures and quantities. Coherence, being the gateway to the logic that guides our reasoning, and turns on the premise that realities are so together, thus accurate descriptions of said realities — truths — must equally fit with one another as they must also fit with realities. This means, rational thought is a main tool (and increasing our power of rational thought is a key motive and end) of Mathematics.

Further to this, I see a key application of the logic of being.For, key Mathematical entities, while abstract, are necessary, framework components of any possible world. Which immediately gives them enormous power and depth, as well as being a source of the aesthetic pleasure excited by well done Mathematics — its beauty. Order, intricacy, organising principles reflecting verisimilitude. So, Mathematics can be enriching, enjoyable and en-noble-ing. All of which are highly relevant to education and praxis. Also, the involvement of the appearance and substance of truth (with logical accountability and duties of prudence) brings out an ethical dimension, the other side of axiology.

Mathematics is a value-rich environment.

For example, ponder the compact, powerfully integrative insights locked in Euler’s expression:

0 = 1 + e^i * π

Going beyond, I find that a survey of key structures such as von Neumann’s exploration of the natural counting numbers, N, will help flesh this out, also teaching us the style of creative, insightful exploration that draws out the insightful creativity you are seeking to promote:

{} –> 0
{0} –> 1
{0,1} –> 2
. . . [HUGE!]
{0,1,2, . . . } –> ω

From this we may rapidly access the “mirror-image” additive inverses, thus the Integers Z. Ratios bring us to the rationals, Q. Infinite continued convergent sums of rationals give us the reals, R. Complex numbers C come in as rotating vectors (which then extend to basis vectors, the ijk system, general vectors, quaternions, matrices, tensors thus also groups, rings, fields and algebras). The transfinite ordinals, transfinite hyperreals

and the catapult through 1/x gets us to infinitesimals, here 1/K.

The Surreals come knocking at the door.

Valid infinitesimals give us an insight into Calculus.

With this in hand as a structured survey, all sorts of gateways for exploration are open, including a sound appreciation of sets, mathematical foundations, topology and more. Worthwhile in itself but also obviously relevant to the Calculus you wish to explore. Also, pointing to the world of computing.

We then gain an insight on axiomatisation and how it is subtly shaped by exploration and discovery of key mathematical facts (especially, necessary entities present in the framework of any world). So, we see how axioms may need to be plausible and if well phrased allow us to spin out abstract logic-model worlds that may speak to this and other possible worlds. Where, computing allows us to use machines in that exploration. More broadly, modelling is seen as a powerful but potentially misleading approach. Thus, issues of validation and testing lurk.

We could go on, but I think we see a vision.

While I am at it, Mr Shallit’s sneer falls to the ground, once we see the reality of necessary entities in world frameworks, the relevance of truth, beauty, prudence and more as well as the power of mind to have insight, to intuit, to perceive and to draw insights that transcend the capabilities of inherently blind, dynamic-stochastic, GIGO-limited computational substrates. Reppert, again, draws out the point decisively:

. . . let us suppose that brain state A [–> notice, state of a wetware, electrochemically operated computational substrate], which is token identical to the thought that all men are mortal, and brain state B, which is token identical to the thought that Socrates is a man, together cause the belief [–> concious, perceptual state or disposition] that Socrates is mortal. It isn’t enough for rational inference that these events be those beliefs, it is also necessary that the causal transaction be in virtue of the content of those thoughts . . . [But] if naturalism is true, then the propositional content is irrelevant to the causal transaction that produces the conclusion, and [so]

we do not have a case of rational inference. In rational inference, as Lewis puts it, one thought causes another thought not by being, but by being seen to be, the ground for it. But causal transactions in the brain occur in virtue of the brain’s being in a particular type of state that is relevant to physical causal transactions.

Mathematics is an exercise of the human spirit, which points to that Spirit who is the greatest Mathematician of all. Manifest, in the Mathematical frameworks of our world. A point long since articulated by founders of modern science who saw themselves as seeking to think God’s creative and world-sustaining thoughts after him. >>

Thoughts? END

F/N: I further follow up [replacing a lost as logged out comment . . . WP that's a bug not a feature!], using the role of more radical streams of thought on constructivism in Mathematics education:
http://www.pat-thompson.net/PDFversions/2013ConstMathEd.pdf 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.
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. KF kairosfocus
F/N: To get a picture of some concerns, ask what is missing here:
https://ptgmedia.pearsoncmg.com/imprint_downloads/merrill_professional/Van_de_Walle_9780132824828.pdf Teaching Mathematics for Understanding Teachers generally agree that teaching for understanding is a good thing. But this statement begs the question: What is understanding? Understanding is being able to think and act flexibly with a topic or concept. It goes beyond knowing; it is more than a collection of in­ formation, facts, or data. It is more than being able to follow steps in a procedure. One hallmark of mathematical understanding is a student’s ability to justify why a given mathematical claim or answer is true or why a mathematical rule makes sense (Council of Chief State School Officers, 2010) . . . . Adding It Up (National Research Council, 2001), an influential research review on how children learn mathematics, identifies the following five strands of mathematical proficiency as indicators that someone understands (and can do) mathematics. • Conceptual understanding: Comprehension of mathematical concepts, operations, and relations • Procedural fluency : Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately • Strategic competence: Ability to formulate, represent, and solve mathematical problems • Adaptive reasoning: Capacity for logical thought, reflection, explanation, and justification • Productive disposition: Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own effi­ cacy ( Reprinted with permission from p. 116 of Add- ing It Up: Helping Children Learn Mathematics, 2001, by the National Academy of Sciences, Courtesy of the National Academies Press, Washington, D.C.) This report maintains that the strands of mathematical proficiency are interwoven and interdependent—that is, the development of one strand aids the development of others
Notice, the lack of the overall picture? Notice, failure to deal with the logic of structure and quantity thus creation of logic model worlds that frame a domain of thought (leading to theorems etc), or the creation of models that then apply to RW settings through in-common elements? And of course the missing computer integration. I think a good case in point is the structuring of numbers on a line centred on zero, extended to hyperreals and involving both transfinites and infinitesimals. An obvious context is Calculus, but it goes beyond to things like what continuum means and the linked mileposting using counting numbers and ordinals etc. The idea of space and dimensions can then be brought into play. From this ideas of relationships, operations, variables, functions etc can be developed. Insightful understanding has to do with grasping integrated logically coherent frames of thought. Including, how one reality logically entails another, thence proof sequences, derivations, etc. In short, what is the STUFF of Mathematics. And of course, seeing the infinity next door to familiar counting etc in a structured way that identifies actual ranges, named numbers and values, then catapults using 1/x to give infinitesimals is a starting point. KF kairosfocus
F/N, FYI/FTR: As a reminder (given the insistent pretence that design theorists do not publish research) I note for the thread :
KF kairosfocus
Brother Brian:
If I were going to establish a centre for educational reform, I sure as hell would install senior management with education and/or extensive experience in the education field if I wanted it to be taken seriously. They are not hard to find. What credibility do these two have with respect to education?
They aren't closed-mined liberals. That is a great start. An intellectual would attack their record and what they say, if it could be so attacked. A lowlife punk attacks the person without even regarding anything else. Imagine never hiring anyone with a fresh take on what is going on and the leadership skills to pull it off. ET
F/N: While it is a bit off to the side from this thread's focus, let's draw a little from the DI Ed centre and see how it speaks to education transformation. Let me pick a point of interest, addressing the governance challenge at the heart of what has been going wrong with education in the US:
https://www.discovery.org/education/2019/07/23/transform-education-from-the-ground-up/ At the center of the problem with the current approach to public schools is that a centralized government has great difficulty fixing localized problems. As Jay W. Richards, Senior Fellow at Discovery Institute, points out in his book, Money, Greed, And God, “The problem isn’t that government workers are stupid or uncaring. The problem is all about information and incentives. A centralized government knows less about individual problems than does practically anyone closer to the problem.” He continues, “replacing a family or a neighborhood or a local church with a federal program for helping the down-and-out is like trying to have an official in the Department of Commerce guess how much I should pay, right now, for a new pair of size-9 Asics running shoes.” A more bureaucratic centralized institution faces a guessing game—namely the need to predict what is best for children in a particular local community. The local philanthropist is in a much better position to understand his particular community and the interests of those within the community. To this point Richards adds that “all of these organizations have one thing in common: they define their mission in part by what the government does or doesn’t provide. If the government weren’t occupying most of the charitable ecosystem, charities would be profoundly different.”
KF
BB, you were specifically asked to address substance. You have tried to distract attention and attack the man. Again. That speaks — again also — to lack of seriousness. Duly noted. KF
If I were going to establish a centre for educational reform, I sure as hell would install senior management with education and/or extensive experience in the education field if I wanted it to be taken seriously. They are not hard to find. What credibility do these two have with respect to education? Brother Brian
kairosfocus- Brother Brian is a waste of bandwidth. If, by now, you don't realize that he is not interested in an honest and open discussion, I don't know what it would take to do so. ET
Brother Brian:
But it was really difficult getting past the SENIOR FELLOW AND PROGRAM DIRECTOR, AMERICAN CENTER FOR TRANSFORMING EDUCATION, who’s expertise involves selling lamps.
That is a lowlife thing to say about someone who He currently serves as Chairman of Lumenal Lighting, LLC., a business he purchased in 2004. Lumenal Lighting is in the commercial lighting business and serves companies, schools, government facilities and non-profit organizations in the Pacific Northwest. And a program coordinator with experience coordinating is a bad thing... :roll: You're kind of a punk, you know that? ET
BB, you were specifically asked to address substance. You have tried to distract attention and attack the man. Again. That speaks -- again also -- to lack of seriousness. Duly noted. KF kairosfocus
Brother Brian:
Is it as productive and effective as BioComplexity, the Discovery Institute’s peer reviewed ID journal?
It is more productive than blind watchmaker research. ET
KF
BB, why don’t you go look and see what their analyses and suggestions are, then actually do the diligence of responding on the merits.
I read them. But it was really difficult getting past the SENIOR FELLOW AND PROGRAM DIRECTOR, AMERICAN CENTER FOR TRANSFORMING EDUCATION, who’s expertise involves selling lamps. Or the PROGRAM COORDINATOR, AMERICAN CENTER FOR TRANSFORMING EDUCATION, who’s “experience includes managing campaigns for elected officials as well administrative roles with elected officials at the local, state, and federal levels of government. He holds a B.A. in Politics and Government from Pacific Lutheran University.” What seems to be missing in the DI’s centre for education transformation is any background in education. Brother Brian
BB, why don't you go look and see what their analyses and suggestions are, then actually do the diligence of responding on the merits. Further to that, I suggest that DI's efforts regarding the design inference have objectively succeeded; the ideological, locked in intensity of objection to the obvious and well warranted point that many things show strong and reliable, tested and found consistently true signs of being designed ironically speaks for itself. But, that is tangential for this thread. The point for here is, serious independent thinking on the need for educational transformation is going on. Such will ultimately prevail, but I fear a terrible, needless price will be paid by our civilisation first. KF kairosfocus
KF
F/N: It seems Discovery Institute has a centre for education transformation:
Is it as productive and effective as BioComplexity, the Discovery Institute’s peer reviewed ID journal? Brother Brian
F/N: It seems Discovery Institute has a centre for education transformation: https://www.discovery.org/education/ KF kairosfocus
Mathematica on the Raspberry Pi https://projects.raspberrypi.org/en/projects/getting-started-with-mathematica More, example based: https://www.raspberrypi.org/magpi/wolfram-mathematica-raspberry-pi/ note issue 67 https://www.raspberrypi.org/magpi/issues/67/ kairosfocus
As an example of plans to change how math is taught see here: https://www.thecollegefix.com/trending-educators-work-to-combat-racism-whiteness-in-math/ LoneCycler
Raspberry Pi tablet https://makezine.com/2014/01/07/how-i-built-a-raspberry-pi-tablet/ kairosfocus
No specifics about the Raspberry Pi. Just selective quotes. kf isn't able to answer. No sense in my continuing to ask Time to sink back into retirement. hazel
Hazel, scientism. Notice from 37:
T Williamson in NYT blogs, 2011: https://opinionator.blogs.nytimes.com/2011/09/04/what-is-naturalism/ . . . One challenge to naturalism is to find a place for mathematics. Natural sciences rely on it, but should we count it a science in its own right? If we do, then the description of scientific method just given is wrong, for it does not fit the science of mathematics, which proves its results by pure reasoning, rather than the hypothetico-deductive method. Although a few naturalists, such as W.V. Quine, argued that the real evidence in favor of mathematics comes from its applications in the natural sciences, so indirectly from observation and experiment, that view does not fit the way the subject actually develops. When mathematicians assess a proposed new axiom, they look at its consequences within mathematics, not outside. On the other hand, if we do not count pure mathematics a science, we thereby exclude mathematical proof by itself from the scientific method, and so discredit naturalism. For naturalism privileges the scientific method over all others, and mathematics is one of the most spectacular success stories in the history of human knowledge. Which other disciplines count as science? Logic? Linguistics? History? Literary theory? How should we decide? The dilemma for naturalists is this. If they are too inclusive in what they count as science, naturalism loses its bite. Naturalists typically criticize some traditional forms of philosophy as insufficiently scientific, because they ignore experimental tests. How can they maintain such objections unless they restrict scientific method to hypothetico-deductivism? But if they are too exclusive in what they count as science, naturalism loses its credibility, by imposing a method appropriate to natural science on areas where it is inappropriate. Unfortunately, rather than clarify the issue, many naturalists oscillate. When on the attack, they assume an exclusive understanding of science as hypothetico-deductive. When under attack themselves, they fall back on a more inclusive understanding of science that drastically waters down naturalism. Such maneuvering makes naturalism an obscure article of faith. I don’t call myself a naturalist because I don’t want to be implicated in equivocal dogma. Dismissing an idea as “inconsistent with naturalism” is little better than dismissing it as “inconsistent with Christianity.”
Why, so hot to push it under the Science-Naturalism umbrella? Similarly, I remind from US NAS in 33:
See it again, here from US NAS? Those are only a few pointers to the problem. KF kairosfocus
I agree with you 100% about widespread computing education and the value of inexpensive technology such as the Raspberry Pi. We are in agreement there. How does this have anything to do naturalism or scientism? I can't see how naturalists and others such as theists would have any disagreement about this on philosophical grounds. Yes, you have been specific about the Raspberry Pi. No, you haven't been at all specific about how this has anything to do with naturalism. hazel
KF
BB, you have clearly shown persistent lack of seriousness. KF
No. I have asked a very simple question. All you have to do is answer yes or no. Should atheist be allowed to teach our children? Brother Brian
Hazel, the specifics are right there, down to recommended technologies and a specific rated Mathematics application on a powerful cost effective platform. Also, I point out that this is an outline curriculum framework which also engages the ethical import of warrant as key to knowledge. Along the way, scientism is also dealt with. KF kairosfocus
BB, you have clearly shown persistent lack of seriousness. KF kairosfocus
kf, how does quoting yourself add anything to the conversation? And where are some specifics about teaching math? For instance, why can't you address just this question: "How does introducing computing early and using inexpensive technology such as the Raspberry Pi have anything to do with naturalism?" hazel
KF
August 27, 2019 at 5:50 pm BB, it is obvious that you have not addressed the substantial approach but have resorted to red herrings, straw men and ad homs.
Am I allowed to say that you are full of fecal matter? Any time you are presented with real arguments you throw out the red herring, strawman, ad hom accusations. If you can’t address the actual issues, just say so. Don’t dismiss them with your prepackaged dismissive nonsense. Just answer my questions. Would you prefer a system where atheists are not allowed to be teachers? Where theists vette who can teach? We both know that this is a rhetorical question. Brother Brian
Hazel, the excerpted definitions are right there, simply read please. Scientism is a key component of naturalism; which in turn is manifestly dominant in the academy and has been for many decades. Notice, especially, how SEP tries to navigate around the challenges. KF PS: For your convenience, I excerpt 42:
The first step to solution is awareness, sensitisation. Next, we need to have a serious discussion on the issues and on our responsibilities. Then, across time — wonder why it took 20+ years to break the slave trade and altogether 50 to end slavery? [Hint: paradigms tend to shift one funeral at a time and a generation is about 30 years] — we need to build a new approach as our understanding grows. That approach will probably shift from N and R to N* and R* with results from the surreals, giving a better picture of numbers. C* will come in through the rotating vectors view. Thus, we have a better picture of core quantities, which will make infinitesimals and Calculus more natural. The logic of being will allow us to see why key structures and quantities are relevant to any possible world, with the principle of distinct identity and its corollaries drawing out the role of logic and how it disciplines our study; self-evident first truths shape any domain of study and any viable worldview will build on them — answering a weak foundations problem across our education and linked selective hyperskepticism and even trollishness. BTW, already, distinct identity of a possible world gives us 0,1,2 inviting the von Neumann construction so we see that numbers and linked structures and relationships are integral to any possible world . . . answering Wigner’s amazement. A natural synergy with computing will need to be encouraged and we have to get used to a world where all mechanical calculations are correct as a matter of course so assessment does not turn on red X’s for errors, instead facility with mathematical thinking, modelling and framing then perhaps simulating with logic model worlds or interfacing with RW systems and investigations. Right now I am impressed with the Raspberry Pi, which comes with Mathematica. This world will not subtly impose self-refuting physicalism as a dominant and domineering regime — by that time, it will be a case study on what went wrong. In that world, there will not be the sort of sneeringly dismissive unjustifiably contemptuous attitude to theism and theists that we see in Mr Shallit’s ugly remarks in reaction to Mr Bartlett’s book on a fresh approach to Calculus (using Hyperreals BTW) — immediately, an indicator of serious ethical issues . . . . I should note, when I first began to find the treasure-house of Russian textbooks, I came to learn that every High School kid did 5 years of Physics and four of Calculus based Mathematics there. With IPS in 3rd form and then O and A level Physics, that’s about what I did but as a specialist in the Sci-Tech track.
Do, tell us whether or no this outlines a curricular vision with roots in Elementary stages, going through high school years and into early College; with connexion to the ethical in the main through the challenge of warrant. kairosfocus
kf, I have known many math teachers, and I have worked on math curriculum at local, state, and even the national level. I have never heard anyone try to "shoe-horn Mathematics in under the prestigious umbrella, Science." As I mentioned above 8 above, (perhaps the only post in this whole thread that addresses the question in the OP), students first must be taught about the logical structure and content of math. They must also be taught applications of math, which can be to science, or business, or carpentry, or just running a home, or a myriad of fields. Math is not a subset of science. Math is a tool that science uses, but so do many other fields. Again, this has nothing to do with naturalism. I don't see how a naturalist and a theist would disagree on the points I've made (and I have taught side by side with both and seen absolutely nothing that would distinguish the teaching of the two.) Again (and maybe for the last time - who knows?), you can think of nothing but your very general and pervasive concerns about how the world is being ruined by naturalism, but you can't mention one concrete specific example of that in respect to real teaching. And for the life of me I don't see how introducing computing early and using inexpensive technology such as the Raspberry Pi has anything to do with naturalism, and I'll bet you can't, in any specific way, explain. hazel
PaoloV, you raise much wider issues tied to civilisational reform. That is a job for the churches as embassies of a Kingdom from Beyond. Such goes far beyond mere Mathematics education reforms, though soundly reared kids will naturally thrive in transformational education. KF PS: A "good time" might convey sexual connotations in English under stated circumstances. kairosfocus
Hazel, kindly scroll up and read the excerpts on the hidden curriculum impacts of that naturalism. A good point to start with is the straining to shoe-horn Mathematics in under the prestigious umbrella, Science. Ponder the lurking demarcation issues and the failure of such arguments. That alone will tell much about what is broken. Mathematics is a logic-disciplined study. It is not a science and that is no problem because scientism is dead, having skewered itself. KF kairosfocus
KF @48: Yes, agree with what you suggest. The learning of logic should help to engage in serious discussions about everything that is within and around us. Mathematics also helps to develop a disciplined approach to abstract and serious thinking. But it isn't sufficient. Just necessary. Much more is needed. My concern is that above all the learning at home and in school, children and young people should experience (proactive) love, first as beneficiaries of receiving it and then as beneficiaries of giving it to others. But unfortunately what seems more common is the reactive kind of love, which is the natural kind. That won't lead us to any desired destination at the end of the day. Any long and winding road won't lead us to the right door if it is not the right road. Communication between human beings is very poor and the tremendous advance of technology hasn't helped to make the communication better. There seems to be a huge disrespect of contextual meaning of words and statements. One gets the impression that nobody cares about anything important. Entertainment (killing time) seems to be a high priority. I know a person who was riding an elevator in a cruise ship and another person in the elevator asked him "are you having a good time?" to what the person I know responded "what do you mean by 'good time'?" The other person was shocked and ran out of the elevator as soon as the door opened. We're not used to seriously profound questions. We're not prepared to hear them. Specially in an elevator in a cruise ship. We use convoluted phrases to express superficial ideas. In the above anecdote maybe the questioner was simply trying to say "hi!" in a more cheerful manner? The expression OMG is heard very often all around in completely irrelevant situations. Turn on HGTV and watch a program where they show a house or an apartment and you hear OMG too often. Does that imply a poor vocabulary? Don't we know how to express feelings, surprise, etc using more appropriate terms? Perhaps Math at early age would help to setup a more serious mindset to appreciate value, to discern good from bad, to search for wisdom and to express ourselves better. But as it was said before, most important is (proactive) love. PaoloV
I keep intending to drop out of this discussion, but kf writes,
As someone who has designed innovative degree programmes for technology, I am looking hard at what is happening with Mathematics. It is pointing to needed reforms that will take us beyond the current broken naturalistic paradigm and extensions of post-/ultra- modernism. It will include pervasive computing, starting with things like the Raspberry Pi ...
I can't imagine at all how math reforms (some of which kf, JohnnyB, and I agree about) have anything to do with a "broken naturalistic paradigm and extensions of post-/ultra- modernism," nor how pervasive computing (which I agree with) has anything to do somehow addressing this "naturalistic paradigm." kf continually throws out all these generalities without being able to point to one specific of math education issue or math reform that has anything to do with naturalism. I am reminded of the old Texas saying, "All hat, no cattle." :-) hazel
BB, it is obvious that you have not addressed the substantial approach but have resorted to red herrings, straw men and ad homs. That is a telling sign that you have no answer. Let's start again, distinct identity and logic of being delivers the key sets, but we go to the hyper sets as more useful -- and in fact they were lurking behind the number lines we were taught. With Non-Standard analysis in hand infinitesimals are back, valid. The Raspberry Pi and successors delivers transformational technology. We now have to engage a world where mechanical calculations will be correct as a matter of course. Likewise, we are framing from self-evident first principles, delivering framework structures and quantities present in any world. The epistemology and required warrant have been delivered, providing worlds of mathematical fact that are antecedent to grand axiomatisation, as a learning experience. Paradigm shift, with abstracta as familiar as empiricals. KF PS: your phrasing above is odd, are you discussing curriculum design at programme level? On the technology side a key issue 20+ years ago was to see significance of mechatronics, mass customisation [to a lesser extent] and linked ICT's. kairosfocus
KF
BB, As someone who has designed innovative degree programmes for technology, I am looking hard at what is happening with Mathematics.
I have done my share of programming as well. In fact, my current position relies extensively on my programming. I still don’t see how my ideology, or yours, makes any difference in the viability of this programming. Brother Brian
KF, Agree with the correction you made in my comment @44. Thanks. PaoloV
BB, As someone who has designed innovative degree programmes for technology, I am looking hard at what is happening with Mathematics. It is pointing to needed reforms that will take us beyond the current broken naturalistic paradigm and extensions of post-/ultra- modernism. It will include pervasive computing, starting with things like the Raspberry Pi and going on from there; I envision a dockable 2 in 1 [~ 10 inch screen I think] with an array of interface ports for experimental and industrial work carrying a built in library of live resources with others living on the web. I simply outlined reasonable reforms above and you blew up, trying turnabout projections on reigns of terror. I simply point out that such, since the 1790's, have been characteristic of radical revolutions. KF kairosfocus
PV
Mathematical logic should be a required course for all undergraduate majors at the university level. Actually, a basic version of it should be a required exam at the lyceum (high school) too.
I agree. Is there anyone who really disagrees with this? Brother Brian
PaoloV, with the UK starting programming based Computing at the 5 - 7 stage, it needs to go to elementary school level. Raspberry Pi is a response to the same 2012 Furber Royal Society Report on Computing in schools, and now a viable Linux machine on a card is from US\$10 - 35 depending on needed features. This is a different world, going forward. KF kairosfocus
PV
Who is trying to impose their religion [–> or IDEOLOGY, KF]on others? Can you name them? Thanks.
KF. He would obviously prefer that we adopt his ideology voluntarily, but I’m not convinced that he wouldn’t accept the ends justifying the means. Please convince me that I am wrong. For example, would he be opposed to the government banning abortion? Or contraceptives? Or same sex marriage? Or requiring the teaching of Christianity in public schools? Brother Brian
Mathematical logic should be a required course for all undergraduate majors at the university level. Actually, a basic version of it should be a required exam at the lyceum (high school) too. PaoloV
BB, with all due respect, you have your answer, building from first self evident principles in a coherent comprehensive, well warranted powerful framework and going through a generation length reformation rather than your projection of the sad heritage of the French Revolution: reigns of terror. I did point to the shift to Hyper-integers (I simply spoke to extending N to N*) and hyperreals as replacing the current focus on R. That's what R* is about, bringing in transfinites and infinitesimals. FYI, knowledge based on warrant starting with well framed first principles is inherently about fulfilling intellectual duties and onward educational ones. Notice, where a world in which a Raspberry Pi SBC with Mathematica, Python and Java costs less {~US\$80] than a HP 50 calculator [US\$125] has to be utterly different from the old one. KF kairosfocus
Brother Brian,
I strongly oppose anyone who tries to impose their religion on others. That is evil.
Who is trying to impose their religion [--> or IDEOLOGY, KF]on others? Can you name them? Thanks. PaoloV
KF, does anyone know what the hell you are saying? I asked a simple question. What is the solution to your fabricated problem? Indoctrination? Brainwashing? Re-education camps? I am not trying to be reactionary, but I get the impression that you simply won’t be satisfied until our education system is governed by a theocracy. And, to be honest, I would fight you on this at every step. Not because I think that religion is evil, but because I strongly oppose anyone who tries to impose their religion on others. That is evil. Brother Brian
KF, all I see is you raising warnings but have proposed no solutions. What do you see as a solution to the dire straights you think we are in? Ban atheists/naturalists from being teachers? Re-education camps for atheists/naturalists? Have all teachers vetted through the church? I have had a Handful of excellent teachers during my formative years. A couple were atheist, a couple others were Christians and one was a Siek. They never hid their personal beliefs, but they also never let their beliefs bias their teaching. Brother Brian
BB, start with knowledge as requiring justification or warrant as true and go on from that to inescapable duties to truth, right reason, prudence [including warrant], sound conscience [aka integrity], justice etc. Immediately you will see that while Mathematical knowledge and education are striking cases in point, ethical considerations and moral government pervade all of our intellectual life. This is antecedent to particular worldviews though it does factor in in which views fare best on comparative difficulties. On education and the wider domain of the academy, I would suggest to you that the dominance of and lack of concern over a dominant worldview and linked cultural agenda which is demonstrably self-refuting, undermining of knowledge and of morality should be of serious concern to any responsible educator. Including Mathematics educators. The implications of hidden curriculum and the deep incoherence in the heart of the academy, here, make for a dangerously toxic environment for education. For the record, I suspect, too, that many of us who believe in God or even advocate that the design inference is well warranted per inductive principles, are also going to be tainted -- we breathe the same atmosphere, and have done so throughout our education, formal and informal. My point here, is profound, across the board need for reformation not the insinuation of party-spiritedness that is the patent subtext of your remark. KF kairosfocus
PPS: Dossey of ISU on Math and teaching:
https://storage.cet.ac.il/SharvitNew/Storage/939935/424648.pdf Perceptions of the nature and role of mathematics held by our society have a major influence on the development of school mathematics curriculum, instruction, and research. The understanding of different conceptions of mathematics is as im- portant to the development and successful implementation of programs in school mathematics as it is to the conduct and interpretation of research studies.
kairosfocus
KF
I note from above on “hidden” matters: >>the presuppositions that lurk in the above will seep out unconsciously in all sorts of ways, starting with the terms we use, the tones in our voices and linked body language, the way a lesson, curriculum or textbook seems “right” and much more. Philosophically loaded, cultural agenda driven hidden curriculum is a real issue and it goes far beyond the politically correct questions on “gender” and curriculum, etc.>> Such are full of ethical significance.
I am having a hard time trying to figure out what you see as the problem with math teaching. Everyone, including teachers, have personal beliefs, opinions and philosophies. I expect that you have no concerns with people who believe in God and ID teaching math, but you somehow have a problem with people who believe in evolution and "naturalism (whatever that is) teaching math. Surely a person who believes in naturalism can be just as capable at teaching math as a creationist. If this were a case of a strict theist teaching evolution, or an evolutionist teaching theology, you might have a point. It would be difficult for them to keep their beliefs separate from their teachings. But math (and physics, and chemistry) are far less prone to being influenced by the philosophical bias of the teacher. I simply don't see the problem here. Brother Brian
F/N: I note from above on "hidden" matters: >>the presuppositions that lurk in the above will seep out unconsciously in all sorts of ways, starting with the terms we use, the tones in our voices and linked body language, the way a lesson, curriculum or textbook seems “right” and much more. Philosophically loaded, cultural agenda driven hidden curriculum is a real issue and it goes far beyond the politically correct questions on “gender” and curriculum, etc.>> Such are full of ethical significance. KF PS: T Williamson in NYT blogs, 2011:
https://opinionator.blogs.nytimes.com/2011/09/04/what-is-naturalism/ . . . One challenge to naturalism is to find a place for mathematics. Natural sciences rely on it, but should we count it a science in its own right? If we do, then the description of scientific method just given is wrong, for it does not fit the science of mathematics, which proves its results by pure reasoning, rather than the hypothetico-deductive method. Although a few naturalists, such as W.V. Quine, argued that the real evidence in favor of mathematics comes from its applications in the natural sciences, so indirectly from observation and experiment, that view does not fit the way the subject actually develops. When mathematicians assess a proposed new axiom, they look at its consequences within mathematics, not outside. On the other hand, if we do not count pure mathematics a science, we thereby exclude mathematical proof by itself from the scientific method, and so discredit naturalism. For naturalism privileges the scientific method over all others, and mathematics is one of the most spectacular success stories in the history of human knowledge. Which other disciplines count as science? Logic? Linguistics? History? Literary theory? How should we decide? The dilemma for naturalists is this. If they are too inclusive in what they count as science, naturalism loses its bite. Naturalists typically criticize some traditional forms of philosophy as insufficiently scientific, because they ignore experimental tests. How can they maintain such objections unless they restrict scientific method to hypothetico-deductivism? But if they are too exclusive in what they count as science, naturalism loses its credibility, by imposing a method appropriate to natural science on areas where it is inappropriate. Unfortunately, rather than clarify the issue, many naturalists oscillate. When on the attack, they assume an exclusive understanding of science as hypothetico-deductive. When under attack themselves, they fall back on a more inclusive understanding of science that drastically waters down naturalism. Such maneuvering makes naturalism an obscure article of faith. I don’t call myself a naturalist because I don’t want to be implicated in equivocal dogma. Dismissing an idea as “inconsistent with naturalism” is little better than dismissing it as “inconsistent with Christianity.”
See what is being hinted at, and how loaded it is with ethical import -- including for educators who presumably traffic in knowledge? In that light look at prelim Q 0 above again. kairosfocus
Hazel, hidden curriculum is an issue that affects every teacher in principle; though relatively few are equipped or have enough institutional influence to do big things about it. KF kairosfocus
Not a thing here about the average teacher in the classroom, though. My guess is that most (virtually all?) classroom teachers, if asked to articulate there thoughts about the nature of math, are Platonists of some sort. hazel
ET, I think that we are often unconscious of latent but fundamental and sometimes unwelcome issues. KF kairosfocus
F/N: A bit of stimulation from the half-hidden web:
To be "mined" in onward discussion. Take 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. Clues to the naturalism agendas (where it is known that US NAS has long been a principal advocate of naturalism and its imposition on education). kairosfocus
PS: Later, SEP raises:
6.1 Natural Science as the Arbiter of Ontology On its first reading, ontological naturalism in the philosophy of mathematics is a straightforward consequence of methodological scientific naturalism. It states that the ontology of mathematics is the mathematical ontology of our best natural science. Scientific platonists claim, following Quine and Putnam, that this ontology is platonist [--> ironically!], as do mathematical-cum-scientific platonists (e.g., Burgess and Rosen (1997)). Resistance to scientific platonism and the associated indispensability argument has been mounted on several fronts (e.g., Field 1980, Sober 1993, Maddy 1997, ch. II.6, Paseau 2007). Consult Colyvan (2011) for a detailed discussion. 6.2 All Entities are Spatiotemporal The second reading of ontological naturalism, according to which all entities are spatiotemporal, amounts to a version of anti-platonism in the philosophy of mathematics. The position subdivides. On a reductionist view, mathematics is taken at logico-grammatical face value but its objects (numbers, functions, sets, etc.) are taken to be spatiotemporal. This view is advocated for sets in Armstrong (1991) and more generally in Bigelow (1988). Non-reductionist views are manifold. They include taking mathematics as meaningless symbol-manipulation (formalism), or as the exploration of what is true in all structures obeying the axioms (structuralism), or as the exploration of what is true in all possible structures obeying the axioms (modal-structuralism). Bueno (forthcoming) discusses various nominalisms, i.e., views which countenance only spatiotemporal entities. Since many of these nominalisms are compatible with non-naturalist as well as ontologically naturalist motivations, we do not discuss them here. We concentrate on a handful of issues relating mainly to reductionist versions of ontological realism. Reductionist ontological naturalism and non-modal structuralism about set theory face an immediate problem: there are apparently far fewer entities in spacetime than there are sets. Even on the most liberal assumptions (spacetime points and arbitrary regions thereof exist, some smallish infinity of entities may be collocated at any of these points or regions), the size of spacetime and the objects in it is a relatively low infinite cardinality (surely no more than [Beth_sub_omega]—even that is generous). Thus there are not enough spatiotemporal entities to interpret set theory literally nor to make a structural interpretation of set theory non-vacuously true, and hence to ensure that set-theoretic falsehoods come out false rather than true. See Paseau (2008) for discussion of this and other problems for set-theoretic reductionism.
Such issues point to huge hidden curriculum questions. Also, these issues point to the effectiveness of the summary naturalism ~ evolutionary materialistic scientism. kairosfocus
hazel's inability to follow along makes it difficult to have a discussion... ET
Hazel, I don't doubt that you do not perceive any impositions/hidden curriculum elements, or that most of your colleagues would be dubious of such a suggestion. However, this is part of the context of discussion in this Blog, including earlier interactions on Math topics as well as more immediately Mr Shallit's accusations against Mr Bartlett on the occasion of publishing a fresh approach Calculus textbook. That Accusation includes the dismissive remark that "Creationists" cannot do Math as they are irrational, with evidence that they doubt big-E Evolution, implying evolutionary materialistic scientism. Where, too, as a simple pedagogical issue, it is relevant that perhaps 85% or more of say the US population by such a framework, count as presumed irrational "Creationists". So, on the face of the matter, there is a question to be addressed regarding ideological imposition. Let's continue, by asking some framework questions i/l/o discussions in this Blog and elsewhere:
I believe these issues will help us clarify concerns in context. SEP on Naturalism in Mathematics, gives some further context:
Contemporary interest in naturalism stems from Quine, whose naturalism is prominent in his later works. A representative quotation is that naturalism is ‘the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described’ (Quine 1981, 21). Another major influence is Hilary Putnam, who articulates his scientific naturalism as follows: …it is silly to agree that a reason for believing that p warrants accepting p in all scientific circumstances, and then to add ‘but even so it is not good enough.’ Such a judgement could only be made if one accepted a trans-scientific method as superior to the scientific method; but this philosopher, at least, has no interest in doing that. (Putnam 1971, 356) Thus from this perspective mathematics is judged by scientific standards because everything is. Moreover, Quine and Putnam maintain that these standards sanction platonist mathematics because mathematics and its platonist construal are an indispensable part of our best scientific theories.
Thus, the dominance of naturalistic presuppositions and linked agendas is relevant. (Indeed, the presuppositions that lurk in the above will seep out unconsciously in all sorts of ways, starting with the terms we use, the tones in our voices and linked body language, the way a lesson, curriculum or textbook seems "right" and much more. Philosophically loaded, cultural agenda driven hidden curriculum is a real issue and it goes far beyond the politically correct questions on "gender" and curriculum, etc. Ponder the implications of the term transfinite as opposed to infinite and why I find the former typically more appropriate, as apparently Cantor did.) KF kairosfocus
I don't think Hawking shows up in the general HS and college math classes. hazel
I have no doubts that a dedicated materialist would find a way. Something about how Hawking was right, no doubt. ET
And can you offer an example of how an imposition of naturalism will show up in a math class, ET? hazel
I know you are sure that such pervades society, but why can't you be more specific? Where in a high school or college math class would any hint of naturalism appear? How could teaching polynomials, or logs, or trig, or the product rule, or any of the vast numbers of topic we teach, even mention, much less impose naturalism. Why can't you give an example? Or is all you can do is offer your standard general indictments? hazel
Hazel, there is a widespread imposition of what is (in the bad sense) an ideological orthodoxy. KF kairosfocus
What imposition? You give no evidence whatsoever that math teachers are "imposing naturalism." Yes, math teachers have some opportunity to discuss real-world ethical issues that students might encounter as they start to use math in conjunction with real-world situations, and I think we agree that this is a good thing to incorporate into teaching. But math teachers themselves are not "imposing naturalism", and the teaching of pure math itself does not reference any particular perspective about the nature of math. Advanced students, or as an enrichment for regular students, may have an opportunity to learn about the philosophy of math at some point, but as a survey of various perspectives, not as a claim that one perspectrive is right or wrong. But that is a different matter than just teaching math. Ona daily basis, "naturalism", pro or con, does not come up at all in any way. hazel
Hazel, you just gave some fairly relevant cases, as that imposition is currently undermining the core of knowledge and the framework for integrity. KF kairosfocus
What I meant was that math teachers aren't teaching any particular ethical philosophy, any more than we are teaching any particular religious or philosophical one. Of course we teach, explicitly and implicitly, lots of ethical and other character principles; honesty (don't cheat on your work), responsibility, punctuality, engagement in learning etc. Also, most stats teachers talk about ways in which stats can be used to mislead: missing portions of axes, confusions about what probabilities means, absolute vs relative percent changes, poor sampling methods, etc, and illustrate these with real world examples. This is non-controversial. Also, yes indeed, college students should have some exposure to ethical situations associated with their field of study, including whole courses in some cases, such as medicine, business, environmental studies, etc.. It is kf's "imposition of naturalism" idea that I strongly object to. There is nothing in the teaching of basic math that has anything to do with "imposing naturalism." hazel
BO'H: thanks for thoughts. I think the matter is that we cannot split up the problem. Yes, we believe the core of Mathematics including statistics is reasonably reliable, but that reflects the ethics of epistemology, a prudence, integrity and accountability issue. Similarly, it is advisable for educators -- this is our context -- to address the cognitive, affective and psychomotor domains (using a common framework). So, that figures don't lie but liars can figure is a relevant concern [and for much more than Statistics], as is the matter of due diligence on degree of warrant. Ethical considerations and maturation will need to be reinforced from multiple contexts, they will not thrive in an era inclined to hyperskeptical dismissal if they are neatly packaged away in a unit or special event isolated from the general technical part of studies. And that highlights how our isolation of the ethical in an era dominated by naturalism is counterproductive. KF kairosfocus
kf -
Hazel, already above we looked at ethics in statistics. A simple, obvious case: bias, cherry picking, reproducibility of results etc.
They're not in the mathematical part of statistics, though. They're issues in science, and the application of mathematical statistics. Bob O'H
It doesn’t show up at all in the general high school or college math classroom. I also don’t see what ethical issues show up.
There had better be some ethical issues, otherwise I'm going to feel a bit of a berk sending my grad students on an ethics course. Bob O'H
Hazel, already above we looked at ethics in statistics. A simple, obvious case: bias, cherry picking, reproducibility of results etc. This bleeds over into what may one fairly infer and what risks will one take -- this happens to be relevant to ID but is much broader. Try debates over opinion polling and statistical norms. Questions on warrant for conclusions. We have debated axiomatisation and facts or truths. . Issues of integrity, prudence (so, diligence in warrant and on limitations) are ethical issues. Ethics of the true and fair view. Recognising the implications of Godel and the like are worldviews issues. The general naturalistic climate is an issue and more. KF kairosfocus
It doesn't show up at all in the general high school or college math classroom. I also don't see what ethical issues show up. Unless you can give an example, I'll not bother any further response to your standard concerns about this topic. hazel
Hazel, the imposition of naturalism (often dressed in the lab coat) is a general problem in our intellectual culture; including Math, its foundations, linked issues in philosophy of Math, logic etc. It also seeps out in education, and here, ethics is a relevant point. KF kairosfocus
I just read a book about the history of infinity in mathematics. I don't see how it has been "downplayed": it's been a critical topic for centuries. I also had a long conversation with you about your notation ideas. I wound up seeing your point in ways, but again, I don't see how this has to do with any philosophical points about naturalism. I think you and kf are seeing an issue where there just isn't one. hazel
Hazel - Naturalism appeared in mathematics as the downplay of infinities, and actually, in my opinion, led mathematics astray for a bit. See this article here. johnnyb
kf, I see we agree on a lot of pedagogical ideas. However, I have no idea how the "subtle imposition of naturalism" can come up in a math class. hazel
Hazel, thanks for thoughts. I think there is need for knowledge of content, insight on intellectual strategies, building up of intellectual skills. Also, cultivation of integrity, here truth, right reason [so, logic], prudence, justice. I would use Ponzi schemes as a case study on ethics of Math, likewise something on statistical manipulation. Cases on right use such as epidemiology and cholera in London may help. The Euler identity in its more usual form (and related ideas on rotating vectors) is a good case study on coherence. I would use it to set a context for the Godel result. Of course the rotating vectors view also gets us complex numbers without the fuss and bother, as a natural result. I would reduce the fuss and bother to a footnote, identifying the power of that vectors view in bridging to applications. Math labs and skills make sense. Something as simple as avoiding parallax errors counts. Likewise, learning to stabilise the hand by putting the little finger to rest on a work surface. Even, setting a point of pencil or compass where one wants it "on the tilt" then rolling upright. Practical measurement skills and the empirical link between calculation and observable fact count. Try, volume of a parabolic wine glass by solid of revolution to go with the cone. Doing the cone by calculus too, seeing how logic and empirical reality connect thus logic of being. Mechanics, astronomy, navigation as application contexts. For example the bridge from a graph of uniformly accelerated motion, to the classic Galilean results then drawing out kinetic energy is a powerful integrative exercise. So is simple ballistics (and computers can bring in air resistance). Exposure to foundations of computing. With JB's revelation of the free Mathematica with Raspberry Pi, use of computing instruments and mathematical software. Operating in a world where grading work and students by red X's on faulty calculations is increasingly irrelevant and outdated. I suggest, subtle imposition of naturalism is just as much a religious and/or ideological imposition as any other. Worldview + cultural agenda = ideology ideology + power = regime Consequences follow. KF kairosfocus
JB, Thanks for thoughts. I agree, the core ideas in hyperreals are easy enough that we should teach students the main domains of number. Surreals as a construction are hard to grasp as presented, but as setting out the "zoo" of numbers, I see no reason why we cannot refer to them. The concept of legitimate infinitesimals re-opens some of the advantages the early pioneers had and used. Certainly, as to concepts. I also agree that the pretence of neutrality (while smuggling in naturalism) does a lot of harm. Including, robbing students of insights. I am also serious on the point that Mathematics as to praxis is a study of the logic of structure and quantity. Thus, key concepts, strategies and skills of reasoning need to be on the table, perhaps even a bit on the philosophy, foundations and ethics of Mathematics that brings out the challenges of accountability before and commitment to truth, right reason, integrity, justice. The dirty games played with statistics and finance, come to mind. Here, truth as accurate description of reality counts, so deliberately untrue and manipulative misrepresentations are destructive. Maybe, why Ponzi-type schemes fail as a case study? Similarly, the concept of modelling, building logic-model worlds and how some entities are framework to any world thus generally applicable will be relevant. Including, core Mathematical structures. Here, the gap between model and reality may be relevant too, but the power of simplified or rather artificial models can be helpful. This is a contact with physics and other sciences, engineering, computer science [virtual machines etc], management, economics. Something as simple as flat geometry vs surface of earth may open vistas. And more. KF kairosfocus
KF - great topic! Glad you moved your comment to a full post. It deserved it. Hazel - I actually teach hyperreals in my high school intro calculus course. My book, Calculus from the Ground Up, is based on this. I've skimmed Knuth's surreal book, but don't totally understand them. Hyperreals are actually really easy and really cool. I've actually toyed with the idea of just teaching hyperreals as an extension to regular math, and just watch calculus magically fall out of it. I actually think that having opinionated books is more helpful for students. It helps them engage. We should teach students not to blindly trust textbook authors, but *engage* with them, and then allow textbook authors to be opinionated. That is where real learning and interaction take place. I also think that the infinite, generally, is a place where we really do need to teach high school students, because so much thinking is based on infinities. Most sets that are logically-defined are actually infinite. That is, "even numbers" are logically-defined. "Countries" is a logically-defined construct. There are not infinite *existing* countries, but the number of possible configurations which fit the construct is infinite. Going over Hilbert's Hotel and Cantor's diagonalization argument is a fun mind-blowing experience for high-schoolers, and it helps pave the way for understanding many of the arguments for design in the future. johnnyb