Algebra vs. geometry/trig

[drjuliadc]

Here’s another thing I never hear about, it is that people in general will be better at either algebra OR geometry/trigonometry. I do believe there are mathematically minded people who are good at both, but in general, people seem to be much better at one than the other.  I don’t know why.

Maybe calculus would be more on the algebra side. If I remembered enough math I might know that.

I personally found geometry and trigonometry much easier than algebra and calculus. Maybe they just have simpler concepts, but I seem to hear individuals say they are much better at algebra but struggled with geometry/trig.

So individual people will tell me THEY were much better at one than the other, almost universally, but I never hear educators, or anyone else talking about math, verbalize this as a tendency. 

[Not_a_Number]

51 minutes ago, drjuliadc said:

So individual people will tell me THEY were much better at one than the other, almost universally, but I never hear educators, or anyone else talking about math, verbalize this as a tendency. 

I've heard that, but it's odd to me, and I don't really understand what it means. 

For me, algebra is a tool. I used algebra ALL THE TIME with pretty much every single other mathematical subject I studied, and that included geometry. I wouldn't have been able to do geometry without being able to do algebra, because you often do have to manipulate equations with variables and solve for unknowns. 

Now, I'll say that geometry was a strength of mine back when I did math contests -- I was better at it than most people. But I wouldn't be able to tell you if I were better at geometry or algebra, because to me, it's like asking whether I prefer swimming or peaches -- they aren't in the same category. 

By the way, I think of calculus as incredibly visual subject 😉 . 

[daijobu]

I agree with NaN here.  I also don't get the good at algebra/good at geometry dichotomy.  And I also agree that calculus is very geometry oriented, especially finding those areas and volumes (my favorite part of calculus). 

In high school, my friends and I joked about the minor freakouts we'd experience when we encountered new notation.  There was always one thing each year.

Freshman year was f(x).  Whenever we saw f(x) we'd joke that we'd need to take a deep breath and calm down.  Sophomore year was the sigma sign.  Deep breath, you can do this.  I think junior year, it was combinations nCr, and senior year it was probably d/dx or the integral sign.    We'd get over it and joke about it eventually, but there was always some new notation that would throw us at first.  

[kiwik]

I can see an argument for calculus OR statistics OR geometry  but algebra is foundational.  You can't do chemistry, physics, geometry, statistics or calculus without a least some algebra.

[SoCal_Bear]

Hmmm...both DH and I both agreed geometry was the one area neither of us liked very much. Everything else was completely fine - algebra, trig, calc, stats. I think number theory was a bit of a challenge for me simply because I was unused to thinking about math that way. I see the same thing in my son whenever we get to geometry type problems as well. 

 

[Carol in Cal.]

I thought geometry was incredibly hard and algebra was challenging but fundamentally easy.  DH is just the opposite.  DD hated all math but was generally better at geometry.  I think it is really common to find one easy and the other hard, just like chemistry vs. electronics.

[lmrich]

I am tutoring a kid in geometry who is also taking Algebra 2 (both Derek Owens' courses). He has a 98 in Algebra 2! He had 71 in geometry when we started and was way behind!  He has pulled it up to an 89 and has one test left before the semester exam. He just didn't really understand that you have to memorize the definitions and theorems. He also figured math has always been so easy for him that he did not really have to do the practice sets for homework... he now knows better!

[Not_a_Number]

6 minutes ago, lmrich said:

I am tutoring a kid in geometry who is also taking Algebra 2 (both Derek Owens' courses). He has a 98 in Algebra 2! He had 71 in geometry when we started and was way behind!  He has pulled it up to an 89 and has one test left before the semester exam. He just didn't really understand that you have to memorize the definitions and theorems. He also figured math has always been so easy for him that he did not really have to do the practice sets for homework... he now knows better!

I wouldn’t precisely memorize the theorems 😉 

[lmrich]

Just now, Not_a_Number said:

I wouldn’t precisely memorize the theorems 😉 

True - but once you learn one, you need to remember it for the next chapter.

[Not_a_Number]

7 minutes ago, lmrich said:

True - but once you learn one, you need to remember it for the next chapter.

True. 

[EKS]

I suspect that people who are good at following procedures but don't necessarily understand the concepts will prefer algebra and people who have gaps in their algebra knowledge but strong logical reasoning skills will prefer geometry.  

 

[EKS]

21 minutes ago, lmrich said:

He just didn't really understand that you have to memorize the definitions and theorems.

You don't.  When I taught my kids geometry I photocopied the list of theorems etc in the back of the book and let them use it as needed.  

[Not_a_Number]

34 minutes ago, EKS said:

I suspect that people who are good at following procedures but don't necessarily understand the concepts will prefer algebra and people who have gaps in their algebra knowledge but strong logical reasoning skills will prefer geometry.  

 

I actually think it’s partially just visual processing.

I do think that anyone who wants to get good at math needs to somehow learn to deal with both. But I do have some kids online who are fairly slow with calculations (usually both arithmetic and algebra) who are lightning fast with primarily visual questions. It’s very noticeable that the ordering of the kids largely stays the same between arithmetic and algebra but shifts for pure geometry.

[EKS]

3 hours ago, Not_a_Number said:

I actually think it’s partially just visual processing.

I do think that anyone who wants to get good at math needs to somehow learn to deal with both. But I do have some kids online who are fairly slow with calculations (usually both arithmetic and algebra) who are lightning fast with primarily visual questions. It’s very noticeable that the ordering of the kids largely stays the same between arithmetic and algebra but shifts for pure geometry.

Dyslexics will behave this way.

[Not_a_Number]

11 minutes ago, EKS said:

Dyslexics will behave this way.

Super interesting. Didn't know that, thanks. 

I'm also very fast with the visual questions, but I'm equally fast with the arithmetic/algebra. So I'm not an example of what I mean 🙂. 

Edited December 18, 2020 by Not_a_Number

[drjuliadc]

7 hours ago, Carol in Cal. said:

I thought geometry was incredibly hard and algebra was challenging but fundamentally easy.  DH is just the opposite.  DD hated all math but was generally better at geometry.  I think it is really common to find one easy and the other hard, just like chemistry vs. electronics.

Ooh ooh yes.  I was so good at so much of science except I struggled more with concepts of electricity. OK and organic chemistry, much harder for me than any other chemistry. 

[Carol in Cal.]

45 minutes ago, drjuliadc said:

Ooh ooh yes.  I was so good at so much of science except I struggled more with concepts of electricity. OK and organic chemistry, much harder for me than any other chemistry. 

ITA about organic chemistry vs. the rest, and I think it's because it involved so much less logic and so much more memorization.  Inorganic chemistry just plain makes sense.  Organic chemistry is more about memorizing things and developing an almost artistic sense of what will happen.  I was fortunate to have two runs at each of my O Chem classes--I dropped the first one partway through the quarter for personal reasons, and then started over the next quarter, and I missed the final in the second class due to getting mono and had to take the class final the next year, which (since it was a different professor) meant that I needed to audit all the lectures to learn his approach to the material.  I did fine in both of them but it was touch and go there for a while.

[Momto6inIN]

I think it's because most people tend to like one better over the other and think one is easier than the other. And we all tend to think we're better at things we like than things we don't.

I had a very rote, nonconceptual, follow the algorithm math education, and I liked algebra much, much, much better. It was so easy to just follow the algorithm. Geometry proofs were pointless, in my oh so informed and mature teenage opinion, because you just told me it was true, why on earth should I have to prove something you just told me was true? We never did proofs in any other math class than geometry. 

Now that I've been through a conceptual algebra and geometry program along with my kids, I can appreciate the beauty of a good proof, but I still don't like geometry as well. 

[Not_a_Number]

2 minutes ago, Momto6inIN said:

Geometry proofs were pointless, in my oh so informed and mature teenage opinion, because you just told me it was true, why on earth should I have to prove something you just told me was true? We never did proofs in any other math class than geometry. 

I really HATE that people only see proofs in geometry in lots of US programs. Proofs are basically a chance to work one's logic muscles. They should be all over math. And they should be in WORDS, not two columns. 

I've had DD8 do proofs since basically the beginning. I think it's a good way to go -- start verbally, proceed from there 🙂 . 

[kiwik]

On 12/18/2020 at 4:30 PM, Carol in Cal. said:

ITA about organic chemistry vs. the rest, and I think it's because it involved so much less logic and so much more memorization.  Inorganic chemistry just plain makes sense.  Organic chemistry is more about memorizing things and developing an almost artistic sense of what will happen.  I was fortunate to have two runs at each of my O Chem classes--I dropped the first one partway through the quarter for personal reasons, and then started over the next quarter, and I missed the final in the second class due to getting mono and had to take the class final the next year, which (since it was a different professor) meant that I needed to audit all the lectures to learn his approach to the material.  I did fine in both of them but it was touch and go there for a while.

I struggled with organic chemistry in first year because they wanted me to memorize reactions.  The next year they covered how the reactions worked and I could stop memorising again.  I can memotise lots of things easily but not chains of O, C and H.  And I refuse to memorize 6 equations when I can learn 1 or 2 and derive them as needed. It makes my brain hurt.

[Carol in Cal.]

On 12/18/2020 at 6:14 PM, Not_a_Number said:

I really HATE that people only see proofs in geometry in lots of US programs. Proofs are basically a chance to work one's logic muscles. They should be all over math. And they should be in WORDS, not two columns. 

I've had DD8 do proofs since basically the beginning. I think it's a good way to go -- start verbally, proceed from there 🙂 . 

I love two column proofs because they are a way to keep track of what you know already and the logic patterns and your progress at the same time.  Once I learned that technique in algebra, I used it for everything STEM from then on.  It was particularly helpful in working out complex iterative chemical engineering and spectroscopy problems.

[Carol in Cal.]

7 hours ago, kiwik said:

I struggled with organic chemistry in first year because they wanted me to memorize reactions.  The next year they covered how the reactions worked and I could stop memorising again.  I can memotise lots of things easily but not chains of O, C and H.  And I refuse to memorize 6 equations when I can learn 1 or 2 and derive them as needed. It makes my brain hurt.

I found that I could not derive them fast enough for our crazy long tests.  So I needed to both remember the logic and do some judicious memorizing.  Whereas deriving had always worked for me before, and I prefer it, so this was quite an adjustment.

[Not_a_Number]

2 hours ago, Carol in Cal. said:

I love two column proofs because they are a way to keep track of what you know already and the logic patterns and your progress at the same time.  Once I learned that technique in algebra, I used it for everything STEM from then on.  It was particularly helpful in working out complex iterative chemical engineering and spectroscopy problems.

In my experience, two column proofs make kids think mathematical logic isn’t like other logic, and that’s false. I’m glad it didn’t affect you like that, but I’ve personally been much happier with the AoPS tack of having kids write down proofs in paragraphs.

[Carol in Cal.]

1 minute ago, Not_a_Number said:

In my experience, two column proofs make kids think mathematical logic isn’t like other logic, and that’s false. I’m glad it didn’t affect you like that, but I’ve personally been much happier with the AoPS tack of having kids write down proofs in paragraphs.

What I like about it is that you’re documented what you know for sure.  As you progress through the problem, there are a bunch of subproblems to solve, and what you know for sure, if it’s annotated off to the side, can easily be used later for the overall problem, rather than hunting back through a bunch of verbiage.  It’s succinct, it’s listed in one place, and it’s out of the way of the rest of the the calculations.

I actually use that method in building the logic of presentations that are not mathematical as well.  It is superbly helpful in organizing material and driving to conclusions.

 

[Not_a_Number]

1 hour ago, Carol in Cal. said:

What I like about it is that you’re documented what you know for sure.  As you progress through the problem, there are a bunch of subproblems to solve, and what you know for sure, if it’s annotated off to the side, can easily be used later for the overall problem, rather than hunting back through a bunch of verbiage.  It’s succinct, it’s listed in one place, and it’s out of the way of the rest of the the calculations.

I actually use that method in building the logic of presentations that are not mathematical as well.  It is superbly helpful in organizing material and driving to conclusions.

Interesting. I can imagine that being useful if you already understand the logic of math. However, the kids I knew who learned it seemed to think the format was required for the logic to make sense. 

It's like kids I taught in college who thought the only kind of proof was "induction." That was the only kind of proof they had ever seen before, so they didn't think of other logical arguments as proofs...

The other thing about it is that it's really hard to read. Human brains do better with reading actual words than a simple list of formulas. 

Edited December 22, 2020 by Not_a_Number

[Dmmetler]

I have a visual-spatial processing disability and for some reason, my brain Really struggles to make the jump from equations to graphs (and to tell if an equation matches a graph or not) but I can handle geometry and from picture to statement and conclusion. My DD is the opposite-she grasps algebra very easily, can easily "see" the equations that go with a graph in calculus or algebra or statistics, but finds geometry and trigonometric proofs frustrating because so much is obvious. She has a much easier time with AoPS proofs than with two column ones because they are more logical to her as opposed to breaking everything down. We had a lot of discussions as to why "well, DUH!" Was not an acceptable statement in a formal proof :). 

[Not_a_Number]

7 minutes ago, Dmmetler said:

She has a much easier time with AoPS proofs than with two column ones because they are more logical to her as opposed to breaking everything down. We had a lot of discussions as to why "well, DUH!" Was not an acceptable statement in a formal proof :). 

Yeah, that's how I feel 🙂. 

 

8 minutes ago, Dmmetler said:

I have a visual-spatial processing disability and for some reason, my brain Really struggles to make the jump from equations to graphs (and to tell if an equation matches a graph or not) but I can handle geometry and from picture to statement and conclusion.

I'm sure this is related to your processing disability, but I actually found that people on average have trouble matching equations to graphs. 

[ieta_cassiopeia]

A lot depends on how each was taught, since often there will be different teachers and different methods for each field. For me, I was a lot better at geometry/trignometry than algebra... ...until I found someone who was able to remediate my algebra. Now I'd say I'm about as good at all of them (though I am still strongest in statistics and mechanics - both because I have more use for those outside the classroom than for "pure" algebra or its geometric/trignometric applications).