tag:blogger.com,1999:blog-7251686825560941361.post2725186691006310680..comments2017-07-14T11:52:32.996-04:00Comments on Launchings by David Bressoud: Beyond the Limit, IMathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-7251686825560941361.post-8749738909972991902014-08-24T04:07:49.095-04:002014-08-24T04:07:49.095-04:00Does any one really over use the formal limit defi...Does any one really over use the formal limit definition anymore? Even in my day (1970's) limits didn't break the course, even though just about everything stated in David's post was true. The crisis I would see is not teaching limits at all. You know, the albeit algebraically challenging stuff.:)<br /><br />This is a great discussion, but sadly, the real crisis right now in AP calculus is not limits, it is students that can't add fractions.Bob Hansenhttp://k12sense.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-7251686825560941361.post-91740993920673460662014-08-24T04:02:01.977-04:002014-08-24T04:02:01.977-04:00It has been known for some time that the "for...It has been known for some time that the "formal definition" of limits is not comprehended by Calculus I students, if only for the fact that this is their first (and only) exposure to formal mathematics. What I fear though is that teachers are confusing the situation with the formal definition of limits with all of the very necessary algebraic exercises students go through to find limits. I can't produce a better argument for those informal algebraic limit exercises than Oehrtman's 11 questions.Bob Hansenhttp://k12sense.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-7251686825560941361.post-22667216591950051782014-07-15T15:50:00.676-04:002014-07-15T15:50:00.676-04:00Thanks David.
Your comments make sense as well as...Thanks David.<br /><br />Your comments make sense as well as bringing some educational studies and theory into the discussion. Your conclusion that the current long standing conviction that treating limits extensively at the beginning of a first calculus course is mistaken has been a thesis I have supported for over 30 years. Most current STEM students are still developing an understanding of numbers and functions when they start a course in calculus. This growth can be nurtured further, but cannot be forced along by premature introduction of concepts that took hundreds of years to formulate at current standards of rigor. More experience with estimation and some of the language that supports the understanding of limits is what is needed at this early stage, not more rigor and formalities.<br /><br />In an editorial published during the heyday of calculus reform I remarked on three central themes for a beginning calculus course- differential equations, estimation, and modeling. [See Flashman, Martin. "A Sensible Calculus," The UMAP Journal, Vol. 11, No. 2, Summer, 1990, pp. 93-96. http://users.humboldt.edu/flashman/EdSenCalculus.htm ]<br /><br />My own materials on calculus (still developing after over 30 years) have followed an approach that focuses attention on these three themes while reducing and/or eliminating more formal and unhelpful treatments that spend time and student energy on a calculus of limits. [See The Sensible Calculus Program - http://users.humboldt.edu/flashman/senscalc.Core.html ]<br /><br />I have not been alone in recognizing and avoiding the obsession with limits. Several of the "reform" calculus texts have also removed those lengthy treatments of limits from their contents while refocusing the treatment on more thematic approaches.<br /><br />It is a fiction that first course in calculus for the general STEM student should be as rigorous as a first course in analysis for a mathematically intense student and that this approach has benefits that make it essential. <br /><br />What is unfortunate is the continued dominance of textbooks that maintain this fiction. This dominance will likely continue until the faculty who select and control these texts and syllabi reorient their choices to available and developing resources for learning calculus that better suit their students. <br /><br />I look forward to your next column and hope that you will include some comments on existing materials that already avoid "many of the pitfalls surrounding limits."Martin Flashmanhttp://users.humboldt.edu/flashmannoreply@blogger.comtag:blogger.com,1999:blog-7251686825560941361.post-42961255795392007312014-07-14T13:48:40.375-04:002014-07-14T13:48:40.375-04:00Limits are not at all required in calculus. Both t...Limits are not at all required in calculus. Both the derivative and integral are well defined in the New Calculus. There is no use of any of the ill-formed concepts: infinity, infinitesimal or limits. The New Calculus is the first and only rigorous formulation of calculus in human history.<br /><br />If you must teach the limit idea, then you'd be better off learning about Gabriel's New Limit theorem for mainstream calculus:<br /><br />http://johngabrie1.wix.com/newcalculus#!Gabriels-New-Limit-Theorem-replaces-all-previous-limit-theorems-in-mainstream-calculus/c9q/D150FC37-5EFE-412D-B2A1-604F7F7E842E<br /><br />http://thenewcalculus.weebly.comJohn Gabrielhttp://www.blogger.com/profile/08536758763788584665noreply@blogger.comtag:blogger.com,1999:blog-7251686825560941361.post-73891911610464554592014-07-03T07:17:35.262-04:002014-07-03T07:17:35.262-04:00Dear David,
This was some of the most interesting...Dear David,<br /><br />This was some of the most interesting I have read in years! At the Univ of Oslo I teach a class on the mathematically difficult parts of the K-12 curriculum, which in Norway includes a fair bit of calculus. I will add some of the Oehrtman questions to my list for next year. Thanks!Helmer Aslaksenhttp://www.blogger.com/profile/11282699304612402004noreply@blogger.comtag:blogger.com,1999:blog-7251686825560941361.post-11139550908469037322014-07-01T21:37:08.311-04:002014-07-01T21:37:08.311-04:00As Paul Lockhart wrote, in calculus we "explo...As Paul Lockhart wrote, in calculus we "explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned."<br /><br />Heaven forbid we show calculus to students the way Barrow and Newton saw it: an intuitive (albeit algebraically challenging) description of nature.<br /><br />Mathematicians were working out methods for finding tangent lines around 1660. The "epsilon-delta" formalism is from when -- 1820? We expect calculus students to make a 160-year jump in a week or so.<br /><br />Starting a calculus course with all of the formalism of limits is like teaching arithmetic beginning with Peano's Axioms.Math Tweephttp://www.twitter.com/mathtweepnoreply@blogger.comtag:blogger.com,1999:blog-7251686825560941361.post-73383547684848374292014-07-01T09:30:28.901-04:002014-07-01T09:30:28.901-04:00Thanks for posting this! The Oehrtman questions at...Thanks for posting this! The Oehrtman questions at the end are especially awesome.<br /><br />I am a non-AP calculus teacher and after years of paring down limits more and more (for the reasons you cite), last year I finally decided to take the plunge and get rid of limits for the sake of limits, and focus even more on their conceptual meaning.<br /><br />I blogged about the intro activity that gets them thinking about the big ideas:<br />http://samjshah.com/2012/11/01/what-does-it-mean-to-be-going-58-mph-at-203pm/<br /><br />and then how I structured the course so it worked:<br />http://samjshah.com/2013/12/02/i-got-rid-of-limits-in-calculus-almost-entirely/<br /><br />Always,<br />Sam ShahSameerhttp://www.blogger.com/profile/10743884978903636387noreply@blogger.com