Why is a negative times a negative a positive?

[fairfarmhand]

@Not_a_Number

I may be showing my ignorance here, but it occurred to me that I memorized the rule but have no idea why it's mathematically true for a negative times a negative to be a positive number. 

I even asked my engineering math nerd dh and he looked at me with a blank stare and said ."I have no idea. I've never thought of that before."

So, break it down for a dummy like me. Why is a negative times a negative a positive number?

[Not_a_Number]

1 minute ago, fairfarmhand said:

@Not_a_Number

I may be showing my ignorance here, but it occurred to me that I memorized the rule but have no idea why it's mathematically true for a negative times a negative to be a positive number. 

I even asked my engineering math nerd dh and he looked at me with a blank stare and said ."I have no idea. I've never thought of that before."

So, break it down for a dummy like me. Why is a negative times a negative a positive number?

I’m going to be super annoying here and ask you some more questions first 😉 .

First of all, in your opinion, why is a positive times a negative a negative?

[egao_gakari]

Bump! I'm very curious.

[fairfarmhand]

6 minutes ago, Not_a_Number said:

I’m going to be super annoying here and ask you some more questions first 😉 .

First of all, in your opinion, why is a positive times a negative a negative?

You would do that... 🙂

Explaining that for me is to explain multiplication as repeated addition.

3 times -3 is the same as -3+-3+-3= -9

When you take a positive times a negative, you are essentially saying you have multiples of that negative number so of course, the answer will be negative. Or at least that's how I think of it. But that same mindset blows up my brain when I try to take a negative times a negative.

Edited February 23, 2021 by fairfarmhand

[Not_a_Number]

1 minute ago, fairfarmhand said:

You would do that... 🙂

Explaining that for me is to explain repeated addition.

3 times -3 is the same as -3+-3+-3= -9

When you take a positive times a negative, you are essentially saying you have multiples of that negative number so of course, the answer will be negative. Or at least that's how I think of it. But that same mindset blows up my brain when I try to take a negative times a negative.

Oh, totally. At some point, I made up a reason for it in terms of repeated addition and a careful mental model of multiplication, but that's not really how I think of it. 

OK, second question, and then I'll explain. Can you explain why

(3+4)*5 = 3*5 + 4*5? 

That is, can you explain the distributive property? 

[ieta_cassiopeia]

I'd have said it's because negative is opposite to positive. At that point, it's possible to make an analogy using manipulatives.

 

Take some books. If the front cover is facing up, they're positive and if the back cover faces up, they're negative. Start by having them all face front cover upwards, since we live in a world that uses positive numbers more often than negative numbers.

 

2 books x 3 books = 2 groups of 3 books = 6 books. No books were flipped, so they're all still positive.

-2 books x 3 books... ....there's still going to be 6 books. But which way do they face?

There is a request here to do the opposite of 2 books x 3 books once. So flip all the books over once. What do you get? All 6 books have the back cover facing upwards. Therefore, we have -6.

 

-2 books x -3 books = -2 groups of -3 books.

There is a request here to do the opposite of 2 groups x 3 books twice. So, flip all the books over twice. They are now all facing front-cover up, which means we have 6 books.

So proof, via manipulatives, that a negative means "the opposite of positive" and how to know how different uses of negative signs affect the signing of the answer.

[Not_a_Number]

2 minutes ago, ieta_cassiopeia said:

-2 books x 3 books... ....there's still going to be 6 books. But which way do they face?

I don't think you're multiplying books by books, since that would require doing books squared. Rather, you're multiplying a unitless -2 by 3 books. 

I think it's POSSIBLE to model this one, but this is one of the few things I chose not to model for my DD8. And I'm compulsive about modeling in general and had her model absolutely every other negative number thing. But this one I find to be easier as a "pure property" thing. 

[ieta_cassiopeia]

(3+4)*5 = 3*5 + 4*5? 

 

Get 35 books.

3 books in one row.

4 books in another row.

Do this 5 times, so you have your 5 groupings of 3 + 4 books.

Move the other book groups, such that you have a row with 5 lots of 3 books, above a row of 5 lots of 4 books.

 

The total number of books is the same.

Thus, distributive property applies. (In practice, it would likely take more examples than just the one for this to be learned via this method).

[fairfarmhand]

13 minutes ago, Not_a_Number said:

Oh, totally. At some point, I made up a reason for it in terms of repeated addition and a careful mental model of multiplication, but that's not really how I think of it. 

OK, second question, and then I'll explain. Can you explain why

(3+4)*5 = 3*5 + 4*5? 

That is, can you explain the distributive property? 

I"m still thinking about this. Struggling with concentration this morning....

[Not_a_Number]

1 minute ago, fairfarmhand said:

I"m still thinking about this. Struggling with concentration this morning....

Hint: count in 5’s. 

[ieta_cassiopeia]

2 minutes ago, Not_a_Number said:

I don't think you're multiplying books by books, since that would require doing books squared. Rather, you're multiplying a unitless -2 by 3 books. 

I think it's POSSIBLE to model this one, but this is one of the few things I chose not to model for my DD8. And I'm compulsive about modeling in general and had her model absolutely every other negative number thing. But this one I find to be easier as a "pure property" thing. 

*facepalm* You're right. It's possible to do that model provided, with the flips being used to indicate signing... ...but it should definitely be written as 2 x 3 books, -2 x 3 books etc.

[sweet2ndchance]

I know you asked NotaNumber but I just wanted to add how I think of it.

Take -3 x -3.

Think of it as money. You owe 3 dollars to 3 people.

If those 3 people forgive your debt, you now have -3 debts to people. Those debts have essentially been taken away.

So your -$3 x -3 = $9. You would be in the black and not in the red.

I probably didn't explain it the best but it makes sense to me and my kids.

[sweet2ndchance]

Someone, I think maybe it was @wendyroo, explained different operations with negative numbers with digging holes. It was clever and it made sense but I know I couldn't do that explanation justice. Hopefully, someone can provide that explanation again.

[Not_a_Number]

5 minutes ago, sweet2ndchance said:

I know you asked NotaNumber but I just wanted to add how I think of it.

Take -3 x -3.

Think of it as money. You owe 3 dollars to 3 people.

If those 3 people forgive your debt, you now have -3 debts to people. Those debts have essentially been taken away.

So your -$3 x -3 = $9. You would be in the black and not in the red.

I probably didn't explain it the best but it makes sense to me and my kids.

Yeah, that's the model that makes the most sense to me 🙂 . 

 

I also like thinking of it as what you need to be true for the distributive property to work. Since you want 

(3 + (-3))*(-3) to be 0, 

you want 

3*(-3) + (-3)*(-3) to be 0.

And that requires (-3)*(-3) to cancel out the -9, which makes it 9. 

 

At some point, I realized that when I was modeling negatives as debt, it really only applied to the second number being multiplied and not so much the first. So the first one is more like "copies," and I guess "negative copies" means "you owe that many copies." So then it's like you owe a negative amount, and if you owe a negative amount, you need to give someone a negative amount, which means... really, they owe you money. 

But I think that rephrasing works best if you already think of negatives as debt. 

 

 

 

[Not_a_Number]

4 minutes ago, sweet2ndchance said:

Someone, I think maybe it was @wendyroo, explained different operations with negative numbers with digging holes. It was clever and it made sense but I know I couldn't do that explanation justice. Hopefully, someone can provide that explanation again.

I've seen that explanation and it's VERY clever, but it's not how I ever think of negative numbers when I work with them. And I prefer keeping my mental models consistent. 

[Swimmer1112]

Multiplying by a negative is repeated subtraction. When we multiply a negative number times a negative number, we are getting less negative.

for example:

Imagine we represent multiplication as jumps on a number line.

3 times 3 on the number line

For 3 × 3, we draw 3 groups of 3 moving to the right. Both the number of groups and the direction of each group are to the right.

But what about 3 × -3? Now we have 3 groups of the number still, but the number is negative.

3 times -3 on the number line

If we find -3 × 3, the size and direction of the number we multiply are the same, but now we are finding -3 groups of that number. One way to think of this is to think of taking 3 groups of the number away. Another is to think of -3 times a number as being a reflection of 3 times the same number.

-3 times 3 on the number line

So -3 × -3 is, therefore, a reflection of 3 × -3 across the number line.

-3 times -3 on the number line

In one sense though, this visual argument is just mathematical consistency represented using a number line. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection.

 

 

[Not_a_Number]

I think the interesting thing is that all these explanations are right, but the one that makes sense will depend on HOW one thinks of negatives.

[wendyroo]

1 hour ago, sweet2ndchance said:

Someone, I think maybe it was @wendyroo, explained different operations with negative numbers with digging holes. It was clever and it made sense but I know I couldn't do that explanation justice. Hopefully, someone can provide that explanation again.

This is how I have always modeled it for my kids.

[Roadrunner]

1 hour ago, wendyroo said:

This is how I have always modeled it for my kids.

This is now my favorite explanation. Thank you!

[wathe]

I've used Wendyroo's model with my kids.

I've also copied language from Jousting Armadillos to reinforce:

(+)  x (+) = gaining gains

(+) x (-) = gaining losses

(-) x (+) = losing gains

(-) x (-) = losing losses

And then they played around with language and double negatives in speech.  This would get very silly (You must not not clear your dirty dishes!  You must not not go to bed now! Then not not not, then not not not not etc), but somehow helped reinforce the point.

 

[frogger]

I've always explained it kind of like wathe's summary of wendyroo's explanation but with money and debts and gains. 

[Not_a_Number]

As another question, do you think your kids would be able to explain this one to you? I’ll ask DD8 and see if she remembers why... last I checked, she remembered all the other options but not this one.

[Not_a_Number]

Out of curiosity, I asked DD8 about why a negative times a negative is a positive. Here's what she said verbatim: 

 

"So you have a negative number, which is a debt. And if you have a debt of a debt would mean that you owe someone a debt. That means you actually get something, which means they have to pay you something, not you. And it's a certain number of debt, so you get even more than if you just have the first debt for you." 

 

So, apparently she did internalize it as "a debt of a debt," not just a pure property! Interesting. I know she didn't remember why for a while, but apparently now she does. 

[Porridge]

I think of numbers as vectors, having both magnitude (the absolute value), and direction (the sign). The negative sign can be thought of like undoing something. When you multiple two negatives, you undo what had been undone, resulting in a net positive.

[Porridge]

this is a great thread, by the way. I'm learning a lot.

[Not_a_Number]

Just now, JHLWTM said:

I think of numbers as vectors, having both magnitude (the absolute value), and direction (the sign). The negative sign can be thought of like undoing something. When you multiple two negatives, you undo what had been undone, resulting in a net positive.

Except we never do multiply vectors, do we? 😄 

When did you start thinking of numbers as vectors? I'm always fascinated when people's number model is primarily length-based. That hasn't been true for most kids I've worked with, so I'm curious when that flips. 

[03loretta]

This is a perfect way to explain to my daughter.

[Not_a_Number]

Just now, 03loretta said:

This is a perfect way to explain to my daughter.

Which one do you like? There's really a smorgasbord of riches in this thread! 

[03loretta]

5 minutes ago, 03loretta said:

This is a perfect way to explain to my daughter.

I think of numbers as vectors, having both magnitude (the absolute value), and direction (the sign). The negative sign can be thought of like undoing something. When you multiple two negatives, you undo what had been undone, resulting in a net positive.

[frogger]

2 hours ago, Not_a_Number said:

Out of curiosity, I asked DD8 about why a negative times a negative is a positive. Here's what she said verbatim: 

 

"So you have a negative number, which is a debt. And if you have a debt of a debt would mean that you owe someone a debt. That means you actually get something, which means they have to pay you something, not you. And it's a certain number of debt, so you get even more than if you just have the first debt for you." 

 

So, apparently she did internalize it as "a debt of a debt," not just a pure property! Interesting. I know she didn't remember why for a while, but apparently now she does. 

My son likes money and understood it from a young age. If any mathamatical concept can be explained by money, we can get there pretty quick. 🤣 I'm pretty sure he learned about percentages long before any curriculum taught him. I can't remember a specific time but pretty sure he learned it in the lego aisle or something similar.🤣

[Not_a_Number]

47 minutes ago, frogger said:

My son likes money and understood it from a young age. If any mathamatical concept can be explained by money, we can get there pretty quick. 🤣 I'm pretty sure he learned about percentages long before any curriculum taught him. I can't remember a specific time but pretty sure he learned it in the lego aisle or something similar.🤣

We did negatives as “owing” when she was 5, but it was often phrased in apples!!

[Porridge]

22 hours ago, Not_a_Number said:

Except we never do multiply vectors, do we? 😄 

When did you start thinking of numbers as vectors? I'm always fascinated when people's number model is primarily length-based. That hasn't been true for most kids I've worked with, so I'm curious when that flips. 

i think you can multiply vectors? But you end up with a vector at right angles to both original vectors? But my signs analogy breaks down here, so it's not a perfect explanation! https://www.mathsisfun.com/algebra/vectors-cross-product.html

The holes of dirt or debts explanation is cleaner 🙂 

I'm not sure why I think of them as vectors. I think I was explaining force or speed to my kids one day and the topic of vectors came up, then when we talked about absolute value later, it seemed natural to relate it to the concept of vectors.

Edited February 25, 2021 by JHLWTM

[Not_a_Number]

15 minutes ago, JHLWTM said:

i think you can multiply vectors? But you end up with a vector at right angles to both original vectors? But my signs analogy breaks down here, so it's not a perfect explanation! https://www.mathsisfun.com/algebra/vectors-cross-product.html

The cross product isn’t really a proper product. It’s weird, it’s not commutative, and it only works in 3D... It’s kind of a definition of convenience that’s not related to anything else.

The dot product is much cleaner, it works for all dimensions, and is in fact the usual product in 1D, but it doesn’t return a vector — it returns a real number, no matter how many dimensions you are in.

 

15 minutes ago, JHLWTM said:

The holes of dirt or debts explanation is cleaner 🙂 

I'm not sure why I think of them as vectors. I think I was explaining force or speed to my kids one day and the topic of vectors came up, then when we talked about absolute value later, it seemed natural to relate it to the concept of vectors.

Interesting!

[fairfarmhand]

Thanks for these thoughts guys! Stuff happened the day I posted it and I forgot to come back to this thread! 

[Not_a_Number]

2 hours ago, fairfarmhand said:

Thanks for these thoughts guys! Stuff happened the day I posted it and I forgot to come back to this thread! 

No worries!! Any of our musing help?? 

[Spring Flower]

I like to put a toy car on a number line with the car on zero and headlights facing the positive numbers. Driving "forward" is positive. Driving in "reverse" is negative. Rule number 1: forward is positive, reverse is negative. Pretty basic.

Now I introduce a rule number 2: flipping the car around changes the sign. For example: I flip the car around so that the headlights face the negative numbers. Now "forward" is negative and "reverse" is positive.

Now the tricky part. If I start with on zero with my headlights facing toward the positive and drive forward, I am moving in a positive direction. If I flip my car around so that my headlights are facing the negatives and I drive in reverse, I am STILL moving in a positive direction. So flipping my car around (negative) and driving in reverse (negative) creates a "positive" number. 

 

 

[Roadrunner]

I like the visual with dirt piles so much more than debt. Debt is more abstract, and lots of young kids have a harder time visualizing debt.  But it’s easy to dig through the piles of dirt in the ground and fill them up! I showed that explanation to my DS and we think it’s the best one we have ever seen. 

[Not_a_Number]

42 minutes ago, Roadrunner said:

I like the visual with dirt piles so much more than debt. Debt is more abstract, and lots of young kids have a harder time visualizing debt.  But it’s easy to dig through the piles of dirt in the ground and fill them up! I showed that explanation to my DS and we think it’s the best one we have ever seen. 

I don’t like it because it doesn’t get integrated with your other understanding. You can have debt in anything — apples, pigeons, money, whatever. But do you normally think of numbers as piles of dirt? Me neither. 

The fallacy here is that “explanations” matter. Explanations don’t matter. Reasons matter. A reason that something is true is something broad you can get a feel for. Debt is very broad and is a reason. It’s a consistent model of negative numbers.

The difference between “reasons” and “explanations” meant that I had no illusions whatsoever that DD8 would understand negative number manipulations the first or second or tenth time she saw them. They are a hard idea. We had to think hard about them. And that was a good thing. She thinks about them this way many years later. She can tell me exactly why any of the manipulations work. 

Sorry, I’m feeling a bit ornery today. But I don’t particularly judge things on whether they are easy to internalize or understand the first time I see them. That’s never the point for me.

[wendyroo]

51 minutes ago, Roadrunner said:

I like the visual with dirt piles so much more than debt. Debt is more abstract, and lots of young kids have a harder time visualizing debt.  But it’s easy to dig through the piles of dirt in the ground and fill them up! I showed that explanation to my DS and we think it’s the best one we have ever seen. 

I think debt is very hard for my kids to grasp because it is a verbal agreement, hence the intersection of two of my kids' weaknesses: language and people. (Plus they are not interested in or motivated by money at all.) Trying to use debt as a mathematical model means they need to sort through all the word-y and interpersonal aspects of the problem before they can grapple with the numbers. Instead, for us, piles of dirt are much purer numbers. There are no words or people or motivations to trip over, just pure physical manifestations of "a positive unit" and "a negative unit".

[Not_a_Number]

6 minutes ago, wendyroo said:

I think debt is very hard for my kids to grasp because it is a verbal agreement, hence the intersection of two of my kids' weaknesses: language and people. (Plus they are not interested in or motivated by money at all.) Trying to use debt as a mathematical model means they need to sort through all the word-y and interpersonal aspects of the problem before they can grapple with the numbers. Instead, for us, piles of dirt are much purer numbers. There are no words or people or motivations to trip over, just pure physical manifestations of "a positive unit" and "a negative unit".

I didn’t do it with money. I did it with apples. But I can see how verbal agreements would be a problem, although that’s not how we started.

[Not_a_Number]

The distinction I'm making (and I never manage to convince people of this) is that with debt or something LIKE it, you can model it in your head very well in a general way. And then you can get a feeling for what is happening via as much guided experimentation as is needed. 

When I was thinking about how to do this with my Zoom class kids, I was thinking about using red poker chips as negatives, and come to think of it, maybe I'll program a "Don't Break the Bank" game like that sometime 😄 . Again, the idea wouldn't be necessarily be that this would be easier to understand quickly -- the point is that if you can have hands-on experience with what happens when you add and subtract negative numbers, you get more of an intuition for what's going on. And if you see it in the context of a game, that's what happens -- you get lots of hands-on experience, without the initial pressure to generalize or calculate. That's the part that seems valuable to me. 

What I did with DD8 is that we modeled lots of things. She didn't have the language issue, so I can see how that wouldn't work for everyone... but anyway, we modeled what happened if she had 6 to start with, and then got a debt of 2. What did that mean? How much did she have, really, at the end? We worked through it for a few months. But again, the point was that she got to do a lot of exploring in contexts that were integrated with her other mathematical experiences. It didn't feel "new," per se. 

I think we'll probably wind up doing the poker chip model for DD4, because I've been using poker chips for her, and honestly I like it better than the purely visual stuff we were doing with DD8 (although I can't really argue with the outcome.) But again, the point will be doing lots of hands-on examples via modeling, and it'll be integrated with what she does normally already. 

If I wanted to use dirt piles, I'd probably have to spend quite a lot of time talking about numbers as dirt piles 😛 . We'd have to think about it what it would mean to take away 3 dirt piles from 2 dirt piles. We would have to think about how that worked for things that AREN'T dirt piles to generalize. And once we were convinced that dirt piles were in fact representative of all things, we'd move on. (For instance, the problem with dirt piles is that there's an infinite amount of dirt. Why aren't we counting that dirt?)

I know I'm being too critical here, and it IS a good visual explanation; I absolutely grant that. It's just that over my time teaching math, I've lost faith in the power of explanations. I don't think I can explain anything to anyone in a way they will retain, no matter how good they feel about the explanation right after hearing it. All I can do is lead them to the model and guide them through it to gain intuitions. And once they are comfortable with the model, we can talk about those intuitions and summarize them and figure out how they fit together. But I always think that the model comes first. 

[wendyroo]

22 minutes ago, Not_a_Number said:

I didn’t do it with money. I did it with apples. But I can see how verbal agreements would be a problem, although that’s not how we started.

Even owing someone apples is still a verbal agreement. They aren't handing the apples over right now, so they are agreeing that sometime in the future they will give you hypothetical apples. And that very much hinges on communication, rational thinking, planning for the future and holding up your end of the bargain. That is a lot for some kids to wade through to get to the numbers.

Even as an adult, thinking about debt makes negatives less clear in my mind. Trying to think about "getting a debt" is incredibly unintuitive to me. I have to ponder over those words for a long time before I can figure out what situation they would model. Over the years, as I have worked with debts, I have rote memorized that "get a debt" means add a negative, etc. But it's been 34 years now, so I don't think subtracting debts or having a negative number of debts will ever help me develop intuitive understanding.

[Not_a_Number]

6 minutes ago, wendyroo said:

Even owing someone apples is still a verbal agreement. They aren't handing the apples over right now, so they are agreeing that sometime in the future they will give you hypothetical apples. And that very much hinges on communication, rational thinking, planning for the future and holding up your end of the bargain. That is a lot for some kids to wade through to get to the numbers.

Yeah, I’m not insisting on debt. It’s what worked for us. But I would definitely want something integrated into our models of number. Which will probably mean poker chips this time 😉 . Stay tuned!! 
 

6 minutes ago, wendyroo said:

Even as an adult, thinking about debt makes negatives less clear in my mind. Trying to think about "getting a debt" is incredibly unintuitive to me. I have to ponder over those words for a long time before I can figure out what situation they would model. Over the years, as I have worked with debts, I have rote memorized that "get a debt" means add a negative, etc. But it's been 34 years now, so I don't think subtracting debts or having a negative number of debts will ever help me develop intuitive understanding.

Getting a debt is pretty easy. You get a piece of paper that say IOU 5 dollars. You can see that this means you have to pay 5 dollars. Taking away debts is a little trickier.

I don’t know that it would help you get an intuitive understanding at this point, because your model is already pretty internalized. But I don’t know that it WOULDN’T have helped if you started out that way.

[Not_a_Number]

I told DD8 about the "piles of dirt" model and she just found it confusing. She said, and I quote "yes, it's easier to visualize, but I don't really need a visualization -- I need an explanation." 

I think that's her way of getting at the issue I'm having as well -- she has built deep intuitions about how negative and positive numbers interact using "debt." New explanations that don't easily fit into how she thinks about numbers aren't clarifying, because they aren't obviously the same thing. We could probably convince her that they ARE the same thing, but it's not obvious to her. 

The fact that she has deep intuitions about WHY things work is also why she's so able to organize her thinking linearly and why she moved so easily into writing proofs for her math. And that's why, for me, a really easy visual is just not essential. The model is a very small part of the learning, and my main requirement for a model is that it fits in well with everything else we understand. 

[Alte Veste Academy]

19 hours ago, Not_a_Number said:

I know I'm being too critical here, and it IS a good visual explanation; I absolutely grant that. It's just that over my time teaching math, I've lost faith in the power of explanations. I don't think I can explain anything to anyone in a way they will retain, no matter how good they feel about the explanation right after hearing it. All I can do is lead them to the model and guide them through it to gain intuitions. And once they are comfortable with the model, we can talk about those intuitions and summarize them and figure out how they fit together. But I always think that the model comes first. 

 

2 hours ago, Not_a_Number said:

I told DD8 about the "piles of dirt" model and she just found it confusing. She said, and I quote "yes, it's easier to visualize, but I don't really need a visualization -- I need an explanation." 

I think that's her way of getting at the issue I'm having as well -- she has built deep intuitions about how negative and positive numbers interact using "debt." New explanations that don't easily fit into how she thinks about numbers aren't clarifying, because they aren't obviously the same thing. We could probably convince her that they ARE the same thing, but it's not obvious to her. 

The fact that she has deep intuitions about WHY things work is also why she's so able to organize her thinking linearly and why she moved so easily into writing proofs for her math. And that's why, for me, a really easy visual is just not essential. The model is a very small part of the learning, and my main requirement for a model is that it fits in well with everything else we understand. 

I'm having trouble following, because it seems like you're talking in circles here regarding explanations and intuitive understanding. When a student truly has intuitive understanding of a concept, it should be easy enough to extrapolate to other valid explanations/models. In fact, I would say solid understanding of any concept is demonstrated by the ability to quickly pick up on or explain other valid examples of that concept. It's when students (of any age!) have a shakier, beginner's verge-of-understanding that they have trouble doing so.

My best teachers have been those who can change gears quickly when explaining something I am not getting. They can do that because they get it, while I am just grasping at it. LOL Negative numbers should be understood as a mathematical concept, obviously, but also as myriad real-life examples, not just debt or just dirt piles or.... This thread started as someone asking for an example to help seed understanding, and it's ok if one example is not perfect for everyone at the seed level. But if they are valid, they should all be easily understood by those at the intuitive level. 

[Not_a_Number]

8 minutes ago, Alte Veste Academy said:

I'm having trouble following, because it seems like you're talking in circles here regarding explanations and intuitive understanding. When a student truly has intuitive understanding of a concept, it should be easy enough to extrapolate to other valid explanations/models. In fact, I would say solid understanding of any concept is demonstrated by the ability to quickly pick up on or explain other valid examples of that concept. It's when students (of any age!) have a shakier, beginner's verge-of-understanding that they have trouble doing so.

Well, she can extrapolate it, and I'm sure she could have figured out how it works. But it was more confusing than what was originally in her head for her. And same goes for both of her parents, who are both mathematicians with fancy PhDs 🤷‍♀️. 

 

10 minutes ago, Alte Veste Academy said:

My best teachers have been those who can change gears quickly when explaining something I am not getting.

Yes, I can generally explain things from about 100 different perspectives, because I have a firm grasp on math, but I still don't think my biggest job as teacher is to explain. My job is to provide a model that allows them understand it themselves, and then I'll help them use that model. 

As an example, let's say you wanted to explain WHY 3/4 divided by 1/2 is 3/2. What would you do? 

[Alte Veste Academy]

11 minutes ago, Not_a_Number said:

Well, she can extrapolate it, and I'm sure she could have figured out how it works. But it was more confusing than what was originally in her head for her. And same goes for both of her parents, who are both mathematicians with fancy PhDs 🤷‍♀️. 

Yes, I can generally explain things from about 100 different perspectives, because I have a firm grasp on math, but I still don't think my biggest job as teacher is to explain. My job is to provide a model that allows them understand it themselves, and then I'll help them use that model. 

As an example, let's say you wanted to explain WHY 3/4 divided by 1/2 is 3/2. What would you do? 

Well, sure, fancy PhDs still have individual preferences, just like common folk. 

A model is a means of explaining. And if one model doesn't work for a particular student, then moving on to another seems like a nice, flexible way of educating. 

LOL, no. I am not playing. 

[Not_a_Number]

Just now, Alte Veste Academy said:

Well, sure, fancy PhDs still have individual preferences, just like common folk. 

A model is a means of explaining. And if one model doesn't work for a particular student, then moving on to another seems like a nice, flexible way of educating. 

My point is that the best thing you can do for a student is to work with THEIR model and to make their specific model cohesive and linear in their heads. 

 

Just now, Alte Veste Academy said:

LOL, no. I am not playing. 

I wasn't trying to "play." I can think of like 5 ways to explain that one, and none of them is better than any other, and depending on what's in a kid's head, some will make sense and some won't.  

[Alte Veste Academy]

1 minute ago, Not_a_Number said:

My point is that the best thing you can do for a student is to work with THEIR model and to make their specific model cohesive and linear in their heads.

I wasn't trying to "play." I can think of like 5 ways to explain that one, and none of them is better than any other, and depending on what's in a kid's head, some will make sense and some won't.  

I'm glad to hear you're flexible in model selection.

I would expect so, as a math instructor! It would have been play for me, something I do not have the time or itch for today.

[Not_a_Number]

1 minute ago, Alte Veste Academy said:

I'm glad to hear you're flexible in model selection.

I would expect so, as a math instructor! It would have been play for me, something I do not have the time or itch for today.

No worries. Come back later if you're interested, and I'll attempt to show what I mean. I know that I come at this from a weird perspective.