The math equivalent of the WTM approach to LA

[forty-two]

I really, really like WTM's LA approach, which to me boils down to teaching explicit skills through living books (a Charlotte Mason thing, which I think of as books that present good ideas in a good - accurate, logical, and aesthetically pleasing - way; in short, good stories, well told). I think of it as having a skills progression list for each LA component (spelling, grammar, writing), a few flexible tools (copywork, dictation, narration, outlining) through which you can teach the skills, using any raw material that you so choose - with the caveat that said material is worth imitating.

 

I've tried for a while to figure out what the conceptual equivalent would be for math. A skills progression is not hard to come by, but what are living books for math? Most "living math books" are twaddle-y stories with rote math shoved in :glare: - an offense to both literature and math. But even with the good ones - ones that present true math in an intriguing way - you still have the question of how do you teach the essential skills through them in a systematic way?

 

With LA, *all* the LA skills are used in any given living book - you can teach whatever spelling rule, whatever grammar rule from any old book you have lying around. Not so with living math books, at least the ones I've seen - you'd need a careful book progression to make it work.

 

But maybe the "book" part of "living book" doesn't really fit with math. Think about it, the end goal of all those LA skills is to be able to read and write well - in effect, to be to comprehend high level written work and to produce written work of your own that, if not high level, is at the very least competent. (An aside: as I think about it, WTM's LA isn't *just* part-to-whole; rather you are immersed in the whole - reading living books - while you systemically learn how that whole is built up; you are getting both the context and the specifics the whole way through.)

 

The end goal of math, however, is *not* to be able to read math in story form and write about math in story form. The end goal of math is either engineering math, the ability to apply math to "real world problems", both to follow others' math and to do the math yourself, or pure math, the ability to comprehend math proofs and write proofs of your own.

 

Thinking of it that way, "living math" would consist of solved problems and proofs, ones that could be comprehended by the student at their current level (probably with help), but not necessarily ones they could solve themselves.

 

Now where do we find such things? And how do we use them to teach our progression of skills?

 

As to finding such things, there's the standard unschooling/living math technique of using daily life to provide problems - talk them out as you solve them. You can solve out of level problems for/with you dc, explaining as you go; I read about one mom who did an SAT math practice problem each day with all her dc from a youngish age, talking them through and asking leading questions. Probably any collection of interesting problems could be used this way - math competition problems would be a great source, I think. Plus you could look for interesting problems in history. I'm just brainstorming, here - I only just thought of this angle.

 

Proofs are a little more interesting ;) - have to get creative here. I have a book of visual proofs that's interesting, though not sure how accessible it is to youngish kids. You can demonstrate proofs with cuisenaire rods and other manipulatives. Geometric constructions might be good. Math history might be a good source.

 

But how to make sure they are *good* problems, not tedious or twaddly? And how to find elegant proofs for the elementary set?

 

As for how to teach math through these sources - what flexible tools to use - I have no idea :tongue_smilie:. Yet ;). But at the least we could go the WTM LA-with-textbooks approach easily enough - where you spend time with living materials, but use textbooks to teach the skills. So that would be basically adding working and talking through out-of-level problems and proofs, preferably interesting or intriguing ones, to a regular math program - letting them see what math can be like, what it can do. Which would be valuable, I think.

 

But I'd love to figure how to actually teach math through those sources (in a way that avoids the problems of new, new math ;)) - the key would be, I think, teaching the skills explicitly and systemically (which progressive math avoids). But what would be the math equivalent of copywork, narration, outlining - the tools by which we interact at an ever higher level with our source material? Given that The Elements is basically nothing more than a collection of solved proofs, and was used as a text for millennia, I bet the traditional approach there would be illuminating - I just haven't yet been able to find it :glare:.

 

Anyway, that's probably enough to be getting on with - thoughts?

[wapiti]

Skimming your post, it sounds like you're looking for a math curriculum (which teaches skills in a systematic way) that's heavy on both concepts and problem-solving. Is that accurate?

 

Most people would recommend SM for that. I like MM. There may be others. I have used neither for the very early grades - for the very early grades (preschool/K/1) I'd go heavier on manipulatives, as you are doing.

 

For post-elementary math (prealgebra and up) that's heavy on concepts and problem-solving, I'd look at AoPS, but it looks like you have a ways to go before you need to worry about that.

 

For "living math" resources, there have been a few threads on that - try a search. I think there's some sort of living math website someplace. Also, for some reason Kitchen Table Math (there are books on the AoPS website) sticks in my mind as possibly involving "living math" though I'm not sure.

 

There are other problem-solving supplementary materials for elementary - the Zaccaro books come to mind first.

 

That's my two cents :). FWIW, I'm always looking for the opposite, trying to apply a math approach to LA :lol:

[Carol in Cal.]

Great post, and good question! I actually think that the grammar of math is arithmetic--four function math facts, and fractions/decimals/percentages and their manipulation. Real math itself doesn't start until Algebra, IMO, although introducing it conceptually (as in with your proofs) is a great assistance to slogging through all the arithmetic. The really sad thing about math is how much arithmetic underlays it, making it really difficult to know whether or not you like or have an affinity for it until you're pretty far along.

[stripe]

Japanese math style consists of heavy dose of working through problems rather than a lecture. Is this what you mean?

 

The UCSMP books for grade 7-9 are based on this approach. The elem books don't look massively different from Singapore, which doesn't mean the classroomstyle of teachers isn't distinct.

 

There are other books that emphasize clear definitions like MEP and Russian math. The UCSMP grade 1-3 math books from USSR are not very talky.

 

I'm not sure it's wise to think one should style elementary math as advanced math style about simple topics, but maybe harkening back to Socrates would be illuminating? Guided discovery?

[forty-two]

I'm not actually looking for a math curriculum in the "here is you set of materials", but in a "here is your plan of attack" way. I have this anti-textbook thing, see - they're not "authentic" :tongue_smilie:. (Also, totally expensive - I just bought 10 books off Amazon for my little homemade world-cultures-through-fairy-tales unit for $60; any curriculum would have added $25-30 - that's cost of *half* of my living books - so imo I gain a lot by learning how to teach without a curriculum.) WTM lays out a great progression for teaching LA without much in the way of curriculum (nothing, if you already knew the skills in question, which sadly, I do not), thus saving you time and money and allowing greater focus on living books - the "real" stuff.

 

So I was theorizing about how to apply the underlying principles of WTM's LA approach to yield a textbook-free, living approach to math. Most of the living approaches I've see are a bit loose for me - I'm not skilled enough to teach to the level I want without a bit more handholding. Plus they mostly still use textbooks, just differently and with extensive enrichment. But mostly their enrichment is not the type I'm looking for.

 

But now at least I know what I *am* looking for :). And I've got back-up plans - a MEP/CSMP combo is my fall-back plan. And a very good plan it is :) - I just have this dream of finding a way to do it without curriculum. And so I'm looking at WTM's LA progression, awesome as it is, for inspiration in my quest :tongue_smilie:.

[forty-two]

WRT problem-based vs lecture - well, sort of. I'm not entirely sure what it is I want - this thread is so I can talk it out :tongue_smilie:.

 

WRT arithmetic vs real math - I've seen arithmetic developed axiomatically (proofy algebra, basically). And Hands-On Equations shows that kids can do algebra, properly presented, earlier than you'd think. So why not a proofy HoE? And various New Math stuff introduces proofy-ish thinking in K-3 with cuisenaire rods (Miquon has some, Math Made Meaningful has a great progression). So I don't want to consign K-6 math to procedural arithmetic entirely. But I'm not sure how to go about it.

 

I guess what I want is to learn math through primary sources, of some sort. But unlike LA, there's not exactly a wealth of primary sources or even quality secondary sources for the younger set. Natural, since they didn't teach it to the younger set.

 

Looking at it from another angle, the LA progression equips you with the tools to go forth and learn from books on your own. What skills are needed to go forth and learn math on your own? Let's say you are working through someone's math-heavy journal article - what do you do? Read through it and work carefully through the math till you think you have it, I suppose. Being able to reproduce the proof from memory, with understanding, would be evidence of having learned it on one level. Being able to apply it to whatever you are doing (why you read the thing in the first place, unless it was just for curiosity; similar to putting different levels of effort in studying a book based on why you read it) would be evidence of learning it on another level.

 

The LA progression kind of has baby steps in learning to interact with the material - what would be the baby steps in learning to interact with new math information?

[forty-two]

To further refine - in LA, you get to interact with the good books while you are learning the skills to tackle the great books. In pure WTM, you actually learn those skills *through* the good books.

 

So what is the math equivalent of the good books? And is it possible to actually learn math skills *through* interacting with "good math"? And if so, how?

[EmilyGF]

I think I understand what you are asking. But I think for the answer you want, you need a slightly different question.

 

You teach language arts through literature because that's where a lot of good language arts are used. Teaching math through language arts (don't throw tomatoes!) doesn't make much sense because that isn't where *good* math is used. The FIAR "math" activities are pure baloney, IMHO.

 

Math can be used for exploration. Read the blog Math For Love (http://mathforlove.com/blog/page/3/ is a good place to start) to understand math as more than what-you-use-to-find-change-at-the-store. The author, in earlier posts, talks a lot about what turns kids off math.

 

A "living" way to teach math might be through science-fair type projects. My hubby and son are currently determining how size, mass, and drop height affect cratering in materials like flour and cornmeal. Math really happens there. Seeing a write-up by someone else (or an example by dad) provides a model that can be used by the child for the child's own calculations. How about the TOPS labs that focus on math and data collecting? You could provide an example to copy then have your child continue. We do this all the time.

 

We feel comfortable providing examples of good math to our children, which is why we don't need a curriculum to follow when we do these projects (but we do use RightStart on a day-to-day basis).

 

My husband and I have always loved math.

 

Emily

[forty-two]

You teach language arts through literature because that's where a lot of good language arts are used. Teaching math through language arts (don't throw tomatoes!) doesn't make much sense because that isn't where *good* math is used. The FIAR "math" activities are pure baloney, IMHO.

:iagree: This, exactly! Only I just realized it today :tongue_smilie:. (Well, the fact that math through LA was not it is a new thought; I've noticed for a while the craptasticness of most "math lit", which is neither math nor literature :glare:.)

 

Thank you for the blog link and examples! Science does make a great avenue for math - very much how history works for LA - will have to think more on how to approach it that way.

[Carol in Cal.]

Real Science 4 Kids Physics does this to some extent. It introduces pretty simple equations. Algebra skills are not required, but the idea that you can develop a mathematical model for motion or force is a really good one to convey early.

[stripe]

I'm not actually looking for a math curriculum in the "here is you set of materials", but in a "here is your plan of attack" way.

 

I was recommending in my post that you examine the Japanese style of math teaching, not the UCSMP books. They don't particularly address a wide range of math, but their approach -- heavy focus on one problem per topic, for example, helps demonstrate a certain academic style. I was going to refer you to the book Learning Gap, the scope & sequences from the Japanese govt, and a number of other resources but I think I would be better served not spending my time on that. But do feel free to search for my posts as I've introduced multiple math resources to the boards (three books and many electronic) that may eventually prove useful to you. I'll leave the thrill of the hunt for those who are interested.

 

Personally, despite the fact that I am competent in math, I am not comfortable inventing my own curriculum for elementary and beyond. Several other members on here have said the same thing, so I don't think it's just personal modesty on my part. It is important to have some insight into how children actually learn math. Understanding math is not the same thing. This is part of what makes Japanese teachers' use of lesson study so professionally valuable.

 

I will stop now. I understand being asked to be quiet.

[Hunter]

I didn't read the whole thread, but have you read the chapter on math in The Core?

 

Have you checked out Waldorf math?

[forty-two]

If "good math" is analogous to the good books, then part of that is surely firing the imagination. If we think of "good math" as the sort of math that inspired the people who grew up to create the "great math", then what does that give us? From what I've read, it gives us fun problems, games, and building/designing toys.

[wapiti]

It's always a good thing to step back and consider your "philosophy" of learning a given subject (perhaps you'd appreciate the Liping Ma book?), though I really believe that going without any math curriculum would mean reinventing the wheel.

 

I like the science idea. Teaching within a context (such as science) is always a helpful approach. To a large extent, that's what word problems attempt to do, albeit in a less tangible way.

 

Other discussions of "living math" might help you, e.g.:

 

What living math books would you recommend?

 

Living Math...what sequence do you use to teach concepts?

[forty-two]

I was recommending in my post that you examine the Japanese style of math teaching, not the UCSMP books. They don't particularly address a wide range of math, but their approach -- heavy focus on one problem per topic, for example, helps demonstrate a certain academic style. I was going to refer you to the book Learning Gap, the scope & sequences from the Japanese govt, and a number of other resources but I think I would be better served not spending my time on that. But do feel free to search for my posts as I've introduced multiple math resources to the boards (three books and many electronic) that may eventually prove useful to you. I'll leave the thrill of the hunt for those who are interested.

 

Personally, despite the fact that I am competent in math, I am not comfortable inventing my own curriculum for elementary and beyond. Several other members on here have said the same thing, so I don't think it's just personal modesty on my part. It is important to have some insight into how children actually learn math. Understanding math is not the same thing. This is part of what makes Japanese teachers' use of lesson study so professionally valuable.

 

I will stop now. I understand being asked to be quiet.

Oh, I don't want to chase you away :(. And I've heard lots of good things about Japanese math - just haven't had a chance to look at it. And resources on how to be a better teacher are always a good thing.

 

And, honestly, I will probably not go textbook-free - I don't have the skills for it. Wish I did, but oh well. It's like the multi-generation thread - *I* will have to use curriculum, but maybe my dc will have the skills to not have to, even if they choose otherwise. And at least the better math curricula reflect their creators' love for the subject, so they are probably darn close to good math themselves.

 

But I want to have a framework for going tb-free, even if I don't use it. Besides, already this thread has solved several problems I'd been wrestling with wrt how to approach math.

[Guest]

You might find this book extremely interesting:

 

Out of the Labyrinth: Setting Mathematics Free, by Robert and Ellen Kaplan

 

The Kaplans were the founders of the original Math Circles in Boston, which were NOT the Olympiad-contest-course-focused math circles that have largely taken their place today. Rather, they think of math, ideally, as "the highest form of intellectual play" that can and should be engaged in by people of all ages. Problem-solving plays a role, but that is not the only aspect of mathematical thinking that engages them; in their math circle classes, they want kids to learn to THINK about mathematics in a way very different than the typical "solve it" class. Hard for me to explain; but I loved their book, and bought a follow-up of a sort about the kinds of problems and activities they like to set for their students. I can find the title later if you are interested; I'm about to run out the door.

 

PM me if you'd like the title -- I don't know whether I'll remember to come back to this thread as I don't often post on the K-8 board.

[woolybear]

This thread is also very helpful, particularly look at posts #2 and #11.

 

http://www.welltrainedmind.com/forums/showthread.php?t=294827&highlight=living+math

 

As I posted on that thread, it seems (for my brain, at least) like it would be a lot of work, but very doable. Living math books can certainly be part of a living math education, but are not the whole thing.

 

I think it would make sense to find a basic math scope and sequence, that you like. Then build from there, finding books,games, activities to support and teach the concepts.

 

You also might want to check out the Living Math Yahoo group.

 

I did not address the WTM approach to Living Math because I just don't quite get the connection. I say, maybe, do Living Math and forget the WTM aspect. :001_smile:

[choirfarm]

Here is Living Math in a 4 year cycle:

http://www.livingmath.net/LessonPlans/C1PrintableBooklists/tabid/1046/language/en-US/Default.aspx

 

It even uses SOTW.

 

Christine

[kirstenhill]

When I was in about 8th grade, I loved "On Numbers" by Isaac Asimov: http://amzn.com/0517371456

 

It looks like it is out of print, but probably something a larger library would have. 20+ years ago my small town library had it. While a work of non-fiction, it totally inspired me to think about numbers in new ways so I would say it qualifies as a book of "living math".

 

Of course, I still did not turn out like my husband, who literally sits with a legal pad doing math "for fun" (usually somehow related though to his professional work doing R & D at a medical device company), with pages and pages of things that I barely understand even though I took 2 semesters of calculus in college!

 

When I think of "living math" I think what comes to mind is how my husband describes he was raised. He worked on all kinds of home projects with his dad, and his dad was always challenging him to figure out the math needed (often in his head) or his dad would just ask him questions to really make him think (for example, asking him how he would estimate the surface area of a lake they were driving by). Now my husband is the type of guy who instead of just measuring and cutting "as you go" to find the length of boards needed to build a wooden fence gate with a friend, he will sit with a pad of paper and work out the geometry given the size of the opening, and show up to work on the project with all the needed lumber dimensions in hand.

 

While are a totally using a math curriculum (though Right Start is pretty far from the traditional methods used when I was in public school!), I am also trying to help my kids see how math is woven into life. I think we're all "doing math" on a regular basis, and I am trying to remember to think out loud to my seven year old and challenge her to think about math beyond what she is learning in her math lessons in RS.

 

Anyway, maybe a different train of thought than where you are headed, but maybe there is something in there for you. :-)