Theoretical math question RE: conceptual vs. traditional

[Crimson Wife]

Singapore Discovering/Dimensions Math definitely has it, but that's pricey and there isn't a lot of guidance for the teacher.

 

Foerster's algebra books are supposed to have really good explanations but I haven't personally seen them.

[Corraleno]

THAT is my current frustration with math curriculums I have tried for my younger students. The conceptual treatment seems too much too soon, at least for my children. It seems like classical math would be stronger in facts with basic conceptual explanations in the younger years and go deeper conceptually as the child gets older.

 

But I think this goes back to the question of whether "math concepts" are sort of optional extra bonus materials, that you can just tack on later, or whether "math concepts" are in fact what math is. Why not teach the "why" with the "how" and the "what"? Without the "why," math just seems like a bunch of totally random rules to memorize and apply, without rhyme or reason. 

 

I've used Singapore & MM, and I've read through Miquon, MUS, and many other programs, and I think they generally do introduce concepts in very gentle and developmentally appropriate ways, building deeper and more complex understanding of math concepts incrementally as the program progresses. I can't think of a topic or explanation that I've seen in MM where I thought "Wow, that's going to be totally over my kids' heads." They don't always get it on the first reading; sometimes we need to go over something, play with it, do some problems, before the light bulb goes on. But that light bulb moment is what I'm after, not just 2 pages of problems with 90% correct.

[ElizabethB]

So what is out there that does this that is not AoPS? Something a little more direct? Or what could I perhaps add to Saxon Algebra 1 to get some of that in?

Of course I like this idea, and I believe in it. I saw it work well with SM. But AoPS is just too much for my kid at this point. He prefers to have things explained rather than trying to figure every single point out. And I think that is ok. He is 12. KWIM?

 

AoPS can't possible be the ONLY thing out there.

 

Maybe you can answer this. I guess it is directed at anyone who is "listening".

The old Dolciani Pre-Algebra and Algebra. I am finally understanding the whys behind algebra and pre-algebra just like Singapore helped me see some of the whys of elementary math.

 

And, I did well in Algebra and tutored algebra, but the Dolciani really lays it out in a way that makes you understand it as a system. I was taught all the bits and pieces, including the set theory and properties, but didn't understand how unified it all was.

[ElizabethB]

This is baffling to me. I have an old copy of Dolciani pre A. I basically used it for extra practice. I didn't feel like there was anything all that mind blowing about it.

What year? Did it start with set theory?

[ElizabethB]

1985

 

I'd have to find the book.

That is too late, they cut out the set theory. Mine is from late 1960s or early 1970s. This is the cover:

 

http://www.amazon.com/Modern-School-Mathematics-Mary-Dolciani/dp/0395143853/ref=sr_1_11?ie=UTF8&qid=1394240617&sr=8-11&keywords=Dolciani+Pre+algebra

 

And this is the Algebra, you can sometimes find it cheaper, I got mine for $15 or $20.

 

http://www.amazon.com/Modern-Algebra-Structure-Method-Book/dp/B000ZF3NFO/ref=sr_1_18?ie=UTF8&qid=1394240740&sr=8-18&keywords=Dolciani+algebra+structure+and+method

[Corraleno]

So what is out there that does this that is not AoPS?  Something a little more direct?   Or what could I perhaps add to Saxon Algebra 1 to get some of that in?

Of course I like this idea, and I believe in it.  I saw it work well with SM.  But AoPS is just too much for my kid at this point.  He prefers to have things explained rather than trying to figure every single point out.  And I think that is ok.  He is 12.  KWIM? 

 

AoPS can't possible be the ONLY thing out there. 

 

A math program doesn't have to use the "discovery method," like AOPS, in order to teach conceptual understanding. Foerster's teaches math concepts quite clearly and explicitly, and includes lots of really good word problems which test conceptual understanding (as opposed to just offering lots of problems in the same format as the sample problems, where students can plug-&-chug without understanding what they're doing). Most of the algebra texts recommended here on the boards (Singapore, Foerster, Jacobs, Dolciani, Lial, Larson) do a pretty good job of explaining math concepts. 

[stripe]

I've been hanging around these boards for a year or so now, and one of the biggest trends I see here is the switch from 'regular' math (like we had, lol) to conceptual maths like MEP and Singapore. 

I don't really identify with this. I did not receive a "regular" or "traditional" math instruction, either at home (where I learned quite a bit) or at a Montessori school. I have seen old math books, and they seem to have lots of dull problems about calculating interest, and problems where milk cost 2c a bottle. I have relatives who calculated their way through activities like flying airplanes over enemy territory, so I am not saying old fashioned wasn't good, but it just doesn't seem like using the ability to use a slide rule and a book of tables is that important. And yes, one can learn this independently. I own a slide rule and a book of how to use one, and I have made it available to my oldest child to use. But I'm not going to make that my major mode of instruction. 

 

Given that MEP was developed in the UK based on Hungarian math and has been available for years, I don't see it as anything "new." But then, I've been using it myself for over 4 years. And the website looked about 10 years out of date when I started using it. Also, I haven't observed that many people on here who actually use MEP. Singapore, yes. But MEP, no. Not at all. MEP is not an "Asian" program. [Can I mention now how much I hate the idea that all Asians, and only Asians, use some "conceptual" mode of thought when studying math? Look up who's won the Field's Medal. It's not really Asian-dominated.] MEP is available in Spanish and two South African languages, by the by.

 

Singapore makes my eyeballs hurt. I haven't seen MEP, but if it is anything like Singapore, or if it depends on manipulatives, it wouldn't be on my short list, either.

I don't think MEP and Singapore are particularly similar (MEP certainly doesn't have a high budget for graphic designers, unlike the cute, cartoony kids in Singapore), and MEP definitely doesn't "depend" on manipulatives. When they suggest them (in year 1, I think, is all I can really think of), they are not anything special. They suggest the students use things like rocks, buttons, and acorns as counters. There is a small amount of Cuisinare rod type things, but they just have paper squares in a row, not a special physical object you must buy.  In fact, the student is expected to do two digit addition and subtraction with carrying/borrowing in their head, prior to the introduction to the vertical style. I have taught two children this way using MEP. Paperless is about as manipulative-free as it gets!

 

 

But I think this goes back to the question of whether "math concepts" are sort of optional extra bonus materials, that you can just tack on later, or whether "math concepts" are in fact what math is. Why not teach the "why" with the "how" and the "what"? Without the "why," math just seems like a bunch of totally random rules to memorize and apply, without rhyme or reason. 

 

 

Exactly.

 

As a result of my own studies, I do not see anything personally appealing about promoting the idea math that is just about memorizing things. I prefer to think that I am more familiar with what upper level math looks like, rather than being arithmetic-focused. At one point in my life, I saw how people who'd never studied math past about algebra looked at the world, mathematically speaking, and I could tell they were missing something very important conceptually. Calculus gives a student experience with both very large and very small numbers, and the concept of a term dominating as the variable approaches infinity, whereas the person who's just studied algebra tends to think of x as being as large as 10, or maybe 20. [And therefore the idea of what term "dominates" in x^2 + 5x + 90 is not clear at all.]

 

So as far as I am concerned, the more familiar one is with mathematics, the less likely it will be that one thinks it's all just a neat hodge podge of discrete, meaningless concepts.

Here's an interesting article: 

http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/

[MomintheMountains]

But I think this goes back to the question of whether "math concepts" are sort of optional extra bonus materials, that you can just tack on later, or whether "math concepts" are in fact what math is. Why not teach the "why" with the "how" and the "what"? Without the "why," math just seems like a bunch of totally random rules to memorize and apply, without rhyme or reason. 

 

 

Did you read my history analogy?  

[wapiti]

This is baffling to me.  I have an old copy of Dolciani pre A.  I basically used it for extra practice.  I didn't feel like there was anything all that mind blowing about it.

 

That was the trade-off there, losing the set theory but enjoying the easier-on-the-eyes mid 80s format.  Plus, my (1973, 1970) prealgebra, the same as the one at Elizabeth's link, happens to have a musty smell...

 

If it helps, the Dolciani Algebra 1 of the same vintage (mid-late 80s) *does* start off with set theory.  It's one of my fun recent textbook discoveries, LOL - simliar straightforward page format to Prealgebra, An Accelerated Course.  I think it looks a smitch more rigorous than Algebra Structure and Method Book 1 of the same late 80s era, or at least it seemed so when I was poking around in them a couple weeks ago.  I haven't done a careful comparison, though, nor have I compared it to the (1973, 1970) Modern Algebra Structure and Method; I'd expect some differences.

 

Eta, what I don't know is whether the late 80s alg book might unify the concepts in the same way that Elizabeth is talking about - I usually look at them on a per topic basis.  FWIW, I definitely feel like AoPS has shown me a perspective on math that I never knew existed, though that may be a problem-solving angle as opposed to a concept angle (apart from the discovery approach).

[regentrude]

I strongly disagree that everything understood will be retained, although it may be true for people who are very gifted in a given area. The best results in educational studies result from good explanation followed up by significant targeted practice. Countries that generated these conceptual math programs (Singapore, Japan, Russia) supplement them with drill, often in amounts that would make Americans weak in the knees.

 

I never said that practice was unnecessary. Of course the student also has to practice procedures so that certain things become automatic! I can't have a student pause and think about how to multiply two polynomials, they just have to be able to do it quickly and accurately.

I hope I did not give the impression that practice was unnecessary- far from it. In fact, very often thorough conceptual understanding is only developed when the student is forced to apply the concept in different ways to a number of problems. The understanding emerges as the result of the practice (but that does not mean that all drill results in understanding). It requires carefully designed problems that make the student examine the concept from all angles and use it in different ways.

[AimeeM]

But you are using Miquon, which is highly conceptual and nontraditional.

 

 

 

Yes, but given the abysmal state of math education in this country, it's clear that most people are not able to just intuit the concepts behind the math, and do need a program dedicated to teaching it that way.

 

I use Miquon as a supplement, true. I wouldn't use it as my core. We haven't done it for weeks. I also do not find it comparable to Singapore or MM, frankly. Maybe I'm weird like that, lol. I'll definitely admit to coming to this site, several years ago, and falling - I wanted to try what everyone else raved about (MM, Singapore, Miquon). I tried it. Don't see the hoopla, frankly.

 

RE your second comment. That isn't what the OP asked, unless I misread. She asked if a student COULD be prepared for college with a traditional math. Of course they can. It's worked for many, many, many years. I did caveat my statement by acknowledging that some of that may just be students who intuitively understand the concepts, without the explicit instruction in them. I'm not sure.

[Corraleno]

Did you read my history analogy?  

 

Yes, you said that you provide conceptual background for historical events that are developmentally appropriate to the age, and that you think math should also provide "basic conceptual explanations in the younger years and go deeper conceptually as the child gets older."

 

And my point is that this is exactly what math programs like Singapore and MM do — they provide basic, developmentally appropriate conceptual explanations, which build incrementally and get deeper as the child progresses through the program. I'm not sure how the explanations in the earliest grades could be any more "basic" and still be conceptual.

[MomintheMountains]

Yes, you said that you provide conceptual background for historical events that are developmentally appropriate to the age, and that you think math should also provide "basic conceptual explanations in the younger years and go deeper conceptually as the child gets older."

 

And my point is that this is exactly what math programs like Singapore and MM do — they provide basic, developmentally appropriate conceptual explanations, which build incrementally and get deeper as the child progresses through the program. I'm not sure how the explanations in the earliest grades could be any more "basic" and still be conceptual.

 

Well, as an example, my second grader doesn't like to change 7 + 5 into 10 + 2.  It frustrates him.  I think it is ok to know 7 + 5 = 12 and move on from there.  My fifth grader decided he "loved" long division after I taught him the old fashioned way I learned in fifth grade.

 

I'm not going to argue.  My children don't do well with MM, so maybe they are dullards!

[EndOfOrdinary]

I don't think anyone is claiming your children are dullards. They are trying to assert that the memorizing 7+ 5= 12 works very well now, when the problem is 7+5. The concept that there is no fixed number, merely a constant regrouping of objects in infinitely more or infinitely less complex ways is what the reworking into 10 + 2 is trying to get at. It is trying to assert that there is no such thing as a permanent number. That concept becomes much more important when you are dealing with larger more complex numbers. At a certain point you can't memorize. You can use a stacking algorithm, but then again you are working with fixed number. It becomes limiting. I believe this is what many people were talking about with the topping out.

 

The stacking algorithm works. 7+5 does equal 12. This is nothing that makes your children wrong or stupid. It might just give them a limited view later which could be difficult if they want to go into abstract mathematics.

 

This applicable with everything from nutrition to Latin. You can either teach someone how to do something or you can teach them why they are doing it. Neither is right or wrong. They are different approaches. In the end, as long as they can function as citizens then you have done your job.

[Crimson Wife]

Well, as an example, my second grader doesn't like to change 7 + 5 into 10 + 2.  It frustrates him.  I think it is ok to know 7 + 5 = 12 and move on from there.  My fifth grader decided he "loved" long division after I taught him the old fashioned way I learned in fifth grade.

 

I'm not going to argue.  My children don't do well with MM, so maybe they are dullards!

The reason to practice "making tens" is that it helps mental addition with larger sums become much easier. 57 + 5 = 50 + 10 + 2 = 62 is easy for the child who is comfortable with making tens.

 

I do think drilling the facts is important and one thing I wish Singapore more explicitly built in to the program is drill (the assumption is that the child is doing drill outside of class, which is true in Singapore but not necessarily true of U.S. users of the program).

[SorrelZG]

I don't recall how I learned math but this debate has confused me to death since I first encountered it. I just couldn't conceive of basic arithmetic without comprehension of what was going on. I eventually encountered a living example of someone with no conceptual comprehension of math and I do get now that I could never recommend this person use a math curriculum that doesn't explicitly teach concepts thoroughly. One may be much more comfortable with a "traditional" method but their comfort doesn't *necessarily* translate to the teaching or learning of math.

 

Actually, though I have no recollection of how I was taught, I think it was "traditional" because I recall describing my advanced math class in my senior years as actually explaining what on earth the things we learned in regular math class were actually about (where they came from, why they were the way they were, etc. but well beyond arithmetic).

[stripe]

I don't know why SM is for "mathy" kids. 

I agree, and I have never felt that MEP is only for smart kids. There is a ton of review built in, and lots of slow steps towards the destination. Frankly, I believe the clearer and more explicit something is, the easier it will be for non-geniuses to grasp it.

[strange_girl]

Wow; I've been away for a couple of days and was excited to see all the responses when I got back!

 

Thanks to all who have discussed here. Reading different experiences and points of view has given me a lot to think about.

 

I think that I must be one of those people who can intuit basic math concepts like making tens and regrouping, etc. Even though my own math education was dismally algorithm based, I naturally knew how to find the answers mentally. Now, looking at programs like Singapore and MEP...it really seems like a lot of overkill. But maybe that's because I already know how to use their methods? So less conceptual maths seem more straightforward and more basic...which may be their draw as a teacher.

 

Now I'm rambling, but I think I understand my own feelings on the subject better now.

[AmandaVT]

For some kids, conceptual math is make it or break it. Barring that, you're probably okay with algorithms. I got good math grades with algorithms, but math terrified me. I was smart enough to realize that an A doesn't mean I'll be able to do math in college, but people thought I was limiting myself for no good reason. I didn't know how to explain how a person gets A's in math but doesn't understand it. That's a lousy position to be in when you're leaving high school and trying to figure out a career--"hey, I got an A in math, but I don't understand it. So, does that mean my A in biology is also built on a total lack of understanding, or could I be a good biology major? Oh wait, they take math too, let's just forget about the whole biology thing. How about computer programming? Oh snap, they take math too." Meanwhile, your college advisor simply thinks you are mentally ill for discounting careers in classes that you got A's in during high school.

 

I will strongly state that I do NOT believe that a math teacher who cannot understand conceptual math has any business whatsoever stepping into a classroom and being allowed to teach. I know many fine math people who like algorithms. I have yet to meet one that can explain math to a struggling student who is struggling because they lack the conceptual information--they simply trot out algorithms and look at you like you have three heads if you still don't get it. There is too much at stake to put these folks in charge of someone's math education even for a year. I had an algebra 2 teacher who didn't understand conceptual math, and it limited my future math possibilities a lot. A lot of mathy kids who intuitively understood math struggled in her class, so it's not just me that had a problem. Being a rural area (our high school was the biggest in the county with 500-ish students in 4 grades), she was the only algebra 2 teacher in the school. I had a friend whose dad taught calculus in the same school, and the algebra 2 teacher would mark down my friend's papers for doing things CORRECTLY the way that her calculus-teacher father taught her because the TEACHER couldn't understand it. That is ridiculous. If you went to the algebra 2 teacher for help, she would just show you the same ONE way to do the problem that she did it in class...over and over and over. No variation. It was like reaching a self-help line that tells you to keep pushing 2 to get to a live person, only the live person was never there. I lived for the days we had a sub, only when the sub was there, I was so far behind in my understanding, it was like trying to drink from a firehose.

 

This was me - I got an A- in AB Calculus in high school, but I didn't understand it at all. And I can't remember much about math past pre-Algebra. I was an elementary ed major and had a great time with Math for Elementary school teachers 1 &2 though!

[stripe]

I think that I must be one of those people who can intuit basic math concepts like making tens and regrouping, etc. Even though my own math education was dismally algorithm based, I naturally knew how to find the answers mentally. Now, looking at programs like Singapore and MEP...it really seems like a lot of overkill. But maybe that's because I already know how to use their methods? So less conceptual maths seem more straightforward and more basic...which may be their draw as a teacher.

Well, I think most basic things look totally obvious, as if they require little instruction. That is part of what makes teaching small children difficult. They don't see the world the same way we do. They don't have any experience.

 

 

This was me - I got an A- in AB Calculus in high school, but I didn't understand it at all. And I can't remember much about math past pre-Algebra. I was an elementary ed major and had a great time with Math for Elementary school teachers 1 &2 though!

 

Yep! It wasn't until my final math class in college that I suddenly understood the whole epsilon/delta thing. I had been so confused by my awful Calculus class in high school, when my teacher appeared to be suffering a mental and physical breakdown, that I just accepted it as fact and proceeded as if I understood it. I did fine, but I never really got it. I was so disgusted when I got it, like, why wasn't this explained to me, because it really wasn't that hard! 

[strange_girl]

Well, I think most basic things look totally obvious, as if they require little instruction. That is part of what makes teaching small children difficult. They don't see the world the same way we do. They don't have any experience.

This is exactly what I was trying to say. MM is tedious for me because it just seems like overkill...but it doesn't seem that way to DD.

[ladydusk]

I still haven't read any convincing evidence here that young children (let's say K-3rd) need a deep conceptual treatment of math. Of course math is conceptual and a basic understanding of "why" or a visualization (counters for example) is helpful, but my boys are frustrated by too many different explanations (as in their Math Mammoth program).

 

Also, I think there is probably a big difference when teaching a child who has a natural mathematical aptitude and is destined for a STEM career vs teaching a child who needs functional math. I, for example, am really good a calculating and estimating. I am good at accounting. But accounting doesn't require much higher math, it is much more functional. I did well with "traditional" math in school. As with other subjects, I guess choosing the appropriate math curriculum really depends on the child's aptitudes.

 

I was always good at math in school.  Not the best student in the class, but not far from the top - I was always in the top math group.  I could plug and chug math pretty well.  I HATED math.  If that's all there was to it, plug and chug, what a waste of time. Bo-RING.  A calculator can do that.  This started in 1st grade (no math in Kindy when I was there, boy am I old). 

 

When I started homeschooling, I was encouraged to read Liping Ma's Knowing and Teaching Elementary Mathematics. It opened my eyes to why math didn't work for me in elementary school ... my teachers did not teach they whys and wherefores.  May not have even known or understood them themselves.

 

Teaching MEP has been eye-opening to me.  I've taught R, Y1, Y2, and am about halfway through Y3.  The number sense I have gained (and that my children have absorbed) is remarkable.  The patterns, logic, beauty - yes beauty - I see in math now makes it my favorite subject to teach.  Which is good, since MEP is very teacher intensive and I spend upwards of two hours a day teaching it to three children. 

 

I have a hard time imagining coming into this kind of thinking, reasoning, understanding math starting in 4th grade.  The building blocks are slowly and firmly planted and stacked upon one another.  I'm not saying it can't be done, I'm saying that changing horses midstream would be incredibly difficult.

 

That being said, I do think conceptual math can be done poorly.  My brother (Chemical Engineer, Math Minor, all around genius) has his daughter in Kindergarten this year. Just this week, he posted this to Facebook:

 

 

I realize that it really doesn't matter at all, but I find it difficult not to be frustrated by kindergarten "math homework". Last week, [dd] had a couple problems marked wrong because she wrote "13=8+5" when the expected (but not in the instructions) answer was "13=10+3". This week, a couple are marked wrong because of "14=4+10" rather than "14=10+4".

 

Perhaps it's my advanced arguing-with-the-professor skills, but I find it silly to mark correct mathematical statements wrong.

 

 

My niece is in Kindergarten in a rich, excellent school district. I'm sure they're trying to teach conceptually, but this is awful.  My DH - my greatest support - wanted to post "Homeschooling FTW" but bit it back because he loves me.

 

I think my brother is wrong - it *does* matter, because my niece is being taught at 5 years old that right answers are wrong and that it's OK that her teacher marked them wrong. 

 

Here's the thing.  I want to give my children a Classical Christian Education. This means inculcating in them an understanding of what is True, and Good, and Beautiful - that points them to God's unchanging character, order, and logic.  Conceptual mathematics does this.  In spades.

 

ETA: I do supplement with Calculadder (drill) and Primary Challenge Math (story problems)

[Corraleno]

Well, as an example, my second grader doesn't like to change 7 + 5 into 10 + 2.  It frustrates him.  I think it is ok to know 7 + 5 = 12 and move on from there. 

 

But that exercise is not about arbitrarily "changing" one expression into another, it's showing the child that when you add 7+5 you get 1 ten and 2 ones. Then, when the child starts adding multidigit numbers in columns, he understands that 7 ones + 5 ones = 2 ones that stay in the ones column, and 1 ten that goes into tens column. Kids who don't understand that just think "write down the 2, carry the 1" and don't really understand what they're doing.

 

Then when they start doing multidigit multiplication, kids who automatically think of the "1" in 12 as a ten, understand that when you multiply by that "1", you're actually multiplying by 10, and that's why there's a zero in the ones column. To some people that's obvious, but to many kids it's not. In the Liping Ma book linked by a PP, some of the teachers claimed that the zeros were just there as "placeholders" to keep the columns straight and weren't really zeros! In fact, one teacher even had her students draw apples or other shapes in place of the zeros so they wouldn't be "confused" and think there should actually be zeros there! I'm sure those teachers would argue that their "traditional" math educations served them just fine, but that's clearly not the case.

[MomintheMountains]

But that exercise is not about arbitrarily "changing" one expression into another, it's showing the child that when you add 7+5 you get 1 ten and 2 ones. Then, when the child starts adding multidigit numbers in columns, he understands that 7 ones + 5 ones = 2 ones that stay in the ones column, and 1 ten that goes into tens column. Kids who don't understand that just think "write down the 2, carry the 1" and don't really understand what they're doing..

I understand the point of the exercise. My point is getting lost, though. Let's just say that I think some children do better with "traditional" math. And by traditional math I don't mean facts without concepts.

[Meriwether]

My kids use Saxon up to Saxon 7/6. They understand concepts. Yesterday, Ds7 and I were talking about x9 for the first time. As he was figuring out the answers, he mentioned how easy they were, because they were +10, -1. Not the most impressive of examples, perhaps, just the most recent. My kids have all been able to figure out these types of strategies for themselves. Ds9 recently learned to divide fractions. I set it up like fraction/fraction and explained why and how we would multiply the bottom part by the reciprocal. He interrupted me at that point with, "Mom, if you multiply the bottom by that, you have to multiply the top by that, too." Again, that is just a recent example. The kids say and do things all the time that show me they understand the math they are doing. It is not just rote memorization. Dd10 had no trouble doing the CWP. She is doing well in AoPS Pre-Algebra.

 

OP, pick a program that works for you and your kids. Whatever program you use, you are their most important resource. You are going to be the one monitoring whether they understand a concept or whether they know their facts, not the book.

[ElizabethB]

I was always good at math in school. Not the best student in the class, but not far from the top - I was always in the top math group. I could plug and chug math pretty well. I HATED math. If that's all there was to it, plug and chug, what a waste of time. Bo-RING. A calculator can do that. This started in 1st grade (no math in Kindy when I was there, boy am I old).

 

When I started homeschooling, I was encouraged to read Liping Ma's Knowing and Teaching Elementary Mathematics. It opened my eyes to why math didn't work for me in elementary school ... my teachers did not teach they whys and wherefores. May not have even known or understood them themselves.

 

...

 

Here's the thing. I want to give my children a Classical Christian Education. This means inculcating in them an understanding of what is True, and Good, and Beautiful - that points them to God's unchanging character, order, and logic. Conceptual mathematics does this. In spades.

)

Liping Ma was helpful in that way for me, too. I worked as a statistician but didn't understand some of the basics of elementary math as a system. We just covered binary and base 5 in Pre-Algebra. I supplemented with a few videos of octal, hexidecimal, and binary. I now understand how my computer is thinking!! I had worked with other bases before and could get the right answer, but only now do truly understand them.

 

If you have not read it yet, I think you will enjoy the book "Mathematics is God Silent."

 

http://www.amazon.com/Mathematics-God-Silent-James-Nickel/dp/187999822X/ref=sr_1_1?s=books&ie=UTF8&qid=1394299581&sr=1-1&keywords=mathematics+is+god+silent

[ladydusk]

If you have not read it yet, I think you will enjoy the book "Mathematics is God Silent."

 

http://www.amazon.com/Mathematics-God-Silent-James-Nickel/dp/187999822X/ref=sr_1_1?s=books&ie=UTF8&qid=1394299581&sr=1-1&keywords=mathematics+is+god+silent

I own it but havent read it yet. :-)

[ElizabethB]

I own it but havent read it yet. :-)

He has a website, too, I just found it looking up the book for you.

 

Here is a good one to start with:

 

http://biblicalchristianworldview.net/documents/whitherMathematics.pdf

 

The page:

 

http://biblicalchristianworldview.net/mathematical-circles.html

[stripe]

But that exercise is not about arbitrarily "changing" one expression into another, it's showing the child that when you add 7+5 you get 1 ten and 2 ones. Then, when the child starts adding multidigit numbers in columns, he understands that 7 ones + 5 ones = 2 ones that stay in the ones column, and 1 ten that goes into tens column. Kids who don't understand that just think "write down the 2, carry the 1" and don't really understand what they're doing.

 

 

...I'm sure those teachers would argue that their "traditional" math educations served them just fine, but that's clearly not the case.

I have observed with my own kids, that they are able to do lots of unit conversions and extrapolate to multi-digit numbers, because of a solid understanding of place value. Having done it twice now, I can say that MEP takes FOREVER!!!! to get through 0-20, but once it does, the child understands things in a very detailed way and it is easy to move on to larger numbers because the concept is very clear. (Liping Ma talks about this too.) 

 

Another one that scared me was the teacher(s?) in the middle of the geometry unit, who couldn't say off the top of their heads what the equation is to calculate the perimeter of a rectangle. If you can't even talk your way through that one, it's scary.

 

Also what does it mean to say one's education served them just fine? Who are all these people of my age who have a great, solid math education based on memorizing stuff without understanding? I haven't met them, personally. I did not receive such an education. The only person still living I might know who might have studied "traditional" math is my grandma, who's never mentioned liking any part of her education (although she's mostly just told me how awful her history classes were) and she has unfortunately not been an example of great mathematical understanding.  I either meet people who are engineer types with a great understanding of math OR people who are utterly terrified of it, are easily confused, and use their calculators incorrectly to calculate all sorts of things. People have thought my husband is a math genius because he can estimate 10%  of a number in his head.

[Monica_in_Switzerland]

I've been sick for the last 24 hours and away... what an interesting discussion! 

 

I want to just add two points that I haven't seen:

 

- Fact memorization- Yes, it is helpful to just "know" that 7+5=12 or 8x4=32.  It is certainly better to know facts than know nothing at all.  I taught a year of high school math- algebra, geometry, alg 2, and pre-calc.  None of my students knew their facts.  When I announced that I would not be allowing calculators in the class, you would have thought I'd shot everyone's pet dog.  And by the end of the year, I was obligated to let them use calculators because I could not write a test sufficiently long enough to cover a chapter, but short enough for them to muddle through calculating their "facts" in more complex problems.  All of these kids claimed they had been using a calculator since third grade.  Why?  Because by that time, students have "memorized their facts" and are going on to procedural problems- multi-digit multiplication, long division, etc.  So why not let them use a calculator?  My 7 year old can run laps around those kids in terms of mental math, and I'm talking about my A students, not just my average students. 

 

- Having said that, if I were to say to him next year when he will be in SM3, "Now you can use a calculator because you have mastered mental math"- his mental math capacity would be utterly gone by the end of the year. 

 

The cognitive scientist Willingham (Why Students Don't LIke School) has the following interesting research that I think applies:

 

- He measured math recall 10, 15, and 20 years after end of last math class a student passed.  (I'm going from memory, but this is essentially an accurate summary).  He looked at recall of Algebra 1 skills.

- He looked at students who received high passing grades in Algebra 1.  Conclusion:  Those students who stopped after Algebra 1 had almost no recall of those skills.  Those students who went on through Calculus had extremely high retention of algebra 1, despite that all students studied had passed algebra 1 with similar grades. 

 

- His point:  The constant review of prior concepts is ESSENTIAL to long-term gains in memory and understanding. 

 

I think this applies quite well to "facts"- Because at Time A, a student has facts memorized does not mean they will still be memorized at Time B.  There must be a constant review.  You can review by drill and kill, or review by advancing in math.  Your child will certainly stand a better chance of reaching adulthood with facts in tact if you keep a calculator out of their hands! 

---------------

 

Next point that I haven't seen mentioned:

 

- We are all talking as if math were a purely skill subject, but I think, like reading, it must be considered a SKILL and a CONTENT subject.  If we only attack math from a skill mindset, we are denying an entire realm of mathematics, akin to thinking the tip of the iceberg is the iceberg itself.  It also makes math incredibly boring and I think it a main factor in why so many children dislike math.  Teaching one concept from multiple angles brings content into contact with skill.  Learning the algorithm for a problem type may be the most efficient and logical choice for a student when faced by a problem- let's call this skill the tip of the iceberg.  Knowing why is the rest of the icerberg. 

 

It is one thing to know the Pythagorean equation.  It is another to derive it.  You do not need to derive it to estimate TV dimensions based on a corner-to-corner measurement in your local Best Buy flyer.  However, in one case you know content, and in another, you know skill.  We can't pretend these are equivalent depths of knowledge, even if the practical outcome in day-to-day life is the same. 

 

The other implication is for people who feel that an algebra level education is sufficient for day to day living- it is!  BUT, to maintain that level of understanding, several more levels of math should be taught to give the algebra time to cement into place in long-term memory. 

 

Finally, viewing math as a content subject makes conceptual math VERY relevant to grammar stage work.  See my recent thread on the Gen Ed board about Hirsch and the Knowledge Deficit.  I think a lot of parallels can be made between the 4th grade reading slump and the high school higher math slump.  Skill can float the boat for a long time, but if you get to higher level math and the content knowledge isn't there, you will be hard-pressed to keep climbing.  The analogy made earlier to phonics vs whole world applies as well.  Some kids can intuit phonics, but (almost) all kids can do direct instruction phonics.  Some kids can intuit math concepts from procedural math, but (I believe) (most) kids will do well from direct instruction of concept with adequate practice determined on a child-to-child basis.  I am NOT a fan of discovery math in the lower years, this is akin to whole word reading in my mind, and I think it's a great waste of time.  I think we need to teach kids to think before they can do it on their own, possibly at the algebra/AOPS stage. 

 

-------------------------

 

Manipulatives:  My DS is in SM2, and he is technically a first grader.  We take out manipulatives at the beginning of each chapter for one lesson or so, and even that is no longer guaranteed.  We also take them out  if he seems to be struggling with a particular problem that previously he understood, but has not seen recently.  Sometimes I think the impression is given (due to certain manipuatlive heavy programs) that conceptual math must equal manipulatives all the way through elementary school, or for every problem for a lesson, etc.  Some kids may need that, but I seriously doubt we will be pulling out the C-rods next year during SM3.  I could see a brief review of our base 10 blocks, as SM3 introduces thousands, and a foray into fraction manipulatives for appropriate chapters, but all math programs are going to have some manipulative work for fractions, or at the very least, drawings.  Manipulatives can allow a child to travel faster and farther without requiring the (over) use of fine motor skills for those who are still young enough to find writing and drawing tiring.  There is a heavy use of manipulatives in the first level of most conceptual math programs, and a heavy reliance on them by most students.  Now, my DS can watch me work a problem with manipulatives to explain a mental math technique, and then move directly to the technique without ever touching the manipulatives himself.  The c-rods in particular are excellent for illustrating a mental process, which is hard to do verbally, and once again accommodates a younger child's less developed verbal skills. 

 

------------------

 

So that's just a few thoughts that I didn't see mentioned.  :-) 

 

 

[myfunnybunch]

Why wouldn't a traditional math also be conceptual?

 

I :001_wub: Ellie.

 

The problem isn't that "traditional" math isn't conceptual. the problem lies in the teaching of traditional math that relies only on teaching algorithms and not why they work. Do this, do this, do this, check your answer in the back of the book,and Wallah!...I mean Viola!....ha, ha, just kidding....Voila! you arrive at the correct answer! It's Mathemagic!

 

There's nothing at all wrong with teaching mathematical concepts using a traditional approach.

 

FWIW, we use Singapore. I tried Saxon and HATED using it. But conceptually, it was a good solid math program. My sister's children used Saxon and do quite well in their college math classes.

 

Cat

[SorrelZG]

The one thing that concerns me is that some parents go with a "traditional" program that looks most like how they were taught and they have no idea that they don't know what they are teaching. The conceptual heavy programs seem foreign to them for this reason. They think it's a "foreign math" like the world has foreign languages. I don't believe these programs ARE (just) for the "mathy" students but (especially) for the very ones for whom it appears foreign.

[kiana]

There are some examples of 'traditional' non-conceptual math teaching in Liping Ma's book. For example, where the teachers were discussing the traditional method of multiplying, to multiply 123 x 456. They were asked how they would help a student who was forgetting to move the partial sums over. Many teachers (but more of the US teachers than Chinese) indicated that they would just have the student put something there as a 'placeholder' to remind them to move the sums over. Some of them even specifically indicated that they would use something other than 0's (like asterisks) because 'those aren't 0's, those are just placeholders'. I can't remember what order they were multiplying them in, so I picked one.

 

For example, they would expect to see something like (dots added for spacing):

 

.....123

....x456

______

.......738

.....605.*

...492.*.*

_________

 

and then the sum at the bottom.

 

This is a completely crazy answer to give students. The zeroes are there because what we are doing is multiplying 123x6, 123x50, and 123x400, and then adding them all up. Any sort of conceptual program (including, btw, many of the pre-New-Math programs -- being old/traditional does NOT make a program non-conceptual) is going to give an explanation that involves place value in some way, shape, or form, that is not just 'memorize the algorithm because I told you to'.

 

It is perfectly possible to teach a traditional program conceptually or to teach a conceptual program in a non-conceptual manner. This is where the teacher variability comes in. Some children will also intuit the rules just from being taught the algorithms.

 

Some programs also stretch students a lot more than others in the *application* of concepts that they have learned. The problems in (for example) AOPS, Singapore, the starred problems in some programs, etc., will expect students to go further using the concepts they have learned than in some other programs I have used. Since most students entering college will NOT have this preparation, it is perfectly possible to enter and do just fine at most universities without this preparation. However, one of the biggest issues with students who want to major in these areas is that, although they are fine at computation, they possess little understanding of what or why they are computing -- even the majors. Trying to get students to make connections across their discipline and even outside of their discipline is a huge topic right now, and it's one reason the 'introduction to proofs/transition to advanced mathematics' course tends to be such a bloodbath. It's also why a fair number of math majors change to a different major and a math minor after introduction to proofs -- because they found out that math really wasn't just about computing and finding the right answer.

[boscopup]

This is a completely crazy answer to give students. The zeroes are there because what we are doing is multiplying 123x6, 123x50, and 123x400, and then adding them all up. Any sort of conceptual program (including, btw, many of the pre-New-Math programs -- being old/traditional does NOT make a program non-conceptual) is going to give an explanation that involves place value in some way, shape, or form, that is not just 'memorize the algorithm because I told you to'.

 

I think every traditional program I've ever seen does explain how multi-digit multiplication works as you mention here. I just checked a PDF of Saxon 5/4 that I found online somewhere, and sure enough, they start out by showing this for multiplying 34x12:

 

34 x 2 = 68 partial product

34 x 10 = 340 partial product

--------------------------------------------

34 x 12 = 408 total product

 

It then shows how to use this algorithm vertically, mentioning that you can put the zeros or leave them out, but that the zeros are there because the 1 (in 12) is actually 10.

 

It's all there. It's conceptual. That's Saxon. I don't care for Saxon myself, but all these traditional programs do teach concepts like this. Which programs don't teach this?

 

I can see where a teacher who doesn't understand the above might gloss over the instruction and/or not reiterate the instruction, and the student might come out only remembering the algorithm. But the concept *was* taught initially.

 

[NASDAQ]

I never said that practice was unnecessary. Of course the student also has to practice procedures so that certain things become automatic! I can't have a student pause and think about how to multiply two polynomials, they just have to be able to do it quickly and accurately.

I hope I did not give the impression that practice was unnecessary- far from it. In fact, very often thorough conceptual understanding is only developed when the student is forced to apply the concept in different ways to a number of problems. The understanding emerges as the result of the practice (but that does not mean that all drill results in understanding). It requires carefully designed problems that make the student examine the concept from all angles and use it in different ways.

 

I may be coming at this from a very place-specific situation. Right now we've gone head-over-heals into conceptual math in my province (and several other provinces) and we're trying to pull back. So most of the people that I meet day-to-day are suffering because the teacher thinks that multiplication tables are optional niceties that parents might or might not drill at home, not because they didn't get the concept explained. They probably did get the concept explained, but they're so underwater they can't internalise it.

 

http://blog.scs.sk.ca/schmitz/math%20makes%20sense%205.pdf

 

Think less Singapore Math and more Everyday Math.

[kiana]

I think every traditional program I've ever seen does explain how multi-digit multiplication works as you mention here. I just checked a PDF of Saxon 5/4 that I found online somewhere, and sure enough, they start out by showing this for multiplying 34x12:

 

34 x 2 = 68 partial product

34 x 10 = 340 partial product

--------------------------------------------

34 x 12 = 408 total product

 

It then shows how to use this algorithm vertically, mentioning that you can put the zeros or leave them out, but that the zeros are there because the 1 (in 12) is actually 10.

 

It's all there. It's conceptual. That's Saxon. I don't care for Saxon myself, but all these traditional programs do teach concepts like this. Which programs don't teach this?

 

I can see where a teacher who doesn't understand the above might gloss over the instruction and/or not reiterate the instruction, and the student might come out only remembering the algorithm. But the concept *was* taught initially.

 

This is why dividing math into conceptual/traditional is really a bit of a red herring. Most programs DO teach concepts, at least to some extent. They may not assign problems that require the understanding of these concepts to the extent of some others, but they teach them. Similarly, most programs DO drill. It's just that some drill more than others. Trying to make them fit into neat little boxes is an exercise in futility -- they are really more on a continuum.

[kiana]

I may be coming at this from a very place-specific situation. Right now we've gone head-over-heals into conceptual math in my province (and several other provinces) and we're trying to pull back. So most of the people that I meet day-to-day are suffering because the teacher thinks that multiplication tables are optional niceties that parents might or might not drill at home, not because they didn't get the concept explained. They probably did get the concept explained, but they're so underwater they can't internalise it.

 

http://blog.scs.sk.ca/schmitz/math%20makes%20sense%205.pdf

 

Think less Singapore Math and more Everyday Math.

 

This is a big problem with Everyday Math. Some teachers do teach it well (and it's not a TERRIBLE program if drill is included) but many people have been given the impression that drill is unnecessary at best and damaging at worst if the understanding is there.

 

This is codswallop. It is horsefeathers. It is complete and utter male bovine excrement.

 

I do believe that drill is rather pointless if the understanding is not there, because the subject will be forgotten as soon as the drilling is stopped. And certainly some students need more drill than others. But drill and understanding MUST go hand in hand.

 

[TracyP]

I think every traditional program I've ever seen does explain how multi-digit multiplication works as you mention here. ...

 

Which programs don't teach this?

 

 

 

CLE teaches zero as a placeholder in 2 digit by 2 digit multiplication.

[Crimson Wife]

It's all there. It's conceptual. That's Saxon. I don't care for Saxon myself, but all these traditional programs do teach concepts like this. Which programs don't teach this?

I looked over MCP Math one time when I was up at the HS bookstore and remember thinking it was very similar to the textbooks I had growing up (not sure which program was used): an example at the top of the page of the procedure with no conceptual teaching followed by a zillion practice problems.

 

A teacher with good conceptual understanding herself could certainly add to what's in the textbook, but mine didn't.

[boscopup]

CLE teaches zero as a placeholder in 2 digit by 2 digit multiplication.

 

Do they not explain what's going on, partial products, etc? They usually do that stuff in the earlier levels. Looks like 403 is where they teach 2x2 multiplication. I don't have that level (I have 1, 2, an 5).

[Matryoshka]

This is a big problem with Everyday Math. Some teachers do teach it well (and it's not a TERRIBLE program if drill is included) but many people have been given the impression that drill is unnecessary at best and damaging at worst if the understanding is there.

 

This is codswallop. It is horsefeathers. It is complete and utter male bovine excrement.

 

I am not in the least an Everyday Math fan.  Quite the opposite.

 

However, I do have to point out in its defense that my mother, a retired 2nd grade teacher and EM convert (sigh), told me that in her EM training, they said that EM itself says all math facts should be learned to automaticity by the end of 2nd grade.  Actually, she didn't just tell me, she showed me where that was said in black and white in her EM TM. 

 

In her 2nd grade math class (taught with EM) the kids all had timed facts drill, and she gave out T-shirts to everyone who could finish all their fact pages in I think under a minute each.  She taught them in groups like doubles, doubles +1, etc., very similar to they way I've seen MUS group them.

 

She used to pester me to use her EM fact drill sheets with my kids (who used Singapore Math), but timed drills made my kids freeze up and cry. :glare:

 

But the EM teacher training and materials do in fact say that fact drill is an integral part of the program.  I'm really wondering how so many teachers have apparently missed that rather vital piece of info...

[NASDAQ]

I am not in the least an Everyday Math fan.  Quite the opposite.

 

However, I do have to point out in its defense that my mother, a retired 2nd grade teacher and EM convert (sigh), told me that in her EM training, they said that EM itself says all math facts should be learned to automaticity by the end of 2nd grade.  Actually, she didn't just tell me, she showed me where that was said in black and white in her EM TM. 

 

In her 2nd grade math class (taught with EM) the kids all had timed facts drill, and she gave out T-shirts to everyone who could finish all their fact pages in I think under a minute each.  She taught them in groups like doubles, doubles +1, etc., very similar to they way I've seen MUS group them.

 

She used to pester me to use her EM fact drill sheets with my kids (who used Singapore Math), but timed drills made my kids freeze up and cry. :glare:

 

But the EM teacher training and materials do in fact say that fact drill is an integral part of the program.  I'm really wondering how so many teachers have apparently missed that rather vital piece of info...

 

I think the problem with EM is that it's a huge math program even without throwing in "btw, do all the math facts." And a lot of it advised drill via games, which is tricky to organise with one kid, let alone thirty.

 

AFAIK Math Makes Sense, our Canadian folly, doesn't include drill.

 

[boscopup]

However, I do have to point out in its defense that my mother, a retired 2nd grade teacher and EM convert (sigh), told me that in her EM training, they said that EM itself says all math facts should be learned to automaticity by the end of 2nd grade.  Actually, she didn't just tell me, she showed me where that was said in black and white in her EM TM. 

 

Sounds a bit like Singapore. :lol: People complain that there isn't enough drill in Singapore, but the teacher is supposed to be providing it, and the TM mentions at different spots that children should have their facts memorized. It just isn't in the daily lesson plans to do xyz speed drill or flash cards each day like Saxon or CLE or some others would do.

[Matryoshka]

I think the problem with EM is that it's a huge math program even without throwing in "btw, do all the math facts." And a lot of it advised drill via games, which is tricky to organise with one kid, let alone thirty.

 

My mom was a huge fan of the games for conceptual understanding (which I may have only bothered to learn a couple of, even though she gave me all the books and the card deck... :tongue_smilie:), but all her drill was on paper with pencil.  A page o' facts, set the timer.  I think they may have even been black-line masters from the EM book, but I'm not completely sure; it's been years now - my kids are in high school!

 

[TracyP]

Do they not explain what's going on, partial products, etc? They usually do that stuff in the earlier levels. Looks like 403 is where they teach 2x2 multiplication. I don't have that level (I have 1, 2, an 5).

 

Here is the wording: Now you will multiply by two tens, but first you must use a zero as a placeholder in the ones place. Write a zero under the 8 in 138. That is every word of that part of the teaching section. Considering how they started the first sentence, it would have been so easy to explain the zero. I agree that they are much better about teaching conceptually at the early levels (through the 300s).

[boscopup]

Here is the wording: Now you will multiply by two tens, but first you must use a zero as a placeholder in the ones place. Write a zero under the 8 in 138. That is every word of that part of the teaching section. Considering how they started the first sentence, it would have been so easy to explain the zero. I agree that they are much better about teaching conceptually at the early levels (through the 300s).

 

Ok, so they did at least say they were multiplying by 2 tens. That's technically the conceptual part. Yeah, it's minimal and not emphasized like Math Mammoth (where she has you take each individual number and do a partial product, so your 2x2 ends up with 4 partial products so you really, truly understand for sure what's going on :lol:), but it's there.

 

I agree that the "zero as a place holder" makes me twitchy. ;)

[NASDAQ]

I have my third grade math book from the nineties downstairs. Maybe I'll look this up.

[kiwik]

Here's an example of how ignorant I discovered myself to be in terms of math having come out of a totally procedural approach: until I started homeschooling, I seriously had no clue why the double-digit multiplication algorithm has the user place a 0 in the second line. I knew to put the 0 there and could quickly calculate the correct answer but it was never explained to me (nor did it dawn on me independently) that the 0 was there because I was multiplying by whatever multiple of ten (20, 30, etc.) rather than by the number.

 

It sounds so stupid in hindsight but here I was an "A" student in math up through calc 3 and scoring >700 on the math portions of the SAT & GRE yet not understanding such a basic concept.

I had similar problems. I have done calculus and statistics to first year university and applied maths and modelling above that but am still now I have kids realising how much stuff I learnt by rote and really didn't understand - carrying the one for heavens sake why didn't anyone say "put the extra tens in the tens column?". It is not precisely that I didn't know just that I didn't think about it. The funniest one was I was how long it took to realise that the line in the fraction was a division line. I knew 1/2 was 1 divided by 2 I just didn't click. I did read through a lot of school though.

[Corraleno]

The funniest one was I was how long it took to realise that the line in the fraction was a division line. I knew 1/2 was 1 divided by 2 I just didn't click.

 

I suspect that MOST people in this country don't understand what a fraction is. Nor do they understand that a decimal number is a fraction. People think of fractions as "pieces of pizza" and decimals as money; pizza and money are different things, so fractions and decimals must be different things. When they learn to divide by fractions, they just learn "invert & multiply," and when they learn division by decimals they learn to "move the decimal," and they have no clue why either of those "tricks" works, let alone that they're doing the exact same thing in each case — creating equivalent fractions by multiplying the numerator & denominator by the same number. To most people, it just seems like two completely separate, totally random rules you have to memorize. And then they end up inverting the wrong fraction, or inverting both, or only moving the decimal in the divisor, because they have no idea what they're doing.

[Meriwether]

Saxon 3 shows has the kids write division problems three ways, including as fractions. They are regularly given fractions in the next levels that they change to long division problems to solve.