Jacobs Geometry: Will the Proofs Ever End?

[JumpyTheFrog]

DS (9th grade, ADHD) is doing the parent-graded version of Derek Owen's Geometry course, although we are using the 2nd edition of the book rather than the 3rd. He is almost done with chapter 11 and it looks like the DO course goes through chapter 15. I think he has about reached his limit with proofs for the year. The remaining chapters are about circles, concurrence theorems, regular polygons, and geometric solids. Is it really necessary to continue doing so many proofs? My vague memory of high school geometry was that had only a chapter or two with proofs, whereas this course has used proofs since just a few weeks in.

[cintinative]

You are not encouraging me! My youngest with ADHD-inattentive still needs to make it through Geometry. 😃

We felt similarly with Jurgensen's (with my oldest). He's my compliant kid and he was complaining by the end.

As far as what people "normally" cover, I am surprised at the breadth of it.  My nephew had proofs in his accelerated public school class. My niece, also accelerated, had no proofs in the same class two years later.  My friend who also covered Geometry in homeschool said all their proofs are in a couple chapters of their book, but in mine, they were spread out throughout. 

I hope some of our math savvy folks chime in on this. In the meantime, I hear you!!

[EKS]

You don't have to assign all of the proofs that are in the second edition.  Look to see which ones support DO's teaching and output expectations, and just assign those.

[Not_a_Number]

Proofs are awesome and build logical reasoning 😛 . 

What's bugging him about the proofs? Is he understanding them well? 

[Clarita]

Sorry no help. I hated proofs. I went into engineering instead where I can study applied calculus instead of mathematics where more proofs.

[J-rap]

High school geometry is a strange thing, because I think your brain has to be at the right place in order to absorb in correctly, and I also think that it varies a lot depending on your math age (I think different brains move through math ages at different times) and also specific brain types.  My brain either wasn't ready at that age, or probably just didn't naturally lean toward that way of thinking, even though I was a very good student.  I barely passed.  When I taught it years later to my kids while homeschooling, I first took a course in it to make sure I knew enough to teach it!   I "got it" then, and actually really enjoyed it and did really well in it.

 

[Not_a_Number]

I feel like the way proofs are mostly taught, they seem totally arbitrary and pointless 😕 . 

DD8 totally slid into proofs on her own at age 7, because it was how we talked about math. I never focused much on rigor, just on clear arguments and good logic. That's all that a proof is. 

[theelfqueen]

Jacob's is VERY proof heavy. If you feel he has a solid grasp of proofs I'd assign fewer. 

[Frances]

3 hours ago, Clarita said:

Sorry no help. I hated proofs. I went into engineering instead where I can study applied calculus instead of mathematics where more proofs.

I liked proofs, except in geometry. But I never got to do them anywhere except geometry until college. So for a long time I thought I hated them.

[Not_a_Number]

52 minutes ago, Frances said:

I liked proofs, except in geometry. But I never got to do them anywhere except geometry until college. So for a long time I thought I hated them.

So what's worse about geometry proofs? 😄 

And I have no idea why geometry is the one and only repository of proofs until college! I did proofs with DD8 from the very start, and they weren't in geometry... 

[Frances]

49 minutes ago, Not_a_Number said:

So what's worse about geometry proofs? 😄 

And I have no idea why geometry is the one and only repository of proofs until college! I did proofs with DD8 from the very start, and they weren't in geometry... 

I just never liked geometry very much. It’s actually one of the few math classes all the way through grad school that I didn’t like, and I don’t think it had much to do with the teacher as my high school was so small we only had one math teacher. My husband and I are the opposite. I loved algebra and he loved geometry. 

[UHP]

10 hours ago, Frances said:

I liked proofs, except in geometry. But I never got to do them anywhere except geometry until college. So for a long time I thought I hated them.

I think there is something different about high school geometry.

[EKS]

10 hours ago, Not_a_Number said:

And I have no idea why geometry is the one and only repository of proofs until college!

Perhaps since (I believe) geometry was the terminal math class in high school back in the day, they decided that if students made it that far, that they should get a taste of real math.  Then it just stayed that way after universal high school became a thing.

[Not_a_Number]

29 minutes ago, EKS said:

Perhaps since (I believe) geometry was the terminal math class in high school back in the day, they decided that if students made it that far, that they should get a taste of real math.  Then it just stayed that way after universal high school became a thing.

Maybe that’s it, I’m not sure.

But I feel like starting proofs with something kids are more familiar with, like arithmetic operations, would be better. Kids don’t necessarily have good geometric intuitions, which really gets in the way. 

[EKS]

17 minutes ago, Not_a_Number said:

But I feel like starting proofs with something kids are more familiar with, like arithmetic operations, would be better. Kids don’t necessarily have good geometric intuitions, which really gets in the way. 

But they can actually see a lot of the geometry stuff with diagrams.  It's more concrete in that sense.

[Not_a_Number]

23 minutes ago, EKS said:

But they can actually see a lot of the geometry stuff with diagrams.  It's more concrete in that sense.

They can't REALLY see it. Practically all geometry proofs require a congruence or similarity criterion, and having taught AoPS geometry, kids do NOT internalize those criteria in a good way. I just spent something like 4 months with DD8 constantly visualizing things until the criteria genuinely felt internalized and robust. It was a long process. 

Whereas with arithmetic operations, I actually think kids have a decent intuition for things from the beginning, so the proofs don't feel arbitrary. Congruence or similarity proofs just feel random to kids -- I constantly had kids try to use things like "AAAA similarity for quadrilaterals," because they genuinely didn't have any intuition for what worked. 

Edited July 9, 2021 by Not_a_Number

[EKS]

10 minutes ago, Not_a_Number said:

Congruence or similarity proofs just feel random to kids

Interesting.  I never had this experience myself or with the (two) students I've taught.

[JumpyTheFrog]

15 hours ago, Not_a_Number said:

What's bugging him about the proofs? Is he understanding them well? 

He doesn't really like to work hard mentally. He is an above average math student (tests in the 90 something percentile), but he doesn't like to work hard at it. We only made it through AOPS Intro to Alegbra (chapters 1-13) because I eventually wound up sitting and doing the problems with him. After that I decided I didn't have the time or willingness to sit and do each course with him.

Some of the lessons have 10+ proofs in them. Most have at least 5-7. It often takes him 10-20 minutes per proof, and this is after doing 25-30 non-proof problems in the text plus about 30 minutes of Derek Owens videos. It can take him 1.5-3 hours per lesson. To get through 15 out of 17 chapters (what DO covers) there are 113 lessons plus 15 tests. This is simply too many hours of math to get through for him.

15 hours ago, J-rap said:

High school geometry is a strange thing, because I think your brain has to be at the right place in order to absorb in correctly, and I also think that it varies a lot depending on your math age (I think different brains move through math ages at different times) and also specific brain types.  My brain either wasn't ready at that age, or probably just didn't naturally lean toward that way of thinking, even though I was a very good student.  I barely passed.  When I taught it years later to my kids while homeschooling, I first took a course in it to make sure I knew enough to teach it!   I "got it" then, and actually really enjoyed it and did really well in it.

I remember my high school geometry teacher saying at the start of the chapter(s) on proofs that we would either get it or not. Sure enough, after correcting the tests he announced that everyone in the class either got an A or an F, except for one student who got a C.

[Not_a_Number]

Just now, EKS said:

Interesting.  I never had this experience myself or with the (two) students I've taught.

Maybe you took things at a more reasonable pace? Maybe you paused and explained the criteria when you ran across them again, until they were internalized? I'm really not sure, but it was really more common than not in AoPS classes. And it really did take us a good long time for DD8 to learn to actually visualize things and not just memorize the acronyms that "work." 

Oh, and no one ever remembered SAS similarity. Ever. 

[Not_a_Number]

1 minute ago, JumpyTheFrog said:

Some of the lessons have 10+ proofs in them. Most have at least 5-7. It often takes him 10-20 minutes per proof, and this is after doing 25-30 non-proof problems in the text plus about 30 minutes of Derek Owens videos. It can take him 1.5-3 hours per lesson. To get through 15 out of 17 chapters (what DO covers) there are 113 lessons plus 15 tests. This is simply too many hours of math to get through for him.

I'd probably simply assign fewer proofs and make sure the ones he does are very nicely understood and clearly explained. That sounds like it would overload most students! 

[bensonduck]

You could also have him do maybe 1-2 formal proofs per lesson, then 1-2 paragraph style proofs, and then the rest he could just verbally explain to you why it is so. 
 

even though geometry is weird in that it is so proof-heavy, you’re definitely proving things in other HS math classes. For example, DD had a problem in Foerster precalculus today that was something like, “Show that the cross product of these two vectors is perpendicular to this other vector.” (I don’t have the actual problem in front of me and it was a couple hours ago, so I may be misremembering.) But there was a calculation piece as well as understanding the concept of cross products and how to show they are perpendicular. When she was done with the problem, it looked basically like a paragraph proof. 

[Not_a_Number]

Just now, bensonduck said:

You could also have him do maybe 1-2 formal proofs per lesson, then 1-2 paragraph style proofs, and then the rest he could just verbally explain to you why it is so. 

Is a "formal proof" a 2-column proof? Because there's literally no reason to study those at all. 

 

Just now, bensonduck said:

even though geometry is weird in that it is so proof-heavy, you’re definitely proving things in other HS math classes. For example, DD had a problem in Foerster precalculus today that was something like, “Show that the cross product of these two vectors is perpendicular to this other vector.” (I don’t have the actual problem in front of me and it was a couple hours ago, so I may be misremembering.) But there was a calculation piece as well as understanding the concept of cross products and how to show they are perpendicular. When she was done with the problem, it looked basically like a paragraph proof. 

Yes, basically any logical explanation is a proof. 

[bensonduck]

1 minute ago, Not_a_Number said:

Is a "formal proof" a 2-column proof? Because there's literally no reason to study those at all. 

Yes, that’s what I meant. I have done Jacob’s 3rd edition and personally worked through all of Jurgensen and they are both heavy on the 2-column style if I remember correctly. 

[Not_a_Number]

8 minutes ago, bensonduck said:

Yes, that’s what I meant. I have done Jacob’s 3rd edition and personally worked through all of Jurgensen and they are both heavy on the 2-column style if I remember correctly. 

I really don't understand the point of those. I don't think removing language makes kids follow things better -- rather the reverse, for almost all kids. 

[EKS]

4 minutes ago, Not_a_Number said:

I really don't understand the point of those.

I suspect it's because they are easier to grade.

[Not_a_Number]

2 minutes ago, EKS said:

I suspect it's because they are easier to grade.

Or maybe because it's easier to learn by rote without understanding. In the same way that when I taught college calculus, we'd only give test problems that were just like homework problems with numbers changed. 

It's a vicious cycle. We don't effectively teach our students, and we then have to pass them, so we give them tests that don't adequately assess if they've learned the material (since they haven't), and then they are not ready for the next class, and so on, so forth. 

Edited July 9, 2021 by Not_a_Number

[EKS]

11 minutes ago, Not_a_Number said:

Or maybe because it's easier to learn by rote without understanding.

I don't see why this should be.  Isn't the difference mostly just formatting?

[cintinative]

15 minutes ago, EKS said:

I don't see why this should be.  Isn't the difference mostly just formatting?

+1  For Jurgensen, I felt like the paragraph proofs were just as detailed as the two column ones. So same question.

 

Incidentally, in my son's Algebra II book there are proofs (Dolciani). 

[Not_a_Number]

1 hour ago, EKS said:

I don't see why this should be.  Isn't the difference mostly just formatting?

Cause there are fewer symbols to remember!

[EKS]

15 minutes ago, Not_a_Number said:

Cause there are fewer symbols to remember!

So the symbols make it "real math"?  I would have thought it was the thinking and logical presentation.

[Not_a_Number]

35 minutes ago, EKS said:

So the symbols make it "real math"?  I would have thought it was the thinking and logical presentation.

No, lol. The point is that it’s easier to memorize without understanding in columns. 

[EKS]

44 minutes ago, Not_a_Number said:

No, lol. The point is that it’s easier to memorize without understanding in columns. 

I wonder how often people are able to memorize their way through proofs like that though--successfully I mean.  I could see doing it for really basic stuff,  but not for something that requires more than a few "moves."

[Not_a_Number]

2 hours ago, EKS said:

I wonder how often people are able to memorize their way through proofs like that though--successfully I mean.  I could see doing it for really basic stuff,  but not for something that requires more than a few "moves."

You memorize the moves 😛 . And you get a tiny bit of understanding so you can connect them 😛 . I mean, that's the way I've seen most people pass calculus tests on "related rates" or "optimization" -- kids would literally memorize the specific steps to the questions we did in class (the ladder falling along a wall is everyone's favorite example!), which is why on tests they could only do those questions with numbers changed... 

[Ausmumof3]

18 minutes ago, Not_a_Number said:

You memorize the moves 😛 . And you get a tiny bit of understanding so you can connect them 😛 . I mean, that's the way I've seen most people pass calculus tests on "related rates" or "optimization" -- kids would literally memorize the specific steps to the questions we did in class (the ladder falling along a wall is everyone's favorite example!), which is why on tests they could only do those questions with numbers changed... 

this is how I got through year 12 advanced maths the year we had the teacher that didn’t teach.  With a bit of help from the review guides.

[domesticidyll]

I have wondered if the introduction of proofs in geometry is the reason UC's ridiculous a-g requirements include more stringent requirements for geometry than for other math classes.

 

Not related to Jacobs, but apparently the way people eat corn correlates to their preference for algebra or geometry:

 

http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eating.html?m=1

[Not_a_Number]

1 hour ago, domesticidyll said:

I have wondered if the introduction of proofs in geometry is the reason UC's ridiculous a-g requirements include more stringent requirements for geometry than for other math classes.

 

Not related to Jacobs, but apparently the way people eat corn correlates to their preference for algebra or geometry:

 

http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eating.html?m=1

I wouldn't say that "algebra" and "analysis" map to high school algebra and geometry! Awesome article though... I just ate corn today, and now I'm forgetting how I eat it. In spirals, I think? 

Edited July 11, 2021 by Not_a_Number

[kiana]

On 7/9/2021 at 4:23 PM, EKS said:

I wonder how often people are able to memorize their way through proofs like that though--successfully I mean.  I could see doing it for really basic stuff,  but not for something that requires more than a few "moves."

I've seen people memorize every step for abstract algebra proofs, and be able to duplicate them, but completely unable to explain a step or recognize a similar problem that required the same technique. They could write it out, but they didn't understand why we might make a specific statement.

I'm not sure what the relative proportion was, compared to the general population; these, unfortunately, were math majors who had made it almost all of the way through by being excellent at memorizing. It didn't help that the person who taught intro to proofs (the prerequisite) that year really wasn't good at it.

[Not_a_Number]

Just now, kiana said:

I've seen people memorize every step for abstract algebra proofs, and be able to duplicate them, but completely unable to explain a step or recognize a similar problem that required the same technique. They could write it out, but they didn't understand why we might make a specific statement.

I'm not sure what the relative proportion was, compared to the general population; these, unfortunately, were math majors who had made it almost all of the way through by being excellent at memorizing. It didn't help that the person who taught intro to proofs (the prerequisite) that year really wasn't good at it.

I've actually always been dubious about Intro to Proofs as a class. I feel like doing proofs in the context of something you care about works better... whereas the Intro to Proofs class, at least where I've taught it, was often discrete math no one cared about. Kids would come out of it thinking there were 4 different types of proofs, one of which was always induction 😛 . They didn't really get that a proof was just a logical argument. 

I remember giving kids an exercise in linear algebra where they were supposed to find wrong steps in other proofs, and they really had no idea how to do this 😕 . They weren't able to think about logical statements in that way. 

[EKS]

1 hour ago, domesticidyll said:

I have wondered if the introduction of proofs in geometry is the reason UC's ridiculous a-g requirements include more stringent requirements for geometry than for other math classes.

I doubt it.  Most geometry classes are light on proofs.

[EKS]

40 minutes ago, Not_a_Number said:

I've actually always been dubious about Intro to Proofs as a class.

Interesting.  I am supposed to take a class called Bridge to Abstract Mathematics next term which I think is essentially Intro to Proofs.  I was kind of looking forward to it (and if they focus on discrete math, that's great as far as I'm concerned).  

46 minutes ago, Not_a_Number said:

They didn't really get that a proof was just a logical argument. 

How could anyone miss this?  Were these math majors? 

[kiana]

12 hours ago, Not_a_Number said:

I've actually always been dubious about Intro to Proofs as a class. I feel like doing proofs in the context of something you care about works better... whereas the Intro to Proofs class, at least where I've taught it, was often discrete math no one cared about. Kids would come out of it thinking there were 4 different types of proofs, one of which was always induction 😛 . They didn't really get that a proof was just a logical argument. 

I remember giving kids an exercise in linear algebra where they were supposed to find wrong steps in other proofs, and they really had no idea how to do this 😕 . They weren't able to think about logical statements in that way. 

Interestingly, discrete math/intro to proofs was when I changed my major TO math. Before that, it was just computation, and it wasn't hard and it was fun to solve problems, but it was kind of a snoozefest.

Yes, I love exercises where they need to find wrong steps in proofs. I also love the exercises Fraleigh uses in his algebra textbook where students are to write a one-sentence or two-sentence summary of the proof.

[kiana]

11 hours ago, EKS said:

Interesting.  I am supposed to take a class called Bridge to Abstract Mathematics next term which I think is essentially Intro to Proofs.  I was kind of looking forward to it (and if they focus on discrete math, that's great as far as I'm concerned). 

It will be your college's intro to proofs. Sometimes it is taught more as a discrete math course, sometimes it is taught to briefly introduce concepts that will be covered in more depth in abstract algebra/analysis. Either way, though, given the posts I've seen you make over the years, I expect you to enjoy it.

[Not_a_Number]

3 hours ago, kiana said:

Interestingly, discrete math/intro to proofs was when I changed my major TO math. Before that, it was just computation, and it wasn't hard and it was fun to solve problems, but it was kind of a snoozefest.

Yes, I love exercises where they need to find wrong steps in proofs. I also love the exercises Fraleigh uses in his algebra textbook where students are to write a one-sentence or two-sentence summary of the proof.

Oh, I love discrete math… I’m a probabilist. So it’s not a knock on it… just that people don’t necessarily wind up caring if they haven’t seen it before.