annegables 49 Report post Posted Wednesday at 07:47 PM My kid is in AoPS geometry this year and he just finished ch 13. I love AoPS and I love that their geometry is heavy on proofs, but good heavens, we are both sick of proofs! I think I am just going to have him say the proofs to me for half the remaining proof problems. All I want is some answers with honest-to-goodness numbers in them.🤣 2 Share this post Link to post Share on other sites

square_25 699 Report post Posted Thursday at 12:17 AM Technically, all of math should be proof-based ;-). 2 1 Share this post Link to post Share on other sites

daijobu 3,948 Report post Posted Thursday at 02:51 AM In AoPS Intermediate Algebra we were getting quite sick and tired of polynomials. I distinctly remember one day ME: "Ah! Finally we're done with our 3 chapters of polynomials!" <turns page> "Chapter 9: Factoring Multivariable Polynomials" Oy! 3 Share this post Link to post Share on other sites

drjuliadc 144 Report post Posted Thursday at 12:56 PM I remember loving proofs when I was in geometry in school (in 1981 in public school, so no comparison to AOPS). I have no memory of what a proof actually is, just that I really liked them. I was disappointed to hear from the developer of Videotext algebra that they don’t teach proofs in school anymore. Is that true? I’m not a skeptical person, but I like to hear something from more than one reliable source. My kids are young and still in the arithmetic stage. My profession doesn’t require anything other than arithmetic so I have to rely on you guys for answers to these questions. Share this post Link to post Share on other sites

annegables 49 Report post Posted Thursday at 02:18 PM 13 hours ago, square_25 said: Technically, all of math should be proof-based ;-). And this is why I love AoPS. And also why I am an engineer by training and not a mathematician😁. 3 Share this post Link to post Share on other sites

RootAnn 11,335 Report post Posted Thursday at 02:43 PM 21 minutes ago, annegables said: And this is why I love AoPS. And also why I am an engineer by training and not a mathematician😁. Yep--another engineer here. I want to use the formulas to build something or make something safer not write/think through proofs. But I'm glad others out there do like proofs. Just not my thing. Share this post Link to post Share on other sites

lewelma 13,785 Report post Posted Thursday at 07:54 PM 5 hours ago, annegables said: And this is why I love AoPS. And also why I am an engineer by training and not a mathematician😁. Yep -- scientist here. I use math as a tool to answer my questions. Glad there are those who love the proofs, my older son included! 1 Share this post Link to post Share on other sites

xahm 2,952 Report post Posted Thursday at 07:54 PM 6 hours ago, drjuliadc said: I remember loving proofs when I was in geometry in school (in 1981 in public school, so no comparison to AOPS). I have no memory of what a proof actually is, just that I really liked them. I was disappointed to hear from the developer of Videotext algebra that they don’t teach proofs in school anymore. Is that true? I’m not a skeptical person, but I like to hear something from more than one reliable source. My kids are young and still in the arithmetic stage. My profession doesn’t require anything other than arithmetic so I have to rely on you guys for answers to these questions. You made me curious, so I looked up the common core standards for high school geometry, and proofs are included in the requirements. 1 Share this post Link to post Share on other sites

kbutton 7,279 Report post Posted Thursday at 10:46 PM We are using AOPS geometry, and so far, it's not traditional proofs with two columns. It's more like show your thinking proofs most of the time. Is there a chapter where proofs start coming out of the woodwork? Share this post Link to post Share on other sites

square_25 699 Report post Posted Thursday at 10:58 PM 11 minutes ago, kbutton said: We are using AOPS geometry, and so far, it's not traditional proofs with two columns. It's more like show your thinking proofs most of the time. Is there a chapter where proofs start coming out of the woodwork? That IS a traditional proof. The two column proof is a bizarre abomination... 2 Share this post Link to post Share on other sites

drjuliadc 144 Report post Posted yesterday at 01:29 AM 5 hours ago, xahm said: You made me curious, so I looked up the common core standards for high school geometry, and proofs are included in the requirements. Ok, thank you. I was thinking, “How could they have done that?” ... but I thought that about cursive too. 1 Share this post Link to post Share on other sites

annegables 49 Report post Posted yesterday at 03:56 AM 5 hours ago, kbutton said: We are using AOPS geometry, and so far, it's not traditional proofs with two columns. It's more like show your thinking proofs most of the time. Is there a chapter where proofs start coming out of the woodwork? I just went through his book and I think I have a case of confirmation bias. He has been getting the problems with number solutions pretty "easily", but there is a disproportionate amount of angst over the proofs (which he can do, he just hates the writing part of them). I also seem to think that review sections have lots of "prove obscure things" , and my son seems to have a phobia of drawing diagrams at a size where they are actually useful for comprehension. Math tends to start out with "Mom! I dont get what this is asking!" Me: "Did you draw it out at a size you can easily label and understand?" Crickets. 5 minutes later, "Now I get it." Share this post Link to post Share on other sites

kbutton 7,279 Report post Posted 13 hours ago 11 hours ago, annegables said: I just went through his book and I think I have a case of confirmation bias. He has been getting the problems with number solutions pretty "easily", but there is a disproportionate amount of angst over the proofs (which he can do, he just hates the writing part of them). I also seem to think that review sections have lots of "prove obscure things" , and my son seems to have a phobia of drawing diagrams at a size where they are actually useful for comprehension. Math tends to start out with "Mom! I dont get what this is asking!" Me: "Did you draw it out at a size you can easily label and understand?" Crickets. 5 minutes later, "Now I get it." Lol! Yeah, my son doesn't like writing them out either. Some problems are easy enough to do with color though--he can color code angles, write it out with the colors, and then sub in the ABD and DBE types of labels in afterwards so that he can think about the concept, not go nuts about the letters. Depends on how frustrating the letters are on any given day. I agree that the non-math ones are less inherently motivating! 1 Share this post Link to post Share on other sites

square_25 699 Report post Posted 13 hours ago Being a bit less snarky, what I mean by "all math should be proof-based" is that I think it's important to think of everything in math as justified and to expect a student to be able to verbalize the reasons for most of the math facts they use at least once in a while. I think that seriously assists in fluency. I have this expectation of my 7 year old, even though she never actually has to write down a proof. But she's started saying things like "We need to prove that!" if I say "we've noticed a pattern, but we aren't sure it always works." Share this post Link to post Share on other sites

EKS 12,209 Report post Posted 12 hours ago (edited) 17 hours ago, square_25 said: That IS a traditional proof. The two column proof is a bizarre abomination... Yes, but the two column kind are easier to grade, or so I hear (I've only graded the two column variety). It's sort of like my son's English teacher who made the kids write their essays not only in the five paragraph format, which was bad enough, but on a form that had all the parts called out--intro, thesis, topic sentence, supporting details, etc--so she could just check things off. I wonder when the two column proof was invented. Was it coincident with the push for high school for all? I looked it up just now and came up with this article called Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century by Patricio G. Herbst, which says this about the first use of the two column proof (emphases mine): The next remarkable change in making proof explicit was pioneered by the second edition of Arthur Schultze and Frank Sevenoak’s geometry textbook (1913). Shibli (1932) says those authors were the first to write proofs in two columns of statements and reasons divided by a vertical line (see Figure 1). Shibli (1932) adds, "[This arrangement in two columns] seems to emphasize more strongly the necessity of giving a reason for each statement made, and it saves time when the teacher is inspecting and correcting written work." ... It is intriguing how widely used this format became and how long it has endured as an icon of proof in school mathematics. The graphical arrangement that the format imposed on the elements of a proof involved yet another move toward establishing a norm for the production and control of proofs by students. The format also established more clearly what was considered important in students’ performance by providing an image of what the finished product should look like. ... Teachers had to take proactive steps to ensure that the course served its purpose. The argument was common at the time and addressed what educators thought of students as learners. D.E. Smith (1911, p. 70) suggested that to give the opportunity to prove might not be enough because the diversity of students in geometry classes made it unrealistic for teachers to expect all students in their classes to be enthusiastic “over a logical sequence of proved propositions.” But whereas it was not reasonable to expect that all students would “discover new truths,” proving truths stated by somebody else was something that all students should be able to do (ibid., p. 160). The task of ensuring that all students would do proofs was one that the teaching profession had to take on. ... For the “rank and file” not “to become discouraged and hopelessly lost in the so-called ‘originals’... the grading [had to be] carefully done, and steps of difficulty [had to be] kept down to a very reasonable lower limit” (Slaught et al., 1912, p. 95). ... The two-column proving custom was an accomplishment of geometry instruction in the sense that it helped comply with a mandate. But that accomplishment did not come for free. It brought to the fore the logical aspects of a proof at the expense of the substantive role of proof in knowledge construction. END OF QUOTATION In other words, the two column proof did to the generation of proofs what writing programs do to the essay writing process. In order to make "proof" accessible to the greatest number of students, it reduced it to form and procedure and removed creativity--or "knowledge construction," as the author terms it--from the mix. Edited 12 hours ago by EKS 1 Share this post Link to post Share on other sites

square_25 699 Report post Posted 12 hours ago 25 minutes ago, EKS said: In other words, the two column proof did to the generation of proofs what writing programs do to the essay writing process. In order to make "proof" accessible to the greatest number of students, it reduced it to form and procedure and removed creativity--or "knowledge construction," as the author terms it--from the mix. I wonder if it's really in order to make things accessible or whether it's to reduce the number of man-hours you need to give feedback. I know that both teaching writing and teaching math have been very teacher-intensive for me, because I don't use programs -- I just have kids write or do math and then I troubleshoot and talk to them about their work. It's all a very interactive, human endeavor. I don't think this approach, while optimal, is possible in a large classroom. 2 Share this post Link to post Share on other sites

EKS 12,209 Report post Posted 12 hours ago 1 minute ago, square_25 said: I wonder if it's really in order to make things accessible or whether it's to reduce the number of man-hours you need to give feedback. I know that both teaching writing and teaching math have been very teacher-intensive for me, because I don't use programs -- I just have kids write or do math and then I troubleshoot and talk to them about their work. It's all a very interactive, human endeavor. I don't think this approach, while optimal, is possible in a large classroom. I think it's both. On the one hand, I think that coming up with original ideas that conform to what is already known and expand upon it ("knowledge construction") is something that takes a certain amount of brain power to do--in fact, I'd argue that it takes a whole lot of brain power to do. So if you're going to ensure that a particular percentage of the population is going to be successful, and that percentage includes folks that don't have that kind of brain power, you need to do something to make it accessible to them. And the way to make it accessible is to remove the really difficult thing, which is the idea generation aspect. Part of the reason that idea generation is difficult is that it cannot be taught directly. I think that with a lot of time and effort you can coach it, but not teach it. And if you can't teach it, how can you expect an average student to actually do it? And then, of course, there is the feedback issue--you go from right or wrong to degrees of quality. You can slap a grade on it, but, like you say here, real feedback requires real human interaction combined with lots of time, something in short supply in a typical classroom. 3 Share this post Link to post Share on other sites

square_25 699 Report post Posted 11 hours ago 13 minutes ago, EKS said: I think it's both. On the one hand, I think that coming up with original ideas that conform to what is already known and expand upon it ("knowledge construction") is something that takes a certain amount of brain power to do--in fact, I'd argue that it takes a whole lot of brain power to do. So if you're going to ensure that a particular percentage of the population is going to be successful, and that percentage includes folks that don't have that kind of brain power, you need to do something to make it accessible to them. And the way to make it accessible is to remove the really difficult thing, which is the idea generation aspect. Part of the reason that idea generation is difficult is that it cannot be taught directly. I think that with a lot of time and effort you can coach it, but not teach it. And if you can't teach it, how can you expect an average student to actually do it? And then, of course, there is the feedback issue--you go from right or wrong to degrees of quality. You can slap a grade on it, but, like you say here, real feedback requires real human interaction combined with lots of time, something in short supply in a typical classroom. I really don't know about whether idea generation can be taught, because I can't teach WITHOUT requiring idea generation. I've never been interested in teaching procedures, so for me teaching requires engaging someone's mind. The levels at which I engage them varies, but so far, I've been able to do that for most of the kids I've come in contact with. I would guess that most people can be taught to write what I think of as normal proof. That is, I would assume a majority of people are able to do this, although I wouldn't be able to tell you how large a majority. I don't think the ability to engage in a logical argument is a particularly unusual skill. For most of the college kids I've taught, they could use this skill outside mathematics but not within it, which I thought was a great shame. Math is such a great tool for honing one's logic, if you allow it to be... 1 Share this post Link to post Share on other sites

lewelma 13,785 Report post Posted 10 hours ago (edited) 3 hours ago, square_25 said: Being a bit less snarky, what I mean by "all math should be proof-based" is that I think it's important to think of everything in math as justified and to expect a student to be able to verbalize the reasons for most of the math facts they use at least once in a while. I think that seriously assists in fluency. I have this expectation of my 7 year old, even though she never actually has to write down a proof. But she's started saying things like "We need to prove that!" if I say "we've noticed a pattern, but we aren't sure it always works." I like the idea of getting kids to verbalize the reasons. I do that, but never verbalized that I get kids to verbalize. LOL. Given our different backgrounds, I thought you would appreciate that I say "Let's investigate that". 🙂 But she's started saying things like "We need to prove that!" if I say "we've noticed a pattern, but we aren't sure it always works." This is awesome! Edited 10 hours ago by lewelma 1 Share this post Link to post Share on other sites

lewelma 13,785 Report post Posted 10 hours ago 2 hours ago, EKS said: In other words, the two column proof did to the generation of proofs what writing programs do to the essay writing process. In order to make "proof" accessible to the greatest number of students, it reduced it to form and procedure and removed creativity--or "knowledge construction," as the author terms it--from the mix. Fascinating. Thanks for doing the research. I'm going to think on this and the ramifications to my different levels of students, from very high to very low. 1 Share this post Link to post Share on other sites