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I was talking to @JazzyMom about this on her thread, but then I thought I'd just make a separate post about it. 

I've noticed from talking to people about how we do place value that I'm not describing what we do or how it works particularly well. As a result, I've been working on a series of blog posts to describe it... so far, I just have the first, introductory one: 

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

What would be helpful to see in my next blog post, before I start working on it? I'd like to be less confusing than I apparently usually am 🙂. (To be clear, I don't confuse kids that I teach this way! But I seem to confuse parents on here when I explain what we do.) 

Edited by Not_a_Number
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Your blog post says:

"So what can we do? We can present them with tools that let them explore the immediate consequences of the mental model itself."

 

What does that look like exactly? What tools are you presenting? I know in past posts you have discussed using poker chips. Is this the only manipulative you use? How does that look exactly? Do you use them for building numbers? So do you have a child make the number 5,784 with poker chips and then... ? Do you use anything besides poker chips? Am I completely off base in thinking that you are even referring to using some sort of manipulative to teach? 

It is interesting because my DD who is my strongest math student really does seem able to visualize a lot of math, which gives her a deeper understanding of what is happening and allows her to solve problems in creative, yet accurate, ways. Using a 10-frame was a big help for her as well as base 10 blocks. However, despite using the same methods for my younger DD, she does not seem to visualize as easily. The tools are all available to her, but it doesn't seem to "click" the same way. I just put it down to different brains working differently, but maybe I am simply presenting the material incorrectly. 

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Like the video posted recently of Gattegno teaching children using Cuisinaire rods, it would be really helpful to see a video of you teaching students.  A transcript is fine, but a video really drives the point home.  

The abstract notion that we don't teach place value correctly is fine, but I want to see how to teach it in action.  

I'm also happy to volunteer to watch one of your zoom classes to see if I can discern what you mean.  

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6 minutes ago, daijobu said:

Like the video posted recently of Gattegno teaching children using Cuisinaire rods, it would be really helpful to see a video of you teaching students.  A transcript is fine, but a video really drives the point home.  

The abstract notion that we don't teach place value correctly is fine, but I want to see how to teach it in action.  

I'm also happy to volunteer to watch one of your zoom classes to see if I can discern what you mean.  

I don’t think my teaching is very special. Honestly. For me, part of the point is that providing tools for exploration is more important than what I say.

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16 minutes ago, amiesmom said:

What does that look like exactly? What tools are you presenting? I know in past posts you have discussed using poker chips. Is this the only manipulative you use? How does that look exactly? Do you use them for building numbers? So do you have a child make the number 5,784 with poker chips and then... ? Do you use anything besides poker chips? Am I completely off base in thinking that you are even referring to using some sort of manipulative to teach? 

I just use poker chips in my classes :-). With DD8, I only used dots and boxes, which was less good but worked with her. I think the specific manipulative isn’t super important, although I prefer it to be easily subitised and in “units” and not in “groups.” Basically, I prefer it to be identical in level of abstraction to actual base 10 numbers.

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22 minutes ago, Not_a_Number said:

Basically, I prefer it to be identical in level of abstraction to actual base 10 numbers.

I don't understand this, can you give an example? Thank you for taking the time to explain. 

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I'll share what worked well for my DC. (Qualified by - I'm not a math teacher, and I don't know whether this would work for all kids).

From when they were very young, preK age, I'd print or draw out a large number on a sheet of paper, and we'd put a sticker on the number to match the number. For instance, if the number was 1, we'd put one sticker on the number. If the number was 5, we'd put 5 stickers on it. When we got to numbers >9, I made these stickers that had 10 dots on them, to represent a unit of 10 or a bundle of 10 ones. I told them that the tens live in the lefthand column, the ones live in the right hand column. So for the number 10, they'd put one of the 10 stickers (the sticker that had 10 dots) on the 1, and no stickers on the zero. For the number 53, they'd put five of the ten stickers on the 5, and three of the single stickers on the 3. You get the idea. They both seemed to have no trouble with the concept of place value. 

I never did stickers with numbers > 100 because it was too much work for me 🙂 and they didn't seem to need it. From the stickers, we were able to move to those place value cards and base ten picture cards, and the concept seemed pretty intuitive to my kids by then. (we bought them from Right Start but ended up being a Beast and Singapore family)

I'll say that though we jumped ship from RIght Start, I did think one of the strengths of the that program (level A/B, I haven't seen the higher levels) was the way they presented place value in so many different ways, to help make number sense more intuitive.

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2 hours ago, amiesmom said:

I don't understand this, can you give an example? Thank you for taking the time to explain. 

Pretend you know NOTHING about base 10. Then if you look at a number, what do you see? Say 524... you see a 5, a 2, and a 4. You don't see hundreds. You don't see tens. You don't see ones. You just see those three digits. 

It takes considerable experience and facility to interpret those digits as counting copies of bigger and bigger quantities. We don't see it that way anymore, because we're used to it, or we think that writing it out as 500 + 20 + 4 helps a lot, but really, it doesn't. If you see 500, you STILL only just see 3 numbers -- a 5, a 0, and a 0. You don't see any hundreds. 

So what I prefer is to have counters or poker chips or something else where when you see a 524, you get 5 of something, 2 of another something, and 4 of another something. I use poker chips -- in my class, kids would get 5 yellow poker chips, 2 green poker chips, and 4 blue poker chips. This is equally abstract to the written form of the number. 

At that point, what you need to know is how to operate with these different colored poker chips. That's what teaching place value looks like over here. I tell them how much each poker chip is worth in terms of the poker chip before it, and then we use that. Over and over again. Until they are completely comfortably trading up and trading down WHENEVER they need to. For addition. For subtraction. For multiplication. For division. 

And in my experience, that takes years, not days and not weeks. Place value takes serious integration into one's head to use well. 

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Posted (edited)
1 hour ago, WTM said:

From when they were very young, preK age, I'd print or draw out a large number on a sheet of paper, and we'd put a sticker on the number to match the number. For instance, if the number was 1, we'd put one sticker on the number. If the number was 5, we'd put 5 stickers on it. When we got to numbers >9, I made these stickers that had 10 dots on them, to represent a unit of 10 or a bundle of 10 ones. I told them that the tens live in the lefthand column, the ones live in the right hand column. So for the number 10, they'd put one of the 10 stickers (the sticker that had 10 dots) on the 1, and no stickers on the zero. For the number 53, they'd put five of the ten stickers on the 5, and three of the single stickers on the 3. You get the idea. They both seemed to have no trouble with the concept of place value. 

I don't think kids usually do have trouble with place value in this specific way. It's actually being able to USE it whenever needed that's a problem. 

Let me give an example. DD5 can add any numbers you give her, with or without carrying. (She may need some manipulatives, but often she doesn't.) However, maybe a week ago she was still getting stuck with questions like 

17 + square = 62. 

Why? Because she could see she had more ones (blues) starting out than she did at the end, after adding, and this required not only being comfortable trading (which she was!), but also remembering that you CAN trade, and that this can mean that you wind up with fewer blues than you started with. Working backwards takes a higher level of comfort and a higher level of integration than just knowing that you CAN do something, and actually DOING something takes more comfort than knowing the names of the quantities represented by the digits. 

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8 minutes ago, Not_a_Number said:

I don't think kids usually do have trouble with place value in this specific way. It's actually being able to USE it whenever needed that's a problem. 

Let me give an example. DD5 can add any numbers you give her, with or without carrying. (She may need some manipulatives, but often she doesn't.) However, maybe a week ago she was still getting stuck with questions like 

17 + square = 62. 

Why? Because she could see she had more ones (blues) starting out than she did at the end, after adding, and this required not only being comfortable trading (which she was!), but also remembering that you CAN trade, and that this can mean that you wind up with fewer blues than you started with. Working backwards takes a higher level of comfort and a higher level of integration than just knowing that you CAN do something, and actually DOING something takes more comfort than knowing the names of the quantities represented by the digits. 

That makes sense. The base 10 abacus was good for visualizing what you describe

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3 hours ago, Not_a_Number said:

I don’t think my teaching is very special. Honestly. For me, part of the point is that providing tools for exploration is more important than what I say.

I think you miss my meaning.  I mean, it will be easier to understand your intention if you could actually demonstrate what how you'd like to see children taught.  We didn't watch the Gattegno film because there was anything special about Gattegno.  We watched because we wanted to see his techniques in action.  Anyone could have demonstrated those techniques.  So if not you, then someone else.  Or even an animation, you know?  

I contrast this with the Galikov paper.  I clicked the link and immediately knew this was not something I was going to read.  Paragraphs are too long and too wide.  The first section "practice vs. understanding" sounded too abstract to be of any use and therefore skippable.  More text and no number formatting like Latex makes it hard to read.   

The old adage applies here:  Show, don't tell.  

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1 hour ago, Not_a_Number said:

Pretend you know NOTHING about base 10. Then if you look at a number, what do you see? Say 524... you see a 5, a 2, and a 4. You don't see hundreds. You don't see tens. You don't see ones. You just see those three digits. 

It takes considerable experience and facility to interpret those digits as counting copies of bigger and bigger quantities. We don't see it that way anymore, because we're used to it, or we think that writing it out as 500 + 20 + 4 helps a lot, but really, it doesn't. If you see 500, you STILL only just see 3 numbers -- a 5, a 0, and a 0. You don't see any hundreds. 

So what I prefer is to have counters or poker chips or something else where when you see a 524, you get 5 of something, 2 of another something, and 4 of another something. I use poker chips -- in my class, kids would get 5 yellow poker chips, 2 green poker chips, and 4 blue poker chips. This is equally abstract to the written form of the number. 

At that point, what you need to know is how to operate with these different colored poker chips. That's what teaching place value looks like over here. I tell them how much each poker chip is worth in terms of the poker chip before it, and then we use that. Over and over again. Until they are completely comfortably trading up and trading down WHENEVER they need to. For addition. For subtraction. For multiplication. For division. 

And in my experience, that takes years, not days and not weeks. Place value takes serious integration into one's head to use well. 

Ok, this is helpful. Thank you. 

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1 hour ago, daijobu said:

I think you miss my meaning.  I mean, it will be easier to understand your intention if you could actually demonstrate what how you'd like to see children taught.  We didn't watch the Gattegno film because there was anything special about Gattegno.  We watched because we wanted to see his techniques in action.  Anyone could have demonstrated those techniques.  So if not you, then someone else.  Or even an animation, you know?  

I just wonder if watching a video is actually particularly helpful, you know? I'm not sure if it's obvious what's going on. 

I definitely wouldn't post videos of my kids, I think, since I'm relatively protective of their privacy. I have videos of them as babies for my babywearing channel, and I'm already feeling kind of iffy about that... and I figure the older they get, the less good I feel about it. 

I'll definitely post transcripts. I know you said that's not as good, but I'm definitely much more likely to do it. I do plan to do that consistently at some point, but I keep putting it off, because I'd like to get the basics out of the way, and those take some thought! 

 

1 hour ago, daijobu said:

I contrast this with the Galikov paper.  I clicked the link and immediately knew this was not something I was going to read.  Paragraphs are too long and too wide.  The first section "practice vs. understanding" sounded too abstract to be of any use and therefore skippable.  More text and no number formatting like Latex makes it hard to read.   

Yes, unfortunately, his blog isn't all that readable 😕 . I really like it, because I have good focus for stuff I'm excited about, but I can't pretend it's written in a helpful way! That's why I'm asking for input, really -- to get an idea of how to communicate the ideas effectively. This thread is definitely helping. 

Does anything other than a video help? Anything that's almost as good that doesn't involve putting my kids online? 

 

2 hours ago, WTM said:

That makes sense. The base 10 abacus was good for visualizing what you describe

Does it do grouping or units for 10s? I haven't used RightStart, so I don't know. 

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20 minutes ago, Not_a_Number said:

Yay, glad to help out! Which bit of that helped, if it was anything in particular? (I'm still trying to get ideas for the blog, lol.) 

Giving the specific details of how, why, and for how long, you use the poker chips. I think from your past posts about poker chips, I had always assumed they had denominations on them, and I thought,  ok, I'm doing that already. I use base 10 blocks, or Singapore's place value disks or whatever,  but by explaining that the chips have no significance until you tell the students their value, that to me is what is different from what I am doing. 

I see that you are not saying one must use poker chips, but I do find it helpful to list one specific tool and how you have successfully used it.

I guess the one other thing I would want to know is how or when you transition from chips to numbers. In other words, do you tell the kids that these chips in front of you have a value of 524 at some point? Do they figure it out themselves? Or is it from day 1? 

Sorry, I feel like neither my question or response is very clear.

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53 minutes ago, amiesmom said:

Giving the specific details of how, why, and for how long, you use the poker chips. I think from your past posts about poker chips, I had always assumed they had denominations on them, and I thought,  ok, I'm doing that already. I use base 10 blocks, or Singapore's place value disks or whatever,  but by explaining that the chips have no significance until you tell the students their value, that to me is what is different from what I am doing. 

I see that you are not saying one must use poker chips, but I do find it helpful to list one specific tool and how you have successfully used it.

I guess the one other thing I would want to know is how or when you transition from chips to numbers. In other words, do you tell the kids that these chips in front of you have a value of 524 at some point? Do they figure it out themselves? Or is it from day 1? 

Sorry, I feel like neither my question or response is very clear.

I think Singapore place value disks would actually be totally identical to what I'm doing 🙂 . I think the only difference is that I expect to use them for a long time and don't expect kids to remember shortcuts other than "this disk is worth 10 times as much as the other one" for quite a while. I come back to this definition over and over again until it's really, really, really firm. 

I don't love base 10 blocks because they are more like groupings and less abstract. For kids who are able to generalize well, I'm sure they work fine, though. I do have one kid for whom the fact that the sizes of all the poker chips (and place value disks, I'd guess!) are the same is actually a big deal. She has trouble generalizing as is, so having different sized representations would mess with her mental model. But the same is not true for my kids -- some kids form mental models easier than others. 

I also do prefer having one main manipulative as opposed to a lot of them. 

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12 hours ago, amiesmom said:

What does that look like exactly? What tools are you presenting? I know in past posts you have discussed using poker chips. Is this the only manipulative you use? How does that look exactly? Do you use them for building numbers? So do you have a child make the number 5,784 with poker chips and then... ? Do you use anything besides poker chips? Am I completely off base in thinking that you are even referring to using some sort of manipulative to teach? 

I took NaN's advice about poker chips for place value in March, and I've been really pleased with them. I wrote a little bit of detail about my first experience using them (with my 6-year-old daughter), not quite publicly, and I looked it up again a few days ago because I'm about to introduce them to a relative. I'll paste it below. 

I've been using some handmade "poker chips," each of them is a square inch that I cut out of red or orange or blue construction paper.

March:

I gave her a light load of worksheets this morning, and told her that when she finished we would play a game with little paper squares I had made. These red squares are 1, and these orange squares are 10. "Then the blue squares must be 100!" Yes.

These won't be her only "model" for three-place numbers, she already has experience with pirate numbers (from BA), plastic cubic centimeters that stick together, the number line, the numerals themselves. But I had reviewed some questions over the last week like "451 is the same as 3 hundreds, __ tens and 1 one" and they weren't effortless.

One way the cardstock squares are worse than plastic poker chips: when you give someone a stack of 8 or 9 of them it takes a little bit of dexterity to separate and count them.

Anyway I set it up like this. She has a blank sheet of paper and pencil, and I slide some orange and red squares to her. "I want you to write down the value of these squares that I gave you, as a two-digit number." She didn't need any explanation, she counted the two orange squares as "ten, twenty", looked at the six red squares, and wrote 26.

"Now here are some more. Don't mix them together yet. How many did I give you in this group?" and she wrote 29.

When I asked her to add them together, she did it in her head by saying "forty fifteen." That's a joke that she likes that I don't always discourage. "That's kind of a good answer as a joke, but you have to remember that 15 isn't a digit. What do you really mean by 'forty fifteen'?" She knows to answer 40 + 15.

"Good, you can remember 40+15 for a minute. Before we talk more about it I want you to write "problem 1" on your paper, and then write 26+29 since that's the problem you're doing." I let her guess a little about the letters in 'problem' before spelling it for her. Then she wrote 26 + 29 = ▢. I hadn't asked her to write the equals sign or the box, and didn't mind that she did.

"OK what was 26+29 again?" 40 + 15, which is 55. "Good. Now, you still have 4 orange squares on your side of the table, and 15 red squares on your side of the table. Is there a way we can make that look more like 55, without changing the value?"

This stumped her, she couldn't guess what I wanted her to say. "Here's a rule for the game. A red square is 1 and an orange square is 10, so whenever you like you can trade me a 10 red squares for 1 orange square. Let's try that now." She counted out ten of her red squares, slid them to me and I handed her one orange square.

"Now, how many tens do you have? How many ones do you have?" She shouted back, I think excited because she had put together what we were doing, "Fifty-five!"

Next problem. "OK, write problem 2 on your paper. No, don't hand those back to me, this time you're going to start with those 55. How many do you have there, five oranges and five reds? Can you write that as a two-digit number." She wrote "PROBLeM 2," underlined it, and wrote 55 underneath.

"Here are some more for you." I gave her 1 blue square and 1 orange square. She knew I had given her 110, I had her write down 55+110, and she knew that was 165. I didn't make her prove it to me by counting her squares.

"OK, now write Problem 3. You can keep those 165. Now I want you to tell me: what's 165 minus seven. Can you use your paper squares to find out?" She fussed with her squares for a little bit and then asked me "can I have two more reds?" I told her no, I don't want to change the value of the number you're starting with. "But you can give me one orange square and I'll give you back ten red squares. That won't change the value and it will help you."

After the trade she was reluctant to give up the seven red squares she was taking away to the "bank." I told her it was fine, she could keep them on her side of the table, she just had to get them out of the way of the squares she ended up with. "When you subtract you take seven away from the number you start out with, so take those away, put them over there, and don't count them when you tell me what you end up with."

She didn't count at first, she was pretty sure she had 148. I asked her why, she said "because 10 minus 2 is 8." Well, that's true and it's a good observation for your problem but let's look at how many orange squares you have, remember you started out with six and you only traded one of them to me, I think you have five. "I'll write 158."

I thought that was the right place to stop for the day.

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I love the way Math U See describes place value, and if you go to ChristianBook.com they have that video in their sample for their primer (probably have it on the main Math U See site too).  You couldn't use it exactly of course but it might give you some ideas.

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3 minutes ago, goldenecho said:

I love the way Math U See describes place value, and if you go to ChristianBook.com they have that video in their sample for their primer (probably have it on the main Math U See site too).  You couldn't use it exactly of course but it might give you some ideas.

We don't use MUS but I've always use Decimal Street as an analogy for place value. All my kids have a great grasp of place value and how to use it using Decimal Street as an analogy for how they do their trades. Love this video for explaining Decimal Street and so do all the kids (my own and others) that I have shown it to.

I always assumed that NaN's poker chips had denominations on them as well, similar to place value chips. Arbitrarily assigning values to colors definitely wouldn't work for the kids I've taught/teach.

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Posted (edited)
4 hours ago, sweet2ndchance said:

We don't use MUS but I've always use Decimal Street as an analogy for place value. All my kids have a great grasp of place value and how to use it using Decimal Street as an analogy for how they do their trades. Love this video for explaining Decimal Street and so do all the kids (my own and others) that I have shown it to.

I always assumed that NaN's poker chips had denominations on them as well, similar to place value chips. Arbitrarily assigning values to colors definitely wouldn't work for the kids I've taught/teach.

Why wouldn’t it work? It’s not arbitrary, it’s fixed. I’m not against writing the numbers on them, but I haven’t had to — same as with the C-rods, kids get used to what each color is worth. I can imagine kids not being able to do it, but I didn’t have that happen with any of the 30 kids I had in my classes.

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3 hours ago, goldenecho said:

I love the way Math U See describes place value, and if you go to ChristianBook.com they have that video in their sample for their primer (probably have it on the main Math U See site too).  You couldn't use it exactly of course but it might give you some ideas.

Ideas for what?

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We actually found a Montessori checkerboard to be brilliant for place value exploration.  The physical movement of up and diagonal helped cement the exponential idea quite a bit, and allows exploration of place value in different bases.  Mostly because the board use can be extended to show the exponential value more than one place value up, and then move it diagonally to the proper spot, giving a kid extra steps using only one column's worth at first and then going for more.

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Hmmmm. So what would be the most useful thing to include in my blog post?? A transcript? The basic idea?

In my experience, kids need the mental model for basically the entirety of elementary math. It can either be accessed mentally or via manipulatives, but it needs accessing. How long it takes to make it purely mental REALLY depends on the kid and also the consistency with which the model is accessed — learning shortcuts too early can wind up detrimental and slow kids down later.

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11 minutes ago, Not_a_Number said:

Hmmmm. So what would be the most useful thing to include in my blog post?? A transcript? The basic idea?

I'm very interested in a transcript. 

I think you should take a break from writing the theory of what you're doing. Write ten thousand words about the practice — "case studies" or just lab notes. If you could go into very fine detail about one of your tutoring sessions, every day for two weeks (impossible — but even longer would be better), I think some people would be interested and afterwards you would have a lot of material to crib from and link to, for synthesizing a more theoretical explanation.

In media res is no problem: you don't have to remember how it all started and try to catch us up. Just tell what happened yesterday, but go into twice as much detail as you think is overkill.

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6 minutes ago, UHP said:

I'm very interested in a transcript. 

I think you should take a break from writing the theory of what you're doing. Write ten thousand words about the practice — "case studies" or just lab notes. If you could go into very fine detail about one of your tutoring sessions, every day for two weeks (impossible — but even longer would be better), I think some people would be interested and afterwards you would have a lot of material to crib from and link to, for synthesizing a more theoretical explanation.

In media res is no problem: you don't have to remember how it all started and try to catch us up. Just tell what happened yesterday, but go into twice as much detail as you think is overkill.

Interesting idea. I was doing that before in here, I know. Was that more helpful? I felt like people would have a hard time visualizing the lessons if I didn’t explain how I introduced things in the first place, though?
 

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2 minutes ago, Not_a_Number said:

I felt like people would have a hard time visualizing the lessons if I didn’t explain how I introduced things in the first place, though?

I don't think so: "how things started" and "what things look like now" are separate reading problems (and more importantly: separate writing problems for you). We won't have trouble visualizing anything that you describe vividly. We don't have to know right away why you're describing it, why you did it the way you're describing and not some other way, or how you got to the point that you were able to do what you described.

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3 minutes ago, UHP said:

I don't think so: "how things started" and "what things look like now" are separate reading problems (and more importantly: separate writing problems for you). We won't have trouble visualizing anything that you describe vividly. We don't have to know right away why you're describing it, why you did it the way you're describing and not some other way, or how you got to the point that you were able to do what you described.

Do you think it'd be a good idea to do a few weeks of "what we've done so far"? Honestly, that was kind of the idea of doing these more theoretical posts, although they are definitely taking more thought than I was hoping. But maybe I can just bang them out quickly and get going on the examples, starting from the start? I do have a very good record in the other thread that I can use to pull transcripts from. 

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3 minutes ago, Not_a_Number said:

Do you think it'd be a good idea to do a few weeks of "what we've done so far"?

To me, it sounds like you could get writers block right away.

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1 minute ago, UHP said:

To me, it sounds like you could get writers block right away.

You mean it'd be harder to get over the hump? Yes, I've been finding that. I suppose it doesn't matter if I do things out of order, you're saying? I could link them up into some sort of timeline later? 

But I could almost copy and paste from the thread, which would be an easy way NOT to get writer's block. 

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21 minutes ago, Not_a_Number said:

I suppose it doesn't matter if I do things out of order, you're saying? I could link them up into some sort of timeline later? 

Yes, I think doing things in order doesn't matter on a blog. Even less than it mattered in the Iliad or in Pulp fiction.

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3 hours ago, Not_a_Number said:

Why wouldn’t it work? It’s not arbitrary, it’s fixed. I’m not against writing the numbers on them, but I haven’t had to — same as with the C-rods, kids get used to what each color is worth. I can imagine kids not being able to do it, but I didn’t have that happen with any of the 30 kids I had in my classes.

Did you choose that yellow equals 100, blue equals 1 and so on? What made you choose which color represented what value? That's what I mean by arbitrary. They may be fixed but there is no rhyme or reason as to which color chip represents what value. They just are because you said they are. Children with certain learning disabilities would have just as much trouble discerning the abstract color system as they do the abstract concept of place value. Rather than tax their already taxed memory functions, I find it works better to go from concrete to abstract by using something like base 10 blocks where they can physically see that 10 is 10 times bigger than 1 and 100 is 10 times bigger than 10 and so on.

I also deal with color blindness within my own family. C-rods still work with color blindness because they can physically see that each block is proportional to another such as ten cubes is equal to the longest rod even if they cannot see what color they are. Sure, you could buy poker chips with the value printed on them or write the value in Sharpie or something but now you are basically working with the same abstract concept of place value as you were before you introduced the manipulatives. 

I've used place value chips or cards in my teaching before but they are not my favorite manipulative. Especially for introducing the concept of place value. To me, the whole point of manipulatives is to make concepts more concrete so it is easier to understand the abstract concept they represent. Making the manipulatives equally as abstract as the concept seems counterintuitive to me.

I have apparently run into more neurodiversity than you have in my years of teaching and tutoring a variety of children. IME,  the best manipulative to use is the one that makes the most sense to the child in front of you. That's why I introduce several different manipulatives to any child that doesn't latch on to the first one I use. Once they latch on to a certain representation, we use that same representation consistently. 

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8 minutes ago, sweet2ndchance said:

Did you choose that yellow equals 100, blue equals 1 and so on? What made you choose which color represented what value? That's what I mean by arbitrary. They may be fixed but there is no rhyme or reason as to which color chip represents what value. They just are because you said they are. Children with certain learning disabilities would have just as much trouble discerning the abstract color system as they do the abstract concept of place value. Rather than tax their already taxed memory functions, I find it works better to go from concrete to abstract by using something like base 10 blocks where they can physically see that 10 is 10 times bigger than 1 and 100 is 10 times bigger than 10 and so on.

I also deal with color blindness within my own family. C-rods still work with color blindness because they can physically see that each block is proportional to another such as ten cubes is equal to the longest rod even if they cannot see what color they are. Sure, you could buy poker chips with the value printed on them or write the value in Sharpie or something but now you are basically working with the same abstract concept of place value as you were before you introduced the manipulatives. 

I've used place value chips or cards in my teaching before but they are not my favorite manipulative. Especially for introducing the concept of place value. To me, the whole point of manipulatives is to make concepts more concrete so it is easier to understand the abstract concept they represent. Making the manipulatives equally as abstract as the concept seems counterintuitive to me.

For kids with color blindness, I'd obviously label the chips or something. But it doesn't really change the idea. 

For me, the whole POINT is that the manipulative is just as abstract as the concept. And you're right that this means that at the beginning, kids aren't proficient with the manipulative. In fact, I've found that it takes years to get fully proficient with the manipulative, just like it takes kids years to get proficient with place value. 

The difference is that kids can USE the manipulative even when they aren't fully able to grok the abstraction. The experience of using a manipulative that lets one experience the abstraction firsthand is what creates the fluency, in my experience. I feel like the experience of immersion in an abstraction is a very specific one that creates fluency. That's how I teach, anyway, and I've been very happy with it. 

I am obviously not communicating what we do very well, but I can assure you that I haven't only worked with "advanced" kids or something. What would help communicate what it is that we do? It really did work well with my very wide range of homeschooled kids -- the only ones who weren't ready were the ones who weren't really able to conceive of quantities as units quite yet, and I tend to think that at this stage kids aren't actually ready for place value. 

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15 minutes ago, sweet2ndchance said:

I have apparently run into more neurodiversity than you have in my years of teaching and tutoring a variety of children. IME,  the best manipulative to use is the one that makes the most sense to the child in front of you. That's why I introduce several different manipulatives to any child that doesn't latch on to the first one I use. Once they latch on to a certain representation, we use that same representation consistently. 

I think the question of what it means to "make sense" is a really interesting one, actually... with new mental models, I actually expect the sense-making phase to be quite slow. Fluently using new ideas is HARD. 

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Anyway, I don't want to argue about my methodology. What I'm curious about is how to explain it so people understand what I mean, because every time I try to explain, I tend to be told that it can't possibly work. But I can absolutely assure you that it works for a very large chunk of kids -- I mean, it's possible all the kids I got in my classes were brilliant, but I really did NOT have that impression. I've so far not confused any kids whatsoever and have confused almost every single adult I tried to explain this to, which is a pretty serious disconnect! And I'm curious how to bridge that -- what can I say or do to communicate? Would examples of lessons be the best thing? 

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47 minutes ago, Not_a_Number said:

That's how I teach, anyway, and I've been very happy with it.

And that's fine. What works for you, works for you. But what works for you, won't work for everybody. We obviously have different pedagogies. Neither is better, just different. That's all I was pointing out.

I'm always interested in different ways to represent ideas which is why I was reading your thread in the first place. I like to have a wide variety of tools at my disposal so that I can teach as flexibly as possible. Unless there is some something that you haven't mentioned, your method seems very similar to place value chips or cards, which like I said isn't my favorite representation but to each their own. If it works for you and the students in front of you who are ready to tackle the idea, then great. It's just not what I would use as a first exposure technique and I tried to explain why. And yes I agree, it can take years to become fluent with anything. It is what it is which is why picking one representation and sticking with it has value. But when a student has obvious issues with the representation I'm giving, I'm not going to beat my head against a wall to make them understand my representation, I just pull out a different tool to teach with until they have that "Aha!" moment. This technique has never failed to work for me but obviously, your mileage may vary.

I have no problem understanding what you are saying, so unless there is something you are leaving out, I don't think it is a lack of comprehension on the adults part or explaining on your part. And I'm not saying your system "can't possibly work", I'm not sure where you got that idea. Just that it won't work for all kids and I gave a couple of examples where a different method or slight modification would be necessary in my opinion. Personally, I would rather change my teaching method to fit the child but again, many teachers teach the same method to all kids with success. 

But you've said that you aren't interested in critiques of your system in this post, so I will bow out. Best of luck.

 

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17 hours ago, Not_a_Number said:

Does it do grouping or units for 10s? I haven't used RightStart, so I don't know. 

If I recall correctly, I think it does both - one side is grouping, the other side is represented as units.

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I think walking us through an actual lesson would be really helpful.   Especially a comparison between what you do to introduce the concept and what you do when the topic is more "integrated", for example when you are using it to do addition or multiplication.   Because it is clearly central to you in terms of visualization of concepts, and yet for those who aren't used to having it as central, it is difficult to see what you mean.   Because in many curriculums place value is basically one unit in a year, and if you understand that unit, it is an underpinning to other concepts that come later, but not central.  So trying to visualize how you are teaching this in a different manner so that it is the integral concept is just hard to picture. 

In addition,  walking through what you do with a child who is "getting it"  versus what you do with a child who is struggling would be helpful as well. 

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17 minutes ago, sweet2ndchance said:

And that's fine. What works for you, works for you. But what works for you, won't work for everybody. We obviously have different pedagogies. Neither is better, just different. That's all I was pointing out.

I'm always interested in different ways to represent ideas which is why I was reading your thread in the first place. I like to have a wide variety of tools at my disposal so that I can teach as flexibly as possible. Unless there is some something that you haven't mentioned, your method seems very similar to place value chips or cards, which like I said isn't my favorite representation but to each their own. If it works for you and the students in front of you who are ready to tackle the idea, then great. It's just not what I would use as a first exposure technique and I tried to explain why. And yes I agree, it can take years to become fluent with anything. It is what it is which is why picking one representation and sticking with it has value. But when a student has obvious issues with the representation I'm giving, I'm not going to beat my head against a wall to make them understand my representation, I just pull out a different tool to teach with until they have that "Aha!" moment. This technique has never failed to work for me but obviously, your mileage may vary.

I have no problem understanding what you are saying, so unless there is something you are leaving out, I don't think it is a lack of comprehension on the adults part or explaining on your part. And I'm not saying your system "can't possibly work", I'm not sure where you got that idea. Just that it won't work for all kids and I gave a couple of examples where a different method or slight modification would be necessary in my opinion. Personally, I would rather change my teaching method to fit the child but again, many teachers teach the same method to all kids with success. 

But you've said that you aren't interested in critiques of your system in this post, so I will bow out. Best of luck.

I actually don't think you quite follow what I'm saying 🙂 . But if you're not interested, I'm obviously not going to make you follow the thread!!! We can talk about it some other time if you're ever interested. 

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1 minute ago, Not_a_Number said:

I actually don't think you quite follow what I'm saying 🙂 . But if you're not interested, I'm obviously not going to make you follow the thread!!! We can talk about it some other time if you're ever interested. 

Never said I wouldn't follow the thread, just that I wouldn't critique your method any further in this thread. 

So what am I not following in your opinion? What makes you think that I and other adults aren't following what you are saying?

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10 minutes ago, NorthernBeth said:

I think walking us through an actual lesson would be really helpful.   Especially a comparison between what you do to introduce the concept and what you do when the topic is more "integrated", for example when you are using it to do addition or multiplication.   Because it is clearly central to you in terms of visualization of concepts, and yet for those who aren't used to having it as central, it is difficult to see what you mean.   Because in many curriculums place value is basically one unit in a year, and if you understand that unit, it is an underpinning to other concepts that come later, but not central.  So trying to visualize how you are teaching this in a different manner so that it is the integral concept is just hard to picture. 

Yes. I think you put your finger on it. I don't think of place value as one unit in a year -- I think of it as central to everything we do in a numerical system. All of your algorithms are place value writ large. 

 

10 minutes ago, NorthernBeth said:

In addition,  walking through what you do with a child who is "getting it"  versus what you do with a child who is struggling would be helpful as well. 

OK, so that's a REALLY good point. I don't tend to believe that there's "getting it" versus "not getting it" for most kids who are ready to start place value at all. There's really just varying levels of comfort. So my more comfortable kids would be able to think backwards about trading, for example. And the number of trading STEPS one is willing to do tends to change depending on level of comfort. 

So, the least comfortable thing is trying to do 14 + 18, winding up with 12 blue poker chips (1s, over here), and saying "I have no idea what to do next." But that's not a PROBLEM or anything. It's just the earliest stage, before the idea of trading has been absorbed in any way, shape, or form. It just requires a reminder. 

 

23 minutes ago, sweet2ndchance said:

Just that it won't work for all kids and I gave a couple of examples where a different method or slight modification would be necessary in my opinion.

Right, but the modifications don't actually change the idea at all. The color or lack of labeling or kind of manipulative is actually basically not in any way central. 

 

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2 minutes ago, sweet2ndchance said:

Never said I wouldn't follow the thread, just that I wouldn't critique your method any further in this thread. 

So what am I not following in your opinion? What makes you think that I and other adults aren't following what you are saying?

Well, honestly, some of the things people are mentioning 🙂. Like, I don't care whether the chips are colorful or labelled particularly. I wouldn't mind if some of them were bigger than others (DD8 drew boxes for tens, which were in fact bigger than the dots we drew for ones!) if that helps kids distinguish, and I wouldn't have trouble adjusting in this way, although I haven't had to with any of the kids I've worked with yet. But I'm not averse and it doesn't change the point. 

Maybe @NorthernBeth is getting at the crux of the matter -- I think of place value as really central and not as a single unit in a single year, and the way I tackle it with the manipulatives treats it that way? Because for me, the point of the manipulatives is really just to make the abstraction usable, and I think this particular abstraction is a really, really enduring thread in elementary mathematics. You really only stop needing it past when you've done decimals and percents. 

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3 hours ago, Not_a_Number said:

Anyway, I don't want to argue about my methodology. What I'm curious about is how to explain it so people understand what I mean, because every time I try to explain, I tend to be told that it can't possibly work. But I can absolutely assure you that it works for a very large chunk of kids -- I mean, it's possible all the kids I got in my classes were brilliant, but I really did NOT have that impression. I've so far not confused any kids whatsoever and have confused almost every single adult I tried to explain this to, which is a pretty serious disconnect! And I'm curious how to bridge that -- what can I say or do to communicate? Would examples of lessons be the best thing? 

Again, I implore you to show don't tell.  If it does work, show me that it works.  Show me what you are doing.  Even if it's an animation, along the lines of 3Blue1Brown (his s/w is freely avaliable I think), you need to show us that it works.  

If you tell me, then I will reflexively disagree.  

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Just now, daijobu said:

Again, I implore you to show don't tell.  If it does work, show me that it works.  Show me what you are doing.  Even if it's an animation, along the lines of 3Blue1Brown (his s/w is freely avaliable I think), you need to show us that it works.  

If you tell me, then I will reflexively disagree.  

But I don't know how to show it! It takes years to work. It's not one or two lessons, it's a program, and I don't think any particular step in the program is all that inspiring. 

I don't even LIKE things like 3Blue1Brown, because they lead to a false sense of understanding. It's like reading solutions instead of doing the problems -- you can think you get it and you really, really don't. 

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2 minutes ago, daijobu said:

Again, I implore you to show don't tell.  If it does work, show me that it works.  Show me what you are doing.  Even if it's an animation, along the lines of 3Blue1Brown (his s/w is freely avaliable I think), you need to show us that it works.  

If you tell me, then I will reflexively disagree.  

So, on a more positive note, HOW would one show it? Would following my kids for a few years do it? 

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Just now, daijobu said:

It's like you are pointing to a cardboard box and explaining to use what is inside and how it's so much better.  Why not just open up the box?  

Because I feel like the box is open and everyone is acting like it's closed!! 😂

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Posted (edited)
1 minute ago, daijobu said:

So when I open up the cardboard box, I see... poker chips.  

Right, and that's obviously not what you should be seeing, lol. But I don't know how to demonstrate. I don't want to post videos of my kids, and anyway, I'm not sure that would be helpful... 

If I post a few lesson transcripts a week for a few years, would that do? 

Edited by Not_a_Number
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