I actually misread the title - thought the post was about banning models-of-determining-healthy-weight that were based on BMI. I was several posts in before I figured out what the title was *supposed* to mean :lol:.

I'm 34, and I wear earrings 24/7 (the *same* pair 24/7). The whole "able to sleep in" thing combined with being very sensitive to any metal but gold really limited my earring choice. For years I used to wear small studs just to keep my holes open, but a few years ago I fell in love with a pair of 2" spirals (made from implant-grade surgical steel) and now they are what I wear everyday. They are fairly distinctive and I get compliments on them, plus comments wondering how I can wear dangly earrings with small children (answer - because they are thicker than typical earrings, so the kids can tug all they want and it doesn't hurt, let alone tear the hole). I'm not much of a girly-girl or jewelry wearer, but I do love my spirals.

Here's a link to the AP edition of the poetry book (Perrine's Sound and Sense), and as far as I can tell (with a very quick comparison of the contents), the main/only difference is the inclusion of AP writing prompts: http://swcta.net/orapello/files/2013/03/Poetry-Text-1.pdf. I would expect that holds true to the literature book as well (it's not all that easy to find the toc for the ap edition online - the publisher's site has the toc online for the regular editions, but not the ap editions).

From the New Math of the 60s, Dolciani's "Modern Introductory Analysis" and Allendoerfer and Oakley's "Principles of Mathematics" were both rec'd on the board in past years. (I have them both on my shelf, but haven't used them; my grandpa actually taught from the Dolciani book (I have his teacher's edition) - turns out he was instrumental in getting his school to switch to New Math.)

  :lol: My kids are PKs, too, but fortunately they haven't suffered from being in the fishbowl (yet anyway). It may be because I'm fairly oblivious to unspoken expectations plus I apparently intimidate people, so no one ever tells me all the ways I'm screwing up to my face and I go on in blissful ignorance ;). It may be differing traditions, but I don't agree that all anyone needs is "the Holy Spirit working through me and my Bible". In a sermon of dh's, he talked about how the Bible is meant to be interpreted within the Church - it's "the Holy Spirit working through me and my Bible and the cloud of witnesses". Just as God works through parents to raise and teach kids (kids raising themselves doesn't tend to work well ;)), He works through faithful pastors and teachers (including parents) to teach His Church. The Bible wasn't written in a vacuum, and the Bible isn't interpreted in a vacuum - God's work is no less God's work for being done through means instead of directly. ETA: I don't think the only (or best) way to experience living ideas is unmediated contact between a learner and the text. A teacher can both convey living ideas and can guide the understanding of a text's living ideas, without draining ideas of their vitality or diminishing the impact of direct contact with those ideas. I subscribe more to the classical idea that a novice benefits greatly from a knowledgeable guide, who can not only facilitate contact with living ideas but also guides the novice in how those ideas fit into the world. Learning directly from the source doesn't have to mean *unmediated* learning.

I think learning about the lives of the saints (which would of course include the saints in the Bible), including discussing applicable character traits, and emulating their example as part of learning to live a life of faith can be beneficial. But I'm hesitant to give those character traits the force of God's Law. I realize your example was spur of the moment, but using the Creation story, which shows how even creation itself is a gracious gift of God to us, as a springboard for teaching orderliness or creativity does seem out of context. And I'm hesitant to say that either orderliness or creativity, good traits though they may be, are part of God's Law. And if they are, the creation story isn't what establishes it. And this is one reason why I'm not a fan of teaching Biblical morality through character traits - most Biblical narratives were not written to tell us the traits of a righteous person, and the Biblical texts that *do* tell us what a righteous person looks like tend to not be narratives, nor do they mesh well with the sorts of character trait lists that tend to be used. Generally it seems people take a list of character traits that are *consistent* with the Bible, instead of *drawn* from the Bible, and start matching them up with a list of Bible stories that seem to somehow mention the trait. This may show that the traits are indeed consistent with the Bible, that living by these traits is consistent with living out a life of faith, but it doesn't show that living by these traits *is* living a life of faith. And I think that's what some of the other posters were getting at - that when you use the Bible to teach character traits it is so very easy for it to come across as "living out character traits" = "living out the faith". Only that's *not true*. Teaching character traits is *not* teaching what it means to live the faith. Heck, even teaching God's Law, drawn straight from the Bible, is still only teaching a small part of what it means to live the faith. My tradition (Lutheran) teaches the 10 Commandments as the summary of the Law, of how God's people are to live - like you are using traits, we use the commandments as our framework for discussing right behavior - but that is only one of the six chief parts of the Small Catechism, and taught alone it leads to legalism. The teaching of right behavior needs to be *explicitly* grounded in an overall Biblical view of how the Law and Gospel work in the life of a Christian.

I grew up south of Houston, and I now live in northern IL. When I first moved here, I couldn't tell the difference between 45 degrees and 20 degrees - it was all just "cold". I got mocked my first winter for overbundling for 45 degrees - it wasn't because I'm a delicate southern magnolia ;), but because I had no idea it had "warmed up" overnight - it still felt "cold" to me. But just as in the south you can tell the difference between how 85 and 100 and 110 degrees feel (northerners here can't do that; "hot" is anything over 80, and no one has any idea how to be active outdoors safely in higher temps, because it rarely is an issue here), I now can differentiate between 45 (and genuinely see it as not-winter weather - coming out of winter it's a heat wave, baby!) and 25 and 15 (and thanks to last year, I now know what 0 and -20 feel like ;)). Second the rec's for good winter gear - not all gear is rated for all temps. Stuff for 20 degrees worn when it's 0 degrees will not provide the same protection ;) - have to plan for that. I had to learn how to be safe in freezing temps, because the southern approach of huddling in our houses until it goes away isn't exactly viable ;). (And I don't want to be like our softball team that held a practice when it was in the high 90s, and didn't think to hydrate more than they would when it's 80 - these temperature differences, they do matter ;).) It surprises me the number of families, life-long northerners, who keep their kids inside whenever it's below 20 degrees; it's like southerners who don't let their kids out whenever it's above 90 - it traps you indoors for a significant period of time. I personally prefer to learn the safe ways of being out in the more extreme temps that are usual for a place. Getting used to the time it takes to bundle everyone up before getting into the car was an adjustment, and we all cheer and ditch the coats for jackets as soon as it's above freezing ;). But the hardest thing for me has been the lack of light in the winter - living in the south all my life, I had no idea that would be an issue. Getting enough light is a major part of dealing with winter for me.

This rule (or "rule") seems to be the go-to example for so-called phonics rules that aren't. "It's only true 43% of the time!" People use it to show why "phonics doesn't work", and phonics advocates are quick to throw it under the bus as an example of crappy phonics teaching. Which is why I was shocked to see that LiPS, an intensive, research-based remedial program, actually teaches it. What in the world?!? But I thought about how LiPS does it - they first teach the primary spelling(s) for the additional vowel sounds beyond the short/long vowels (au/aw, oo, oi/oy, and ou/ow), and only use the "two vowels go walking" rule on other two-vowel phonograms, as a guide to their most *common* sound (not their *only* sound). And when you do that, the accuracy rate goes way up. There are very few big exceptions: namely, eu/ew, which is only used for /OO/ and /U/ (and so is outside the scope of the rule), and "ie", which is the only true exception, because "the second letter does (most of) the talking". There are a few more phonograms where another bit of phonics knowledge is needed to complement the "two vowels" rule (usually wrt foreign spellings). The big one is "ey"/"ei"; the primary sound is the *Latin* sound for "e" (/A/) and it is used most in words of Latin origin; the next most common sound for these phonograms is long-e (and actually, for the suffix -ey, /E/ is the *only* sound). "ui"/"ue" both have second sounds with consistent, easily identified criteria (-ue is silent at the end of words of French origin, and in some words that start with g, the "u" in "ui" is silent, and serves to separate the "g" from the "I", making it clear it's a hard "g"). So I ran the numbers (using The ABCs and All Their Tricks as the source of all my numbers), and I'm starting to think that this much maligned rule doesn't deserve its bad rap. For the following list of phonograms (basically the Spalding two-vowel phonograms, excluding au/aw, oo, oi/oy, ou/ow, and eu/ew as explained), here's how the "two vowels go walking" rule of thumb works out: When you include the known outlier "ie", it identifies the correct sound 90% of the time. When you *exclude* the known outlier "ie", it identifies the correct sound 95% of the time. If you don't include second sounds with consistent, easily identified criteria (ue/ui), to look only at true outliers, it identifies the correct sound 98% of the time. I think that's a pretty darn good rule of thumb :). I think a key thing is to see "two vowels go walking" as a *rule of thumb*, a guide to which sound to try first, to the *most common sound* for phonograms that aren't primarily used for non-short/long vowels, instead of as some absolute, "it's *always* this sound" rule. Within those parameters, it's quite accurate and useful :thumbup:. Here's my numbers, for the phonics geeks ;): ai: long-a, 98%; other, 2% (long-a: 308; other: 5) ay: long-a, 99%; other, 1% (long-a: 143; other: 2) ea: long-e, 67%; short-e, 32%; other, 1% (long-e, 325; short-e, 156; other, 5) ee: long-e, 99%; other, 1% (long-e: 307; other: 2) ei: Latin long e (long a), 73%; long-e, 15%; other: 12% (Latin long-e: 54; long-e: 11; other: 9) ey: in base words, Latin long-e, 85%; other, 15% (Latin long-e: 17; other: 3) as a suffix, long-e, 100% (43, no exceptions) ie (the outlier): within words, long-e, 97%; other, 3% (long-e: 77; other: 2); at the end of words, long-e, 63%; long-i, 37% (long-e: 17; long-i: 10) oa: long-o, 99%; other, 1% (long-o: 132; other: 1) oe: long-o, 83%; other, 17% (long-o 15; other: 3) ue: long-u, 51%; silent at the end of words from French, 49% (long-u: 44; silent: 43) ui: long-u, 62%; silent u, 38% (long-u: 15; silent u: 9)

So I ran the numbers (using The ABCs and All Their Tricks as the source of all my numbers), and for the above list of phonograms (excluding ow), here's how the "two vowels go walking" rule of thumb works out: When you include the known outlier "ie", it identifies the correct sound 90% of the time. When you *exclude* the known outlier "ie", it identifies the correct sound 95% of the time. If you don't include second sounds with consistent, easily identified criteria (ue/ui), to look only at true outliers, it identifies the correct sound 98% of the time. I think that's a pretty darn good rule of thumb :thumbup:. Here's my numbers, for the phonics geeks among us ;): ai: long-a, 98%; other, 2% (long-a: 308; other: 5) ay: long-a, 99%; other, 1% (long-a: 143; other: 2) ea: long-e, 67%; short-e, 32%; other, 1% (long-e, 325; short-e, 156; other, 5) ee: long-e, 99%; other, 1% (long-e: 307; other: 2) ei: Latin long e (long a), 73%; long-e, 15%; other: 12% (Latin long-e: 54; long-e: 11; other: 9) ey: in base words, Latin long-e, 85%; other, 15% (Latin long-e: 17; other: 3) as a suffix, long-e, 100% (43, no exceptions) ie (the outlier): within words, long-e, 97%; other, 3% (long-e: 77; other: 2); at the end of words, long-e, 63%; long-i, 37% (long-e: 17; long-i: 10) oa: long-o, 99%; other, 1% (long-o: 132; other: 1) oe: long-o, 83%; other, 17% (long-o 15; other: 3) ue: long-u, 51%; silent at the end of words from French, 49% (long-u: 44; silent: 43) ui: long-u, 62%; silent u, 38% (long-u: 15; silent u: 9)

Actually, you can simplify it - remove all the phonograms that are primarily used to spell the non-short/long vowels sounds (au/aw, oo, ou/ow*, oi/oy), plus the outliers ew/eu, which are only used for OO and long-u - and the "two vowels go walking" rule of thumb correctly identifies the most common sound for all the common vowel phonograms left (except for the already mentioned "ie", and Latin sound of "ey/ei"). (Afaik, this is how LiPS does it; certainly it teaches the above spellings (au/aw, oo, ou/ow, oi/oy) as the primary spellings for /au/, /oo/, /OO/, /ou/, and /oi/ long before it brings up "two vowels go walking".) (*Asterisked ow/ou, because they are the primary spellings for /ow/ as in cow, but half the time, "ow" has the long-o sound.) Phonogram list: ai/ay: most common sound by far is long-a ea: most common sounds by far are long-e (325 times in ABCs and All Their Tricks list) and short-e (156) ee: most common sound by far is long-e ei: most common is the Latin long e (long a; 44) and second most common is long-e (11) ey: in base words, most common is the Latin long-e (17); as a suffix, the most common is long-e (43) ie: the outlier, within words the most common sound is long-e (77); at the end of words, the most common is long-e (17), followed by long-i (10) oa: most common sound by far is long-o oe: most common sound is long-o (ow: most common sound is long-o (130), but almost as common is ow-as-in-cow (122)) ue: most common sound is long-u ui: most common sound is long-u Really, I think the key thing is to see "two vowels go walking" as a *rule of thumb*, a guide to which sound to try first, to the *most common sound* for phonograms that aren't primarily used for non-short/long vowels, instead of as some absolute, "it's *always* this sound" rule. Within those parameters, it's quote accurate and useful :thumbup:.

I'd heard so much negativity wrt the "when two vowels go walking" rule that I was shocked to see it taught in LiPS (an intensive, research-based remedial program). But while it fails when applied indiscriminately to *all* two-vowel phonograms, it actually *does* work as a rule of thumb for which sound to try first for ambiguous phonograms. Most two-vowel phonograms are fairly consistent, with one primary sound and very few exceptions (most of which are in common words, so for unfamiliar words the default sound is very likely to be right). Applying the "two vowel goes walking" rule would indeed fail for many of these - but actually that rule is entirely *unnecessary* for those phonograms the first place, especially in phonogram-teaching programs - because those phonograms have one primary sound, and it is learned directly, *prior* to ever being introduced to the "two vowels go walking" rule of thumb. That's how LiPS does it, at any rate, and it cuts way down on the error rate for that rule of thumb. For ambiguous phonograms, the only one it doesn't work for is "ie" (long-e is slightly more common than long-i). However, it *does* work straight-up for "oe", "ue", "ui", and "ea" (for "ea" the two primary sounds are the two sounds of "e", short-e and long-e), and works in conjunction with other bits of phonics knowledge for "ey"/"ei" (Latin sound for "e", and used most in words of Latin origin; next most common sound is long-e) and "ou" (/ou/ as in shout is by far the most common sound for "ou", but long-o is tied with /ou/ as in soup for the next most common, and the "ou as in soup" is a French sound, found in words of French origin). It surprised me, but the "two vowels go walking" rule of thumb, used not to determine *the* sound for all phonograms, but to pick the *most common* sound for otherwise *ambiguous* phonograms - it's quite accurate in that context. There's actually a legit use for it.

We screen all our calls (no caller id; just listening to the message as they come in), so no one picks up unless it's for them and they want to talk - "taking messages" is letting someone know there's a message for them on the machine ;).

My mom always thought Martha got a bad rap - it always seems to turn into "doing = bad, being still = good". I thought of her when I read a new perspective on that story, in a book on spirituality (Grace Upon Grace, by John Kleinig), and she'd appreciated me passing it on. This post reminded me of that. The author said that the issue with Martha wasn't that she was busy preparing food or that she was failing to sit at the feet of Christ like Mary - but that she yielded to anxiety, that she was annoyed with Mary. He maintains that they were both engaged in meditation - Mary by listening to Jesus and Martha by cooking for Him - because the core of Christian meditation is nothing more or less than focusing on Christ. Martha's problem came in when she lost her focus on Christ - *that* was the most needful thing. The author concludes that whether we are activists like Martha or contemplatives like Mary, Jesus must be the focus of our meditation; anything else is distraction.

Another Confessional Lutheran, and ditto - we have service on Maundy Thursday and I've always considered it to be a "big deal" service, equal to Good Friday - I wouldn't miss it.

:grouphug: When my ILs visit (planned and welcomed) it's still pretty stressful. I tend to do the "mi casa es su casa" thing - here's the kitchen, feel free to use it whenever - just clean up when you're done :). And I go hide whenever in my bedroom when I need some peace and quiet. My grandma is an introvert, and my grandpa is an extravert, and there's *always* people dropping in and out of their house. Grandma used to do the hostess thing for everyone, fixing food and cleaning up and entertaining (and in retrospect probably martyring herself). But after major surgery, she wasn't able to do anything for months, and Grandpa learned how to take care of her and the house and all the visitors, instead of it always being Grandma taking care of everyone else (everyone, including Grandpa, said it was good for him ;)). Anyway, now she's a lot more relaxed, and only does what she's able to do without it being a problem. You want food - there's the kitchen, help yourself :). And she doesn't have a problem going to take a rest and letting people entertain themselves. And nobody minds :). Anyway, just throwing that out there - if you need rest, rest - everyone else will figure something out ;).

I'm not doing so hot on working it out practically myself, either :grouphug:. I do think most programs are coming from the same basic place as regentrude: the fact math works is empirically verifiable, and that's all that's necessary to use math with confidence. Feel free to layer on whatever philosophical or theological reasoning you may have as to *why* math is empirically verifiable, but it's not necessary - the key bit is that math *is* empirically verifiable, and learning how to verify it yourself is what matters. (You see how this appears to make a given program usable for people of vastly different theological and philosophical approaches.) That's been the dominant view for a few centuries, displacing everything else, and though I question its basis, I really have no idea (yet) what in the world math education from a more unified perspective looks like :(. It's been largely lost to us, and regaining it is hard. Especially because we can't turn back the clock and adopt pre-modern views wholesale, like the intervening centuries didn't happen. Because they did, and any mining from the past has to interact with the present. On the helpful side, most churches that have held onto the idea of transcendent truth already have an implicit sense that empirical reality isn't everything - any teaching rooted in that is going to somewhat counteract empirical assumptions. So knowing and conveying how learning math fits into your theological worldview (which it seems you have a basic sense of), is itself necessary and of great value. As the teacher, you are providing the context for learning math, and that makes a great deal of difference. People are forever lifting random ideas out of context and using them for quite contradictory purposes from what their original creator intended, all quite inadvertently, and while it's not ideal, it often works out a lot better than you'd think. Having an intentional, conscious worldview rooted in your core beliefs, that you intentionally and purposefully live out as best you can in every aspect of your life - that will make up for a *lot* of inadvertent use of contradictory stuff :thumbup:. (Let us all give thanks to God for that :).) For a piece of random speculation wrt squaring empirical math with a transcendent worldview while using empirical programs: clearly separating the study of the natural world from philosophy has accomplished a lot. Even though I think its legacy is a mixed bag, it's so entrenched in our Western society that I don't think writing it all off is a good idea (even if it were possible). So it's a matter of finding a new way to integrate science into an overall view of life. I'm wondering if, in the absence of anything else (that I know of), if it would work to acknowledge that the whole of reality involves both physical and metaphysical truths intertwined and in harmony together. While they cannot be separated in reality (which is the default modern/secular assumption), it can be beneficial to temporarily look at only the natural aspect. The key is remaining aware that this is only a mental fiction, not a reflection of underlying reality. In practical terms, I guess it would be using a regular program within a life that works hard to generally and deliberately integrate physical and metaphysical truth. The Circe Institute certainly tries to craft an education that unifies truth, goodness, and beauty in everything; Norms and Nobility (haven't read yet) talks about an education that keeps the moral dimension of life unified with the physical dimension; Beauty for Truth's Sake: On the Re-enchantment of Education, by Stratford Caldecott (also haven't read yet) is trying to re-integrate physical and metaphysical truths in math/science (idk if I'm going to agree with him, but I think he's gone the farthest in trying to bring the philosophy of the past into the present, in the area of math/science education, that I know of). The Catholic and Orthodox worldviews of the above sources can be tricky as a Protestant if you don't have a good handle on your faith and where it differs (I've had problems there), but they are the ones with a past to draw on, one that Protestants share in, too. C.S. Lewis also tries to re-unify the physical and moral dimensions of life, in the face of an empirical worldview - The Abolition of Man, among others, discusses this. Help any? Clear as mud? ;)

Was this directed at my post #58? I ask because we cross-posted, and that post was a response to the post before yours, but what with the crossposting and my lack of quoting it wasn't clear (I went back and edited it to add clarity). In any case, I wasn't in any way meaning to suggest that "proofs in elementary school" was the only valid way to teach in light of a belief in absolute truth and that math provides a vital tool in finding that truth :grouphug:. Eta: It's a change in thinking for me, but I now agree with you that arithmetic is a worthy subject in its own right, communicating truth. And I thought your question to regentrude was a good one - I'm interested in the answer, too.

I'm sort of seeing two separate things conflated in your post - that humans cannot know *any* absolute truth for certain (e.g. a proof is true now but might not remain so) and that humans cannot know *all* absolute truth for certain (e.g. the existence of aliens as an open question). I would agree with the latter, but not the former. (FWIW, the belief that absolute truth exists but humans cannot know any of it for certain is consistent and a relatively common belief right now.) But truth that's only our best approximation of truth, and constantly subject to change, is different from truth that's Truth. I mean, embracing "the best truth we have right now" - generic you might stake your life on it in the everyday, acting as if it is true sense (we'd go mad otherwise :willy_nilly:), but it's not truth you'd stake your life on in the sense of being willing to die for it, is it? (Or even more stringent, truth you'd want your spouse or children to die for rather than go against it - that one gives me the heebie-jeebies.) I mean, truth that is always in flux, always subject to change - nothing is sacred, everything is potentially up for reassessment - that's a very different sort of truth to base one's life on than truth that's unchanging. Isn't it? Genuine question - it seems like it to me, but lived realities can be quite different things to philosophical abstractions ;). The "would I die rather than recant this belief" question is my litmus test here - it kind of demands a 100% all-in commitment ;) - and I personally can't fathom doing so over a "good enough" truth, one that I believe has the potential to change. To me, any truth a person would be willing to die for rather than abandon is one they believe they know well enough to be certain it won't change in any way that matters. Which isn't really compatible with a belief that no truth can be known with certainty - is it? If a person would die rather than kill an innocent person - well, that's a pretty firm belief, if an unconscious one, that murder is always wrong, isn't it? Otherwise, if in theory everything was up for grabs, when your imminent death was on the line, you'd be pretty motivated to find a loophole you could live with, yes?

Yes, even the choice and framing of the core questions is itself a philosophical position, and a contested one - people don't all agree on the starting point, and the particular starting point does indeed make a difference. Totally agree re: how one answers a particular classic question directly impacts one's view of math. (Is sort of my point - how can one's view of the purposes of math *not* impact how one teaches it?) Will take a stab at an example: Something I've seen debated is this: is arithmetic math? (Common answer is *no*.) Related is the question of whether anything in the typical K-12 math sequence actually resembles math as mathematicians conceive of it, except of proof-based geometry. Are proofs necessary in order for math to be *math* in a meaningful sense, instead of applied math? As you said, for those who think humans can *know*, math is a vital tool for that understanding. And ime, math in that sense is commonly seen as limited to proof-based math. And that radically changes how you'd approach teaching math, since the usual sequence doesn't introduce proofs until upper-level undergraduate courses in math, which are taken by very few people (math majors, certainly, but not engineering; idk about physics and other sciences). So if you believe that math is vital to knowing the world, and that means proof-based math, you somehow want to bring proofs into the K-12 sequence, in a major way, for *all* your students, so far as possible. There's not many choices for that, and most are for mathematically gifted kids. But if proofs are vital to knowing, and seeking to know is essential to being fully human, you're far more likely to do whatever it takes to enable your students, all of them, to work through those programs, instead of using a more standard program. And how do you view arithmetic? Do you try to bring proofs into it somehow (as many New Math programs in the 1960s tried to do), or do you see it as a ultimately pointless but necessary intermediate step before you can get to *real* math. The latter, in addition to flavoring all your arithmetic teaching (I don't think you can underestimate the impact of that), often leads to rushing through arithmetic as fast as possible, just hitting whatever is needed in order to be able to start a proof-based algebra sequence (the beginning of math that matters). Which has clear implications for teaching. Does that help any? ETA: And if we turn it around - for those who believe that humans *cannot* know fundamental truths of the world with certainty - then math loses its place as a major tool for that knowing. At that point, math is about its practical usefulness only; conceptual vs procedural math is a matter of which allows any given student to master whatever math they need for their life. If math no longer is a path to finding (abstract) truth, the importance of learning to think abstractly is materially lessened; understanding math is reduced to being able to use math skills to solve the practical problems of life - concepts are only important inasmuch they contribute to this. And proofs are back to being only for particular college students. They might teach really great thinking skills, but so do lots of other things - math has lost its unique place.

I am so woefully underequipped, but I'll take a stab at explaining (just explaining, not defending or trying to persuade; iow I'm just trying to explain the what without much in the way of the why): If I am understanding you correctly, the natural world works as it works, whatever the answers to the Big Questions of life. No matter what the answer to the Big Questions, nothing about the natural world would be remotely different. So how in the world do answers to the Big Questions have anything to do with math, or teaching math? Because the Big Questions have nothing to do with the natural world, they can have nothing to do with learning about the natural world, or teaching about the natural world, so they can have nothing to say about math, or the teaching of math. The thing is, the idea that the natural world is separable from the philosophical/metaphysical/supernatural realm - that's a relatively recent idea, and is far from universal. It's actually quintessentially modern; the physical and metaphysical were united in pre-modern thought, and post-modern thought is trying to figure out ways to re-unify them (albeit in very different ways from pre-modern thought). I can't explain the whys or wherefores of why people reject the strict separation of the physical and metaphysical here, not so it would help, but for the purposes of the discussion, accept that such people exist (and generally includes those who are seeking truth/goodness/beauty), people who think the answers to the Big Questions of life are inseparable from the nuts and bolts of *living* that life. (And historically there's nothing weird at all about that - modernism's separation is the outlier.) In which case, for them, *everything* about life is impacted and informed by those Big Questions. And part of the everything is education, and part of that is math. It's not so much needing to have detailed answers to abstract math philosophy issues to teach math, but that one's overall philosophy of life narrows the scope of what schools of math philosophy are compatible and which ones aren't, *and* that philosophical questions and their answers are seen as explaining the nature of the natural world, too. In this view *all* ways of teaching math have some underlying assumption about the philosophical nature of math (and wrt teaching, additional assumptions about the nature of man, and the purposes of education), because it's unavoidable (physical and metaphysical truth are intertwined, not separate) - and you want to find the ways that are compatible with your views. ETA: I think it's a common belief today that for a given set of answers to the Big Questions to be valid, nothing in them can go against scientific fact. Historically, that went the other way, too - physical truth and philosophical truth needed to be in harmony, and just as physical truths could point out philosophical falsities, philosophical truths in turn could point out physical falsities. For them, just as the natural world limits what can be philosophically true, the philosophical realm limits what can be physically true. Sounds weird to modern ears, but it's a basis for thinking the Big Questions have direct impact on something as apparently mundane as teaching math. ETA2: And really, modernism's position that philosophical truths don't impact the natural world is a *philosophical* position ;), one that does indeed limit what can be physically true.

Thank you for clarifying :). My overall point stands, I think - there is indeed a difference between that view of math and a "seeking truth/goodness/beauty" view of math, and it involves whether it is good or not to separate the philosophical questions of math from the discipline or practice of math.

Yes, that was a sloppy phrase. I think my point is that modern math separates, or allows for the separation of, the doing of math from the "philosophical debate of whether a mathematical entity even exists in the 'real world'". I mean, currently, what a mathematician believes about the metaphysical connection of mathematical truth to the rest of the universe is seen as irrelevant to the mathematics he does, yes? Two totally separate issues, and work done on one does not materially impact work done on the other; or at least the metaphysical question does not impinge on the math itself, though math itself might impinge on the metaphysical question. Is that more accurate?

I would love it if you could explain this some more... pretty please? I'll try :). Can't find the original source :(, but a series of googlings and memory jogging has (hopefully) enabled me to say something intelligible ;). The original context was about "art for art's sake", but I believe it applies to "math for math's sake", or any sort of "learning for learning's sake" or "creating for creating's sake". "Art for art's sake" says that the highest, purest motivation for making art is for the sake of art itself. Art itself becomes its own end; it no longer has a "self-transcending purpose". Pure art only has meaning as art; it does not have inherent meaning in the world outside itself. But the point of seeking truth, beauty and goodness is because of the belief that those things embody transcendent truth. Art is not good in and of itself, but by whether it conforms to, reflects, transcendent truth. But art for art's sake rejects the idea that artists should conform to anything outside the demands of art itself - embodying transcendent verities is no longer seen as the purpose of art. In math, my understanding is that in Ancient Greece, the first principles, the axioms, of math were meant to be fundamental truths, and finding the best and purest axiom system meant finding the one that best reflected transcendent truth. But from what I understand in modern math, the best axiom system is the smallest, most conceptually elegant one, and it makes no difference whether those "conceptually elegant" axioms reflect "real world" truth or not - it's rather beside the point. "Math for math's sake" has math as its own end, judging mathematical truth and beauty apart from any universal sense of transcendent truth (or the idea that there's any inherent moral qualities to math itself, whereas "seeking truth/goodness/beauty" believes all those qualities are unified in everything (goodness meaning moral goodness), and educators should preserve that unity in how they teach). Does that make any sense? (Hopefully I didn't make too many gross math, history, or philosophy errors ;).)

Here's how I think of it: my approach to math is likely to reflect my overall educational approach, and my educational approach reflects my educational goals; in turn, my educational goals reflect my overall goals in life. What does it mean to live a good life? What is my purpose in life? How do I live a meaningful life? Those questions are fairly core to a person's worldview; sometimes they are answered by a person's religious beliefs, but there are non-religious answers to those questions, too. Anyway, here's a personal example of how the two intersect. I used to be in the "math for math's sake" camp - education is about seeking the true, the good, and the beautiful, and so math should be presented and taught so that the inherent truth/goodness/beauty of math is brought out, and so I only looked to educators/mathematicians/curricula that were in line with this view, which vastly simplified things. Concepts before procedures; understanding before memorizing, with memorization primarily in context; proofs, not calculations, as the focus of math; focus on flexible thinking and seeing multiple ways to solve a given problem - all answered because of the choice of approach. And I would be equipped to modify programs to fit kids, because I knew what my ultimate goal was. However, I recently learned that my religious beliefs and my educational beliefs (re: education being about seeking the true, good, and beautiful) may actually be at odds. Because the idea that education should be about seeking the true/good/beautiful has as its basis that belief that the highest aspiration in *life* is the contemplation of the true/good/beautiful. And my religious tradition, as it turns out, does *not* consider contemplation of the divine as the highest human aspiration, or as the life God means us to live. (In fact, I recently realized that I actually have *no idea* what my religious tradition teaches about the highest goal of life, or how our beliefs impact daily life, or anything of that nature. Which is why I was borrowing from other, apparently contradictory, traditions. The funny thing is that *all* the traditions in question are rooted in the classical tradition :lol:.) Anyway, a re-think in my approach to the "Big Questions" of life had direct impact on my theoretical approach to math. "Math for math's sake" is no longer an option (and actually, after many years of equating them, it turns out that "math for math's sake" is at odds with "seeking truth/goodness/beauty", something else I didn't know till recently), and I haven't replaced it with anything yet, because the answers to the big questions are still in flux. And it is totally unsettling. Because right now I'm just doing math because it's probably necessary no matter what ;), but I'm not aiming at anything in particular (other than mastering the content in the books we have), because I no longer know what to aim for - I'm just keeping the status quo until I know where to go next.

Yeah, structure is both needed and hard for me to implement. (I love to plan, but making plans that are actually doable by the real me (as opposed to the ideal me), well, it's a work in progress ;).). Teacher intensive stuff inevitably becomes twice a week instead of every day, and I want to nip that in the bud wrt the three Rs. And this thread is helping me organize my thoughts and evaluate what really does get done and what doesn't. That said, I really do think there is *something* different with dd8. The writing only blossomed this week, really. And she's spelling based on pure visual memory - and that's just blossomed this week, too. But she can't segment anything beyond a cvc word, and that only just. And she's noticed a difference between herself and others when it comes to spelling and writing. And whether flighty and normal or flighty and exceptional she needs to learn to calculate - I just want to feed her interest in math at the same time, kwim? But idk, hitting dd8's sweet spot wrt skills practice is *hard* and always has been, in a way that it just isn't for dd6. Not so much a matter of intelligence but a matter of having a narrow optimum window, and non-optimum is *really* not optimum. Oh, idk, I really don't think I'm making it all up, but at the same time it all might be taken care of by steady daily taking one's medicine skills practice. Which needs to be done in any case, so it makes a good focus. (Is it better or worse that the majority of the programs on my shelf were bought ahead of time and have just never been used instead of bought, tried and discarded? Math has been SM with varying levels of modification and/or LoF, and spelling has been copywork, LiPS, and going through the Let's Read (minus two disastrous weeks with R&S 2 last year). I'm flighty, but not nearly as flighty as my list of curricula owned makes me sound ;); I stress *buy* at the drop of a hat, but I don't jump ship nearly as quickly. Although they aren't all useless - I do regularly pull ideas from them and incorporate them into my teaching. And it mostly has worked. It's just a fair bit of effort. And I'd sort of like to be able to do a program pretty much as written and have it pretty much work. Or at least have the *option* to do so. SM is a constant dance of flying ahead or parking for weeks, and it's getting to be a bit much at the moment. I don't think it's *all* me, but it's probably partly me. Idk, the winter kills me and it all comes to a head in Feb/Mar. Whatever, I'm now armed with a plan that can be effective whether it's all in my head or not ;).