Theoretical math question RE: conceptual vs. traditional

[strange_girl]

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 am slogging through making a workable plan for my K-12 curriculum choices, and the math is just so confusing. So here is my question for you wise teachers:

 

Can a student be prepared for college-level math (should they choose to go to college) using traditional math such as Saxon?

 

I find I do not like conceptual math (perhaps because it is foreign to me?) and would be extremely happy teaching traditional math straight through. I don't think my kids need to be math whizzes, but I would like them to be prepared when the time comes. What do you think?

[regentrude]

Can a student be prepared for college-level math (should they choose to go to college) using traditional math such as Saxon?

 

 

Yes.

And I am saying this as somebody who hated Saxon with a passion (as did my kids).

Many people on these boards have used Saxon successfully for their students, and they were well prepared for college.

 

A very different question is whether Saxon is a good fit for the student (it was not for mine). Some students do really well with the incremental spiral, whereas others require a mastery approach.

 

[Crimson Wife]

Plenty of us did just fine in college despite having had very procedural math growing up. If you've got a good memory and can remember what to do, you'll be able to quickly calculate the correct answer despite not having a clue why the algorithms work.

 

That said, I think I was done a disservice by not being taught the why's of math. I got good grades up through calc 3 and strong SAT & GRE scores. But I wound up learning an embarrassing amount from Right Start B. It's been a humbling experience to relearn math the Asian way, but I think I'm much better off as a result. I find that I actually enjoy math, whereas before I feared it.

[Crimson Wife]

BTW, if Math Mammoth is confusing, take a look at Right Start. It's totally scripted and I found it pretty much open-and-go.

[Ellie]

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 am slogging through making a workable plan for my K-12 curriculum choices, and the math is just so confusing. So here is my question for you wise teachers:

 

Can a student be prepared for college-level math (should they choose to go to college) using traditional math such as Saxon?

 

I find I do not like conceptual math (perhaps because it is foreign to me?) and would be extremely happy teaching traditional math straight through. I don't think my kids need to be math whizzes, but I would like them to be prepared when the time comes. What do you think?

 

Yes, of course.

 

Why wouldn't a traditional math also be conceptual? :confused1:  Millions of people have learned math traditionally and done things like build skyscrapers and pyramids and computers and all sorts of stuff, for crying out loud!

 

Back in the early 80s, process maths (those which use manipulatives such as c-rods, such as Miquon, or manipulatives of pretty much any kind, such as Making Math Meaningful) were popular such that some would imply that if you weren't using manipulatives there was sin in your life. :glare: Which is hogwash, of course. Some children learn better with manipulatives, some don't. (Process vs )traditional is still the first thing I think of when categorizing math materials, not spiral versus mastery, but that's another story). So this discussion isn't really a "trend;" there are just more opportunities to see the discussions. :-)

 

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.

 

If you like traditional math, and your children are doing well, then I see no reason to jump on the Singapore/MEP bandwagon. :-)

[Rosie_0801]

Plenty of us did just fine in college despite having had very procedural math growing up. If you've got a good memory and can remember what to do, you'll be able to quickly calculate the correct answer despite not having a clue why the algorithms work.

 

That said, I think I was done a disservice by not being taught the why's of math. 

 

Yeah. I had a good memory until I had kids, then all the procedural maths fell out of my head and I didn't have any maths skills to speak of left in there. That taught me to believe in the value of conceptual maths! 

 

But as the others said, people can live well and happily with only procedural maths. I just figure since I'm here, I might as well give it a go. There are enough maths phobic people in my family and I'm convinced it is an "enemy" that can be conquered. :p And, you know, there are people like maths and people who think it is *beautiful.* This is weird stuff and I'd hate to miss out if I could have that too. I learned to like sushi because I felt like I was missing out and maths gives me no concerns about bad tastes or squidgy textures.  :lol:

 

But for context, I have a 6 yo with Echolalia and no one seems to know how to teach maths to kids with Echolalia, so we'll have done 3 Kindergarten maths curriculums before I feel safe moving on to grade one. Meaning, I know bugger all about what I'm doing, but am doing it anyway and it seems to be working.

 

If we keep at it, whatever we do, we're bound to get somewhere. If you prefer procedural maths and your kids do too, and you stick at it for 12 years, you're bound to have kids who can pass their college classes.

[strange_girl]

To clarify: I do not find Math Mammoth confusing, just the number of choices and math philosophies in the homeschooling world.

 

I'm really glad to hear that people still think a non-Asian math is a viable option. I know it probably works very well for many kids, but I have an unecplained antipathy toward the whole thing. :laugh:

[Rosie_0801]

Mm. We've used MEP, which is Hungarian, and Miquon and CSMP which are both American. Do I need a program from each continent? That'd be geeky, wouldn't it? :D

[strange_girl]

Mm. We've used MEP, which is Hungarian, and Miquon and CSMP which are both American. Do I need a program from each continent? That'd be geeky, wouldn't it? :D

 

 

"If I choose a math from each country my kids should be pretty well covered, right?" :lol:

 

I also have an antipathy to doing more than one math program at a time. I want to find one that just teaches math the good old fashioned way and be done with it.

 

Guess I'm showing my age now :ph34r:

[kbutton]

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.

[Stars]

I'm so glad the OP asked this question!  We chose Saxon based exclusively on TWTM recommendation, and while I've been pleased and DS6 has been pleased (but not challenged), these boards have made me wonder if I'm doing him a disservice by not switching to another program.  It's such a relief to have reassurrance from some of you truly mathy people!  I'm reluctant to try conceptual math because I fear I might not understand it well enough to teach it...

[SorrelZG]

I think there is also I kind of child that, in spite of "traditional" math, intuitively *gets* what is going on and really does not need anyone to draw bar diagrams for them or break every possible conceptual way of approaching a problem into baby steps. Perhaps those were the ones building skyscrapers and pyramids. (or not, but maybe :D)

[Monica_in_Switzerland]

An answer to the original question:  Yes!  I know a 13 year old who has only done Saxon, and is now part of a very competitive university program for gifted middle and high school math students. 

 

Having said that, I have always loved math.  I have a degree in physics.  But, like Crimson Wife, I learned more teaching Right Start B and Singapore 1 and 2 than in 16 years of previous education... 

 

So I am a huge fan of "Asian math".  I add drill to Singapore because I  think we can have our cake and eat it, too.  :-) 

 

If you choose an Asian math, just do your homework.  Read the teacher's manuals.  Watch the videos at EducationUnboxed.  Understand before you teach.  Otherwise it's just the blind leading the blind, so to speak! 

 

I have only been HSing a few years.  But it has become clear to me that the program you teach best is more important than the program that matches your student's learning style.  If you can't teach Asian math no matter how hard you want to or try to, then don't do it.  Teach the program where you can confidently answer your child's questions and give additional information if necessary. 

 

And now, having said THAT, I don't think Asian math is that hard to grasp for most of us.  It's not like learning a new language.  Reading the teacher's manuals is more like hitting yourself in the forehead constantly and saying, "Why didn't anybody teach me this way?  Why am I just now spotting this connection?!"

 

 

[Arcadia]

I think there is also I kind of child that, in spite of "traditional" math, intuitively *gets* what is going on and really does not need anyone to draw bar diagrams for them or break every possible conceptual way of approaching a problem into baby steps. Perhaps those were the ones building skyscrapers and pyramids. (or not, but maybe :D)

Like Vi Hart's math doodles on YouTube which my boys rather watch than sleep *sigh*

What if Vi Hart design a skyscraper :)

[Ellie]

I think there is also I kind of child that, in spite of "traditional" math, intuitively *gets* what is going on and really does not need anyone to draw bar diagrams for them or break every possible conceptual way of approaching a problem into baby steps. Perhaps those were the ones building skyscrapers and pyramids. (or not, but maybe :D)

 

I would put it a different way: Traditional math does teach concepts; it just doesn't use bar diagrams or show children every possible way of solving a problem. :-) 

[Ellie]

I had traditional math until grade 5.  ime, what is taught depends on the provider.

 

I don't mean to be obtuse, but by "provider," do you mean "publisher"? Or teacher?

[MomintheMountains]

I am so happy that this thread was started! I have been looking for a more "straightforward" or "traditional" math for my kids. We have been doing Math Mammoth and time4learning.com. All of my school age boys, 10, 8 and 6 think MM is confusing. They just want to know how to do it and then take the shortest route to the answer; which is funny because I always thought boys would do better with conceptual math. My oldest, who is now 16 and doing well with math public high school, did Math U See and I never liked that one, either, even though I stuck with it for him.

 

I am a little confused by the attraction to conceptual math on a classical board. I have been thinking about math as it relates to classical education and it seems to me that traditional math would be more consistent with the idea of Grammar in the Trivium, moving on to more conceptual math when the child is ready (Logic). It seems like particularly in the early grades, children should be mainly learning their facts. As with any other subject, it seems like it would be more effective and efficient to teach the concepts when they are older and developmentally ready.

[regentrude]

I am a little confused by the attraction to conceptual math on a classical board. I have been thinking about math as it relates to classical education and it seems to me that traditional math would be more consistent with the idea of Grammar in the Trivium, moving on to more conceptual math when the child is ready (Logic). It seems like particularly in the early grades, children should be mainly learning their facts. As with any other subject, it seems like it would be more effective and efficient to teach the concepts when they are older and developmentally ready.

 

I completely disagree. I see the purpose of classical education to train students to think, not merely to regurgitate memorized information. Memory work has its important place, but I am not under the impression that classical education advocates memory work to be the only thing in the elementary grades.

 

A normal elementary age student is certainly developmentally ready to understand arithmetic with positive integers conceptually, especially since it is very concrete, close to every day experience, and can be made tangible  through manipulatives.

Requiring young children to approach math as an assortment of facts that have to be memorized does them a great disservice and prepares the ground for math aversion, because at some point memorizing won't cut it anymore. (It is those students that have been

conditioned to see math as a series of tricks to memorize that will flounder in algebra and higher math)

 

It will be much more effective if kids are shown from the beginning that there is are reasons behind those facts and algorithms, and that they should be able to reconstruct everything by thinking.

[Rachel TX]

I would put it a different way: Traditional math does teach concepts; it just doesn't use bar diagrams or show children every possible way of solving a problem. :-) 

 

I agree.

 

Whether a student is being taught with a traditional or conceptual method, the goal is the same: that the student will be able to understand and do math. 

 

Some people go through traditional math without fully understanding concepts, but someone can go through a conceptual program without enough practice to solidify their understanding. And that's where the teacher comes in. Homeschooling is great for making sure that your child can both understand and do math.

[MomintheMountains]

I completely disagree. I see the purpose of classical education to train students to think, not merely to regurgitate memorized information. Memory work has its important place, but I am not under the impression that classical education advocates memory work to be the only thing in the elementary grades.

 

A normal elementary age student is certainly developmentally ready to understand arithmetic with positive integers conceptually, especially since it is very concrete, close to every day experience, and can be made tangible through manipulatives.

Requiring young children to approach math as an assortment of facts that have to be memorized does them a great disservice and prepares the ground for math aversion, because at some point memorizing won't cut it anymore. (It is those students that have been

conditioned to see math as a series of tricks to memorize that will flounder in algebra and higher math)

 

It will be much more effective if kids are shown from the beginning that there is are reasons behind those facts and algorithms, and that they should be able to reconstruct everything by thinking.

Of course one of the goals of classical education is to teach children how to think, but there is a path to achieving that. I would like to have this discussion, but I think either I didn't use proper terms or you misunderstood my post. I didn't say that memory work would be the only thing that elementary aged children should be doing. I do see that my 6 and 8 year olds right now do best with concrete math, facts and computation. My ten year old is starting to grasp more of the concepts.

[Monica_in_Switzerland]

I would put it a different way: Traditional math does teach concepts; it just doesn't use bar diagrams or show children every possible way of solving a problem. :-) 

 

I think this is a pretty big oversimplification of conceptual math.  Bar diagrams are a useful tool, but they do not define conceptual math, and many conceptual math programs do not even teach them. 

 

It is also a false dichotomy to think that students either do conceptual math or memorize facts.  Most students doing conceptual math are also memorizing math facts, though it is true that they first learn several ways to quickly calculate before moving on to flashcards or another system for speed drill. 

 

Knowing 4 different ways to do an addition problem serves a purpose.  If all the methods are understood, it means the student has a very strong number sense that has been built in.  For me, the entire beauty of math is that, as long as you follow the "laws", it doesn't matter HOW you solve a problem, you will get a correct answer.  Learning in this way is clearly not slowing down the student- most Asian math programs appear to be a half year ahead of traditional counterparts. 

 

I'm not an Asian math pusher, but I am an Asian math lover.  Ultimately, you must teach the math that you are most able to teach and most enthusiastic about teaching.  A good teacher is far more important than a good curriculum.  But the right curriculum might just turn you into a better teacher if you are willing to put in some study time!

[VinNY]

I was attracted to a Classical type education because of the emphasis of sequencing, observation and/or memorization throughout the early grades with all subject matter. I  used the traditional type math programs (I've used so many with 8 kids, like Abeka, CLE, Horizon, MCP and Saxon) as our spine  but we also strengthened observations of math through patterns, manipulatives, cooking, problem solving etc, games. I ended up adding  Miquon Math to the mix and my older children used Developing Mathematical Reasoning Through Verbal Analysis (Critical Thinking Press...no longer in print). So I tried to do more with math instruction observation wise.It wasn't perfect because I don't think it was my strength though I had the desire. I am definitely stronger (or find it more natural) in preparing my children for Language  arts/History. Most of my children have gone on to private high schools and college and did not struggle in math, they did well (A's). Their teachers always said they were well prepared and most importantly had the perseverance to problem solve. 

[Ellie]

I think this is a pretty big oversimplification of conceptual math.  Bar diagrams are a useful tool, but they do not define conceptual math, and many conceptual math programs do not even teach them. 

 

It is also a false dichotomy to think that students either do conceptual math or memorize facts.  Most students doing conceptual math are also memorizing math facts, though it is true that they first learn several ways to quickly calculate before moving on to flashcards or another system for speed drill. 

 

Knowing 4 different ways to do an addition problem serves a purpose.  If all the methods are understood, it means the student has a very strong number sense that has been built in.  For me, the entire beauty of math is that, as long as you follow the "laws", it doesn't matter HOW you solve a problem, you will get a correct answer.  Learning in this way is clearly not slowing down the student- most Asian math programs appear to be a half year ahead of traditional counterparts. 

 

I'm not an Asian math pusher, but I am an Asian math lover.  Ultimately, you must teach the math that you are most able to teach and most enthusiastic about teaching.  A good teacher is far more important than a good curriculum.  But the right curriculum might just turn you into a better teacher if you are willing to put in some study time!

 

I'm not an educated person.

 

Many years ago, I read an article published in the Teaching Home that talked about two basic methods of math: process and traditional. Being an uneducated person, this became my paradigm. :-) The author said that process math is one that depends on manipulatives; Miquon and MathUSee would be examples. Traditional math does not; it may use visuals and diagrams and occasional hands-on activities, but it does not depend on them.

 

I don't even know how to describe the difference between "conceptual math" and "traditional math," because in my limited knowledge, traditional math does explain concepts. It does it differently, perhaps, from the way Singapore does, but it still does. Some children learn concepts better with c-rods and counting beans and abacus and whatnot; some children don't need them.

[boscopup]

My oldest son and I both do/did better with pictorial presentation rather than manipulatives. I understand math conceptually very well. Singapore's mental math techniques are how I naturally think. A program such as Right Start would frankly drive me nuts (and even my visual middle son didn't get much out of RS A - he did better with a combination of C-rods and pictorial examples).

 

Most (all?) traditional math programs do teach conceptually (the "why"), but many don't continue to reinforce the conceptual piece once it has been taught the first time, and some kids will forget why they do something. That is where the teacher is very important. I use CLE with my younger two kids, but I emphasize things such as place value even if CLE isn't reminding them of it. I'm not worried about my kids only having rote math skills. they know and understand math conceptually because I understand math conceptually and have taught them as such.

 

I know plenty of kids who went to a school that uses Saxon, and the kids did well in college math and became engineers. I have no doubt that a Saxon kid can go into a STEM career. While I don't like Saxon personally, it is a solid math program, and it does teach the "why" the first time they teach a topic. They also use a lot of manipulatives in the lower grades.

[lewelma]

I'm going to go out on a limb here, and suggest that a student who has worked through a program like Saxon would be fine in first year university math classes, but would have a rude awakening if he/she wanted to be math major. ( I'm not talking science or engineering majors)

 

Aops has really opened my eyes to what difficult math looks like. My son has experienced what Andrew Wiles described. Solving a math problem is like entering a dark room and stumbling around in the dark bumping into furniture for six months, until you find the light switch. My son does not take six months, but a single problem can take five hours and quite a lot of stumbling. Saxon and programs like it teach students to know how to solve every problem encountered, and this is not the way mathematicians work.

 

Would love to see this aspect discussed by a mathematician, as I am a scientist and only use math as a tool.

 

Ruth in NZ

[Crimson Wife]

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.

[KSinNS]

I agree with the statement that traditional vs. conceptual is a false dichotomy. All kids really should be aiming to get a "profound understanding of elementary mathematics." (I've forgotten the woman's name who writes about this stuff.) By that she means, the hows and whys of arithmetic. There are many ways to do that. Both my husband and I (mathy people to the core) got that out of our very traditional math educations, but some of our classmates didn't. They memorized how to do the problems. Conversely, many kids coming out of the more "conceptual" programs being taught in our local public schools now are simply confused by the multiple ways to do things, and never really learn "the right way" or even "a right way" to do their calculations. And some don't drill enough to be good at problems, which leads to the endless problems of university math placement tests. Anyway, there are lots of right ways to teach kids math. I think as a parent (and/or teacher), the goal is to make sure they understand math, and don't just memorize or wander in confusion. How you do that is really going to depend on you and the kid in question. Just my 2 cents.

[NASDAQ]

I grew up with traditional math and understood all this stuff. Maybe my mother was very bright.

 

Conceptual math makes explicit what traditional math may not. Some people don't need explicit, and skilled teachers have always demonstrated the rationale behind the procedure. Unfortunately elementary school teachers are frequently weak on math.

 

I'm not at all sure that a deep understanding of arithmetic is necessary for higher math.

 

In general I think people fail at math for want of practice, not for want of explanation.

[kbutton]

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.

 

This is an excellent example of what I think of when I think about conceptual math vs. algorithms or traditional math. BTW, my son figured out how to go from doing problems like 123 x 2 to multiplying 12345 x 12345 in about fifteen minutes (with a couple of well-placed questions to help him realize that he needed to add that zero). That's the power of the kind of conceptual math that I am thinking about--it's not just manipulatives, bar models, etc. He would not have intuited how to do that without having learned the concepts for how multiplication works, place value works, etc. (though I supposed some kids will if it occurs to them). In fact, when he was in school, he struggled with the steps for algorithms. When we explained the concepts at home, he could rebuild the algorithms when he got stuck (or the teacher misspoke, which happens).

 

 

[regentrude]

I'm not at all sure that a deep understanding of arithmetic is necessary for higher math.

 

Oh yes, it is. I teach physics at a university and get to observe students doing higher math on a daily basis. If a student never conceptually understood the arithmetic concept of how fractions are added or divided, he will not be able to manipulate fractions when they do not consist of simple numbers, but complex symbolic expressions in the numerator and denominator . I can tell which students have memorized "I must flip" and which have understood why they have to flip what and in which situation. (Typical mistake: dividing a complex fraction by an integer number; I can't tell you how often that number ends up on top instead of in the denominator, because they see a division and remember "flip".)

There are plenty of more examples where a lack of very basic conceptual understanding hinders students in higher math.

 

In general I think people fail at math for want of practice, not for want of explanation.

 

I disagree. Anything that has been conceptually understood will not be forgotten, and can be reconstructed years later. Not using the quadratic formula or binomial formulas will cause most students to forget eventually; those who understood how to derive them will easily be able to reconstruct them. It is a lot more effective to remember the concepts than attempting to memorize every single formula through practice.

 

ETA: And this conceptual memory is an amazing thing. Just last week, I re-derived an identity from my multivariable calculus class that I had not used in 25 years and could not recall - but because you don't ever forget concepts, I was able to put it back together. THAT is the kind of math instruction I want my kids to have.

 

another ETA: I have encountered many adults who had memorized from their school math 2*pi*r and pi*r^2 and know that one of these is for the area and the other for the circumference of the circle, but have trouble remembering which is which. Very very common. Yet one glance at those formulas and the understanding that a circumference has to have the dimension of a length, and area the dimension of a length squared would tell them immediately which is which. Sadly, most attempts to explain this fall on deaf ears, because this concept had never been introduced in elementary/middle school, and they are not used to thinking in those terms. they have been drilled to memorize, and I have no doubt that this has been cemented with numerous computational exercises.

[lewelma]

Just curious what you are basing that suggestion on. 

 

 

Well being a scientist, I know I have no scientific proof.  So just a hypothesis with enough behind it that it would be worth investigating. 

 

So my suggestion is based on :

 

1) Personal experience.  I used tradition programs in high school, not exactly plug and chug but leaning towards that.  And when I got to Duke and took 3rd semester calculus, I found a lot of proofs were required.  I needed to have the persistence and experience to stumble in the dark, and I completely failed at it. 

 

2) Reading about mathematicians like Andrew Wiles who emphatically stated that mathematics is wandering in the dark.  It is not plug and chug.

 

3) My personal comparison of Saxon vs AoPS textbooks.  The style of problem could not be different and clearly they teach a very different way of thinking about mathematics.  Saxon does not line up with what Andrew Wiles describes; AoPS does.

 

4) My experience teaching math in High school and now tutoring students.  I have seen so many students that have no idea *why* they are doing something, but are quite happy and good at plug and chug.  Depending on the student, they fall over somewhere between algebra and calculus.

 

5) My eldest child.  He is set to be a mathematician, and in our search for higher level mathematics, have found problems that require him to search in the dark, for hours, days, even weeks.  His hardest problem so far took 20 hours of desk time in addition to 2 weeks of just thinking.

 

6) Hearsay. I have heard numerous people discuss how strong high school math students are in for a rude awakening when they get to university math.

 

But still, I would love to hear from a mathematician, as I am a scientist without personal experience as a math major.

 

Ruth in NZ

 

 

[regentrude]

Well being a scientist, I know I have no scientific proof.  So just a hypothesis with enough behind it that it would be worth investigating. 

 

So my suggestion is based on :

 

1) Personal experience.  I used tradition programs in high school, not exactly plug and chug but leaning towards that.  And when I got to Duke and took 3rd semester calculus, I found a lot of proofs were required.  I needed to have the persistence and experience to stumble in the dark, and I completely failed at it. 

 

2) Reading about mathematicians like Andrew Wiles who emphatically stated that mathematics is wandering in the dark.  It is not plug and chug.

 

3) My personal comparison of Saxon vs AoPS textbooks.  The style of problem could not be different and clearly they teach a very different way of thinking about mathematics.  Saxon does not line up with what Andrew Wiles describes; AoPS does.

 

4) My experience teaching math in High school and now tutoring students.  I have seen so many students that have no idea *why* they are doing something, but are quite happy and good at plug and chug.  Depending on the student, they fall over somewhere between algebra and calculus.

 

5) My eldest child.  He is set to be a mathematician, and in our search for higher level mathematics, have found problems that require him to search in the dark, for hours, days, even weeks.  His hardest problem so far took 20 hours of desk time in addition to 2 weeks of just thinking.

 

6) Hearsay. I have heard numerous people discuss how strong high school math students are in for a rude awakening when they get to university math.

 

But still, I would love to hear from a mathematician, as I am a scientist without personal experience as a math major.

 

Ruth in NZ

 

I am not a mathematician, but a theoretical physicist -  which is pretty close ;-)

I tend to agree with your statements.

 

[Korrale]

I'm going to go out on a limb here, and suggest that a student who has worked through a program like Saxon would be fine in first year university math classes, but would have a rude awakening if he/she wanted to be math major. ( I'm not talking science or engineering majors)

 

Aops has really opened my eyes to what difficult math looks like. My son has experienced what Andrew Wiles described. Solving a math problem is like entering a dark room and stumbling around in the dark bumping into furniture for six months, until you find the light switch. My son does not take six months, but a single problem can take five hours and quite a lot of stumbling. Saxon and programs like it teach students to know how to solve every problem encountered, and this is not the way mathematicians work.

 

Would love to see this aspect discussed by a mathematician, as I am a scientist and only use math as a tool.

 

Ruth in NZ

I actually have a friend who got her BA in math, although she is currently working as an analysts.

She and her other math degree friend who went into secondary math teaching both are advocates for traditional math. It is what they learnt, what they knew and what they excelled in.

I don't think it was detrimental to them in any way to learn math that way. Maybe, if they went into a more math intensive career? But possibly people that do already understand the how. Arthur Benjamin speaks of this.

 

 

I kept reading, and I think limbs are good. Limbs open discussion. :)

[My3girls]

My only issue with "traditional" math is that you're pretty much taught one way to do the math then it's drill and kill. If the kid doesn't "get it", it's torture. I like the conceptual math because they learn multiple ways to work the problem then they can choose the way that works best for them. I tell my girls all the time to work it however they want as long as they get the right answer.

 

Back to the original question... Either way will get them to college, but their conceptual knowledge or lack there of will limit them. In calculus based physics, I could work the algorithms all day long, but then when it came to applying them to real world problems and knowing which ones to use... Hello big fat D and moving from college of engineering to college of arts and sciences. Lol. The real world doesn't give you the algorithm and tell you to solve it; it gives you a problem and tells you to solve it. It's up to you to know which algorithm to use and why.

 

Some kids can just absorb the concept and some kids need to be expressly taught. My girls never thought of working 9+3 by breaking 3 into 1 + 2. The ps had them just memorize the math fact 9+3=12. I have taught them the other way, and now they can use that method with any number that ends in 9 like 39+3. It's made a world of difference for them.

[lewelma]

I should add that my younger son, although using a conceptual program (singapore DMCC), will not be using AoPS.  I want his math skills strong enough for a STEM career if he chooses, but I don't think that being an actual mathematician is in his future. Just don't want to scare people into thinking AoPS is the only way.

 

I also want to reiterate that I am NOT saying that Saxon will not prepare you for university.  I *am* saying that students who do well in Saxon will not necessarily do well in a *math* major, because Saxon focuses on different skills than mathematicians focus on. 

 

Ruth in NZ

[MomintheMountains]

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.

[Arcadia]

4) My experience teaching math in High school and now tutoring students. I have seen so many students that have no idea *why* they are doing something, but are quite happy and good at plug and chug. Depending on the student, they fall over somewhere between algebra and calculus.

Being public schooled, there are kids that just want to get their school work done so they can do other things they like more. Then there are kids that are like my hubby who didn't have great teachers who can explain well the concepts so the teachers teach to the math exams (IGCSE). He did get great engineering math lecturers. Majority of my math teachers are great and I have cousins and a private math tutor go as deep as I want.

Same singapore math curriculum in same country but different outcomes.

 

I see the same scenario here for both public and private schools. I do agree on everything else in your post.

 

ETA:

As for your point 6, I guess strong is use differently by different people. Some teachers say strong math students to mean highly likely to get As for math exams. Other teachers mean that the kid is conceptually strong in math. While there are other teachers who use strong to mean kids good for math competition.

Of course the three "groups" do intersect.

[boscopup]

I'd bet the vast majority of math majors in US schools were taught some sort of traditional math. I don't think Saxon would cause someone not to be a math major. If they're going to be a math major, they will likely have some natural talent in math and probably intuit some of the conceptual understanding, as I did. I know I had a better understanding of things like unit conversions than most of my friends. I found them to be easy, because I understood exactly how they worked. I had been taught traditional math. Some of my friends just didn't get it, and I could do unit conversions all day long (the kind where you convert from A to B to C to D to E to F in one big fraction multiplication).

 

In fact, my sister was a math major (I keep forgetting that, as she ended up working as an editor for a university press, never to use her math degree :lol:). She had the same traditional math education that I did. It wasn't Saxon, but it was basic, traditional math. We had good teachers in high school who did explain why, but I'm also sure many of the students didn't retain the why because it wasn't drilled into them like it is in something like AoPS. Students using traditional math programs in school can get access to "hard problems" via math teams starting in middle school.

 

I agree that conceptual understanding is necessary to really do well in higher math. I just don't agree that using Saxon or some other traditional math program relegates you to a life of mediocrity... becoming just an engineer instead of a mathematician. ;) I also agree that good math teachers are essential.

[lewelma]

I'm going to bow out of this conversation.  I know that my experience with my older son is incredibly unusual, and I think it is clouding my judgement.  I don't want to advise anyone because it really depends on the student. 

 

Clearly, there are different levels of math talent, and I can assure any parent that if your child has talent like mine, Saxon would never ever fit.  And you would absolutely know. Absolutely. So if Saxon is working, keep with it.

 

Ruth in NZ

 

ETA: Wow! I hope that did not sound arrogant or condescending.  Not my intention.

[AimeeM]

Yes, yes they can. I'm not a fan of singapore, math mammoth, etc - and neither is my husband (who holds advanced degrees in science related fields - physics and engineering). He thrived with traditional programs and so do our "mathy" children.

 

ETA: many people are able to understand the concepts behind the maths, without needing a program dedicated to teaching it that way. My husband is an amazing engineer; he most certainly understands the "whys" behind the math that he does so instantly in his head. Every child is different, though, so I digress.

[EndOfOrdinary]

I am a little confused by the attraction to conceptual math on a classical board. I have been thinking about math as it relates to classical education and it seems to me that traditional math would be more consistent with the idea of Grammar in the Trivium, moving on to more conceptual math when the child is ready (Logic). It seems like particularly in the early grades, children should be mainly learning their facts. As with any other subject, it seems like it would be more effective and efficient to teach the concepts when they are older and developmentally ready.

If you actually study classical texts - Euclid, Newton, Kelper, Gallileo, they are so heavily conceptual it is quite boggling.  The application of their conceptual ideas to theorum is what is so fantastical about them.  So if we are really getting back to Classicsal education and the masterful works, the conceptual is almost the most significant.  Without a conceptual idea, the student cannot understand foundational materials.  So the concept is the part that the student needs the earliest.  The concepts are the facts which are important.  The algebra cannot be applied without those concepts.  It is not the memorizing of 4 x 6 = 24 that is important.  It is the memorizing that multiplication is commutative.  That 4 x 6 gives you four rows of six objects or six rows of four objects.  It is the ability to build those types of scenerios which is truely important during the Grammar Stage.  Those are the true foundational facts of Classical mathematics and the classical math scholars.  Later, those bits of information are what build into the application of algebra or geometry.  4x6 = 24 isn't going to get you very far with classical masters.

[msrift]

Personal example regarding the difference between conceptual and traditional math: I was always considered a mathy kid. Traditional math came easy to me and I was encouraged to go into engineering. Took 2.5 years of engineering classes in college (including most of the required math classes) before I finally decided that this stuff bored me to tears and switched to biochemistry. I tried to get my engineering math classes counted toward a math minor and was denied as it 'wasn't real math'. These were advanced math classes (matrix algebra, differential equations, etc.) but as they were taught from a practical (engineering) and not theoretical standpoint, they weren't considered pure math.

Do to a fluke with credits, I found myself taking one of those 'pure' math classes. What a difference!! That class kicked my butt and proved to me that there really is a difference in math. 'Calculus is calculus no matter where you take it' is absolutely not true. Even at the same university, the same math topics can be taught to very different levels.

[Corraleno]

I think that when people talk about "conceptual math" as if it's a different type of math from "regular" or "traditional" or "procedural" math, it creates a false impression of what "conceptual math" actually refers to.  Calling it "Asian math" has the same effect, as if choosing between "Asian math" and "American math" is somehow like choosing between French and Spanish for your foreign language: two different but equivalent things, and the choice between them is just a matter of preference. But the difference between "procedural" and "conceptual" math is the depth of the understanding, not the "variety" of math being taught or the country it comes from. 

 

How many people say "I just want my kids to memorize a bunch of names and dates for history, so they can get As on multiple choice tests; it's really not important if they know how or why things happened the way they did"? Do people say "When we study literature, I just want my kids to be able to decode the words on the page, label the correct parts of speech, and know if they're spelled correctly; it doesn't matter if they understand the style or structure or meaning of the story"? Yet, to me, that's what people are saying about math when they say they just want their kids to have a procedural knowledge, and skip all that confusing conceptual stuff. 

 

Math is a language for expressing ideas about the world. Limiting oneself to a procedural understanding of math is like learning to imitate grammatical sentences in a foreign language, without being able to actually read the sentences or understand their meaning. IMHO, that really misses the whole point of learning the language.

[MomintheMountains]

 

 

How many people say "I just want my kids to memorize a bunch of names and dates for history, so they can get As on multiple choice tests; it's really not important if they know how or why things happened the way they did"? Do people say "When we study literature, I just want my kids to be able to decode the words on the page, label the correct parts of speech, and know if they're spelled correctly; it doesn't matter if they understand the style or structure or meaning of the story"? Yet, to me, that's what people are saying about math when they say they just want their kids to have a procedural knowledge, and skip all that confusing conceptual stuff.

.

I think there are a couple of tangential discussions going on here. For me personally in this conversation, I am focusing on younger children. You have brought up an excellent analogy here. You are correct, I don't have my young children memorizing history facts out of context. I am giving them some context *to the extent that they are ready for it*. I can give my second grader some basic context for the civil war, but not at the same depth as with my fifth grader. My fifth grader can understand the civil war on a deeper level, but certainly not at the same depth as my tenth grader. 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.

[Corraleno]

If you actually study classical texts - Euclid, Newton, Kelper, Gallileo, they are so heavily conceptual it is quite boggling.  The application of their conceptual ideas to theorum is what is so fantastical about them.  So if we are really getting back to Classicsal education and the masterful works, the conceptual is almost the most significant.  Without a conceptual idea, the student cannot understand foundational materials.  So the concept is the part that the student needs the earliest.  The concepts are the facts which are important.  The algebra cannot be applied without those concepts.  It is not the memorizing of 4 x 6 = 24 that is important.  It is the memorizing that multiplication is commutative.  That 4 x 6 gives you four rows of six objects or six rows of four objects.  It is the ability to build those types of scenerios which is truely important during the Grammar Stage.  Those are the true foundational facts of Classical mathematics and the classical math scholars.  Later, those bits of information are what build into the application of algebra or geometry.  4x6 = 24 isn't going to get you very far with classical masters.

 

:iagree:

 

Not to mention the fact that the whole concept of young children being incapable of little more than "poll parrot" memorization is the invention of one person, based on a sample size of exactly 1: herself. This concept is not true of my children, was not true of me, was not true of the vast majority of kids I taught in 2 yrs as a preschool teacher, and it is not supported by developmental research. Young children are absolutely capable of understanding the "how" and "why" of math — in fact, I think if more kids understood math conceptually, fewer of them would hate it.

[MomintheMountains]

:iagree:

 

Not to mention the fact that the whole concept of young children being incapable of little more than "poll parrot" memorization is the invention of one person, based on a sample size of exactly 1: herself. This concept is not true of my children, was not true of me, was not true of the vast majority of kids I taught in 2 yrs as a preschool teacher, and it is not supported by developmental research. Young children are absolutely capable of understanding the "how" and "why" of math — in fact, I think if more kids understood math conceptually, fewer of them would hate it.

I don't think anybody is advocating poll parrot math. I certainly am not.

[NASDAQ]

Oh yes, it is. I teach physics at a university and get to observe students doing higher math on a daily basis. If a student never conceptually understood the arithmetic concept of how fractions are added or divided, he will not be able to manipulate fractions when they do not consist of simple numbers, but complex symbolic expressions in the numerator and denominator . I can tell which students have memorized "I must flip" and which have understood why they have to flip what and in which situation. (Typical mistake: dividing a complex fraction by an integer number; I can't tell you how often that number ends up on top instead of in the denominator, because they see a division and remember "flip".)

There are plenty of more examples where a lack of very basic conceptual understanding hinders students in higher math.

 

 

I disagree. Anything that has been conceptually understood will not be forgotten, and can be reconstructed years later. Not using the quadratic formula or binomial formulas will cause most students to forget eventually; those who understood how to derive them will easily be able to reconstruct them. It is a lot more effective to remember the concepts than attempting to memorize every single formula through practice.

 

ETA: And this conceptual memory is an amazing thing. Just last week, I re-derived an identity from my multivariable calculus class that I had not used in 25 years and could not recall - but because you don't ever forget concepts, I was able to put it back together. THAT is the kind of math instruction I want my kids to have.

 

another ETA: I have encountered many adults who had memorized from their school math 2*pi*r and pi*r^2 and know that one of these is for the area and the other for the circumference of the circle, but have trouble remembering which is which. Very very common. Yet one glance at those formulas and the understanding that a circumference has to have the dimension of a length, and area the dimension of a length squared would tell them immediately which is which. Sadly, most attempts to explain this fall on deaf ears, because this concept had never been introduced in elementary/middle school, and they are not used to thinking in those terms. they have been drilled to memorize, and I have no doubt that this has been cemented with numerous computational exercises.

 

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.

 

My own kids do Singapore, but my daughter is still not particularly strong in math. I think people overestimate the extent to which the texts make a difference.

 

I believe that attempts to run trials with Singapore math in American schools were similarly uninspiring.

[NASDAQ]

If you actually study classical texts - Euclid, Newton, Kelper, Gallileo, they are so heavily conceptual it is quite boggling.  The application of their conceptual ideas to theorum is what is so fantastical about them.  So if we are really getting back to Classicsal education and the masterful works, the conceptual is almost the most significant.  Without a conceptual idea, the student cannot understand foundational materials.  So the concept is the part that the student needs the earliest.  The concepts are the facts which are important.  The algebra cannot be applied without those concepts.  It is not the memorizing of 4 x 6 = 24 that is important.  It is the memorizing that multiplication is commutative.  That 4 x 6 gives you four rows of six objects or six rows of four objects.  It is the ability to build those types of scenerios which is truely important during the Grammar Stage.  Those are the true foundational facts of Classical mathematics and the classical math scholars.  Later, those bits of information are what build into the application of algebra or geometry.  4x6 = 24 isn't going to get you very far with classical masters.

 

Not taking sides on what children do or do not do in the grammar stage (because I don't care), but Euclid &c. were intended to be studied later on in education, and thus I don't think they make your point here.

 

I think knowing that 4*6 is itself useful, in addition to understanding that multiplication is commutative, but ymmv.

[Crimson Wife]

many people are able to understand the concepts behind the maths, without needing a program dedicated to teaching it that way. My husband is an amazing engineer; he most certainly understands the "whys" behind the math that he does so instantly in his head. Every child is different, though, so I digress.

This is analogous IMHO to the debate over teaching reading via phonics vs. "whole language". Some kids when taught via a "whole language" approach will naturally intuit the phonics rules themselves. I apparently figured out phonics for myself when I was 3. My mom read aloud to me constantly and I learned to read by matching the words I saw on the page to the stories I was familiar with. My mom thought my "reading" was merely reciting from memory until I mispronounced some word with a silent letter. By the time I entered kindergarten, I was reading long chapter books so the private school I attended just sent me off to a corner to read independently while my classmates had their small-group reading instruction.

 

When I was an adult and encountered the phonics & spelling rules, it was a very strange experience because it was the first time I'd seen them explicitly laid out, yet they were things I had internalized just through exposure to books.

[Corraleno]

I'm not a fan of singapore, math mammoth, etc - and neither is my husband (who holds advanced degrees in science related fields - physics and engineering). He thrived with traditional programs and so do our "mathy" children.

 

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

 

 

ETA: many people are able to understand the concepts behind the maths, without needing a program dedicated to teaching it that way. My husband is an amazing engineer; he most certainly understands the "whys" behind the math that he does so instantly in his head. Every child is different, though, so I digress.

 

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.