Grading *Geometric Proofs* for AoPS, Jacobs, Lials, BJU, etc...

[dereksurfs]

I am very curious how parents grade geometric proofs fairly, consistently and accurately for any of these various programs. I am not at all interested in programs which do not cover proofs or do so very lightly as I think the answer is obvious. Rather I am wondering about the curriculum which does address proofs in any kind of substantial way, whether that be through using the two column approach or a more narrative style. Since proofs can be answered in a variety of ways I'm not quite sure I want to do that myself even though I did well in Geometry way back in the day.

 

I am also not convinced at all that computer based programs could grade proofs correctly such as TT or Kinetic Books, etc... because of all the possible ways a student could answer. But that really is off topic as I want to know how parents perform the grading themselves. And would programs such as AskDrCallahan help with the grading at all or at least provide direction?

 

As an alternative I'm strongly considering outsourcing a la Derek Owens, Jann in Texas, etc... leaving the grading up to them in this case.

 

What say you hive members who have traversed these waters before?

 

Thanks,

[8filltheheart]

I actually do the proofs alongside my student. Then when I grade them, I can easily follow the logic if it is different from my own b/c I know what they needed to prove.

 

FWIW, this really isn't that difficult to do. I am typically way faster than my student. ;) (and I am not that strong in math!) And my kids love it when their proof is far more elegant than my own!! (btw, I have never done this w/AoPS and never plan to!! :p )

[Kendall]

I actually do the proofs alongside my student. Then when I grade them, I can easily follow the logic if it is different from my own b/c I know what they needed to prove.

 

 

I was going to say this, but hesitated because I have a degree in math, and more importantly, taught Geometry for 4 years. So what she says holds more weigh,t and I think she is absolutely correct.

 

Each Geometry book has a slightly different set of building blocks, and even I have to keep up with what each book has taught in order to know how to do the proofs with what has been given to the student.

 

Kendall

[regentrude]

I am simply not grading proofs.

I can check through my student's proof and check the logic, find mistakes in reasoning - but I do not grade proofs on a percent scale. It is either a correct proof or it is not. (It may be a clumsy proof or particularly elegant one). But I would be hard pressed to establish criteria when a proof should warrant a grade of B or C. Either it is proven, or not. If it is not proven, student goes back and does it again.

 

So, we teach proofs, practice proofs, I have my students narrate proofs for me, make sure my students master proofs. But this does not require grading.

 

Btw, I do not believe that an automated system is possible that is able to evaluate proofs, because there are so many different ways - even with the same logical flow, there are different ways to write down the same proof. No computer program is that smart yet.

[Jen the RD]

I am very curious how parents grade geometric proofs fairly, consistently and accurately for any of these various programs. I am not at all interested in programs which do not cover proofs or do so very lightly as I think the answer is obvious. Rather I am wondering about the curriculum which does address proofs in any kind of substantial way, whether that be through using the two column approach or a more narrative style. Since proofs can be answered in a variety of ways I'm not quite sure I want to do that myself even though I did well in Geometry way back in the day.

 

I am also not conviced at all that web based programs could grade proofs correctly such as TT or Kinetic Books, etc... because of all the possible ways a student could answer. But that really is off topic as I want to know how parents perform the grading themselves. And would programs such as AskDrCallahan help with the grading at all or at least provide direction?

 

As an alternative I'm strongly considering outsourcing a la Derek Owens, Jann in Texas, etc... leaving the grading up to them in this case.

 

What say you hive members who have traversed these waters before?

 

Thanks,

Grading proofs was exactly the reason that I hired out geometry. I was never as good at geometry as I was algebra, trig, & calculus, therefore I did not feel comfortable with the grading. I know proofs don't always have one right way and I wasn't sure I could follow the logic of the proof and grade it accurately & objectively if it was different than an answer key. My dd has used DO Geometry and I have nothing but high praise for him. (If you'll search on my name, you'll see my recent comments about Derek Owens.) Honestly, I don't think you could go wrong with Derek or Jann. She is also wonderful and you'll find many, many positive reviews for her classes. It all comes down to whether or not you want someone else to do the teaching/grading. It has been a great experience for my dd to do an online format, scan homework/tests, ask a teacher questions via email, etc. I think sweethomealabama has used Jann for algebra & geometry if you'd like to pm her. HTH, Jennifer

[Julie of KY]

I don't grade. My son must master the proof in a logical stepwise fashion. He usually does this orally. Oftentimes, he sounds just like the AoPS solution guide; other times he approaches a problem from a completely different perspective but is just as correct.

[dereksurfs]

I am simply not grading proofs.

I can check through my student's proof and check the logic, find mistakes in reasoning - but I do not grade proofs on a percent scale. It is either a correct proof or it is not. (It may be a clumsy proof or particularly elegant one). But I would be hard pressed to establish criteria when a proof should warrant a grade of B or C. Either it is proven, or not. If it is not proven, student goes back and does it again.

 

So, we teach proofs, practice proofs, I have my students narrate proofs for me, make sure my students master proofs. But this does not require grading.

 

Btw, I do not believe that an automated system is possible that is able to evaluate proofs, because there are so many different ways - even with the same logical flow, there are different ways to write down the same proof. No computer program is that smart yet.

 

Thanks, Regentrude. This is helpful in understanding how you grade or correct proofs. I think the term 'grade' can actually mean a number of things. For example does one give partial credit for a proof or as you do simply mark it as correct or incorrect? Maybe 'evaluate' proofs would be another way to to put it. Regardless it must be corrected in some way. And that is really what I am referring to when say grade them.

 

I know there have been other discussions on this forum where some parents have asked you about your 'grading' system for proofs which might include more credit for more steps or more credit for a more elegant solution. But as you describe both here and in those threads using correct logic is more important than elegance, number of steps, etc... I also think it makes sense to give full credit whether it is concise or wordly for example. I'm a little torn about giving no credit if everything else is there except for one simple mistake for example. But that's digressing a bit from the main point. Giving full credit or no credit would definately make my life easier in this case, as long as I could follow the student's logic.

 

Just as an aside this is actually very similar to the work I do daily as a software engineer. There are many logical paths an engineer can take using the same language to achieve similar functionality in their code. Elegant code is always preferred. But messy software can potentially perform the same function.

 

Kendall also brought out a very good point. Even as a Geometry teacher she has to keep up with what the book teaches to better assess the student's work. I am thinking that is part of the key or a common thread being stated here. 8FillTheHeart seems to take a similar approach working the proofs along with her kids. I guess that's part of the extra work required on the part of the parent to better serve in this role as Geometry instructor. I know AoPS and other programs teach directly to the student. But still I don't think one can get away from reviewing over the materials and possibly working some of the proofs as a minimum. Does that sound about right or par for the course?

 

Thanks all for sharing your experiences!

[regentrude]

Thanks, Regentrude. This is helpful in understanding how you grade or correct proofs. I think the term 'grade' can actually mean a number of things. For example does one give partial credit for a proof or as you do simply mark it as correct or incorrect? Maybe 'evaluate' proofs would be another way to to put it. Regardless it must be corrected in some way. And that is really what I am referring to when say grade them.

 

I know there have been other discussions on this forum where some parents have asked you about your 'grading' system for proofs which migh include more credit for more steps or more credit for an more elegant solution. But as you describe both here and in those threads using correct logic is more important than elegance, number of steps, etc... I also think it makes sense to give full credit whether it is concise or wordly for example. I'm a little torn about giving no credit if everything else is there except for one simple mistake for example. But that's digressing a bit from the main point. Giving full credit or no credit would definately make my life easier in this case, as long as I could follow the student's logic.

 

Why do we need to think in terms of "giving credit" at all? What is the "credit" good for?

The goal of geometric proofs is to teach students to think logically through a mathematical problem, to introduce them to formal writing of math, to make them think mathematically. In order to do so, it is not necessary to "give full or partial credit" - it is necessary to check whether they have mastered the proof, or whether their logic has holes, their writing is not coherent, their argument does not flow, and help them to correct these.

 

The only reason I give tests at all is to have a tangible measure for my assignment of a grade (*I* can certainly ascertain my child's mastery through daily observation), and the only reason I assign a grade is for the transcript. So, if we need a geometry grade, I find it perfectly fine to teach proofs to mastery, but put on the grade determining exam all those other things that have solutions one can more easily evaluate.

(Along a similar line: I do not give grades for science labs either. Lab and lab report have to be completed to my satisfaction. i.e. to mastery, but the course grade is determined by exams only)

 

I know AoPS and other programs teach directly to the student. But still I don't think one can get away from reviewing over the materials and possibly working some of the proofs as a minimum. Does that sound about right or par for the course?

 

Oh, with AoPS, it is absolutely necessary that the parent can work the problems (not necessarily that the parent also actually does work them!) if the parent wants to attempt to critique the solutions! Merely consulting the solution manual will not suffice to determine whether the student's alternative proof is correct or not - the evaluator must be able to work through the student's proof and check it.

I have spent hours on certain star problems in the geometry book, LOL - even though,as a theoretical physicist, I am rather good at math.

But I have never actually worked every single problem; it is sufficient that I am capable of verifying whether a proof is correct or not.

In none of the books this is as important as in the geometry book, because it will be very likely that the student's solution is not identical to the proof in the solution manual, and that consulting the solution book will not give the student the necessary feedback to determine whether HIS proof is correct, even though different.

[Kendall]

Ah, the importance of definition! (I'm trying to work through Kreeft's Socratic Logic:))

 

When I responded to the thread, I took grade to mean "see if the student has proven it in a logical way", the idea of giving a grade, never entered my mind. Regentrude took it to mean giving a grade.

 

As a teacher I only gave a grade for proofs as part of the grade for the test. They got points for the parts that were logical and points off for the parts that weren't. I don't give Geometry tests here at home.

 

I like what regentrude said about teaching to mastery and grading on easier to evaluate things. This year my son has done very well on his Biology tests. It is easy to see that all of his tests are in the A range. I have quit wasting my time determining if the grade should have been a 97 vs. a 94 based on how complete some paragraph answers were. If he fluctuated more I would try to grade more precisely.

[dereksurfs]

Yes, defining terms is obviously important! ;) My original question was probably a little ambiguous because I am interested in all of these things. But primarily I'm asking about evaluating the proofs as you've described. Whether I give them 80 or 90 percent on a test is another matter all together and most likely not as important.

 

The main question I have as well as many other parents reading this thread I think is can I fairly and effectively evaluate the proofs for correctness without being a math major, instructor, etc...?

 

Even though both you and regentrude are instructors your input is no less valid. In fact since you do teach these kinds of things more regularly you are also more keenly aware of what is required to perform such evaluations of the student's work. I also find it interesting to hear your different takes when you put on the parent/instructor hat vs. the school hat. I would imagine assigning grades and giving points on tests is much more mandatory in the formal classroom setting depending somewhat on the institution of course. However at home, so far anyway, our kids still take math tests and are used to seeing 'their scores' on the test. Maybe with Geometry this will look a bit different. Actually, I take that back. Since we just started with AoPS intro to Algebra with ds11 our approach toward grading and test taking may be different sooner rather than later. :D

[kiana]

Ah, evaluating for correctness really does require you to understand the material quite well. I do not think that you can fairly and effectively evaluate the proofs for correctness without that. If you do not have time to work the problems I would seriously consider outsourcing geometry.

 

When I put proofs on exams I usually use a 5-point scale.

 

5 - correct proof, even if clumsy. If it was clumsy and there was a much more elegant way to do it I will point that out but not deduct points.

4 - almost correct, very minor error -- usually in phrasing and not in logic, but an incorrect statement did result at some point.

3 - correct in the main but some errors in logic.

2 - good approach taken but went off the rails.

1 - bears a resemblance to a correct answer.

0 - huh???

 

But obviously this is just how I do it.

[regentrude]

The main question I have as well as many other parents reading this thread I think is can I fairly and effectively evaluate the proofs for correctness without being a math major, instructor, etc...?

 

 

 

Yes, you can - IF you have a through mastery of the subject.

You do not have to be a math major to have a thorough mastery of high school geometry. In fact, I have never done any geometry since middle school (when it is taught in Germany, including proofs); it never recurred in my math courses at the university.

But you do need to be a few steps ahead of the student to have enough perspective; a person who is scrambling to follow a proof in the solution manual will probably not be able to discern the validity of a differing approach.

 

Even though both you and regentrude are instructors your input is no less valid. In fact since you do teach these kinds of things more regularly you are also more keenly aware of what is required to perform such evaluations of the student's work. I also find it interesting to hear your different takes when you put on the parent/instructor hat vs. the school hat. I would imagine assigning grades and giving points on tests is much more mandatory in the formal classroom setting depending somewhat on the institution of course.

 

 

This is a very good point. Actually, I would prefer to give fewer tests and grades in my college courses, too. In my class of 80 students, I can make a pretty good guess as to mastery and final grade about three weeks into the semester, just from seeing their homework and class interaction. There are occasional surprises, but usually not. The tests are more of a stick to get them to do their work than a mere evaluation tool. If I were allowed to do what we had back home, an oral comprehensive final would be by far the most accurate assessment (albeit time consuming). Our final math grade in college came solely from a comprehensive oral exam over the material of five semesters of math classes.

 

For a homeschooled student, an oral exam would be brilliant. I'd do that in a heartbeat, if I did not have the feeling I need some tangible proof of evaluation in case somebody questioned my grades. I guess a video of the exam would not quite work as well...

[dereksurfs]

Ah, evaluating for correctness really does require you to understand the material quite well. I do not think that you can fairly and effectively evaluate the proofs for correctness without that. If you do not have time to work the problems I would seriously consider outsourcing geometry.

 

When I put proofs on exams I usually use a 5-point scale.

 

5 - correct proof, even if clumsy. If it was clumsy and there was a much more elegant way to do it I will point that out but not deduct points.

4 - almost correct, very minor error -- usually in phrasing and not in logic, but an incorrect statement did result at some point.

3 - correct in the main but some errors in logic.

2 - good approach taken but went off the rails.

1 - bears a resemblance to a correct answer.

0 - huh???

 

But obviously this is just how I do it.

 

 

Wow,

 

That's really a good breakdown Kiana for those who want to assign points for each proof. Thank you for sharing this. Just out of curiosity do you teach classes or your kids or both?

 

Thanks again,

[8filltheheart]

yep, when I read the question, I read it as evaluate, not assign a letter grade.

 

I only grade tests in math and since we progress at mastery, their tests are pretty much always close to 100%. (the advantage of sitting beside them and working the problems at the same time. I know exactly what they know and I understand exactly what they don't understand.)

 

[kiana]

Wow,

 

That's really a good breakdown Kiana for those who want to assign points for each proof. Thank you for sharing this. Just out of curiosity do you teach classes or your kids or both?

 

Thanks again,

 

 

Math classes at university.

 

(actually, self-brag alert -- I just got a fulltime job doing that) :D :D

[Dana]

 

 

Math classes at university.

 

(actually, self-brag alert -- I just got a fulltime job doing that) :D :D

 

Congrats!!

[dereksurfs]

 

Math classes at university.

 

(actually, self-brag alert -- I just got a fulltime job doing that) :D :D

 

Contrats! That is quite an accomplishment!!! :thumbup:

[lewelma]

I do not grade proofs, but then again I have a very self-motivated boy who loves proofs. So there is really no point. I do keep up with my son by working through the AoPS textbooks concurrently (but not with him) which requires about 3 hours per week of my personal effort (until this term, see siggy). When we first started geometry proofs, I went over every single one of his for a few chapters to make sure he understood what it was all about. I discussed why things needed to be included and how to set up the proof. But now, I just ask him to show me 2 proofs per chapter that I look over to make sure he is making progress, and when I find a particularly hard proof, I ask to see what he has done to make sure that he is putting in the effort that he needs to. I also let him evaluate my proofs, because you learn a lot by teaching!

 

Because my son will be taking a 1-month long proof based exam in July, I asked a university math postdoc to review a few of my son's proofs. We gave him an example proof from geometry, combinatorics, number theory, and algebra - they were from the previous year's exam. I knew that many of my ds's solutions were not elegant, in fact a few really went the long way around. Two of them actually made little sense to me, but I did not understand the questions very well, so could not evaluate the level of his proof. After reading them, the postdoc said that he would have received either a 6 or 7 out of 7 for each proof. It was really nice to have that confirmation. He said that issues of elegance and design that I saw were maturity issues not logic issues and that they would develop as he wrote and read more proofs. So I guess it is like teaching a child to write. You may give them a really easy topic which they can write on but their style will still be immature. It just takes time.

 

At this point, we are fully involved in learning to write proofs in preparation for the exam. And we are using a bunch of materials (not listed in my siggy) to learn together what constitutes a good proof in many areas of math. We are spending 5 hours per week on proof writing only. All I can say is WOW. I had no idea what real math is all about. I now think that proofs are incredibly important to learn, and I would encourage you to continue with your efforts!

 

Ruth in NZ

[quark]

Derek, I have nothing smart to say lol...I outsource it to a tutor who works more like a mentor/ math buddy with my son than a tutor.

 

But I'm writing to say I found another thread from a few years ago that might be helpful to you and others following this (ETA: I used to do something similar to post#11 in that link before handing the instruction over to his tutor).

[dereksurfs]

...

At this point, we are fully involved in learning to write proofs in preparation for the exam. And we are using a bunch of materials (not listed in my siggy) to learn together what constitutes a good proof in many areas of math. We are spending 5 hours per week on proof writing only. All I can say is WOW. I had no idea what real math is all about. I now think that proofs are incredibly important to learn, and I would encourage you to continue with your efforts!

 

Ruth in NZ

 

Ruth, thanks for the encouragement. And I must say your signature looks insane! :tongue_smilie: Why would you go through all those AoPS texts at once? Is he working toward an early doctorate or something?

[regentrude]

 

Math classes at university.

 

(actually, self-brag alert -- I just got a fulltime job doing that) :D :D

 

Congratulations! That's fabulous.

[dereksurfs]

Derek, I have nothing smart to say lol...I outsource it to a tutor who works more like a mentor/ math buddy with my son than a tutor.

 

But I'm writing to say I found another thread from a few years ago that might be helpful to you and others following this (ETA: I used to do something similar to post#11 in that link before handing the instruction over to his tutor).

 

Quark, thanks for the link. Geometry is obviously one which many parents are perfectly happy to outsource. I'm still undecided. But I have been considering taking a Derek Owens course at some point as I like the idea of an outside course experience for a variety of reasons. Though I must admit AoPS Geometry sounds intriguing in a somewhat masochistic, difficult sort of difficult way. :D

[quark]

For a child who loves math I don't think it odd to do several books/ programs at once. We are currently juggling three and a few small bunny trails here and there as well. There is a HUGE disadvantage with using Derek Owens when you end up liking his method...it's hard to say goodbye when the course comes to an end! At least for this parent. I will miss him. If we didn't already like our math tutor so much (and were not limited by cost) we would jump at the chance to do a math course with DO too. :001_smile:

[lewelma]

And I must say your signature looks insane! :tongue_smilie: Why would you go through all those AoPS texts at once? Is he working toward an early doctorate or something?

 

 

He has a real chance to go to the NZ Math Olympiad camp this year, probably a better chance than next year even though next year he would be further along in math. They like younger kids because they have more years to train them for the international Math Olympiad. So I have given him 4 months to study math only, and then he has to go back to a balanced schedule after the exam. He won't be able to score well unless he knows all 4 areas in the olympiad - number theory, counting, algebra, and geometry - hence the long list of books. There are not enough people in NZ to bother with the AMC and AIME. Basically, the exam he is taking is equivalent to the USAMO and is all proof based.

 

Ruth in NZ

[TravelingChris]

Well let me tell you how I did it. I took geometry in a public school where I had a good teacher but then I got very sick and had to be homebound for most of the latter part of the year, where I basically had a homebound teacher who came by once or twice a week and didn't really teach geometry. I was an algebra type person, not a geometry type one so although I did much more math in high school and college and grad school, I was hesistant about teaching geometry. It wasn't hard for me. It helped that my first student, my son, loved geometry and hardly made any mistakes. BUt I could follow the logic and check whether the proof was right or not. That gave me confidence and I haven't had any problems teaching the next two- one for whom geometry was difficult and one for whom it was again easy.

 

Just like regentrude, I never graded the proofs with a grade- they were right or they were wrong. The wrong ones were to be corrected. On tests, I would give back half the possible points for corrected work. All of them got A's, though if I had been doing + and - , one would have had an A- but I didn't so they all had A's.

[mathwonk]

when i grade proofs i simply ask myself whether every statement has been justified and whether the statements present suffice to convincingly establish the result. But Professor Sybilla Beckmann of UGA has written a "holistic grading scale" for scoring proofs that I will try to post here. Of course this browser does not allow me to post much of anything so it is tough to do.

 

I thought so. As with every other attempt, I get : "this upload failed". What gives?

[mathwonk]

trying again:

 

 

A Holistic Scale

for scoring explanations in MATH 5001, 5002, 5003

 

 

by Sybilla Beckmann

 

 

A major goal of the MATH 5001, 5002, 5003 courses is for you to learn

to write clear and convincing explanations about mathematics. Since an

explanation is best viewed as a whole, the following holistic scale for scoring

explanations has therefore been devised. This scale is based on other holistic

scales for assessing problem solving in mathematics as well as on holistic

scales for assessing writing (where holistic scales are much more common).

Please keep this scale in mind when you write your explanations. Think of

your explanations as short essays that will be graded in a way that an English

or History professor might grade: what counts is not only factual correctness

but also explanatory power and clarity of expression.

 

Exemplary, 5 points: An exemplary explanation could be used as a model

for other students and is characterized by the following:

 

1. The explanation is factually correct, or nearly so, with only minor

flaws (for example, a minor mistake in a calculation).

2. The explanation addresses the specific question or problem that

was posed. It is focused, detailed, and precise. There are no

irrelevant or distracting points.

3. The explanation is clear and convincing. A clear and convincing

explanation is characterized by the following:

(a) The explanation could be used to teach another college student,

possibly even one who is not in the class.

( B) The explanation could be used to convince a skeptic.

© Key points are emphasized.

(d) If applicable, supporting pictures, diagrams, and/or equations

are used appropriately and as needed.

(e) The explanation is coherent.

(f) Clear, complete sentences are used.

Competent, 4 points: A competent explanation shows some weakness in

 

meeting the criteria for an exemplary explanation but must also display

 

notable strengths.

 

1

 

 

Basic, 3 points: A basic explanation shows more serious weakness in meeting

the criteria for an exemplary explanation than does a competent

explanation. To be considered basic, an explanation must be largely

correct.

 

Emerging, 2 points: An emerging explanation shows a genuine, thoughtful

attempt at the problem but shows serious deficiencies in meeting

the criteria for an exemplary explanation. In order for the work to

be considered emerging, some aspects of the explanation must be correct

and must show at least one possible “starting point†for a good

explanation.

 

Unsatisfactory, 0 points: Work is considered unsatisfactory if it is characterized

by any one of the following three criteria:

 

•

it is sloppy or haphazard, or

•

it does not show a genuine, thoughtful attempt at the problem, or

•

does not display any emerging ideas that could become part of a

good explanation.

 

 

 

This was followed by some examples which however did not reproduce correctly here for some reason.

[dereksurfs]

Thanks for taking the time to post this MathWonk. It's interesting to consider this holistic grading scale when evaluating proofs.

[lewelma]

Derek, one of the things I have learned this past month about formal proofs is that they are the final stage of a larger thought process. First you investigate -- you find patterns, trial different tactics, and determine the best proof type to use (induction, contradiction etc). Then you often write an informal proof - this is written in the language of the question rather than in mathematical terms. Finally you write a formal proof. In the beginning, my ds thought that his investigation was the proof. Then he thought that his informal proof was the proof. And only now does he understand that he is after a formal proof.

 

We have also found that the investigation phase is very important and repetition is the key to success. So to allow him to get through lots of different types of problems, I allow him to just write messy, informal proofs that get to the point eventually but might have extraneous information. But then every 5 proofs or so, I ask him to edit and copy it over. This has been a good balance.

 

Ruth in NZ

[mathwonk]

Let me give more credit where due. Here is Professor Beckmann's web site. She has done it all: pure mathematics research, outstanding teaching, teacher education, author of books on math teaching, national meetings on the topic, mother of gifted children,....

 

http://www.math.uga.edu/~sybilla/