Math Talk #01 - Language in Arithmetic Education

[mom2bee]

So, the role that language plays in teaching arithmetic, particularly in primary/elementary school level, interests me a lot. It's something that I've been thinking about quite a bit for a while now.

If you had to give each of the operations a simple, plain-English definition (or interpretation) to be used in 1st-8th grade arithmetic, what would it be? Please note the range there. If your definition is only good enough for a sub-set of that range, can you think of a way to improve it? (It's okay if the answer is "no")

To be clear, the arithmetic operations are:

• Addition
• Subtraction
• Multiplication
• Division
   

I'm especially hoping some of our Math-minded homeschooling parents will chime in on this, but everyone is invited to chime in.
If you feel that you're elementary math series of choice gives a really good definition for the operations, please share the definitions/meanings that the series uses. (Hint: you may need to check the teachers guide to get the definitions/explanations as opposed to just the student text).

 

Edited May 26, 2020 by mom2bee
adding tags

[maize]

Hm, off the top of my head:

Addition means joining quantities or values together, the addends are the parts we want to combine, the sum is the combined value. For young children, this should first be demonstrated concretely: 2+3 means I have two items (say, cookies) on one plate and three on another. How many cookies are there altogether? If I want to teach the concept of equation, I might present two cookies on one plate, three cookies on another, then put down a third plate after an equals symbol and have the child figure out how many cookies need to go on the third plate so that there are the same number of cookies on each side of the equals sign.

That's all I have time for right now, I have to go do dishes 🙂

[Terabith]

This seems like a question for @square_25

[Monica_in_Switzerland]

Addition means "put together"

Subtraction means "the difference between"

Multiplication means "of"

Division means "shared equally between" (this one isn't quite as good as the others)

Equal means "balances with"

 

I didn't invent these, I learned about them from educationunboxed.com.  But they work quite well, especially when trying to determine which operation to use in word problems.  The "of" for multiplication is especially helpful.  How many times have I heard myself saying to my kids:

- What do you need to find?

- 3/20 of 60.  How do I do that?

- What does "of" mean in math?

- Oh yeah!  

[Monica_in_Switzerland]

Like @CuriousMomof3, we also spend a lot of time going over if we know a part or a whole, which can help with picking out the right operator.  

- Ok, so with this information do we know the whole or are we trying to find the whole?  Remember, when we know the whole and are looking for parts, order matters in the operation.

- Are these parts?  Are we trying to find the total and we know all the parts?  

 

Also, I discovered early on that several of my kids did not know the difference between "whole" and "hole" 🤣  Be sure to clear that up with little kids!!!  

[Monica_in_Switzerland]

2 hours ago, square_25 said:

Similarly, for multiplication, as adults we know that multiplication is commutative: that it doesn't matter what order we multiply numbers in. However, I do not tend to introduce that to kids as I introduce multiplication. I define 3*5 as "adding three copies of 5" and 5*3 as "adding five copies of 3." Those are not obviously the same thing! 

I actually made a mistake with DD7 (back when she was 5, I think) of trying to teach her about the commutativity of multiplication before she had really internalized what multiplication was. That was one of my first clues that before you can teach properties of an operation or a symbol, you need to let the child internalize ONE unambiguous definition of the symbol. After this is internalized, it is much easier to add onto that structure by noticing other properties of the symbol. However, I firmly believe that there needs to be a primary definition that the kid can operate with confidently. (Eventually, the secondary ones also get absorbed, and that's something I've worked on explicitly with DD7. However, she's always had lots of confidence that she understood what the symbols meant because she had a solid primary understanding. It has helped her feel that she can figure out anything out.)

 

While I deeply respect everything you have to say about math around here, my experience with the commutative property of multiplication has been very different, so far for my first three kids, including my struggling student.  We used cuisinaire rods for the first week of multiplication, constantly showing that a 2x6 rectangle is the same as a 6x2 rectangle in terms of area, and so on.  So maybe I could say that we defined multiplication as area before defining it as repeated addition?  Anyway, we had no issues.  Every time I read a mulitpilcaiton problem as we introduce the topic, I always read it as "3 times 4.  That's 3 copies of 4 or 4 copies of 3.  Which do you think you could do most easily?" and then let them choose how to proceed.  I do the exact same when teaching adding.  And every time we do subtraction or addition, I say, "Remember with subtraction/division, order matters.  We need to start with the whole amount."

I don't like defining subtraction as taking away, because it is also the operation of comparing.  I try to distinguish early on these two types of subtractions.  Problems like "Jim is 123cm and Ryan is 140cm, how much taller is Ryan?" are not obviously subtraction since no one is taking away from anyone.  We always draw bar diagrams for these to see better what the whole and 2 parts are.    

[Terabith]

I wonder if it makes a difference developmentally when you introduce the terminology?  With my older child, who is much more math intuitive, we started addition and subtraction by maybe four and multiplication at five.  And...she needed a primary definition of the operation at first, although with multiplication I did introduce commutativity almost immediately because she was familiar with the concept from addition.  We did do other descriptions of subtraction later, but that first introduction was “taking away the second number from the first group.”  
 

If you weren’t introducing it until age six or seven, maybe you could be more nuanced from the beginning?  Although I think that is confusing to have too many options at first.  

[Shoes+Ships+SealingWax]

Addition: Combining parts into a whole.

Subtraction: Removing parts from the whole or comparing wholes. 

Multiplication: Combining equal groups of parts into a whole. 

Dividing: Removing equal groups of parts from a whole. 

[EKS]

12 hours ago, square_25 said:

Do you all know how to extend that definition to fractions so it's consistent?

How you describe this further down the thread is essentially how how Math U See introduces the concept of fraction division.  4 ÷ 1/2  can be thought of as "how many 1/2s are in 4?"  When you think of it that way it's obvious there are 8.  Likewise, 1/3 ÷ 2/3 can be thought of the same way.  How many 2/3s are in 1/3?  Half of 2/3 can "fit into" 1/3.  2/3 becomes the whole.  With fractions with different denominators, you have to make a common denominator before you can see the answer this way.  So 3/7 ÷ 1/3 would turn into 9/21 ÷ 7/21 or "how many 7/21s are in 9/21?"  One whole 7/21 and 2/7 of that whole 7/21--so, 1 2/7 or 9/7.  They have these ingenious manipulatives that show this really well.

At least I think this is what you were saying...

[EKS]

4 hours ago, square_25 said:

Yes, that's the definition I've always thought of, which is the quotative definition. However, the other one is the partitive definition extended to fractions, which I had literally never used before and had no feeling for. But apparently it's not actually that hard, because DD7 internalized it very quickly :-).

I obviously need to go back over your explanation 😄

ETA:  Ok--are you talking about the problems like "Ms Smith owns 1 3/5 acre of land.  She wants to divide it into plots of 3/16 acres each.  How many plots will there be?  How much land will be left over?"  

ETA2:  In thinking about it, I'm pretty sure this is the same as my original example.  I will look at Liping Ma's book and get back to you.

Edited April 20, 2020 by EKS

[EKS]

13 minutes ago, square_25 said:

Let me start with the question from before, but this time, could you answer it? That is, if 2/3 of a container has 7 quarts of water, how much does 1 container have?

I do problems like this by thinking that if 2/3 of the container has 7 quarts, 1/3 has 7/2 quarts, so 3/3 will have 21/2 quarts.  The actual calculation I do is 7/2 times 3 or 7 ÷ 2 x 3.  

[Monica_in_Switzerland]

On 4/20/2020 at 9:00 PM, EKS said:

I do problems like this by thinking that if 2/3 of the container has 7 quarts, 1/3 has 7/2 quarts, so 3/3 will have 21/2 quarts.  The actual calculation I do is 7/2 times 3 or 7 ÷ 2 x 3.  

 

This is also how I'd do it.  Bar models are great for making this comprehensible.  

[Monica_in_Switzerland]

Another little comment on the "How much does one have?" idea: I use this all the time when determining rate.  Rate is typically "over 1," that is miles per (one) hour, meters per (one) second, revolutions per (one) minute, and so on.  

Another way of thinking of 7 / (2/3) would be, "A bird flies 7m in 2/3 of a second.  What is their speed in m/s?"  And I always think of these problems as ratios, so I'd do something like

7 : 2/3 as ___ : 1

which I'd set up as fractions, 7/x = (2/3)/1 = 2/3, then cross multiply.