Math wars and long division

[Roadrunner]

We have been working through Beast 4B lately and I was struck how different their approach was to long division. After a little research I figured out it's called Partial Quotients method. Is this the dreaded fuzzy math? I am asking because I came across this video from Everyday Mathematics

http://everydaymath.uchicago.edu/teaching-topics/computation/div-part-quot/

and I know they are suppose to be the leaders in fuzziness.

My kid knows the old fashioned algorithm, but truthfully I like this method for mental math.

[Roadrunner]

I'm so confused about long division...I don't think our kids do it! At least, not that I've come across. and I've taught K-9 at home...

Now you have me confused. :) you mean no long division in PS or at home?

[IsabelC]

Long division was actually taken out when the National Curriculum was introduced. So now, no Australian school teaches it at all unless they choose to. (Not without some complaints.) I found this out recently when I was trying to establish why my Mr. 10 had not had it in school. I decided to keep going with it at home anyway though. We've just done it in MM4B and will be going over it again in MM5, and I quite like the way she presents it, because the student learns why it works as well as how it works.

[kiana]

Long division was actually taken out when the National Curriculum was introduced. So now, no Australian school teaches it at all unless they choose to. (Not without some complaints.) I found this out recently when I was trying to establish why my Mr. 10 had not had it in school. I decided to keep going with it at home anyway though. We've just done it in MM4B and will be going over it again in MM5, and I quite like the way she presents it, because the student learns why it works as well as how it works.

 

wtf, how do they learn to do it in algebra then?

[kiana]

We have been working through Beast 4B lately and I was struck how different their approach was to long division. After a little research I figured out it's called Partial Quotients method. Is this the dreaded fuzzy math? I am asking because I came across this video from Everyday Mathematics

http://everydaymath.uchicago.edu/teaching-topics/computation/div-part-quot/

and I know they are suppose to be the leaders in fuzziness.

My kid knows the old fashioned algorithm, but truthfully I like this method for mental math.

I wonder why she's subtracting 70 twice instead of 140 once?

 

This is how I do division without a calculator and always has been. I actually got moved out of the advanced math class in school (before we started homeschooling) because I didn't like long division (it didn't make sense, until I took algebra and learned polynomial long division -- when I suddenly realized -- this is what they were trying to teach me before! just the x's were all 10s before!). The only difference is that I actually write it out as a series of fractions. So my scratch work would look like:

 

165/7 = 140/7 + 25/7 = 20 + 21/7 + 4/7 = 20 + 3 + 4/7 = 23 and 4/7.

[Roadrunner]

I wonder why she's subtracting 70 twice instead of 140 once?

 

This is how I do division without a calculator and always has been. I actually got moved out of the advanced math class in school (before we started homeschooling) because I didn't like long division (it didn't make sense, until I took algebra and learned polynomial long division -- when I suddenly realized -- this is what they were trying to teach me before! just the x's were all 10s before!). The only difference is that I actually write it out as a series of fractions. So my scratch work would look like:

 

165/7 = 140/7 + 25/7 = 20 + 21/7 + 4/7 = 20 + 3 + 4/7 = 23 and 4/7.

I learned the traditional algorithm and I am glad I got to learn this way along with my kid. That's why I am wondering now if I had been on the wrong side of math wars. :)

 

She is subtracting 70 twice because it's easier for kids to multiply on 10.

[Crimson Wife]

The "partial products" method for multiplication is part of "fuzzy math" (it's in that famous Youtube rant against Every Day Math and TERC Investigations by the meterorologist lady). So while I haven't seen BA 4B yet, it sounds similar.

[sunnyday]

I learned the traditional algorithm and I am glad I got to learn this way along with my kid. That's why I am wondering now if I had been on the wrong side of math wars. :)

 

Here's how I reconcile this in my head. Up until the 1850s, mathematics was about getting right answers to complex calculations, plugging in x to get y, etc. Conceptual underpinnings of these calculations were always understood by mathematicians but were not the important thrust of the field. After the mid-1800s this changed as the understanding of functions got flipped on its head and the mindset of the professionals in the field began to shift, emphasizing concepts and de-emphasizing computation. Functions as abstract mappings, non-intuitive results, and a mathematical understanding of very abstract ideas, you know? In the mid-1960s, during the Cold War's STEM explosion, the Russians were bringing a more conceptual math to their students and so the Americans tried to copy, but they swung way too far in the conceptual direction (especially too far for concrete little elementary kids) and that was the New Math. It crashed and burned in schools even though it was actually based on a sound understanding of mathematics as a field and the exciting revolution that had happened in mathematical thought since Riemann and Dirichlet.

 

I understand that Everyday Math is a reboot of New Math, another try to get exciting concepts in at the ground floor. But it's even worse in the "concepts and understanding are more important than answers" department, and I hear that it ignores standard algorithms entirely. There's nothing *inherently* wrong with Everyday Math, but I don't think it's the best way to teach, especially in schools. There's much better curriculum out there.

 

I think that Singapore Math balances computation and concepts very nicely and that Beast Academy does an even better job. I'd be surprised if they *never* teach standard long division, especially since it's a 6th grade standard, but this partial quotients method does sound interesting and intuitive!

 

Anyway, so that's why I feel like I can agree that "fuzzy math" has been a mistake in schools while still appreciating conceptual elementary math. :)

[wapiti]

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Edited January 12, 2016 by wapiti

[kiana]

 

 In the mid-1960s, during the Cold War's STEM explosion, the Russians were bringing a more conceptual math to their students and so the Americans tried to copy, but they swung way too far in the conceptual direction (especially too far for concrete little elementary kids) and that was the New Math. It crashed and burned in schools even though it was actually based on a sound understanding of mathematics as a field and the exciting revolution that had happened in mathematical thought since Riemann and Dirichlet

 

Also much too far for their teachers, who in many cases were computationally proficient but lacked the theoretical understanding necessary to really teach this properly.

 

I actually think a lot of these curricula could have worked much better than they did, had the elementary teachers understood what they were going after. The same problem appears in current conceptual curricula.

 

(This is not intended as a slam on teachers in general, rather on the implementation of these curricula).

[Farrar]

We did that chapter.  It is probably the Beast chapter that I have the most mixed feelings about (though I felt similarly about multiplication).  NOT because of partial quotients, which I think is a GREAT introduction to the concepts that bolster long division, but because it was a huge step backwards in many ways since we had already covered a lot of that in Miquon.  He basically understood that stuff and wanted to do things in his own way, having learned the algorithm, but Beast forced him to back up...  which was good in some ways since Beast goes so much deeper with everything, especially the simplest things, but also frustrating and didn't really help him move forward that much in his understanding of division and ability to do new division problems overall.

 

There's nothing fuzzy about partial quotients IMO.  It's a good mental math strategy and helps show the ways in which long division, which seems so inexplicable to so many students as a set of steps to be learned, actually works.  Especially the way Miquon introduced it - through examples that were really intuitive (this is why I always loved Miquon so much) - it really helps lead kids into long division in a more natural way.

 

And, I'm with anyone who is saying why in the world would any curriculum completely skip long division?!?  What's the long term mathy rationale there?  Is something like 14294 divided by 94 really something that some students don't learn to do without a calculator at all at any point in their math education?!?

[wapiti]

I understand that Everyday Math is a reboot of New Math, another try to get exciting concepts in at the ground floor. But it's even worse in the "concepts and understanding are more important than answers" department, and I hear that it ignores standard algorithms entirely. There's nothing *inherently* wrong with Everyday Math, but I don't think it's the best way to teach, especially in schools. There's much better curriculum out there.

 

I suspect that there might actually be things inherently wrong with EM, considering that the mathematicians abandoned the development team in the 80s, leaving the education department to complete it (e.g., the lattice stuff was supposed to be just a sidebar, etc.)

 

I haven't noticed anything in the few New Math books I have that are like the fuzziness of the post-1990 NCTM programs like EM (not that I have any EM), maybe because the fuzzy programs seem to focus so much more on having a different instructional method (group discovery?) than the New Math books.

 

I think that Singapore Math balances computation and concepts very nicely and that Beast Academy does an even better job. I'd be surprised if they *never* teach standard long division, especially since it's a 6th grade standard, but this partial quotients method does sound interesting and intuitive!

 

It'll be interesting to see what happens in BA5 with long division.  I don't see any more division covered in BA4 and I don't see the standard algorithm at all - is there a TOC for 5 somewhere? (the Prealgebra text comes after 5)  I don't know how I'd feel about abandoning the algorithm - I'm glad my kids learn it elsewhere.

 

Long division aside, sometimes, looking at BA and knowing where this all goes for the Prealgebra text, I feel that BA is sort of a simpler version of a lot of the topics covered in the Prealgebra text.

[sunnyday]

I suspect that there might actually be things inherently wrong with EM, considering that the mathematicians abandoned the development team in the 80s, leaving the education department to complete it (e.g., the lattice stuff was supposed to be just a sidebar, etc.)

 

I haven't noticed anything in the few New Math books I have that are like the fuzziness of the post-1990 NCTM programs like EM (not that I have any EM), maybe because the fuzzy programs seem to focus so much more on having a different instructional method (group discovery?) than the New Math books.

 

Oi, thanks for the clarification. I definitely want to do learn more about this whole history of math education stuff -- New Math sounds like it was terrific for those who really understood how to implement it, while the 1980s versions not so much. My mom was pretty aghast at some of the methods used in my brother's Open Optional elementary school program in the 80s, both math and language arts.

 

 

It'll be interesting to see what happens in BA5 with long division.  I don't see any more division covered in BA4 and I don't see the standard algorithm at all - is there a TOC for 5 somewhere? (the Prealgebra text comes after 5)  I don't know how I'd feel about abandoning the algorithm - I'm glad my kids learn it elsewhere.

 

Long division aside, sometimes, looking at BA and knowing where this all goes for the Prealgebra text, I feel that BA is sort of a simpler version of a lot of the topics covered in the Prealgebra text.

 

They don't seem to make the TOC available until shortly before publication. The very rough S&S for the rest of 4 can be found of course -- I wonder if they use dividing fractions in 4D as another conceptual underpinning before going to the full algorithm somewhere in 5? I swear I've seen the topics list for 5 somewhere, hm...

[freesia]

Oh my goodness--that is the exact method I was taught in the 70s. I remember staring at the board COMPLETELY confused about what the teacher was doing! ACK! I've never met anyone who knew what I was talking about when I tried to explain it. Now, I do have a strong conceptual understanding of math--but who's to say if that was the New Math I was taught or natural proclivity (coming from a family rife with financial advisers and engineers.)

 

I teach long division with 10s and 1s blocks in order to teach what we are doing. We do it over and over and over again until they can do it with mental pictures in their heads.

[Paige]

I have no problem with partial quotients being used when introducing the concept of division. It works. It makes sense. It makes it easy to divide large, relatively easy problems in your head. I think that it is terrible to teach partial quotients and leave it at that and never teach long division. Long division is really just partial quotients organized well. It makes difficult numbers easy to work with. Why wouldn't you teach it? It's not that hard! I wonder how children are expected to work through decimal problems w/partial quotients? 

 

Hopefully BA is using partial quotients in 4B with the intent to teach long division in another book.

[wapiti]

Oi, thanks for the clarification. I definitely want to do learn more about this whole history of math education stuff -- New Math sounds like it was terrific for those who really understood how to implement it, while the 1980s versions not so much. My mom was pretty aghast at some of the methods used in my brother's Open Optional elementary school program in the 80s, both math and language arts.

 

FWIW, the 1980s versions of texts by the New Math gurus (e.g. Dolciani) tend to be quite straightforward, a good mix of concept and procedure, and relatively rigorous, although they took a lot of the set theory out (it's still in some of the algebra texts).  I don't really know, but my guess is that the move to fuzzy math was mostly the result of educators (as in the story behind EM and the stuff coming out of NCTM) making a sort of faux attempt to teach concepts that weren't well understood by some of the teachers (the reason New Math failed to begin with, AFAIK).  It is an interesting history to read about, like any other subject with each writer coming from their own perspective.  

 

I agree about SM being a nice balance, which IMO demonstrates that it isn't impossible.  What amazes me is how easily that balance gets messed up in more mainstream PS programs that claim to be like Singapore (e.g. Envision? with the spiral and data-driven aspects).

 

They don't seem to make the TOC available until shortly before publication. The very rough S&S for the rest of 4 can be found of course -- I wonder if they use dividing fractions in 4D as another conceptual underpinning before going to the full algorithm somewhere in 5? I swear I've seen the topics list for 5 somewhere, hm...

 

I too could swear I saw a topic list for 5 someplace but I cannot find anything.

[Roadrunner]

We did that chapter. It is probably the Beast chapter that I have the most mixed feelings about (though I felt similarly about multiplication). NOT because of partial quotients, which I think is a GREAT introduction to the concepts that bolster long division, but because it was a huge step backwards in many ways since we had already covered a lot of that in Miquon. He basically understood that stuff and wanted to do things in his own way, having learned the algorithm, but Beast forced him to back up... which was good in some ways since Beast goes so much deeper with everything, especially the simplest things, but also frustrating and didn't really help him move forward that much in his understanding of division and ability to do new division problems overall.

 

This has been our exact experience! He has faught hard to do it his way, but lost the battle. I am so glad Beast introduced us to this, because I have never seen this method before and I do think it's an excellent way to start out doing the division. Yep, we are backing up. :)

After this chapter though I wonder what else I have never been taught at school.

[Roadrunner]

It'll be interesting to see what happens in BA5 with long division. I don't see any more division covered in BA4 and I don't see the standard algorithm at all - is there a TOC for 5 somewhere? (the Prealgebra text comes after 5) I don't know how I'd feel about abandoning the algorithm - I'm glad my kids learn it elsewhere.

 

Long division aside, sometimes, looking at BA and knowing where this all goes for the Prealgebra text, I feel that BA is sort of a simpler version of a lot of the topics covered in the Prealgebra text.

This is their tentative schedule for grade 5 from facebook posting:

3D Geometry

Integers

Expressions and Equations

 

Patterns and Sequences

Fractions

Data and Statistics

 

Fractions

Ratios and Rates

Decimals

 

Percents

Square Roots

Exponents

[jennynd]

DS does not do math with his homeroom in PS but still has to take grade level standardized test. one of the review questions before the test sent home by the homeroom teacher was kinda like that, i can't understand what exactly the question asked so i ask DS to double check with the teacher. apparenbtly, the teacher was very very confused too and took couple try to get the answer ... lol..

 

I will certainly not let my kids use this as long division method. "HOWEVER" i do think it is valuable to learn. it really is a reverse of distributive property

[Farrar]

This has been our exact experience! He has faught hard to do it his way, but lost the battle. I am so glad Beast introduced us to this, because I have never seen this method before and I do think it's an excellent way to start out doing the division. Yep, we are backing up. :)

After this chapter though I wonder what else I have never been taught at school.

 

If you have a heck of a lot of C-rods as well as some base 10 hundred flats (and maybe some thousand cubes), the long division method at Education Unboxed is really neat and also leads in from partial quotients.

[Roadrunner]

If you have a heck of a lot of C-rods as well as some base 10 hundred flats (and maybe some thousand cubes), the long division method at Education Unboxed is really neat and also leads in from partial quotients.

My kids never took to C-rods, but I was like a kid in a candy shop when I saw her factor the polynomials. Talk about a lightbulb moment!

[Korrale]

I love how Montessori introduces division. I wonder if it is the same as education unboxed (that site streams too slowly for me to enjoy to the fullest)

[IsabelC]

wtf, how do they learn to do it in algebra then?

 

Sorry, I should have said it was taken out of Primary School (grade 1-6). That's the bit of the Curriculum I've read, because my eldest child is in 5th grade. However, I understand that long division is covered in some way (indirectly?) in the context of algebraic calculations when students reach grade 10.

 

Unfortunately, our new curriculum is seen as a bit of a step backward by many people. It's full of phrases like "investigate strategies to... " and "use technology to..."  Instead of learning specific skills, students are supposed to "become self-motivated, confident learners through inquiry and active participation in challenging and engaging experiences".

 

The justification for ditching long division was apparently that there are

“many methods of computing such an answer and the long-­division algorithm is but one. What is essential is that students ... perform the calculation (using a method of their choice, including a calculator) and judge the reasonableness of the solution obtained based on their ­estimation and the context or situation.â€

 

So yeah, the official position is why work it out manually when they can use a calculator. I don't let Mr. 10 use a calculator yet though, as I'm following Maria Miller's suggestion.

[IsabelC]

But isn't it interesting how there are all different ways to do things, and how they come in and out of fashion. Ten years ago, I would never have thought that math was a pretty fixed sort of thing, that nothing really changed, and it wasn't really subject to fads, differences of opinion etc. The first glimmer came when I encountered equal addends subtraction and went  :confused1: . 

[Crimson Wife]

New Math sounds like it was terrific for those who really understood how to implement it, while the 1980s versions not so much.

 

CSMP is New Math and it's really an excellent program *IF* the teacher can wrap her brain around how to teach it. I've dabbled it in a bit and really wish I were better able to teach it because I think my DS especially would love it.

[RootAnn]

If you have a heck of a lot of C-rods as well as some base 10 hundred flats (and maybe some thousand cubes), the long division method at Education Unboxed is really neat and also leads in from partial quotients.

 

My dd#2 who hates math eventually resonated really well with Ed. Unbox's way of tackling long division. While she started out with all the c-rods/flats we had, she eventually switched to using the abacus until she felt comfortable with going full-bore with the algorithm. 

[Dana]

CSMP is New Math and it's really an excellent program *IF* the teacher can wrap her brain around how to teach it. I've dabbled it in a bit and really wish I were better able to teach it because I think my DS especially would love it.

 

They're the same people who did the EM curriculum.

[Roadrunner]

They're the same people who did the EM curriculum.

Oh Dana. You had to introduce me to a new math curriculum. :)

Now I want to ask for a review.

[Dana]

It's good!

I used the texts in middle school in the MEGSSS program. Saw some stuff that I didn't see again until my undergrad math courses. I managed to get a copy of the texts and have used a couple with ds.

He did the first EM course in conjunction with the text I have. The online course leaves a few things out (not as many examples for some problems, some sections that are dated have been removed).

If I hadn't already bought the books, I would definitely have ds doing their courses.

eIMACS owns the rights to the books. They also have a logic course online that's too pricey for me, but if it's based on the text we used, it's a formal mathematical logic course that clearly sets up proofs.

 

We've decided to focus this year on AoPS, but I'm really torn about whether we'll include the EM texts.

[Roadrunner]

It's good!

I used the texts in middle school in the MEGSSS program. Saw some stuff that I didn't see again until my undergrad math courses. I managed to get a copy of the texts and have used a couple with ds.

He did the first EM course in conjunction with the text I have. The online course leaves a few things out (not as many examples for some problems, some sections that are dated have been removed).

If I hadn't already bought the books, I would definitely have ds doing their courses.

eIMACS owns the rights to the books. They also have a logic course online that's too pricey for me, but if it's based on the text we used, it's a formal mathematical logic course that clearly sets up proofs.

 

We've decided to focus this year on AoPS, but I'm really torn about whether we'll include the EM texts.

Do they sell the books separately? I can't find it.

[Dana]

Do they sell the books separately? I can't find it.

 

I doubt they do anymore.

I bought them before the EM materials were online & had the background of having used them as a kid. Even so, I can't get a teacher's guide, tests, or answers, so for the books we've gone through, I've had to work all the problems as well. Borrowed some of my sister's books that she'd still had to check her answers in comparison. We doodled a lot :)

 

And I would be okay using just online if I didn't own the texts. I just can't justify re-paying for them.

But they are excellent and I really was (am) torn between them and AoPS.

[Gil]

Who is  the author and what is the name of the book/series?

I doubt they do anymore.

I bought them before the EM materials were online & had the background of having used them as a kid. Even so, I can't get a teacher's guide, tests, or answers, so for the books we've gone through, I've had to work all the problems as well. Borrowed some of my sister's books that she'd still had to check her answers in comparison. We doodled a lot :)

 

And I would be okay using just online if I didn't own the texts. I just can't justify re-paying for them.

But they are excellent and I really was (am) torn between them and AoPS.

 

[kiana]

Who is  the author and what is the name of the book/series?

 

The publisher was CEMREL, the authors were various, but Robert Exner did a lot of them.

 

Here's some details (you need to scroll down to the late '60s):

 

http://www.imacs.org/about/news/burt-kaufman.html

 

[happyhome]

It's good!

I used the texts in middle school in the MEGSSS program. Saw some stuff that I didn't see again until my undergrad math courses. I managed to get a copy of the texts and have used a couple with ds.

He did the first EM course in conjunction with the text I have. The online course leaves a few things out (not as many examples for some problems, some sections that are dated have been removed).

If I hadn't already bought the books, I would definitely have ds doing their courses.

eIMACS owns the rights to the books. They also have a logic course online that's too pricey for me, but if it's based on the text we used, it's a formal mathematical logic course that clearly sets up proofs.

 

We've decided to focus this year on AoPS, but I'm really torn about whether we'll include the EM texts.

Oh boy, just when you think it's safe to hit the "new content" button.....I thought we had our AoPS plan all set, now I look at EM and questions are swirling again. I'm going to start a new thread.

 

Regarding BA, the standard algorithm is most probably going to be taught at some point because books later in the sequence reference it. Also, AoPS is the curriculum of choice for math competition kids so they are always introducing faster methods of problem solving for those timed competition settings. From what I've seen, these are not the fuzzy math tricks (we pulled our kids from school because of Everyday Math and then the Envision "improvement"). These are concrete problem solving strategies based on conceptual understanding. That said, my daughter could not do her current AoPS Algebra work without the algorithm, so if they don't introduce it in the later BA books, that would be a big oversight. I just don't see that happening.

[kiana]

I doubt they do anymore.

I bought them before the EM materials were online & had the background of having used them as a kid. Even so, I can't get a teacher's guide, tests, or answers, so for the books we've gone through, I've had to work all the problems as well. Borrowed some of my sister's books that she'd still had to check her answers in comparison. We doodled a lot :)

 

And I would be okay using just online if I didn't own the texts. I just can't justify re-paying for them.

But they are excellent and I really was (am) torn between them and AoPS.

 

Man, I hope not. This set of books has always been on my "to buy someday when I have cash" list :/