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Ktgrok

Long division in 2nd grade???

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My sister mentioned her daughter was working on long division. Her daughter is in 2nd grade in a public school.  I'm not crazy - that's REALLY early, right???

Edited by Ktgrok
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That does seem super early. And kind of silly, because getting a good sense of what you’re doing when you’re dividing seems like something you ought to do before doing this algorithm. Like, when did this kid learn to multiply??

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It's not impossible if you have a math accelerated student. Pretty sure that's what my son was doing in 2nd grade because he was doing SM 5A at the end of 2nd grade.

So, it is early and not typical.

Edited by calbear
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I also do not think it is super early.  Early, yes, for some students, but on par for others. My 3rd dd was doing long division last yr, but she started this yr using a 4th grade text. 

She started multiplication in her 2nd grade text. I cannot remember exactly when division is introduced (in the 2nd or 3rd grade books...not sure.)  

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If this is a traditional classroom situation, she is probably doing division with a bracket and calling that long division.  So maybe something simple like 40 /2 in a bracket set up in which the divisor goes evenly into each digit.  If she is doing longer problems that require regrouping, then yes, that's probably a year ahead in common core-maybe she's in an accelerated or gifted class, or maybe she's in a gifted pull-out/enrichment.  If this is a homeschool situation, then she is probably accelerated, but only a year or two.  I wouldn't call it crazy.

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25 minutes ago, 8FillTheHeart said:

I also do not think it is super early.  Early, yes, for some students, but on par for others. My 3rd dd was doing long division last yr, but she started this yr using a 4th grade text. 

She started multiplication in her 2nd grade text. I cannot remember exactly when division is introduced (in the 2nd or 3rd grade books...not sure.)  

 

Now I'm curious: how long do kids usually do division before they see long division with regrouping? 

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If she's in public school I know that would be early for the public schools here- even in the GT classes. I have a friends with a dc in 1st grade and they're just now learning to add up to 10 and things like parts to whole- they are no where near that level. That would be a huge jump in a year, but Texas isn't CC technically so maybe we're behind? 

If it's homeschool it could definitely happen, but I would think that would be much sooner than average. 

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In the traditional public school progression (at least in the past), long division was a 4th grade topic. According to the World Book Typical Course of Study, multiplication and division of any kind are not even listed in 2nd grade math topics, and in 3rd grade, multiplication and division is with 1- and 2-digit numbers ("up to 100"), which indicates that longer long division (lol) begins in 4th grade.

Obviously, if the student is accelerated, then the student may be encountering long division in 2nd grade. 🙂

Side note: Does it really matter when your niece starts doing long division? It's not a race or a competition. 😉

Just our experience: DS#2 with mild LD did not click with long division until 5th grade. We introduced simple long division at the end of 3rd grade; beginning of 4th grade; middle of 4th grade. Each time it was a major bust, so we set it aside and tried again later. Eventually his brain matured into being able to click with long division, but in his own timing. DS#1 who likes math and is math minded was doing long division in 3rd. In the end, it all got learned, and with a lot less stress and tears by providing the right amount of challenge for each child in his own timing.

 

Edited by Lori D.
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8 minutes ago, Æthelthryth the Texan said:

If she's in public school I know that would be early for the public schools here- even in the GT classes. I have a friends with a dc in 1st grade and they're just now learning to add up to 10 and things like parts to whole- they are no where near that level. That would be a huge jump in a year, but Texas isn't CC technically so maybe we're behind? 

If it's homeschool it could definitely happen, but I would think that would be much sooner than average. 

Having just moved from Texas to NYC and having heard reports from other people: yeah, I think Texas is considerably behind some other places in the country. 

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I should have clarified, she's in public school. She does go to a gifted pull out so maybe? My sister made it sound like she had long division homework she was working on (and finding difficult). 

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10 minutes ago, Ktgrok said:

I should have clarified, she's in public school. She does go to a gifted pull out so maybe? My sister made it sound like she had long division homework she was working on (and finding difficult). 

 

That seems unfortunate for an activity in a gifted program... there are so many more things that are more fun than long division :-). Remind me of something I heard from a friend of mine whose kid is in a public school: the fact that he's "gifted" means that he gets the same kinds of problems, just more of them... Way to make someone dislike being smart. 

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12 minutes ago, square_25 said:

 

That seems unfortunate for an activity in a gifted program... there are so many more things that are more fun than long division :-). Remind me of something I heard from a friend of mine whose kid is in a public school: the fact that he's "gifted" means that he gets the same kinds of problems, just more of them... Way to make someone dislike being smart. 

And then when the rest of the class gets to long division the pull-out kids are even further ahead and more bored.  I have fond memories not my parochial school pull-out teacher teaching us french, and not-so-fond memories of being treated as an unpaid teacher's aid in a public school.

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Well, I think it depends.
I think some programs introduce "long division" in a way where it's like "take out groups of 5....take out more groups of 5....take out more groups of 5...add them up."  It was one of the main complaints of Everyday Math and TERC from parents.
Or it can be introduced as block play/reverse order multiplication: sort the hundreds flats into groups, sort the tens into groups, sort the units into groups..

It is early, but it's not outside the realm of possibility if kids are concretely looking at how numbers work, you know?  I'd be intrigued to see what her teacher is doing in the classroom and how she's making that work for a group of 2nd graders.

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My kids did it in or before 2nd grade, but we were using Singapore 1-3 levels above grade level.  It think Singapore math does it in 3rd.  I currently volunteer with some students who are doing it in 4th, which I think is when I was taught to do it. The biggest impediment seems to be knowing the multiplication/division facts well enough to do the problems.  When helping, I have struggled to tell when they don't understand the problem and when they get so distracted by having to skip count to see how many times 3 goes into 28 that they forget what they're doing.  I see no reason to rush teaching it - there is no advantage to students spending a lot of time being frustrated by long division - but if students know the underlying math facts, I see no reason to put it off until a particular grade.  

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In my experience that is early for a public school.  I was taught long division in 4th grade back in the day.  My uncle, a 5th grade public school teacher, says that long division is one of the main math topics he teachers - and from what he says, he does so at a very procedural level, just drilling the algorithm rather than teaching conceptually.

As others have said, I would not be surprised at all by a second grader being taught "long division" of the 624 / 2 sort, especially in a gifted pull out.

For a homeschooler, OTOH, I don't think second grade is unusually early for long division.  Both of my older boys entered second grade with a strong concrete, conceptual understanding of long division (and a lot of practice doing long division problems via a partial product method).  During second grade, they were ready to tackle the algorithm and learn how to quickly and accurately solve long division problems with one and two digit divisors.

Wendy

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I'm pretty sure all of my kids have done some long division by the time they were 8. My oldest was comfortable with it when she was 7. The others were at least doing things like 144/3 at 8. But they are homeschooled and get personal tutoring. I would not expect a public school to be much into long division until 4th grade.

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7 hours ago, Meriwether said:

I'm pretty sure all of my kids have done some long division by the time they were 8. My oldest was comfortable with it when she was 7. The others were at least doing things like 144/3 at 8. But they are homeschooled and get personal tutoring. I would not expect a public school to be much into long division until 4th grade.

 

Would they be using long division for 144/3?

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8 minutes ago, square_25 said:

 

Would they be using long division for 144/3?

I'm not sure at what you are getting at, but long division is one way for an 8 yr old to approach 144/3. Another might be to realize that 144 is divisible by 3 followed by 12x12 equals a 144 and factor out a 3 and multiple 4 x 12, etc.

If the implication is that 2nd graders doing long division are only learning an algorithm without any understanding of what they are doing then that assumption is false.  

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16 minutes ago, square_25 said:

 

Would they be using long division for 144/3?

I just made up an example, but, yes, they could? Unless I am not understanding what is meant. How many times does 3 go into 14? 4, so subtract 12. How many times does 3 go into 24? 8, so 144 divided by 3 is 48. That seems about like what they started with? My 3rd grader is now using long division for longer, harder problems, but that is about the difficulty of what they do in 2nd grade to learn the algorithm. 

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7 minutes ago, 8FillTheHeart said:

I'm not sure at what you are getting at, but long division is one way for an 8 yr old to approach 144/3. Another might be to realize that 144 is divisible by 3 followed by 12x12 equals a 144 and factor out a 3 and multiple 4 x 12, etc.

If the implication is that 2nd graders doing long division are only learning an algorithm without any understanding of what they are doing then that assumption is false.  

Thanks. I wasn't sure what the question meant. I fully admit to often having self-doubts about how I'm doing with the kids, but I can state confidently that they have good number sense and are fully understanding any algorithms I have them use. They've all picked up long division easily and naturally because they understand what it is doing.

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1 minute ago, Meriwether said:

Thanks. I wasn't sure what the question meant. I fully admit to often having self-doubts about how I'm doing with the kids, but I can state confidently that they have good number sense and are fully understanding any algorithms I have them use. They've all picked up long division easily and naturally because they understand what it is doing.

I understand. My kids all have great number sense and are excellent math students. 🙂 Horizons and all!

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1 hour ago, 8FillTheHeart said:

I'm not sure at what you are getting at, but long division is one way for an 8 yr old to approach 144/3. Another might be to realize that 144 is divisible by 3 followed by 12x12 equals a 144 and factor out a 3 and multiple 4 x 12, etc.

If the implication is that 2nd graders doing long division are only learning an algorithm without any understanding of what they are doing then that assumption is false.  

 

I wasn’t implying that, but yes, I think spending a lot of time with an algorithm can make the original intuitive understanding harder to access. I’d definitely expect them to do multidigit divisions by hand for quite a while before doing algorithms, but I’m not sold on algorithms as a way to learn math in the first place. But I’m probably too algorithm-phobic :-).

My daughter is only 6, so I’m really not sure whether by 8 I’ll think long division is worth introducing or not, though. Personally, I don’t see the rush. It’s not all that interesting or illuminating.

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1 hour ago, Meriwether said:

Thanks. I wasn't sure what the question meant. I fully admit to often having self-doubts about how I'm doing with the kids, but I can state confidently that they have good number sense and are fully understanding any algorithms I have them use. They've all picked up long division easily and naturally because they understand what it is doing.

 

I’m sorry, I really wasn’t trying to make you doubt yourself!! I was really just asking whether it was actually long division. I’d probably teach it as splitting into piles first: like, do 40 40 40, then split up the rest. It comes to the same thing as long division, really, just more hands on. I was wondering if you meant actual long division format or not. 

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11 minutes ago, square_25 said:

 

I wasn’t implying that, but yes, I think spending a lot of time with an algorithm can make the original intuitive understanding harder to access. I’d definitely expect them to do multidigit divisions by hand for quite a while before doing algorithms, but I’m not sold on algorithms as a way to learn math in the first place. But I’m probably too algorithm-phobic :-).

My daughter is only 6, so I’m really not sure whether by 8 I’ll think long division is worth introducing or not, though. Personally, I don’t see the rush. It’s not all that interesting or illuminating.

And I think the bolded gets to the point. Why is it rushing if a child is ready?  Just b/c you might think it is rushing does not mean it is. I can be as simply as the next logical progression in developing mathematical skills.

I did have a chuckle, though, when I read the suggestion that learning long division at 8 is rushing bc it isn't interesting or illuminating.  The last time I was accused of rushing a student was at a college visit when an UG adviser accused my ds of rushing through physics without slowing down to understand what the study of physics actually is.  He couldn't possibly understand physics at that level. 

Slower progression does not equate to laudable anymore than advanced equates to deficit.  There is just too much individual variance.

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31 minutes ago, square_25 said:

Would they be using long division for 144/3?

I think that would be great problem to solve when a child is first learning the algorithm because there are so many other ways to double check their solution.

Obviously, by the time I introduce the algorithm, the child already has a firm grasp of conceptual division, and has a lot of experience physically dividing manipulatives into piles, and fully understands division by partial products.

Then I move to problems like 846 / 2 where they just have to divide each place value evenly.

Then, for us, the next step is 144 / 3.

Obviously, I could withhold the algorithm and continue to make them work out more complex division problems via other methods...but why?  By that point, they have already figured out the correct procedure, so all I am really teaching is the written convention.  And while if forced to choose, I would prioritize conceptual understanding over the algorithm, it is not an either / or choice, and my ideal is a student who BOTH has strong number sense and conceptual understanding AND can efficiently and correctly use the algorithm as a short cut.

I don't see the algorithm as an absolute necessity, but certainly as beneficial.  Knowing that algorithm is going to make prime factorization way less tedious.  And having a lot of experience with prime factors makes LCM and GCF intuitive.  And LCM leads nicely into adding fractions with unlike denominators.  All of those are topics that we are going to naturally reach in ~third grade, so having had 6 months to get comfortable with the long division algorithm will be a huge asset for the student.

Wendy

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6 minutes ago, 8FillTheHeart said:

And I think the bolded gets to the point. Why is it rushing if a child is ready?  Just b/c you might think it is rushing does not mean it is. I can be as simply as the next logical progression in developing mathematical skills.

I did have a chuckle, though, when I read the suggestion that learning long division at 8 is rushing bc it isn't interesting or illuminating.  The last time I was accused of rushing a student was at a college visit when an UG adviser accused my ds of rushing through physics without slowing down to understand what the study of physics actually is.  He couldn't possibly understand physics at that level. 

Slower progression does not equate to laudable anymore than advanced equates to deficit.  There is just too much individual variance.

  

It’s not how I’d choose to progress at that stage given my experience with older kids is all. I don’t think it’s an important skill for later understanding, although it’s a neat computational trick.

I’m sure I could teach it to my daughter and she’d understand the reason it works. However, most algebraic properties don’t stem from the algorithm so I’d rather spend more time with the definition. It’s all an opinion and an experiment.

Hah, as for progressing myself, I was the idiot who decided to take third year abstract algebra as soon as I got to college and who graduated in 3 years. I was ready in the sense that I hit high grades and got into a good Ph.D program. Was it actually a good idea in retrospect? No.

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I think it depends on the math program the school is using.  It is early if the school follows that traditional method of addition in first, subtraction in second, etc.  It is not that early, especially for an advanced child if the school takes a more integrated maths approach.

My children were introduced to the concept of long division by the end of second grade.  They were not necessarily expected to use it, but it was a tool they colud use. Their math program had them solve problems that were set up like long division but worked doing repeated subtraction.   Using 144/3 - subtract 10 3s, 114 remains.  Subtract 10 more 3s, 84 remains.  Subtract 10 more 3s 54 remain.  Subtract 10 3 again, 24 remain.  There are 8 3s in 24.  4 10s plus 8 equals 48.  144/3=48.  When the child realizes that he can subtract larger numbers than 3x10,  there are 40 3s in 120, subtract 120 from 144 to get 24, 8 3s in 24, 40+8=48.  Then explain that the algorithm is a shorthand way of doing the later.

 

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2 minutes ago, square_25 said:

  

It’s not how I’d choose to progress at that stage given my experience with older kids is all. I don’t think it’s an important skill for later understanding, although it’s a neat computational trick.

I’m sure I could teach it to my daughter and she’d understand the reason it works. However, most algebraic properties don’t stem from the algorithm so I’d rather spend more time with the definition. It’s all an opinion and an experiment.

Hah, as for progressing myself, I was the idiot who decided to take third year abstract algebra as soon as I got to college and who graduated in 3 years. I was ready in the sense that I hit high grades and got into a good Ph.D program. Was it actually a good idea in retrospect? No.

You completely miss the point.  That WAS delving deep and not skimping on theoretical understanding.

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3 minutes ago, 8FillTheHeart said:

You completely miss the point.  That WAS delving deep and not skimping on theoretical understanding.

Not sure what you mean, sorry. What was?

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3 minutes ago, square_25 said:

Not sure what you mean, sorry. What was?

My ds's physics progression that had him graduating from high school at such a high level that the UG advisor could not accept that he was actually at that level.   (But, hey, he took alg at 10, too.....equally not rushing.)

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1 minute ago, 8FillTheHeart said:

My ds's physics progression that had him graduating from high school at such a high level that the UG advisor could not accept that he was actually at that level.   (But, hey, he took alg at 10, too.....equally not rushing.)

 

I've had experience with this, too. My husband was held back at school for no reason at all because he couldn't possibly be at the level he is actually at. I've seen this play out in public school a number of times now as well, like when they didn't let my daughter read the books she was actually reading. It's super frustrating and ridiculous, I agree. I'm sorry if I implied your DS's situation was analogous to mine, I just got reminded of it. But it sounds totally different. 

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5 minutes ago, HomeAgain said:

I'm not sure why long division is considered nothing more than a computational trick.

 

Sorry, I misunderstood. 

What is it if it's not a computational trick? Understanding it requires understanding base 10 but using it does not. 

Edited by square_25

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16 minutes ago, wendyroo said:

Obviously, by the time I introduce the algorithm, the child already has a firm grasp of conceptual division, and has a lot of experience physically dividing manipulatives into piles, and fully understands division by partial products.


Personally, I would rather milk the definition for all of its properties before doing the algorithm. I'd want them to know it's the opposite of multiplication and be fluent with that, I'd want them to know that 120/3 is both splitting into 3s and splitting into 3 piles and be able to explain why those are the same thing, I'd want them to do things like 120/(120/3) very fluently, and I haven't even thought about whether it would make sense to do division by fractions first or not, to make the analogies more immediate or not. 

I'm sure I'll teach it at some point. It's a cool trick and makes multidigit division faster. I certainly wouldn't expect my daughter to do multidigit division quickly until she learns it. 

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5 minutes ago, square_25 said:

 

Sorry, I misunderstood. 

What is it if it's not a computational trick? Understanding it requires understanding base 10 but using it does not. 

Do you make the same argument about all basic operations?  I'm being serious here.  Why is division simply a trick?

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Just now, HomeAgain said:

Do you make the same argument about all basic operations?  I'm being serious here.  Why is division simply a trick?

 

It's not, just the long division algorithm is. Yes, I think of all the standard algorithms (addition with carrying, subtracting with borrowing, multiplication by stacking) as computational tricks, but I don't think the definitions of the operations as computational tricks. I'm not against computational tricks but they aren't a priority for how I'm choosing to teach math at the moment. 

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5 minutes ago, square_25 said:


Personally, I would rather milk the definition for all of its properties before doing the algorithm. I'd want them to know it's the opposite of multiplication and be fluent with that, I'd want them to know that 120/3 is both splitting into 3s and splitting into 3 piles and be able to explain why those are the same thing, I'd want them to do things like 120/(120/3) very fluently, and I haven't even thought about whether it would make sense to do division by fractions first or not, to make the analogies more immediate or not. 

I'm sure I'll teach it at some point. It's a cool trick and makes multidigit division faster. I certainly wouldn't expect my daughter to do multidigit division quickly until she learns it. 

I guess I wonder why on earth you think that students learning long division DON"T know that it is "the opposite of multiplication and be fluent with that" or that is splitting into groups of and why that is the same thing, etc.

 

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6 minutes ago, 8FillTheHeart said:

I guess I wonder why on earth you think that students learning long division DON"T know that it is "the opposite of multiplication and be fluent with that" or that is splitting into groups of and why that is the same thing, etc.

 

I assume some do and some don't? Certainly I've made the mistake of teaching algorithms and formulas far too early in my teaching career. It's about the most common teaching mistake I see. 

I know my daughter isn't perfectly fluent with these things despite being mathy. She'd understand them if I explained them to her and she uses some of them better than she did when I just told her about division but it has surprised me how long it has taken her to get complete intuition about it all. I'm pretty sure if I gave her 120/(120/3), for example, she'd do 120/3 on the bottom first, then do 120/40. This isn't wrong, obviously, but I'd want to talk to her about why this isn't the most efficient thing and I feel like playing around with the definition leads to growth of intuition in a way that algorithms do not. 

Now, my daughter is 6, and I assume it takes older kids less time to have sufficient intuition to have the algorithm make sense. All I know is that my sense of how long it took kids to "absorb" an intuition had been flawed in the past.  It's possible it's now also flawed in the wrong direction. Maybe I'll wind up in the middle somewhere :-). 

Edited by square_25

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8 minutes ago, 8FillTheHeart said:

I guess I wonder why on earth you think that students learning long division DON"T know that it is "the opposite of multiplication and be fluent with that" or that is splitting into groups of and why that is the same thing, etc.

 

Here's a question: what exactly is the benefit of learning long division? Why is that the natural next step after learning what division is and practicing it conceptually? I know that's the standard sequence and it helps with larger calculations, but why is that where you go next? 

Edited by square_25

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11 minutes ago, square_25 said:

 

Here's a question: what exactly is the benefit of learning long division? Why is that the natural next step after learning what division is and practicing it conceptually? I know that's the standard sequence and it helps with larger calculations, but why is that where you go next? 

Well in our case I taught it next because that is what DS commonly sees in tests, math books and computer programs, math videos, pretty much everything math. Its a long accepted method so if he's going to be able to understand what he sees he needs to learn it.

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1 minute ago, Sarah0000 said:

Well in our case I taught it next because that is what DS commonly sees in tests, math books and computer programs, math videos, pretty much everything math. Its a long accepted method so if he's going to be able to understand what he sees he needs to learn it.

 

Yeah, that makes sense to me. If we had to do standardized testing every year, say, I'd have to teach it on target as well. Since I'm making things up as I go along and I have issues with the standard sequence, I've decided to see what happens if I don't for now.

But then I have a pretty good idea of what I want our math program to look like for the next year or so and none of it requires long division, per se. If I were using outside materials more it would probably be different. 

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26 minutes ago, square_25 said:

 

It's not, just the long division algorithm is. Yes, I think of all the standard algorithms (addition with carrying, subtracting with borrowing, multiplication by stacking) as computational tricks, but I don't think the definitions of the operations as computational tricks. I'm not against computational tricks but they aren't a priority for how I'm choosing to teach math at the moment. 

You have lost me completely and entirely.  I look at how my kids were taught division and how it was imprinted on their minds and nothing you say makes sense.  I would encourage you to look deep into elementary mathematics and see all that is out there and how it compares with your perception.  I know you have a background in higher math, but it's really the beauty of these foundational skills that make the rest shine.  Kids are taught the understanding, the manipulation, and how to intuit the algorithm to make the rest of division (like groups of over 10 and fractions) easier.  They are not thrown a procedure and told to do it no matter what.

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10 minutes ago, HomeAgain said:

You have lost me completely and entirely.  I look at how my kids were taught division and how it was imprinted on their minds and nothing you say makes sense.  I would encourage you to look deep into elementary mathematics and see all that is out there and how it compares with your perception.  I know you have a background in higher math, but it's really the beauty of these foundational skills that make the rest shine.  Kids are taught the understanding, the manipulation, and how to intuit the algorithm to make the rest of division (like groups of over 10 and fractions) easier.  They are not thrown a procedure and told to do it no matter what.

 

I'm not sure what to tell you. I have a lot of experience of what kids bring from their elementary years into the higher years. I don't think the foundational skills that prepare them for higher math are these algorithms. The understanding behind these algorithms, yes, that is incredibly important. But the whole point of an algorithm is to be able to use it without thinking about it. I'd go insane if every time I had to divide by a fraction I had to think about exactly why (a/b)/(c/d) = ad/bc, say. Again, that's not during learning it, that's after you've learned it. Do you disagree with that? 

Edited by square_25

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8 minutes ago, square_25 said:

 

Here's a question: what exactly is the benefit of learning long division? Why is that the natural next step after learning what division is and practicing it conceptually? I know that's the standard sequence and it helps with larger calculations, but why is that where you go next? 

I feel like this conversation is becoming rather pointless.   Learning estimations.  Real world problems. Understanding the relationship between divisors, remainders and fractions, repeating decimals...….

I guess the same question could be asked, why not?  If a student has a firm understanding of multiplication, regrouping, and simple division, why not learn long division?  

FWIW, just b/c you have witnessed gaps in understanding in upper level students does not mean that the flaw is in the sequence in what they learned.  If these are mostly classroom kids, it could be that they progressed without mastery of understanding and subsequently continued to build on the vague understanding of what they were doing. 

FWIW, at 6, no, I would not see long division as the next logical step in math.  But, what students can master in a single yr can be quite a lot.  My ds at 6 had just started formal education b/c I do not teach preschool academics at all.  He taught himself the concept of multiplication through playing with Legos and observing groups of things in nature/construction/and cookies on a pan. 😉 He equally understood fractions the same way.  When he was younger, I had no idea that his math progress would be what it was.

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3 minutes ago, square_25 said:

 

I'm not sure what to tell you. I have a lot of experience of what kids bring from their elementary years into the higher years. I don't think the foundational skills that prepare them for higher math are these algorithms. The understanding behind these algorithms, yes, that is incredibly important. But the whole point of an algorithm is to be able to use it without thinking about it. I'd go insane if every time I had to divide by a fraction I had to think about exactly why (a/b)/(c/d) = ad/bc, say. At some point, you use an algorithm without thinking about and that's the whole point. Again, that's not during learning it, that's after you've learned it. Do you disagree with that? 

So what you're then saying, is that teaching the algorithm is important so that students can use it without thinking about it and exactly why it works - they've already spent time understanding exactly why it works and can now use the algorithm quickly and efficiently.


Which is what I'm saying about how long division is taught when it is done well.  Of course the students should be taught how and why it works, and given practice in it.  That is a solid foundation.

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40 minutes ago, square_25 said:

Now, my daughter is 6, and I assume it takes older kids less time to have sufficient intuition to have the algorithm make sense. All I know is that my sense of how long it took kids to "absorb" an intuition had been flawed in the past.  It's possible it's now also flawed in the wrong direction. Maybe I'll wind up in the middle somewhere :-). 

I think there is a false dichotomy inherent in the bolded idea.  On one hand, you have a 6 year old who has played around with the ideas a bit, but doesn't yet have the intuition.  On the other hand, you are supposing that if an 8 year old played around with the ideas that they would develop intuition more quickly.

But, what about the 8 year old who was that 6 year.  It a child is playing around with the ideas at 6, and continues for the next two years to grapple with them in more and more complex ways, then they haven't developed the intuition quickly, but they have developed it thoroughly.

As for why I teach the algorithm, mostly because my kids want to know it.  They love math; they explore it and use it and practice it recreationally.  They choose to read so many math books, and watch so many math YouTube videos, that it would actually be very hard to avoid them knowing about the algorithm.  And I'm not sure why I would want to hide it from them.  Once I am absolutely sure that they are rock solid on the concepts underlying the algorithm, then I want them to master it and start using it so that they can then devote their time and attention to higher level concepts rather than wasting their time reinventing the wheel every time they need to solve a non-intuitive multi-digit division.

Wendy

Edited by wendyroo
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My kids do fractions and long division concurrently.

 

As to why I would introduce an algorithm (and please read this as friendly discussion, not argument) - I have five kids. I spend a good 5 hours on math each school day. Once Dd9 understands something, a "trick" is fine. I need time to play with math manipulatives with my 4 year old and work through Geometry and Algebra with my 14 and 12 year olds. I don't spend much time with my 15 yo, because she is mostly self-taught.

I've got a kid who does band at the school an average of 4 times per week, a kid who volunteers one morning per week, two kids with piano lessons on morning, and three kids with play practice one afternoon. Plus, my Dd15 does odd jobs occasionally during the school day and is going to regularly start giving piano lessons next week. I am the only driver.  My school day usually starts at 7:00 and ends at 5:00, because I need that much time to read aloud, teach, discuss, and DRIVE! Once a kid truly understands something, we do whatever is fastest and easiest. Dd9 hasn't worked sheets and sheets of long division problems. We do a few long division  lessons (with about 8 problems each) and then review them a problem or two at a time. Then she uses them for story problems or whatever. She knows very well when dividing 144/3 that the 4 she puts on the top means 40 and the 12 she subtracts is 120, so I flat out don't have time to do manipulatives with her. The algorithm is fast and easy.

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2 minutes ago, wendyroo said:

I think there is a false dichotomy inherent in the bolded idea.  On one hand, you have a 6 year old who has played around with the ideas a bit, but doesn't yet have the intuition.  On the other hand, you are supposing that if an 8 year old played around with the ideas that they would develop intuition more quickly.

But, what about the 8 year old who was that 6 year.  It a child is playing around with the ideas at 6, and continues for the next two years to grapple with them in more and more complex ways, then they haven't developed the intuition quickly, but they have developed it thoroughly.

As for why I teach the algorithm, mostly because my kids want to know it.  They love math; they explore it and use it and practice it recreationally.  They choose to read so many math books, and watch so many math YouTube videos, that it would actually be very hard to avoid them knowing about the algorithm.  And I'm not sure why I would want to hide it from them.  Once I am absolutely sure that they are rock solid on the concepts underlying the algorithm, then I want them to master it and start using it so that they can then devote their time and attention to higher level concepts rather than wasting their time reinventing the wheel every time they need to solve a non-intuitive multi-digit multiplication.

Wendy

To be fair, she's been able to divide conceptually for maybe a year now. But yeah, I'm sure her level of abstraction will skyrocket by age 8, and I'm not going to be particularly surprised if we do learn it by then. I don't think your kids are average 8 year olds, though, and she's also very proficient for her age. 

My daughter is mathy in the sense that it's easy for her but she's not into it. What she's really into is reading and building. I wouldn't be at all surprised if she winds up in some sort of science (or even math) but she isn't fascinated by it the way your kids are. I'm a bit jealous, honestly: I was a very puzzle-oriented kid. Although apparently my hubby wasn't and he wound as a mathematician, so who knows? :P 

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