tag:blogger.com,1999:blog-2084641237928654834.comments2016-11-14T21:24:51.018-05:00Algebra, EssentiallyEmily Allmanhttp://www.blogger.com/profile/08966304042607333303noreply@blogger.comBlogger86125tag:blogger.com,1999:blog-2084641237928654834.post-9604510014755720622016-11-14T21:24:51.018-05:002016-11-14T21:24:51.018-05:00Found this essay while I was trying to find a conv...Found this essay while I was trying to find a convincing argument for a student who makes mistakes about 50% of the time, largely because he is convinced that if he is doing math on the computer, he "is supposed to do it in [my] head," regardless of what parents and teachers tell him. (He's doing it on the computer because a book "is too much and I'll never be able to get through all of that" and he (for unknown reasons) believes it has to be fully, 100% completed before he'll be able to do anything ever again. Yes, he has some issues, but he's a great young man, for all that.<br /><br />In his case, it would really benefit him to put it on paper. I think that's true for a large number of students. It's the convincing them of that that is the difficult bit for many of us. Lauranoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-63976399522682101732016-01-20T21:08:17.395-05:002016-01-20T21:08:17.395-05:00yah!yah!StephanieDailey785http://www.blogger.com/profile/16909463652952814987noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-13140805909530342682016-01-20T21:08:00.865-05:002016-01-20T21:08:00.865-05:00This comment has been removed by the author.StephanieDailey785http://www.blogger.com/profile/16909463652952814987noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-14700145702134835102016-01-02T10:35:03.390-05:002016-01-02T10:35:03.390-05:00look. I am a kid at school and I think that you sh...look. I am a kid at school and I think that you shouldn't waste time getting us to show their workings out all of the time. you should definitely show us how to do it and make us practice a bit, but all of the time is not needed. I think that if we write our answers and we are right, than great, their way of doing it is working. if we are wrong, then you can go over with us and show us another method. I know the main reason is to see where they are going wrong but the main thing is that they get the answer right and as said before, just show us another method if we are wrong.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-21383091814851151472015-12-23T03:26:39.202-05:002015-12-23T03:26:39.202-05:00Thank you, I got it. Thank you, I got it. Lourdes Rodríguez Fernándezhttp://www.blogger.com/profile/17366095005520960157noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-367655712295863342015-12-11T04:52:57.484-05:002015-12-11T04:52:57.484-05:00Hello Emily. Thank you for your blog. I just want ...Hello Emily. Thank you for your blog. I just want to comment that I have tried to grab your bottom to include it on my site but apparently it is not working. If you could please check it, it would be nice. <br />Thank you again for your work.Lourdes Rodríguez Fernándezhttp://www.blogger.com/profile/17366095005520960157noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-81123763978925624992015-10-06T21:00:29.184-04:002015-10-06T21:00:29.184-04:00Thank you, Christopher Danielson, for pointing me ...Thank you, Christopher Danielson, for pointing me to your post about logs and non-commutative binary operations: https://christopherdanielson.wordpress.com/2011/12/07/enough-with-the-logs-already/<br /><br />I think this is getting me somewhere. Somehow, the fact that exponents are not commutative will throw off the ease with which inverse functions are grasped. For example, it is true that f(x) = x+3 and f(x) = 3+x will both yeild the same inverse function f-1(x) = x-3. What casues problems is this: if g(x) = x - 3, then g-1(x) = x+3... BUT if g(x) = 3-x, then g-1(x) is NOT 3 + x. Since the original g function is not commutative, we cannot use the same 'thought process' to invert them when the arguments are reversed. I like.Emily Allmanhttp://www.blogger.com/profile/08966304042607333303noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-75758926194937683142015-10-06T20:41:13.909-04:002015-10-06T20:41:13.909-04:00Thanks Mike. I like how you ask another question t...Thanks Mike. I like how you ask another question that's also interesting. I HAVE done the compositions: f(g(x)) and also g(f(x)) and see of course that they are not x (although, you are right, they CAN be, which IS interesting). My students have only a rudimentary understanding of inverses and compositions at this point, so it would be useless to use this method as a rationale for them. They were satisfied with simply finding a counterexample: g(f(3)) is not equal to 3, so this cannot be the inverse. But to them, the xeth root of n seemed like such the logical choice for the inverse. After all, x^n does indeed yield nth root of x as its inverse. How can I explain this quirk to them so that they feel secure in their understanding of inverses overall (which, BTW, i think is excellent)?Emily Allmanhttp://www.blogger.com/profile/08966304042607333303noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-40867257693602501132015-10-06T19:04:00.418-04:002015-10-06T19:04:00.418-04:00For the sake of being able to write this out in a ...For the sake of being able to write this out in a comment, let's call your inverse function g(x). If they're truly inverses, then g(f(x)) = x. The reason that this doesn't work becomes more obvious when you build the composite statement. Fleshing out the composite function, we get that (n^x)√(n) = x (if it isn't obvious, the n to the x-eth root of n equals x). Looking at it this way, it's clearly a very rare scenario where this would work. I think the more interesting question given those to functions is to ask when, if ever, g(f(x)) = x.Mikehttp://www.blogger.com/profile/17209841472729555694noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-55505947528523457652015-02-24T12:25:25.300-05:002015-02-24T12:25:25.300-05:00I don't know if this is true that "only s...I don't know if this is true that "only some people understand". I think its more correct to say that some people have not developed the skill of showing work. The skill is similar to elaborating on ideas. When I was a kid I might have just answered "yes" if someone were to ask me if I liked Disneyland. Today after learning how to express my thoughts in a fuller way would go on to explain why I liked it and what things I liked about it. The "yes" is the answer and the rest of the stuff is showing work. If you haven't practiced and learned how to do it its going to be hard. <br /><br />I was one of those kids who didnt want to show work. All through middle school and high school I just put the answer -no work. Eventually after I grew up and matured I realized that I thought I knew more that my teachers. Well I didn't! In my mid-20s when I went back to school, I listened and trusted that the math teacher probably knew better than I did and followed their directions. It worked. Once I started to do what they asked of me it was much more comprehensible. <br /><br />The teachers were always saying the same thing. The difference was in me; I chose to listen. . Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-90039675125811266032015-01-30T15:57:26.437-05:002015-01-30T15:57:26.437-05:00Hi Emily,
Nicely said! I can totally identify wit...Hi Emily,<br /><br />Nicely said! I can totally identify with the grumblings of the mid-year slump. In my classroom, and with my tutoring students, it seems like they push through this for one more big chunk of effort that lasts until about late April. Do you agree?<br /><br />Thanks,<br /><br />-DawnDawnhttp://www.gradesquad.comnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-58451061138055922712014-11-19T14:05:33.126-05:002014-11-19T14:05:33.126-05:00I stumbled across this while trying to plan a less...I stumbled across this while trying to plan a lesson on inverse functions. My kids loved Joe "Math Guy" and were really excited to help Joe keep his job when he messed up :)Michelle Millerberghttp://www.blogger.com/profile/08510831358593548996noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-58028520099071348912014-05-15T16:13:42.354-04:002014-05-15T16:13:42.354-04:00I can relate and enjoyed your article. Also, I fou...I can relate and enjoyed your article. Also, I found this article (http://www.byrdseed.com/to-show-or-not-to-show-work/) encouraging. Maybe you will too. I'm still learning as a math teacher and tutor, and didn't go higher than college algebra with an intro. to trig. and calc., as a student, so I am still learning from people more experienced in math, to help me prepare my elementary and middle school students. <br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-77772207022066750682014-01-04T04:59:53.358-05:002014-01-04T04:59:53.358-05:00This comment has been removed by a blog administrator.D Phanchaladhttp://www.blogger.com/profile/11644673187095497433noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-86298228687024865322013-12-01T10:39:03.779-05:002013-12-01T10:39:03.779-05:00Thank you! We need to make sure we teach the math/...Thank you! We need to make sure we teach the math/algebra/calc content along with the science that employs it so that students can use the math/etc concepts to concrete applications they encounter in school (and life!). As a science teacher I hear too many "that's Math, I don't do well in math...How am I supposed to know how THAT works in YOUR class?!". The only other comment that I get more frequently is about using complete sentences, subject/verb agreement in their science writings....Elisa Cousinshttp://www.blogger.com/profile/13120338315155431188noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-4293982914940997872013-09-17T22:12:44.651-04:002013-09-17T22:12:44.651-04:00The thing is I can do work In my head also and wel...The thing is I can do work In my head also and well I actually couldn't write it out, and still can't. That's the problem with most students they can do it but can't write it out. Its something that only some people understand and can relate with.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-58506423830159478122013-06-19T11:34:16.274-04:002013-06-19T11:34:16.274-04:00Aha, touché! Spoken like a true teacher in June. T...Aha, touché! Spoken like a true teacher in June. Thanks for reading, and I appreciate your comments. -EmilyEmily Allmanhttp://www.blogger.com/profile/08966304042607333303noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-87686876010448763612013-06-19T11:25:24.998-04:002013-06-19T11:25:24.998-04:00I suspect that most students will say that the BES...I suspect that most students will say that the BEST solution is the FASTEST.MrsFaxonnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-1416632833514149612013-06-19T11:21:11.599-04:002013-06-19T11:21:11.599-04:00I stumbled across your Blog while searching for Al...I stumbled across your Blog while searching for Algebra 1 Essential Questions. I have been trying to come up with questions similar to these for weeks now. I think more teachers need to go through this exercise. I love the idea of a "second set" of objectives. Uncovering these is a difficult and delightful process. I think it's important for students to see this second set objectives....maybe more important than seeing the first set! This is why we teach math. To inspire students to ask these questions in their every day life and have practice coming up with reasonable, logical answers to them. Thank you!MrsFaxonnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-85144328419270420252013-05-08T13:30:44.521-04:002013-05-08T13:30:44.521-04:00Interesting. Those who teach math should certainly...Interesting. Those who teach math should certainly go thru this post. This has helped me. Worksheet Wizardshttp://www.blogger.com/profile/04325089914174392127noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-61898895802348473722013-03-05T20:53:29.214-05:002013-03-05T20:53:29.214-05:00I am interested to hear your perspective on standa...I am interested to hear your perspective on standardization.<br /><br />The company I work for Learn Bop (http://learnbop.com) is striving to adhere to the common core standards while empowering teachers to have better relationships with their students.<br /><br />Traditionally technology doesn't permit for solid personalized instruction because its very simply oriented and non-adaptive. From what I infer in your article here beyond simple arithmetic its very difficult to make procedural, non-linear math instruction or abstractions. <br />learnbopfredhttp://learnbopfred.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-61960247680756114072013-02-09T17:37:34.188-05:002013-02-09T17:37:34.188-05:00Very articulate and to the point. Thanks for shari...Very articulate and to the point. Thanks for sharing.Scipihttp://www.blogger.com/profile/04540662041195568946noreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-14817902526166819532013-01-28T13:05:38.431-05:002013-01-28T13:05:38.431-05:00Good points to follow. Being a teacher I will try ...Good points to follow. Being a teacher I will try to remember and follow the tips. Thanks.Math Tutorhttp://newdelhihometuition.comnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-59452315089026455802013-01-28T13:00:32.630-05:002013-01-28T13:00:32.630-05:00Written with excellency. I liked it very specially...Written with excellency. I liked it very specially :Do I really have to memorize all these rules and definitions?, and Isn't there an easier way?.Math Tutorhttp://newdelhihometuition.comnoreply@blogger.comtag:blogger.com,1999:blog-2084641237928654834.post-68040056229927552972013-01-13T11:14:17.816-05:002013-01-13T11:14:17.816-05:00>Arithmetic involves counting and manipulation ...>Arithmetic involves counting and manipulation of quantities where algebra relies more heavily on reasoning and generalizing the patterns that are observed from arithmetic procedures.<br /><br />Sheer nonsense. Generalization should start with arithmetic, but because such misconceptions plague teachers, once students start algebra, they too get confused. And so the cycle of weak math understanding is perpetuated.Anonymousnoreply@blogger.com