tag:blogger.com,1999:blog-7775513617406888811.comments2021-01-22T20:47:39.667-05:00Math Horizon's AftermathMathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger54125tag:blogger.com,1999:blog-7775513617406888811.post-65767543585057321842014-05-13T14:02:29.993-04:002014-05-13T14:02:29.993-04:00I agree! One advantage to having students do prese...I agree! One advantage to having students do presentations in an <a href="http://maamathedmatters.blogspot.com/2013/05/what-heck-is-ibl.html" rel="nofollow">inquiry-based learning</a> (IBL) environment is that they get to regularly practice their public speaking in a safe environment. Every semester, students in my IBL classes comment that getting up in front of their classmates to present and defend their ideas has been extremely beneficial to them.Anonymoushttps://www.blogger.com/profile/18425048303220563633noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-27185846394189373582013-04-24T15:18:45.038-04:002013-04-24T15:18:45.038-04:00The value of a Masters degree or any degree in mat...The value of a Masters degree or any degree in math depends upon if employers recognize the school you attended as having good mathematics credentials. Read the book plutocrats. You need to attend a top school to have a shot at a data-driven analyst job. I got my degree from a university which has no alumni news from math grads since mid 2000's and now is it 2013. Apparently the alumni didn't accomplish much. My BS in Math came from a university that had some good life science programs, but was largely recognized as a drinking school.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-76263497441586039732013-04-04T16:11:12.246-04:002013-04-04T16:11:12.246-04:00While I appreciate the notion of good reasoning, I...While I appreciate the notion of good reasoning, I do think mathematical reasoning is different than reasoning in other subject, so I'll have to respectfully disagree. One of the things that I think is interesting is the way in which mathematics standards (such as the NCTM standards or the Common Core State Standards in Mathematics) separate "practices" or "processes" from "content" when for me they are so interwoven, thus habitual. Part of "understanding" congruence or functions is reasoning about them appropriately and using precise language in discussing them (to choose two from those standards of mathematical practice). So, I am not talking about good thinking skills in general, but in mathematics. There are many students who have great thinking skills that they never apply when doing mathematics because they are not part of their mathematical habits of mind but their "not mathematical" habits of mind. Karen Kinghttps://www.blogger.com/profile/04794040454775778784noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-65168732009719829712013-04-04T15:49:15.314-04:002013-04-04T15:49:15.314-04:00It should be clear from the link I provided in my ...It should be clear from the link I provided in my comment that the habits of mind you describe are just habits of good thinking that can be taught without recourse to any branch of mathematics. The word "mathematical" should be removed from the question, instead of being inserted in the answer. To allow for a slight exaggeration, one can have great thinking skills without even knowing the multiplication table. This should not be confused with the fact that some math knowledge might be handy in various circumstances that require good thinking skills.<br /><br />I am sorry your setup does not allow (or I fail to find out how) to leave a signed comment for those who are not members of the selected networks. The least of all I would want is to leave an anonymous comment.<br /><br />A. BogomolnyUnknownhttps://www.blogger.com/profile/01272365600951800811noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-90391275521333976482013-04-04T15:14:29.211-04:002013-04-04T15:14:29.211-04:00Thank you for your comment. I think the focus that...Thank you for your comment. I think the focus that I was trying to take is on the word "habits" as opposed to "practices" and the power that the "habits" metaphor brings. So, instead of just calling them reasoning skills, framing these reasoning skills as mathematical habits of mind, "start thinking about these ... as habits of mind" affords us a different set of instructional strategies that can help support students replacing "bad"/unproductive habits of mind for mathematical reasoning with "good"/productive habits of mind for mathematical reasoning (some might call these right/correct). <br /><br />I think instead of omitting "mathematical" in the first sentence, it should be added in the second sentence (and was probably removed as I was way over the word limit on my early drafts). Karen Kinghttps://www.blogger.com/profile/04794040454775778784noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-39065000114339049192013-04-04T07:57:49.789-04:002013-04-04T07:57:49.789-04:00Dear Karen, in the following paragraph I see a log...Dear Karen, in the following paragraph I see a logical misstep:<br /><br />"Why do I keep referring to reasoning skills as “mathematical habits of mind”? Because I believe that if we start thinking about these unproductive practices as habits of mind, it opens up a different set of strategies for addressing the problem."<br /><br />The first sentence poses a question which the second sentence is supposed to answer, but it does not. It would, had you omitted the word "mathematical". <br /><br />The habits of mind you describe are the best possible tools that students could carry out of school, but their acquisition, though, may not necessarily happen in math classes. See, for example,<br /><br />http://www.cut-the-knot.org/ctk/NatureOfProof.shtml<br /><br />A. BogomolnyUnknownhttps://www.blogger.com/profile/01272365600951800811noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-41742907497111412252012-11-01T09:49:57.825-04:002012-11-01T09:49:57.825-04:00Nice piece. Your personal story of discovering th...Nice piece. Your personal story of discovering the power of algebra resonates with me. I guess all of us who love math have had, and continue to have, those experiences. As teachers, we want students to have those experiences, too.<br /><br />Unfortunately, arguments about "learning to learn" and transferability aren't convincing to those who don't already understand or appreciate math. "Transfer" is a nebulous idea to begin with, and couldn't many subjects be the vehicle to "learn to learn"?<br /><br />Let's hope that the public at large continues to see the value of mathematics education. <i>We</i> know it's worthwhile!Patrick Honnerhttp://www.mrhonner.comnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-37624557742840754392012-09-14T15:09:52.725-04:002012-09-14T15:09:52.725-04:00Unless one is looking at a new university (e.g., U...Unless one is looking at a new university (e.g., University of California at Merced) or another unusual situation (lots of faculty members left or were fired), the percentage of women in T/TT positions tends to reflect the effects of many years of hiring, thus many years of past hiring practices and PhD production. For example, some current faculty members were hired in the 1970s and and 1980s. In the 1970s, women earned about 10% of the PhDs in mathematics. Very few of them were hired at top research universities during that decade. Despite that, women's share of PhDs in mathematics increased to about 20% in the 1980s, and to about 30% in the late 1990s and 2000s. <br /><br />For an illustrative example and current statistics, see my article for the Association for Women in Science magazine posted <a href="http://mathedck.wordpress.com/2012/02/15/the-pipeline-and-the-trough/" rel="nofollow">here</a>. Cathy Kesselhttp://mathedck.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-52516066933473549982012-07-20T21:55:30.458-04:002012-07-20T21:55:30.458-04:00I've been tutoring math for now almost a decad...I've been tutoring math for now almost a decade and I've seen the deterioration in the value of Math master degree. Basically this has been brought about due to introduction of many IT oriented job. This really saddens me but none the less I still insist Math degrees are very helpful even for moving to other fields like CSE, ECE etcTutor For Mathhttp://tutorformaths.comnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-51121023865283102432012-05-02T16:49:39.641-04:002012-05-02T16:49:39.641-04:00D'OH! Don't ask me why I thought Middlebu...<b>D'OH!</b> Don't ask me why I thought Middlebury College was in Connecticut. Ah well, make it "must be something going on in New England" then. You heard about <a href="http://mitadmissions.org/blogs/entry/i-have-smashing-news" rel="nofollow">what MIT did with their admissions decisions</a> this year, didn't you?Joseph Lindenberghttps://www.blogger.com/profile/17937849128943764513noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-35650087572734118162012-05-01T15:56:16.915-04:002012-05-01T15:56:16.915-04:00I was thrilled to see your Math Horizons column ab...I was thrilled to see your Math Horizons column about tau, Stephen. There must be something going on in Connecticut. One of your fellow Connecticutters(?) Ethan Brown is trying to set a world record this weekend for memorizing the most digits of tau.<br /><br />You might find some of the graphics on my website helpful in convincing your fellow mathematicians about tau. It's at <a href="http://sites.google.com/site/taubeforeitwascool" rel="nofollow">sites.google.com/site/taubeforeitwascool</a>. Feel free to use/redistribute anything there you find useful. And let me know if there's any other way I can help.<br /><br />Joseph LindenbergJoseph Lindenberghttps://www.blogger.com/profile/17937849128943764513noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-15357844461104131802012-03-27T21:50:06.390-04:002012-03-27T21:50:06.390-04:00So instead of $\sin x$, I should type $\sin \left(...So instead of $\sin x$, I should type $\sin \left( x \right)$? And although I prefer $\sin^2 x$, I encourage my students to write $\left( \sin x \right)^2$, now you're telling me to write $\left( \sin \left( x \right) \right)^2$. What we need is a definitive guide from a typesetter to let us know what's proper. We all have peeves (inverse notation for me), but we need a consistent style guide for both written and typeset work.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-48312232218314933862012-03-05T11:18:06.244-05:002012-03-05T11:18:06.244-05:00Tommy, I've been ranting about this for YEARS....Tommy, I've been ranting about this for YEARS. Way to go! I don't know if I agree about \sin^2 and \sin^{-1}, since they are so ubiquitous, but it's way better than the alternative of sin x.Karl-Dieter Crismannoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-38356683844812618272012-02-12T22:07:53.832-05:002012-02-12T22:07:53.832-05:00Apropos Tommy Ratcliff's lament for explicit (...Apropos Tommy Ratcliff's lament for explicit ( & ) vis-a-vis implicit - - - Back in the pre-transistor days - - - when FORTRAN first came out - - - at the Bell Labs the failure to make explicit in FORTRAN - - - multiplication that wasn't so in the math expression - - - was the major source of user coding errors for some time - - - a few years later - - - when discussing with Jonh Kemeny of Dartmouth the Fortran alternative he was developing - - - BASIC - - - we thrashed over several times this issue of implicit operations in math expressions having most likely to be annoyingly explicit in any programming language - - - Duncan Morrill, Merrimack, NH - - - WV1J@QSL.NETAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-59086196195064511442012-02-09T15:22:12.792-05:002012-02-09T15:22:12.792-05:00I have been a math professor for three years. I h...I have been a math professor for three years. I have been writing arcsin(x), arccos(x), arctan(x), etc. ever since one of my own undergraduate professors pointed out to me the ambiguity in the sin^{-1}(x) notation. Thank you for pointing out the need to rid ourselves of this unnecessary distraction for our students.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-44626418754216268462012-02-05T09:55:45.248-05:002012-02-05T09:55:45.248-05:00Not since the invasion of Iraq has someone had a m...Not since the invasion of Iraq has someone had a more mistaken premise. Statistics is not now, nor has it ever been, a branch of mathematics. I shudder to think of how many prospective statisticians have been turned off by this instructor's approach.<br /><br />Frank Harrell<br />Chairman, Dept. of Biostatistics<br />Vanderbilt UniversityFrank Harrellhttps://www.blogger.com/profile/15263496257600444093noreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-55926613318321435812012-01-24T10:00:54.574-05:002012-01-24T10:00:54.574-05:00See http://community.amstat.org/AMSTAT/Blogs/BlogV...See http://community.amstat.org/AMSTAT/Blogs/BlogViewer/?BlogKey=784e55b4-cb6c-499a-a2da-391e548d36b2.Emil Friedmanhttp://www.statisticalconsulting.orgnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-88511370836863543122012-01-18T17:40:30.456-05:002012-01-18T17:40:30.456-05:00The best explanation I've seen of the differen...The best explanation I've seen of the difference between statistics and mathematics was given by the famous statistician George Box in Technometics (1990). He wrote, <br /><br />"Statistics is, or should be, about scientific investigation and how to do it better, but many statisticians believe it is a branch of mathematics.... Now I agree that the physicist, the chemist, the engineer, and the statistician can never know too much mathematics, but their objectives should be better physics, better chemistry, better engineering, and in the case of statistics, better scientific investigation. Whether in any given study this implies more or less mathematics is incidental."Jim Higginshttp://www.k-state.edu/statsnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-58090153504056203772012-01-18T16:24:55.053-05:002012-01-18T16:24:55.053-05:00I hated stats courses until I finally took one tha...I hated stats courses until I finally took one that contained math. Before that it felt like a confusing cookbook of ideas to memorize - extremely boring. After finally running into graduate stats courses (with math) I switched from studying graduate level math to stats.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-65298601955154378862012-01-18T16:22:25.270-05:002012-01-18T16:22:25.270-05:00http://community.amstat.org/AMSTAT/Blogs/BlogViewe...http://community.amstat.org/AMSTAT/Blogs/BlogViewer/?BlogKey=784e55b4-cb6c-499a-a2da-391e548d36b2Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-31254284721868669092011-10-24T07:43:07.614-04:002011-10-24T07:43:07.614-04:00I too agree with Josh as i am also a student.........I too agree with Josh as i am also a student...........so i can say what a student will prefer..........'cuz v get borred if v r only left out wid books...........inaranoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-73512071404747925422011-09-21T13:49:33.231-04:002011-09-21T13:49:33.231-04:00For a somewhat hand-wavey but easily 7th-grade exp...For a somewhat hand-wavey but easily 7th-grade explanation of the negative reciprocal slope thing: try constructing a square. Pick a starting place on the Cartesian plane; that's your first vertex. Go a units right and b units up; that's your second vertex. Go b units left and a units up; that's your third vertex. Go a units left and b units down; that's your fourth vertex. Go b units right and a units down; you're back at your first vertex.<br /><br />Looking at any two adjacent sides, it's easy to show that their slopes are negative reciprocals.<br /><br />(For 7th-graders, you'd want to use actual positive integers in the place of a and b.)<br /><br />The "hand-wavey" aspect here, of course, is: how do we know it's a square? It's visually "obvious" if you draw it on graph paper, but I think that for an actual proof you'd want to use the Pythagorean Theorem, as per Wiley Williams' comment above. (Or, I suppose, the dot product, but that's usually not covered in the 7th-grade curriculum...)<br /><br />- AliceAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-89643143179755215512011-09-16T23:33:36.626-04:002011-09-16T23:33:36.626-04:00As to the slopes of perpendicular lines being nega...As to the slopes of perpendicular lines being negative reciprocals, you can derive it from the dot product. If u=(u1,u2) and v=(v1,v2) are perpendicular u.v=0=u1v1+u2v2. (c.f. the law of cosines) Then solve to get v2/v1=-u1/u2. Since u2/u1 and v2/v1 are the slopes of u and v, it is proved.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-47423754760326750912011-09-15T15:40:59.324-04:002011-09-15T15:40:59.324-04:00Concerning the statement that the slopes of perpen...Concerning the statement that the slopes of perpendicular lines must be negative reciprocals (assuming the lines aren’t vertical and horizontal): <br /><br />I’m not sure this is “geometric” enough, but here’s a reason using the Pythagorean Theorem:<br /><br />We may assume both perpendicular lines pass through the origin (after translation if necessary). Then one line (with slope m) goes through a point of the form (1, m) and the other line (with slope n) goes through a point of the form (1, n). Set up the statement of the Pythagorean Theorem for the triangle with vertices at these two points and the origin. Simplifying leads to mn = -1 as desired.Wiley Williamshttp://www.math.louisville.edu/undergrad/index.htmlnoreply@blogger.comtag:blogger.com,1999:blog-7775513617406888811.post-7499238782294015482011-09-15T15:31:15.960-04:002011-09-15T15:31:15.960-04:00Be careful about making statements like “anything ...Be careful about making statements like “anything to the 0 power is 1” to students. Is 0^0 = 1? No. The pattern that 0 ^ (any positive power) = 0 would lead to 0^0 = 0. So one pattern leads to an answer of 1, the other to an answer of 0! For that reason 0^0 is not defined.<br /><br />I think a better reason for defining 0 ^ (any non-zero power) = 1 is so that the basic rules of exponents that work for positive exponents also work for non-negative exponents (when 0^0 is ruled out).Wiley Williamshttp://www.math.louisville.edu/undergrad/index.htmlnoreply@blogger.com