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Jane in NC

Talk, talk, talk. Literature, history, why not math?

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Hi all,

 

There have been a number of threads pertaining to math these days as parents plan for next year or regroup for this. One thing, though, that I have noticed is that while many of us promote discussion as a necessary part of our Great Books education, I do not see parents mention discussing mathematics.

 

I know that many parents feel some insecurity regarding mathematics. But honestly many feel the same insecurity regarding the Great Books as well, yet they read these books with their kids and enjoy the resulting conversations. In recent months, my husband and I each read a book that we haven't read since high school, Inferno is his case, Canterbury Tales in mine. Talk about new perspectives coming to these great works of literature as an adult!

 

It also seems that adoloscent minds are well connected some days and off in the hinterlands on others. At least my dear son's brain functions this way. Yesterday he began his Dolciani Algebra II chapter on Exponential Functions and Logarithms with a unit on rational exponents. These were not hard problems and given what he already knows about radicals, this should have been easy. But yesterday was one of those days when his synapses were not firing--he was equally klutzy physically on the ice when he played hockey last night. So I literally stood next to him as he did his math and walked him through things, answered his questions and gave him the necessary approvals (without popping him on the bean as I was tempted to do!)

 

Let's see where his brain is today.

 

Anyway, I wonder if some things like geometric proofs or chapter five or whatever it is in Lial's that bogs many kids down which just get a bit easier if the material were discussed. I'm not suggesting that parents lecture on the math, but ask the same kinds of questions that we ask regarding literature and history: why would you need to do that, does this always work, can you see a different way of setting this up, I don't understand the graph you drew, you didn't draw a picture??? You get my drift.

 

Jane the Fearless Homeschooler (at least at the moment!)

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I think parents sometimes think that they can just hand over the math text and all will go smoothly. This is true of HS and PS; my sister wasn't even aware if her daughter's Geometry book included proofs. My 13 year old son still gets daily instructions with me on his Algebra, and then I'm nearby for questions. Fortunately, he's a talker, and seems to enjoy explaining things. He's also entering that age where he'll discuss his studies with me, just to see if he can prove me wrong somwhere.

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but it is discussion, and usually not with the textbook in front of us. It is usually after we've done math, and it often goes very interesting places.

 

For math instruction itself, I have one student who believes that the less explanation, the better. The other two learn at the openings to their ears. We talk and question and discuss, and "what if" a lot. This is very difficult for me, as I tend to be a "teach at the end of my pencil", visual learner. :eek:

 

Thanks for bringing up this thread.

 

V

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It forces me to discuss the math. I also work each example problem out on paper for them, rather than just pointing in the book. That forces me to talk my way through the problem for them. NEM is rather good at forcing me to do this, too, by having class activity sections. I've never worried about NEM/PM needing more teacher involvement because I know my children would quickly become a mass of misconceptions if I weren't an active part of their math learning. It is only by talking that I can tell where their thinking has gone astray. They see patterns that work fine in the examples shown, but won't work for more complicated things that they haven't gotten to yet. I have to watch for that and explain. I also have to remind them of the bits they've forgotten, and make sure they aren't just memorizing the algorithms but really putting the pieces together into the big picture, and make sure they understand how things can be applied. NEM helps me to do all these things.

-Nan

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Hi all,

 

There have been a number of threads pertaining to math these days as parents plan for next year or regroup for this. One thing, though, that I have noticed is that while many of us promote discussion as a necessary part of our Great Books education, I do not see parents mention discussing mathematics...

 

Anyway, I wonder if some things like geometric proofs or chapter five or whatever it is in Lial's that bogs many kids down which just get a bit easier if the material were discussed. I'm not suggesting that parents lecture on the math, but ask the same kinds of questions that we ask regarding literature and history: why would you need to do that, does this always work, can you see a different way of setting this up, I don't understand the graph you drew, you didn't draw a picture??? You get my drift.

 

Jane the Fearless Homeschooler (at least at the moment!)

 

We discuss math, economics, philosophy (and how math affects it) pretty regularly. We teach the boys a lot through the projects we do at home. They built surveying equipment this past weekend, and ds9 spent the weekend surveying the landscape for the new grade and drainage. They've designed and built jigs for the woodwork they need to do when we don't have the optimal tools for the job. They design circuits, wiring diagrams, power systems and proofs - not from books, but from projects, from "what if" discussions, from their rich imaginations. We get in a lot of Life Is Math around here.

 

The kids know more about the theory of math, the applications of math, and the history of math at the ripe old ages of 9, 7 and 4.5 than I did at 20. (I've learned a little bit here and there, too. ;))

 

Really, I don't mention it so much, because I don't think about it when I read threads. It's not something we have a curriculum for, it's just something we "do". Sort of like with literature (and if you look at my posts, I don't generally contribute to those threads, either, mostly for two reasons: I'm not entirely sure what we do is going to "work", and to be honest, I'm not entirely sure what we're doing.) We just go, one step at a time.

 

Sometimes I still want to thump them on the head. But I think that's just the nature of the beast. (Me being the beast.)

 

Anyhow, great question. Some of us do discuss these things, but we forget to mention that here. :o

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We did have to do this with our son when he was going through Algebra I and II. The funny thing was, we often didn't even say anything any different, really, than what the book said. But after we'd walk him through it, he seemed to calm down, or for whatever reason felt able to do the work. I'm still not certain I understand why, except that with him, I think maybe he has an auditory processing glitch and that he just needed more time to process.....

 

Regena

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How I wish I'd done a better job of talking about math from the beginning! I'm just not "wired" that way, though. I guess I needed a "fostering math skills through discussion for dummies" book.

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I'm not sure that I "discuss" math much, but I do discuss it in one way -- when my kids don't come up with the right answer, I ask them if they should have known that the answer is wrong. This gets them to think about the answer -- what should the answer look like?

 

For example, my 6th grader is doing compound interest, and she was asked to figure out how much needed to be repaid given a principle of X. Her answer was smaller than X! She just took a quick look and gasped -- and realized that she was hopelessly wrong. She looked at her equations and realized what she had done wrong -- and all I did was ask her why her answer was wrong!

 

my older kids internalize the process at some point. Asking the kids to think about the answer does not eliminate all errors, but it at least means that it will be in the right units, the right order of magnitude, etc.

 

An ability to do back-of-the-envelope calculations should be encoouraged, and asking our kids to estimate helps develope this ability.

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How I wish I'd done a better job of talking about math from the beginning! I'm just not "wired" that way, though. I guess I needed a "fostering math skills through discussion for dummies" book.

 

I guess we'd call that book "Deconstructing Permutations" ....

 

Regards,

Kareni

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Hi all,

 

There have been a number of threads pertaining to math these days as parents plan for next year or regroup for this. One thing, though, that I have noticed is that while many of us promote discussion as a necessary part of our Great Books education, I do not see parents mention discussing mathematics.

 

Anyway, I wonder if some things like geometric proofs or chapter five or whatever it is in Lial's that bogs many kids down which just get a bit easier if the material were discussed. I'm not suggesting that parents lecture on the math, but ask the same kinds of questions that we ask regarding literature and history: why would you need to do that, does this always work, can you see a different way of setting this up, I don't understand the graph you drew, you didn't draw a picture??? You get my drift.

 

Jane the Fearless Homeschooler (at least at the moment!)

 

I definitely should do this more often! Sometimes I do discuss the lesson with the kids, and they do comprehend the material so much better that way. I tend to go through this process more when I'm correcting their math. I admit, however, that I don't always explain the math lesson to them---they usually read the material themselves (the older two watch their lessons on the DVDs/DVTs and then read through the material). So, I tend to catch them on the "back end" of things after they've misapplied the rules or obviously misunderstood some key point in the lesson. I know that's not optimum, though. When I do explain material to them, I don't give them the answer----I try to lead them through the thinking process until they can reach the right conclusion themselves.

 

What I understand, from the main thrust of your post, is that you are looking more for a way to teach math with the classical method. This is very interesting----I hope you pursue this post with further thoughts on the topic!

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I guess we'd call that book "Deconstructing Permutations" ....

 

Regards,

Kareni

 

I would buy that book!

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that we ask regarding literature and history: why would you need to do that, does this always work, can you see a different way of setting this up, I don't understand the graph you drew, you didn't draw a picture??? You get my drift.

Anyway, I wonder if some things like geometric proofs or chapter five or whatever it is in Lial's that bogs many kids down which just get a bit easier if the material were discussed. I'm not suggesting that parents lecture on the math, but ask the same kinds of questions

 

Jane the Fearless Homeschooler (at least at the moment!)

 

Jane, it sounds like you might be talking about taking a "philosophical" approach to math. From what I have seen this is in fact how mathematicians conduct class at the higher levels and whenever I've had questions about something it's how people interact with me as well.

 

Are you familiar with "The Moore Method" by the way? It was the teaching style of RL Moore.

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Are you familiar with "The Moore Method" by the way? It was the teaching style of RL Moore.

 

Most of my graduate math classes were more traditional lectures, but there was some incorporation of this method occasionally. Some professors wanted us to do our problem sets in isolation, but most wanted us working together at the chalkboards in each other's offices so that we could offer up our often counter intuitive ideas in a "safe" environment. It was interesting to see how personalities could work together, the big idea people with the those hung up on minutia. My husband actually hung out with a crowd who would write out proofs while playing bridge (the dummy had to do something!)

 

I drive my son nuts sometimes (hah! often!) by not answering his specific question but countering with either another question or a hopefully brief lecture on the big picture. Kids are often "answer" focused when they should be focused on methodology. This is why they can get good grades on a chapter test but fail the material on the final or on a placement exam. By asking some big picture questions, students are reminded of how the current puzzle piece fits in with the whole.

 

This is also where the use of graphing calculators can drive me crazy. Gwen touched on this in her post--students need to have a common sense idea of what an answer can or cannot be or what a graph can or cannot look like. For example, if a student's sketch of a cubic polynomial looks parabolic, a red flag should be flying in their brain. Discussing those polynomial graphs as a student sketches them gives the parent/teacher the opportunity to address the degree of the polynomial and the anticipated outcomes in their graph. This can be self discovery for both student and parent if the parent doesn't remember what graphs of polynomials look like.

 

Pull back and talk is my suggested mantra. Ask just what the heck is going on here. And try some family proofs. A chalkboard in every kitchen or school room with a suggested theorem to prove! Why didn't I think of this before?

 

Jane

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...I TA it. I don't strictly teach anything. I just show up with a low tolerance for bull****, an expectation that people will listen to me and take me seriously, and possibly a grimace. I imagine R L Moore to be that way. In fact, the one overriding theme I see in these legendary thinkers is a low tolerance for bull****, at least when it comes to their subject matter.

 

But, then again, I am not strictly in charge of the daily lessons, either (which may be a good thing). Occasionally, we do have those moments that sounds like it came straight out of one of Plato's dialogues which makes it all worth doing....

 

Frankly, we all know that it really all just comes down to what kind of problems you can do. I'm just here to make sure they do them.....

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Most of my graduate math classes were more traditional lectures, but there was some incorporation of this method occasionally. Some professors wanted us to do our problem sets in isolation, but most wanted us working together at the chalkboards in each other's offices so that we could offer up our often counter intuitive ideas in a "safe" environment. It was interesting to see how personalities could work together, the big idea people with the those hung up on minutia. My husband actually hung out with a crowd who would write out proofs while playing bridge (the dummy had to do something!)

 

LOL! My office was the "bridge office", too! Didn't you say your husband went to Ohio State? I'm starting to realize now that I think I would have gotten along with Mityagin, infamous at OSU. My understanding was that he only takes the stick out of his arse when he needs to beat one of his students. He actually flunked his own student on their Comprehensive exams (which, Mityagin, himself, recommended the guy for). But, however legendary his brutality may sound, I really think he just has standards -- too high of standards for OSU. But, you sure could learn some functional analysis from the guy. (His lineage goes: Kolmogorov -> Gelfand -> Shilov -> Mityagin!) Higher standards is something OSU needs.

 

Had I known then what I know now, I would have stayed and tried to become his student.

 

This is also where the use of graphing calculators can drive me crazy. Gwen touched on this in her post--students need to have a common sense idea of what an answer can or cannot be or what a graph can or cannot look like. For example, if a student's sketch of a cubic polynomial looks parabolic, a red flag should be flying in their brain. Discussing those polynomial graphs as a student sketches them gives the parent/teacher the opportunity to address the degree of the polynomial and the anticipated outcomes in their graph. This can be self discovery for both student and parent if the parent doesn't remember what graphs of polynomials look like.

 

The cure for that kind of BS is harder problems. Use your calculator all you want -- it won't help you. In fact, that's why I could almost endorse the widespread use of calculators or even computers -- it's just going to cause me to make my section harder, giving you series that converge on the computer but not in reality, problems that only involve symbols and not any numbers, and so on. We could actually start doing real math eventually....

 

Pull back and talk is my suggested mantra. Ask just what the heck is going on here. And try some family proofs. A chalkboard in every kitchen or school room with a suggested theorem to prove! Why didn't I think of this before?

 

Jane

 

My motto seems to have been: "Grill them hard for justification when they get the answer wrong and grill them harder when they get the answer right."

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we do need to discuss math with our dc, and beginning at a very early age.

 

In addition to going through the lesson with my dc when I can, I also try to ask them stretching questions -- questions that are just outside the realm of the lesson (e.g. when you're done discussing adding 2 digit numbers, casually ask if they would know how to solve 123 + 456. You're getting them to think about the relationships and how they would apply in another similar case.

 

I also find myself doing what Gwen mentioned -- constantly asking my children, "Does this answer seem right to you?" I figure if they hear me say it a couple of times a day for 8 or 10 years, it might become automatic.

 

I think that numbers are a kind of language of their own. They have some rules and patterns that need to be understood in order for an individual to speak "math" fluently. I was thinking that you'd never hand your child a Spanish textbook and tell them to go read the lesson and do the exercises, then expect a fluent speaker of Spanish as the result. I feel like we need to guide our dc through the language of math -- teach them the inherent beauty of numbers, patterns, geometry, etc. Understanding these deeply lets them go beyond the lesson and apply the math to real life.

 

Getting off my soapbox,

Brenda

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LOL! My office was the "bridge office", too! Didn't you say your husband went to Ohio State?

 

 

 

Nope--Chapel Hill. Every math department probably has the bridge office!

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Interesting point, I just assumed everybody spent lots of time talking about Math with their kids. We discuss math, the history of math, and famous mathematicians at our home. In fact I spend more time teaching/discussing math than any other subject. I want to ensure that my dc understands what we are working on before we move on. I may be misguided but I feel like if they don't grasp every concept in the history that we are studying it is okay because we will hit it again in 4 years, but they must master whatever we are studying in math. My dh has degrees in Math (and does proofs for fun and relaxation) and dd works as a math tutor at the cc so we probably put more of an emphasis on it than other families. Math invades evey part of our life, my dc will often write a math joke on the white board in the kitchen i.e. "The i's have it".

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"If I've told you once, I've told you k + 1 times."

 

That is officially my new favorite quote! Thanks, Ann

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talking through math with our children (i.e., not handing out answers but requiring them to think through the problems) is part of the larger issue of the necessity to talk through all of these subjects with our children? A great amount of the classical method should be spent in discussion, don't you think? Yet, I realize that I probably don't do nearly enough of it! :o I guess in a sense this is an application of the Socratic method at home, only being employed in the context of math?

 

I wonder how the ancient mathematicians taught math. Did Pythagoras teach math by a question-and-answer method? I wish we could travel back in time for a moment to find out.

 

Speaking for myself, and possibly for some others on these boards, I know that I'm still trying to figure out what exactly a classical education is, not having been a recipient of one myself! :rolleyes:

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I wonder how the ancient mathematicians taught math. Did Pythagoras teach math by a question-and-answer method? I wish we could travel back in time for a moment to find out.

 

 

 

Their mathematics is not our mathematics. The foundations are there in Euclid, but the algebraic notation that we use, Cartesian coordinates, Calculus, etc. are much newer. This is not to say that a discussion of mathematics or the use of proof is irrelevant. It is just that mathematics as a field has blossomed in the last four hundred years--particularly the last one hundred. Yet many people think that Math is like Latin--a dead language. The problem is that high school students rarely have a glimpse at any math of the last couple hundred years, although a math major will. Working your way to the 20th century practically requires graduate school.

 

I guess that I see a classical education encompassing the study of history, Latin and literature, but also including challenging work in Mathematics and Science since those subjects rest on the shoulders of the great men of the past. What becomes clear through discussions on this board is that what constitututes, say, a geometry course for one person is totally different for another. And the unfortunate reality is that success now seems measured by performance on a standardized test of some sort--not true knowledge or ability.

 

This thread has gone to some non-anticipated places. When I posted initially I was thinking about the dissatisfaction expressed with curricular materials on this board and wondering how much of it is caused because parents expect their children to do mathematics independently, that parents have bought into the notion that the point of math is getting "the right answer" when the point should be methodology and an understanding of systems--not just algorithms.

 

I'll make another confession: I studied mathematics because I found it to be beautiful. It breaks my heart that students are not exposed to aspects of axiomatic geometry because it does not seem to serve a pragmatic purpose. My mentor professor in undergraduate school felt that studying mathematics and philosophy gives us purpose beyond our day to day lives. Is this any different than reading Shakespeare?

 

Jane

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Their mathematics is not our mathematics. The foundations are there in Euclid, but the algebraic notation that we use, Cartesian coordinates, Calculus, etc. are much newer. This is not to say that a discussion of mathematics or the use of proof is irrelevant. It is just that mathematics as a field has blossomed in the last four hundred years--particularly the last one hundred. Yet many people think that Math is like Latin--a dead language. The problem is that high school students rarely have a glimpse at any math of the last couple hundred years, although a math major will. Working your way to the 20th century practically requires graduate school.

 

I guess that I see a classical education encompassing the study of history, Latin and literature, but also including challenging work in Mathematics and Science since those subjects rest on the shoulders of the great men of the past. What becomes clear through discussions on this board is that what constitututes, say, a geometry course for one person is totally different for another. And the unfortunate reality is that success now seems measured by performance on a standardized test of some sort--not true knowledge or ability.

 

This thread has gone to some non-anticipated places. When I posted initially I was thinking about the dissatisfaction expressed with curricular materials on this board and wondering how much of it is caused because parents expect their children to do mathematics independently, that parents have bought into the notion that the point of math is getting "the right answer" when the point should be methodology and an understanding of systems--not just algorithms.

 

I'll make another confession: I studied mathematics because I found it to be beautiful. It breaks my heart that students are not exposed to aspects of axiomatic geometry because it does not seem to serve a pragmatic purpose. My mentor professor in undergraduate school felt that studying mathematics and philosophy gives us purpose beyond our day to day lives. Is this any different than reading Shakespeare?

 

Jane

 

 

I study math because it is brutal and ugly. I find joy in having hapless engineering students driven before me in chains, brought to their knees with my indomitable epislons and deltas! WHOO!

 

(just kidding)

 

Actually, I just wanted to point out an interesting side issue that I have come to realize while looking through Apostol's calculus.

 

"The fundamental concept on which the whole of analysis ultimately rests is that of the limit of a sequence."

-- Courant (from his famous calculus text)

 

One interesting aspect of Apostol's approach is that he puts integration first as opposed to most modern texts that put differentiation first and older texts like Courant that do them almost at the same time. (Although, Courant does technically do integration first.) What is so ingenious about doing something like what Apostol is doing is that to develop the integral as he does it, all he needs is the Archimedean Property to do it with. He doesn't need a full blown limit and indeed hasn't even covered limits by that point. In fact, all the limit really is is a couple applications of a general tactic based on the Archimedean property. So, I will say that, in fact, if you are really clever about it, you can arrange a lot of it to be done that way -- with the Archimedean Property rather than the limit. Or, in other words, it isn't the limit that is the fundamental concept but the Archimedean property or more specifically, a particular application of it that is more or less equivalent to saying that the lim(1/n)=0 (which is, as a matter of fact, the very first limit discussed in Courant).

 

The reason I bring all this up (saving my conclusion for the very end to build maximum suspense) is because who really first articulated that along with the method of exhaustion? No -- it wasn't actually Archimedes which would probably make the point anyway if it were. It was Plato's student Eudoxus.

 

So, even our precious modern invention of Calculus really does (no really) go back to none other than Plato's Academy (or at least the key insight of it -- what makes the difference between a mere finite sum and an actual integral, for instance).

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Their mathematics is not our mathematics. The foundations are there in Euclid, but the algebraic notation that we use, Cartesian coordinates, Calculus, etc. are much newer. This is not to say that a discussion of mathematics or the use of proof is irrelevant. It is just that mathematics as a field has blossomed in the last four hundred years--particularly the last one hundred. Yet many people think that Math is like Latin--a dead language. The problem is that high school students rarely have a glimpse at any math of the last couple hundred years, although a math major will. Working your way to the 20th century practically requires graduate school.

 

I hope I didn't divert the intention of your thread. It just made me begin thinking, first of all, how much I have expected my children to do math on their own, only to find out later they've gotten themselves into a ditch. Today I spent much more time on explanations. I'm not sure if I could explain a proof in algebra (which shows the deficiencies in my own education) in mathematical terms, but I think I could intelligently explain why the steps must happen in a certain order.

 

Your post started me thinking in general about the necessity of teaching our children to think, and math should be no different. You're right---math should have more meaning to our children than just getting the right answer on a standardized test. How much of our educational system is geared in that direction, though? In our school district, the state-wide testing that takes place at the end of each school year literally dictates everything that the public schools teach. Granted, some very intelligent, gifted kids come out of the school system and go on to college ready to learn. However, I think the large majority of kids are not really learning how to think.

 

Your post was excellent, and I hope I didn't divert the topic too much!

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So, even our precious modern invention of Calculus really does (no really) go back to none other than Plato's Academy (or at least the key insight of it -- what makes the difference between a mere finite sum and an actual integral, for instance).

 

I did an independent study on the History of the Calculus using in part some materials that had been produced by the British Open University back in the '70's. That particular program laid the foundations of calculus with the Greeks as you have pointed out. Refinements are certainly post-Newton/Leibniz, but I did a number of classic geometric proofs in that independent study. So I suppose that one who has nothing else to do could recreate it all from its ancient roots. But just how much time do you have on your hands? And do you wish to risk the extreme wrath of your adolescent child?

 

Look, I'm just excited that in the days ahead The Boy will be proving that a function with an inverse is one to one as well the converse. My husband says that I'm just an "analyd" at heart.

 

Cheers,

Jane

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This is one thing I really gained by having an oldest child who needed extra help with math. We had to sit down and discuss it, do multiple problems together, and do it every day. It conditioned me to expect to spend lots of time teaching math.

 

Going through Lial's with dd#2 who has a much easier time of math, I've done the same thing. We read each lesson aloud, and work all of the examples on the whiteboard together. Even some of the problem set, although I do give her most as her assignment.

 

I don't know that she would really get it if we didn't do this. She might, but this way I can be sure and we can stop and see where our potential mistakes lie before going any further. I've found this invaluable for the chapter reviews as well - anything that might still be fuzzy by then definitely isn't by the time we're finished with it, so she's ready for the test. Not that she's gotten excellent grades on every test, but I feel like we've put a good effort in. For a few tests that she scored in the 70-80% range, we re-did the problems and hopefully it will 'take' that time.

 

I had talked to her about using Chalkdust or another video program for Alg 2 & up, and she told me that she really feels like she understands it when I explain it. That was *news* to me, because I feel like I am muddling along with her - but I think her point was that doing the lessons together and talking about it was what was really helping.

 

I am trying to study up on Geometry and Alg II now. We probably will need some other instruction to help us sooner rather than later, but I will try to keep the same approach, just using the instructional videos first. I was surprised to see the posts about Chalkdust needing the parent to still be familiar with the subject matter - but that does make sense.

 

Looks like I'll be getting a good math education over the next few years! It's pretty intimidating, but I'm going to try to keep up as best I can, for a few more years at least. I don't make any assumptions that I'll be able to muddle through calculus with her even with something like Chalkdust - we've got university classes available for that. Maybe it will head off senility and keep my brain agile.

 

It is really hard to devote this much time to math and still have enough to devote to long discussions about history and literature. I think I can do it, though. I'm going to try, anyway!

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Yes, I think this is a great idea. We do, but not necessarily by formal arrangement. Often when they're learning a lesson. This is also one of the reasons my eldest and I will be going back to Gelfand's. Even if you never do the problems, reading it aloud together can help start the thinking and conversation. Not sure what to use for younger ages. Also, when I read most of the book Mathematical Innumeracy (recommended on the old boards somewhere) I started working with them so that they really understand the difference between numbers (like a million to a billion to a trillion). There are some kids' books like this, but beyond that. Also, about odds, etc. How people are so terrified of certain things where the odds of it happening are about 1 in a million but don't think twice about doing things where the odds of danger are far, far higher.

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Another point on the importance of discussing math-- we're just trying out a new online math class by the Art of Problem Solving, and the class starts tonight (Intro to Number Theory), but at least there will be a group of kids, with a teacher, talking through and discussing math. We're excited about it-- I hope it is as helpful as we wish!

 

http://www.artofproblemsolving.com/Classes/AoPS_C_About.php

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This has been the first year that I really discuss math with my three boys. The difference is huge... they're doing so much better and enjoying math more. Whatever I can't discuss too well (algebra word problems in Foerster's) my husband will take. You're right, Jane... math is just as important as history and lit. to talk over with children. It's less abstract if words are connected, and with words, math (frequently) becomes (just) a logic puzzle.

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My husband's away so I get to go through old threads for self- (and hopefully my children's) education. I hope you don't mind an add-on at this late date.:)

 

Since my second son recently got his Swiss Maturite (like a French baccalaureate - exam at the end of high school), I got to watch the process.

 

There was an oral as well as a written component of the math exam. And he had to stand alone in front of several unknown teachers from another canton (=state) and solve a difficult math problem.

 

To practice for this, through high school, he would periodically have to go in front of the class and solve problems, and he had oral components of his major tests from about 10th grade on.

 

I have to say that I'm in admiration (because I would be absolutely tongue and brain tied) and let him get his younger brother in front of him to solve problems. But that fell by the wayside and I think I'll have to resurrect it.

 

Thanks Jane for this thread!

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My husband's away so I get to go through old threads for self- (and hopefully my children's) education. I hope you don't mind an add-on at this late date.:)

 

 

 

Thank you, Joan, for resurrecting this thread. It was great fun to read through it and realize how much I miss Charon's and Myrtle's contributions (sniff, sniff). I erased part of your post in the quote, but I found it interesting when you wrote about the oral components of your son's math education in Switzerland. Someone in another thread mentioned that students in Britain are expected to be able to articulate their ideas verbally. Perhaps this is what is lacking in American education? I think of my husband who works with some people who feel that communication is all about being the LOUDEST person in the room. I have often commented that most engineers work in teams and hence must be effective written and oral communicators. This often comes as a shock to students who are are whiz kids in math and physics, the gang that feels that they do not need to know how to write. Ha! We should add that oral presentations are equally important.

 

Well to make a report on the math that was discussed in this home today: I had to remind my son of the Fundamental Theorem of Algebra, the implication of which is that every nth degree polynomial of a single variable with complex coefficients has precisely n roots (taking multiplicity of roots into account). He is solving equations over the field of Complex numbers and had initially failed to understand why every odd degree polynomial with real coefficients has to have at least one real root.

 

Anyone else have any interesting math conversations today?

 

Jane (sending virtual hugs to Joan)

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Jane, you bring up great point, like usual!

 

My high schooler is done doing math at home, I have entrusted her to the nice professor at the local community college!

 

But with her younger siblings I found that they are pretty happy to work alone as long as they understand what they are doing. When the material is new and confusing I like to do the whole thing with them and instead of notebook paper their math is worked out on the whiteboard. Something about Mom doing it with them and seeing the work larger than life on the whiteboard is a big help to them.

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so I have to make a real effort to communicate verbally, unless there is an enormously pressing subject that enables me to synthesize the ideas into manageable sound-bites. But the desire to not disable my children through lack of discussion exposure is driving me into action and why I'm trying to do more of the work orally. I have to say that I had scheduled only about 5-10 minutes for math questions with my 15 yo as that was when I was going to do English with my 10 yo. I have to rethink this schedule now. Maybe that white board really does have to be in the kitchen!

 

(I wouldn't mind a "bring-up" file of old posts to periodically remind us of good ideas that go by the wayside when schedules get overloaded or life's emergencies take precedence and then get forgotten completely.)

 

My children are at my side encouraging me to put in smilies (put in the car, mommy) in the text and noodles in the pot for dinner!

 

Waving over the ocean (and some of the continent) to Jane!

 

Joan

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We're using an online curriculum that has short video lessons, but I go through each lesson with him. I do the practice problems alongside him, and we compare notes as we go. We also pause the lesson whenever I get the sense that he's not fully absorbing the information so that I can check in with him, answer questions and talk him through the problem.

 

He took the CogAT last year, and reading his profile sparked understanding for me that this kiddo NEEDS interaction and "story" or narrative in order to learn. He loves metaphors and connections and anecdotes. It helps him to remember and understand what he's learning. So, that's how we do math now.

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I think parents sometimes think that they can just hand over the math text and all will go smoothly.

 

I am one who hands over the math books.

And have done so since he was 7.

I don't know what the definition of smoothly is, but he does average a book a year on his own, smoothly or not ;)

 

:seeya:

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I gave up the teaching and discussion to the video teacher; that was a first for us and it bombed. My ds needed to hear and discuss math as well as his other subjects which is what we did all through homeschooling. He zoned out with the long video lessons in CD alg. I didn't like sharing the algebra book (no teacher book) so I left him to learn algebra on his own. BIG mistake! :lol: No harm was done because that was in grade 8. He's doing algebra 1 again in grade 9 at his private high school.

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T

 

 

 

This thread has gone to some non-anticipated places. When I posted initially I was thinking about the dissatisfaction expressed with curricular materials on this board and wondering how much of it is caused because parents expect their children to do mathematics independently, that parents have bought into the notion that the point of math is getting "the right answer" when the point should be methodology and an understanding of systems--not just algorithms.

 

 

Jane

 

Hi Jane,

 

Although I agree with you in theory...my problem is how to "discuss Algebra II with my 8th grader, Key to Percents (which I have been we had to pull it out when TT7 wasn't enough on this topic) and then 1st grade math. I already discuss TOG history and literature with them... discuss Biology with 8th grader, Zoology with 6th and 1st grader, etc... At some point, I need SOMETHING that I can just hand to them. I just can't discuss EVERYTHING all the time.. My brain is exploding...

 

Christine

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I do discuss math with my kids, but nearly as much as other subjects due to my lack of understanding! I wish I could see the art and beauty in math and be able to communicate that to my kids, but alas I am blind :001_huh:

 

I do much better at discussing things with them when I take the time to really understand the concept myself ahead of time.

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Ds, 20, loves to talk mathematics until my head spins. The other day he enthusiastically explained Dd, 14's Algebra 1 question until she cried. He loves math, she hates math, and his enthusiasm added to her frustration. She doesn't blame him for her tears, but she doesn't see the point in advancing any further in math. She has only recently "hit the wall" and we are now letting her work through "Math at Work" for a breather. Dh and Ds, 22, have mathematical minds as well. When the three men get to talking in depth, I put my fingers in my ears. :tongue_smilie:

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... students in Britain are expected to be able to articulate their ideas verbally. Perhaps this is what is lacking in American education? I think of my husband who works with some people who feel that communication is all about being the LOUDEST person in the room. I have often commented that most engineers work in teams and hence must be effective written and oral communicators.

 

Now that dinner and violin concert are over..:)

 

A long time ago someone on a thread noted that architects need to be able to communicate with and listen to their clients. There are lots of professions that would serve us better if people listened more carefully (like doctors) - but then in this speedy modern world people don't think they have the time or are under deadlines, etc.

 

Anyway, over here I am impressed with the way that people will formulate questions in discussions at the Geog Society meeting for example. I have begun to realize that I have a subconscious tendency to be cryptic and my youngest son is even worse.

 

Back to Swiss high school - they had to do oral exams as well as written not only for math, but also for physics (because that was his specialization), French, English, German, philosophy, and the presentation of their high school research paper - all with unknown professors. For a previously quiet boy, it did help develop his articulation skills. So I have a tall order to match.

 

Which reminds me that I better get back to work!

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Jane,

 

When I first started reading the recent posts I thought, "I don't discuss math with my kids." But then I read your post and realized that I had a similar conversation with my 11th grader today. He put 5 as the possible complex roots for a function of degree 5. So I guess I do discuss, but I see it as answering questions and explaining.. I was thinking that discussing was something different.

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Jane,

 

When I first started reading the recent posts I thought, "I don't discuss math with my kids." But then I read your post and realized that I had a similar conversation with my 11th grader today. He put 5 as the possible complex roots for a function of degree 5. So I guess I do discuss, but I see it as answering questions and explaining.. I was thinking that discussing was something different.

 

If the answer to a question is "2", I would not call that a discussion. But to ask your student what is the difference between this equation and that one can be a discussion. When you are "discussing" a novel, do you "explain" it or have some back and forth exchange (which may include some prompting of where the discussion is going)? The same can be applied to math. Ask the leading question, prompt the student for his observations. "Can a generalization be made?... Do you see a counter example?...Remember when we did (such and such) in the rocketry example. We were using a similar equation... So--what's the point here?" I'd call this a discussion. If one person is doing all of the talking, then it is a lecture.

 

Jane

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Reraising a thread.

 

So I've been reading tons of old posts on logic - traditional logic vs. symbolic logic - learning the differences between the two and realizing that I seem to "get" things better through words, and struggle more with symbols. I've read that some people think that the study of symbolic logic comes easier after studying traditional logic, and I think I get why now, after realizing I do better with understanding a concept through words. So, all that reading led me back to my saved posts by Jane in NC on Dolciani math, because it seemed like Dolciani was described by Jane as more verbal-oriented, in explaining math concepts. Which led me to this thread, so I'm resurrecting it. :D

 

students in Britain are expected to be able to articulate their ideas verbally.

 

Anyone else have any interesting math conversations today?

 

Jane, with the British students, did you mean that they were expected to articulate their ideas *about math* verbally? How did they learn to do that, if so?

 

How do you *have an interesting math conversation*?? Where does a math-phobic person like me start to understand the concept of having a math conversation? (again, feeling really stupid to be asking these things)

 

I wish I could see the art and beauty in math and be able to communicate that to my kids, but alas I am blind

 

Again, how does one see this about math? Do the older Dolciani books open up this world?

 

But to ask your student what is the difference between this equation and that one can be a discussion. When you are "discussing" a novel, do you "explain" it or have some back and forth exchange (which may include some prompting of where the discussion is going)? The same can be applied to math. Ask the leading question, prompt the student for his observations. "Can a generalization be made?... Do you see a counter example?...Remember when we did (such and such) in the rocketry example. We were using a similar equation... So--what's the point here?" I'd call this a discussion. If one person is doing all of the talking, then it is a lecture.

 

Jane

 

This makes sense to me. But how would I know where to begin? SWB gives a whole workshop on how to have literature discussions, complete with open-ended questions to get things going. And this appeals to me, who enjoys words. But how do you begin this with math?

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We do sit at the table together trying to do the problems...

 

We watched How to be a Superstar Student over the summer. The teacher gave us a great idea in it. He said the kids should read the lesson before the lecture, even take notes, so that the lecture is a second pass at the material. We use Chalkdust and are trying this. So far, so good. It is helping dd to know what the teacher is talking about ahead of time, and usually she can go straight to the problems without a lot of "figuring it out" time.

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Jane, with the British students, did you mean that they were expected to articulate their ideas *about math* verbally? How did they learn to do that, if so?

 

 

Colleen,

 

Rereading the thread, I see I had referred to a post in another thread, something now lost from my feeble memory. Perhaps Laura Corin could weigh in if you asked?

 

 

How do you *have an interesting math conversation*?? Where does a math-phobic person like me start to understand the concept of having a math conversation? (again, feeling really stupid to be asking these things)

 

 

 

Math is about more than answers. Ask about process. Start with a simple word problem. You can ask your student to draw a picture and explain it. Determine an alternate way of finding a solution together. My son will say "I like my way." Great. I will say "Well, I tend to see things geometrically, so I prefer to draw a picture and set it up this way."

 

Question all statistics. I think that the newspaper is a great way to start discussions on logic and critical thinking. In one of the earlier posts, I noted that my husband works with a man who is boorish. Instead of having give and take at a meeting, his strategy is to repeat his argument--only louder. (Apparently he is not alone in believing that debate consists of raising one's voice--not solid arguments.) Some thirteen year olds will be very frustrated by your inability "to see" their point. Instead of asking for a restatement, I would ask for more information on a specific point of their argument. This is probably not much different than those literary conversations you mentioned--now you are focusing on a proof, an algorithm, a scientific hypothesis.

 

I also think there is nothing wrong with admitting ignorance and telling a child that we will figure this out together. Having a write board with a hard problem on it can encourage discussion. (The same white board can have a new vocabulary word on it another day or some "research" question. "What lives in the hadal zone? Since photosynthesis cannot occur at this depth, how do you speculate animals in the hadopelagic live?") I think the key is to talk, talk, talk about everything.

 

A microbiologist with whom I am good friends gave me an article from a scholary journal to read this summer. Did I understand all of it? Heck no. But I read it, penciled some questions in the margin, we talked about philosophical ideas in modern microbiology. My friend had to reduce his vocabulary to mine since I am not a microbiologist. Later friend's wife told me how much he enjoyed that discussion. I think that he found a way of sharing a glimpse of his world with me--something that I appreciated.

 

Keep talking.

Jane

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Rereading the thread, I see I had referred to a post in another thread, something now lost from my feeble memory.

 

Even though the idea of being able to *talk* about math is new to me (as opposed to just learning formulas and solving problems with them for some unknown-to-me reason), I'm thinking that the answer is probably yes.

 

Math is about more than answers. Ask about process. Start with a simple word problem. You can ask your student to draw a picture and explain it. Determine an alternate way of finding a solution together. My son will say "I like my way." Great. I will say "Well, I tend to see things geometrically, so I prefer to draw a picture and set it up this way."

 

See, I used to hear this (what I bolded) and think that people meant that it didn't matter that 7+3=10, as long as the child was playing around with the counting sticks and *trying* to figure out how to combine 7 sticks and 3 sticks. I thought that was wrong, because I think after they play around with the sticks for a few months/years, they still need to memorize 7+3=10 at some point. But I think you mean math as opposed to basic arithmetic facts and processes?

 

When I took algebra and geometry and dropped pre-cal after months of trying, I could never see why I needed these classes, except that they were on the college-prep track, and I wanted my options open. But from what you're saying (and I'm going to have a browse in the "mathematics" sections of my World Book and Britannica encyclopedias), learning various branches of mathematics and the theories/concepts will help me to solve various problems in life? Help develop new ways of thinking about something? See that there may be more than one way of solving something? Is *this* why math is discussable? Because it has ideas and those ideas can be translated back into words? And then back into symbols? If *that's* the case, I think I could see my way into discussing more (or finding a mathematician tutor to discuss with them) as they get older, as opposed to handing over a math text and hoping they figure it out enough get a good grade to impress universities.

 

Some thirteen year olds will be very frustrated by your inability "to see" their point. Instead of asking for a restatement, I would ask for more information on a specific point of their argument. This is probably not much different than those literary conversations you mentioned--now you are focusing on a proof, an algorithm, a scientific hypothesis.

 

Ah, ok. I'm still not too sure what algorithm and proof mean (but I did look them up in the dictionary and re-read your explanation in another thread about proofs and axioms), but I think I see the correlation. I think I just have to get a better handle on what these math terms mean and have a look through some algebra and geometry texts to get a better feel. Hmmmm, math could turn out to be interesting for me when my kids get to high school! This is an exciting thought for me!

 

Having a write board with a hard problem on it can encourage discussion.

 

I'm getting the feeling from this and other posts about white boards that there is something more to using them that just something to write on (I use lots of scrap paper every day to illustrate math concepts - hey, maybe I *do* "discuss math" or at least arithmetic with my kids! - and to diagram sentences with my kids). Maybe the white board would somehow make things more interesting or understandable just because it's bigger and can have colourful markers.

 

My friend had to reduce his vocabulary to mine since I am not a microbiologist.

 

My children's allergy doctor is like this. He explains everything to me, and answers my questions with great detail that I can understand. When we first started going to him, one thing that really impressed me was when he whipped out a scrap paper and pen in response to a question, and *drew a picture* for me so that I could understand why my son could have eggs-in-baked-goods but not plain cooked egg. It was a very cool scientific explanation, and I was so glad he knew how to communicate it to a non-scientist.

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I tend to discuss math, science, history and not literature. I love to hear people discuss and analyze literature but I am sadly lacking in that quality. Give me a math problem, a bill, or an issue to discuss any day.

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Have you ever seen this book:

 

Good Questions for Math Teaching: Why Ask Them and What to Ask (Grades 5-8) by Lainie Schuster and Nancy Canavan Anderson. (There is also a K-6 version, but I'm not sure about the high school years.) It is essentially a collection of math problems to supplement any curriculum, but designed to be "good" questions.

 

A "good" question:

 

-help students make sense of mathematics

-are open-ended (multiple answers/multiple approaches)

-help unravel misconceptions

-make connections and generalizations

-lead students to wonder more about a topic

 

Anyway, I use these questions as "class openers." It works really well to generate discussion.

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We discuss math, economics, philosophy (and how math affects it) pretty regularly.

 

Anyhow, great question. Some of us do discuss these things, but we forget to mention that here. :o

 

Same here. I don't know much about math, but dh knows tons, and loves math. So there is much natural discussion.

 

But even with my limited knowledge ... if I hand my elementary student his Singapore book and he has problems, we sit down and talk through the problems. But that's just ... what we do. If I get stumped as well, we take it to Daddy, who discusses the problem at length ;-)

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