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Math Talk #03 -- Take Action with Fractions


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I just wanted a snazzy sounding title ūüôā¬†

In my experience, I have found that for many students fractions are the make-or-break concept for their mathematical literacy and competency in higher level mathematics and I've been working to build out my delivery and scaffolding of fractions.

I have found that many students struggle with fractions because often fractions are the first topic in mathematics where context is as meaningful as the numbers and operations themselves. It can be very difficult to work through fraction problems without understanding what the symbols mean.

There are a few things that make fractions seem complex to young students, such as:

an incomplete understanding of the basic operations +, -, * and / can be uncovered when students begin to work with rational numbers and fractions. Because this is the first time that a student may be exposed to a more nuanced meaning of the operations, their confusion and thus struggle may mistakenly be attributed to rational numbers.

the need to be mindful of context and thus unit-blindness

the algorithms for calculating fractions can seem very random/arbitrary when you don't understand the operations or the meaning of fractions themselves (or both).

an under-developed understanding of fractions and their various context-based interpretations.

the widespread use of the notation a/b for different concepts.

It doesn't matter what curriculum you use in your homeschool or tutoring. Fractions are a concept that many kids struggle with.

In my own teaching, I've found that really developing strong unit-awareness while still working with whole-numbers greatly helps to ease the transition to fractions. Another practice that I've found helpful is getting students very familiar with numerical expressions.

 

Anyway, I want to open the floor now to discussion of strategies for teaching the concepts with-in the fraction concept in such a way that students are not lead-in ever-widening circles of confusion and despair--which is how some kids really seem to feel when it comes to fractions. Most students want fractions to make sense, and when they feel as though fractions don't they can turn to the dark-occult practice of Algorithm Worship.

 

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What a great thread!

I was thinking the other day that I need to find a good way to teach multiplying fractions.

Suppose I'm multiplying 1/2 x 1/3. 

In my mind, I explain this to myself as "one half of one third". Visually, I can see that half of a third is one sixth. 

I can also see that one third of a half is a sixth.

What I cannot really explain well is why we can arrive at that answer by multiplying the numerators and the denominators. I'm kind of embarrassed by that lack of understanding : (

I mean, I guess that that's what we're doing in any multiplication really. Like if I did 2x3 I'd really be multiplying 2/1 x 3/1. But I still feel a little unsatisfied by this.

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14 hours ago, mom2bee said:

  

the algorithms for calculating fractions can seem very random/arbitrary when you don't understand the operations or the meaning of fractions themselves (or both).

 

 

This resonated with me.  I refuse to use the phrase "cross multiply" because I think that's an unnecessary level of abstraction, even with my students in calculus.   I will always, always say, "we multiple both sides by 3" or something similar.  It harkens back to first principles: do the same thing to both sides and equality still holds.  

When dividing fractions, I like to tell younger students, "Remember, our new definition of division is to multiply by the reciprocal."  Or I'll ask them "What is our new definition of division?"  I want students to be clear on what has been proved and what has been defined.  

Great topic!

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30 minutes ago, daijobu said:

 

This resonated with me.  I refuse to use the phrase "cross multiply" because I think that's an unnecessary level of abstraction, even with my students in calculus.   I will always, always say, "we multiple both sides by 3" or something similar.  It harkens back to first principles: do the same thing to both sides and equality still holds.  

When dividing fractions, I like to tell younger students, "Remember, our new definition of division is to multiply by the reciprocal."  Or I'll ask them "What is our new definition of division?"  I want students to be clear on what has been proved and what has been defined.  

Great topic!

 

I agree with everything here.  I find "cross multiplying" to be a shortcut that is not worth the sacrifice in understanding.  We multiply in two steps for a long time before I point out (if it hasn't occurred to them) that we can do it in one "double step."

Similarly, there are two phrases I drilled in grade 5 math:

- Subtracting a number is the same as adding its opposite.

- Dividing by a number is the same as multiplying by its reciprocal.  

Knowing these two phrases which can convert subtraction to addition and division to multiplication makes moving to pre-algebra SO much easier.  The first phrase feels pretty intuitive, and the second phrase needs a bit more practice with real problems.  

 

What I find amazing about fractions is how we can move from pies and cakes to things like graphing (x^2)/(5-x) in so few years.  It's truly remarkable!!! 

 

 

 

 

 

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2 hours ago, Little Green Leaves said:

What I cannot really explain well is why we can arrive at that answer by multiplying the numerators and the denominators. I'm kind of embarrassed by that lack of understanding : (

You shouldn't be embarrassed. Multiplication and division are uniformly mistaught all over the nation and have been generations.
When we combine the Multiplication Lie with a fuzzy, incomplete understanding of Fractions, wait a couple of years and voila! you have an entire nation of educators, the vast majority of whom do not have a good understanding of fractions.

When I was in elementary school, I had a teacher explain to the class that fractions were one of the reasons she believed whole-heartedly in god. Because even though fractions make no sense, they do work. She said it was one of the ways we should know god works in mysterious ways. ūüôĄ

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55 minutes ago, Gil2.0 said:

You shouldn't be embarrassed. Multiplication and division are uniformly mistaught all over the nation and have been generations.
When we combine the Multiplication Lie with a fuzzy, incomplete understanding of Fractions, wait a couple of years and voila! you have an entire nation of educators, the vast majority of whom do not have a good understanding of fractions.

When I was in elementary school, I had a teacher explain to the class that fractions were one of the reasons she believed whole-heartedly in god. Because even though fractions make no sense, they do work. She said it was one of the ways we should know god works in mysterious ways. ūüôĄ

Thanks for letting me off the hook there : )  I'm just glad to be thinking this out ahead of trying to teach it. I remember being so frustrated as a student because it felt like so much of math had to be accepted on faith. I don't want my kids to experience that.

 

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2 hours ago, Little Green Leaves said:

Thanks for letting me off the hook there : )  I'm just glad to be thinking this out ahead of trying to teach it. I remember being so frustrated as a student because it felt like so much of math had to be accepted on faith. I don't want my kids to experience that.

To understand the multiplication algorithm for fractions you have to be aware of a few things .

(1) Multiplication is not repeated addition. Multiplication is more in line with proportionately stretching or shrinking a thing. In other words, it's a scaling operation.

Multiplication is not repeated addition. It never was. It never will be. The statement "Multiplication is Repeated Addition" is a lie. (It might be perpetuated in ignorance, or good-intentions or anything else, but it's a lie)

(2) Fractions are uniform portions, pieces or parts of something. You never just have 1/3. You will have 1/3 of something.
1/3 of a pancake, 1/3 of a mile, 1/3 of an apple slice, 1/3 of a lb, 1/3 of a unit.
This is of course in line with whole numbers. You never just have 6. You have 6 of something.
There can be 6 toys, 6 blue markers, 6 spider-man toys, 6 children or 6 chairs. But there is no puddle of "6-ness" just floating around in the world.

(3) When you are multiplying fractions you are scaling a quantity up or down appropriately and the result is compared back and framed in-fractional parts of the original thing. This is kind of like regrouping whole-numbers.
When you compute 38 + 75 it's true that you get 10-tens and 13-units, but we always frame this result in terms of standard decimal-numbers and call it 1hundred-thirteen 113.

When you compute 3/4 x 5/7, we re-frame the result in terms of the original unit and it is 15/28 of the original unit.

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53 minutes ago, Gil2.0 said:

To understand the multiplication algorithm for fractions you have to be aware of a few things .

(1) Multiplication is not repeated addition. Multiplication is more in line with proportionately stretching or shrinking a thing. In other words, it's a scaling operation.

Multiplication is not repeated addition. It never was. It never will be. The statement "Multiplication is Repeated Addition" is a lie. (It might be perpetuated in ignorance, or good-intentions or anything else, but it's a lie)

(2) Fractions are uniform portions, pieces or parts of something. You never just have 1/3. You will have 1/3 of something.
1/3 of a pancake, 1/3 of a mile, 1/3 of an apple slice, 1/3 of a lb, 1/3 of a unit.
This is of course in line with whole numbers. You never just have 6. You have 6 of something.
There can be 6 toys, 6 blue markers, 6 spider-man toys, 6 children or 6 chairs. But there is no puddle of "6-ness" just floating around in the world.

(3) When you are multiplying fractions you are scaling a quantity up or down appropriately and the result is compared back and framed in-fractional parts of the original thing. This is kind of like regrouping whole-numbers.
When you compute 38 + 75 it's true that you get 10-tens and 13-units, but we always frame this result in terms of standard decimal-numbers and call it 1hundred-thirteen 113.

When you compute 3/4 x 5/7, we re-frame the result in terms of the original unit and it is 15/28 of the original unit.

Okay. I feel like I do understand this.  

So in terms of why the algorithm works...

we multiply the two denominators to determine the size of the fractional parts (since we are further subdividing the original whole)

and we multiply the two numerators to determine how many of the new fractional parts we're talking about.

 

 

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2 hours ago, square_25 said:

However, for fractions, I do think ‚Äúscaling‚ÄĚ is the way to do it.¬†

When we multiply a number by r, we can think of it as being scaled by r. It’s easiest to visualize using lengths.

So, for example, if we multiply by 1/2, we’re scaling by 1/2. But what does scaling mean? It means that the proportionality between x and xr is exactly the same as the proportionality between 1 and r.

That’s all very abstract, though :-). Let’s see an example. The proportionality between 1/2 and 1 is the same as between 1/2*1/3 and 1/3. So do you expect 1/2*1/3 to be more or less than 1/3? What would you expect the relationship between the lengths to be?

 

Okay, I like that. And I can visualize it, too, which is nice.

I feel a little more solid already but not 100 percent yet. I guess...maybe I am overthinking things. I am still trying to understand why it is that multiplying fractions is so much easier than adding fractions. Is that because of some fundamental quality of multiplication? Why don't we have to find a common denominator BEFORE we multiply fractions? Is it just that the method of multiplying numerators and the denominators is a shortcut to finding the lowest common denominator? or is there something conceptual that I'm missing?

Ugh if anyone has a suggestion of books I can use to get a better understanding of this, I'd love that.

 

 

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2 minutes ago, square_25 said:

I can probably walk you through this, I promise :-). 

Can you answer my question of what 1/2*1/3 should be, conceptually? And why?

Ooh I'd love that.

 

Conceptually...does this answer the question?

I'd expect 1/2 * 1/3 to be smaller than 1/3. Two times smaller. I usually think of 1/2 * 1/3 as "one half of one third". Or, half as much as one third of one.

 

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1 minute ago, square_25 said:

 

Exactly! Perfect :-). 

So, 1/2*1/3 is "one half of 1/3". And 1/3 is, by definition, what happens if you split a unit into three pieces. 

Ok, so we have a third, which is what happens when we split a unit into three pieces. And then we take half of that. How many of the resulting number would we need to make a unit? (This is easier with a picture.) 

Six.

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2 minutes ago, Little Green Leaves said:

One sixth

Okay. So when I multiply the denominators, I am basically saying

I had already split my unit into three pieces, hence the 1/3. Now I'm splitting each of those thirds into two more pieces.

And when I multiply the numerators, I'm saying, I'm going to scale the number of pieces by (in this case) one.

 

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2 minutes ago, square_25 said:

Right :-). 

Let's try it with another multiplication. Say you wanted to do 1/3*1/4. That means we're taking a third of 1/4, or in other words, splitting 1/4 into three pieces. Here's a visual of 1/4 for you: 

_ _ _ _ 

Can you tell me what you get for 1/3*1/4, using the same kind of reasoning? (If you don't mind writing out the whole sequence of reasoning, that would help.) 

To get a third of one fourth, I'll need to divide the fourth into three pieces. Each of those pieces represents one twelfth of the original whole. So one third of one fourth is a twelfth.

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1 minute ago, square_25 said:

Right :-).

So then what’s 2/3*1/4?

This is so helpful : ) 

 

Okay. So I am trying to find two thirds of one quarter. I divide my quarter into three parts. Each of those parts is equal to a twelfth of the original whole. This time, I take two of those parts. Two twelfths, or one sixth.

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Just now, square_25 said:

I actually think you’re pretty much there. Let’s do one more. Can you extend this idea to 

2/5*3/4?

Let me try : )

I want to get two fifths of three quarters.

I'll divide each of the quarters into fifths. Each fifth is equal to a twentieth of the original whole. So that'll give me 15 twentieths. 

One fifth of that would be three twentieths. Two fifths would be six twentieths, or three tenths.

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55 minutes ago, square_25 said:

Perfect. You’re a natural.

So can you answer your own question? Why don’t we do a common denominator here?

You're a great teacher.

 

Would it be fair to say that when we multiply fractions,  we are not actually working with two different fractions? We are just working with one fraction which is being scaled by a factor?

 

I mean, in multiplying, we are operating on one fraction by a factor which is expressed by the other fractio ? I don't know whether I¬†put that clearly ūüôā

 

While with addition or subtraction,  we are working with two fractions which need to be put into the same form (common denominator) so that we can find their sun or difference? 

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13 minutes ago, square_25 said:

 

Aw, thank you :-). Much appreciated. 

 

Right! Because we're scaling, there's no reason to express both fractions as fractions with the same denominator. 

When we're adding, it's easier to add if we have the same denominator. So, for instance, 

1/3 + 1/13 = 13/39 + 3/39,

and that makes life easier, because both are now some number of 39th's, so you can simply figure out the total number of 39th's, and voila, you get the fraction. 

Whereas if you're doing 1/3*1/13, you're scaling the 1/13 by 1/3, and while it's helpful to write 1/13=3/39 so it's easy to take a third of it, there's no reason to convert the 1/3, since it won't help you scale anything at all :-). 

I get it!!

Thank you so much. That was really really helpful!! 

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4 hours ago, square_25 said:

Well, if you disagree, we could perhaps have a discussion about it. I've heard people say that and I never really understand what their point is. Is your point that if you start by defining multiplication as repeated addition of the same number, you have to redefine it later? This definition certainly does give the right answer for positive integers. So what's wrong with it? 

"Multiplication is repeated addition" is a false statement, so that's what's wrong with it. It's a POPULAR statement, but so is that old tale about Santa and we all know that's not true.

Obviously, multiplication gives the same result as repeatedly adding a positive integer to itself but that doesn't make the operation of multiplication repeated addition.

This is a crude analogy, but I hope it helps illustrate my point.
A hammer is a tool to drive nails into, or pry nails out of, wooden structures.
Hammers are occasionally used to maim or kill, but that doesn't mean that a hammer is a weapon.
Just because a hammer can serve the purpose of a weapon, doesn't mean that we should begin teaching kids hammers are weapons to maim or kill.

The truth (for the real numbers) is that the operations of addition and multiplication are not one and the same. To my mind, it's not really sound to teach a known lie for the sake of quick results now only to confuse the kids in a couple of years when you have expose them to the truth, (Also, side rant of mine: Of all the series that teach The Multiplication Lie from 2nd and 3rd grade, I don't think I've ever seen one that goes back and addresses this directly in 4th-5th grade when it's time for kids to begin multiplying fractions. IMO, It's only fair that if you go out of your way to convince your kids that Santa Claus is real, you should at least take the time to clear up the falsehood when you're ready to let them in on the truth. Not just drop the pretense and hope for the best.)

Additionally by telling The Multiplication Lie on such a scale, we have to uniformly limit how much of the general truth about numbers they can really access (artificially restricting the "scope and sequence") of elementary school mathematics.

By teaching "Multiplication is Repeated Addition" we as a nation don't have to concern ourselves on developing the young childs number-sense and training their quantity awareness early and often. We can instead systematically ignore it. Their number sense only needs to be developed enough to add and subtract, because we're going to screw them on the back-end with The Multiplication Lie.

If instead, we taught kids to understand what scaling is, and helped them develop a "feel" for scaling, the way we do for greater/less, or quanties in general, then we could teach them that "Multiplication is a scaling operation" and they can practice multiplication with whole and rational numbers from day 1.

Many kids will notice on their own that when I make something half-as-big, it's just cutting it in half, but when I make something twice as big, it's doubling it. And similar for making something a-third as big or triple it's size, etc..." In my experience, helping kids access the forwards and backwards of an operation early and often always off nicely.

Then we can prompt 2nd, 3rd and 4th graders with specific exercise sets that enable us to discuss questions such "Do you notice anything about multiplying by whole numbers?" and "When is multiplying not like repeated addition?"

But by mis-teaching multiplication as repeated addition on such as massive scale, it restricts the type of numbers we can expect young students to work with. Children are typically taught fractions in only the weakest way in K-3, because we can't  have fractions exposing The Multiplication Lie we start dishing out typically sometime in 2nd grade.

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5 minutes ago, square_25 said:

I think it's reasonable to say that multiplication IS repeated addition when you restrict your definition to positive integers. It's actually a pretty common mathematical trick: you define an operation on a certain subset of objects, and then you extend the definition to other objects using properties you want that operation to have. One sees it all the time in more abstract mathematics :-).

Yes, but the rub is that most young children who are being introduced to multiplication have no idea about positive integers. They aren't being told and they don't typically have the mathematical exposure or experience to know that "Multiplication is repeated addition" is a conditional statement.

Any time I've seen a definition in a higher level mathematics that is only valid for a subset, it typically begins with
"For all real numbers, n,...." or
"Given that k is a whole number,..." or
"Yada, yada, yada....provided that n is a rational number.

In other words, it's made clear out of the gate that we're making a conditional definition. To be clear, I'm not a professional mathematician, so there could be this practice of giving definitions that are only true conditionally without ever making it clear that it's conditional, but I have never encountered it.

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17 hours ago, square_25 said:

 

Yeah, you'd usually be clear what numbers it applies to. But to be fair, at the ages I teach multiplication, we don't tend to have defined any numbers OTHER than positive integers. We did wind up extending this to negative integers at a very young age (DD7 was very interested in them and learned them before 6), but mostly via the commutative and distributive properties. (And scaling isn't a perfect way to think of negative multiplication, anyway: I can make a story that makes it make sense as scaling, but it doesn't feel intuitive.) 

I'm generally OK with definitions that turn out to not tell the full story, because part of my motivation of introducing operations is to create sufficient experience with an operation to create strong intuitions. Once intuitions are formed, it is easier to expand one's understanding. And I try to be VERY upfront about updating models and seeing why our new model is the same as our old one.

So, with this it brings us back to: Ok, whatever you say.

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17 hours ago, square_25 said:

On the other hand, I am genuinely curious how you would start out defining multiplication as scaling :-). How would you explain what "scaling" means to a little kid? I do teach multiplication very young, so I'd need this to be accessible to an accelerated 5 year old or an average 6-7 year old. 

I do not try and teach children (especially outliers) that I haven't met and assessed myself. Experience has taught me it's not a great use of time for anyone involved.

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11 minutes ago, Gil2.0 said:

I do not try and teach children (especially outliers) that I haven't met and assessed myself. Experience has taught me it's not a great use of time for anyone involved.

Ok, but how did you teach your child(ren) about multiplication as scaling and avoid the words "repeated addition". I've got a fuzzy idea in my head of how it could be done but I'm really curious to hear from someone who has actually done it. I think the idea is fascinating and a good one but I've never seen this kind of teaching in action so I'm really curious how you have done it. So instead of answering the question, "How would you teach a hypothetical child?", could you answer "How did you teach multiplication to your children as scaling and at about what age were they when you did this?" 

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19 hours ago, Gil2.0 said:

"Multiplication is repeated addition" is a false statement, so that's what's wrong with it. It's a POPULAR statement, but so is that old tale about Santa and we all know that's not true.

The truth (for the real numbers) is that the operations of addition and multiplication are not one and the same. To my mind, it's not really sound to teach a known lie for the sake of quick results now only to confuse the kids in a couple of years when you have expose them to the truth, (Also, side rant of mine: Of all the series that teach The Multiplication Lie from 2nd and 3rd grade, I don't think I've ever seen one that goes back and addresses this directly in 4th-5th grade when it's time for kids to begin multiplying fractions. IMO, It's only fair that if you go out of your way to convince your kids that Santa Claus is real, you should at least take the time to clear up the falsehood when you're ready to let them in on the truth. Not just drop the pretense and hope for the best.)

 

 

Thank you for raising this issue.  I remember having a discussion about the lies we sometimes tell our children with Bernie Nebel and users of BFSU.  For example, we often tell our students the lie that electrons rotate around the nucleus like planets in a solar system.  Then one day, it's like Santa Claus, and now electrons occupy a probability distribution.  I felt uncomfortable telling my students the lie of electron orbits versus orbitals, but some argued that orbitals are just too much for younger students.  (I still don't really get it.)  

Do you have other examples of lies, in math or other subjects?  

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34 minutes ago, square_25 said:

That's rather rude. 

As blunt as I am, I didn't mean it to be rude. I'm just trying to spare myself time. Great teachers rarely change their tactics on a whim. When you've considered and decided against other approaches on purpose, than you're not about to flip on a whim, abandon a method you've spent months (or years) developing and using successfully.

 

I knew I wasn't going to say anything that would change your mind and I wasn't about to tell you anything about teaching Multiplication in elementary school that you hadn't already considered, discussed and decided against. *shrug* I felt that typing out all that wasn't going to be a great use of my time, and lo and behold it wasn't.

You're willfully informed about math pedagogy, not willfully ignorant.

 

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43 minutes ago, daijobu said:

 

Thank you for raising this issue.  I remember having a discussion about the lies we sometimes tell our children with Bernie Nebel and users of BFSU.  For example, we often tell our students the lie that electrons rotate around the nucleus like planets in a solar system.  Then one day, it's like Santa Claus, and now electrons occupy a probability distribution.  I felt uncomfortable telling my students the lie of electron orbits versus orbitals, but some argued that orbitals are just too much for younger students.  (I still don't really get it.)  

Do you have other examples of lies, in math or other subjects?  

In math? YES! The "Math Lies" are kinda a pet-peeve for me! We'd better start a separate thread...

As for other subjects, not off the top of my head, no. The natural sciences are not a personal strength of mine so I wouldn't know how to separate the lies from the truths.
Of course history is littered with them and we tell a bundle of lies when we do a sloppy half-job of teaching English phonics/spelling....

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1 hour ago, square_25 said:

Multiplication as repeated addition is not inaccessible to an average 6-7 year old. That's why I'm asking whether the same thing applies to scaling. It was a genuine question, because I was musing whether it'd be possible to define multiplication as scaling from the beginning and if so, how I would explain it. How would you define a quantity like 3*4? I assume even if you change your underlying model, you would wind up calculating that as 4+4+4? Or what would you do? I'm trying to get a picture of how this works in practice. 

I am not very hard-core about this, but I do try to word even early multiplication as scaling rather than "arbitrary" repeated addition.

So, a model for 3*4 might be, "A dog has 4 legs.  How many legs total do three dogs have?"  I would not use the word scaling, but that is what I am describing: scaling up from one dog to a group of dogs.  To solve this we would "figure out" 3 groups of 4 (I avoid suggesting adding the three 4's together).  Initially, I always make three groups of 4 with manipulatives or on the abacus and then physically rearrange them to show that we have one group of 10 plus 2 more.  Eventually each of my kids has realized the shortcut of repeated adding, and I don't stop them from doing that, but I continue to use the language of "figure out how many are in x groups of y", leaving a lot of flexibility as to how we figure that out.

Then it is a pretty easy leap to go to "A dog has 4 legs.  How many legs does half a dog have?" = "figure out how many are in 1/2 group of 4".   I draw my dog splitting picture such that each half gets two legs, but it only takes a couple problems for my kids to realize that if we split into a top half and a bottom half then one will have 0 legs and the other will have 4.  So then I introduce the language of "how many legs (on average) does half a dog have."

And then "Each alien only has 1/2 a leg.  (We draw an alien standing on one bloody stumpy leg - this catches my boys' attention.)  How many legs (on average) does 1/4 of an alien have? (We illustrate bisecting our alien through two vertical orthogonal planes.)" = "figure out how many are in 1/4 group of 1/2".

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26 minutes ago, square_25 said:

But here's the question. What do we mean by half a dog? That's sort of skipping the problem of what 1/2*dog really means... 

And what does 0.45 of a dog mean, say? How do we define that? I feel like the above description basically just assumes that kids know what "of" means, which... has frankly been my experience, but it doesn't solve the problem. 

For what it's worth, verbally I say that "4*3 is four 3s," which isn't very different. But it's still repeated addition, basically, if kids have any kind of model of addition. 

I have never once worried about my post-toddler-aged children knowing what "of" means.  Of is used in a lot of ways, but one very common one is defining a "whole".  Itty bitty children understand "you can have part of my cookie" (my cookie being the whole), "most of the leaves have fallen now" ("the leaves", presumably on one tree or in one area, being the whole), "only one of us can fit" (us being the whole).

In multiplication, of is used the same way, because you can only define x groups of y, if first you define what a whole y-sized group is.  So if you say 5 groups of monkeys, we can all vaguely picture it in our head, it is a valid multiplication, but we cannot calculate the total number of monkeys because we have not adequately specified the size of our "whole".  The "of" is showing us the size of one "whole group", but we lack the context to define that number.  On the other hand, if you say we have 5 groups of a dozen, we both share the background knowledge to know what "whole" the of is defining.

So when we say half of a dog, the of is simply defining our frame of reference: for this calculation, "a dog" = 1 whole.  We could just as easily say half of this pack of dogs.  Or half of a dog's spleen.  And once we have defined our whole, then we know that 1/2 means splitting that whole into 2 equal pieces and "keeping" one of them.

So talking that through, I guess the difference between "three 4's" and "3 groups of 4" is a subtle distinction between treating the 4 as 4 individual objects or as a cohesive "whole".  Which is why when you shift to fractions, the "x y's" language kind of falls apart; what is "1/3 18's"?  But since the group language was already treating the 4 as one whole entity, it can deal with fractions: "1/3 of a group of 18"...so the "of" defines 18 as one whole, which then gets split into 3 equal parts, and we "keep" one part, or 6 individual items.

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On 5/27/2020 at 11:59 PM, square_25 said:

(And scaling isn't a perfect way to think of negative multiplication, anyway: I can make a story that makes it make sense as scaling, but it doesn't feel intuitive.

Since you have a story already in mind that works for integers and fractions, what is it about your story that you don't like?

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45 minutes ago, square_25 said:

It’s not one single definition that generates a way of calculating. It’s just a bunch of different definitions I’ve glued together.

Basically, I think my story contains a ‚Äújump‚ÄĚ in reasoning. And I prefer making models without jumps.

Which definitions are you using?

How many definitions is "a bunch"?


 

 

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On 5/29/2020 at 5:26 PM, mathmarm said:

But is there a feasible solution to propose @Gil2.0? How do elementary teachers (who usually don't know any advanced mathematics themselves) avoid "the multiplication lie" when teaching multiplication?

I'm not in any position to say what elementary teachers should do. I don't have any interest in trying to minister to the public school system. None. I've never been into converting other adults to my way of anything.

*deleted the rest*

Edited by Gil
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On 5/29/2020 at 3:03 PM, square_25 said:

With DD7, I definitely piggybacked off of the fact that 1/n of something is that number divided by n. But why is that? We never discussed that. And to be totally transparent about the meaning of multiplication (or if you prefer, the meaning of the word "of"), you'd have to explain WHY that is. 

I've given this a lot of thought, and I have concluded that I really don't care if a grammar stage student doesn't know why a fraction behaves like a fraction.  I mean on a purely computational level:  1/n of something = 1/n * something = (1 * something)/n = something/n means something divided by n.  But my goals for grammar stage math doesn't go any further than that.

I guess in my mind, one of the hallmarks of grammar stage learning is being allowed to build upon foundations even if you are not in a position to understand why or how the foundations are true.  I see my job as summing up the grammar, the definitions, the foundations of the subjects into correct, versatile, simple axioms that the child can learn and expand upon.  So, for example, I teach a definition for a noun.  With that in hand, the child can learn to identify nouns, use nouns as nouns and verbs and adjectives, diagram nouns, use nouns symbolically in poetry, etc.  They can build all of that from a simple, true definition of a noun...but they don't necessarily have a clue why the definition is true.  Why do we group people, places, things and ideas all into one group?  Would we have to do it that way?  Do all language define nouns that way?  Why don't we have to worry about noun cases like in Old English?  Do all nouns behave the same way, even those that are actually verbs or adjectives being used as nouns?  Why doesn't our language use gendered nouns like most of its ancestors?  That is all incredibly interesting information which can deepen a student's knowledge and intuition of grammar, but it would just serve to confuse the issue during the grammar stage.

So during the grammar stage I keep things simple.  My kids all have/had significant speech delays, so their first math definitions were actually hand gestures I teach them.  By the time they finish arithmetic, they have a whole host of definitions that they can call back on.  I might prompt a struggling child with, "A fraction is...", and they will fill in, "a division problem."  If I say, "The denominator tells us...", they can quickly recite, "how many equal parts we are dividing something into."  They don't know why that is the case any more than they know why sound moves in a wave...they know it does, and they can build upon that information, but they can't explain or defend the foundational concept...yet.

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Just a few comments on the general discussion : 

- I don't really think of "lies we tell our children", I think of layers of understanding.  In a few ways, the solar system model of the atom is useful.  Adding and correcting that model over many years may be easier than trying to teach a more accurate model at age 6, or alternatively leaving the child with no model until high school.  Similarly, knowing that George Washington did certain factual things to create the USA in grade 2, that he was a slave owner in grade 5, and that he was, like many humans both deeply flawed and brilliant in middle/high school allows kids to have "hooks" to continue to hang knowledge on, and that knowledge is constantly clarified, corrected, and refined over time.  I would argue the same for things like telling a young child, "You can't take away more than you have" (with regards to subtraction) is accurate so far as that child is likely to need subtraction (taking cookies from the plate and so on).  As the child gets older and has an expanded understanding of things like debt, below sea level, and so on, they have more hooks to hang a more complicated definition of subtraction.  Some math curriculums do a good job of creating visual or concrete examples so that some concepts can be learned earlier than life-experience would easily allow for.  

- I have begun using fraction notation interchangeably with division symbol much earlier for my second kid.  For one, I prefer it, and for two, I realized she could understand it easily.  Maybe the interchangeability of these should be made explicit earlier.  

- I can remember when I learned how to divide into the decimals, and thinking "Well, remainders are just for baby math!"  I now try to make it really clear to my kids that remainders are real, valid answers to problems, and list off examples- like .45 of a dog is meaningless unless we're feeding dogs to lions, and needing 2.3 busses to get all the kids to the zoo is meaningless as well.  

I've never taught skip-counting, and my kids are regularly confused by the entire concept of skip counting when their friends mention it.  I don't see what purpose it serves, aside from encouraging using fingers and toes to count things.  I teach multiplication as:

x2 is doubling

x3 is tripling (this is the hardest one)

x4 is doubling the double

x10 is shifting everyone one place value 

x5 is halving the x10

x6 is doubling the triple

x8 is the double of the double of the double

x9 is one copy less than 10 copies

x7 is 5 copies and 2 more copies, or, at this point, just memorize 7x7=49 and figure out all the other using commutative property

Eventually, doing the calculations becomes as fast as rote fact recall.  

I don't know if that counts as teaching scaling or not.  We do all of it with manipulative (c rods), so it certainly looks like areas doubling or tripling or etc.  I've never said "repeated addition" to my kids.  

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