Help me understand conceptual vs. traditional math

[Meadowlark]

I don't really understand the big difference here. Today. I taught my 2nd grader subtraction with borrowing. We are currently using Singapore, but CLE is on the way because I'm just finding Singapore hard to teach. I keep hearing about how Singapore, Rightstart, MIF, etc. are "conceptual" and CLE, Saxon, etc.are more traditional.

 

So, can someone spell this out for me? Within the context of the lesson above, how would CLE teach it vs. Singapore? We're only on 2A but so far Singapore seems very traditional to me, but maybe I'm missing something? Now I'm worried that CLE will lack the conceptual thing...ugh, but maybe if I can get a grasp in what it IS, maybe I can add and supplement with more conceptual things.

[Farrar]

First, it doesn't have to be either/or. When people talk about that, it's often just shorthand and oversimplication about programs or methods for the sake of brevity.

 

I would suggest you read Knowing and Teaching Elementary Mathematics by Liping Ma. It's a short, quick, compelling read and it answers some of this question.

 

For the sake of keeping it relatively simple though... Subtraction with regrouping... In a traditional program, it might be called subtraction with borrowing. There might be models and manipulatives and number lines and so forth, but the focus of the teaching would be on learning and practicing the algorithm. Stacking the numbers, crossing out each number in turn if there's not "enough" to subtract from, etc.

 

In a more conceptual leaning program, it would never be called "borrowing" because the term is misleading. You're not borrowing anything, you're simply rearranging the numbers. If you're subtracting 19 from 42 and you have to regroup the 42 to become 3 tens and 12 units instead of 4 tens and 2 units, then that's the exact same amount. Nothing was borrowed really. While the end goal may be the ability to do the algorithm as mentioned in the traditional program, the focus would be on understanding why it works to "regroup" the numbers like that and how to regroup them.

 

Many conceptual programs also focus on mental math. Mental math requires being able to do multiple methods. So in our 42-19 example, you could subtract 43-20 instead, which requires understanding that subtraction is difference and if you add one to each number they'll still be the same distance away. It also requires knowing that this will be a much easier problem to do in your head. But for another problem, it might be right to subtract in chunks or do simply do the traditional algorithm in your head. Having a deeper understanding *should* allow kids to be able to choose the method that will work quickest and easiest.

 

Many of us like conceptual programs because we want our kids to have that deeper understanding where they can figure out the problem without pen and paper to mark out and "borrow" numbers. Even if that's what they do the majority of the time, we want them to be able to explain why it works.

 

On the other hand, many parents like more traditional programs because they're often pared down and simpler, which they believe will help the child see the math more clearly than in a program teaching lots of different ways to conceptualize something. Some people may also believe that many (or even most) kids cannot understand something such as subtraction with regrouping until they've done it many, many times. So it's through doing this traditional algorithmic practice that they'll come to have a greater understanding of concepts.

[vaquitita]

I second reading Liping Ma's book. It will help you know what conceptual teaching is. You can use any good math program, as long as you the teacher, teaches them what/why they are doing what they're really doing, not just move your number over. Ie: do they understand what they are doing when they 'borrow'? Or do they just know that they are supposed to cross out this number here, write that number there?

[kiwik]

Are you using the home instructors guide? If you use Singapore materials but teach how you are taught you will get whatever you were taught.

[MiMi 4under3]

These 2 examples helped me to begin to see the difference between teaching math conceptually vs. teaching the process.

 

 Thanks for the link!  :)

[Ellie]

I don't understand it, either.

 

When I'm looking at math materials, the first thing I want to know is whether it's a *process* math or a *traditional* math. A process math is one which depends on manipulatives such as c-rods or base 10 blocks; a traditional math does not. After that, I try to imagine myself teaching it and my dc learning it. "Conceptual" isn't on my radar at all (and neither is spiral vs mastery).

[Meadowlark]

I don't understand it, either.

 

When I'm looking at math materials, the first thing I want to know is whether it's a *process* math or a *traditional* math. A process math is one which depends on manipulatives such as c-rods or base 10 blocks; a traditional math does not. After that, I try to imagine myself teaching it and my dc learning it. "Conceptual" isn't on my radar at all (and neither is spiral vs mastery).

Do you mind me asking why it's not on your radar? And spiral vs. mastery too? You sound so confident in your choice so I'm genuinely curious why ( as I try to pave our math future...)

[Ellie]

Do you mind me asking why it's not on your radar? And spiral vs. mastery too? You sound so confident in your choice so I'm genuinely curious why ( as I try to pave our math future...)

 

Because it just sounds like gobbledygook to me. It sounds like public school people trying to make their math sound different or better when it's really just the 60s New Math all over again even though it turns out that traditional math, the kind that has been taught for hundreds of years, actually works and the new stuff doesn't. It reminds of people trying to repackage the sight reading from the 50s, which was a massive fail, into the whole language of the 80s, which wasn't any better, instead of sticking with tried-and-true phonics. Or like Scott Foresman trying to get a toe in the penmanship market with their D'Nelian product by referring to traditional manuscript as "stick and ball." 

 

I feel the same away about spiral vs mastery. You have to decide on process or traditional math first, not spiral vs mastery (and people don't always agree on which product is spiral and which is mastery, so...).

 

IMHO, traditional math *does* teach concepts.

[AimeeM]

I do not find CLE lacking conceptually - at all. We're doing CLE 1 with my middle son, and it definitely builds conceptually before pushing memorization. For example, in the TM for grade one, it has you using manipulatives such as counters, toothpicks, pencils, and pennies to teach the concept of addition; it uses manipulatives and a place value chart to teach place value; it has the parent teach these things ALONGSIDE memorizing math facts. 

I can honestly say the concept of "how" and "why" are very well laid out in CLE. I believe the same can be said for most "traditional" math programs.

[kiana]

I won't even try to answer your question because I'm also confused about this but I want to share an interesting article about the characteristics of a good elementary math curriculum. ftp://math.stanford.edu/pub/papers/milgram/milgram-msri.pdf

 

He's got some very good points.

 

Some in particular: "My impression is that too many K–8 teachers in this country, and consequently too many

students as well, see mathematics as lists of disjoint, disconnected factoids, to be memorized,

regurgitated on the proximal test, and forgotten, much like the dates on a historical time-line.

Moreover, and more disturbing, since the isolated items are not seen as coherent and connected,

there does not seem to be any good reason that other facts cannot be substituted for the ones

that are out of favor."

 

This viewpoint is definitely prevalent and very worrying. It is especially prevalent among developmental students, and trying to train them to see the bigger picture, rather than attempt to memorize separate algorithms for each type and sub-type of problem, is a huge issue. 

 

I also strongly agree with him about the lack of precision in definitions. Again, this is not something that I'm applying to advanced undergraduates or as a mathematician, but the fundamental lack of understanding I see of students in developmental math classes. They'll look at an equation like "2x = 14" and not understand whether they're supposed to subtract 2 from both sides or divide by 2 on both sides, because they don't really understand what 2x means. 

 

Another comment: "For the common stages of school mathematics, students must practice with numbers. They

must add them until basic addition is automatic. The same for subtraction and multiplication.

They must practice until these operations are automatic. This is not so that they can amaze

parents and friends with mathematical parlor tricks, but to facilitate the non-verbal processes

of problem solving. At this time we know of no other way to do this, and I can tell you,

from personal experience with students, that it is a grim thing to watch otherwise very bright

undergraduates struggle with more advanced courses because they have to ï¬gure everything

out at a basic verbal level. What happens with such students, since they do not have total

fluency with basic concepts, is that – though they can often do the work – they simply take

far too long working through the most basic material, and soon ï¬nd themselves too far behind

to catch up."

 

If someone is attempting to learn algebra, and wants to factor something like x^2 - 17x + 42, but cannot reliably factor 42 multiple ways without a calculator, this will be a tremendous hindrance. The fluency with basic calculations is very necessary for higher-level performance in any sort of math. It is possible for an extraordinarily bright person to achieve without it, but it is far more difficult. 

[shinyhappypeople]

This is kind of a rabbit trail, but I wanted to point out that there are those of us with kids who are "whole to parts" thinkers (my younger DD).  For some kids "Show me the algorithm and then I'll figure out how we got there later" works best.  My younger DD, who is very bright, got lost in the details of MM, but give her a quick, simple explanation of what we're trying to accomplish and how to get there (the algorithm) and she's a happy girl.  As she internalizes the algorithm, she internalizes the concept.  What we think of as conceptual math is not best.... for her.

 

I have the other type of thinker, too.  Older DD loves how MM builds concepts, one small conceptual building block at a time until you finally get to the traditional algorithm.   In her case, if we go too quickly to the algorithm (TT I'm looking at you...) she'll plug in the numbers but usually have no idea what she's doing or why she's doing it.  Traditional math is not best... for her.

 

I think public schools have the awful task of picking a road to walk down and hoping that most of the children can learn the way the chosen curriculum teaches (at the moment, that usually means concept-heavy approaches, b/c of CCSS).  Homeschooling is so much better for children in this regard.  If your kids are learning and you're comfortable teaching, that's the best math approach.

[Targhee]

Singapore 1A-2B are simple enough many just teach from their experience, rather than from the HIG - make sure you offer students a concrete - visual - abstract sequence of models, and that you try all the problem solving methods in the book. The differences are subtle, but these methods in themselves demonstrate and emphasize a conceptual reason for performing the algorithm so that students understand the process well enough to adjust/alter/mentally calculate/apply/problem solve/estimate more easily when needed. Only after the ability to conceptually manipulate are they shown the standard algorithm. At that point they are free to manipulate problems in their mind with greater facility than if they had been married to the algorithm.

 

Even then, I think the big difference between Singapore and other maths comes in level 3.

[kiana]

This is kind of a rabbit trail, but I wanted to point out that there are those of us with kids who are "whole to parts" thinkers (my younger DD).  For some kids "Show me the algorithm and then I'll figure out how we got there later" works best.  My younger DD, who is very bright, got lost in the details of MM, but give her a quick, simple explanation of what we're trying to accomplish and how to get there (the algorithm) and she's a happy girl.  As she internalizes the algorithm, she internalizes the concept.  What we think of as conceptual math is not best.... for her.

 

I have the other type of thinker, too.  Older DD loves how MM builds concepts, one small conceptual building block at a time until you finally get to the traditional algorithm.   In her case, if we go too quickly to the algorithm (TT I'm looking at you...) she'll plug in the numbers but usually have no idea what she's doing or why she's doing it.  Traditional math is not best... for her.

 

I think public schools have the awful task of picking a road to walk down and hoping that most of the children can learn the way the chosen curriculum teaches (at the moment, that usually means concept-heavy approaches, b/c of CCSS).  Homeschooling is so much better for children in this regard.  If your kids are learning and you're comfortable teaching, that's the best math approach.

 

I absolutely agree that for some kids beginning with procedural fluency (the algorithm) and then figuring out how they got there later is a far superior method to trying to front-load the concepts.

 

I definitely also agree with you about the difficult task the public schools have. Don't forget, also, to add in teacher competence -- because the conceptual curricula, while I much prefer them, are extraordinarily terrible with a teacher who does not understand math at a fundamental level. If I were choosing curricula for an elementary school with teachers whose understanding of math was shaky themselves, frankly, I would use Saxon (even though I do not care for it) because I think that the heavy emphasis on procedural fluency would be much better for the students than a half-understood conceptual curriculum. It is much easier to teach concepts later to students who are procedurally fluent than to try to teach concepts and procedures to students who have understood little and retained less. 

[Ad astra]

I'm working through Elementary Mathematics for Teachers which is published by Singapore. I had to do a subtraction problem using regrouping and I was stumped. I knew how to use borrowing to calculate the answer but could not figure out how to solve it using regrouping. I searched on Youtube for videos explaining regrouping with 3 digit subtraction and found 2 videos.

 

The first one seems more procedural to me. It shows you how to regroup but does not explain what you are doing.

 

 

This one is from Khan Academy and while it solves the problem in the same way as the first video, it explains what you are doing when you are regrouping.

 

 

These 2 examples helped me to begin to see the difference between teaching math conceptually vs. teaching the process.

 

I watched the video and it's exactly how I was taught to do it in my home country. How do you do it differently in "traditional" math??

[kiana]

It's been so long since I was taught subtraction that I can't quite remember how to do it the "traditional" way but I'm certain that the second video does not represent how I was taught back in elementary school. The first video (the one that does not really explain the concept) is probably how I was taught although the word "re-grouping" would not have been used. Instead, "borrowing" would have been used.

 

The biggest difference that I see is that while the method is the same in both videos, the second one shows how the 3 digit number is broken up into hundreds, tens and ones which explains what you are doin when you "borrow" or "re-group." I was not taught to look at numbers like that.

 

Yes. And Liping Ma's book discusses the results of similar lack conceptual understanding when she discusses teachers who were asked how they would deal with a student who always forgets to move over the partial products when doing a sum such as 123 x 456. 

 

If someone really, deeply understands that the standard algorithm is really doing 3x456 + 20x456 + 100x456 and then adding up the results, they should never make this mistake. Yet many of the teachers said they would teach the students to put asterisks in the space because there were no numbers there, or that they would teach the students to put 0s there but remind them that the 0s weren't really there. Of course they are there! It is just shorthand to leave them out. 

[Targhee]

123 x 456.

 

If someone really, deeply understands that the standard algorithm is really doing 3x456 + 20x456 + 100x456 and then adding up the results, they should never make this mistake. Yet many of the teachers said they would teach the students to put asterisks in the space because there were no numbers there, or that they would teach the students to put 0s there but remind them that the 0s weren't really there. Of course they are there! It is just shorthand to leave them out.

I remember being in about 4th grade and asking WHY you put the asteric there and no one could tell me! Not my teacher, not my dad, not my mom - I was frustrated and hung up on this point. It finally clicked, long after calculus and grad level stats, when I was instructing my child in Singapore math.

[EmseB]

I found this thread very helpful...I'm having a discussion on another board about a monologue that Greg Gutfeld did on a CC implementation video that, to me, just looked horrendous.  The video is here, in case anyone wants to watch it (yes, it's fox news, for those of you who don't want to click there, but I find the analysis kind of funny, and the meat of what I'm talking about is in the first two minutes of the clip).  Breaking down 9+6 seems so...unnecessary!!  Instead of doing one computation that can easily be learned as 9 things + 6 things = 15 things (and then memorized via repitition), they have the kids making the problem into 10+5 via 9+1 and 6-1. 

 

But the thread on the other board really challenged me on why I think traditional is better than conceptual.  I really identify with what Ellie said above (traditional does teach concepts, and likening it to the whole word debacle), as well as the issue kiana brought up about teacher competance.  If teachers are really having to watch instructional videos like the one I linked above in order to explain conceptual math to their students, then they should probably not be teaching conceptual math.

[mohini]

Just adding to pp - One thing that I think is important to consider is also the child's readiness to "understand" and manipulate numbers conceptually. It is easy for me to do 28 + 37 as 30 + 40 = 70 and 2+3=5 so 70-5= 65 so 28+37=65. But for a child who is just being introduced to addition, and has to calculate every little fact, it is not easy to manipulate the numbers this way OR to understand WHY you would want to manipulate the numbers this way.

 

Classical education emphasizes the importance of repetition and fact acquisition during the grammar stage. We have our kids doing copy work and dictation without knowing *why* but we know that the value is in the repetition, practice and internalization of the writing they do. IMO the current trend towards "conceptual" math is pushing kids into logic stage thinking before it is relevant to them. We all want little Johnny to know why 28+37=65 but in order for him to really understand why he has to have his basic skills mastered. Grammar stage is about memorizing and repeating - about building a base of knowledge to draw from when the question turns from WHAT to WHY.

 

I don't mean to say that all the illustrations, examples and strategies in "conceptual" math programs are bad. In fact, they are fascinating and, taught properly, really demonstrate the LOGIC of math. However, the current push to prioritize that "understanding" above mechanics, fact acquisition and memorization is as counter productive as drill and kill ever was. Kids need both approaches to have real "understanding." To manipulate the numbers they must know their facts and be proficient in the 4 operations but proficiency alone does not a mathematician make. Yet, I say that the time to solidify facts is during the grammar stage and the time to pull back the curtain on the math wizard - that comes later.

 

I don't ask my 6 yr old to postulate about the socio-economic factors that may have influenced the fall of the Roman Empire. The whole conversation would be referentially opaque to her.  Instead, I tell her "Rome fell in 476 AD." I expect that in a couple of years, she can read Gibbon's account of the decline of the Roman Empire and come up with a terribly compelling thesis about how and why Rome fell. She's super smart - It's not that I don't think she's capable of understanding those things. She does not have the frame of knowledge to analyze them yet. But the more facts she tucks away, the deeper and more original her analysis will be when it does come.

 

Anyway, IMO it's the same with math. We're so worried about teaching the why that we forget the natural learning process. That's why Singapore reduces so many kids to tears. Not because their parents are poor teachers or because they are not as bright as other kids - simply because they are not ready for that discussion yet.

 

I know a lot of people use Singapore et al successfully and that's great - but I can only assume that those people are supplementing to teach the facts and operations or that their children are naturally inclined toward maths or have already bloomed into logic stage students.

 

And one last thing - many of the concepts being used here to illustrate "conceptual" math actually are taught in the more traditional programs. CLE teaches expanding numbers (356 = 300 +50+6) it also teaches number bonds (though they don't call it that and it's a triangle, not a triangle made of circles.) It teaches why borrowing works etc.... It's just that the "traditional" programs don't put the focus on the why, they put it on the what.

 

[kiana]

Breaking down 9+6 seems so...unnecessary!!  Instead of doing one computation that can easily be learned as 9 things + 6 things = 15 things (and then memorized via repitition), they have the kids making the problem into 10+5 via 9+1 and 6-1. 

 

While I totally agree that breaking down something like this seems unnecessary, the whole point of teaching kids to break down something like 9 + 6 (where it's easy) is so that they can break down problems like 199 + 106, and then continue to problems like 1979 + 176. Learning a new mathematical skill where the problems are very difficult and the numbers are bigger tends to lead towards frustration as well.

 

Again, though, I think that if the teacher doesn't understand why they're doing this, or why the curriculum wants them to learn this -- if the teacher doesn't see where it's going later on -- it's going to be little more than a different kind of rote memorization for the students. 

[Hunter]

 

Classical education emphasizes the importance of repetition and fact acquisition during the grammar stage. We have our kids doing copy work and dictation without knowing *why* but we know that the value is in the repetition, practice and internalization of the writing they do. IMO the current trend towards "conceptual" math is pushing kids into logic stage thinking before it is relevant to them. We all want little Johnny to know why 28+37=65 but in order for him to really understand why he has to have his basic skills mastered. Grammar stage is about memorizing and repeating - about building a base of knowledge to draw from when the question turns from WHAT to WHY.

 

 

:iagree:  :iagree:  :iagree:

[wapiti]

Classical education emphasizes the importance of repetition and fact acquisition during the grammar stage. 

 

FWIW, I think this depends entirely on how one defines "classical" education - it sounds more like a "Lost Tools of Learning" stage-based neo- version than true classical ed.  (I haven't even had coffee yet so I'm not ready to go there, except to say that repetition and fact acquisition without understanding why would not be a part of my own personal goals for early elementary ed.)  In any case, I don't think even the "traditional" programs aim for memorization with no understanding in early elementary.  I think there's a continuum of emphasis on understanding the why behind the algorithm.

[LikeToListen]

Just adding to pp - One thing that I think is important to consider is also the child's readiness to "understand" and manipulate numbers conceptually. It is easy for me to do 28 + 37 as 30 + 40 = 70 and 2+3=5 so 70-5= 65 so 28+37=65. But for a child who is just being introduced to addition, and has to calculate every little fact, it is not easy to manipulate the numbers this way OR to understand WHY you would want to manipulate the numbers this way.

 

Classical education emphasizes the importance of repetition and fact acquisition during the grammar stage. We have our kids doing copy work and dictation without knowing *why* but we know that the value is in the repetition, practice and internalization of the writing they do. IMO the current trend towards "conceptual" math is pushing kids into logic stage thinking before it is relevant to them. We all want little Johnny to know why 28+37=65 but in order for him to really understand why he has to have his basic skills mastered. Grammar stage is about memorizing and repeating - about building a base of knowledge to draw from when the question turns from WHAT to WHY.

 

I don't mean to say that all the illustrations, examples and strategies in "conceptual" math programs are bad. In fact, they are fascinating and, taught properly, really demonstrate the LOGIC of math. However, the current push to prioritize that "understanding" above mechanics, fact acquisition and memorization is as counter productive as drill and kill ever was. Kids need both approaches to have real "understanding." To manipulate the numbers they must know their facts and be proficient in the 4 operations but proficiency alone does not a mathematician make. Yet, I say that the time to solidify facts is during the grammar stage and the time to pull back the curtain on the math wizard - that comes later.

 

I don't ask my 6 yr old to postulate about the socio-economic factors that may have influenced the fall of the Roman Empire. The whole conversation would be referentially opaque to her.  Instead, I tell her "Rome fell in 476 AD." I expect that in a couple of years, she can read Gibbon's account of the decline of the Roman Empire and come up with a terribly compelling thesis about how and why Rome fell. She's super smart - It's not that I don't think she's capable of understanding those things. She does not have the frame of knowledge to analyze them yet. But the more facts she tucks away, the deeper and more original her analysis will be when it does come.

 

Anyway, IMO it's the same with math. We're so worried about teaching the why that we forget the natural learning process. That's why Singapore reduces so many kids to tears. Not because their parents are poor teachers or because they are not as bright as other kids - simply because they are not ready for that discussion yet.

 

I know a lot of people use Singapore et al successfully and that's great - but I can only assume that those people are supplementing to teach the facts and operations or that their children are naturally inclined toward maths or have already bloomed into logic stage students.

 

And one last thing - many of the concepts being used here to illustrate "conceptual" math actually are taught in the more traditional programs. CLE teaches expanding numbers (356 = 300 +50+6) it also teaches number bonds (though they don't call it that and it's a triangle, not a triangle made of circles.) It teaches why borrowing works etc.... It's just that the "traditional" programs don't put the focus on the why, they put it on the what.

 

Well said.  And this is why I chose McRuffy over singapore or RS for K level.  McRuffy does every operation on a  number line before moving to using pictorial representation and introducing arithmetic symbols.  Seeing the relationships on a numberline (Mcruffy is traditional math) helps her to figure out the fact families/relationships herself, and introducing the fact families later felt like giving shape to what she had been observing on a numberline.  It could be MY approach and I could be wrong, but I wasn't comfortable introducing fact families (with picture counting) to help them see the relationships.   It just felt like it could lead to the infamous "conceptual leaps" at a later stage.

 

DD is my first child so my experience is very limited.  Just my 2cents....

[Erin]

I've used/researched a number of math programs over the years, both as a homeschooling mom and a public school teacher.  
I honestly can't think of ANY that aren't "conceptual."  

 

Near as I can tell, they all start out teaching the concept, before moving on to an algorithm to represent it.  I think the only real variance is how long they do JUST the concrete before moving on to reliance on the algorithm.  As well as how often a previously taught skill comes back into the lesson.  

 

If someone says, "________ isn't conceptual" what that usually means to me is that they didn't get it, not that the text didn't try to teach it.  

I remember being in college in my math methods classes many moons ago, learning concrete teaching methods for things like adding, multiplication, algebra, etc. and telling my mom it's too bad this stuff wasn't taught when I was in school, I might have done better at math.  
She gave me this puzzled look and said it WAS, then drug out an old math book to show me.  Sure enough.  It's just that the only part that clicked with me were the algorithms, so that's the part I remembered to the exclusion of the "concept."  

[mohini]

FWIW, I think this depends entirely on how one defines "classical" education - it sounds more like a "Lost Tools of Learning" stage-based neo- version than true classical ed.  (I haven't even had coffee yet so I'm not ready to go there, except to say that repetition and fact acquisition without understanding why would not be a part of my own personal goals for early elementary ed.)  In any case, I don't think even the "traditional" programs aim for memorization with no understanding in early elementary.

 

Obviously I wasn't implying that there should be NO understanding or that any program would teach memorization with NO understanding. Obviously.  I'm saying that the emphasis on understanding is premature - not that there should be no understanding but that the emphasis should be on learning, repeating and internalizing the facts and mechanics of math. Of course I expect comprehension - but the emphasis on strategy vs. mechanics goes beyond comprehension into analysis.  It's the same thing grammar stage students do with language. They read, repeat, memorize rules and mimic structures. Of course we still expect them to comprehend what they reading, but the amount of analysis is limited. In math though, we expect students to analyze numerical relationships before they have a basic proficiency with the mechanics which means that many students never get an opportunity to build a knowledge base from which to draw later.

 

And there is no "true" classical ed. anymore. By virtue of the systems and philosophies that have influenced education in our day and age there is probably nobody who is practicing "true" classical education. However, the basic precepts of "classical," trivium based education define a grammar stage where a young student is exposed to ideas, rules, facts and processes and a Dialectic stage where a student begins to analyze those processes, rules and facts and to discuss causation, formulate arguments and study logical syllogism. Much of the strategy taught in "conceptual" math programs falls under the heading of logical syllogism.

 

9+6 -

If I take 1 from the 6 and add it to 9 then the 9 becomes 10 and the 6 becomes 5. 

10+5 = 15

Therefore 9+6=15

 

This is akin to if P then Q and more appropriate to the dialectic stage. Just saying :)

[EKS]

I don't know how CLE would teach it, however, the difference between conceptual and *procedural* math is this.

 

A conceptual lesson would emphasize place value.  When we regroup we are decomposing a ten into ones.  You'd probably use manipulatives or pictures to explain (using base ten blocks, you'd take a ten bar and break it into unit cubes and put the unit cubes in the ones place.

 

A procedural lesson would tell the student to cross out the number in the tens place, take away one, write it down above it and put a one next to the number in the ones place.  There would be little or no discussion of place value or why one is able to do such a thing.  This is how I learned math.  

 

Using Singapore math (and other conceptual programs like RightStart and MUS) and learning the conceptual approach has absolutely enriched my life.

[EmseB]

While I totally agree that breaking down something like this seems unnecessary, the whole point of teaching kids to break down something like 9 + 6 (where it's easy) is so that they can break down problems like 199 + 106, and then continue to problems like 1979 + 176. Learning a new mathematical skill where the problems are very difficult and the numbers are bigger tends to lead towards frustration as well.

 

Again, though, I think that if the teacher doesn't understand why they're doing this, or why the curriculum wants them to learn this -- if the teacher doesn't see where it's going later on -- it's going to be little more than a different kind of rote memorization for the students. 

 

I get that, but I also think that it shouldn't be the way 9+6 is taught to kids, if that makes sense.  In other words, I think the concept is important, and to teach the concept it's easier to show it with smaller numbers at first, but I don't think that's the way they should be teaching kids to learn or be "comfortable" with single digit addition in the early elementary years.  So I think what you're talking about -- breaking numbers down to make problems easier -- is a different lesson than teaching a 1st grader that 9+6=15.

[mohini]

I get that, but I also think that it shouldn't be the way 9+6 is taught to kids, if that makes sense.  In other words, I think the concept is important, and to teach the concept it's easier to show it with smaller numbers at first, but I don't think that's the way they should be teaching kids to learn or be "comfortable" with single digit addition in the early elementary years.  So I think what you're talking about -- breaking numbers down to make problems easier -- is a different lesson than teaching a 1st grader that 9+6=15.

 

 

:iagree: YUP.

[kiana]

I get that, but I also think that it shouldn't be the way 9+6 is taught to kids, if that makes sense.  In other words, I think the concept is important, and to teach the concept it's easier to show it with smaller numbers at first, but I don't think that's the way they should be teaching kids to learn or be "comfortable" with single digit addition in the early elementary years.  So I think what you're talking about -- breaking numbers down to make problems easier -- is a different lesson than teaching a 1st grader that 9+6=15.

 

The problem with this, though, is that once they get accustomed to just memorizing facts instead of breaking numbers down, it is virtually impossible to convince them to do it in any other way than the one they are comfortable with. They are very, very resistant to going back and re-learning how to do it in another way that is less efficient at first, even if it is more efficient in the long run. 

 

I'd rather see kids working more slowly and with smaller numbers in the elementary grades, but learning mathematical strategies like this, than working faster through more material, but learning only fewer strategies.

 

Edited: To me, teaching math facts as something to be memorized seems more like teaching words as something to be memorized, rather than teaching phonics rules so that children can figure out words that they don't know by sight. Yet I have been told by reading teachers that it was too difficult for children to learn phonics and that whole words were "better for them", and that if they needed phonics they could learn it later. As with math, children (and adults) who have learned to read inefficiently with whole words are frequently very resistant to going back and starting with phonics. 

[mohini]

The problem with this, though, is that once they get accustomed to just memorizing facts instead of breaking numbers down, it is virtually impossible to convince them to do it in any other way than the one they are comfortable with. They are very, very resistant to going back and re-learning how to do it in another way that is less efficient at first, even if it is more efficient in the long run. 

 

I'd rather see kids working more slowly and with smaller numbers in the elementary grades, but learning mathematical strategies like this, than working faster through more material, but learning only fewer strategies.

 

Edited: To me, teaching math facts purely as something to be memorized seems more like teaching words as something to be memorized, rather than teaching phonics rules so that children can figure out words that they don't know. 

 

That is like saying that if kids do copy work they will be resistant to learning to formulate compositions later. It's not true - but the concepts must be introduced at the right stage of course. They will not have to re-learn anything but will be enlightened to the complexity and number relationships with which they are already familiar.

 

Nobody here is saying that we should be teaching PURELY math facts, it is a matter of where the emphasis is placed. And FWIW, we actually do teach kids to memorize phonics rules precisely so that they can decode different words later. We don't start out by emphasizing all the different exceptions to the rules, we build up a set of rules for them to use so that, as they become proficient in reading, they can analyze them to decode and later to spell. Then, in the logic stage, we teach them WHY there are so many different phonics rules (different linguistic roots, colloquialisms, strange etymologies etc....) but we don't teach them WHY the phonics rules are the way they are from the beginning. That would be far too much analysis for a grammar stage student.

[wendyroo]

Just adding to pp - One thing that I think is important to consider is also the child's readiness to "understand" and manipulate numbers conceptually. It is easy for me to do 28 + 37 as 30 + 40 = 70 and 2+3=5 so 70-5= 65 so 28+37=65. But for a child who is just being introduced to addition, and has to calculate every little fact, it is not easy to manipulate the numbers this way OR to understand WHY you would want to manipulate the numbers this way.

 

Classical education emphasizes the importance of repetition and fact acquisition during the grammar stage. We have our kids doing copy work and dictation without knowing *why* but we know that the value is in the repetition, practice and internalization of the writing they do. IMO the current trend towards "conceptual" math is pushing kids into logic stage thinking before it is relevant to them. We all want little Johnny to know why 28+37=65 but in order for him to really understand why he has to have his basic skills mastered. Grammar stage is about memorizing and repeating - about building a base of knowledge to draw from when the question turns from WHAT to WHY.

 

I don't mean to say that all the illustrations, examples and strategies in "conceptual" math programs are bad. In fact, they are fascinating and, taught properly, really demonstrate the LOGIC of math. However, the current push to prioritize that "understanding" above mechanics, fact acquisition and memorization is as counter productive as drill and kill ever was. Kids need both approaches to have real "understanding." To manipulate the numbers they must know their facts and be proficient in the 4 operations but proficiency alone does not a mathematician make. Yet, I say that the time to solidify facts is during the grammar stage and the time to pull back the curtain on the math wizard - that comes later.

 

FWIW, I just asked Peter (5.5 years old, most of the way through Singapore 1B) how much 28+37 is (I wrote it on the whiteboard so he could see the numbers).  Without missing a beat he said, "We want the 28 to be 30, so we need to give it two more from the 7.  So, 30 plus 30 is 60, plus the 5 left from the 7 is 65."  While he talked it through, I counted Mississippis and figure the whole thing took him about 7 seconds.  Yes, he's advanced in that he is finishing 1B before he turns 6, but I think many (most?) students who have completed Singapore Essentials and 1A&B would be able to do the same calculation equally adeptly.  Peter has never seen the addition algorithm, but that certainly has not hampered his ability to accurately add numbers together.  I would be much more worried if he had to rely on the algorithm and did not have strong mental math skills.

 

I don't understand the idea that conceptual math programs downplay fact acquisition.  Singapore math fully expects students to master the fact families during 1A.  Obviously, in order to do that problem mentally "the Singapore way" Peter had to know that 10 - 8 = 2 and 3 + 3 = 6 and 7 - 2 = 5.  There also can be a lot of repetition in Singapore (depending on how much the student requires and how the teacher chooses to utilize the program components).  Peter fully understands the concept of addition with regrouping, but he can do a problem like that quickly and accurately because we have practiced A LOT of them.  

 

To me, one of the things that makes conceptual math unique, is that the algorithms are introduced very late, and almost as an afterthought.  First the kids master mental math both conceptually and procedurally.  Then the algorithm is eventually introduced, but since it is slower than mental math and it requires a pencil and paper, the students often come to view it as a tool, and not a particularly great one, as opposed to the tool.

 

Wendy

[Roadrunner]

FWIW, I just asked Peter (5.5 years old, most of the way through Singapore 1B) how much 28+37 is (I wrote it on the whiteboard so he could see the numbers). Without missing a beat he said, "We want the 28 to be 30, so we need to give it two more from the 7. So, 30 plus 30 is 60, plus the 5 left from the 7 is 65." While he talked it through, I counted Mississippis and figure the whole thing took him about 7 seconds. Yes, he's advanced in that he is finishing 1B before he turns 6, but I think many (most?) students who have completed Singapore Essentials and 1A&B would be able to do the same calculation equally adeptly. Peter has never seen the addition algorithm, but that certainly has not hampered his ability to accurately add numbers together. I would be much more worried if he had to rely on the algorithm and did not have strong mental math skills.

 

I don't understand the idea that conceptual math programs downplay fact acquisition. Singapore math fully expects students to master the fact families during 1A. Obviously, in order to do that problem mentally "the Singapore way" Peter had to know that 10 - 8 = 2 and 3 + 3 = 6 and 7 - 2 = 5. There also can be a lot of repetition in Singapore (depending on how much the student requires and how the teacher chooses to utilize the program components). Peter fully understands the concept of addition with regrouping, but he can do a problem like that quickly and accurately because we have practiced A LOT of them.

 

To me, one of the things that makes conceptual math unique, is that the algorithms are introduced very late, and almost as an afterthought. First the kids master mental math both conceptually and procedurally. Then the algorithm is eventually introduced, but since it is slower than mental math and it requires a pencil and paper, the students often come to view it as a tool, and not a particularly great one, as opposed to the tool.

 

Wendy

Liking this wasn't enough.

 

I can't believe there are people who blindly have kids memorize "math facts."

[EmseB]

The problem with this, though, is that once they get accustomed to just memorizing facts instead of breaking numbers down, it is virtually impossible to convince them to do it in any other way than the one they are comfortable with. They are very, very resistant to going back and re-learning how to do it in another way that is less efficient at first, even if it is more efficient in the long run. 

 

I'd rather see kids working more slowly and with smaller numbers in the elementary grades, but learning mathematical strategies like this, than working faster through more material, but learning only fewer strategies.

 

Edited: To me, teaching math facts as something to be memorized seems more like teaching words as something to be memorized, rather than teaching phonics rules so that children can figure out words that they don't know by sight. Yet I have been told by reading teachers that it was too difficult for children to learn phonics and that whole words were "better for them", and that if they needed phonics they could learn it later. As with math, children (and adults) who have learned to read inefficiently with whole words are frequently very resistant to going back and starting with phonics. 

 

A) I don't teach math by having my kids "just memorizing facts".  You've set up a straw man here.  I (and most traditional programs I know) teach my kids the concepts via number lines, manips, and drawings along with flashcard practice.  And it's not about re-learning in any case.  Like I said before, once they know that 9+6=15, then it's easier to teach them why 9+6 is also 10+5 and so on.  The lesson of teaching place values and breaking down larger numbers comes later, and it is built on the idea that 9 things + 6 things = 15 things, along with the idea that 6-1=5, 1+9=10, therefore 10+5=9+6.  Again, I think we're talking about two separate lessons, the latter built off of mastery of simple addition.  If they don't know that 9+6=15, they can't know that 9+6=10+5.

 

B ) I teach reading comprehnsion to my kids as well.  But I also have them memorize phonics and spelling rules, as well as basic grammar concepts.  There are however, things I don't ask them to do in 1st grade that I will ask them to do later built off of  the basic skills and rules they've memorized and internalized via practice in their 1st grade year.  It's not about convincing them to do it some new way, it's using the skills they've developed, the facts they've mastered via doing them, along with the new development in their brains in order to do more complex tasks.

 

C) The problem with simply going more slowly is that 1) that's not what the schools in my area are doing, they are simply teaching stuff in a different way at the same pace and 2) some kids are not ready for the algebraic thought process that is required to get from 9+6=10+5 precisely because they have to know what 9+6 is first before moving on to applying it to different equations.  And some kids will not develop this kind of dialectic or logical thought until they are a few years past the 1st grade.  So going more slowly doesn't help.  But later on, they will get it in a flash.

 

I actually think your analogy holds, but rather in reverse.  I see this "conceptual math" much like the whole word trends of the past.  Educators noticed that kids who mastered reading weren't sounding words out, they simply looked at the words to read them.  So they started trying to teaching kids to read by just "seeing" the word.  As it turns out, the kids who were good readers had already mastered and surpassed the phonics of reading and were very good at applying it to words they didn't know, so it looked from the outside like they were just seeing the word and reading it.  With the type of math we're talking about in this thread, I think educators noticed that kids fluent in math could do the 10+5=9+6 type of manipulations fluently and thought that must be what's making them good with math, so let's teach it that way from the beginning.  In reality, the kids that were good in math had already mastered the idea that 9+6=15 and that's why they could manipulate it so easily.  So trying to teach it this way is putting the cart before the horse in the same way that whole word reading tried to.  Mastery of simple concepts is required before more difficult ones.  Some kids easily and quickly master the simple parts, making it look like they've skipped it entirely, but have really just internalized it.  At least, that's my opinion after having read around a lot about conceptual math and having two kids learning elementary math right now.

 

What you're talking about here with re-learning seems to me to be akin to saying we can't teach a kid algebra because he's already learned to do addition, subtraction, multiplication and division a certain way and it will be very, very difficult to teach him a new way of doing math.  Obviously that doesn't make any sense, because he needs to master those skills to move on to algebra, and I'd say it's likewise with the manipulations of numbers in "conceptual" math.  They need to master the basic skills before doing the manipulation with those skills.

[EmseB]

I can't believe there are people who blindly have kids memorize "math facts."

 

Can I ask who you're referring to here?  Or which program?  Maybe I skimmed too much to see someone who is advocating this?

[mohini]

Liking this wasn't enough.

 

I can't believe there are people who blindly have kids memorize "math facts."

 

Aw. See, I thought we were trying to have a productive conversation. Nobody ever said anything about "blindly memorizing 'math facts'," But I guess you only read the posts you agree with. The PP makes a point of mentioning that her ds must have memorized certain facts in order to complete the problem. As I said before, kids need both approaches - the discourse was about how and when to introduce the discussion of WHY. For the record, my ds who has done primarily "traditional" math (as it suited his disposition) is very quick with mental math and has DISCOVERED the Singapore strategies largely on his own as he learned to manipulate numbers. Yes, I taught place value and sometimes suggested that it might be easier to work with a ten ect... but he is coming into his conceptual understanding through his knowledge of the "facts."

 

And, just because a kid can replicate a strategy does not mean that they understand WHY it works or why we do it that way.

 

I just tire of the mathematical snobbery. Even SWB says that Singapore is introducing logic stage thinking at a time where it may not be appropriate. People want to think that they are accelerating their kids by pushing them into that thought process too early but *sometimes* they are actually decelerating them. The boards are full of Asian math users who complain that their kids have not mastered their facts.

[EmseB]

Obviously, in order to do that problem mentally "the Singapore way" Peter had to know that 10 - 8 = 2 and 3 + 3 = 6 and 7 - 2 = 5.  There also can be a lot of repetition in Singapore (depending on how much the student requires and how the teacher chooses to utilize the program components).  Peter fully understands the concept of addition with regrouping, but he can do a problem like that quickly and accurately because we have practiced A LOT of them.  

 

And I think this is key.  And yes, your son is advanced doing this at 5.5.  I also have a 5.5 year old and it is taking him much longer to learn these concepts and this application would confuse him, whereas my older DS was like Peter and mastered it easily. The point being, Peter mastered and knew what those single digit facts were in order to break down the larger numbers and apply them to a more difficult problem.

 

But what I've quoted here what I was trying to point out in my last post.  He had to learn all those facts and repeat them to be able to master the manipulation easily.  And, IM(limited)E, that is not what public schools are doing with CC implementation of conceptual math, so they are losing a lot of what you have gained by having your son learn math this way.  And if a teacher is not as well versed as you in teaching this method or as fluent in math as they should be, then this is going to be even more difficult to impart to the student.

 

I think Singapore is a great program for some kids, but not all.  And what got me into this discussion originially was the CC instruction video showing teachers how to teach 9+6 as 10+5 to kids who aren't "comfortable" with the idea that 9+6 is 15.  So, it's trying to teach conceptual math without mastering the facts that you're talking about above and I think that's where a lot of the pushback is coming from.

[Roadrunner]

As long as nobody is having their kids drill 7 + 9 = 16 without understanding it, all is good. If anybody is blindly memorizing facts than they should reconsider. I have seen enough posts on this board that sound like blind memorization. I want to caution against that approach (lots of people read this board). No reason to take things personally.

 

Sound mathematical advise doesn't equal mathematical snobbery. Please refrain from name calling.

SM math (not for everybody) is just a good math program. Saying that using SM math sometimes decelerates kids is a statement not supported by evidence.

[wendyroo]

And I think this is key.  And yes, your son is advanced doing this at 5.5.  I also have a 5.5 year old and it is taking him much longer to learn these concepts and this application would confuse him, whereas my older DS was like Peter and mastered it easily. The point being, Peter mastered and knew what those single digit facts were in order to break down the larger numbers and apply them to a more difficult problem.

 

But what I've quoted here what I was trying to point out in my last post.  He had to learn all those facts and repeat them to be able to master the manipulation easily.  And, IM(limited)E, that is not what public schools are doing with CC implementation of conceptual math, so they are losing a lot of what you have gained by having your son learn math this way.  And if a teacher is not as well versed as you in teaching this method or as fluent in math as they should be, then this is going to be even more difficult to impart to the student.

 

I think Singapore is a great program for some kids, but not all.  And what got me into this discussion originially was the CC instruction video showing teachers how to teach 9+6 as 10+5 to kids who aren't "comfortable" with the idea that 9+6 is 15.  So, it's trying to teach conceptual math without mastering the facts that you're talking about above and I think that's where a lot of the pushback is coming from.

 

I mostly agree with your first paragraph.  It is a bit misleading to say that Peter memorized the single digit addition facts, because he only knows the facts up to 10, and he just picked them up along the way.  When we started Singapore Essentials he was counting on his fingers a lot.  When we started 1A he was still using Cuisenaire rods to figure out the 7, 8, 9, and 10 fact families  We never drilled the facts, but by the start of 1B he had slowly mastered them.

 

In your second paragraph I completely agree that many public school teachers are not able to teach conceptual math well.  In my experience, many public school elementary teachers are nearly math illiterate.

 

As for the final paragraph...for Peter (and me), 9+6 falls into the same category as 28+37.  Both of those are addition with regrouping and get treated the same way.  Sure, he could carry out that process on 9+6 earlier than 28+37, because 9+6 has less digits, but he did not memorize it.

 

I was the valedictorian of my high school class, I went to MIT, earned all A's and graduated with a Bachelors and Masters in engineering.  I am very good at math, and yet when faced with the problem 8+5 I still regroup and add in my head.  It is fast, lightening fast, but I still go through the process rather than having it memorized.

 

I absolutely do not agree that you need to know that 9+6 = 15 if you want to know that 9+6 = 10+5.  If I have a cup half full of water and a bottle half full of water and I pour some water from the bottle into the cup, I am 100% sure that I still have the same amount of water total even though I don't have the faintest idea how much that is.  This is the exact concept I am working on with my 3 year old in math right now: if you have 5 counters and then split them into a pile of 2 and a pile of 3, you still have 5 counters.  And then if you move one counter from your 2 pile and add it to your three pile, you now have a 1 pile and a four pile and you still have 5 counters.  I am confident that by the time he is ready for Singapore 1A that he will have no trouble seeing that 9+6 = 10+5.

 

(This is not to say that Singapore, or any conceptual math, is right for every child.  Just that it is working very well for our family in all its conceptual glory.)

 

Wendy

[Alte Veste Academy]

And, just because a kid can replicate a strategy does not mean that they understand WHY it works or why we do it that way.

 

People want to think that they are accelerating their kids by pushing them into that thought process too early but *sometimes* they are actually decelerating them. The boards are full of Asian math users who complain that their kids have not mastered their facts.

 

I don't agree that people are using conceptual math programs as a way of pushing/accelerating their kids. I think most people who choose math programs with an emphasis on conceptual understanding are doing so because they want their children to have strong conceptual understanding as an underpinning for the end game, efficient use of algorithms with understanding. It is much easier to replicate an algorithm without understanding than it is to replicate a strategy without understanding.

 

You are right that these boards are full of Asian math users complaining about facts mastery. I think most of the time, the problem is that the complainers didn't train themselves in methods and/or are not using all of the components of their programs. As someone who has successfully used SM K-6 twice over (I'm now on my 3rd child), the SM users who are not using the HIG and complain about lack of facts practice make me batty. It's in there. The teacher just has to do the reading and prep as a teacher. ETA: This is separate from the issue of kids with learning differences that make it hard to memorize math facts. That is a completely separate issue from conceptual vs traditional math.

[Alte Veste Academy]

I found this thread very helpful...I'm having a discussion on another board about a monologue that Greg Gutfeld did on a CC implementation video that, to me, just looked horrendous.  The video is here, in case anyone wants to watch it (yes, it's fox news, for those of you who don't want to click there, but I find the analysis kind of funny, and the meat of what I'm talking about is in the first two minutes of the clip).  Breaking down 9+6 seems so...unnecessary!!  Instead of doing one computation that can easily be learned as 9 things + 6 things = 15 things (and then memorized via repitition), they have the kids making the problem into 10+5 via 9+1 and 6-1. 

 

A) I don't teach math by having my kids "just memorizing facts".  You've set up a straw man here.  

 

Perhaps you do not recall, but at about the mid point of the video you linked (after the "meat"), as they begin their anecdotal discussion (devoid of any actual dialogue about mathematical concepts or input from a math professional) of this example of conceptual instruction, the commentators sit and talk about how they just flat out memorized the facts when they were young. And, LOL, the last woman they asked said, "You're asking a girl who took algebra 1 three times. So I didn't know what they were saying under traditional....I can't imagine being in common core." Well, maybe if she had been taught with the kind of conceptual teaching they were mocking, she would have had understanding that would have led to success in algebra 1. :lol:

[EmseB]

Okay, I can't get multi-quote to work on the board and the wyswyg editor is giving me fits when I try to reply to parts of posts, so sorry if this is a mess when I post.

 

Roadrunner said:

As long as nobody is having their kids drill 7 + 9 = 16 without understanding it, all is good. If anybody is blindly memorizing facts than they should reconsider. I have seen enough posts on this board that sound like blind memorization. I want to caution against that approach (lots of people read this board). No reason to take things personally.

 

 

I wasn't taking it personally, I thought you were commenting on something in the thread and I was trying to figure out what the context of your statement was as it related to the rest of the conversation.

Wendyroo said:

It is a bit misleading to say that Peter memorized the single digit addition facts, because he only knows the facts up to 10, and he just picked them up along the way.  When we started Singapore Essentials he was counting on his fingers a lot.  When we started 1A he was still using Cuisenaire rods to figure out the 7, 8, 9, and 10 fact families  We never drilled the facts, but by the start of 1B he had slowly mastered them.

 

I'm pretty sure I didn't say he memorized them, I think in the portion you quoted I used the same word that you are using (mastered).  I think the nitpick is important, because kids who master math facts in traditional programs don't just memorize them either.  No, they don't regroup them every time they do a computation (single-digit), but they know that 9 things + 6 things = 15 things and it isn't just a fact without meaning.  I think we might be talking past each other, because I feel like much of the beef I have with conceptual math is a lack of mastery that I see when it's taught by people who don't know that the kids still have to master the facts (as your son did).  And when people have a beef with traditional, they think that we're just making our kids "blindly memorize" stuff without learning meaning.  Neither of those caricatures is wholly true.

 

As for the final paragraph...for Peter (and me), 9+6 falls into the same category as 28+37.  Both of those are addition with regrouping and get treated the same way.  Sure, he could carry out that process on 9+6 earlier than 28+37, because 9+6 has less digits, but he did not memorize it.

 

I was the valedictorian of my high school class, I went to MIT, earned all A's and graduated with a Bachelors and Masters in engineering.  I am very good at math, and yet when faced with the problem 8+5 I still regroup and add in my head.  It is fast, lightening fast, but I still go through the process rather than having it memorized.

 

 

I absolutely do not agree that you need to know that 9+6 = 15 if you want to know that 9+6 = 10+5.  If I have a cup half full of water and a bottle half full of water and I pour some water from the bottle into the cup, I am 100% sure that I still have the same amount of water total even though I don't have the faintest idea how much that is.  This is the exact concept I am working on with my 3 year old in math right now: if you have 5 counters and then split them into a pile of 2 and a pile of 3, you still have 5 counters.  And then if you move one counter from your 2 pile and add it to your three pile, you now have a 1 pile and a four pile and you still have 5 counters.  I am confident that by the time he is ready for Singapore 1A that he will have no trouble seeing that 9+6 = 10+5.

 

 

Yes, but he has to know that both of those are 15, or else, what's the use of knowing they are the same?  At some point you have to get to the actual sum and not just know there's water in the glass.  He has to know that there are 15 counters in either case, no?  He has to know that 9+1=10 6-1=5 so 9+6=10+5.  There is value in knowing how to manipulate these facts, I'm not saying there isn't.  All I'm saying is that at some point a kid has to know the sums (and with fast recall, otherwise regrouping 8+5 any time you need to know becomes painful), or the manipulation is meaningless and confusing.

And, honestly, for single digit addition, I can't imagine having to regroup every time I wanted to do quick math in my head.  I can do it for larger problems if I need to, but 8+5 just pops up as 13 in my brain when someone asks.  That doesn't mean that I can't also recognize that it is 10+3, but for everyday situations I can't see the value in having to regroup in my head in order to know it at the single digit level.

 

[EmseB]

Perhaps you do not recall, but at about the mid point of the video you linked (after the "meat"), as they begin their anecdotal discussion (devoid of any actual dialogue about mathematical concepts or input from a math professional) of this example of conceptual instruction, the commentators sit and talk about how they just flat out memorized the facts when they were young. And, LOL, the last woman they asked said, "You're asking a girl who took algebra 1 three times. So I didn't know what they were saying under traditional....I can't imagine being in common core." Well, maybe if she had been taught with the kind of conceptual teaching they were mocking, she would have had understanding that would have led to success in algebra 1. :lol:

 

I took Algebra 1 twice, but I don't recall ever specifically memorizing math facts.  :D  Part of my problem was (I believe) that I was accelerated in math and I had a fall birthday, and when I got up to algebra, I just wasn't developmentally ready for it.  Then again, maybe if I'd had an Asian math program I would have been fine.  But when I went to college as an adult and took math, I loved it and I had no trouble.

 

But, if you asked my kids if they have to memorize math facts, they would hopefully answer yes.  Of course, that's not all we do, either.

[mohini]

 

I absolutely do not agree that you need to know that 9+6 = 15 if you want to know that 9+6 = 10+5.  If I have a cup half full of water and a bottle half full of water and I pour some water from the bottle into the cup, I am 100% sure that I still have the same amount of water total even though I don't have the faintest idea how much that is.  This is the exact concept I am working on with my 3 year old in math right now: if you have 5 counters and then split them into a pile of 2 and a pile of 3, you still have 5 counters.  And then if you move one counter from your 2 pile and add it to your three pile, you now have a 1 pile and a four pile and you still have 5 counters.  I am confident that by the time he is ready for Singapore 1A that he will have no trouble seeing that 9+6 = 10+5.

 

(This is not to say that Singapore, or any conceptual math, is right for every child.  Just that it is working very well for our family in all its conceptual glory.)

 

Wendy

 

This is an excellent point and you are right that it's not necessary to memorize 9+6=15 in order to see that it is equivalent to 10+5. Basically, this is the beginning of number bonds - re-arranging the parts of a number into a whole. But, where many children may *see* the equivalence when using manipulatives, they will not be able to "do" number bonds without the manipulatives until they have memorized the different ways to make the whole. You are especially right in saying that repetition will build that skill. And then you have accomplished the real goal of teaching both the "conceptual" and "factual" aspects of the problem.

 

By the same token, you do not need to be taught that the process for finding 9+6 is to turn it into 10+5 in order to understand that the 2 are equivalent and there is a natural progression as math facts are mastered that leads the student into the process of making 10's to solve problems like this. When I teach using a traditional program,  I start with the whole, 15 and memorize the different parts (9+6, 10+5, 7+8) etc... Once students have these facts down, they can "fill in" the number bond mentally. The only real difference is that the  manipulation of 9+6 into 10+5 comes later.

 

I was actually trained to teach Singapore, Everyday Math and MIF. The first 2 while teaching in PS and the last in a private international school. When I first saw Singapore I loved it  - and I thought it would be a fantastic, exciting way to learn. I used a TM when teaching it in a BM school and, when my ds was in K-1 I used it with the HIG (later switching to MIF.) But after seeing how most kids (my own ds included) were unable to progress at the speed of the program (granted this may not be an issue when Hschooling) and had little retention, I found myself reaching for more and more traditional supplements.

 

The reason that I even engaged in this conversation is because I think there is a misconception that "conceptual" programs are somehow more advanced, better or more "mathematical." They are not - they constitute a different approach which is highly effective for *some* students - and not others and it does demand logic stage thinking sooner. The key is to know your student and yourself as a teacher - also to understand that the ultimate goal, from both angles, is conceptual and procedural mastery.

 

Taught equally well (and completely,) I don't think that one or the other method is superior to the other - just that the "facts first" approach is more classical in nature and can be as effective in promoting mathematical thinking as the conceptual programs.

 

 

 

 

 

[Alte Veste Academy]

I do not agree that programs like SM require logic stage thinking in the early grades. Concrete, pictorial, abstract. There is nothing in there that requires logic stage thinking, just understanding, practice, and drill.  

 

You said in an earlier post...

 

I know a lot of people use Singapore et al successfully and that's great - but I can only assume that those people are supplementing to teach the facts and operations or that their children are naturally inclined toward maths or have already bloomed into logic stage students.

 

Does this then include the entire nation of Singapore? LOL The daily instruction is what teaches the operations. Drilling of facts is not supplemental. It is an elemental part of the program, as I assume you know since you say you were trained in the use of SM. Drilling of facts does not replace or precede conceptual understanding (which is what you seem to be saying requires logic stage thinking?). It complements instruction. 

[Alte Veste Academy]

Thanks for this. When I began reading this board a few years ago (I lurked long before I joined) I got the impression that the *real* math programs were Miquon and Singapore and everything else was inferior. (Obviously I didn't read every thread and probably missed many of the discussions about other curricula so that's an unfair characterization of this forum.)

 

So of course when it came time for me to begin homeschooling (only kindergarten so this is just baby steps), I wanted "Asian-style" math too so I bought c-rods and Miquon. But DD doesn't like rods and hates the Miquon worksheets but loves her Horizons workbook and loves to count. I felt guilty for ditching Miquon but Miquon felt like a round peg in a square hole for my DD and me as the teacher.

 

Well, Miquon is just odd, even for conceptual math. :tongue_smilie: I loved it, and I used it, but it was enough to make you break out in a cold sweat when it arrived. There used to be a thread a week with people posting that they received it and were freaking out. What to do with it?! It is VERY different, even in a group of conceptual math products. Horizons is strong too. I'm not one to think Asian math is the be all and end all, but I also don't like to see conceptual math in general judged falsely. 

[mohini]

I do not agree that programs like SM require logic stage thinking in the early grades. Concrete, pictorial, abstract. There is nothing in there that requires logic stage thinking, just understanding, practice, and drill.  

 

You said in an earlier post...

 

 

Does this then include the entire nation of Singapore? LOL The daily instruction is what teaches the operations. Drilling of facts is not supplemental. It is an elemental part of the program, as I assume you know since you say you were trained in the use of SM. Drilling of facts does not replace or precede conceptual understanding (which is what you seem to be saying requires logic stage thinking?). It complements instruction. 

 

It's true that Singapore does have "drills" which (in the standards editions) they call "Mental Math" because it is done orally. In my experience USING those drills with students, they were not sufficient. The computational strategies were so cumbersome for many students that facts above the +2's were unruly for regrouping.... I would say with the exception of the doubles (and for some kids also the d+1 facts) Of course there were some students who flew through. Now  take it with a grain of salt because in a classroom it is difficult to tailor the pace of a program to meet students where they are - you can differentiate a little bit but are required to cover material according (basically) to the school's scope and sequence.  I also reiterate that I am talking about GRAMMAR stage students here (up to 4th grade or so.)

 

In the classical model, memorization, repetition and fact acquisition do precede ANALYSIS (not conceptual understanding) of a particular subject and its ABSTRACTION. Nobody is advocating teaching facts in a vaccuum - just suggesting that teaching the why and the how simultaneously is difficult for many students. SWB in WTM says that Singapore requires LOGIC stage thinking sooner. This is because Singapore requires students to create a syllogistic understanding of computational problems. That may be appropriate for some students but it is not universally "better."

 

There are many countries besides Singapore that approach maths education the way that Singapore does. The system works well there and can work well here too. However, those countries have much more cross-curricular support for the kind of thinking that is required in Singapore. In my experience living in an Asian country (where math is taught in a similar way) I saw elementary students who were in school from 7 or 8 AM until their tutoring finished at 6PM. They also had full on university style final exams for all their subjects including computer and 2 foreign languages - I mention this to point out that the entire culture of education is different (probably better but different nonetheless.)

 

I wholly maintain that Singapore requires Logic stage thinking from Grammar stage students - some of them my benefit from that and some may not. If you had actually read my pp, you would have understood that I think both the "conceptual" and "factual/ procedural" areas of math should be mastered by all students. The process by which mastery is achieved may be different - I just think that it is a fallacy to say that only a conceptual approach can produce a robust understanding of mathematical concepts and facility in approaching mathematical problems.

 

Edited to remove my own passive-aggressive nastiness. Sorry Alte Veste Academy - you were right and I was wrong :blush:

[wendyroo]

BTW usually when somebody refers to something as concrete .... they mean to say that it is not abstract... so I'm not sure what you mean when you call Singapore both concrete and abstract.....

 

The whole philosophy of Singapore math, as I understand it, is the progression from concrete, to pictorial to abstract.  

 

The first time a student encounters a new concept it is via exploration with manipulatives.  These are the lessons outlined in the HIG that come before you even get to the textbook.  Next the student sees the same concept illustrated pictorially in the textbook with pictures of objects or manipulatives accompanying the example problems.  Finally, the pictures disappear and the student sees how the concept is shown abstractly in mathematical symbols.

 

Wendy

[Alte Veste Academy]

It's true that Singapore does have "drills" which (in the standards editions) they call "Mental Math" because it is done orally. In my experience USING those drills with students, they were not sufficient. The computational strategies were so cumbersome for many students that facts above the +2's were unruly for regrouping.... I would say with the exception of the doubles (and for some kids also the d+1 facts) Of course there were some students who flew through. Now  take it with a grain of salt because in a classroom it is difficult to tailor the pace of a program to meet students where they are - you can differentiate a little bit but are required to cover material according (basically) to the school's scope and sequence.  I also reiterate that I am talking about GRAMMAR stage students here (up to 4th grade or so.)

 

In the classical model, memorization, repetition and fact acquisition do precede ANALYSIS (not conceptual understanding) of a particular subject and its ABSTRACTION. Nobody is advocating teaching facts in a vaccuum - just suggesting that teaching the why and the how simultaneously is difficult for many students. SWB in WTM says that Singapore requires LOGIC stage thinking sooner. This is because Singapore requires students to create a syllogistic understanding of computational problems. That may be appropriate for some students but it is not universally "better."

 

There are many countries besides Singapore that approach maths education the way that Singapore does. The system works well there and can work well here too. However, those countries have much more cross-curricular support for the kind of thinking that is required in Singapore. In my experience living in an Asian country (where math is taught in a similar way) I saw elementary students who were in school from 7 or 8 AM until their tutoring finished at 6PM. They also had full on university style final exams for all their subjects including computer and 2 foreign languages - I mention this to point out that the entire culture of education is different (probably better but different nonetheless.)

 

I wholly maintain that Singapore requires Logic stage thinking from Grammar stage students - some of them my benefit from that and some may not. If you had actually read my pp, you would have understood that I think both the "conceptual" and "factual/ procedural" areas of math should be mastered by all students. The process by which mastery is achieved may be different - I just think that it is a fallacy to say that only a conceptual approach can produce a robust understanding of mathematical concepts and facility in approaching mathematical problems.

 

BTW usually when somebody refers to something as concrete .... they mean to say that it is not abstract... so I'm not sure what you mean when you call Singapore both concrete and abstract..... But I did super appreciate the condescending, belligerent tone of your post ;) It totally proves how much righter you are than me :)

 

:confused:

 

You read my tone completely wrong. For what it's worth, I did "actually read" your pp, and I did understand that you think both conceptual and procedural math should be mastered.

 

What I meant by concrete, pictorial, abstract is that SM teaches with the concrete (manipulatives), then moves to pictorial representation, then to abstract (algorithms). I am confused by your confusion since you say you are trained in the teaching of SM and MiF, which both teach with this progression. No snark intended, really. It is how SM is taught. It is elemental to the program.

 

We simply disagree. No biggie. I am not all worked up about it. LOL

 

[Erin]

The whole philosophy of Singapore math, as I understand it, is the progression from concrete, to pictorial to abstract.  

 

The first time a student encounters a new concept it is via exploration with manipulatives.  These are the lessons outlined in the HIG that come before you even get to the textbook.  Next the student sees the same concept illustrated pictorially in the textbook with pictures of objects or manipulatives accompanying the example problems.  Finally, the pictures disappear and the student sees how the concept is shown abstractly in mathematical symbols.

 

Wendy

 

 

I've never seen a math series that doesn't work this way...  

 

Seriously, unless you're talking something from a century ago, they're ALL conceptual.  

[mohini]

:confused:

 

You read my tone completely wrong. For what it's worth, I did "actually read" your pp, and I did understand that you think both conceptual and procedural math should be mastered.

 

What I meant by concrete, pictorial, abstract is that SM teaches with the concrete (manipulatives), then moves to pictorial representation, then to abstract (thinking problems through and executing the algorithms). I am confused by your confusion since you say you are trained in the teaching of SM. No snark intended, really. It is how SM is taught. It is elemental to the program.

 

We simply disagree. No biggie. I am not all worked up about it. LOL

 

Yeah - I did misread you :blush: sorry. I guess I'm all up in a huff...   -

 

So to address your question more properly I will say this - the manipulative aspect of Singapore was fantastic - (working with unifix cubes for number bonds, we also used counters and 10's frames, counters alone, physical number bonds with paper plates etc.. ) but - the leap from doing the concrete into the modeling was difficult for many students who were then lost later when they were required to manipulate the numbers mentally. Because the basic facts (including pairs to 10) were not memorized, kids had a terrible time translating the strategies into actual computation and applying them. SOME students excelled, loved the program and were able to manipulate numbers with incredible speed and a mathematical maturity beyond their years as the pp described of her own ds.

 

I worked with 1st and 3rd graders and in my experience they not only had trouble remembering the facts but also determining which strategies to use.  This is where logic stage thinking comes into play. If I told the students "we are going to regroup to have 10" they were fine but there was a huge lag for many students in being able to determine which strategy to use (and why.) I found that explaining the why was :banghead: for students who could not fundamentally understand that it is easier to add 10 to a number than say 7. I supplemented with C-rods and did a ton of written drills to improve their factual knowledge - which I admit gave me another bunch of students who could add 9+6 but could not explain it. But, it's easier to model a strategy for a student when they have the facts down and they can translate that knowledge into the conceptual/ abstract relationships of the numbers.

The thing is that demanding that a student express several different ways of solving a problem before they have a base of knowledge from which to draw is essentially putting the why before the what. Even those early "tell different stories about this picture" exercise require a kind of analysis. A lot of kids require much more time than is "scheduled" into the program to make these analytical leaps.

[Roadrunner]

I wholly maintain that Singapore requires Logic stage thinking from Grammar stage students -:

Can you give me an example? And can you tell he how non-conceptual (or Asian) programs approach a similar thing? I didn't grow up in Asia, but the math program we used as kids wasn't that different. Examples can help me understand what you mean.