Math is Figure-Out-Able

[dovrar]

Has anyone here heard of Pamela Harris? I just did a free 3 day challenge webinar series that she offers and it was amazing! The biggest takeaway for me was the idea that teaching kids math in a way that requires them to think through what they're doing is an excellent way to begin to teach logic to young children and promote independent thinking. Love it!

[HomeAgain]

I can't say I've heard of her, but I think anyone who promotes math as something to think through is a good person to follow!

[daijobu]

@Not_a_Number coined the idea of "learned helplessness" in mathematics.  The way many students learn math is that they are unable to logic their way out of unfamiliar territory.  But math is ALL about this.  I'm also not familiar with Harris, but agree with anyone who is combating this learned helplessness. 

[Not_a_Number]

12 minutes ago, daijobu said:

@Not_a_Number coined the idea of "learned helplessness" in mathematics.  The way many students learn math is that they are unable to logic their way out of unfamiliar territory.  But math is ALL about this.  I'm also not familiar with Harris, but agree with anyone who is combating this learned helplessness. 

I don’t think I coined it, but I love the phrase 🙂 

[8filltheheart]

I have never heard of her either, but the premise behind the excitement made me laugh.  Sometimes the absolute obvious is overlooked and has to be pointed out as........ obvious.  Our entire modern educational system's pedagogy is designed around simple knowledge as master.  (fill in the blank, matching, xyz correct answer)   There is a reason why we homeschool.  🙂  

Edited September 10, 2021 by 8filltheheart

[dovrar]

1 hour ago, 8filltheheart said:

I have never heard of her either, but the premise behind the excitement made me laugh.  Sometimes the absolute obvious is overlooked and has to be pointed out as........ obvious.  Our entire modern educational systems pedagogy is designed around simply knowledge as master.  (fill in the blank, matching, xyz correct answer)   There is a reason why we homeschool.  🙂  

So true!

[dovrar]

I have always thought it important to teach the whys behind math and to struggle through for an answer. Like 8filltheheart said, I just had never heard it put so plainly. Pam Harris helps by giving teachers strategies to work through problems in ways that get kids thinking outside of the box.

[HomeAgain]

34 minutes ago, dovrar said:

I have always thought it important to teach the whys behind math and to struggle through for an answer. Like 8filltheheart said, I just had never heard it put so plainly. Pam Harris helps by giving teachers strategies to work through problems in ways that get kids thinking outside of the box.

Sometimes there's a difference between knowing something and having the strategies to apply it.  I'm still working on the second. 🙂  My weekend right now is spent trying to figure out ways to teach strategies of long multiplication in micro-steps for one of "my" kids.  Another one is working on math vocabulary and another is working on slowing down his brain to be able to verbalize what he is doing - (I know him, he's going to hit a wall of being unable to figure it out if he just "knows" the answer to the easy problems but not how he got there).  I think we're going to play the Montessori banker's game next week because it'll work on all three of those skills if I add in some specifics.

[Not_a_Number]

2 hours ago, HomeAgain said:

Sometimes there's a difference between knowing something and having the strategies to apply it.  I'm still working on the second. 🙂  My weekend right now is spent trying to figure out ways to teach strategies of long multiplication in micro-steps for one of "my" kids.  Another one is working on math vocabulary and another is working on slowing down his brain to be able to verbalize what he is doing - (I know him, he's going to hit a wall of being unable to figure it out if he just "knows" the answer to the easy problems but not how he got there).  I think we're going to play the Montessori banker's game next week because it'll work on all three of those skills if I add in some specifics.

Let me know if you need suggestions 🙂 

[HomeAgain]

47 minutes ago, Not_a_Number said:

Let me know if you need suggestions 🙂 

I'm always open to ideas!  Kid 2 is doing what he needs to, I think.  He gets confused looking at a word problem, but when he reads it aloud or has it read to him, he can hear the vocabulary cues and take the next step of circling or underlining them and writing it out "in math".  It's just the process now of interpreting what he's reading into the verbal (visual?) memory cues so he can picture it as he reads silently.

[LMD]

On 9/11/2021 at 1:57 AM, HomeAgain said:

I'm always open to ideas!  Kid 2 is doing what he needs to, I think.  He gets confused looking at a word problem, but when he reads it aloud or has it read to him, he can hear the vocabulary cues and take the next step of circling or underlining them and writing it out "in math".  It's just the process now of interpreting what he's reading into the verbal (visual?) memory cues so he can picture it as he reads silently.

I literally talk to my kids about how we're translating from 'english' language sentence into a 'math' language sentence, it seems to help them find the components. This is in conjunction with talking about maths as just the language of science - math is a pure description of our world, it's not just fiddling around with squiggles on a page. 

[Not_a_Number]

On 9/10/2021 at 8:46 AM, HomeAgain said:

Sometimes there's a difference between knowing something and having the strategies to apply it.  I'm still working on the second. 🙂  My weekend right now is spent trying to figure out ways to teach strategies of long multiplication in micro-steps for one of "my" kids.  Another one is working on math vocabulary and another is working on slowing down his brain to be able to verbalize what he is doing - (I know him, he's going to hit a wall of being unable to figure it out if he just "knows" the answer to the easy problems but not how he got there).  I think we're going to play the Montessori banker's game next week because it'll work on all three of those skills if I add in some specifics.

In terms of long multiplication, what can the kid currently do to multiply, if you don't mind me asking? 🙂 

[HomeAgain]

7 hours ago, Not_a_Number said:

In terms of long multiplication, what can the kid currently do to multiply, if you don't mind me asking? 🙂 

 

He understands the idea of multiplication, has most of his facts memorized, and can find an answer by breaking it into two sets of facts when he doesn't know (like if he can't remember 8x6, he knows (10x6)-(2x6) and can work that way).  He can multiply up to 3 digits by 1 digits without problem, but place value trips him up.  We're starting to move into 2 digits, and we're working on this in two ways currently, both heavy on the place value.

The first is with an array of tokens.  643 x 14 means

4 of the stacks of 3 units

4 of the stacks of 4 tens

4 of the stacks of 6 hundreds

AND

10 of the stacks of 3 units

10 of the stacks of 4 tens

10 of the stacks of 6 hundreds.

We have a gridded worksheet to work each part of the problem separately and then as a stacked problem.

This leads into the checkerboard, where we use the units on the colors for each place value and they can shift over to create a ten and slide in with the rest.

Slowly this all becomes a way to do standard multiplication on paper, working from right to left.  When we get to decimals, it'll be a really good tool to revisit.

-----------

The other part is working on multiplication as area, but we can only do that with smaller numbers (tens, no hundreds).  14 x 13 is laid down on the outside of the cloth napkin, along the edges.  The problem is then worked from the smallest to the biggest: 3 rows of 4 blocks. 3 rows of ten blocks. 10 of the 4s (or 4 of the tens), and 10 of the tens.  Then added together.  We use the same gridded worksheet to get the answers, and it's written out first as miniature area blocks and then as they progress, with the exact same worksheet from the other method.

I want them to see it both ways and be comfortable with the area method for algebra, but know the standard algorithm.  Right now we're still in the beginning stages for this one kid, really working on place value and groups.  It's why I think the banking game will be fun.  They'll have to take turns with each role and work on each multiplication step.

[Not_a_Number]

1 hour ago, HomeAgain said:

 

He understands the idea of multiplication, has most of his facts memorized, and can find an answer by breaking it into two sets of facts when he doesn't know (like if he can't remember 8x6, he knows (10x6)-(2x6) and can work that way).  He can multiply up to 3 digits by 1 digits without problem, but place value trips him up.  We're starting to move into 2 digits, and we're working on this in two ways currently, both heavy on the place value.

The first is with an array of tokens.  643 x 14 means

4 of the stacks of 3 units

4 of the stacks of 4 tens

4 of the stacks of 6 hundreds

AND

10 of the stacks of 3 units

10 of the stacks of 4 tens

10 of the stacks of 6 hundreds.

We have a gridded worksheet to work each part of the problem separately and then as a stacked problem.

This leads into the checkerboard, where we use the units on the colors for each place value and they can shift over to create a ten and slide in with the rest.

Slowly this all becomes a way to do standard multiplication on paper, working from right to left.  When we get to decimals, it'll be a really good tool to revisit.

-----------

The other part is working on multiplication as area, but we can only do that with smaller numbers (tens, no hundreds).  14 x 13 is laid down on the outside of the cloth napkin, along the edges.  The problem is then worked from the smallest to the biggest: 3 rows of 4 blocks. 3 rows of ten blocks. 10 of the 4s (or 4 of the tens), and 10 of the tens.  Then added together.  We use the same gridded worksheet to get the answers, and it's written out first as miniature area blocks and then as they progress, with the exact same worksheet from the other method.

I want them to see it both ways and be comfortable with the area method for algebra, but know the standard algorithm.  Right now we're still in the beginning stages for this one kid, really working on place value and groups.  It's why I think the banking game will be fun.  They'll have to take turns with each role and work on each multiplication step.

So the thing I always focus on with long multiplication is

a) The associative and distributive properties

and 

b) Multiplication by 10. 

So I'd generally make sure kids are really good and intuitive with multiplying by 10, and that they'd also have a great understanding of the fact that 30*something is the same thing as 3 times (10*something.) Then I tend to start splitting things up like 

23*something = 20*something + 3*something = 2*(10*something) + 3*something. 

At least, that's what I've done with my kids and my stronger online class kids. I haven't gotten that far into multiplication with my weaker online class kids -- we're still working on smaller numbers. 

[HomeAgain]

18 minutes ago, Not_a_Number said:

So the thing I always focus on with long multiplication is

a) The associative and distributive properties

and 

b) Multiplication by 10. 

So I'd generally make sure kids are really good and intuitive with multiplying by 10, and that they'd also have a great understanding of the fact that 30*something is the same thing as 3 times (10*something.) Then I tend to start splitting things up like 

23*something = 20*something + 3*something = 2*(10*something) + 3*something. 

At least, that's what I've done with my kids and my stronger online class kids. I haven't gotten that far into multiplication with my weaker online class kids -- we're still working on smaller numbers. 

That's similar to how I'm approaching it.  With using the blocks/place value discs, we've been focusing on the specific vocabulary of multiplication. 3 x 10 is the same as 3 of the tens.  When we have 10 of the tens, it's a new place value.  We are slowly working toward the crossed bar approach where we create the stacks of 10 x 10 x 4 or what have you but it's just a step he's not quite to yet.  He's still grouping physically.  I have a feeling this specific kid is going to be hanging out with place value for a long while, until he gets really, really comfortable.  Once he gets it, we can move that that multiple crossed bars, but there's probably going to be an intermediate step for him of literally bringing out trays and showing groups of trays of blocks/place value discs to be 2 of 10  of something.

 

Feel free to correct me if I'm misunderstanding what you're saying.

[Not_a_Number]

59 minutes ago, HomeAgain said:

That's similar to how I'm approaching it.  With using the blocks/place value discs, we've been focusing on the specific vocabulary of multiplication. 3 x 10 is the same as 3 of the tens.  When we have 10 of the tens, it's a new place value.  We are slowly working toward the crossed bar approach where we create the stacks of 10 x 10 x 4 or what have you but it's just a step he's not quite to yet.  He's still grouping physically.  I have a feeling this specific kid is going to be hanging out with place value for a long while, until he gets really, really comfortable.  Once he gets it, we can move that that multiple crossed bars, but there's probably going to be an intermediate step for him of literally bringing out trays and showing groups of trays of blocks/place value discs to be 2 of 10  of something.

 

Feel free to correct me if I'm misunderstanding what you're saying.

So my current thinking is actually that it's easier to teach kids to take 10 of something than it is to take that number of 10s. With DD9, I definitely had them taking copies of 10, but I realize that it's much easier to see that each stack becomes the "next place" if you take 10 of something. 

So, for 34*10, I'd definitely have them do "ten 34s" and not "34 tens." You know what I mean? Because I found that with DD9, taking 34 10s required a LOT of steps to make it clear why it's 340, whereas if you take ten 34s, there's a very easy way to demo it. 

I really need to start updating my blog again. We've been totally overwhelmed with the move and with all of DD9's behavior issues 😕 . But I think some sample lessons would help a lot more than me just typing randomly. 

[dovrar]

That is the beauty of modeling with blocks. You can use an area model and show that 34 tens is the same as ten 34s.

A good exercise might be to build a model and ask if she can find two ways to represent the model. 

[Not_a_Number]

24 minutes ago, dovrar said:

That is the beauty of modeling with blocks. You can use an area model and show that 34 tens is the same as ten 34s.

A good exercise might be to build a model and ask if she can find two ways to represent the model. 

Well, yes, they are the same, but if you're using place value manipulatives, one of those is easier to think about than the other. 

It's like doing 2*34. THat's the same thing as either 2 34s or 34 2s, but one of those is a LOT easier to think about 😉.

[Not_a_Number]

I actually don't spend much time with the area model. I use it to show kids the commutative property and that's about it. I don't find it nearly as rich as a mix of place value and the definition. 

[dovrar]

Yes, but if the child realizes that they are equivalent then they can choose the easier one to solve. Does that make sense? With my kids they didn't necessarily initially recognize that they had that option.

[Not_a_Number]

Just now, dovrar said:

Yes, but if the child realizes that they are equivalent then they can choose the easier one to solve. Does that make sense? With my kids they didn't necessarily initially recognize that they had that option.

Oh, absolutely. I have to explicitly teach that generally. I've recently taught it to DD5 and she's been making good use of it 🙂 . (I have very mathy kids, so multiplying at age 5 is very much the norm here.) 

[dovrar]

Okay, just went back and reread the initial example of 34 tens or ten 34s. To me it seems easier to find 34 tens. I mean, I for one don't typically count by 34s. But counting by tens is easy peasy. If I know (or can figure out that) ten 10s are 100 then three of those are 300. And 4 tens are simply 40. 

Whatever clicks for the child though. 

[HomeAgain]

3 hours ago, Not_a_Number said:

So my current thinking is actually that it's easier to teach kids to take 10 of something than it is to take that number of 10s. With DD9, I definitely had them taking copies of 10, but I realize that it's much easier to see that each stack becomes the "next place" if you take 10 of something. 

So, for 34*10, I'd definitely have them do "ten 34s" and not "34 tens." You know what I mean? Because I found that with DD9, taking 34 10s required a LOT of steps to make it clear why it's 340, whereas if you take ten 34s, there's a very easy way to demo it. 

I really need to start updating my blog again. We've been totally overwhelmed with the move and with all of DD9's behavior issues 😕 . But I think some sample lessons would help a lot more than me just typing randomly. 

Hmm....I'm going to have to mull this over.  Since we use base ten blocks, I wonder if it's a difference that will require more steps than just a mental process would.

[Not_a_Number]

33 minutes ago, HomeAgain said:

Hmm....I'm going to have to mull this over.  Since we use base ten blocks, I wonder if it's a difference that will require more steps than just a mental process would.

I can explain how I use poker chips for multiplying by 10, if you like! DD5 caught on really quick with this method.

 

2 hours ago, dovrar said:

Okay, just went back and reread the initial example of 34 tens or ten 34s. To me it seems easier to find 34 tens. I mean, I for one don't typically count by 34s. But counting by tens is easy peasy. If I know (or can figure out that) ten 10s are 100 then three of those are 300. And 4 tens are simply 40. 

Whatever clicks for the child though. 

I don’t count by 34s, but taking 10 of a particular “unit” bumps it up to the next place. So it’s actually clearer that everything will go up a place if you arrange ten 34s in rows and then group by place.

I also used to think that it’s easier to count by 10s, but having done this with both kids, the idea that each digit “shifts” into the next place is easier if you take 10 copies of something.

I can make a blog post if anyone’s interested!! I’ve been neglecting my blog, but I’ve been meaning to pick it back up.