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Thoughts on Math Theory/Instruction


stm4him
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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. 

 

What are the "Big Questions" and who decides which ones make the cut?

One of the fundamental philosophical question for many is whether there are rational explanations for natural phenomena and whether humans can, in principle, know - a very old and definitely not modern question.

This is intrinsically linked with math, since for those who answer "yes" to this question, math is a vital tool for this understanding.

 

I still do not see how philosophy would alter the way one teaches fractions, or calculus. Can you give an example so I can understand what you mean?

 

No time for more, gotta run and use math.

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I don't think it's necessarily that the curriculum is flawed.  Perhaps, it is that the curriculum is just wrong for that particular student?  Singapore was a perfect fit for eldest DS, and I was pretty convinced that it would be an appropriate curriculum choice for just about any other average to above average math student.  But it (so far) is not a good match for middle DS.  Its "flaws" are more apparent with him, than they were with DS. 

 

The thing is...the aspects of Singapore that are "flaws" for middle DS...were positives for eldest DS.  

 

May I ask what flaws you see in Singapore?

 

 

 

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. 

 

I just wanted to say thank you for responding to my earlier question, and for continuing the discussion. You have helped me to clarify and order my thoughts.

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What are the "Big Questions" and who decides which ones make the cut?

 

 

One of the trauma recovery rights I was taught is "I have the right to be 'wrong' and do it anyway."

 

There is a school of thought among some trauma recovery "experts" that we all are unique individuals of equal worth and we get to decide for ourselves how we want to live our lives. We don't need permission, and even if we did need permission, it doesn't have to be based on proving that we are "right" about what we want.

 

One weapon commonly used on domestic abuse victims is that they must prove their abuser's belief system to be wrong. A black and white scenario is created where the victim must prove the abuser's belief system wrong or adopt it fully no matter the consequences to the victim.

 

Whether an abuser, a concerned family member, or a society's expectations, I've been taught I have the above right. It's a universal right that is about ME, not who is challenging my beliefs and choices.

 

So, who gets to decide about big questions and cuts? I do! I get to decide for ME. Even if I am "wrong". It's not about proving that I am RIGHT; it's just all about ME.

 

Now, what about the kids of us whack jobs? Statistically our kids that are "rescued" by the public schools and foster care seem to do worse than our kids who are not rescued and left in our incompetent hands, so statistically people should just let us keep being whack jobs even when we have kids. Some of our kids follow us to be whack jobs and others move into mainstream society pretty unscathed and sometimes even brilliant and incentive and ground breaking. I believe society as whole benefits more than it is harmed by whack job families. Even if we don't use Asian maths or teach calculus. Maybe BECAUSE we don't use Asian maths or teach calculus.

 

I once read a quote that said something like this: It doesn't matter what you do, as long as you do what you do with all your might.

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What are the "Big Questions" and who decides which ones make the cut?

One of the fundamental philosophical question for many is whether there are rational explanations for natural phenomena and whether humans can, in principle, know - a very old and definitely not modern question.

This is intrinsically linked with math, since for those who answer "yes" to this question, math is a vital tool for this understanding.

 

I still do not see how philosophy would alter the way one teaches fractions, or calculus. Can you give an example so I can understand what you mean?

 

No time for more, gotta run and use math.

 

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.

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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.

 

See, to me that was never an issue since in the K-12 math education in my home country, proofs are introduced as early as 6th grade and used throughout. For me, there is no dichotomy between "proofless" school math and "real" math reserved for university students. There is nothing about the idea of a geometric proof that can not be understood by a 6th grader - provided the teacher himself understands it (which seems to be a problem in this country). I am still not sure why this is deemed developmentally inappropriate in the US, since I can not fathom why American brains should be developing at a slower rate than brains of children in other parts of the world.

 

A math program that simply hands the student equations to plug and chug instead of deriving (or proving - I don't care in which direction this is done) in my opinion does not teach math; it encourages memorization of seemingly unrelated statements because without proof/derivation, there can be no long term retention. That is not a philosophical issue, but a mere practical one.

 

 

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.

 

My view? Arithmetic with integers is very specific and intuitive and is taught at an age when formal proofs are not age appropriate. Instead of proofs, visualization and manipulatives can make it clear to children that what they are doing makes sense.

With fractions, a certain level of proving/deriving is necessary - otherwise you end with the multitude of students who remember "flipping" but have not the slightest idea why to flip (and thus forget when to flip and have again, no long term retention.

These do not have to be mathematically rigorous proofs, because an axiomatic approach to arithmetic, again, is not age appropriate.

 

 

 

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 truth, 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 disagree, because pure procedural teaching does not lead to long term retention. Only relationships that have been understood, i.e. proved or derived, can be reproduced in the long term.

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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.

 

Ehhh...

 

Ok...this discussion has gone way over my head, lol.  So bare with me if my comments seem out of place.  

 

That said...I take issue with this particular comment.  Just a little though.  

 

Personally, I don't believe that humans can know the fundamental truths of the world with absolute certainty because there are an infinite number of possibilities and it is impossible for us to consider them all.  We may have our belief systems, and believe to our core that those belief systems are truth...but absolute truth?  

 

That doesn't mean I disregard the possibility of absolute truth...certainly, I hold that there MUST be an absolute truth because the alternative is that there is not an absolute truth, and that in and of itself is...an absolute truth!  Unless you consider the possibility that an absolute truth and many truths can occur concurrently.  This belief is an absolute truth as well, though, isn't it?  

 

Anyways...my point.

 

Even though I do not believe that humans can know fundamental truths of our world with absolute certainty, I hold math in no less esteem.  I see math as an avenue for the exploration of the "theories" of the truths of our world.  Proofs have value to me, if only because they reflect our understanding of the world insofar as much as we understand it RIGHT NOW.  But I am open to the idea that there may come a time when a proof is no longer valid.  

 

 

This past school year, the kids and I have been studying the Middle Ages...a time period that began with humans believing 100% that the world was flat and the earth was the center of the universe. By the end of the Middle Ages...everything that we knew, believed, and understood about our world was completely flipped upside down.

 

It does not escape me that such revelations are still possible.  WHAT IF things really were like Hitchhiker's Guide to the Galaxy?  WHAT IF God created this planet AND OTHERS and has multiple "experiments" ongoing?  WHAT IF there is a parallel dimension?  

 

You cannot definitively prove that these things do not exist and/or are not possible.  Thus...the possibility remains that they are possible.  

 

Still...this philosophical approach does not prevent me from embracing what we currently know, and think to be truth.  

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I thought things were retained through intensity, duration, and/or frequency.  So if someone does procedural math (even without full understanding) for long enough and often enough, you still believe they can not retain it?

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Personally, I don't believe that humans can know the fundamental truths of the world with absolute certainty because there are an infinite number of possibilities and it is impossible for us to consider them all.  We may have our belief systems, and believe to our core that those belief systems are truth...but absolute truth?

...

Even though I do not believe that humans can know fundamental truths of our world with absolute certainty, I hold math in no less esteem.  I see math as an avenue for the exploration of the "theories" of the truths of our world.  Proofs have value to me, if only because they reflect our understanding of the world insofar as much as we understand it RIGHT NOW.  But I am open to the idea that there may come a time when a proof is no longer valid.  

 

 

This past school year, the kids and I have been studying the Middle Ages...a time period that began with humans believing 100% that the world was flat and the earth was the center of the universe. By the end of the Middle Ages...everything that we knew, believed, and understood about our world was completely flipped upside down.

 

It does not escape me that such revelations are still possible.  WHAT IF things really were like Hitchhiker's Guide to the Galaxy?  WHAT IF God created this planet AND OTHERS and has multiple "experiments" ongoing?  WHAT IF there is a parallel dimension?  

 

You cannot definitively prove that these things do not exist and/or are not possible.  Thus...the possibility remains that they are possible. 

 

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.)

 

 

 

Still...this philosophical approach does not prevent me from embracing what we currently know, and think to be truth.

 

 

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?

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Just because I believe that math should aid in proving absolute truth doesn't mean that I believe it to be developmentally appropriate to teach proofs in elementary school.  It also doesn't mean that it would be inappropriate for all 11 year olds.  In my mind, if a child can't do basic arithmetic thoroughly (with or without understanding and regardless of whether they just have never been taught or is not developmentally ready), then they should not be taught to do geometric proofs until that base is solid.  But I do believe it is a valid thing to study when the child is ready, as I do logic.  Just because one believes that a certain aspect of math is vital to the goal of math does not mean that it must be implemented at the lowest levels of instruction.  The other levels of instruction should be leading a child to have the ability to do proofs, though, if that is the chosen means to proving absolute truth; however, I think absolute truth can also be communicated in arithmetic.  Or at the very least, attributes of God can be illustrated through arithmetic.  

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Just because I believe that math should aid in proving absolute truth doesn't mean that I believe it to be developmentally appropriate to teach proofs in elementary school.  It also doesn't mean that it would be inappropriate for all 11 year olds.  In my mind, if a child can't do basic arithmetic thoroughly (with or without understanding and regardless of whether they just have never been taught or is not developmentally ready), then they should not be taught to do geometric proofs until that base is solid.  But I do believe it is a valid thing to study when the child is ready, as I do logic.  Just because one believes that a certain aspect of math is vital to the goal of math does not mean that it must be implemented at the lowest levels of instruction.  The other levels of instruction should be leading a child to have the ability to do proofs, though, if that is the chosen means to proving absolute truth; however, I think absolute truth can also be communicated in arithmetic.  Or at the very least, attributes of God can be illustrated through arithmetic.  

 

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.

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It was in reference to your earlier post.  I hope I did not come off as offended.  I actually appreciated your clarifications because they were fuzzy to me in theory and I think I understand now what you are saying (and to a certain degree agree with you.)  I do see that it matters how we teach math and what our fundamental beliefs are, but I honestly don't know how to evaluate which programs do or do not line up with mine.  In other words, I haven't seen a math program that obviously DOESN'T line up with mine.  Do you have suggestions about how to know?  I have seen many people with worldviews that I believe are very similar to mine choose different math programs and I have never considered that perhaps one of us is using a program which doesn't actually line up with our worldview.  

 

Thank you for your help and your deep insight.  I do not feel adequate to engage on this topic at the level some of you are able to, and while I think I understand your underlying point(s), I have no idea how to work it out practically.  

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It was in reference to your earlier post.  I hope I did not come off as offended.  I actually appreciated your clarifications because they were fuzzy to me in theory and I think I understand now what you are saying (and to a certain degree agree with you.)  I do see that it matters how we teach math and what our fundamental beliefs are, but I honestly don't know how to evaluate which programs do or do not line up with mine.  In other words, I haven't seen a math program that obviously DOESN'T line up with mine.  Do you have suggestions about how to know?  I have seen many people with worldviews that I believe are very similar to mine choose different math programs and I have never considered that perhaps one of us is using a program which doesn't actually line up with our worldview.  

 

Thank you for your help and your deep insight.  I do not feel adequate to engage on this topic at the level some of you are able to, and while I think I understand your underlying point(s), I have no idea how to work it out practically.  

 

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? ;)

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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?

 

Actually I don't think modern mathematics does say these things are unrelated.  But there is more than one school of thought among mathematicians about what math is, in a metaphysical sense.  It isn't uncommon to find mathmaticians who are platonists, who would see mathematical concepts as forms that exist in some way in the divine mind. 

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I thought things were retained through intensity, duration, and/or frequency.  So if someone does procedural math (even without full understanding) for long enough and often enough, you still believe they can not retain it?

 

Yes, I know people like that. What I think you can't get under this model is much in the way of creativity.

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I thought things were retained through intensity, duration, and/or frequency.  So if someone does procedural math (even without full understanding) for long enough and often enough, you still believe they can not retain it?

 

Yes. I believe that they will not retain it in the long term.

Just ask around adults who can still use the quadratic formula.. most won't be able to, since they only memorized it without understanding. The ones who understood where it came from can derive it within a few minutes.

 

It is impossible to master math on a memorization basis. The students who attempt that struggle terribly in higher math.

 

This is not to say that a certain automatization in computation is unnecessary; of course it is necessary. But in my job I see plenty of evidence that procedural math without understanding does not lead to long term retention.

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Good golly Miss Molly, some of this is over my head.  :svengo:

 

I  think arithmetic is beautiful, though, like I think phonics is beautiful. I'm a minimalist and I find the greatest beauty in the basics and the plain and simple things in life. Often a very simple pen and ink sketch is more beautiful than a complicated painting, to me. A crusty loaf of fresh bread, some sweet grapes, and piece of perfectly aged Swiss cheese is often more delicious to me than an elaborate dinner. When all the superfluous is stripped away, I feel like we can more clearly discern quality.

 

I don't think we all need the equivalent of a symphony in math. Some of us are content and sated with a little folk music.

 

And ARE all children equally ready for certain topics at the same ages? Maybe not, depending on their genetics, environment, diet, access to medical care, habits, etc. I don't know for certain that they are. Theory is great, but when I have boots on the ground and see that a student is not ready for something or that they need more of something else, I really don't give two hoots about theories of what other students are or are not doing. I gotta just deal with what is right in my face. A person. A unique individual with unique needs. And a very flawed me that is sometimes the only one standing in a huge gap that no one else is willing to fill.

 

The first time I remember being shamed for not being good enough at standing in the gap, I was 3 years old and being blamed for another child's near drowning that left him with brain damage. It guess I didn't respond to his disappearance correctly and quickly enough, I heard one adult telling some others that and no one stuck up for me. It appeared that everyone agreed I was at fault and the negligent one. I've been blamed and shamed so many times for so many reasons since then for not being able to stand flawlwssly in the huge gaps I so often find myself alone in.

 

Math is one of those things I am SO sick of seeing moms being shamed about! Enough is enough of this! There is a place for symphonies and wall sized paintings, but we don't ALL need to work on math of that scale. Life is really short and hard for some of us. We need to triage and get REAL about what is in front of us. 

 

Not all homeschoolers have the backgrounds and resources of the more privileged here. Not all homeschoolers are homeschooling to beat what the local PS are doing, or God forbid what a PS is said to be doing half way around the world. This is a big country and HUGE world. There is a LOT going on that is a LOT bigger than math. Math only takes over our lives and homeschool/tutoring if we let it. It's a CHOICE to let math get that big. It's a choice I refuse to make anymore, no matter who shames me.

 

:001_tt2:  to the idea of STEM centered education.

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Good golly Miss Molly, some of this is over my head.  :svengo:

 

I  think arithmetic is beautiful, though, like I think phonics is beautiful. I'm a minimalist and I find the greatest beauty in the basics and the plain and simple things in life. Often a very simple pen and ink sketch is more beautiful than a complicated painting, to me. A crusty loaf of fresh bread, some sweet grapes, and piece of perfectly aged Swiss cheese is often more delicious to me than an elaborate dinner. When all the superfluous is stripped away, I feel like we can more clearly discern quality.

 

I don't think we all need the equivalent of a symphony in math. Some of us are content and sated with a little folk music.

 

And ARE all children equally ready for certain topics at the same ages? Maybe not, depending on their genetics, environment, diet, access to medical care, habits, etc. I don't know for certain that they are. Theory is great, but when I have boots on the ground and see that a student is not ready for something or that they need more of something else, I really don't give two hoots about theories of what other students are or are not doing. I gotta just deal with what is right in my face. A person. A unique individual with unique needs. And a very flawed me that is sometimes the only one standing in a huge gap that no one else is willing to fill.

 

The first time I remember being shamed for not being good enough at standing in the gap, I was 3 years old and being blamed for another child's near drowning that left him with brain damage. It guess I didn't respond to his disappearance correctly and quickly enough, I heard one adult telling some others that and no one stuck up for me. It appeared that everyone agreed I was at fault and the negligent one. I've been blamed and shamed so many times for so many reasons since then for not being able to stand flawlwssly in the huge gaps I so often find myself alone in.

 

Math is one of those things I am SO sick of seeing moms being shamed about! Enough is enough of this! There is a place for symphonies and wall sized paintings, but we don't ALL need to work on math of that scale. Life is really short and hard for some of us. We need to triage and get REAL about what is in front of us. 

 

Not all homeschoolers have the backgrounds and resources of the more privileged here. Not all homeschoolers are homeschooling to beat what the local PS are doing, or God forbid what a PS is said to be doing half way around the world. This is a big country and HUGE world. There is a LOT going on that is a LOT bigger than math. Math only takes over our lives and homeschool/tutoring if we let it. It's a CHOICE to let math get that big. It's a choice I refuse to make anymore, no matter who shames me.

 

:001_tt2:  to the idea of STEM centered education.

 

Thank you, Hunter.  :)

 

(says the woman who has taught both conceptual and procedural, in varying doses, for various kids, at various times.  It all seems to be working out okay so far. I don't think it has to be all or nothing...)

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Amen Hunter.  Amen!  

 

 

 

And....I agree with Regentrude regarding procedural vs conceptual.  Truth is...without that conceptual understanding AND practice/use of the procedures...the procedures AND concepts will generally be forgotten.  At least...the more complicated math procedures, anyways.

 

Don't believe me?  Hop onto Khan Academy and start their World of Math course.  You might be surprised how many math topics you've forgotten!  

 

Even more humbling...start the 3rd grade math journey.  Don't ask me how I know.  

 

 

But...once again....all of our kids don't necessarily need to retain all of those math procedures into adulthood.  How often do we find equivalent fractions?  How often do we convert fractions to decimals?  Etc.  

 

This has been a fascinating discussion...lol.  

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But...once again....all of our kids don't necessarily need to retain all of those math procedures into adulthood.  How often do we find equivalent fractions?  How often do we convert fractions to decimals?  Etc.  

 

Well, yes and no -- as long as you're sure they've already passed all the math they will ever need in their life, I guess it's ok, but I teach developmental math at a college, and there are a lot of students for whom even my developmental classes are too rapidly paced.

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Well, yes and no -- as long as you're sure they've already passed all the math they will ever need in their life, I guess it's ok, but I teach developmental math at a college, and there are a lot of students for whom even my developmental classes are too rapidly paced.

 

 

Also a valid point.  We really don't have an absolute idea just how much math our kiddos will need as they become adults, choose careers, etc.  

 

There really isn't an easy answer here, is there?  

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Also a valid point.  We really don't have an absolute idea just how much math our kiddos will need as they become adults, choose careers, etc.  

 

There really isn't an easy answer here, is there?  

No. All we can do is the best we can and cross our fingers and hope for the best. If it wasn't good enough, that's what developmental classes are for -- and honestly, a kid who can read and understand what they read can usually catch up, although it might take a few trips through the lowest developmental class.

 

However -- I do believe that if I had a student who just.did.not.get. conceptual teaching, I would absolutely work on procedural teaching with them, although I might continue conceptual work at a far lower level. But I wouldn't stop based on "they won't need this" until it either becomes obvious that college is not in the cards, or they pass a dual enrollment college algebra class (this transfers for most colleges and non-stem majors). And I would default to a procedural (edit -- I mean conceptual) method of teaching unless it became obvious that it really wasn't working for this kid.

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I have also seen kids who were supposedly advanced in math because their parents had focused on conceptual understanding and exposed them to higher level concepts earlier than most. However, it became obvious over time that there were big holes in their foundation. They didn't even know their math facts in fourth and seventh grade but because they understood the facts they moved on. So maybe they understood higher level concepts, but they were still crippled by their lack of drill in the basics.

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But I think it would be very helpful to find kids who did get a solid math education and who weren't naturally good at math and find out what programs their educator(s) used to get them there. I think it is easy to find kids from all math backgrounds and point out where they are lacking and blame it on the way they were taught.

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However -- I do believe that if I had a student who just.did.not.get. conceptual teaching, I would absolutely work on procedural teaching with them, although I might continue conceptual work at a far lower level. But I wouldn't stop based on "they won't need this" until it either becomes obvious that college is not in the cards, or they pass a dual enrollment college algebra class (this transfers for most colleges and non-stem majors). And I would default to a procedural method of teaching unless it became obvious that it really wasn't working for this kid.

 

And to the bolded, I have written this before: it's not just students who go to college who need math.

My HVAC installer told me he needs trigonometry.

That's not some fancy STEM career - it's a trade, which is often suggested here for kids as an alternative to college. But that does not mean they won't need math. Electricians, for example, need a lot of math as well.

 

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But...once again....all of our kids don't necessarily need to retain all of those math procedures into adulthood.  How often do we find equivalent fractions?  How often do we convert fractions to decimals?  Etc. 

 

But we don't know what they want to do with their lives.

That's what irks me about the argument "they won't need xyz". It's parents making assumptions and limiting their kids' choices.

It's not that a kid could not re-learn some math later; it's that this kid likely will not even consider a future in which math could play a role.

 

It's not about everybody needing to have a STEM career - it's that I do not believe it is the parent's call to decide which possible careers a child may or may not aspire to.

Many people don't need to write extensively, but I don't hear people arguing we don't need make sure kids write well "because they won't need that as adults".

 

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But we don't know what they want to do with their lives.

That's what irks me about the argument "they won't need xyz". It's parents making assumptions and limiting their kids' choices.

It's not that a kid could not re-learn some math later; it's that this kid likely will not even consider a future in which math could play a role.

 

It's not about everybody needing to have a STEM career - it's that I do not believe it is the parent's call to decide which possible careers a child may or may not aspire to.

Many people don't need to write extensively, but I don't hear people arguing we don't need make sure kids write well "because they won't need that as adults".

 

I do not believe it's the average parent's responsibility to prepare every child for every career. Parents are just people, not some kind of superhuman robots.

 

No one, not even the rich and connected, can prepare a child for EVERYTHING. Some of us just have to give up on more things earlier.

 

Parents who think they are providing EVERYTHING probably have a limited idea of all the potential careers in this world. At the very least I doubt they are preparing their children to be a coconut picker or seal hunter.

 

https://www.youtube.com/watch?v=6Ds8ihihF6Q

 

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This thread actually got me to finally join the Hive!    Just wanted to share some math thoughts and experiences here FWIW.

 

 

 

I grew up in a rural public school that used Saxon math.   I graduated from high school 20 years ago.   There were only 58 people in my graduating class.    I generally had the same 20 or so people in MOST of my classes.   Back then we were "segregated" (for lack of a better term) by grade performance.   There was accelerated math, English, etc. and thus we nerds were in most classes together.  

 

I remember that after we had taken our first ACT in high school, it was announced in an assembly that 24 people (in my small class of 58) scored above a 32 in the MATH portion of the ACT.    Back then the highest one could make in each of the four areas of the ACT was a 36 (it may be the same now).    I remember this distinctly because a big deal was made of it both at our school and in the education community at large.   Our results had prompted several of the surrounding schools in the tri-county area to convert to Saxon because of our school's performance on the state/college testing IN THE AREA OF MATH. And it wasn't only our class but several of the graduating classes before us who had grown up with Saxon math.   It was becoming more and more obvious each year that Saxon was making a difference; it was just very apparent with my particular class.

 

I honestly don't remember what Math curricula we used in early elementary (grades 1 to 4) but I do remember using Saxon beginning around 5th grade. 

 

I myself was NOT the best at getting the concepts in advanced math (the WHYs) as compared to my nerdy peers.  I was a what my chemistry teacher called a "memorizer" == he used it derogatorily to describe my propensity to (albeit successfully) apply the same "pattern" of solving a problem to all problems of its kind to derive the answer.  As was demanded in Saxon, I was made to show my work which FORCED me to get what was going on conceptually more so than I would have otherwise.     

 

I myself made a 34 on the Math section of the ACT (I didn't make this high in ALL the other areas).   This is quite a feat as math was NEVER my strongest subject (but then again, I was surrounded by some major brainiacs in math and felt like a dunce by comparison and rarely scored as high as they did in the subject over the years).   Most of my these 'nerdy classmates' have very successful careers and most of them chose the math/science fields; I think this is chiefly because our school enjoyed a great math and science program.   

 

I DO remember VIVIDLY (in 2nd grade in particular) doing TONS of the timed math tests.   I remember competing with classmates with these timed math fact tests --- doing them daily, feverishly, competitively !    Math "drill and kill" was most definitely a HUGE component to our early math education.   Again, I don't know if this was so much SAXON as it was that we were lucky enough to have had some old "Battle Axes" / "WarHorses" (my words for great, old-school, no-nonsense teachers).

 

Years later this served us well, because when we were, for example, taking the ACT --- we weren't having to use our fingers to calculate the required "math facts" of the questions and thus having to devote precious time to this "little stuff" that should NOT be subtracting from one's limited time one has for this test! We had been required in our early years to drill in our math facts; thus, while we took the ACT/SAT, we were ABLE to devote the quickly-fleeting time to THINKING about the harder concept that the test was trying to "test" and to use that time-saved to answer all the problems and had time to go back and check over the tough ones.  That's just one example, where the "drill and kill" served us well.    

 

It's not just in the testing arena where that serves one well.    From a practical standpoint,  I'm able to triple/half recipes in my head easily; I never need paper/calculator for such!  I can figure a tip in my head; I can rattle off most math "memory" works to this day (I graduated from high school 20 years ago).   (i.e. "percent times total equals part", etc.) quite quickly.    I use the basic ratio to solve math all the time!  My "everyday" math skills have served me well and I feel I owe a debt of gratitude, in a large part, to Saxon.    And, again (echo echo echo)..I'm NOT even  math-y!

 

My handful of 20-ish classmates and I are where I hope my son ends up in math education.    

 

At our last class reunion, as we sat around chatting past midnight, this subject came up and we agreed that our history education was lacking; it was taught to us by coaches who, except for one, didn't care a hoot about teaching this important subject.   But we all concurred, hands down, about the efficacy of our math teachers and use of Saxon.    

 

Two of my best girlfriends (who are two of these 'nerdy classmates') --- one who has her Master's degree in Math and was employed by one of the nation's largest Naval design firms in the world to calculate weight distribution of war vessels (you have to KNOW math to do this!) and the other who was an civil engineer --- both (as I did) left their respective careers with first pregnancy to be stay-at-home moms and who now homeschool --- we all use Saxon!  (....ignore this run-on sentence here!!!)

 

If my "math" success were just my own, then I wouldn't have felt compelled to write all of this on here; but the fact that so many of my classmates scored ABOVE 32 on the ACT's Math section  --- I think that's significant.      And, this wasn't a private school; this was a public school.   Our parents weren't the phDs, doctors, lawyers --- we weren't the socioeconomically elite.   We were "Archie and Arlene Average" from a small town on the Mississippi Gulf Coast who were blessed by great, caring teachers who had the wisdom, IMO, to agree to use Saxon.  

 

 

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I do not believe it's the average parent's responsibility to prepare every child for every career. Parents are just people, not some kind of superhuman robots.

 

No one, not even the rich and connected, can prepare a child for EVERYTHING. Some of us just have to give up on more things earlier.

 

Parents who think they are providing EVERYTHING probably have a limited idea of all the potential careers in this world. At the very least I doubt they are preparing their children to be a coconut picker or seal hunter.

 

Leaving some things out cuts off far more careers than others.

 

If a child cannot read and understand what they read, they may be able to be a coconut picker or a seal hunter, but there are a lot of other things that are not going to be in the cards without a LOT of remedial work. (and quite honestly, even as a mathematician I *do* consider reading more important and that a lack of reading closes more doors).

 

If a child can't do basic arithmetic, this *also* closes a large number of doors, although nowhere near as many.

 

If a child doesn't hit trigonometry or calculus, this closes far fewer doors. If they *do* need them, there are classes at the CC -- as long as they can do the basic arithmetic/algebraic reasoning necessary to actually succeed in those classes. For a neurotypical kid, almost any program will get them there if it is completed conscientiously and extra help is brought in when a kid is struggling with an area.

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This thread actually got me to finally join the Hive!    Just wanted to share some math thoughts and experiences here FWIW.

 

I grew up in a rural public school that used Saxon math.   I graduated from high school 20 years ago.   There were only 58 people in my graduating class.    I generally had the same 20 or so people in MOST of my classes.   Back then we were "segregated" (for lack of a better term) by grade performance.   There was accelerated math, English, etc. and thus we nerds were in most classes together.  

 

.....

 

If my "math" success were just my own, then I wouldn't have felt compelled to write all of this on here; but the fact that so many of my classmates scored ABOVE 32 on the ACT's Math section  --- I think that's significant.      And, this wasn't a private school; this was a public school.   Our parents weren't the phDs, doctors, lawyers --- we weren't the socioeconomically elite.   We were "Archie and Arlene Average" from a small town on the Mississippi Gulf Coast who were blessed by great, caring teachers who had the wisdom, IMO, to agree to use Saxon.  

 

Welcome CrispyBiscuit!  :grouphug:  :party:

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I do not believe it's the average parent's responsibility to prepare every child for every career. Parents are just people, not some kind of superhuman robots.

 

Hunter, I'm afraid I'm not understanding - maybe you could flesh this out a bit?  What math preparation requires superhuman efforts?  Are you referring to higher levels (which ones?) or curricular depth or something else? (And, while we're at it, what's a STEM-centered education?)

 

With regard to a STEM career, "good enough" is the standard math track as appropriate for the student's ability level.  Whether high school ends at precalc or at calc may affect things like selective college admissions, but not whether the doors are open to a possible STEM career.  Ending before precalc would probably add a layer of logistical difficulty.  Are you chafing at debates about depth levels?  I'm trying to figure out the source of apparent irritation...  I'm probably off-base?

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Hunter, I'm afraid I'm not understanding - maybe you could flesh this out a bit?  What math preparation requires superhuman efforts?  Are you referring to higher levels (which ones?) or curricular depth or something else? (And, while we're at it, what's a STEM-centered education?)

 

With regard to a STEM career, "good enough" is the standard math track as appropriate for the student's ability level.  Whether high school ends at precalc or at calc may affect things like selective college admissions, but not whether the doors are open to a possible STEM career.  Ending before precalc would probably add a layer of logistical difficulty.  Are you chafing at debates about depth levels?  I'm trying to figure out the source of apparent irritation...  I'm probably off-base?

 

I'll try and answer this tomorrow. I have to deal with some IRL stuff today. Sorry.

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BTW, I don’t actually think the textbook used is as important as the teacher’s knowledge and ability to explain.  When I taught, I used Singapore for some kids and Saxon for others.  I really like the vintage Hamilton’s Arithmetic and Wentworth-Smith School Arithmetic series.  But whatever actual curriculum or text I use(d), I actually teach the concepts.  I supplement with word problems (when I taught these were from CWP), puzzles and mind-benders (again Singapore IP is great for that).  I  throw in some living math books such as The Giant Golden Book of Mathematics.  I use methods gleaned from various sources (such as Elementary Mathematics for Teachers).  Point being: I use the textbook as a tool, not as the teacher.  There is not an “ideal†curriculum if you don’t know the content well enough to teach it.  Yes, at a certain point the student can take ownership of their learning and the parent can become a co-learner/facilitator, but the basic groundwork needs to be taught.

 

OK, off to catch up on the other posts I've missed.

 

Would you mind sharing some other examples of living math books?  I suspect DS would greatly appreciate them, lol.  

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That said, I do want to point out that I am struck by the false dichotomy of the choices: arithmetic vs. conceptual, vintage vs. modern, spiral vs. mastery.  You are the teacher: you make what you will of a given set of materials.  In my mind to separate out arithmetic from the greater field of mathematics is a little like the dieticians who tell us to discard the yolks and eat the whites or vice versa.  When I introduce solids to my babies I give them the yolk first.  But, the goal is for them to ultimately eat the whole egg.

 

I just wanted to highlight this part of your long, interesting post.  Program choices are tools and while we can compare the advantages and disadvantages of various tools, it is usually more difficult to compare what the teacher is adding to the instruction (or not adding) or how the teacher is using the tool for the inherently unique individual student.

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And to the bolded, I have written this before: it's not just students who go to college who need math.

My HVAC installer told me he needs trigonometry.

That's not some fancy STEM career - it's a trade, which is often suggested here for kids as an alternative to college. But that does not mean they won't need math. Electricians, for example, need a lot of math as well.

 

Yup.  My husband works for the government in a trade combines aspects of both HVAC and construction among other things. 

 

His position doesn't require a college degree, but he has to understand and daily use a lot of higher math in his job.  He's in a supervisory role and every single time he's hired someone to work in his department he's had to reteach them math to one degree or another.  It used to surprise him, but now he just assumes that he's going to have to give new hires the math they need.

 

I rely on his opinion A LOT when it comes to working on math with the girls.

 

My own minimum goal for my children is to have enough higher math so that they can go to one of our many reputable state universities without requiring remediation.  For us that means Algebra 1, Algebra 2/Trig, and Geometry.   How we get there will look different for each of them (one needs full on conceptual, one needs "show me how before you show me why"), but they'll both be fine. 

 

I guess my theory of math instruction is built on the idea that it's not the journey it's the destination.  As long as they end up with the skills they need to be successful, who cares if they used Saxon or AOPS?  Who cares if all their arithmetic is from Rod and Staff or Singapore or Teaching Textbooks?  Use what works for the individual student, use it consistently, and don't sweat it.  That philosophy is the only way that Math Mammoth (my favorite math) can peacefully co-exist with Teaching Textbooks in our home. :) It's part of the beauty of home schooling.

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I just skimmed through this thread and have had many responses to different sections which have been covered by others. I feel very strongly that it is my responsibility to give my children the very best maths education that I and they are capable of. For my kids that means ploughing through some pretty advanced topics at home but that doesn't mean that I think everyone needs to. But I do want to point out that their high school math education has been far cheaper monetarily than earlier years simply because we use old used textbooks and free classes from MIT and Coursera as opposed to standard curriculum.

 

This thread also made me think of my father who would be 95 now. His education was a bit sketchy overall but in math he was rock solid through geometry which he used in various trades throughout his life. One of his hobbies was investing in the stock market rather successfully. He was a rather gruff man and one of my favorite memories of him was when we spent a day comparing what I (finance and accounting major) had been taught in college about the stock market to his methods. It was fascinating, my dad had notebooks filled with how he had made his decisions. Pages filled with long calculations that matched my quick formulas every single time. We both knew he had been successful but we hadn't realised that he was actually already doing what he sent me to college to learn! Understanding the concepts is what is important!

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Well, now I feel better about choosing either Right Start Activities for the ALAbacus and How to Tutor to either Saxon Intermediate 3 or Strayer-Upton.  Maybe from Strayer-Upton back into Saxon Alg 1/2 or 1.  Either route sounds that it will produce good results.  Right now I am giving my ten year old a choice and he has chosen to stay with Saxon for now.  He likes working independently and then coming to me.  My oldest is open to trying S-U but I need to wait and order another copy so we can each hold our own and she is on a trip right now so I have some time.  We will probably try it in April and see how she does.  My third one is doing S-U and we are practicing speed in easy addition problems.  The fourth one will begin Activities for the ALAbacus when I can order the 2nd edition teacher manual (I already have the workbooks) or find my copy in the garage (which I can't remember which edition I own).  And she will start How to Tutor when I can print off the pages, probably next week.  For now we are practicing writing out and singing our skip counting.  The fifth one will soon begin learning to write her numbers from 1-100 and counting that high.  She is pretty good with the counting but needs help here and there.  When she can write it all out I will make sure she recognizes them individually using number cards and go from there into ALAbacus Worksheets and How to Tutor as well.  

 

Thanks for all of the opinions.  I really liked your post, ltlmrs.  It reminded me of my aspirations.....back out of reality and into my dream.  I need that sometimes too.

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And to the bolded, I have written this before: it's not just students who go to college who need math.

My HVAC installer told me he needs trigonometry.

That's not some fancy STEM career - it's a trade, which is often suggested here for kids as an alternative to college. But that does not mean they won't need math. Electricians, for example, need a lot of math as well.

 

Yes, a lot of not-science related jobs require math.

 

A friend of mine is a welder, and she uses quite a bit of trigonometry as well.  My dad was a merchant sailor on oil tankers, and I have an elderly friend who was a merchant sailor on all kinds of different cargo ships, and both have said that mathematics was a big part of their jobs.  My uncle, who had gone to university an the arts, had to get my grandfather to bring his math up to snuff so that he could go to trade school to become a carpender.  My grandfather was in the navy and never went to university, but worked repairing helicopters - also using a lot of math.

 

I also know quite a few people who found they needed to improve math skills to run their own businesses.

 

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