"Does McDonalds Sell Cheeseburgers?"...grrrrr

[Spy Car]

Within a year (of homeschooling) he'd moved up 2 grade levels, using a strong conceptual math program, and I've never written off any math concept as just being "beyond him." For example, we spent several weeks on the concept of rounding, which for some reason was a big sticking point for him, and when I was finally able to explain it to him in a way he understood he said "Oh! Why didn't you explain it that way to begin with!" :lol:

 

Let's take this dreaded subject :tongue_smilie:

 

We want to divide 10 by 3.

 

We could take an Orange Rod (a Ten value) and ask the child to see how many 3 Rods (Light Green) we could use if we wanted to spilt the 10 into parts of 3. They would then see they could use 3-Three Rods.

 

Is there any part left over? Yes. How much? One. That is what we call the "remainder."

 

That is the gist of it. Not difficult really.

 

I would really urge anyone who thinks that their child "just isn't the conceptual type" to read Liping Ma's book, and to keep trying to find the conceptual explanations that they can understand, instead of assuming that if they don't get the one explanation in the textbook, then they're just not capable of undertanding math concepts.

 

 

 

I'm getting confused on the multi-quotes. Did you write this, or did I? :tongue_smilie:

 

Either way :iagree:

 

Bill

[Guest Dulcimeramy]

As another person who does not get the hype about conceptual math, I do have a question:

 

How do people teach math to children without teaching them why it is true?

 

I'm thinking of the early elementary years. Don't you have to show them with manipulatives and such, exactly what they are doing when they add, subtract, multiply and divide? Don't you have to get out the measuring cups, pint jars, bushel baskets, measuring tapes, etc. to teach math?

 

How can you teach fractions without a knife and an apple, or measuring cups or cuisenaire rods?

 

How can you teach place value without buttons or other counters to show them that you are moving a "10" or a "100" over to the the other column?

 

I thought everybody showed these things. Am I right that they do, and there is some other near-esoteric level of conceptual understanding that goes beyond "acting out" math problems?

 

What is conceptual math and what is non-conceptual math?

[Daisy]

Never mind. I'm feeling low today. Should just get off this computer and go teach more math.

Edited October 7, 2010 by Daisy

[Pamela H in Texas]

in my opinion, the teacher did not explain the 'why' of long division well to my sibling.

 

Did not explain it well or the child didn't learn it in the first week? And the TEACHER is at fault because the child may need another round at it, the next method of instruction or to mature slightly? If YOUR kid doesn't learn something the instant you would like him to, do you claim you failed? Or do you think he probably just needs more time, more maturity, or a different consideration? Of course, you get a break (and SHOULD!), but so should the teacher.

 

This isn't to say that the teacher *could* have failed miserably at teaching it. It certainly wouldn't be the first time. I just think it stinks to ASSUME the teacher did a poor job (if she did it at all).

 

Yes, children should be taught the "why" behind what they are doing and not just the rote memorization of the steps but there will always be a child who looks at you with a deer in the headlights look when you explain mathematical concepts. No matter how long and hard you talk yourself blue in the face trying to explain it in a way that makes sense to them, they don't get the why of what they are doing. But they still need to know how to divide like everyone else. One day, when they are ready they might see the same explanation and the light bulb will come on and they will say "oh now I get it" but until then, they still need to know how to divide.

 

I agree :) Seriously, the kid had to be sent home for supper. So she didn't get it this week. It'll click next week or whatever.

 

I know when I was teaching my daughter long division, there was A LOT of discussion about kids "getting" long division. There was a lot of discussion about HATING it. And then they all got over it. Some took a short time and others took a LONG time. It all worked out, but usually not just in one day. Usually not just in one week.

 

mmmmmmm.... french Fries!! (and more importantly, for my sad gluten free self, they are some of the only fast food variety of fries that I can eat!) Now I want to go get some!!

 

Ummmm, they are not allowable for GF people at the one two towns over from here. Just letting you know if you travel, they aren't all acceptable. In addition, we HAVE found two restaurants at which dd can have them.

Edited October 7, 2010 by 2J5M9K

[Corraleno]

This is exactly what we do with history. When they are young, we tell them history stories (SOTW) and memorize some info. We don't sit around and explain the indepth whys of history to young children.

 

With life science we expose young children to the various kingdoms, but we don't spend a lot of time explaining the processes of biology. For example, we may discuss reproduction, but we don't usually get into the details the first time around.

 

We may introduce grammar through memorization and a little diagraming or parsing, but we do it again year after year. We don't expect young children to understand it all at once and never need to practice with the material.

 

Much of what we do in the elementary grades is fire lighting and peg hanging.

Well, that's not how I teach history or science, even to young kids. I do explain the processes and the connections and the "whys," so I guess that's just a difference in educational philosophies. However, I disagree strongly with the idea that young children aren't capable of understanding the hows and whys and the deeper connections, whether that's in math, science, history, or any other subject.

 

Jackie

[Mandy in TN]

Well, that's not how I teach history or science, even to young kids. I do explain the processes and the connections and the "whys," so I guess that's just a difference in educational philosophies. However, I disagree strongly with the idea that young children aren't capable of understanding the hows and whys and the deeper connections, whether that's in math, science, history, or any other subject.

I would be the Doubting Thomas when people claim that they teach something for the first time and at the same time teach all the hows and whys behind it.

 

Firstly, unless you have a highly gifted student, most people are not capable of gaining and retaining large amounts of new information at one time.

 

Secondly, history is flat out a survey subject. You can spend your whole life studying the hows and whys of just the French Revolution, so it is impossible to present them all to a fourth grader in a week.

 

Thirdly, (and this is more of a philosophy thing) if someone spends that much time attempting to spoon feed all the connections of Roman history or chemical analysis to a first grader, then they are not spending much time lighting a fire.

 

HTH-

Mandy

[Corraleno]

I've heard on these boards many times about homeschool moms who have had those 'aha' moments in math while teaching their own children (and thus are encouraged to teach the understanding to their own children)... but you had this aha moment as an adult when you have so much more maturity and growth in your brains.

I hear this a lot, too — and in every case I've read, the parent has had the "aha" moments because of using a math program or reading a book or participating in a discussion here, that explained the concepts properly. That, IMO, is the key, not age or brain growth. Many many math discussions on this board involve parents saying "if only I'd been taught this way when I was in school, math would have made so much more sense to me." I don't see people writing "I'm so glad I just turned 40 — now I can finally understand math!" :lol:

 

Not trying to argue.. just wondering about child development.... and thinking about things like why we don't start logic until later etc...

The idea of dividing education into three discrete stages is a useful heuristic device and a convenient organizational tool, but they are really not developmental stages. In classical education, Grammar, Logic, and Rhetoric are subjects, not stages. It's a bit like dividing education into the Language, Math, and Science stages, and implying that Language Stage kids aren't really capable of understanding science-level material.

 

Jackie

[Corraleno]

Thinking about child development is a good idea. One needs to ask themselves how to teach children the things they will need to know using means that fit their intellectual development at the moment. But trust me, there are ways to begin teaching math a very early age that address a deep understanding of how our number system works, and how mathematical laws function in ways that make math fun and engaging.

 

Math is usually a painful subject when "you don't get it." But when the early math education is all about making sure the child does "get it" math is fun and meaningful.

:iagree:

And I think this often becomes a vicious cycle — the child doesn't get the early conceptual foundation they need, so they're even more confused at the next level, which leads the parent/teacher/school to assume that this kid is just not good at math, so we'll skip the conceptual explanations and just let them memorize the algorithms so they don't "fall behind."

 

IMO the idea that "conceptual math" is just for "mathy kids" and plug-&-chug arithmetic is good enough for everyone else is backwards. I think the less inherently "mathy" a child is, the more important it is to make sure that math concepts are explained, from the very beginning, in ways they understand, and that they really do "get it" before moving on. I would much rather have a child who's doing Alg I in 10th grade and Alg II in 12th and totally understands it, than a child who scrapes through Calculus in HS by punching #s into a calculator with no real understanding of what they're doing.

 

Jackie

[Spy Car]

I hear you, Bill. It feels overwhelming though when you realize how many years and long agonizing hours you've dedicated to math only to realize that your kids are still behind. I'm worried about my daughter. We spent those years in Singapore and frankly she is still struggling horribly with any kind of math. There has never been that light bulb moment. I feel I've let her down. I loved Liping Ma's book, but feel like I'd need a brain transplant before I'd really be able to convey that knowledge to my children.

 

I wanted my children to love math. I really did. I was so excited to teach conceptual math. Yeah, I feel like a loser, and I really think that is how many homeschool moms feel about it. None of us wants to short-change our kids.

 

Of course you don't want to fail your kids. I want to be encouraging not discouraging. If anything I've written has made you feed bad I'm profoundly sorry for that. And if there is anything I can do to help just ask.

 

In my limited experience it can be the smallest things that could change things around for my son. It is not like he is a super-genius and I'm a super-teacher. I would sometimes try to "explain" things and get that blank look of non-comprehension we all dread.

 

So then it would mean "back to the drawing board" for me. How do I get the idea across in a way he will understand, and that is (hopefully) fun???

 

Sometimes it is the tiniest thing. Just changing the language of the explanations a little. Or working through the problems with base-10 flats and C Rods. And a huge factor has been speaking of numbers as their values in Hundreds, Tens, and Units not just in "English". Sure we use English too, but anytime something feels the least bit "rocky" we return to How many Hundreds? How many Tens? how many Units? And now Thousands too. This does clarify the mind. A small thing, but huge!

 

If I can help you in any way I will. I love your spirit Daisy, and I know you are a great mom.

 

Bill

[FO4UR]

I, for one, passed calc and physics in high school by merely plugging in numbers to formulas and using pneumonic devices like the one in the title of this thread. I had the ability to memorize tons of useless bits of info for short periods of time. For long term mastery, I needed to truly understand the concepts. According to my HS transcript, I was a good candidate for a math/science major, but it was all smoke and mirrors.

 

I think the issue is majoring on a *crutch* to the exclusion of true teaching and learning. The little McD's line will get a child through long division this year, but the bigger issue is that the child is learning to bypass problem solving in order to stuff their brains with yet another bit of trivia.

 

I think it's better to wait on long division until the child has the maturity to understand it. Pacing my own children individually is one of the main reasons I hs. No, individual pacing is not possible in a classroom...and the students pay a price for that.

[Poke Salad Annie]

I think this is why I hit such a wall in algebra during jr. high school. I never had trouble with math until I started that class. I was so used to just following along, doing what the book had shown to do for all of elementary. When it was expected of me to have and use some higher order thinking skills to set up algebra problems, I couldn't do it.

 

Since using MEP at home over the last couple of years, I have had several *aha* moments myself, and I truly wish that I could have used something that made better sense during my elementary school years the way this program does. We are constantly solving for the unknown number (which I call *n*--to set up for algebra later), and we work problems in an orderly fashion, copying the problem onto a dry-erase board, then bringing down each step until we find the solution. Just the fact that MEP teaches solution sets at a very young age is thrilling to me!

[Spy Car]

I would be the Doubting Thomas when people claim that they teach something for the first time and at the same time teach all the hows and whys behind it.

 

Firstly, unless you have a highly gifted student, most people are not capable of gaining and retaining large amounts of new information at one time.

 

 

Not necessarily all it at once, but with exposure to bigger ideas from the outset.

 

What do I mean?

 

Well, a young child playing with rods can "prove" 4 three times is the same value as 1-Ten and 2-Units. They do this a number of times and they become convinced 3x4=12.

 

And they can also "see" that 3 four times is the same as 4 three times. There is the Commutitive Law again.

 

Then once in a wee bit deeper, you could look at 3 nine times. It might be a stumper for a young child to do that in their heads. But what if you ask: Is 3 nine times the same as 3 five times plus 3 four times? Not sure? Let's see. Rod play.

 

Look it is. That is the Distributive Law. We will do more on this law, but for now let work with this. Tell me, what is 3 five times? The same as 5 three times, good you knew that. So what's 5 three times? 15. OK what about the other part? What is 3 four times? 12, that's right. You remember doing that with the rods many times right.

 

OK so now we have 15 plus 12. 15 is how many Tens and how many Units? 1-Ten and 5-Units. And we are adding what? 1 more Ten and how many Units? 1-Ten and 2-Units. So now how many tens and how many Umits are there?

 

2-tens and 7-Units. Very good. So we learned 3x9=27, and in the process we learned about the Distributive Law, and used our old friend the Commutative Law.

 

Now we could have a trick about taking one from the multiplier of 9 being a 2 and needing to add a digit (7) to add up to 9, to get 27, or just memorized the stupid "math fact" because it is "easier" than walking though this process until it gels.

 

But when you 6 year old (or whatever age) can build on these lessons and employ the Distributive Law to solve equations that are much more complex than 3x9 in their minds (because they can apply the laws of mathematics) it is very satisfying.

 

Bill

Edited October 7, 2010 by Spy Car

[Snowfall]

I have no problem with mnemonics, even if I wouldn't have chosen that particular one ;). My issue is that according to my sibling, the teacher did not explain the rationale behind the process. Of course, my sibling could be mistaken. She could have been in the bathroom when the teacher explained it, she could have been giggling in the back row (although I doubt it LOL). Or perhaps, just perhaps, it doesn't stick out in her mind because her teacher actually didn't emphasize it. What the teacher does seem to have emphasized is procedural math. Hence the memorable mnemonic.

 

Personally, I feel _understanding_ the WHY of math is just as important, if not more important, than understanding the _how_ of math. Mandy in TN wrote "some children simply will not get the concept. This doesn't mean that it shouldn't be presented, but dawdling over it can likely bring about confusion and contempt." I politely, but strongly, disagree with this statement. To state that the act of explaining the rationale behind a procedure can actually confuse a child more (perhaps, in the short term, that may be true) and thus, one should simply move along to the procedure because, after all, that's what matters, is a bit mind-boggling to me. Perhaps the explanation itself was inadequate, not the child's understanding. I do think most PS teachers are concerned more about whether a child will get the right answer (and that's obviously not their fault--the emphasis on test scores is ridiculous) than whether a child understands why one, for example, "carries down" the next digit in the number.

 

I have always loved math and done well at it myself. But it wasn't until I sat down and learned the process of math that I truly understood the beauty, the versatility, the potential of math. And if anyone should be conveying that beauty to children, it's the teachers. I think overall, our nation would be far more competitive in engineering, science, and other math-based disciplines if the WHY of math, the beauty of math, were brought more front and center.

 

Of course, many will argue "who has time for that?" and "most kids don't care about that" and "teachers can't teach 30 kids about the beauty of math--they need to teach them how to get the answer". I would argue that that is specifically what is wrong with mathematics education in this country.

 

Okay, off my soapbox.:tongue_smilie:

 

:iagree::iagree::iagree:And couldn't do so more. My dd is not easy to teach. She's difficult, she's petulant, she's whiny, and she doesn't like to work hard at math because she's very easily frustrated (hence the difficult, petulant, whiny behavior - her frustration tolerance it extremely low). That means sometimes she doesn't want to know why - she just wants the answer. BUT when she gets something, when she really, really GETS something, it's this amazing thing. When she was 5 and she said, "There's a zero in every number!" and was so proud of herself for figuring out that any number plus or minus zero does not change, or when she told me (not the other way around) "It doesn't matter how you put the numbers - it's all the same thing!" explaining the Commutative Law, it was amazing, and it DOES matter that she understands that. It makes it easier for her to add 2+8, because she has trouble thinking in terms of adding that big number to that small number, but it's easy for her to think of adding that small number to that larger number. These things do matter, and even kids whose eyes cross when you explain them or lead them to work it out on theit own can get it, if given the opportunity.

[Momling]

At least they're teaching long division at all! Our public school district uses the TERC Investigations math series and children are never taught how to do any algorithm at all.

[Corraleno]

I would be the Doubting Thomas when people claim that they teach something for the first time and at the same time teach all the hows and whys behind it.

 

Firstly, unless you have a highly gifted student, most people are not capable of gaining and retaining large amounts of new information at one time.

 

Secondly, history is flat out a survey subject. You can spend your whole life studying the hows and whys of just the French Revolution, so it is impossible to present them all to a fourth grader in a week.

 

Thirdly, (and this is more of a philosophy thing) if someone spends that much time attempting to spoon feed all the connections of Roman history or chemical analysis to a first grader, then they are not spending much time lighting a fire.

 

HTH-

Mandy

Lol, you're making a whole lot of assumptions about (1) how I teach and (2) how my kids learn. First, I never ever said that I'm teaching ALL the hows/whys/connections in every subject in kindergarten. The fact that one can't teach all of the hows/whys/connections in history and science at a graduate level to an elementary student doesn't mean you can't teach any of them, or that telling stories and memorizing facts are the only alternative. When DS was 5 and asked me if it was possible to add/subtract/multiply/divide infinity, I didn't say "oh, you can't possibly understand that, you're only 5." ;)

 

Secondly, who says I'm spoon-feeding? :confused: How do you know I'm not asking them questions and teaching them how to research the things they're interested in, and helping them design & do experiments to see what happens? The idea that I'm not spending any time "lighting fires" is laughable — that's exactly what I'm doing. My 12 yo's current bedtime reading is Warfare in the Classical World, because he's learning classical history through his own research and interests in the history of warfare and weapons. I bet he knows more about Ancient Greece than 99.9% of adults in this country, and he's "getting it" in a deep and meaningful way — although theoretically he should only be capable of "beginning Logic Stage" thought processes. We have discussions about the differences between Spartan and Athenian culture, and the sacrifices Sparta made in terms of quality of life in order to have the military prowess they had. DS made the comment that Sparta gave up so much in order to focus 100% on their military, and yet Athens was nearly as powerful, militarily, while having (in his opinion) a much better quality of life. We've discussed the irony of, and reasons for, the fact that Spartan women enjoyed much higher status than women elsewhere in Greece. DS's questions as to why we, in the west, seem to idolize Classical cultures that routinely raped, tortured, and slaughtered people, including children, kept slaves, and frequently treated women as barely human, have led to some very interesting discussions as well.

 

Obviously I don't go into the same depth with my 8 yo, but we still talk about why things happen, and how some events can lead to other (often unexpected) events, how people's attitudes and values both affect and reflect what their society does, etc. And the same in science — when DD was 7 and asked why the lines on her arms were blue but if she gets a cut it bleeds red, I didn't just say "the oxygen in the air makes it red." I got out the books, looked up some videos on Discovery Streaming, and we learned about the circulatory system. One of her particular interests is money and economics — she's constantly asking questions like who decides what something costs, where does the bank get the money from, where does WalMart get all the stuff they sell, etc. so I've gotten books for her on the history of money at her level (DK and Betty Maestro), we've had many discussions about where WalMart gets their stock, and how much people are paid in other countries to make things that sell for far more in this country, how the difference between what the store pays for something and what the customer pays for it is how stores make money, etc. We also looked at books like Hungry Planet and How People Live to see the differences in standard of living among different countries, and that led to looking at the atlas to see where most of the richer people live compared to where most of the poorer people live, etc.

 

A parent can choose to explain the hows and whys and connections in history, science, math, etc. to young children (at the child's level), or they can choose to postpone those kinds of discussions to a later date/stage, depending on their own philosophy of education. But to assume that children are inherently incapable of understanding these things, or that anyone who does choose to teach this way is "spoonfeeding" or failing to light fires, is quite untrue.

 

Jackie

[Spy Car]

At least they're teaching long division at all! Our public school district uses the TERC Investigations math series and children are never taught how to do any algorithm at all.

 

"Between two evils, chose neither" -C. H. Spurgeon

 

Bill (who figure someone will get a kick out him quoting Spurgeon :D)

[FO4UR]

"Between two evils, chose neither" -C. H. Spurgeon

 

Bill (who figure someone will get a kick out him quoting Spurgeon :D)

 

:lol::lol::lol:

 

Funniest thing I've heard all day...

[catholicmommy]

Thinking about child development is a good idea. One needs to ask themselves how to teach children the things they will need to know using means that fit their intellectual development at the moment. But trust me, there are ways to begin teaching math a very early age that address a deep understanding of how our number system works, and how mathematical laws function in ways that make math fun and engaging.

 

Math is usually a painful subject when "you don't get it." But when the early math education is all about making sure the child does "get it" math is fun and meaningful.

 

Bill

 

Thank you so much for your thoughtful response. Now you all have me thinking more about math and about my own understanding of it and I'm intrigued :glare: How can I add in some of this great conceptual math to the program I am already using? I can't see me jumping ship on my curriculum mid year, but I'm open to some helpful ideas to add in some of this stuff. How can I enlighten MYSELF about this way of math thinking?

[catholicmommy]

I disagree.

 

"I never see anyone write that their child totally doesn't understand the "concepts" of history, so it's OK for them to just memorize a bunch of names and dates and forget about understanding the "why" of it."

 

This is exactly what we do with history. When they are young, we tell them history stories (SOTW) and memorize some info. We don't sit around and explain the indepth whys of history to young children.

 

With life science we expose young children to the various kingdoms, but we don't spend a lot of time explaining the processes of biology. For example, we may discuss reproduction, but we don't usually get into the details the first time around.

 

We may introduce grammar through memorization and a little diagraming or parsing, but we do it again year after year. We don't expect young children to understand it all at once and never need to practice with the material.

 

Much of what we do in the elementary grades is fire lighting and peg hanging.

Mandy

:iagree:

[catholicmommy]

I hear this a lot, too — and in every case I've read, the parent has had the "aha" moments because of using a math program or reading a book or participating in a discussion here, that explained the concepts properly. That, IMO, is the key, not age or brain growth. Many many math discussions on this board involve parents saying "if only I'd been taught this way when I was in school, math would have made so much more sense to me." I don't see people writing "I'm so glad I just turned 40 — now I can finally understand math!" :lol:

 

 

OH THAT'S my problem... i'm not 40 yet :lol:

 

Seriously though, I can see where you might be right about the curriculum, but now as adults, we already have the structure in our minds to hang the understand on. We know how to do math so it's easier to understand the why now because we aren't trying to learn it all at the same time. (like I said, i'm just trying to think through these things out loud.. I don't have any real experience or knowledge about all of this so don't attack me, but I'm certainly open to getting a better understanding of it).

The idea of dividing education into three discrete stages is a useful heuristic device and a convenient organizational tool, but they are really not developmental stages. In classical education, Grammar, Logic, and Rhetoric are subjects, not stages. It's a bit like dividing education into the Language, Math, and Science stages, and implying that Language Stage kids aren't really capable of understanding science-level material.

 

Jackie

 

Of course I am aware of this and I have the LCC book and know about neo-classical vs real classical (oh the debates homeschoolers can have:D)... but I believe there is truth to the fact that the human mind does need to grow before it can understand and manipulate certain kinds of information... or maybe that's totally outdated? No, it can't be... I have watched my own kids grow slowly through preschool and the elementry years and they do have to mature before they can get certain things... For one, I'm thinking about something simple like a sense of humor and learning to get the irony in jokes.

 

Is this conceptual math thing really that serious? I am truly going to be holding my kids back from some great career if I don't teach them that way? Can you REALLY compare it to breastfeeding vs bottle feeding? I"m interested.. really. I guess I'll have to get the famous book. I wonder if it's in my library.

[Spy Car]

Thank you so much for your thoughtful response. Now you all have me thinking more about math and about my own understanding of it and I'm intrigued :glare: How can I add in some of this great conceptual math to the program I am already using? I can't see me jumping ship on my curriculum mid year, but I'm open to some helpful ideas to add in some of this stuff. How can I enlighten MYSELF about this way of math thinking?

 

If you have not read Liping Ma's book, do so. It eloquently lays out with examples the differences between having only procedurally based knowledge and teaching/learning for conceptual understanding. It will only get you so far, as it is not a "how to", but it should help identify the differences and help establish goals.

 

The Miquon teachers materials are fantastic. Even if one did not use Miquon (which is a program I will be eternally grateful for having started with in preschool) the teaching materials do a magnificent job of teaching parents how to teach mathematical axioms in simple and easily understood ways.

 

Right now there is a Lab Annotations book on the For Sale board for $6 postage paid. The thing is worth it's weight in gold (if used)!

 

The Malcom and Baldrich (sp?) book distributed by Singapore math is very informative.

 

The Activities for AL abacus book by Right Start author Dr Cotter has many good ideas, especially on the value of teaching place value early. I'm not with her on making 5s and some of her other ideas, but still a valuable rescourse from I've stolen a number of good ideas.

 

For making math a fun "activity time" the MEP Lesson Plans, available without cost on-line, are chock full of teaching ideas.

 

These are a few.

 

Bill

[TKDmom]

This is the mnemonic that my half-sister's gifted public school is using to teach her long division (4th grade). It stands for "divide, multiply, subtract, carrydown". Does she understand the why's and how's. Of long division? Er...no. But she does do page after page of division problems to "cement" this mnemonic in her head.

 

My son has been studying long division over the last week or so, and we've spent a lot of time learning WHY exactly one subtracts, why exactly one "drops down"....so that if he does forget,he can really think about what to do next. It's slow going but I really feel strongly about this, and to be honest, I feel irrationally annoyed that my half-sister's "gifted" class is teaching her no more than a silly procedural mnemonic when she certainly has the capacity to understand it on a deeper level.

 

You're preachin' to the choir. :lol: I have been reading Liping Ma's book for the last 3 months. It's going slowly because I can only read about a page before I get too upset at the American teachers' lack of understanding to read any further for a week or two. I was blown away by the accounts of teachers who could not divide by fractions, let alone teach kids how to do it.

[Corraleno]

Is this conceptual math thing really that serious? I am truly going to be holding my kids back from some great career if I don't teach them that way? Can you REALLY compare it to breastfeeding vs bottle feeding? I"m interested.. really. I guess I'll have to get the famous book. I wonder if it's in my library.

It depends on what you mean by "serious." I suspect that the vast majority of people in this country have very little conceptual understanding of math, so most careers are clearly possible without it. Most Americans have little conceptual understanding of the processes of history or science either, and perhaps that's more significant, because all these people vote on issues that have historical and scientific bases.

 

With my kids, a conceptual understanding of math is necessary because DS will definitely be going into science, and DD may end up in a field related to finance, business, or economics. But even if they were headed for majors in the arts or humanities, it would still be important to me that they truly understand math, just as it's important to me that DS truly understands history and DD truly understands science, even though they (probably) won't be majoring in those fields. For me, personally, that's the whole point of homeschooling, and the point of a semi-classical education: the ability to instill a deep and meaningful understanding of the hows and whys of the world, not just the whats and whens.

 

Jackie

[TracyP]

LOL, I have been "offline" for several weeks. It is surreal to come back to the same old debate.:lol:

 

Catholicmommy, I was a Doubting Thomas who has "seen the light" thanks to the Ma book. I highly recommend. Not a how to, like Bill said, but very eye opening. It was *my* a-ha moment.

 

When I hear this procedural vs. conceptual debate I do get confused about one thing. I hope this doesn't sound snarky - It is not meant that way. Why are our (American) kids expected to not "get it" while kids in Finland, Singapore, Japan are expected to actually understand math. Do they have higher IQ's? Are they all "early bloomers"? Or do they have superior teachers with a superior understanding of math who take the time to make sure the students do "get it"? That is the frustration I hear in the OP.

Edited October 7, 2010 by TracyP

[FO4UR]

 

When I hear this procedural vs. conceptual debate I do get confused about one thing. I hope this doesn't sound snarky - It is not meant that way. Why are our (American) kids expected to not "get it" while kids in Finland, Singapore, Japan are expected to actually understand math. Do they have higher IQ's? Are they all "early bloomers"? Or do they have superior teachers with a superior understanding of math who take the time to make sure the students do "get it"? That is the frustration I hear in the OP.

 

There is a link to an article on math Scope & Sequence . It's interesting. Americans try to cover everything from K up at increasingly deeper levels. Asians focus only on the 4 operations early and then their S&S looks like a triangle from there...growing in the number of concepts taught as the children grow.

 

Thinking about the differences in S&S brings to light why American teachers in ps classrooms CANNOT teach like Asian teachers. They cannot teach to mastery every concept covered on the end-of-year tests b/c there is just.too.much.content...and yet their jobs depend on test scores. The solution is the procedural trickery to gain the appearance of mastery. Asian 1st graders master addition and subtraction...American 1st graders memorize the facts quickly so they can cover concepts not asked of Asian children until years later.

 

Geesh, I was going into teaching *music* at the elementary level and was scolded b/c while my lessons were very good, enjoyable and "the kids would never forget...." I couldn't "waste" time covering each topic so thoroughly b/c of THE TEST at the end of the year. In music.:glare: And, I was expected to create a S&S that looks much like the American math S&S...cover everything at the surface and move along, move along...who cares if they retain beyond our school walls. That's someone else's problem.:glare: I decided to SAHM with my babies rather than teach...surprise, surprise...:tongue_smilie:

 

I agree with the sentiment that we need to "hang pegs" for later learning. I believe that the "math pegs" are full and complete comprehension of the 4 basic operations. That is the "grammar" stage level of math if we are going to think in terms of the trivium being developmental stages. In fact, I think that the reason that long division is so hard for most kids is b/c they lack either the understanding of basic +-x *or* they lack the experience of "figuring out hard things" and have grown to expect things to be cut-up and dipped in ketchup before they can digest it. (...yes, I'm running with the McD's theme here...:D) Let a young child work to figure out 7+8 and don't rush to memorize facts...the process is healthy and fruitful. It even results in automatic recall of facts! LOL (Of course, that means setting aside State Standards and maybe even your fave curric in 1st grade.)

[Corraleno]

I agree with the sentiment that we need to "hang pegs" for later learning. I believe that the "math pegs" are full and complete comprehension of the 4 basic operations. That is the "grammar" stage level of math if we are going to think in terms of the trivium being developmental stages. In fact, I think that the reason that long division is so hard for most kids is b/c they lack either the understanding of basic +-x *or* they lack the experience of "figuring out hard things" and have grown to expect things to be cut-up and dipped in ketchup before they can digest it. (...yes, I'm running with the McD's theme here...:D)

:iagree:

I don't know where the assumption came from that the "pegs" we set in elementary school have to come from memorization of names/dates/facts. "Conceptual pegs" can be just as, if not more, effective, whether we're talking about math, history, or science.

 

I was fine with letting DS move ahead conceptually before he'd memorized all his math facts, but I would not have let him move ahead without the conceptual understanding. And, waddya know, now he knows his math facts just from using them. OTOH, he would not have achieved the conceptual understanding just by applying the algorithms over and over. Benoit Mandelbrot managed to "father" the field of fractal geometry without ever memorizing his times tables, and DH has patents in several countries that depend on sophisticated mathematical algorithms he developed himself, despite not knowing his times tables.

 

Jackie

[catholicmommy]

There is a link to an article on math Scope & Sequence . It's interesting. Americans try to cover everything from K up at increasingly deeper levels.

 

I agree with the sentiment that we need to "hang pegs" for later learning. I believe that the "math pegs" are full and complete comprehension of the 4 basic operations. That is the "grammar" stage level of math if we are going to think in terms of the trivium being developmental stages. In fact, I think that the reason that long division is so hard for most kids is b/c they lack either the understanding of basic +-x *or* they lack the experience of "figuring out hard things" and have grown to expect things to be cut-up and dipped in ketchup before they can digest it. (...yes, I'm running with the McD's theme here...:D) Let a young child work to figure out 7+8 and don't rush to memorize facts...the process is healthy and fruitful. It even results in automatic recall of facts! LOL (Of course, that means setting aside State Standards and maybe even your fave curric in 1st grade.)

 

Mmmm ketchup!

I like your explanation of the 'grammar' of math. We started with MUS and i got stressed and scared into switching when i realized all we did in alpha was learn addition and subtraction, amd the other math S&Ss where teaching fractions and calendars and who knows what else. Would you consider MUS a more 'conceptual type program'?

 

When you say get them to work to understand 8+7 does this mean the whole number bonds thing? I have a dd in K right now, and two 4 year olds who will be starting K next year and i'm open to learning a new way to teach math. It's just scary to try to learn something new for myself, kwim? Is singapore easy to teach? What about MM? I ask, not to be lazy, but to be realistic, as i don't have the conceptual understanding, nor do i have loads of time to teach each individual child as i have six kids under ten. (i ditched traditional spelling like spelling workout because it was not providing any conceptual understanding :lol: and switched to AAS, but find it takes so much more time to implement that i have a harder time getting it done. Is a great program that gathers dust still better than a poorer program that you use every day?

[Halcyon]

Can I just interject that I love this board :D?? I started this thread annoyed and vaguely pi**ed off, but now I'm just plain ol' happy.

 

I love passionate discussions ;)

[catholicmommy]

 

For making math a fun "activity time" the MEP Lesson Plans, available without cost on-line, are chock full of teaching ideas.

 

These are a few.

 

Bill

 

Thank you! i will look into MEP as free sounds like my style right now. What i need are things that can be adapted for the needs of a large busy family. Any ideas? I just don't have oodles of time.

 

Btw where does LoF fit into the spectrum?

[forty-two]

Would you consider MUS a more 'conceptual type program'?

 

From what I've read about it, I'd say it hits the surface level whys. MUS seems to consider place value as its fundamental concept (as do most elementary programs), and works hard to relate everything back to place value. So, it *does* explain the whys in that it explains how place value works in recording quantities (that 78, for example, represents seventy-eight units) and how the various procedures work in terms of place value.

 

But afaik MUS doesn't go beyond place value, to explain the field axioms of the natural numbers (communtative law, distributive law, etc.) and show how place value is developed from those axioms and how the various procedures can be derived from those axioms. Place value is certainly fundamental to *calculating*, but it isn't all that fundamental to *math*. The field axioms, however, *are*. And so I consider a "proper" :lol: conceptual arithmetic program to be one which uses the fields axioms as its fundamental concept.

 

From what I see, Miquon does this, and MEP and CSMP seem to do it as well (it certainly underlies their presentation, but I'm not sure how explicit they make it; SM, for example, implicitly teaches it, so that you get practice thinking that way, but does not explicitly develop everything from first principles).

[catholicmommy]

 

From what I see, Miquon does this, and MEP and CSMP seem to do it as well (it certainly underlies their presentation, but I'm not sure how explicit they make it; SM, for example, implicitly teaches it, so that you get practice thinking that way, but does not explicitly develop everything from first principles).

 

What is CSMP?

[forty-two]

Btw where does LoF fit into the spectrum?

Going on samples and WTM discussions (truly impeccable sources ;)), LoF seems pretty darn conceptual to me. Fits my criteria of using the field axioms as its first principles, as far as I can tell. Don't know whether it is more rigorous (in the math sense of being proofy) than AoPS, but it was written by a math PhD, so he *should* know what's what - and from what I've seen of it, he does :).

[forty-two]

What is CSMP?

CSMP stands for Comprehensive School Mathematics Program, and it is a 60s New Math K-6 program (Miquon is also New Math) that is now offered free online. It leads into a *very* proofy secondary program, so it definitely has a pure math bent. Basically it feels like a bunch of math geeks got together and tried to make arithmetic fun :tongue_smilie:. They seem to be using arithmetic as a vehicle to explore math. Full of geeky awesomeness :thumbup:. But *very* non-traditional. I love it :lol: - I plan to use it and MEP as joint spines, with a Miquon/RS lead-in.

[maryanne]

Forgive me if I don't get what all the fuss about conceptual math is all about. I also don't get the accusations that Saxon doesn't teach concepts either. Math is very painful to do if you understand the concept but have no speed or fluency with the procedures. Certainly you can re-derive everything, but that's a long,slow, painful way to get math done.

 

OK, I've gone and read part of the Liping Ma book. I don't have time to read the whole thing right now. Specifically, I read the intro and the chapter on subtraction with regrouping. This was the most relevant chapter for me to read because I have already taught both of my children to subtract with regrouping: my older child using Saxon and my younger child using Singapore.

 

Now that I've read the description of teaching this concept--conceptually as the Asians do, I'm even more befuddled by the accusations that Saxon doesn't teach concepts. I taught both my children essentially the same thing, one using Saxon and pennies, dimes & dollars, the other using Singapore and poker chips. Either way it was the same concept, and they both got it. I don't remember the precise wording in the Saxon script, but I probably didn't follow the script anyway. Singapore doesn't have a script--it's mostly pictoral so I was using my own words in both cases.

 

Saxon emphasizes automaticity (speed and fluency) more that many curriculums, but it also teaches concepts. What it lacks in some areas is variety in applications, so we supplement with Singapore CWP books. Singapore's base text and workbook lack sufficient practice to develop speed and fluency for some and we will supplement as needed with Saxon drill sheets. Both curriculums are very good, neither is perfect.

 

What I did learn from reading that chapter is that apparently, according to the author's research, the US has far too many innumerate (the mathematical equivalent of illiterate) elementary school teachers attempting teach arithmetic and failing rather badly at it. Can you imagine the uproar if an illiterate person was hired to teach reading!

 

I also agree with the previous posters who have suggested that conceptual understanding often follows (as opposed to occuring concurrently with) procedural mastery, sometimes sooner and sometimes later. I also think that the analogy with history is not an unreasonable one. We can introduce and teach procedures and have students memorize processes and formulas. Then we can go back and revisit the ideas later to develop the understanding of the concept. This often occurs in higher level mathematics where a relationship (or formula) is introduced and used, but the proof is not provided until later on after additional mathematics needed for the proof have been covered.

 

Many of us use procedures and formulas successfully without understanding the why regularly in every day life. When is the last time anyone derived the formula for the area of circle or volume of a cylinder. I use these formulas regularly (without a calculator) to determine which round casserole dish I need if I make 1 1/2 or 2 times a recipe. I'm very thankful that I've memorized the procedure and that I don't have to discover it every time I need it, because I don't think I could do that if my life depended on it.

[Spy Car]

 

When you say get them to work to understand 8+7 does this mean the whole number bonds thing? I have a dd in K right now, and two 4 year olds who will be starting K next year and i'm open to learning a new way to teach math. It's just scary to try to learn something new for myself, kwim? Is singapore easy to teach? What about MM? I ask, not to be lazy, but to be realistic, as i don't have the conceptual understanding, nor do i have loads of time to teach each individual child as i have six kids under ten. (i ditched traditional spelling like spelling workout because it was not providing any conceptual understanding :lol: and switched to AAS, but find it takes so much more time to implement that i have a harder time getting it done. Is a great program that gathers dust still better than a poorer program that you use every day?

 

So with things like 8+7 (which happens to be the equation my son was hammered with more than any other to the point where it's mere mention elicits laughter from us both :D) "making tens" per se is more a "strategy" rather than a "concept" but is built on the mathematical understanding that numers can be re-grouped to facilitate mental math.

 

What does the 8 need to become a Ten? What does the 7 become if it gives up 2 to the 8 so the 8 came become a Ten? What is 1-Ten plus 5-units?

 

Lather, rinse, repeat. Does it take more effort that just memorizing? [insert expletive here) yes! Is this the long way? Yes.

 

But later, when they are doing muti-digit addition and subtraction in their minds, as my son can do now, I for one only wish I'd been given this sort of education. And all the effort pays off in a big way.

 

I've drawn on more math programs than I'd care to admit, and have been quite happy with that approach for me. One I have not used (my plate being full) is Math Mammoth, which I know from limited samples and through reading many discussions of that program here. For what you are saying I would tend to think it might be a program well suited to you. Less bouncing around with many books, and Maria Miller seems to have a talent for teaching the same sort of skills one finds in Singapore Math in a way that is accessible to someone who is being "realistic" with her time.

 

If you were feeling more ambitious, I would start the children off using Miquon and dig into the 3 teachers in Miquon. Even if you did not do the whole program there is gold there. And learning mathmatical laws by "discovery" (which is an misunderstood concept and simpy means the children prove the relationships and laws are true using concrete means) means they "own it."

 

MEP, while "free" and higly interesting, might (or might not) throw you for a loop if you are feeling a lack of confidence or a lack of time, and might be less well suited to your needs than something like MM, I think.

 

Bill

[Mandy in TN]

Not necessarily all it at once, but with exposure to bigger ideas from the outset.

Bill-

 

That was a very nice math lesson. I really did read Liping Ma's book. However, there was nothing off-the-wall new to me in her book. I am a huge fan of unmarked C-rods. My dh and I taught our little guy with these, dominoes, dice, playing cards, pattern blocks. This sort of instruction comes to us naturally and we have no problem incorporating it into whatever math program we use.

 

My problem when people talk about conceptual math is that they come back later saying that their child isn't retaining his math facts. Well, this is where balance comes into play. The child may understand a concept when it is presented. However, if he doesn't practice and routinely return to practice more, he forgets the concept and can't perform the procedure with immediacy.

 

If you present a concept and the child immediately understands, that doesn't mean that he doesn't need to practice. Sure- there are exceptions, but most children most of the time benefit from practice.

 

Mandy

[Spy Car]

Bill-

 

That was a very nice math lesson. I really did read Liping Ma's book. However, there was nothing off-the-wall new to me in her book. I am a huge fan of unmarked C-rods. My dh and I taught our little guy with these, dominoes, dice, playing cards, pattern blocks. This sort of instruction comes to us naturally and we have no problem incorporating it into whatever math program we use.

 

My problem when people talk about conceptual math is that they come back later saying that their child isn't retaining his math facts. Well, this is where balance comes into play. The child may understand a concept when it is presented. However, if he doesn't practice and routinely return to practice more, he forgets the concept and can't perform the procedure with immediacy.

 

If you present a concept and the child immediately understands, that doesn't mean that he doesn't need to practice. Sure- there are exceptions, but most children most of the time benefit from practice.

 

Mandy

 

Who said no practice? My son gets quizzed during car trips, we play games, he does workbooks. He knows his "math facts." But every step of the way toward learning these he had to me able to "explain" the operation first. Otherwise, no credit.

 

Simply "memorizing" was not the way "math facts" were taught here. And I think he's as quick as almost any First Grader with his basic math facts (maybe he'd be a bit faster if that's where we put all the effort), but he can also do 573 - 257 in his mind, which impresses me. And I've had to sharpen up because he is gaining on me :D

 

In any case it is not a race, if it took him two years to get where he is now that would be fine. The point is he understands what he is doing as a result of methodical application of method combined with a age appropriate understanding of the mathematical laws behind the method. These are reasonable goals.

 

Bill (who is getting his own math re-education)

[Mandy in TN]

Who said no practice? My son gets quizzed during car trips, we play games, he does workbooks. He knows his "math facts." But every step of the way toward learning these he had to me able to "explain" the operation first. Otherwise, no credit.

 

Simply "memorizing" was not the way "math facts" were taught here. And I think he's as quick as almost any First Grader with his basic math facts (maybe he'd be a bit faster if that's where we put all the effort), but he can also do 573 - 257 in his mind, which impresses me. And I've had to sharpen up because he is gaining on me :D

 

In any case it is not a race, if it took him two years to get where he is now that would be fine. The point is he understands what he is doing as a result of methodical application of method combined with a age appropriate understanding of the mathematical laws behind the method. These are reasonable goals.

 

Bill (who is getting his own math re-education)

This is exactly what I do and what needs to be said.

 

Anyway, I didn't mean you.;)

 

I just see threads where after using conceptual maths and only doing a few problems a day for a period of time someone's child is unable to perform basic multiplication or something. They heard you and others espousing the benefits of conceptual math, but they didn't understand that they also needed to practice those skills outside of the conceptual lesson.

 

You understand that there needs to be balance. After the concept is presented it must be practiced. This is balance.

Mandy

[Spy Car]

This is exactly what I do and what needs to be said.

 

Anyway, I didn't mean you.;)

 

I just see threads where after using conceptual maths and only doing a few problems a day for a period of time someone's child is unable to perform basic multiplication or something. They heard you and others espousing the benefits of conceptual math, but they didn't understand that they also needed to practice those skills outside of the conceptual lesson.

 

Oh yea, and in some countries they beat it into the children with cram courses and the like.

 

I have not found that necessary, and have been more concerned with "automaticity" masking understaning than having the calculations being "slow."

 

But not unlike reading. I tried to make the sounding out of words "music to my ears" (cognitive dissonance) rather than trying to rush to fluency through sight-reading. Same with the math facts. Just had to hunker down and talk them through again and again. And now they are fast, like the reading is fluent. Or close to it.

 

You understand that there needs to be balance. After the concept is presented it must be practiced. This is balance.

Mandy

 

Yes, balance. I just think automaticity can come faster post-understanding, than understanding comes post-memorization. And that memorization can mask problems. So that balance has to be weighed and measured.

 

Bill

[Spy Car]

So the First Grader came home from school and said the teacher was showing 3+5 is the same as 5+3 today. And he evidently raised his hand and said:

 

That's the Commutitive Law

 

Excuse me? Could you repeat that?

 

The Commutitive Law

 

*blink*blink*

 

Well that's some pretty fancy Third Grade Math!

 

:lol::lol::lol:

 

Bill (honorary home-schooler today :tongue_smilie:)

[Corraleno]

I have a dd in K right now, and two 4 year olds who will be starting K next year and i'm open to learning a new way to teach math. It's just scary to try to learn something new for myself, kwim? Is singapore easy to teach? What about MM? I ask, not to be lazy, but to be realistic, as i don't have the conceptual understanding, nor do i have loads of time to teach each individual child as i have six kids under ten.

I think Math Mammoth would be a great option for you. It teaches math conceptually, like Singapore does, but Maria Miller addresses the explanations directly to the child, which eliminates the step of parents having to read a TE/HIG and then teach the concepts themselves. She also breaks the concepts into very small increments and walks the student through the process with clear, explicit, step-by-step explanations and illustrations — so even if the student has questions or needs clarification, the parent can easily "get" the concept being taught by reading the directions themselves. There are plenty of practice problems, including good challenging word problems, and there's more built-in facts drill in the lower grades than you'll find in Singapore. The Light Blue Series books include chapter tests and built-in cumulative reviews, plus you get free downloadable software you can use to generate additional worksheets if a child needs more practice in a certain area. MM is also inexpensive and reusable with multiple kids, which is great for larger families.

 

Here are some recent threads on MM, if you want to learn more about it:

http://www.welltrainedmind.com/forums/showthread.php?t=195462

http://www.welltrainedmind.com/forums/showthread.php?t=213505

http://www.welltrainedmind.com/forums/showthread.php?t=215160

http://www.welltrainedmind.com/forums/showthread.php?t=190849

 

Jackie

[FO4UR]

Mmmm ketchup!

I like your explanation of the 'grammar' of math. We started with MUS and i got stressed and scared into switching when i realized all we did in alpha was learn addition and subtraction, amd the other math S&Ss where teaching fractions and calendars and who knows what else. Would you consider MUS a more 'conceptual type program'?

 

When you say get them to work to understand 8+7 does this mean the whole number bonds thing? I have a dd in K right now, and two 4 year olds who will be starting K next year and i'm open to learning a new way to teach math. It's just scary to try to learn something new for myself, kwim? Is singapore easy to teach? What about MM? I ask, not to be lazy, but to be realistic, as i don't have the conceptual understanding, nor do i have loads of time to teach each individual child as i have six kids under ten. (i ditched traditional spelling like spelling workout because it was not providing any conceptual understanding and switched to AAS, but find it takes so much more time to implement that i have a harder time getting it done. Is a great program that gathers dust still better than a poorer program that you use every day?

 

 

I've never even seen MUS, but I whole-heartedly believe that the early years should focus narrowly in on +-x/. I think I might differ with MUS on other things...I've never really researched that one. Most kids (esp at home with mom) can learn clocks and money and anything that can relate to cooking in a casual and natural way. Whatever is missed by about 3rd grade can likely be caught up quickly. I do a bit of worksheet work with my 2nd grader on clocks and $ and geometry,etc...but I'm very aware that CWP and Miquon are more beneficial to him long-term and so I carefully balance what worksheets will give us the most "bang for our buck" as I plan. I rip apart almost all of our math resources so I can pull from here and there. (I love MM b/c I don't feel guilty using it this way LOL!) I don't care about standardized tests in the least...I don't care what Johnny-down-the-street is doing in 2nd grade math...I've read the threads on EM and feel fairly confident that I can't mess my kid up *THAT* bad! :tongue_smilie:

 

 

I think MM is very good from what I've seen (1-2 grades), and have heard enough rave reviews from posters I respect that I'm confident in trying the other levels as well. My ds7 has vision issues and possibly dyslexia and the MM format is not the best fit for him...it kills me. I do put lots of it up on the white board or we do the work orally, but it's not the same as handing him the book while I listen another child read. (sigh!) I want to just hand him MM and tell him to "do math" on super busy days. My dd5 love-love-loves workbooky work and I plan on starting her on MM soon.

 

Miquon is great to read as a teacher, even if you don't use the program. Find it used and cheap. 1st Grade Diary, Notes, and Annotations.

 

You have to use a program that will actually get *used.* I think the ideas/methods in Miquon can be applied using other workbooks...jmho.

 

So with things like 8+7 (which happens to be the equation my son was hammered with more than any other to the point where it's mere mention elicits laughter from us both :D) "making tens" per se is more a "strategy" rather than a "concept" but is built on the mathematical understanding that numers can be re-grouped to facilitate mental math.

 

What does the 8 need to become a Ten? What does the 7 become if it gives up 2 to the 8 so the 8 came become a Ten? What is 1-Ten plus 5-units?

 

 

 

This is what I mean when I say "let the child work it out." I never just tell him the answer...never. He has C rods, an abacus, and other manipulatives he can use (no calculator though). He played so much with the C rods as a youngster that he visualizes the rods even without them in his hands.

 

It's so much more fun when he can say, "I figured it out, Mom! Look, you just...." Not every math lesson is so happy in my house, but he truly gets a thrill out of solving problems that he thinks are "toughies."

 

8+7 easily translated into 28+7 to 28+57 to 528+127...and so on. b/c he did the tough work up front, figuring out how to add numbers that go over 10, regrouping was a non-lesson. Truly. He told me how to do it before I could teach it to him.:001_huh: He would not have been able to make that leap if I had simply told him 8+7=15 every time he asked...or if I had of merely drilled it into him with flashcards and worksheets.

 

 

 

So the First Grader came home from school and said the teacher was showing 3+5 is the same as 5+3 today. And he evidently raised his hand and said:

 

That's the Commutitive Law

 

Excuse me? Could you repeat that?

 

The Commutitive Law

 

*blink*blink*

 

Well that's some pretty fancy Third Grade Math!

 

:lol::lol::lol:

 

Bill (honorary home-schooler today :tongue_smilie:)

 

:thumbup:

[Corraleno]

My problem when people talk about conceptual math is that they come back later saying that their child isn't retaining his math facts. Well, this is where balance comes into play. The child may understand a concept when it is presented. However, if he doesn't practice and routinely return to practice more, he forgets the concept and can't perform the procedure with immediacy.

 

If you present a concept and the child immediately understands, that doesn't mean that he doesn't need to practice. Sure- there are exceptions, but most children most of the time benefit from practice.

I think we're using the term "math concepts" in two different ways, so I'm not quite sure I understand what you mean. I think that whether a math program teaches a student to just apply the algorithm or whether it aims for a deep conceptual understanding of the processes behind the algorithm, the practice problems are going to look the same. For example, if one program teaches students to "just invert and multiply" when dividing by a fraction, and another takes the time to teach why this works, so the student truly understands what they're doing rather than just applying the "trick," the problem sets in both programs are basically going to look the same: 3/4 divided by 7/16 = ?, etc. Whether a student does 10 of these or 100 isn't going to affect their understanding of the concept behind the algorithm, because what's being practiced is the computation, not the concept. Reinforcement of conceptual understanding comes from a curriculum that continually builds on the concepts previously taught, by creating connections between lower level concepts and each higher level, so that when a student looks at a complicated algebraic equation they can "parse the grammar of it," so to speak.

 

Students are able to progress from Bob books to Moby Dick because they understand that the same rules that allow them to read "The cat sat on the mat" will let them read the sentence in Bill's signature. Yet, because the "grammar" of math is not really taught in this country, many kids see algebra as a totally different animal from basic arithmetic, instead of seeing it as simply a more complex and powerful way of arranging (the mathematical equivalent of) the same "parts of speech" they've been studying since 1st grade.

 

I just see threads where after using conceptual maths and only doing a few problems a day for a period of time someone's child is unable to perform basic multiplication or something. They heard you and others espousing the benefits of conceptual math, but they didn't understand that they also needed to practice those skills outside of the conceptual lesson.

If a student is forgetting basic operations because they haven't been doing enough practice problems, then the problem is that they're not doing enough practice problems. The problem is not that they're using a "conceptual" math program. I've occasionally seen a parent complain that a student using Singapore wasn't learning or retaining their math facts, but the issue invariably turns out to be that the parent didn't realize Singapore expects the student to be drilling facts separately. I remember seeing one post from a parent who complained that her child was having trouble remembering what he had learned in previous years using MUS, but that is more likely to be due to the lack of review in MUS, not the conceptual explanations.

 

You seem to be saying that the inherent weakness of "conceptual" math programs is that they don't involve enough computation practice, but to me that's like saying that the inherent weakness of phonics programs is that they don't include enough reading practice. Whether a parent teaches reading using phonics or whole language, how much reading practice the child gets is a completely independent variable. I've never personally seen a math program that teaches conceptual understanding at the expense of computational fluency — on the contrary, I think that if a student truly understands what they're doing, that will increase computational fluency, especially with word problems.

 

Jackie

Edited October 8, 2010 by Corraleno
typo

[Spy Car]

I think we're using the term "math concepts" in two different ways...

 

What a clear-headed and well-articulated post my dear Jackie :001_smile:

 

Bill

[forty-two]

Students are able to progress from Bob books to Moby Dick because they understand that the same rules that allow them read "The cat sat on the mat" will let them read the sentence in Bill's signature. Yet, because the "grammar" of math is not really taught in this country, many kids see algebra as a totally different animal from basic arithmetic, instead of seeing it as simply a more complex and powerful way of arranging (the mathematical equivalent of) the same "parts of speech" they've been studying since 1st grade.

:iagree::iagree::iagree:

 

That just *perfectly* encapsulates it!

 

The grammar of math is its basic building blocks, the components of its underlying structure (and its *not* math facts :glare: - not any more than the list of helping verbs constitutes English grammar ;)). And as far as school math goes, the basic field axioms *are* those building blocks.

[Guest Dulcimeramy]

Alrighty. I've spent about all the brain power I can spare on this topic today, and here's what I've come up with about how I'm teaching math:

 

Ray's Arithmetic is awesome for the basic four operations and fractions. It has other strengths, but I've used it for all four of my children for those particular aspects. Everything that I've read about foundational work was covered by Joseph Ray, surprisingly enough.

 

I didn't know the fancy names for field axioms until I used Horizons math with one of my dc, which introduces those terms in the fourth grade book.

 

Once I did know what the commutative law was, etc. I did begin using those terms with the dc.

 

(Although I still maintain that anybody with a brain knows those truths about numbers, even if they didn't know what to call them. I certainly didn't learn anything new from Horizons math, other than new vocabulary.)

 

So.

 

Between making sure my children can manipulate tools to prove what they are doing with every part of elementary math, and learning and teaching the proper terms for the things we are doing, I think I am covering concepts as well as I can.

 

I know my children are learning more thoroughly than I learned and I know they will be ready for more advanced math. I also know that they are miles ahead of their public schooled peers in my township (not counting the rich folks' schools in California, but just my poverty-stricken township in the midwest).

 

Liping Ma and Bill would probably still see giant inadequacies in me, but I can't really help that.

 

I appreciate you all taking the time to discuss these things. I want to do my best in educating my children, and I am always willing to learn more. Of course, there comes a point where I have to concede that my best can only be my best. It is often not "the" best.

[Spy Car]

I appreciate you all taking the time to discuss these things. I want to do my best in educating my children, and I am always willing to learn more. Of course, there comes a point where I have to concede that my best can only be my best. It is often not "the" best.

 

:grouphug:

 

I think we are all just trying to do they best we can for our children. I've learned a lot on this forum, have plenty more to learn about a great number of things, and appreciate it when people who have some amount of insight, experience, or expertise share their wisdom with me.

 

Where I feel I have something to share, and Can hopefully help others, I do so. I certainly don't want to make you (or anyone else) feel bad. The spirit of the discussion (from my point of view) is trying to get at what are the ideal goals of a good grammar school math education and how one goes about achieving that end. In no way die I bean the discussion as an attack on anyone for not being "the best."

 

If I have hurt your feelings in anyway I'm sorry for that. It is certainly not my aim.

 

Bill

Edited October 8, 2010 by Spy Car

[Halcyon]

My son also does a great deal of facts practice, but I expect the conceptual understanding to come first. My older knows his times tables cold, and we drill frequently with Flash Master, games and "pop" quizzes. We don't move onto the next topic until he has both concepts anf facts cold. This led us to memorize the table in early 2nd grade, as that was when SM introduced the concept of multiplying (little did I realize that multiplication, as an entire topic, was not taught until third!) My younger is learning his addition facts with the help of printable worksheets (he's a fan of them). But we've used rods, discussion and lots of number bond work to explain the why's of addition as well.

 

This is exactly what I do and what needs to be said.

 

Anyway, I didn't mean you.;)

 

I just see threads where after using conceptual maths and only doing a few problems a day for a period of time someone's child is unable to perform basic multiplication or something. They heard you and others espousing the benefits of conceptual math, but they didn't understand that they also needed to practice those skills outside of the conceptual lesson.

 

You understand that there needs to be balance. After the concept is presented it must be practiced. This is balance.

Mandy

[Beth in SW WA]

:bigear:

 

I read all the posts here and find these math discussions fascinating.

 

My older dc used Saxon at private school and CD pre-alg and alg 1 at home during middle school. They are strong math students. Would they be stronger had we used a different - conceptual - program? Not sure. Will never know.

 

Is Prof. Mosely conceptual or procedural? Ds is the strongest math student in his alg 2 class at private school (they use PH which is rigorous and solid).

 

Thanks to these types of math discussions here, I am switching things up for my younger dds - but they are Asian and seem to think differently anyway.

 

Thanks to the OP for starting this discussion. Keep it going!!

[8filltheheart]

OK, I've gone and read part of the Liping Ma book. I don't have time to read the whole thing right now. Specifically, I read the intro and the chapter on subtraction with regrouping. This was the most relevant chapter for me to read because I have already taught both of my children to subtract with regrouping: my older child using Saxon and my younger child using Singapore.

 

Now that I've read the description of teaching this concept--conceptually as the Asians do, I'm even more befuddled by the accusations that Saxon doesn't teach concepts. I taught both my children essentially the same thing, one using Saxon and pennies, dimes & dollars, the other using Singapore and poker chips. Either way it was the same concept, and they both got it. I don't remember the precise wording in the Saxon script, but I probably didn't follow the script anyway. Singapore doesn't have a script--it's mostly pictoral so I was using my own words in both cases.

 

.

 

While I do not disagree with the vocal majority on this forum that maths like SM do focus on mental math in a way that traditional math programs do not, I do believe that the entire "conceptual" debate on the forum has diminished the value of perfectly solid math programs.

 

For instance, the entire "elevation" of MM. I have thoroughly reviewed several of the MM books (3A&B, 5B, and 6A). MM is very reminiscent of Horizons except that Horizons is spiral and MM is mastery. The same arguments used to "support" MM as conceptual are used to "disprove" that Horizons is.

 

There is no question that the bar diagram method does get children to manipulate complex problems in a concrete way.

 

However, and this a huge however, programs like Horizons do get kids to understand how the exact same manipulations take place via algebraic equations. The process is different in how they get their answers. For example, when kids finish Singapore, they really need a pre-alg program to help them transition to alg. A lot of kids can finish a program like Horizons and jump straight into alg w/o a problem.

 

While I can appreciate enthusiasm for any method that develops critical thinking skills, I think the entire "conceptual" math debate on this forum is distorted.

Edited October 8, 2010 by 8FillTheHeart