Whole to parts math vs. parts to whole math?

[smilesonly]

what would be some math programs that fit into these two categories? i know r&s teaches adding/subtracting whole-to -parts. now i have learned mm teaches parts-to-whole.

 

aside from fitting a dc to a particular learning approach, what would be advantages/disadvantages to each style?

 

which curriculum would be an example of each approach?

 

tia:)

[wapiti]

now i have learned mm teaches parts-to-whole.

 

I disagree. I'm copying from an old post of my own, so please forgive any irrelevant bits:

 

Personally, I would not call MM a parts-to-whole program. On the contrary, I'd call it mostly whole-to-parts. I can't say whether this is true for every single aspect of the curriculum, but generally the lessons go from the conceptual explanation and move to the traditional algorithm. The concept is the big picture, the whole, the why, the context. The traditional algorithm has the parts, the details.

 

For example, look at division of fractions. First we look, pictorially, at how many times the fractional part that is the divisor fits into the dividend. This is important because that is the true meaning of the division, the concept, the big picture. Only then do we learn the easier way to compute, by multiplying by the reciprocal, the details. I think it's a mistake to conclude that the reverse order, learning the algorithm first and then looking at why, is whole to parts. The algorithm is not the big picture. The algorithm is the detail that can be understood after learning the concept has provided the context for it.

 

Another whole-to-parts aspect of MM is problem-solving. Word problems involve looking at the context of the problem to figure out which arithmetic concept is at work and applying it within the framework of the problem. Word problems that involve more than one step, which may be the case more often in the upper levels, provide yet another layer of context, an even bigger picture. Problem-solving skills become very useful in higher-level math. Thinking deeply about a word problem can help cement a concept in a student's mind when they crack the puzzle.

 

Now, there is something about MM that may make it seem parts to whole: it is a very incremental program, so there are many steps, or increments, on the way from the concept to the algorithm. The intellectual leaps between lessons are very small. Some kids, who can fill in the ideas in between for themselves, might be frustrated by the multitude of increments (Singapore might be worth looking at in that regard, from what I have read here on the boards). Other kids need the more explicit connecting of the dots that MM provides. That is why I think a flexible approach is helpful with MM. Its organization makes it easy to skim through ideas that a particular student already understands well.

 

If you have a whole-to-parts learner, IMO you are looking for a curriculum that teaches concepts and not just the traditional algorithm. I'd guess that conceptual curricula might include RS, MUS, Miquon, Singapore and MM, just from what I've read about these programs here. I'm sure there are others.

[smilesonly]

I disagree. I'm copying from an old post of my own, so please forgive any irrelevant bits:

 

Personally, I would not call MM a parts-to-whole program. On the contrary, I'd call it mostly whole-to-parts. I can't say whether this is true for every single aspect of the curriculum, but generally the lessons go from the conceptual explanation and move to the traditional algorithm. The concept is the big picture, the whole, the why, the context. The traditional algorithm has the parts, the details.

 

For example, look at division of fractions. First we look, pictorially, at how many times the fractional part that is the divisor fits into the dividend. This is important because that is the true meaning of the division, the concept, the big picture. Only then do we learn the easier way to compute, by multiplying by the reciprocal, the details. I think it's a mistake to conclude that the reverse order, learning the algorithm first and then looking at why, is whole to parts. The algorithm is not the big picture. The algorithm is the detail that can be understood after learning the concept has provided the context for it.

 

Another whole-to-parts aspect of MM is problem-solving. Word problems involve looking at the context of the problem to figure out which arithmetic concept is at work and applying it within the framework of the problem. Word problems that involve more than one step, which may be the case more often in the upper levels, provide yet another layer of context, an even bigger picture. Problem-solving skills become very useful in higher-level math. Thinking deeply about a word problem can help cement a concept in a student's mind when they crack the puzzle.

 

Now, there is something about MM that may make it seem parts to whole: it is a very incremental program, so there are many steps, or increments, on the way from the concept to the algorithm. The intellectual leaps between lessons are very small. Some kids, who can fill in the ideas in between for themselves, might be frustrated by the multitude of increments (Singapore might be worth looking at in that regard, from what I have read here on the boards). Other kids need the more explicit connecting of the dots that MM provides. That is why I think a flexible approach is helpful with MM. Its organization makes it easy to skim through ideas that a particular student already understands well.

 

If you have a whole-to-parts learner, IMO you are looking for a curriculum that teaches concepts and not just the traditional algorithm. I'd guess that conceptual curricula might include RS, MUS, Miquon, Singapore and MM, just from what I've read about these programs here. I'm sure there are others.

 

thanks for sharing(and, explaining) this. my quoted comment is not my original opinion-it was regurgitating what i've read about mm.:blush:

[Spy Car]

I honest think I have a hole (or is it a whole? :D) in my brain on this subject, because I can never understand what the heck whole-to-parts vs parts-to-whole really means.

 

I admit it. I don't get it.

 

Who's your "math genius" now? :tongue_smilie:

 

Bill

[boscopup]

I honest think I have a hole (or is it a whole? :D) in my brain on this subject, because I can never understand what the heck whole-to-parts vs parts-to-whole really means.

 

I admit it. I don't get it.

l

 

I'm with ya. And I would think MM goes parts to whole, because it introduces the parts and then puts them together to make the whole. The standard algorithm being the parts doesn't make sense to me?

 

So yeah, I'm clueless.

 

MM is conceptual, as is Singapore and some others. Most math programs have at least some conceptualness to them. Some have way more than others.

 

True parts to whole might be Saxon? It gives very incremental pieces and doesn't put those pieces together until later. :confused:

[Embassy]

I think what we do could be considered whole to parts math. But I may be wrong:tongue_smilie: We do math on three different levels. I introduce new topics/concepts above their standard math level. Exposure to concepts through books/videos often motives them to complete the steps in their standard math program. I use a standard math program on another level (Singapore). We go step by step through Singapore. It does focus on concepts before the algorithm which does fit my kids well. The lowest level is math facts. Usually math facts aren't too difficult to learn by this point because they have had lots of practice using them. So in one week a child may be introduced to exponents, complete early multiplication problems, and work on addition facts. It works for us:001_smile:

[smilesonly]

I admit it. I don't get it.

 

Who's your "math genius" now? :tongue_smilie:

 

Bill

 

um, that would be wapiti.:D:lol:

 

i think it may have been through private messaging that someone was explaining how rod & staff teaches addition/subtraction-that it teaches the whole to parts way.

 

for ex; 15 is the whole. 9 and 6 are it's parts.

 

so in the context of adding/subtracting, the dc is taught that whenever he sees those 3 numbers, he knows that they fit together-either by adding or subtracting.

 

i don't get it, either. but how wapiti explained it makes some sense.:)

[Spy Car]

um, that would be wapiti.:D:lol:

 

i think it may have been through private messaging that someone was explaining how rod & staff teaches addition/subtraction-that it teaches the whole to parts way.

 

for ex; 15 is the whole. 9 and 6 are it's parts.

 

so in the context of adding/subtracting, the dc is taught that whenever he sees those 3 numbers, he knows that they fit together-either by adding or subtracting.

 

i don't get it, either. but how wapiti explained it makes some sense.:)

 

But that's whole-parts math. Teaching that number values can be spit into component parts, and that parts can be joined to make a whole, or given a whole and a part one can find the difference which comprises the other part(s).

 

Whole-parts math has nothing to do with "parts-to-whole" or "whole-to-parts" (at least I think that's the case, given I'm clueless about these latter terms).

 

"Whole-parts" math (not whole-to-parts math) is what you find in programs like Primary Mathematics (Singapore), Math Mammoth, RightStart, Miquon, and the like.

 

It builds abstractions on the sort of concrete foundations one gets using manipulatives like Cuisenaire Rods.

 

Maybe the ideas got conflated?

 

Bill

[wapiti]

:lol: :001_huh: I'm not sure I can explain it any better than my last pathetic attempt, but I'll say this: when we say "whole-to-parts" as a method of learning, we mean learning the big picture first. when it comes to a given math topic, ask yourself, what is the big picture? What are we really doing here? What does this mean? What is the purpose? What is the context for what we are doing?

 

Take, for example, multiple-digit multiplication. What is the big picture? What are we really doing? We're multiplying times the various place values and then adding them together. The point of place value is a little harder to see in the traditional algorithm, so we multiply in parts first to demonstrate that a little more clearly:

 

215 x 6 = (200 + 10 + 5) x 6 = (200 x 6) + (10 x 6) + (5 x 6)

 

That is the big picture. After we understand that, we can move on to the traditional vertical algorithm, which sort of collapses the processes into a single "problem." The algorithm includes the details.

 

I know, I know: "multiplying in parts" sounds like parts, and the single problem sounds like the whole. That's backwards. However, the lesson on multiplying in parts provides the context or meaning or concept for what happens in the traditional algorithm. ETA, maybe I can distinguish whole-to-parts as a method of learning/teaching from a mathematical procedure that happens to proceed from parts to whole.

 

*sigh* I don't know if I can explain it any other way, other than concept = big picture, algorithm = details, and concept comes first. Maybe I'll try to come up with some other examples that aren't so backwards :lol:

Edited November 8, 2011 by wapiti

[Spy Car]

:lol: :001_huh: I'm not sure I can explain it any better than my last pathetic attempt, but I'll say this: when we say "whole-to-parts" as a method of learning, we mean learning the big picture first. when it comes to a given math topic, ask yourself, what is the big picture? What are we really doing here? What does this mean? What is the purpose? What is the context for what we are doing?

 

Take, for example, multiple-digit multiplication. What is the big picture? What are we really doing? We're multiplying times the various place values and then adding them together. The point of place value is a little harder to see in the traditional algorithm, so we multiply in parts first to demonstrate that a little more clearly:

 

215 x 6 = (200 + 10 + 5) x 6 = (200 x 6) + (10 x 6) + (5 x 6)

 

That is the big picture. After we understand that, we can move on to the traditional vertical algorithm, which sort of collapses the processes into a single "problem." The algorithm includes the details.

 

I know, I know: "multiplying in parts" sounds like parts, and the single problem sounds like the whole. That's backwards. However, the lesson on multiplying in parts provides the context or meaning or concept for what happens in the traditional algorithm.

 

*sigh* I don't know if I can explain it any other way, other than concept = big picture, algorithm = details, and concept comes first. Maybe I'll try to come up with some other examples that aren't so backwards :lol:

 

I can wrap my head around "big-picture." I like "big-picture."

 

Bill

[boscopup]

I understand what you're saying, wapiti. :) It is interesting to see conceptual math as being whole-to-parts math, since many people want parts-to-whole instruction in most topics, but we want conceptual instruction when it comes to math.