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 thatbrain 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, togethercausethe belief [–> concious, perceptual state or disposition] that Socrates is mortal.It isn’t enough for[so]rational inferencethat these events be those beliefs,it is also necessary that the causal transaction be in virtue of thecontentof those thoughts. . . [But] if naturalism is true, then the propositional content is irrelevant to the causal transaction that produces the conclusion, andwe 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.seento 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**

JB: what is important in math . . . ?

Get a ruler in your hands. Measure things until you start to understand how a ruler works. Measure some stuff and figure out where the center is. Say you measure a book and it’s 7/8″ thick. You look at your ruler and see that every eighth is divided into two sixteenths, so obviously half of 7/8″ is going to be 7/16″. If you write that out you have 1/2 x 7/8 = 7/16. And you notice that 1/2 is divided into 2/4 and then into 4/8 and so on, so you can convert anything to anything by multiplying all the numbers on top and then all the numbers on bottom.

Other rulers are divided into 10 and 100 parts. But an inch is still an inch, so anything on one ruler can be translated to the other ruler. A half inch on one ruler is 5/10 or 50/100 on the other. An eighth inch is just 12.5 marks when you have 100 marks per inch. A metric ruler divides an inch into 25.4 parts, so a half inch would be 12.7 of those parts. Pretty simple, isn’t it? Practice this a bit and people will think you went to wizard school.

Practically NOTHING learnt after grade school about Math is important. You’re sure not going to find any USE for it. After all, TV commercials regularly tell us that the product does “400% MORE” than the competitors. Does ANYONE know what that means? I worked with a woman who had earned a BS in Math. It had ABSOLUTELY no value in government processes run by edicts and impenetrable computer programs. Also, our job was being done by people with no college education at all, and when our “intern” program failed to show that 20 years of following the rules could be successfully replaced by 4 years of college, The Government went back to filling the positions from pools of “warehouse clerks”, who are not known for their THINKING about generalized, as opposed to today’s SPECIFIC, problems.

I read some years back that the Ebonics approach to what passes as Logic and grammar GUARANTEES that no Ebonics speaker will EVER be able to make a coherent statement about number juggling.

As with junk like Physics, there is some TEENY number (half of 1%?) of individuals who should be trained in the Ancient Magic to keep the flame alive, but there ain’t no point in attempting to DISCUSS “Whither the Mathematics” in public.

From David Berlinski’s Advent of the Algorithm:

“Mathematicians have loved mathematics because, like the graces of which Sappho wrote, the subject has wrists like wild roses. If it is beauty that governs the mathematicians’ souls, it is truth and certainty that remind them of their duty. At the end of the nineteenth century, mathematicians anxious about the foundations of their subject asked themselves why mathematics was true and whether it was certain and to their alarm discovered that they could not say and did not know. Working mathematicians continued to work at mathematics, of course, but they worked at what they did with the sense that some sinister figure was creeping up the staircase of events.”

As a land surveyor I established the physical boundaries of municipalities and private property before the GPS system existed. When you’re engaged in using measuring instruments and math to establish the structure and quantity of someone’s land they take an interest in the results. You can wave your State license all you want it doesn’t overcome the emotions that result when someone’s grandfather built a mule barn on someone else’s property but now it and the land it’s on has to go. You can explain the math and the methods used to arrive at your result and have a court uphold the survey but that won’t protect your trucks and equipment from vandalism, either.

I have a feeling that if the property boundary were instead established by a modern GPS system, that is, by a blind GIGO-limited computational substrate, those particular survey results might have been more easily accepted. I have gone back to check this and some other results I arrived at over 30 years ago using modern GPS equipment and it turns out my results were wrong. By 2 centimeters over a distance of 4500 meters; less than the width of a fence post. That personal sinister figure was stopped on its climb up the steps.

When I was in school sweating my way through calculus courses I never had any idea what it would be good for in the real world. School did not teach me that knowledge of math was conducive to becoming anything other than, say, a math professor at some State college, if I was lucky, and that’s a shame. It was just something you had to get through before the finish line. There is a gap between educational institutions and the rest of the world in many things, even more so these days where cultural agenda seems to be the prime focus regardless of the field of study.

I think that math is embedded in the substrate of our world. KF is much better at explaining why mathematical logic explains our duties to the truth, right reason, prudence, sound conscience and justice. The world root is not just causally adequate to our physical world with computational entities but it also grounds responsible rational free morally governed minds. This points to a necessary, inherently good and utterly wise being responsible for all we see.

Only if “grade school” means any and all levels of schooling.

VM, for me, it is very different. KF

LC, a surveyor is routinely using all sorts of Math up to spherical trig, all that stuff about haversines and other transcendentals they don’t usually talk about in grade school or high school. It’s a pity that Calculus (first step to Analysis) is so often so badly taught. But then, something like ballistics with air effects (a gateway to dynamics of flight) would be very politically incorrect today. That tale in a nutshell on the barn over the line is sad but telling, and of course is about triangle power. Yep, many mathematical entities and relationships are in the framework of this or any world, tied to logic of being. The beings in mathematics are those of structure and quantity. And insofar as duties of truth and right reason [= logic, a part of Math], prudence etc obtain, they are tied to how we are morally governed. Come to think of it the issue of integrity explicitly came up in just two courses I ever did, 6th form Physics and Finance — both, heavy users of advanced topics in Math. But it is implicitly there all the time. I guess we are more willing to trust a programmer over in India somewhere we don’t see than a live surveyor toting theodolite and maybe a calculator we do see. That says something. KF

I taught high school math (mostly geometry and Intro to Calculus) in a small semi-urban school district for many years, and also the first year of college calculus and college algebra for a while, so my thoughts here are primarily about college-bound high school students, including their first year of college math. FWIW, many students do not take calculus in college (taking rather college algebra and/or stats), and those that do often take an applied or business calc rather than the more rigorous engineering or math major track.

I think there are three broad primary goals in teaching math:

1. Teach the conceptual theories of the various concepts. A part of this should be to teach an awareness of and appreciation for the way in which math is a logically coherent system which builds on basic premises and logical reasoning. The process of understanding the development of the concepts and using them in problem solving teaches logical problem solving skills, including that such involve following steps each of which can be logically supported.

2. Teach and develop skills in the various conceptual, algorithmic, and procedural skills needed to actually use math correctly. Some examples: recognize a quadratic equation when it appears in a problem and know how to solve in the most appropriate way, know how to solve an equation with fractional exponents, or know how to differentiate a rational function. This includes knowing when it is appropriate to do math by hand, with a calculator, a spreadsheet, or dedicated piece of equipment. This also includes being able to write one’s work clearly in a way that can be communicated to others.

The students have to know how to

dothe math expecting of them correctly and expeditiously.3. Teach a wide variety of ways that math can be applied to the real world. This is critical. Seeing that math can be applied is a motivation to learn it: some students are motivated by the beauty of the math itself, as described in 1) above, and some by just the satisfaction of being able to “do the math” correctly, as in 2) above, but for many applying math to situations they think might be important is what motivates and interests them so that they want to apply themselves in learning. Real-world applications provide a vehicle for students to apply all the conceptual understanding and skills they have learned in an organized, directed way.

4. I also think it is important to provide some history and philosophy of math along the way, but given the practical demands of so much curriculum to cover, this needs to be more of an enrichment area. For instance, I used to do a class on the history of the number system, leading up to complex numbers, culminating with Euler’s Identity e^(i*pi) = -1. I also used to do a class on the nature of math, teaching some about Platonic and non-Platonic views, and then using the question ”is math discovered or invented” as a springboard for a class discussion. I emphasized this was a perennial unsettled question, and that they might enjoy studying some philosophy in college. (Footnote: I also used to do a guest lecture in the high school College English class on Plato’s Parable of the Cave, with the same intention of just introducing them to a range of philosophical views about ontology and epistemology that they had had very little experience with.)

Now a few responses to the OP and comments.

1. kf mentions hyppereal and surreal numbers, but I believe those concepts are justifiably extremely rare in the average calculus curriculum, even in the second and third years of calculus, which I took (long ago) but never taught.

2. It is not the place of a math teacher to offer particular religious or philosophical reflections about math as the correct ones, but rather, as described in 4) above, if addressed, to make the student aware of the general broad views on the topic.

3. I agree with SmartAZ that even a college-bound curriculum needs to include building skills with standard measuring instruments. For instance, I used to do a “math lab” where students computed the volume of a cone using calculus and then checked their work by pouring water into a measured beaker. I taught them how to use a simple calipers, how to account for the “dead space” at the end of a ruler, and many other measuring skills in this and similar other projects. Part of this is teaching them to think about the level of accuracy they can obtain, and take that into account when calculating final answers.

4. Needless to say, I highly disagree with Vmahuna. He seems to have no real conception about how important math is so many occupations.

I will say, however, that it is true that many specific skills and concepts are not used after being taught in skill. Two responses to that: first students incorporate an understanding of concepts that are important for understanding the world as a citizen even if they never have to “do the math” again. For instance, once one learns about general types of curves relating to variables (linear, polynomial, exponential, trigonometric, etc.), they can see and understand different patterns in the changes they learn about in the real world.

Second, as teachers we never know what student will go on to use what skills, so we have to teach broadly so everyone has the skills they need as their occupation become settled. Also, these days students will change as the get older, so knowing how to learn math, and being able to dig back about things they did learn once, can be critical

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.

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

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

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 –

Naturalism appeared in mathematics as the downplay of infinities, and actually, in my opinion, led mathematics astray for a bit. See this article here.

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, 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

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, 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

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.

kf –

They’re not in the mathematical part of statistics, though. They’re issues in science, and the application of mathematical statistics.

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

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, you just gave some fairly relevant cases, as that imposition is currently undermining the core of knowledge and the framework for integrity. KF

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, there is a widespread imposition of what is (in the bad sense) an ideological orthodoxy. KF

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?

Not society, hazel. Academia.

And can you offer an example of how an imposition of naturalism will show up in a math class, ET?

I have no doubts that a dedicated materialist would find a way. Something about how Hawking was right, no doubt.

I don’t think Hawking shows up in the general HS and college math classes.

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:

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

hazel’s inability to follow along makes it difficult to have a discussion…

PS: Later, SEP raises:

Such issues point to huge hidden curriculum questions.

Also, these issues point to the effectiveness of the summary naturalism ~ evolutionary materialistic scientism.

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).

ET, I think that we are often unconscious of latent but fundamental and sometimes unwelcome issues. KF

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, 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

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:

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.

KF

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.

PPS: Dossey of ISU on Math and teaching:

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

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.

BB, the extremism in your projections speaks for itself, tellingly; it is radical revolutions that produce reigns of terror, ever since the French revolution. 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. Unfortunately, the price tag for the lessons will likely be high as say the US is already in early stage 4th gen civil war. KF

PS: 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.

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,

Who is trying to impose their religion

[–> or IDEOLOGY, KF]on others?Can you name them? Thanks.

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

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.

PV

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?

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

PV

I agree. Is there anyone who really disagrees with this?

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

KF,

Agree with the correction you made in my comment @44. Thanks.

KF

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.

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.

I keep intending to drop out of this discussion, but kf writes,

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.” 🙂

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.

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

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.

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, 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:

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.

KF

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.

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?”

BB, you have clearly shown persistent lack of seriousness. KF

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

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?

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,

scientism.

Notice from 37:

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

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,

has it dawned on you that there is a RW to address? That Dorian passed through yesterday, causing both Net access and power headaches? That I had meetings to attend, issues to address elsewhere and more?

As far as the R-Pi is concerned, it is a cost-effective, transformational PLATFORM for education, and in my view transitional to what I really want in the end, here from 50:

“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.”That can probably be done using the R-Pi as mini motherboard, and I raise this as an option as I am thinking a deal had to be struck to get Mathematica embedded with Raspbian Linux. Such a super-tablet platform could do a lot. I would also do a kit dockable to a flat screen TV, converting it into a large screen tablet able to access the Net and to host local resources as well as to teleconference. My ideal size is 50 – 60 inches given current cost patterns. This will shift teachers to facilitators rather than primary content deliverers, as we can then stream high quality presentations from a repository using the best of the best, leveraging the ideas of Khan Academy etc. Already, the regional Exams (and so curricula) syndicate CXC is moving to a hub with rich resources. Think here, online multimedia library.Coming back, I think there is a concept challenge on ethics, knowledge, learning and the rational dimension of our life.

I have repeatedly highlighted something that is effectively self-evident: our rational thought is inescapably morally governed through duties to truth, right reason, prudence [thus, warrant]. sound conscience [eqn: integrity = sound convictions + courage to live by them], neighbourliness [= PaoloV’s love principle], fairness and justice etc. Thus our intellectual lives are shot through with pervasive ethics; moral government . . . which opens out in another space I am currently dealing with, law and governance i/l/o the laws of our morally governed nature. In which context, knowledge — warranted, credibly true (and so, reliable) belief — is therefore similarly ethically pervaded, noting the import and implications of “warranted.”

So,

apart from the sort of things associated with statistricks or ponzi schemes etc,. Which, necessarily includes Mathematics.the whole course of knowledge and education, epistemology and reasoning towards warrant is pervaded with ethical considerationsThat, I add, is a central point in my discussion.

In that context,

the tendency documented from the horse’s mouth [US NAS] to try to shoehorn Mathematics in under the umbrella of science despite its radically different framework of warrant both exhibits the scientism component of imposed, dominant, domineering naturalism, and inadvertently highlights the implied ethical gap. Evolutionary materialistic scientism is self-refuting by undermining the credibility of mind, and it is domineering, demanding genuflection before the vestments of the lab coat. It has unjustly claimed a monopoly on credible, reliable knowledge. So, somehow it has to capture Mathematics, despite the radically differing methodologies; especially as Mathematics is vital to key sciences. (Notice the struggles above over naturalistic ontology, as SEP summarised.)The “solution” is to gerrymander the “boundaries” of Science and claim Mathematics as a science rather than acknowledge something that would blow up the monopolising project: there is no one size fits all, surefire method of warrant that captures all types of sound, sufficiently reliable knowledge to be used freely in responsible contexts.

In short,

something is wrong ethically and epistemologically as well as logically, and.the inappropriate relabelling of Mathematics as a “science” is a key clueFurther to this, I just pointed to three of the five major domains of the REAL root discipline for the academy, philosophy: metaphysics (including Ontology, logic of being), logic, axiology (including ethics and aesthetics), epistemology, Meta-studies of disciplines [Phil-of-X]. In short,

we see a sounder framing of the Academy itselfhere.In that context,

the reform agenda framework I outlined anchors Mathematics in logic of being and logic starting from first principles of right reason applied to possible worlds and framework entities.Axiomatisationsthen reveal themselves as ways to structure abstract logic-model worlds and connect to both set constructions and modelling thus also simulation and computing. The von Neumann construction then allows us to move to the key quantity-sets: N, Z, Q, R, C and hyper-extensions, bringing in transfinites and infinitesimals thus opening up Calculus naturally. The ijk cut down from Quaternions — posit other roots of negative unity — then frames space and opens up Euclidean Geometry, non-Euclidean, Hilbert Spaces, vectors, matrices, tensors [now emerging in computing] and other relevant structures. Integral transforms and complex numbers, tied to frequency domain and transients, with the famous heavy rubber sheet model allows powerful exploration of dynamics of lumped systems. Differential equations come along for the ride, and partials come a knocking especially for fields. State/configuration and phase space frameworks open out more. And much more.All, tied to clearing up the ethics (and flawed epistemology) of trying to shoehorn Mathematics in under “science.”

KF

Raspberry Pi tablet https://makezine.com/2014/01/07/how-i-built-a-raspberry-pi-tablet/

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/

LC, sad, very sad. KF

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/

F/N: It seems Discovery Institute has a centre for education transformation: https://www.discovery.org/education/ KF

KF

Is it as productive and effective as BioComplexity, the Discovery Institute’s peer reviewed ID journal?

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

KF

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:

It is more productive than blind watchmaker research.

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

Brother Brian:

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… 🙄

You’re kind of a punk, you know that?

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.

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?

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:

This is in fact a simple statement of a famous challenge issued by von Mises in the 1920’s, the issue of sound socialist-centralist calculation of values [thus, criteria for rationing scarce resources across competing wants]. This boils down, in turn to a computer architecture problem. When estimators are hard to specify and calculate out side of a market, communication of quantities and priorities to a decision centre to hope for prompt sound, balanced, flexible decision is bound to fail. Just doing a 300 x 300 array national accounting table (at a very high level of abstraction that in practice depends on markets) already implies 9*10^4 elements in the Matrix. Going down to rapid response to very local and rapidly changing circumstances simply will not fit with the decision making lags for central planners. Instead a large array of much simpler planning entities (households and firms) coupled through markets will be far more likely to be rapidly responsive and far less vulnerable to single point failures. So, across time, autonomous, local planning in market networks will provide far superior performance on resource allocation and rationing problems. Including, estimating opportunity costs; the real economic costs for alternatives taken — the next best opportunity foregone.

In education context, major, centralised curriculum decisions are fraught with serious power conflicts and usually decisions are made on horse trading power and influence. As a result, central bureaucracies attract ideologues with agendas who are backed by moneyed interests [which include those who control tax revenue streams]. This tends to lock out sound innovation and leads to decision by real or manipulated crisis.

The disastrous trend is predictable, especially in an age with cultural marxism and post modernism on the march multiplied by the destructive, suicidal tendencies of evolutionary materialism as has been known since Plato warned us in The Laws Bk X, c 360 BC.

In that context, thinktanks that can mobilise money to fund genuinely independent analysis and research leading to publishing of a growing body of knowledge under minority paradigms can have disproportionate positive impact. Here, the impact of the Austrians on Economics in the past three generations is a classic study.

Such a growing body then can guide alternative policy frames, demonstration projects and alternative schools for practitioners. Here, that is directly relevant to education. However, the aggressive hegemony and ruthlessness of ideologised centralised bureaucracies will have to be curbed. Which sounds about right for a Libertarian Thinktank. Which is what the Discovery Institute is.

When paradigms need to be drastically shifted, the critical expertise is going to be innovation, governance and incubating transformational change, not indoctrination in the paradigm in unacknowledged crisis. The technical fellows will come [change comes from idea originators and champions but it needs sponsorship, incubation and Godfathers able to deploy resources and marshals to defend the incubator], you need innovators to focus their efforts towards successful strategic change.

The obvious blunder in your race to dismissal is failure to understand the role of fellows. Who, in highly polarised times, may need to be protected from idea-implementer hitmen, trolls, agit prop media hacks and howling SJW mobs.

So, in that context, just maybe, rethinking Mathematics education is something that can grow legs and go somewhere serious.

JB’s ideas for Calculus education are being put on the table at an auspicious time.

KF

Brother Brian:

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.

F/N, FYI/FTR: As a reminder (given the insistent pretence that design theorists do not publish research) I note for the thread :

KF

F/N: To get a picture of some concerns, ask what is missing here:

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

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:

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:

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