Place Value

[Not_a_Number]

19 minutes ago, Eilonwy said:

The sort of theory I think would be helpful to people trying to use your ideas is probably not really theory at all, but rather discussion of things like how do you think about subtraction/ multiplication/ equations/ division/ variables etc., and why, and how you talk about them with students. I don’t think all of these would be obvious from just reading examples & transcripts, but they should be combined with examples & transcripts to illustrate what it looks like in practice. 

Yeah, that's what I meant, really -- that kind of theory. Basically, posts I could link to when I give examples of work my kids do. 

 

19 minutes ago, Eilonwy said:

It’s not so much a difference in what is logical as that what is logical is more obvious in hindsight rather than in advance, for people without experience.  Like where you’ll hit sudden jumps in difficulty.  

Yes, that's fair. 

Where have you found sudden jumps in difficulty? 

[Eilonwy]

22 minutes ago, Not_a_Number said:

Yeah, that's what I meant, really -- that kind of theory. Basically, posts I could link to when I give examples of work my kids do. 

Yes, to give more background, the bigger picture, and the more subtle points of how to put a program together.

23 minutes ago, Not_a_Number said:

Where have you found sudden jumps in difficulty? 

That a question with more trading is quite a bit harder than one with more digits but fewer trades. This can be very similar on the surface, say 68+23 vs 68+33.  I watch for these more now, but I still make questions that take me by surprise.  Or that 4.5x8 is much more intuitive than 8x4.5.  

[daijobu]

3 hours ago, Not_a_Number said:

I think it's less "theory" and more "big picture"? 

 

Also, I want you to avoid that trip to crazy town that is Garlikov's website, with its preambles and citations and left-margin-to-right-margin formatting.  (It's not a good look.)  

Busy parents don't care about that stuff.  He should delete everything and start with part 4) Representation of Groups, halfway through the second paragraph with "We have these 3 different color poker chips."  Then he should delete part 5 and everything that follows it.  While he's at it, he should insert some more white space into part 4.  His power point is actually quite nice this way.  

I just don't want you to get too caught up in why we should do something.  Show us what you do, and write a little bit about how it's been helpful in your experience.  We can take it from there.  

[Not_a_Number]

43 minutes ago, daijobu said:

Also, I want you to avoid that trip to crazy town that is Garlikov's website, with its preambles and citations and left-margin-to-right-margin formatting.  (It's not a good look.)  

Busy parents don't care about that stuff.  He should delete everything and start with part 4) Representation of Groups, halfway through the second paragraph with "We have these 3 different color poker chips."  Then he should delete part 5 and everything that follows it.  While he's at it, he should insert some more white space into part 4.  His power point is actually quite nice this way.  

I just don't want you to get too caught up in why we should do something.  Show us what you do, and write a little bit about how it's been helpful in your experience.  We can take it from there.  

OK, give me some credit -- my blog does NOT look or read like that 😛 . It's all nice and readable and short and to the point.

Or are you saying it's been too rambly? If so, I'd like to know. 

[Clarita]

53 minutes ago, Not_a_Number said:

OK, give me some credit -- my blog does NOT look or read like that 😛 . It's all nice and readable and short and to the point.

No it does not, you haven't given me all your thoughts about how hideous you think the American education system while telling me about place value. 😜

 

[Not_a_Number]

1 minute ago, Clarita said:

No it does not, you haven't given me all your thoughts about how hideous you think the American education system while telling me about place value. 😜

 

I know. I know. His blog is kind of unreadable. And yet he has some great ideas! He really does. 

[Not_a_Number]

4 minutes ago, Clarita said:

No it does not, you haven't given me all your thoughts about how hideous you think the American education system while telling me about place value. 😜

 

Seriously, though, do I need to be more concise? 

[UHP]

16 minutes ago, Not_a_Number said:

Seriously, though, do I need to be more concise? 

Concision is great but you can only get to it by writing too much and then editing it. Start by writing too much.

[Not_a_Number]

Just now, UHP said:

Concision is great but you can only get to it by writing too much and then editing it. Start by writing too much.

Sure. But how about what I have right now? Too concise, too wordy, just right? I have access to an edit button. 

[Clarita]

21 minutes ago, Not_a_Number said:

Seriously, though, do I need to be more concise? 

I thought it was just fine. My only issue is that when I'm reading a series there is no clear quick link to other parts to that series. On the right I see links to a lot or all the posts you have and on the bottom is just shows previous and next in chronological order - so it may be unrelated post.

[Not_a_Number]

2 minutes ago, Clarita said:

I thought it was just fine. My only issue is that when I'm reading a series there is no clear quick link to other parts to that series. On the right I see links to a lot or all the posts you have and on the bottom is just shows previous and next in chronological order - so it may be unrelated post.

Yes, I'm going to link it up after I finish the series. My counting on series does have links, right? 

https://mentalmodelmath.com/2021/04/20/is-counting-on-obvious/

[UHP]

10 minutes ago, Not_a_Number said:

Sure. But how about what I have right now? Too concise, too wordy, just right? I have access to an edit button.

Don't press the edit button! I haven't been reading you wishing you'd express yourself better, just wishing you would say more.

[Not_a_Number]

2 minutes ago, UHP said:

Don't press the edit button! I haven't been reading you wishing you'd express yourself better, just wishing you would say more.

I've pressed it many a time when I've found misspellings 😂. Honestly, though, I don't mind stylistic input at all. I may not necessarily take it, but I've worked with editors before and don't mind constructive criticism. 

 

[Clarita]

12 minutes ago, Not_a_Number said:

Yes, I'm going to link it up after I finish the series. My counting on series does have links, right?

Yes it does.

[Not_a_Number]

2 minutes ago, Clarita said:

Yes it does.

Is linking at the bottom OK? I wasn't sure if I should do it at the top, but then the preview in the blog is unhelpful. 

Edited July 4, 2021 by Not_a_Number

[daijobu]

2 hours ago, Not_a_Number said:

OK, give me some credit -- my blog does NOT look or read like that 😛 . It's all nice and readable and short and to the point.

Or are you saying it's been too rambly? If so, I'd like to know. 

No, no, no, it doesn't.  I was just worried you were starting go down that route.  

I think my bias comes from not reading well on screen.  If it's a short article, nicely formatted, with gobs of white space them I'm good.  If it's more like a novel then I give up very quickly.

I supposed i'm also irritated because Garlikov has some excellent ideas but he had the gall to make me work to find them.  

[daijobu]

49 minutes ago, Not_a_Number said:

Yes, I'm going to link it up after I finish the series. My counting on series does have links, right? 

https://mentalmodelmath.com/2021/04/20/is-counting-on-obvious/

Okay, so this isn't going to be constructive at all, and I apologize in advance, but...

I found this blog post to be... um... useless?  

Again, I'll admit I don't read well on screen.  But you lead with an anecdote about your dd.  I stop at the word "subitize" because I don't know what that means, and I've never heard anyone use it before.  So then I wonder is that some kind of education jargon, and I start to question the content.  

I found paragraph 2 difficult to read.  Can you use Latex with Wordpress?  I think you delimit it with $latex x^2 $.  

I skip three paragraphs down  you describe how difficult it is for students to understand "counting on" but that was not my experience with my own daughters.

Continuing to skim, I see you prefer option (b), so I look back to read what option (b).  (I don't bother to read option (a) because you aren't advocating for it.)  Then you say you don't like either option (a) or (b) so now I'm a little irritated that you made me read about option (b) for nothing.  Then you put the words "mental model" in a pretty blue/green bold, but then you don't define it.  

So now I feel like I've read a blog post with zero ROI.  

(This also tells you more about me than it does about your writing, lol.)  

[daijobu]

But you're a published author already aren't you?!  Who am I to criticize your writing?  

[Clarita]

2 hours ago, Not_a_Number said:

Is linking at the bottom OK?

Yes I think it makes more sense that way.

[Not_a_Number]

8 hours ago, daijobu said:

Okay, so this isn't going to be constructive at all, and I apologize in advance, but...

I found this blog post to be... um... useless?  

Again, I'll admit I don't read well on screen.  But you lead with an anecdote about your dd.  I stop at the word "subitize" because I don't know what that means, and I've never heard anyone use it before.  So then I wonder is that some kind of education jargon, and I start to question the content.  
 

I’m kind of amazed you haven’t seen this word! People use it on here all the time. I picked it up from the forums.

 

8 hours ago, daijobu said:

I found paragraph 2 difficult to read.  Can you use Latex with Wordpress?  I think you delimit it with $latex x^2 $.  
 

Ah-ha. That’s brilliant, actually — I had no idea this was possible.

 

8 hours ago, daijobu said:

I skip three paragraphs down  you describe how difficult it is for students to understand "counting on" but that was not my experience with my own daughters.

I’m sure it’s not everyone’s experience! Not everyone is going to struggle with the same stuff. But I definitely actually had to teach both my kids to count on — it wasn’t obvious to them. I suppose I could have simply taught them the rule, but I don’t do that, and they certainly didn’t come up with it themselves at the ages we were doing this. (Which is usually age 4.5 or so with my kids, but I had to do this with half my homeschool classes, who were a wider age range.)

 

8 hours ago, daijobu said:

Continuing to skim, I see you prefer option (b), so I look back to read what option (b).  (I don't bother to read option (a) because you aren't advocating for it.)  Then you say you don't like either option (a) or (b) so now I'm a little irritated that you made me read about option (b) for nothing.  Then you put the words "mental model" in a pretty blue/green bold, but then you don't define it.  

So now I feel like I've read a blog post with zero ROI.  

(This also tells you more about me than it does about your writing, lol.)  

I suppose I can’t promise to write no philosophy whatsoever, and that might have low ROI except in the long run 😉 . But skimming it is probably useless.

[Not_a_Number]

And another post... this thread is inspiring me! Thanks, all. 

I think I'll make this part of the "Place Value" series, even though it's kind of not. But it's all integrated in my head, you know? 

https://mentalmodelmath.com/2021/07/04/the-operations/

[Eilonwy]

2 hours ago, Not_a_Number said:

And another post... this thread is inspiring me! Thanks, all. 

I think I'll make this part of the "Place Value" series, even though it's kind of not. But it's all integrated in my head, you know? 

This looks good and seems very clear to me. I don’t think that examples would make sense without this. 
 

There is one typo I saw where you say “no more and more less.”

[Not_a_Number]

13 minutes ago, Eilonwy said:

This looks good and seems very clear to me. I don’t think that examples would make sense without this. 
 

There is one typo I saw where you say “no more and more less.”

Thank you! Will fix. Let me know if you see any typos, of course! 

[Not_a_Number]

41 minutes ago, Eilonwy said:

This looks good and seems very clear to me. I don’t think that examples would make sense without this. 
 

There is one typo I saw where you say “no more and more less.”

Do you think it makes sense to make that post part of my place value series? 

[Eilonwy]

12 minutes ago, Not_a_Number said:

Do you think it makes sense to make that post part of my place value series? 

To me, not exactly, no.  I think both place value and operations definitions could come under a common heading of key mental models, or building blocks, or something like that. But if you think of it as a component of place value rather than it’s own thing, then it should go in the place value series anyway.  

[Not_a_Number]

2 minutes ago, Eilonwy said:

To me, not exactly, no.  I think both place value and operations definitions could come under a common heading of key mental models, or building blocks, or something like that. But if you think of it as a component of place value rather than it’s own thing, then it should go in the place value series anyway.  

No, I do think of it as its own thing. 

Hmmm, should I do a different intro post to the "basics" series? It would be out of order, but I could say something like "I'm going to talk about the basic definitions I use for elementary math. Here are the posts on place value, operations, equals signs, and variables"? 

[EKS]

16 minutes ago, Not_a_Number said:

Do you think it makes sense to make that post part of my place value series? 

I think that you are going to find that the logical order to present things in your blog is the order that you present things to your students.  I'd group this particular post with basic operations (or something like that).  That said, so much (all) of elementary math is interrelated, so posts will very likely have multiple tags.

[Not_a_Number]

Just now, EKS said:

I think that you are going to find that the logical order to present things in your blog is the order that you present things to your students.  I'd group this particular post with basic operations (or something like that).  That said, so much (all) of elementary math is interrelated, so posts will very likely have multiple tags.

Oh yeah, I'm not worried about tags. It's more like I want to make a series of posts and link them up, like I did with the Counting On sequence. Just so that I have somewhere to send people who need an intro to what we do, you know? 

[EKS]

2 hours ago, Not_a_Number said:

Oh yeah, I'm not worried about tags. It's more like I want to make a series of posts and link them up, like I did with the Counting On sequence. Just so that I have somewhere to send people who need an intro to what we do, you know? 

Just make sure it doesn't stifle your writing.  I know that when I'm trying to make things fit into some larger whole, I get stuck.

[Not_a_Number]

19 minutes ago, EKS said:

Just make sure it doesn't stifle your writing.  I know that when I'm trying to make things fit into some larger whole, I get stuck.

I totally get that! I'm just writing at this point... just thinking about the fact that I'd like to have a series, and the recent post doesn't really fit the "place value" series, so I need to reorient the series a bit 😉 . 

I think I also need an "equals sign" and "variable" post, then a few examples of how these work in conjunction in a LOT of detail. 

[Not_a_Number]

OK, I made a new intro to the series, and I linked up all the posts: 

https://mentalmodelmath.com/2021/07/06/the-basics/

As usual, input is welcome. 

[Clarita]

3 hours ago, Not_a_Number said:

As usual, input is welcome.

I don't want to sound like a fan-girl, but I really love this. 

The right time to start generalizing one’s observations about numbers is when one is working with numbers in a hands-on way. And similarly, the right time to build a robust mental model of a variable is when one is doing arithmetic.

I never realized this while I was learning math. The other day while we were working on adding (stage 2 of counting on) he asks what numbers added together make 7, then he totally wanted to do more problems where we were finding a missing addend rather than the sum. Some days I wonder how I'm going to teach this son of mine because he jumps between preschool math topics into more advanced math willy-nilly. So, I'm really thankful for your blog to show me how I can do that.  

[Not_a_Number]

1 minute ago, Clarita said:

I don't want to sound like a fan-girl, but I really love this. 

The right time to start generalizing one’s observations about numbers is when one is working with numbers in a hands-on way. And similarly, the right time to build a robust mental model of a variable is when one is doing arithmetic.

I never realized this while I was learning math. The other day while we were working on adding (stage 2 of counting on) he asks what numbers added together make 7, then he totally wanted to do more problems where we were finding a missing addend rather than the sum. Some days I wonder how I'm going to teach this son of mine because he jumps between preschool math topics into more advanced math willy-nilly. So, I'm really thankful for your blog to show me how I can do that.  

I love missing addend problems! We do a LOT of them. And we interweave them with place value and the different operations... it's a ton of fun. I'll post our worksheets and their motivation very shortly -- I just have a few "basics" things left to post. 

[Not_a_Number]

4 hours ago, Clarita said:

The other day while we were working on adding (stage 2 of counting on) he asks what numbers added together make 7, then he totally wanted to do more problems where we were finding a missing addend rather than the sum

By the way, how's the "counting on" going? I'm always super curious when people try stuff I suggest, because that has rarely happened so far! Mostly, I've been the only one to ever use my ideas... 

[Clarita]

58 minutes ago, Not_a_Number said:

By the way, how's the "counting on" going?

Originally I was the only one doing it and he would still slowly count up the sum of numbers. (After he does his slow count I would do the quick count then counting on to check the answer.) Yesterday was the first day he tried the quick count and counting on. It was a mix of doing it correctly, and quick counting the sum which resulted in the wrong answer.

I think doing it actually peaked his interest in adding, because sometimes he gets discouraged that I can do his "school" work so quickly and him so slowly. This essentially is showing him the steps for him to get to my speed, so he's pretty curious about that. 

[Not_a_Number]

7 hours ago, Clarita said:

Originally I was the only one doing it and he would still slowly count up the sum of numbers. (After he does his slow count I would do the quick count then counting on to check the answer.) Yesterday was the first day he tried the quick count and counting on. It was a mix of doing it correctly, and quick counting the sum which resulted in the wrong answer.

I think doing it actually peaked his interest in adding, because sometimes he gets discouraged that I can do his "school" work so quickly and him so slowly. This essentially is showing him the steps for him to get to my speed, so he's pretty curious about that. 

Cool! Let me know how it goes now that he's starting to work on it. 

I do find that I have to prod kids to change tactics, because they are so used to their own. Although, come to think of it, my kids jump to "rearranging things" relatively fluently after they count on -- like, to do 5+6, they do 5 + 5 + 1= 10 + 1 = 11 with relatively ease after counting on, because they've absorbed that adding 1 is very easy. This is even true for my non-mathy homeschool class kids so far. 

But counting on I've always had to prod for a few weeks/months, or they just keep pointing at every single object. 

Edited July 7, 2021 by Not_a_Number

[Xahm]

I love reading your ideas and integrate them into what I'm doing. 

For your blog: Most people aren't going to totally drop curriculum and develop our own, and most of us probably shouldn't, haha. It would be a ton of time on your part, so I don't expect you to do this, but it would be helpful to me if you read through a few of the most common curricula, like a pretty deep look, and gave your thoughts and ideas for ways to shore up weaknesses. For example, I know you've talked about how Beast Academy relies to much on the x+y = blank formation of question that could lead to kids thinking = means "put your answer here" if that's all they see. It's pretty easy to add in practice in other formats, so a blog post suggesting Beast parents do that would likely be helpful.

Place value is important, I see that, but of I'm incorporating your ideas into curriculum, it would help if you included "it's best to introduce this before your kid starts formal studies" or "you want to be sure to circle back to this just before introducing two digit subtraction."

Other than that, my suggestion would be to include made-up transcripts with multiple students, based on composites of the kids you've worked with. One who tends to get things quickly (and you could show how you move quickly or offer extensions), one who understands well but needs full explanations, and one who misunderstands and needs additional help.

Sorry it's disjointed. I keep reading and intending to post but putting it off when I get in the computer, so I decided to go ahead even though I'm on my phone.

[Not_a_Number]

On 7/7/2021 at 12:16 PM, Xahm said:

I love reading your ideas and integrate them into what I'm doing. 

For your blog: Most people aren't going to totally drop curriculum and develop our own, and most of us probably shouldn't, haha.

Lol, no, I don't think most people should. I did feel like I knew what I was doing (I had strong opinions based on almost a decade of teaching experience), but I've been very pleasantly surprised how well things have worked so far with a fairly wide range of kids. But I do think that's me reaping the benefit of long experience and also a lot of pondering... it wouldn't work in general and wouldn't even work for me in some other subjects. 

 

Quote

It would be a ton of time on your part, so I don't expect you to do this, but it would be helpful to me if you read through a few of the most common curricula, like a pretty deep look, and gave your thoughts and ideas for ways to shore up weaknesses. For example, I know you've talked about how Beast Academy relies to much on the x+y = blank formation of question that could lead to kids thinking = means "put your answer here" if that's all they see. It's pretty easy to add in practice in other formats, so a blog post suggesting Beast parents do that would likely be helpful.

I've definitely looked over many curricula, although I don't know how deep the look was, given that most of them didn't seem to be doing anything like I wanted them to be doing. It's more like I looked through the curriculum, realized they didn't meet almost any of my concerns, then I decided not to even try. 

I'll say that the "=" meaning "put your answer here" thing is basically there in every single curriculum I've ever seen, except maybe Gattegno and Miquon. It's simply standard usage, despite the fact that by algebra we suddenly expect kids to have a more robust understanding of the equals sign -- so much so that we want them to not only be able to apply it to specific numbers but to generalized numbers (variables.) I don't really get it, but that seems to be how it is. My experience is that a lot of these things are kind of faddy -- people do things in a particular order because everyone does it, not because it's the best way according to rigorous testing. I've seen the exact same thing in higher-level math.

I talk about Beast in this context mostly because I got the chance to observe this with my own kid and it freaked me out 😛 . But I already knew that it was a potential issue beforehand -- I just didn't realize how little time with that usage DD8 would need to absorb it as normal.  

 

Quote

Place value is important, I see that, but of I'm incorporating your ideas into curriculum, it would help if you included "it's best to introduce this before your kid starts formal studies" or "you want to be sure to circle back to this just before introducing two digit subtraction."

Most things I do I simply incorporate into all lessons. Like, I've taught kids to use poker chips, and they could use it for as many things or as few things as they like. My own kids are mathy and move away from the actual manipulative fairly rapidly, because their mental model forms quickly. Right now, DD5 can do most forward things in her head, but she uses poker chips to check her answers, and I do think she needs poker chips for "backwards" questions more often. (Like, "find the missing addend" questions.) 

But some of my weaker kids will probably be using the manipulative for a long time. It's basically a tool that assists mental model building. 

 

Quote

Other than that, my suggestion would be to include made-up transcripts with multiple students, based on composites of the kids you've worked with. One who tends to get things quickly (and you could show how you move quickly or offer extensions), one who understands well but needs full explanations, and one who misunderstands and needs additional help.

Hmmm, good idea. Although I'll say that I do most things the same way with strong and weak students -- my general sense is that it's best not to leave gaps, because you never can tell which gaps your strong student will fill in themselves and which ones they won't, especially since I haven't yet had a student who actually did math on their own time -- only students who were extremely strong when taught well. The big thing I see is that my strong students go through things much faster.

 

Quote

Sorry it's disjointed. I keep reading and intending to post but putting it off when I get in the computer, so I decided to go ahead even though I'm on my phone.

Feel free to comment on the blog, too 🙂 . I'm sadly lacking for comments on there! 

Edited July 9, 2021 by Not_a_Number