I love Saxon but my child says she hates math....

[lewelma]

1 minute ago, Not_a_Number said:

If not done carefully, it can be worse than doing nothing.

Agreed. If your methods/teaching lead to hatred of maths, you will have a very long hard road to turn it around. And kids that hate math will never put effort into it. At that point math is done *to* them -- they will not engage.

[Shoeless]

8 minutes ago, lewelma said:

LOL. We would estimate the number of cars, and then count them.  If there were like 400 cars, he very quickly realized why estimating was so useful.  Then, I would show him how to dive the cars into 10 groups, count only the cars in one group, and multiply by 10.  It was clearly a much faster way to see if your estimate was right. I could then bring in the idea of things that are moving, like birds, where you could never actually count them before they would fly off, so estimating them was the only way. At that point we could talk about estimates in time. How many trucks go by here in a day? Well, we weren't going to sit there all day, so I showed him how to count for 5 minutes, and then multiply. Then we could discuss how accurate the estimate would be. That there could be fewer trucks at different times of the day.  That estimates would never be right, but that they were useful.  That maths in general was a way of understanding the world. We walked all the time, in both the city and the woods, and we estimated something almost everyday. There is no way that a 1st grade curriculum could ever be as good. We *lived* maths. We took joy in it. It was a shared endeavour. 

DS12 simply wasn't having it with estimation. At all. lol. He felt like it was the lazy approach, almost like a defect in one's character to estimate rather than do the work and count everything out! 😂

We also had a battle of wills where he was determined to prove to me that the distributive property was utterly useless 😂 

In hindsight, the issue was that he wanted to do harder work, but did not have the maturity to work through harder problems. So he took easier problems and made them unnecessarily hard to challenge himself. He's come around now, and happily estimates and distributes, lol. 

[lewelma]

11 minutes ago, MissLemon said:

So he took easier problems and made them unnecessarily hard to challenge himself.

I still get this from my younger! I kind of just wish he would do the work. But it is sense making for him so critical for owning the content. I've worked hard to change my attitude/approach and embrace his 'investigations'. This is why it is taking us 2 years to get through calculus. So. Many. Investigations!

[lewelma]

13 minutes ago, MissLemon said:

DS12 simply wasn't having it with estimation. At all. lol. He felt like it was the lazy approach, almost like a defect in one's character to estimate rather than do the work and count everything out! 😂

 

And I do hear this. It is one of the reasons that I went after estimating in such a large way when my kids were young. As a statistician, I know that we never really know the answer. I wanted my kids to see math as an endeavour to solve real life questions rather than as a problem with only one correct answer. 

If I give you the measurements of an ice cream tub, and the radius of the ice cream scoop, can you tell me how many scoops can be gotten out of the tub?

The answer is NO. It depends on how big a scoop a person makes, and if they have that hole in the middle of the scoop, and how much ice cream is left in the little corners, and how many different people with different scooping styles are using the tub. And to go even deeper, it depends on the parameters of the distribution that the company uses to fill the ice cream container because the machines are not exact so there is never the exact same amount in each tub. 

Maths is only an estimate of the answer to a real life problem.  

 

[lewelma]

I do find it interesting that Not-a-Number and I have very different approaches based on our own background. As a PhD mathematician, she focuses on the deep theoretical concepts that underpin maths. And as a PhD statistical modeller, I focus on using maths in real life and how there is no truth. We each do both, but we definitely have different focuses. Fascinating, really. 

[Shoeless]

39 minutes ago, lewelma said:

I still get this from my younger! I kind of just wish he would do the work. But it is sense making for him so critical for owning the content. I've worked hard to change my attitude/approach and embrace his 'investigations'. This is why it is taking us 2 years to get through calculus. So. Many. Investigations!

There are a lot of investigations over here, as well. It is taking us longer than expected to work through some material, but I can see that he's engaged very deeply with what he is learning, and that is far, far more important to me than being on or above grade level.  And when he's ready to move on? Boom! Off like a rocket, and I struggle to keep up with him!  DS12 has never tracked neatly with "grade level" or even with the "gifted" kids, which was why we took him out of school. He needed something tailor made for him, and we weren't going to find that at any b & m institution here. 

[lewelma]

42 minutes ago, MissLemon said:

There are a lot of investigations over here, as well. It is taking us longer than expected to work through some material, but I can see that he's engaged very deeply with what he is learning, and that is far, far more important to me than being on or above grade level.  And when he's ready to move on? Boom! Off like a rocket, and I struggle to keep up with him!  DS12 has never tracked neatly with "grade level" or even with the "gifted" kids, which was why we took him out of school. He needed something tailor made for him, and we weren't going to find that at any b & m institution here. 

My kids are the same. My older took 3 full years for AoPS intro Algebra, and took off like a rocket.

My younger I stalled for 2.5 years in PreA because he needed to really master the idea of linear, logical work which was a massive problem for him due to his dysgraphia. We did 5 PreA programs during those years -- Singpore 7/8, AoPS preA (first few chapters), Mathematics a Human Endeavour, MEP for the puzzlers, and Jacob Algebra (the first few chapters). My goal was to do math every day but to keep him excited and eager for more. He tired quickly with a single program, so I was constantly shaking it up. I did not just let him march forward because that was the standard approach. I knew that the lack of logical linear thinking would be the death of math for him in highschool, so I spent the time to shore it up. 

Edited June 15, 2021 by lewelma

[lewelma]

The delay in his math maturity was also caused by his synesthesia, which I didn't know he had until he was 15, but that in hindsight caused *huge* problems for his understanding in math. He has grapheme colour synesthesia so that different numbers are different colours. He kept trying to figure out patterns with the math we were doing and the colours he saw. I would get so cross with him for just sitting there and thinking. And after all that time still not understanding what we were doing. I had no idea he was trying to organize the impossible into some sort of logical link. In addition, his ordinal linguistic personification meant that he had a visceral reaction to negative numbers because he perceived them as evil, and he struggled for a very long time to allow a negative times a positive to equal a negative, because it seemed to him that the evil numbers always won.  He would fight me and argue about these rules and I could not figure out why.  It also had a massive impact on his ability to understand graphing in math, with axes that went from positive to negative because there was a color differentiation overlaid with the personification. PreA was just a b**ch for him.

This just shows that you really really need to not just march forward in some sort of progression laid out by textbook publishers. You need to teach *the child* NOT the curriculum. 

Edited June 15, 2021 by lewelma

[Shoeless]

1 minute ago, lewelma said:

My kids are the same. My older took 3 full years for AoPS intro Algebra, and took off like a rocket.

My younger I stalled for 2.5 years in PreA because he needed to really master the idea of linear, logical work which was a massive problem for him due to his dysgraphia. We did 5 PreA programs during those years -- Singpore 7/8, AoPS preA (first few chapters), Mathematics a Human Endeavour, MEP for the puzzlers, and Jacob Algebra (the first few chapters). My goal was to do math every day but to keep him excited and eager for more. He tired quickly with a single program, so I was constantly shaking it up. I did not just let him march forward because that was the standard approach. I knew that the lack of logical linear thinking would be the death of math for him in highschool, so I spent the time to shore it up. 

Ha, funny enough, I was just flipping through an on-line copy of "Mathematics a Human Endeavor" to decide if I wanted to work through some of it with DS12 or if he would like to read it on his own (I suspect he'd prefer to read it by himself). We fiddled around with AOPS pre-A for a few chapters, and that was a slog. Then we did a bit of Math Mammoth 7, and that was not right, either. I dropped that, and just had him work through puzzles in the Math Perplexors series for awhile.  Now we're in the Jousting Armadillos series, and it is "Just Right", but he's zipping through it quickly, (hurry up and ship the rest of my books, Arbor Mathematics!).  The material is a little easy in the JA books, but he's excited to read them and he's learning how to document his work in a way that others can read.  I could not get him to write a thing down for AOPS pre-A; he had all these little scribbled notes all over the place, so I had no way of knowing whether he was getting any of it right or wrong. He was determined to *not* keep an orderly notebook of his work because I-don't-know. 

Today, he announced with gravitas "I am going to keep this notebook for writing assignments and this one for math. It's better if they are in separate books".  Oh, do tell! 🙄

[Shoeless]

7 minutes ago, lewelma said:

The delay in his math maturity was also caused by his synesthesia, which I didn't know he had until he was 15, but that in hindsight caused *huge* problems for his understanding in math. He has grapheme colour synesthesia so that different numbers are different colours. He kept trying to figure out patterns with the math we were doing and the colours he saw. I would get so cross with him for just sitting there and thinking. And after all that time still not understanding what we were doing. I had no idea he was trying to organize the impossible into some sort of logical link. In addition, his ordinal linguistic personification meant that he had a visceral reaction to negative numbers because he perceived them as evil, and he struggled for a very long time to allow a negative times a positive to equal a negative, because it seemed to him that the evil numbers always won.  He would fight me and argue about these rules and I could not figure out why.  

This just shows that you really really need to not just march forward in some sort of progression laid out by textbook publishers. You need to teach *the child* NOT the curriculum. 

Interesting! DS12 got on a kick for awhile where he was trying to assign Dungeons and Dragons alignments to different numbers and groups of numbers. I don't remember exactly how he sorted all of them out, but I think he decided that 0 was chaotic neutral and negative numbers were lawful evil, lol. 

I'll have to ask him if he ever thinks of numbers in terms of colors. 

[Not_a_Number]

2 hours ago, lewelma said:

As for place value, the best time to introduce it IMHO is when the kid asks about it or when it is a needed tool. This allows the brain to absorb the idea. Slogging through exercises (like with a 1st grade scripted curriculum) will not be nearly as effective as working with the concept when the child needs it to accomplish something they want to accomplish. 

I don't really agree with that. And it's a needed tool basically whenever you do ANY arithmetic above 10 or 20, whether a kid realizes it or not.

I don't think it's necessary to wait for a kid to ask about things, because they might not know what to ask for. I think it's important to make math interesting, relatable, and comprehensible, but I don't think it needs to be discovered. 

Edited June 15, 2021 by Not_a_Number

[Not_a_Number]

1 hour ago, lewelma said:

I do find it interesting that Not-a-Number and I have very different approaches based on our own background. As a PhD mathematician, she focuses on the deep theoretical concepts that underpin maths. And as a PhD statistical modeller, I focus on using maths in real life and how there is no truth. We each do both, but we definitely have different focuses. Fascinating, really. 

I'm technically a Ph.D probabilist 😉. I don't think I focus on the theoretical concepts because I'm a mathematician -- I focus on them because I've seen the inability to work with the concepts trip countless kids up. You only need to reteach so many kids things they were supposed to have learned earlier before you realize that having gaps is a disaster. 

I do lots of math in real life with the kids, although I'll admit to not starting with estimation -- I think questions with exact answers fit kids' early mental models better. But we do estimate eventually when we need to. 

However, my personal motto is to make words and pictures out of math instead of making math out of words and pictures. At least, that's where I start. I find working forwards much easier than working backwards. 

[lewelma]

24 minutes ago, Not_a_Number said:

I don't really agree with that. And it's a needed tool basically whenever you do ANY arithmetic above 10 or 20, whether a kid realizes it or not.

I don't think it's necessary to wait for a kid to ask about things, because they might not know what to ask for. I think it's important to make math interesting, relatable, and comprehensible, but I don't think it needs to be discovered. 

Yup. I completely agree. I just would not introduce place value in an abstracted way. I would link it to something they know. I keep kids in the concrete stage for as long as needed. I also didn't mean they had to *ask*, just that it needed to be related to things they were doing and interacting with. But keep in mind that my experience of teaching place value is quite limited. My older already knew everything by the time I tried to teach him, and my younger perceived of math through is dysgraphia and synethesia so I had to adapt all teaching to this very unusual combination.  So I will bow out of all place value conversations!  🙂 

[lewelma]

26 minutes ago, Not_a_Number said:

I'm technically a Ph.D probabilist 😉. I don't think I focus on the theoretical concepts because I'm a mathematician -- I focus on them because I've seen the inability to work with the concepts trip countless kids up. You only need to reteach so many kids things they were supposed to have learned earlier before you realize that having gaps is a disaster. 

I do lots of math in real life with the kids, although I'll admit to not starting with estimation -- I think questions with exact answers fit kids' early mental models better. But we do estimate eventually when we need to. 

However, my personal motto is to make words and pictures out of math instead of making math out of words and pictures. At least, that's where I start. I find working forwards much easier than working backwards. 

What we really need to do is video ourselves teaching, share our videos, and then have a conversation. I bet we are actually pretty similar. 🙂 

[Not_a_Number]

2 minutes ago, lewelma said:

What we really need to do is video ourselves teaching, share our videos, and then have a conversation. I bet we are actually pretty similar. 🙂 

Yes, that would be interesting 🙂 . I would guess we're locally similar (we make good sense of the math) and we sequence the learning differently. Mostly because I sequence it differently from everyone else I know. 

[regentrude]

20 hours ago, lewelma said:

I do find it interesting that Not-a-Number and I have very different approaches based on our own background. As a PhD mathematician, she focuses on the deep theoretical concepts that underpin maths. And as a PhD statistical modeller, I focus on using maths in real life and how there is no truth. We each do both, but we definitely have different focuses. Fascinating, really. 

And as a PhD physicist and prof at an engineering school, I probably have yet another different approach to math 🙂

ETA: I really dislike the term " in real life". The time dependency of a current in a discharging defibrillator is "real life", too.

Edited June 16, 2021 by regentrude

[Not_a_Number]

45 minutes ago, regentrude said:

And as a PhD physicist and prof at an engineering school, I probably have yet another different approach to math 🙂

ETA: I really dislike the term " in real life". The time dependency of a current in a discharging defibrillator is "real life", too.

I agree there. But it’s not accessible to kids.

[daijobu]

On 6/15/2021 at 3:59 PM, lewelma said:

The delay in his math maturity was also caused by his synesthesia, which I didn't know he had until he was 15, but that in hindsight caused *huge* problems for his understanding in math. He has grapheme colour synesthesia so that different numbers are different colours.

Wait wait wait.  This is the MIT kid?  How does one overcome something like this?  Is it just a matter of telling him that negative numbers aren't actually evil, or is some sort of therapy required?  Is there anyway that this condition is helpful to a math student or is it always a disability?  

[bookbard]

@lewelma that is so interesting re synaesthesia and your son. I have very intense synaesthesia - grapheme-colour, numbers with personalities, time with a certain shape, music-colour. I really struggled with maths, right from primary school, even though I found the ideas interesting and used to read about maths for pleasure. I do wonder if I was always trying to pattern and order things in my own way. I started my kids on maths early as I was worried they too would struggle, but so far they've done well - my daughter definitely has synaesthesia but not sure about my son. 

[lewelma]

3 minutes ago, daijobu said:

Wait wait wait.  This is the MIT kid?  How does one overcome something like this?  Is it just a matter of telling him that negative numbers aren't actually evil, or is some sort of therapy required?  Is there anyway that this condition is helpful to a math student or is it always a disability?  

No, not the MIT kid. It is my younger who has the same level of mathematical intuition but with these overlaying issues has found maths to be difficult to work with . I actually didn't know about it until he was 15 by which point most of the difficulty was past. He never told me because he didn't know it was unusual. I had no idea why he would fight so hard, or refuse the do assignments. Or why he would work so hard to understand something fairly easy for his innate skill - because he was trying to find a correlation between the colours and the math processes. He has overcome the impact of both the dysgraphia and synesthesia on his math through one on one tutoring an hour a day by me, a math tutor. He has never yet worked independently. Only through such intensive remediation has he gotten top marks on the 12th grade NZ national exams. I have looked for and not found an upside to the synthesia or dysgraphia in math.

[daijobu]

On 6/15/2021 at 4:47 PM, lewelma said:

What we really need to do is video ourselves teaching, share our videos, and then have a conversation. I bet we are actually pretty similar. 🙂 

I would totally watch these videos.  

[daijobu]

32 minutes ago, lewelma said:

No, not the MIT kid. It is my younger who has the same level of mathematical intuition but with these overlaying issues has found maths to be difficult to work with . I actually didn't know about it until he was 15 by which point most of the difficulty was past. He never told me because he didn't know it was unusual. 

Thank you for disabusing me of the idea of a learning difference also being a superpower.  

I feel like there's a whole subset of learning differences and other conditions that could be screened with a single question.  Like "Do the numbers appear to have colors?"  or something similar.  You could compile a list of questions for a variety of learning differences and just ask them of even asymptomatic children, just to screen for undiagnosed learning differences and save everyone a lot of pain with little cost.  

[ieta_cassiopeia]

On 6/13/2021 at 7:42 PM, Hunter said:

But I was there long enough to learn that the USA needs more STEM soldiers to continue on the path they have been traveling. It is getting critical, especially since the pandemic. If the USA wants to draft from the lower classes, they are going to have to reduce the financial abuse being inflicted upon the garden they are trying to harvest. That is not one of the ideas being discussed. Everyone just keeps arguing about curriculum and increase the hours of study from children. It is not going to work, inside and outside the public schools.

This is true. People who cannot afford to learn - or who are obliged to be in places which cannot afford to teach - is incompatible with college/university, let alone a STEM career.  Considering the number of STEM people I know who are having trouble getting jobs, perhaps issuing more specific and helpful careers advice (to the government as well as to potential employees) wouldn't hurt either. After all, there are degrees in many fields of science, technology, engineering and mathematics, and there's demand for graduates with specific versions of these degrees, but I've never seen a degree in STEM universally accepted by employers who want "STEM" graduates...

[Not_a_Number]

9 hours ago, lewelma said:

I have looked for and not found an upside to the synthesia or dysgraphia in math.

I have pretty strong number to color associations, actually, but it's never gotten in my way. (3 is red and 4 is green. OBVIOUSLY.) Your son's sounds like it's something much more specific! 

Edited June 17, 2021 by Not_a_Number

[vonfirmath]

On 6/13/2021 at 11:39 AM, Not_a_Number said:

I'm not. But then I do think kids have very different takes on what they are learning and what they find hard. 

DD8 would come home from kindergarten saying "math is hard" because she got tired of drawing every single addition question, including the ones she could do in her head. Kids don't always really know how to describe what's bugging them perfectly. But in my opinion, 1st grade math oughtn't put a kid off of math. If that's happening, I'd drill down and figure out what was going on and WHY they weren't happy with the program. 

I've also heard that Saxon isn't supposed to be accelerated, which is perhaps why I'm suggesting a switch. If a kid hates things taking forever, Saxon seems like it wouldn't work in the long term. 

They did that "drawing how many there are" in my kids' math courses (public school) as well. Neither kid loved it. The teachers were very lenient, allowing circles and then just hatch marks -- as long as it was very clear which went to which item in the problem (My daughter liked using colors to delineate). There were only a few such problems in each lesson (2 or 3) and they managed to move along to NOT having to draw pictures before my kids rebelled.

[vonfirmath]

On 6/15/2021 at 9:11 AM, Not_a_Number said:

I like to do serious place value as early as possible, but I don’t know that any books out there do, lol. 

I don't remember if my kids workbooks did it in first grade. But in second grade and every year afterward (through fifth at least -- after which the workbooks stopped coming home at the end of the year) They would have periodic assignments where you needed to write numbers (in whatever level they were currently working on) in different ways.

 

So if the number was 323 they were looking for

Three hundred twenty-three

300+20+3

3 Hundreds, 2 tens, 3 ones

 

Then when they moved to thousands: 4,236

Four thousand, two hundred thirty-size

4000 + 200 + 30 + 6

4 thousands, 2 hundreds, 3 tens, 6 ones

 

And yes they threw in 0s.  So 20,123 would be

Twenty thousand, one hundred twenty-three  (NO "and" -- "And" is for decimals)

20000 + 100 + 20 + 3

2 Ten thousands, 0 thousands, 1 hundred, 2 tens, 3 ones.

 

[vonfirmath]

On 6/15/2021 at 4:03 PM, MissLemon said:

DS12 simply wasn't having it with estimation. At all. lol. He felt like it was the lazy approach, almost like a defect in one's character to estimate rather than do the work and count everything out! 😂

We also had a battle of wills where he was determined to prove to me that the distributive property was utterly useless 😂 

In hindsight, the issue was that he wanted to do harder work, but did not have the maturity to work through harder problems. So he took easier problems and made them unnecessarily hard to challenge himself. He's come around now, and happily estimates and distributes, lol. 

My son made (still makes, really) a lot of "silly math errors" I finally convinced him that estimation was a way of finding out if the final answer he came out with made sense in terms of the problem. He could catch a portion of his math errors by recognizing the final answer he came up with was WAY off of the estimate he made originally.  And when he went to investigate why, it caused him to "see" his math errors where other methods did not work.

 

[vonfirmath]

On 6/16/2021 at 12:31 PM, regentrude said:

And as a PhD physicist and prof at an engineering school, I probably have yet another different approach to math 🙂

ETA: I really dislike the term " in real life". The time dependency of a current in a discharging defibrillator is "real life", too.

My dad had a saying growing up "Good enough for government work"

But there were other times he needed "Engineer's precision"  (Which was usually paired with "Measure twice, cut once")

 

So I guess I got a sense that sometimes an estimate was fine and other times you needed to be as exact as you could.

 

Edited June 17, 2021 by vonfirmath

[Not_a_Number]

14 minutes ago, vonfirmath said:

I don't remember if my kids workbooks did it in first grade. But in second grade and every year afterward (through fifth at least -- after which the workbooks stopped coming home at the end of the year) They would have periodic assignments where you needed to write numbers (in whatever level they were currently working on) in different ways.

 

So if the number was 323 they were looking for

Three hundred twenty-three

300+20+3

3 Hundreds, 2 tens, 3 ones

 

Then when they moved to thousands: 4,236

Four thousand, two hundred thirty-size

4000 + 200 + 30 + 6

4 thousands, 2 hundreds, 3 tens, 6 ones

 

And yes they threw in 0s.  So 20,123 would be

Twenty thousand, one hundred twenty-three  (NO "and" -- "And" is for decimals)

20000 + 100 + 20 + 3

2 Ten thousands, 0 thousands, 1 hundred, 2 tens, 3 ones.

 

Yeah, that’s not what I mean. Most kids I know can do this. They can’t use it, though. Naming us not a mental model.

[Not_a_Number]

25 minutes ago, vonfirmath said:

They did that "drawing how many there are" in my kids' math courses (public school) as well. Neither kid loved it. The teachers were very lenient, allowing circles and then just hatch marks -- as long as it was very clear which went to which item in the problem (My daughter liked using colors to delineate). There were only a few such problems in each lesson (2 or 3) and they managed to move along to NOT having to draw pictures before my kids rebelled.

They apparently did this the whole kindergarten year. I’d guess it stopped by Grade 1, but then DD8 has been doing algebra for more than a year now, so she’d have found another reason to hate school math 😛 . She’s really not easy when it comes to “boring” work — we also constantly have this issue at home, too.

[regentrude]

39 minutes ago, vonfirmath said:

They did that "drawing how many there are" in my kids' math courses (public school) as well. Neither kid loved it. The teachers were very lenient, allowing circles and then just hatch marks -- as long as it was very clear which went to which item in the problem (My daughter liked using colors to delineate). There were only a few such problems in each lesson (2 or 3) and they managed to move along to NOT having to draw pictures before my kids rebelled.

The assignment that was the straw that broke the camel's back, after which we pulled DD out to homeschool, was a worksheet in SIXTH grade ps where they had to color: There was a box with 20 mice. The instruction was " Color a fifth of the mice." When the kids asked whether they could just write the number 4, they were told, no, they had to color. It was an entire worksheet full of division problems like this one.

Elsewhere in the world, kids are solving linear equations at that age. In the USA, we color the mice. Pathetic.

 

Edited June 17, 2021 by regentrude

[vonfirmath]

16 minutes ago, Not_a_Number said:

Yeah, that’s not what I mean. Most kids I know can do this. They can’t use it, though. Naming us not a mental model.

Could you explain what you mean? I always thought this was a pretty good method of cementing place value.

 

[Not_a_Number]

17 minutes ago, vonfirmath said:

Could you explain what you mean? I always thought this was a pretty good method of cementing place value.

I haven't found that, because it doesn't really get at what place value lets you DO, which is (in Liping Ma's excellent words) "decomposing a unit of higher value." The expanded expression is only helpful if you understand that you can decompose a 5,000 into 5 thousands, and each thousand is 10 hundreds, you know? And really, the fact that 1,000 is 10*100 is actually a consequence of place value and not the definition of it, even though I see people act like they are equivalent. 

I've been working on some place value posts at my blog to explain what I mean, actually, although I've only got one so far. I've also found that people generally just mean "naming what the units were" or at most "writing out the expanded form" when they mean place value, but what I tend to mean is a serious level of comfort and facility with trading the different-valued units. 

Edited June 17, 2021 by Not_a_Number

[regentrude]

5 minutes ago, Not_a_Number said:

 And really, the fact that 1,000 is 10*100 is actually a consequence of place value and not the definition of it, even though I see people act like they are equivalent. 

Doesn't that come down to semantics?

ETA: It's not like 1,000 has been defined differently in a variety of bases before humans have decided to use the decimal system.

Edited June 17, 2021 by regentrude

[Not_a_Number]

9 minutes ago, regentrude said:

Doesn't that come down to semantics?

ETA: It's not like 1,000 has been defined differently in a variety of bases before humans have decided to use the decimal system.

No, I don’t think it comes down to semantics. You’re right that ten hundreds is a thousand no matter how we write them, but 10*100=1000 is true no matter what base we use and is easily derived using the DEFINITION of place value. 

[vonfirmath]

9 minutes ago, Not_a_Number said:

No, I don’t think it comes down to semantics. You’re right that ten hundreds is a thousand no matter how we write them, but 10*100=1000 is true no matter what base we use and is easily derived using the DEFINITION of place value. 

I do remember doing math in different bases -- but it wasn't until 6th or something (I don't remember exactly and if the kids have done it, it did not come home) I loved it because it was another form of puzzle to me. And it does help with understanding why, in the decimal system, it is "ones, tens, hundreds" etc.   Just like learning Spanish helped me learn English. But I'm not sure I'd try to teach it before the kids had basic math facts down. That comfortable knowledge of ones, tens, hundreds, etc can help understand why, in base 8, it is "ones, eights, sixty-fours..."

 

[Not_a_Number]

2 minutes ago, vonfirmath said:

I do remember doing math in different bases -- but it wasn't until 6th or something (I don't remember exactly and if the kids have done it, it did not come home) I loved it because it was another form of puzzle to me. And it does help with understanding why, in the decimal system, it is "ones, tens, hundreds" etc.   Just like learning Spanish helped me learn English. But I'm not sure I'd try to teach it before the kids had basic math facts down. That comfortable knowledge of ones, tens, hundreds, etc can help understand why, in base 8, it is "ones, eights, sixty-fours..."

Oh yes, I wouldn't teach different bases until base ten is fully integrated, although I did teach DD8 binary at age 6 🙂 . But she's a very accelerated kiddo. 

[daijobu]

9 hours ago, Not_a_Number said:

I haven't found that, because it doesn't really get at what place value lets you DO, which is (in Liping Ma's excellent words) "decomposing a unit of higher value." The expanded expression is only helpful if you understand that you can decompose a 5,000 into 5 thousands, and each thousand is 10 hundreds, you know? And really, the fact that 1,000 is 10*100 is actually a consequence of place value and not the definition of it, even though I see people act like they are equivalent. 

I've been working on some place value posts at my blog to explain what I mean, actually, although I've only got one so far. I've also found that people generally just mean "naming what the units were" or at most "writing out the expanded form" when they mean place value, but what I tend to mean is a serious level of comfort and facility with trading the different-valued units. 

With all due respect, I agree with the others.  I really have no idea what you are talking about here, or why it even matters.  

[Not_a_Number]

6 hours ago, daijobu said:

With all due respect, I agree with the others.  I really have no idea what you are talking about here, or why it even matters.  

Oh, yes, I’m aware people generally have no clue what I mean by this. It doesn’t mean that it has no meaning, though 😉 . Just that I’m failing to communicate it.

[HomeAgain]

7 hours ago, daijobu said:

With all due respect, I agree with the others.  I really have no idea what you are talking about here, or why it even matters.  

I think it's easier for adults to see when you go the other direction.

If there is a clear communication of ten of something being a new place value, then it becomes equally clear that dividing by ten moves place value as well.  All the way across, in both directions.

Which means problems like 1/10 x 1/10 x 1/10 are easily absorbed as a tenth of a tenth (one hundredth), moving to one tenth of that ( one thousandth).  It manipulates numbers more easily in the mind and avoids the pitfalls of trying to teach decimals by "count the decimal places in the problem, move your decimal point that far over in the answer".  It gives specific information as to why that works, because the pattern has been ingrained in both directions.

 

Unless I'm reading this conversation all wrong because it's pre-coffee time here.  I've only finished half a cup. 😄

 

Adding more thoughts:
It also helps with long division. 3,514 / 7 requires the child to immediately decompose the thousands into hundreds to understand how to divide the groups from left to right.  35 hundreds are broken into 7 groups, not 3 thousands and 5 hundreds.

Edited June 18, 2021 by HomeAgain

[Not_a_Number]

17 minutes ago, HomeAgain said:

I think it's easier for adults to see when you go the other direction.

If there is a clear communication of ten of something being a new place value, then it becomes equally clear that dividing by ten moves place value as well.  All the way across, in both directions.

Which means problems like 1/10 x 1/10 x 1/10 are easily absorbed as a tenth of a tenth (one hundredth), moving to one tenth of that ( one thousandth).  It manipulates numbers more easily in the mind and avoids the pitfalls of trying to teach decimals by "count the decimal places in the problem, move your decimal point that far over in the answer".  It gives specific information as to why that works, because the pattern has been ingrained in both directions.

 

Unless I'm reading this conversation all wrong because it's pre-coffee time here.  I've only finished half a cup. 😄

 

Adding more thoughts:
It also helps with long division. 3,514 / 7 requires the child to immediately decompose the thousands into hundreds to understand how to divide the groups from left to right.  35 hundreds are broken into 7 groups, not 3 thousands and 5 hundreds.

I’m really not communicating well, to be honest. So I actually can’t tell if we’re saying the same thing or not 😂.

Here’s my first blog post:

https://mentalmodelmath.com/2021/06/09/place-value/
 

And I really like this blog post (not mine, but it’s good):

http://www.garlikov.com/PlaceValue.html

[HomeAgain]

24 minutes ago, Not_a_Number said:

I’m really not communicating well, to be honest. So I actually can’t tell if we’re saying the same thing or not 😂.

Here’s my first blog post:

https://mentalmodelmath.com/2021/06/09/place-value/
 

And I really like this blog post (not mine, but it’s good):

http://www.garlikov.com/PlaceValue.html

I think we're saying the same thing!  The composing and decomposing of units leads to ease of manipulating large problems: anything from where a number of units is taken away that is greater than currently available in the place value those units hold, to working with decimals and division.  This is partly where the Gattegno place value chart has its strength, I think, because it's a very visual way of seeing the move up and down in a base-ten system with every number from 1 to 10.
But when place value is inherently learned as the grouping of X number of items to form a unit, it's easy for the same application to move to base 2, base 3...and so on.

[Not_a_Number]

6 minutes ago, HomeAgain said:

I think we're saying the same thing!  The composing and decomposing of units leads to ease of manipulating large problems: anything from where a number of units is taken away that is greater than currently available in the place value those units hold, to working with decimals and division.  This is partly where the Gattegno place value chart has its strength, I think, because it's a very visual way of seeing the move up and down in a base-ten system with every number from 1 to 10.
But when place value is inherently learned as the grouping of X number of items to form a unit, it's easy for the same application to move to base 2, base 3...and so on.

Ok, yes, we are saying the same thing! I teach place value as grouping, and I’ve found this makes it easy even for fairly non-mathy kids to deal with it (especially when given manipulatives — we’ve been using poker chips, although I don’t have strong preferences here.) And it both allows one to deal with all the operations in the same way and allows one to easily work with other bases. 

I’ve seen many kids who actually have a fair amount of trouble with the “grouping,” or “trading” — the model isn’t integrated well in their heads, so then they have to learn it anew for every single new thing. And it comes up SO MUCH. It’s in every algorithm, and in decimals and percent... so really getting it down is amazing for kids.

Most kids can tell me how many “hundreds” 8,432 contains. Most kids also kind of don’t understand the relevance of those hundreds (and the accompanying ones, tens and thousands.) The grouping idea is what makes this all coalesce. 

I do think Liping Ma’s book is excellent reading on this subject 🙂 .

[UHP]

54 minutes ago, Not_a_Number said:

I’ve seen many kids who actually have a fair amount of trouble with the “grouping,” or “trading” — the model isn’t integrated well in their heads, so then they have to learn it anew for every single new thing.

Can you recall some details, or make them up, of a concrete instance of this kind of trouble?

I'm still sometimes confused by your ideas about this. It sounds like you're saying that a kid who knows that 8,432 is denoting 4 hundreds (and 8 thousands and...), and who also knows that a hundred is the same as 10 tens, might still be missing something important. Something that they'll feel the lack of later on. But I don't know what that is. 

[HomeAgain]

1 hour ago, Not_a_Number said:

Ok, yes, we are saying the same thing! I teach place value as grouping, and I’ve found this makes it easy even for fairly non-mathy kids to deal with it (especially when given manipulatives — we’ve been using poker chips, although I don’t have strong preferences here.) And it both allows one to deal with all the operations in the same way and allows one to easily work with other bases. 

I’ve seen many kids who actually have a fair amount of trouble with the “grouping,” or “trading” — the model isn’t integrated well in their heads, so then they have to learn it anew for every single new thing. And it comes up SO MUCH. It’s in every algorithm, and in decimals and percent... so really getting it down is amazing for kids.

Most kids can tell me how many “hundreds” 8,432 contains. Most kids also kind of don’t understand the relevance of those hundreds (and the accompanying ones, tens and thousands.) The grouping idea is what makes this all coalesce. 

I do think Liping Ma’s book is excellent reading on this subject 🙂 .

I think Ma's book is excellent, too, but I really like how Gattegno takes it a step further and uses exponents to relate to place value in different base systems and reinforces it that way.

So, my husband I got into this same discussion about an hour ago.  I gave him the same problem: 3,514 / 7, and we had very different ideas of how to explain.

Him: "7 can't go into 3 so you move over..."

Me: "right, so now it becomes 35 hundreds to divide..."

Him: "What? No!  You just move over and divide using two numbers!"

Me: "okay, but why?"

Him: "because you do!"

Me: "but WHY?"

🤣

He says thinking of it by decomposing the thousands is too complicated.  And getting into why we have to go from left to right for division instead of right to left, like written multiplication, was just "the way it is".

So..........dh doesn't get to teach math here.

[daijobu]

44 minutes ago, HomeAgain said:

Me: "okay, but why?"

Him: "because you do!"

Me: "but WHY?"

 

So..........dh doesn't get to teach math here.

Yes, I totally agree.   What a terrible conversation to have with a student.  

[daijobu]

3 hours ago, Not_a_Number said:

I’m really not communicating well, to be honest. So I actually can’t tell if we’re saying the same thing or not 😂.

Here’s my first blog post:

https://mentalmodelmath.com/2021/06/09/place-value/
 

 

Oh I got it now.  Thank you.  

[daijobu]

3 hours ago, Not_a_Number said:

I’m really not communicating well, to be honest. So I actually can’t tell if we’re saying the same thing or not 😂.

Here’s my first blog post:

https://mentalmodelmath.com/2021/06/09/place-value/

Also:  is there a way to receive a notification by email when your blog has been updated?  

[vonfirmath]

5 hours ago, UHP said:

Can you recall some details, or make them up, of a concrete instance of this kind of trouble?

I'm still sometimes confused by your ideas about this. It sounds like you're saying that a kid who knows that 8,432 is denoting 4 hundreds (and 8 thousands and...), and who also knows that a hundred is the same as 10 tens, might still be missing something important. Something that they'll feel the lack of later on. But I don't know what that is. 

Actually 8432 has "84 hundreds"  (84.32 actually. But at the point the kids books are asking the question they have not started working with decimals)

The answer being looked for depends on how the question is ask.

"What number is in the hundreds place"

or

"how many hundreds are in the number"

 

(And it does remind me. At the beginning of the year, when the kids were all virtual, I got to listen in on some of the classes (when my daughter didn't have her headphones on).  Reviewing math at the beginning of 4th grade. When they were explaining how they worked out a subtraction problem, the math teacher was requiring them to be precise in their language. They were not borrowing "1" from the tens place. They were borrowing one ten from the tens place to add ten ones to the ones place then doing the subtraction.

 

(Or borrowing ten tens from the hundreds place. etc)

 

I expect this preciseness did not continue all year. But during review she was definitely working on the point.

Edited June 18, 2021 by vonfirmath

[Not_a_Number]

5 hours ago, UHP said:

Can you recall some details, or make them up, of a concrete instance of this kind of trouble?

I'm still sometimes confused by your ideas about this. It sounds like you're saying that a kid who knows that 8,432 is denoting 4 hundreds (and 8 thousands and...), and who also knows that a hundred is the same as 10 tens, might still be missing something important. Something that they'll feel the lack of later on. But I don't know what that is. 

Sure. Say that a kid is dividing 402 by 3. They split up the hundreds between the 3 people and they have 102 left. Now we just have one hundred and 2 ones. How do we keep going? They don’t know.

If you ASKED them how many tens is a hundred, they’d know. They successfully add  and subtract and multiply using a mixture of place value and tricks they remember. But it’s not fully automatic to trade up and down in every question. 

This isn’t even a theoretical example. This constantly happens in my Zoom class. It happened 2 days ago with these numbers.

“Knowing” has different levels. The more you use something, the more automatic and comfortable it is. Lots of kids learn consequences and then don’t think much about the trading. But the trading never leaves one alone. It’s ALWAYS there.

Edited June 18, 2021 by Not_a_Number