I teach mathematics at a community college. This includes teaching a lot of elementary algebra courses. These are pre-college level math courses. In elementary algebra we introduce solving quadratic equations by factoring. In the next course intermediate algebra we teach the quadratic formula.
Students in elementary algebra are not equipped to understand change of coordinates. This is too hard of a concept at that stage in my opinion. The method described in the article is a very nice one and is appropriate for elementary algebra. It ties in nicely with factoring and makes solving quadratic equations quite easy. I will be using this method in the future.
How aren't they equipped for it? Do you mean change of coordinates in its full generality? I'm talking about simple substitution of x' = x + k, connected with the intuition sketched below. They know how to perform simple substitution by then, and they can surely follow this logic:
x 0---1---2---3-->
x' 0---1---2---3---4-->
x' = x + 1
==>
x = x' - 1
y = f(x)
==>
y = f(x' - 1)
If you can talk about Napa being an hour north of San Francisco and San Francisco being an hour north of San Jose, this kind of student should be able to tell how far north of San Jose Napa is.
Can they not even do that? If not, I'd questioning why we teach them to solve quadratic equations before they can do substitution.
Well for one thing the simple substitution you mention is a hard concept. They certainly will have a hard time with f(x-1) and introducing a new variable is a mental block at this stage. At my college we don’t introduce functions until the next course. I suggest your viewpoint is clouded by the fact that you know this stuff so well that you no longer remember what the pain points are for students learning it for the first time.
After learning to solve simple quadratic equations and then the quadratic formula we typically introduce variable substitutions to solve things like x^4 + 5x^2 - 6 = 0.
I freely admit that my viewpoint is clouded by fluency, but I didn't come up with this today. This is how I've done it since I was still in school and I remember being annoyed by the opaque "just memorize this formula" approach from early on, where the concepts just seemed much clearer. I admit to not being the typical math student. But I think my approach was my competitive advantage, not something that makes my experience inapplicable.
To be clear, you're saying that if you tell a student
y = x^2 + b*x + c
and you tell them
x = z + 2
they aren't yet equipped to learn to combine those into
y = (z+2)^2 + b*(z+2) + c.
Is that right? Why do you teach quadratic equations at this stage? I'd consider those to be much more advanced than simple "replace x with (z+2) everywhere you see it" plus some "alice's house to bob's house to carol's house" problems along a single axis for the concepts.
It seems like you're saying they're taught quadratic equations before they're equipped to poke around with them, which seems to be setting up for the black box/memorize-the-formula version of math.
In elementary algebra students can do this for the most part:
Evaluate x^2 + 2x + 4 for x = 3.
Quite a few will struggle with:
Evaluate x^2 + 2x + 4 for x = –3
Almost all will struggle with:
Evaluate -x^2 - 2x + 4 for x = -3
I don't think they'd handle replacing x with z+2.
In elementary algebra they hate fractions. Many struggle with 4 – (–5). I like the approach in the article because of how it relates to factoring and difference of squares. This reinforces those concepts and shouldn't be too great of a leap at this stage. Also, there is a nice geometry behind the (b/2 + z)(b/2 - z) idea. The approach is nice precisely because it isn't a memorization approach. It's an approach that says, "Hey, let's analyze what factoring trinomials is all about and what the relationship between b, c, r, and s are.". It says, we know that r and s have to have the property that r+s is b and rs is c. We know this because of our analysis of multiplying binomials and our experience with factoring trinomials. We are using patterns and pattern recognition and from this we are constructing solutions to an equation that we can't solve by isolating the x like we did with linear equations. This to me, is true mathematics and it's nice to show students this exploration. To show how mathematicians think and approach problems. It's not a black box. It builds upon previous ideas and uses them to solve problems we weren't able to before.
Note that I'll be using this method in elementary algebra from now on but will not be using it to prove the quadratic formula. Indeed I will not even tell them what the quadratic formula is. I will save that for the next class.
'sykick you have my sincere appreciation for your tireless efforts in this thread to communicate both the existence and particulars of real differences in math aptitude among humans. Those on HN who have never attempted to help someone else with math would have had no idea...
I remember discovering this for myself when tutoring math in college. It's amazing how difficult it is to explain a lot of "basic" math without using "advanced" math.
Trying to solve limits without L'Hospital's rule is excruciating.
Comments
I teach mathematics at a community college. This includes teaching a lot of elementary algebra courses. These are pre-college level math courses. In elementary algebra we introduce solving quadratic equations by factoring. In the next course intermediate algebra we teach the quadratic formula.
Students in elementary algebra are not equipped to understand change of coordinates. This is too hard of a concept at that stage in my opinion. The method described in the article is a very nice one and is appropriate for elementary algebra. It ties in nicely with factoring and makes solving quadratic equations quite easy. I will be using this method in the future.
How aren't they equipped for it? Do you mean change of coordinates in its full generality? I'm talking about simple substitution of x' = x + k, connected with the intuition sketched below. They know how to perform simple substitution by then, and they can surely follow this logic:
If you can talk about Napa being an hour north of San Francisco and San Francisco being an hour north of San Jose, this kind of student should be able to tell how far north of San Jose Napa is.Can they not even do that? If not, I'd questioning why we teach them to solve quadratic equations before they can do substitution.
Well for one thing the simple substitution you mention is a hard concept. They certainly will have a hard time with f(x-1) and introducing a new variable is a mental block at this stage. At my college we don’t introduce functions until the next course. I suggest your viewpoint is clouded by the fact that you know this stuff so well that you no longer remember what the pain points are for students learning it for the first time.
After learning to solve simple quadratic equations and then the quadratic formula we typically introduce variable substitutions to solve things like x^4 + 5x^2 - 6 = 0.
I freely admit that my viewpoint is clouded by fluency, but I didn't come up with this today. This is how I've done it since I was still in school and I remember being annoyed by the opaque "just memorize this formula" approach from early on, where the concepts just seemed much clearer. I admit to not being the typical math student. But I think my approach was my competitive advantage, not something that makes my experience inapplicable.
To be clear, you're saying that if you tell a student
and you tell them they aren't yet equipped to learn to combine those into Is that right? Why do you teach quadratic equations at this stage? I'd consider those to be much more advanced than simple "replace x with (z+2) everywhere you see it" plus some "alice's house to bob's house to carol's house" problems along a single axis for the concepts.It seems like you're saying they're taught quadratic equations before they're equipped to poke around with them, which seems to be setting up for the black box/memorize-the-formula version of math.
In elementary algebra students can do this for the most part:
Evaluate x^2 + 2x + 4 for x = 3.
Quite a few will struggle with:
Evaluate x^2 + 2x + 4 for x = –3
Almost all will struggle with:
Evaluate -x^2 - 2x + 4 for x = -3
I don't think they'd handle replacing x with z+2.
In elementary algebra they hate fractions. Many struggle with 4 – (–5). I like the approach in the article because of how it relates to factoring and difference of squares. This reinforces those concepts and shouldn't be too great of a leap at this stage. Also, there is a nice geometry behind the (b/2 + z)(b/2 - z) idea. The approach is nice precisely because it isn't a memorization approach. It's an approach that says, "Hey, let's analyze what factoring trinomials is all about and what the relationship between b, c, r, and s are.". It says, we know that r and s have to have the property that r+s is b and rs is c. We know this because of our analysis of multiplying binomials and our experience with factoring trinomials. We are using patterns and pattern recognition and from this we are constructing solutions to an equation that we can't solve by isolating the x like we did with linear equations. This to me, is true mathematics and it's nice to show students this exploration. To show how mathematicians think and approach problems. It's not a black box. It builds upon previous ideas and uses them to solve problems we weren't able to before.
Note that I'll be using this method in elementary algebra from now on but will not be using it to prove the quadratic formula. Indeed I will not even tell them what the quadratic formula is. I will save that for the next class.
'sykick you have my sincere appreciation for your tireless efforts in this thread to communicate both the existence and particulars of real differences in math aptitude among humans. Those on HN who have never attempted to help someone else with math would have had no idea...
Thank you. That’s very nice of you to say.
I remember discovering this for myself when tutoring math in college. It's amazing how difficult it is to explain a lot of "basic" math without using "advanced" math. Trying to solve limits without L'Hospital's rule is excruciating.