How important is geometry and/or proofs?

[Janeway]

I think it is likely my child will go in to a STEM or otherwise math based career. Personally, I enjoyed doing proofs when I was in school, long ago. They came easy for me. But my husband did not like it. My child does not mind it, it comes easy for him. But I find myself feeling like all the time spent writing out proofs is taking from time one could spend on manipulating numbers and angles and such. We are doing Jurgenson's Geometry. He is complaining about the math, not because it is too hard, but because it is tedious. He is asking how much he really has to do before he can move on to algebra 2. He enjoyed algebra 1 and never complained. To be honest, I get tired of grading proofs. 

 

Would he be short changed if we skimped on the proofs? Or switched to a less proof heavy program? 

[Tsuga]

I think it's important particularly for the kind of algebra you need for computer science and statistics.

[Sunshine State Sue]

Would he be short changed if we skimped on the proofs? Or switched to a less proof heavy program? 

 

I am probably in the minority, but imo your child will not be short-changed by skimping on proofs or switching to a less proof heavy program.  However, I would not recommend TT or MUS.

 

I have a BA in math and computer studies (what would now be computer science).  Geometry was the only math class I hated.  That had to do with proofs and the fact that I never understood the difference between postulates, corollaries, theorems, etc.  I am logical to a fault and have worked as a programmer for 30 years.

 

Ds was a strong math student.  We tried to use Jacob's Geometry, but neither of us understood it.  So, we switched to Michael Serra's Discovering Geometry which is lite on proofs.  Ds scored in the 92%+ range on the Geometry sections of the ACT.  He is a sophomore at a STEM school currently majoring in mechanical engineering.  Cumulative GPA 3.25.

 

My 2c.

[Sunshine State Sue]

I think it's important particularly for the kind of algebra you need for computer science and statistics.

 

Would you elaborate on this?  Or give an example?  I don't understand.

[regentrude]

I think it is likely my child will go in to a STEM or otherwise math based career. Personally, I enjoyed doing proofs when I was in school, long ago. They came easy for me. But my husband did not like it. My child does not mind it, it comes easy for him. But I find myself feeling like all the time spent writing out proofs is taking from time one could spend on manipulating numbers and angles and such. 

 

Manipulating numbers is not math,  just arithmetic.

 

Proofs are vital. Geometry is the easiest area in which formal proofs are introduced, and practicing proofs hones logical skills and the way one thinks about math.

Ideally, in a good math curriculum, everything should be proved or derived; no relationship or formula or procedure should be introduced without proof or derivation - so proofs are not restricted to geometry, but should be the foundation of all math. Only, many curricula skimp and hand students procedures or formulas without derivation... then "math" becomes rote manipulation.

 

If your student intends to go into a math based career, I consider the skills trained through mathematical proofs necessary. He will need logical thinking and the ability to construct a logical argument without holes or unjustified assumptions.

 

It is, however, entirely possible that the curriculum you have chosen dwells on aspects of proofs that are tedious or formalized and which emphasizes format over the actual gist of a proof. I would suspect that much of what is perceived tedious about proofs is not inherently the fault of geometric proofs, but of the presentation and format.

 

We have used AoPS where everything is proved from algebra 1 on, but which never forces the artificial "two column" format on the student or insists on memorizing the names of theorems. In the end,  proofs are simply a logical explanation of why something is true, based on other things that have been shown to be true.

[aggieamy]

I think Regentrude's answer is probably the best.  

 

That said I'll offer up my anecdotal experience.

 

I took calculus in high school and then again in college.  I went to Texas A&M and when I was there it was ranked #3 nationwide for civil engineering, which is what my degree is in.  I have never done a proof in my life.  

[regentrude]

 

I took calculus in high school and then again in college.  I went to Texas A&M and when I was there it was ranked #3 nationwide for civil engineering, which is what my degree is in.  I have never done a proof in my life.  

 

How do they teach mathematics at this university without proofs???

That is disturbing.

 

Calculus should be choke full of proofs. You have to prove that the derivative of a power law is n*x^(n-1). You have to prove the fundamental theorem of calculus. You have to prove Green's Theorem and Gauss' and Stokes'...

 

 

[wapiti]

In your situation, the only reason I'd switch away from Jurgensen would be to try AoPS Geometry.  Jurgensen is tedious and dry in comparison to AoPS; AoPS is more fun but harder.  Caveat:  my ds isn't very far into Jurgensen yet and from what others have written, it seems likely that proofs will get harder later in the book.  I don't mind Jurgensen - I like that it's clearly written - but I loved AoPS (my dd used it last year).  If AoPS isn't likely to be a good idea, I vote to stick with Jurgensen.

 

FWIW, I didn't enjoy writing proofs in high school geometry.  However, the logic practice was useful to me many years later for legal writing.

[Tsuga]

Would you elaborate on this?  Or give an example?  I don't understand.

 

Maybe I am thinking of a different kind of proof, but we had to do proof after proof in linear algebra which was a requirement for all the applied statistics and data science courses I wanted to take. It was challenging but I was glad to have had all the practice in geometry.

 

I use this kind of work all the time in my programming. I have to construct data sets that meet the assumptions of certain statistical theorems in order to provide data to people.

 

The fact is, I work for the government and actually nobody cares because either they are willing to pay taxes or not and nobody gives a hoot how well I do my job since all we are doing is helping poor people and nothing we can ever say will convince legislatures to tax people more, but whatever. Since I like to pretend that we actually care whether or not programs are helping poor people and social science is actually complex, I use statistics which requires an understanding of linear algebra.

 

I could build databases and everything without understanding linear algebra but I could not construct a dataset.

 

This was a required course for all the computer science, engineering, and science courses I took so although I went to statistics for social sciences / "data science", most people in the course were engineers and I assumed most engineers would also have to have this understanding of vectors and different spaces.

 

Could be wrong though. I'm not an engineer, per se... I do data architecture and statistics for analysis and reporting.

 

Okay, an example... I didn't watch this but it's Khan Academy:

 

https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_independence/v/span-and-linear-independence-example

 

So, how that works in statistics is that many statistical theorems and tools only work if the spaces are linearly independent. 

 

And here is something that I have kind of put out of my mind, thanks to tequila:

 

https://books.google.com/books/about/Bayesian_Inference_in_Dynamic_Econometri.html?id=zUejahgxTScC

 

"Bayesian Inference in Dynamic Econometric Models"

 

So, I'm not saying I hold all this in my head all the time as I work on reports counting people for the flipping National Science Foundation, but I do think it was important to understand it and have proved it at some point, yes.

[aggieamy]

How do they teach mathematics at this university without proofs???

That is disturbing.

 

Calculus should be choke full of proofs. You have to prove that the derivative of a power law is n*x^(n-1). You have to prove the fundamental theorem of calculus. You have to prove Green's Theorem and Gauss' and Stokes'...

 

My roommate was a math major and I know she did proofs because she'd talk about it to me.  I only took Calculus 1, 2, & 3 and then Differential Equations so maybe it was taught at a higher math level than I was at.  

 

My mom got her engineering degree from Johns Hopkins in the 70's and I don't think she did proofs then either but she didn't take that much math because her degree was in Industrial Engineering.  

[regentrude]

My roommate was a math major and I know she did proofs because she'd talk about it to me.  I only took Calculus 1, 2, & 3 and then Differential Equations so maybe it was taught at a higher math level than I was at. 

 

But HOW can that work? Did they just serve you a bunch of formulas to use?

I mean, I can't imagine how one could teach even calculus 1 without proving that the stuff was actually true.

 

I have a very hard time envisioning what a proof free calculus sequence could look like - other than an assortment of formulas to plug and chug.

You did not prove that a certain function is another function's derivative by working out the ratio of differences delta f/delta x and taking the limit delta x to zero??? They just served that up on a silver platter without motivation?

trying to wrap my brain around how that can possibly work...

[Tsuga]

Sorry Sue I edited while you were liking. I hope the main thrust of my post was the same.

 

I am with regentrude. I simply can't fathom all that math without proofs. It's like reading Moby Dick and writing an essay based on having memorized a professor's essay. It doesn't make sense. You have to know why by proving why.

[aggieamy]

But HOW can that work? Did they just serve you a bunch of formulas to use?

I mean, I can't imagine how one could teach even calculus 1 without proving that the stuff was actually true.

 

I have a very hard time envisioning what a proof free calculus sequence could look like - other than an assortment of formulas to plug and chug.

You did not prove that a certain function is another function's derivative by working out the ratio of differences delta f/delta x and taking the limit delta x to zero??? They just served that up on a silver platter without motivation?

trying to wrap my brain around how that can possibly work...

 

Those classes were 15 years ago and I haven't thought about them since much.

 

I do remember professors doing problems on the board and then boxing the "answer" and saying that was the formula.  As a student I never had to write out any proofs though for homework or an exam.  Exams and homework would just be solving problems.  

[Tsuga]

Wow. I took calc at a rural high school in 1995, and then at a regional university (known for its engineering program but of course not at the level of MIT or anything, though they had some great environmental engineering tracks) in 1996, and I took linear algebra at a state flagship in 2012 (no math for 15 years and straight to lin alg FTW! I PASSED! Woot me, lol) and we had proofs in every one.

 

This was all in Washington State. We aren't known for our high graduation rates or fast talking but I am pretty sure--in fact, I know for certain--that our children do proofs in geometry and calculus and that they are absolutely required, even now, for the simplest undergraduate STEM degrees. In fact I know our community college requires them because I help faculty analyze statistics for math courses and one of those courses is calculus and yes, there are proofs in the assessments of what students learn.

 

You will not complete calc at our community college without proofs at least, nor will you graduate our high schools without proofs.

 

I can't speak for the entire state but for the schools where I still know the teachers, I know that's the case.

[Janeway]

I suspect Sue and I are talking about the same thing when I say proofs but Tsunga is talking about something different. I am referring to making two columns to solve a problem and on one side, solve it, and on the other, give the property to explain each step you take. Two column proofs with each step being justified with stating the theorem, corollary, postulate, that you use in each line. That is what I am referring to. I am not seeing a single two column proof in either example that Tsunga gave. 

[regentrude]

I suspect Sue and I are talking about the same thing when I say proofs but Tsunga is talking about something different. I am referring to making two columns to solve a problem and on one side, solve it, and on the other, give the property to explain each step you take. Two column proofs with each step being justified with stating the theorem, corollary, postulate, that you use in each line. That is what I am referring to. I am not seeing a single two column proof in either example that Tsunga gave. 

 

The "two column" thing is simply the format.

No, it is absolutely not necessary to write proofs in this particular format. The most rigorous of high school math curricula does not use this format, math instructors and curricula in many other countries do not use this format, and most mathematicians do not use this format when they write proofs ( which is basically what research mathematicians do for a living.)

 

It is, however, absolutely necessary to do proofs. The specific format is a crutch.

 

Just like the "five paragraph essay" is a crutch. It is absolutely necessary that the student learn to write good essays - but it is unnecessary to write formulaic "five paragraph essays".

[MerryAtHope]

He enjoyed algebra 1 and never complained. 

 

In my experience, students generally prefer either algebra or geometry. I loved algebra and trig but didn't care for geometry. My son was the opposite--loved geometry but didn't care for algebra. 

 

I wouldn't change the math up to suit the student's preferences, but I might compare your math program to other high-quality math programs and see if the number of proofs required is comparable. If yours has an inordinate amount compared to other programs, then it may be worth amending some of the lessons. 

[FaithManor]

I agree with regentrude. When I tutor kids who wonder what the of purpose writing proofs is, I tell them that even if they struggle to earn good grades in geometry, the stretching of their critical thinking skills helps their brains mature. I truly believe this. I have seen kids who just seemed to crumble at the beginning of geometry who through sheer hard work matured immensely just grappling with the concepts even when good grades did not follow.

I remind my kjds that there are numerous ways in which logical thinking skills are taught and that in high school it is really important that we explore multiple avenues with students to make sure everyone has a chance to blossom.

Geometry with proof writing is one. Sentence diagramming is another. Literary analysis. Even sewing and electronics (schematic reading and writing), etc.

One thing I do with my own kids is allow them to keep postulates, theorems, etc. on notecards and let them use their notecards when taking chapter quizzes and final exams. It is otherwise an overwhelming amount of memorization which can cause students to have anxiety from the start. Eventually the "old reliables" that are used so often, so handy, like side angle side.become committed to quick memory recall.

[Tsuga]

I suspect Sue and I are talking about the same thing when I say proofs but Tsunga is talking about something different. I am referring to making two columns to solve a problem and on one side, solve it, and on the other, give the property to explain each step you take. Two column proofs with each step being justified with stating the theorem, corollary, postulate, that you use in each line. That is what I am referring to. I am not seeing a single two column proof in either example that Tsunga gave. 

 

That's just the format. I personally wrote my proofs in two columns and referenced the rules because otherwise I would get disorganized, but you can do it without that format.

 

The columns aren't the point. The point is being able to explain the logical relationships between the figures based on the rules.

[AppleGreen]

.

 

One thing I do with my own kids is allow them to keep postulates, theorems, etc. on notecards and let them use their notecards when taking chapter quizzes and final exams. It is otherwise an overwhelming amount of memorization which can cause students to have anxiety from the start. Eventually the "old reliables" that are used so often, so handy, like side angle side.become committed to quick memory recall.

I wish this were the case with our on-line teacher! The volume is so much for my kid. I can see some of those standbys are getting in there, but quizzes are getting tanked with the requirement of memorizing all of those and it is very discouraging for him. He is already quite worried about the mid-term because of this. 

[Angie in VA]

I agree with regentrude. When I tutor kids who wonder what the of writing proofs is, I tell them that even if they struggle to earn good grades in geometry, the stretching of their critical thinking skills helps their brains mature. I truly believe this. I have seen kids who just seemed to crumble at the beginning of geometry who through sheer hard work matured immensely just grappling with the concepts even when good grades did not follow.

 

I remind my kjds that there are numerous ways in which logical thinking skills are taught and that in high school it is really important that we explore multiple avenues with students to make sure everyone has a chance to blossom.

 

Geometry with proof writing is one. Sentence diagramming is another. Literary analysis. Even sewing and electronics (schematic reading and writing), etc.

 

One thing I do with my own kids is allow them to keep postulates, theorems, etc. on notecards and let them use their notecards when taking chapter quizzes and final exams. It is otherwise an overwhelming amount of memorization which can cause students to have anxiety from the start. Eventually the "old reliables" that are used so often, so handy, like side angle side.become committed to quick memory recall.

 

 

Faith, please don't take this wrong, but I just love you!

 

No one ever mistook me for an intellect, but I have preached the importance of proofs for ages. Even wrote my high school geometry teacher a few years back lamenting the lack of proofs in most books. He said he'd spoken to the state dept of ed about it. My kids did proofs!

 

ITA w/ all that you said, but especially w/ the part in bold and have said that (in less articulate words) many times to my dc.

 

I am going to fee smart for 5 whole minutes. :)

[MarkT]

http://mathwithbaddrawings.com/2013/10/16/two-column-proofs-that-two-column-proofs-are-terrible/

 

from comments

"

There are other forms of proof out there. I’ve always liked the tree-style proofs that come up in certain branches (yay, tree puns!) of formal logic, where you start with the thing that you want to prove, and work backwards, spreading out to all of the things that you need to support it, until you get back to some claims that you’re happy to take as axiomatic (or don’t, in which case what you’re proving is probably wrong, or you need to take another approach to getting to it).

"

 

"

That geometry is the class to which proofs are attached seems to be a vestige of early twentieth century and late nineteenth century education where it had such a prominent place in the liberal arts. There really is no reason to stubbornly stick to the subject. Personally I would much rather see discrete mathematics (start with the basics: sets, counting, permutations, combinations, leading to some basic graph theory and probability) used instead. For one, discrete math is a bit more relevant in our digital world compared to Euclidean geometry, and secondly, in discrete mathematics one can reach more quickly to some of those good proofs with obvious steps leading to surprising conclusions (something like Hall’s marriage theorem or Arrow’s impossibility theorem).

"

 

http://art-of-logic.blogspot.tw/2013/08/constructivism.html

 

 

[Chrysalis Academy]

http://mathwithbaddrawings.com/2013/10/16/two-column-proofs-that-two-column-proofs-are-terrible/

 

 

 

 

That was awesome!!!  I was definitely the 9th grade geometry student he mentions.

 

Wait, did that just prove the utility of proofs? He proved that the way we often teach proofs is ridiculous! Using proofs!   :lol:

 

Ok, sorry, the grownups should get back to talking now.  :bigear:

[wapiti]

Janeway, if you'd like to get away from the two-column format, AoPS teaches paragraph-style proofs.  I was afraid the style would be harder, but IMO the paragraph style is actually easier because it makes more sense (though the geometry is harder, it is definitely less tedious).  

 

In light of your other thread about how bored your ds is generally, you might supplement with the geometry topics in Alcumus (just problems, not proofs) even if you don't use the AoPS text.

[daijobu]

 

 

We have used AoPS where everything is proved from algebra 1 on, but which never forces the artificial "two column" format on the student or insists on memorizing the names of theorems. In the end,  proofs are simply a logical explanation of why something is true, based on other things that have been shown to be true.

 

I'm going to take issue with the bold.  I have the aops geometry book in front of me, and in 5 minutes I have found several unproven assertions, in particular the volume of a pyramid, volume of a sphere, and surface area of a sphere. 

 

However, I think being able to quickly derive most formulas is extremely helpful if only because I so often forget them, and my only recourse (other than the internet) is to re-derive them on the spot.  For example, I ALWAYS forget the formula for the sum of a geometric series (finite...infinite is easy).  So I just rederive it EVERY. SINGLE. TIME.  

 

And it still won't stick.  

[regentrude]

I'm going to take issue with the bold.  I have the aops geometry book in front of me, and in 5 minutes I have found several unproven assertions, in particular the volume of a pyramid, volume of a sphere, and surface area of a sphere.

 

Granted. These are, however, exceptions, because these particular proofs would be beyond the level of the course. The authors specifically point out the omission.

Frequently, a proof of a relationship similar to one that is presented in the chapter is relegated to the exercises.

So, I should have said "almost everything". My main point, however, is that the overall philosophy of all the texts is to prove or derive relationships instead of having them fall from outer space - proofs are not treated as a separate category, but as an intrinsic part of math. This is not limited to geometry, but present in the other texts as well. (We just proved and derived oodles of trig identities....)

 

 

Btw, I would not want to prove the volume of the sphere with merely geometrical means. I much rather wait until I have calculus :-)

 

[Caroline]

How do they teach mathematics at this university without proofs???

That is disturbing.

 

Calculus should be choke full of proofs. You have to prove that the derivative of a power law is n*x^(n-1). You have to prove the fundamental theorem of calculus. You have to prove Green's Theorem and Gauss' and Stokes'...

 

My students derived the distance formula (from Pythagorean, which we derived earlier) and the equation of a circle today in class. I thought of you. My students do a ton of proving and deriving in class. I can't imagine teaching math without having the students do that. 

[Lori D.]

I think it is likely my child will go in to a STEM or otherwise math based career. Personally, I enjoyed doing proofs when I was in school, long ago. They came easy for me. But my husband did not like it. My child does not mind it, it comes easy for him. But I find myself feeling like all the time spent writing out proofs is taking from time one could spend on manipulating numbers and angles and such..

 

...Would he be short changed if we skimped on the proofs...

 

Perspective from a Humanities-based family here with no one having taken Calculus -- absolutely think that proofs are very helpful and important in NON math areas (as well as the specific STEM and higher math applications mentioned by previous posters). I think *how* proof Geometry is taught makes all the difference, though.

 

My high school Geometry class was taught more like there was only ONE series of correct steps for a proof -- so it ended up being more about hopeful guessing that you got it right, rather than teaching thinking skills, and making a series of connected, supported statements for an argument  -- which are skills that are so very useful in many areas -- analysis, persuasive writing and argumentative papers, and real-life problem-solving. It wasn't until I did Jacobs Geometry alongside DS#1 that the lightbulb went on for me of what the point of Geometry was and how useful it was for thinking, beyond just the specific math applications. :)

 

 

... We are doing Jurgenson's Geometry. He is complaining about the math, not because it is too hard, but because it is tedious. He is asking how much he really has to do before he can move on to algebra 2...

 

As long as DS is continuing to understand and get it, you might try assigning just the odd or even numbered proof problems from each lesson and moving on. No personal experience with Jurgenson's, but textbooks often have far more problems in them so teachers can selectively assign as many or as few as are needed for student practice, so if you're doing every.single.problem, that might be way overkill. ;)

 

Would he be short changed if we skimped on the proofs? 

 

As long as he's still getting the concepts and his grade average isn't dropping by doing  fewer proofs, if that is a help in keeping alive an interest and enjoyment of math, go for it!

 

 

 Or switched to a less proof heavy program? 

 

JMO: If you're on a typical school year schedule, I would guess you're about a third of the way through the program by now; that's pretty late in the game to switch to a new program, which would require several weeks of getting used to a new presentation method, AND trying to figure out how and when you need to speed up/slow down to not repeat what already has been learned while not accidentally skipping things not already covered. At this stage, unless the student was really struggling with understanding the presentation of the material, I would not switch -- it would more likely take even longer to get through Geometry and on to Alg. 2.

 

BEST of luck in your Math journey, whatever you decide! :) Warmest regards, Lori D.

[Mike in SA]

You know, from my years in managing information technology projects, I can tell you that it is obvious how few employees have a proper mathematical upbringing, complete with both geometry and proof.

 

Geometry isn't as tangibly relevant as it was in ancient Egypt, sure. From 100 years ago to now, though, not much has changed. It teaches one to think differently, to approach problems from second, third, and fourth points of view. Nothing else taught today comes close, with physics being the closest, and completely dependent upon geometry.

 

Proofs teach logical structure and self-rigorous testing. They don't have to be bound to geometry, but that is where one learns to test those secondary and tertiary perspectives.

 

Today's business world suffers tremendously from poorly structured solutions...

[sandalwood]

Just for what it's worth, our geometry was proof heavy.  It was very trying some days (a lot of days) but in the end, I thought it was worth it and I think it really helped in the long run.

 

I will also say that math is one of the subjects, if I can't do it I get a teacher.   This year we are doing an online "live class" precal. course with other students (that there is no way on earth I could teach). 

[Mike in SA]

. I don't remember my son's geometry text as being overly proof-heavy, but he still mentions how it helped him with college courses. He's had to take a Proofs class, and I'm pretty sure most of his other required classes have relied somewhat on proofs: Real Analysis, linear, computational complexity, and abstract algebra. I do know for certainty that calc classes are full of proofs.

 

Hopefully, real analysis and abstract algebra had more than just a little!  :)  It's kind of the point of those courses.

 

Yes, calculus is full of proofs (it is a branch of analysis), but unfortunately, there are plenty of teachers who don't ask the students to perform the proofs themselves.  Business calc, I could get, but it's pretty common among regular calculus, too.  Green's and Stokes' are beautiful, but not all STEM majors have to go that far into calculus...