# Teaching properties of numbers.

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5 hours ago, square_25 said:

Replying to everything else now (I was on my phone before, so didn't feel like typing a lot!): it's good to hear that the puzzles get better! In general, I feel like AoPS is really excellent at providing harder questions, that's a big strength of theirs. We'll probably get the other books as well, then, because the puzzles are really fun as enrichment. (Besides, I'm pretty sure I can get them as freebies, so why not? 😛

I haven't seen the Critical Thinking Company's books, no! I'll have to take a look.

I just asked my daughter "If x stands for a number, and x plus 4 is 7, then what's x?" and she answered without thinking "x is 3," so I see what you mean about variables! I'm going to have to think about whether we should switch to variables or not... I sometimes feel like shapes would help out even college kids conceptualize the idea that "a variable is just a stand-in for a number" (I occasionally feel like tattooing that on my forehead as a constant reminder), but on the other hand, if we started working with variables soon, I could keep reminding her about that... food for thought, thank you for the suggestion.

Did you ever wind up having any conceptual variable issues after introducing them?

Yes that way of introducing variables is good but then you can keep the answer as a variable as well. So if you have an equation with shapes and one variable, one of your shapes can only have an answer with the variable. I don't do my own problems, something similar is in MM and BA.

No there hasn't been any conceptual issues with variables but so far he has primarily done straightforward computation problems with them similar to what you've been doing already. Word problems where he has to identify that something is a variable based on context and logic is trickier. I have to give him hints to define the variable first. And tell him that the problem may be solved even if he doesn't arrive at an actual figure.

I'm not an expert but I figured if my kid can solve an equation using bigger numbers and with multiple operations (in other words he can't just automatically see the answer with mental math) then he does understand basic algebra conceptually (that the equals signs means balanced and how to manipulate numbers as you mentioned initially).

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Sorry, I reread your comment and have something else to add. The question you asked your daughter about x+3=7...that's something I introduce to my kids pretty much right away after learning to subitize. Sometimes x, sometimes shapes, sometimes apple, sometimes something unknown hidden in my hand, it doesn't seem to matter to them at all, even at three years old.

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6 hours ago, square_25 said:

Replying to everything else now (I was on my phone before, so didn't feel like typing a lot!): it's good to hear that the puzzles get better! In general, I feel like AoPS is really excellent at providing harder questions, that's a big strength of theirs. We'll probably get the other books as well, then, because the puzzles are really fun as enrichment. (Besides, I'm pretty sure I can get them as freebies, so why not? 😛

I haven't seen the Critical Thinking Company's books, no! I'll have to take a look.

I just asked my daughter "If x stands for a number, and x plus 4 is 7, then what's x?" and she answered without thinking "x is 3," so I see what you mean about variables! I'm going to have to think about whether we should switch to variables or not... I sometimes feel like shapes would help out even college kids conceptualize the idea that "a variable is just a stand-in for a number" (I occasionally feel like tattooing that on my forehead as a constant reminder), but on the other hand, if we started working with variables soon, I could keep reminding her about that... food for thought, thank you for the suggestion.

Did you ever wind up having any conceptual variable issues after introducing them?

Bolding mine.

What kind of problems do you see with the (older) kids you teach and their understanding of variables?

I feel like the difficulties we tend to see in teens/adults who have been doing algebra and higher maths for a while can be traced back to a lack of number-sense.

It holds true for me, anyhow. I went all the way to pre-calc, still unable to add 5+4 in my head instantly [part of it was a bit of an OCD tic where I had to count the dots in each number]. But I was able to get past that after teaching math to my son. We used the cuisinare rods and they were awesome for him (and I benefited a lot too!) He was not into drawing or building diagrams (not even dot diagrams) so having the rods to function as pre-drawn diagram pieces (and also as building blocks to play with a la miquon style) helped him a lot. It sounds like your daughter might be past the stage of benefiting from them, but it's not like they will hurt anything if you give them a shot.

I taught my son from Singapore Math through 3rd grade, and teaching all of the different ways of thinking about the problems helped me (and him) better understand how the numbers relate to each other.

We're currently using Life of Fred. I love it from a geeky perspective. Planning to go back to Singapore Math once we've finished their intermediate math series.

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.

Edited by Petrichor

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13 hours ago, Petrichor said:

Bolding mine.

What kind of problems do you see with the (older) kids you teach and their understanding of variables?

I feel like the difficulties we tend to see in teens/adults who have been doing algebra and higher maths for a while can be traced back to a lack of number-sense.

It holds true for me, anyhow. I went all the way to pre-calc, still unable to add 5+4 in my head instantly [part of it was a bit of an OCD tic where I had to count the dots in each number]. But I was able to get past that after teaching math to my son. We used the cuisinare rods and they were awesome for him (and I benefited a lot too!) He was not into drawing or building diagrams (not even dot diagrams) so having the rods to function as pre-drawn diagram pieces (and also as building blocks to play with a la miquon style) helped him a lot. It sounds like your daughter might be past the stage of benefiting from them, but it's not like they will hurt anything if you give them a shot.

I taught my son from Singapore Math through 3rd grade, and teaching all of the different ways of thinking about the problems helped me (and him) better understand how the numbers relate to each other.

We're currently using Life of Fred. I love it from a geeky perspective. Planning to go back to Singapore Math once we've finished their intermediate math series.

Good question. I see a variety of difficulties with variables. By far the most common one is the sense that identities involving variables are some sort of magic trick that there's no way to think about conceptually. For example, kids learn that sin(x+y) is not equal to sin(x) + sin(y) in trigonometry, but for the average students, it's not at all clear why. Kids see so many variable identities that don't often have numbers plugged in that they internalize that a variable is its own thing, not a placeholder for a number. As a result, it doesn't occur to people to check whether various identities hold.

The equal sign confusion is another one, although I think it's related. What happens, as far as I can tell, is that students internalize algebraic manipulations not as "we have two equal things, and we can do the same thing to both sides" but as "here are some magic things you can do to an equation (which is also some sort of magic expression), for example, we can move a + x from one side to the other, and in the process it becomes a -x." I think it's actually part and parcel of the same thing: since an x isn't really a number, it's hard for kids to think about things like "What happens if we add x then subtract x?" and therefore they resort to rules about equation manipulations. This really hampers kids once they learn new operations, or when certain arithmetic operations stop being allowed. For example, we can take the conjugate of both sides of an equation with complex numbers, and it's a lot easier to internalize this one if you think of an equation as something telling us two things are equal. And we can't divide both sides by a vector or a matrix and that also trips us the kids who are used to thinking of equations and variables as something not at all concrete.

Am I making sense here? I think it's the standard issue that people complain about, where students treat mathematics as a series of black boxes. This blog post is somewhat relevant:

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