tag:blogger.com,1999:blog-5964889903484807623.post6803817309223874818..comments2023-12-18T04:44:25.358-08:00Comments on Questions?: It Was a Simple Question...David Coxhttp://www.blogger.com/profile/06277427735527075341noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-5964889903484807623.post-50905028843088602902015-01-25T10:13:41.669-08:002015-01-25T10:13:41.669-08:00I appreciate that you have cultivated an environme...I appreciate that you have cultivated an environment where students feel free to explore and wonder. I observed a fourth grade class this week where students were invited to share their wonderings about fractions. These are some of the questions that came up...what's the smallest (largest) fraction? if numbers go on forever, do fractions go on forever? They were contemplating infinitely small, infinitely big...the basis of limits! <br /><br />I see the same curiosity in the question your students posed. Takes me back to that class in college when we talked about delta epsilon and neighborhoods...it's a mind-blowing concept...at the time I wondered what does that mean about numbers and the point that depicts that number...does this mean points have an aura? I never did go much with that wondering. <br /><br />Faced with the definition provided above, I now wonder how do these two ideas work together? or am i connecting dots that shouldn't be connected?<br /><br />The art of teaching is finding that balance between exploring the wonderings our students bring to the table and delivering the content of the course. May this journey be fruitful David!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-41849797752995997442015-01-24T06:11:06.201-08:002015-01-24T06:11:06.201-08:00I have to confess, that question/argument comes up...I have to confess, that question/argument comes up annually in at least one of my classes, but as it's old hat to me, I often just answer and move on these days. When I first taught it, the question surprised me and I allowed the conversation to fully develop... Maybe I need to step back and revisit the old math through young eyes.keninwahttps://www.blogger.com/profile/05922110653128688365noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-20154136059928219772015-01-22T14:38:27.871-08:002015-01-22T14:38:27.871-08:00A function is increasing on an interval if, for an...<b>A function is increasing on an interval if, for any a and b in the interval, when a < b, f(a) < f(b).</b><br /><br />That definition helps a lot, thanks. David Coxhttps://www.blogger.com/profile/06277427735527075341noreply@blogger.comtag:blogger.com,1999:blog-5964889903484807623.post-79915751946951498072015-01-22T13:59:13.732-08:002015-01-22T13:59:13.732-08:00The definition of increasing/decreasing that I tea...The definition of increasing/decreasing that I teach (and that has has been in all the calculus books I've used in the past 15+ years) is not about what's happening at a point — it's about what's happening on an interval:<br /><br />A function is increasing on an interval if, for any a and b in the interval, when a < b, f(a) < f(b).<br /><br />In other words, if the y-values get larger from left to right, the function is increasing on the interval. The largest such interval for the function depicted here is [-4, 6]. It's not about what's happening <i>at</i> any one point.Anonymoushttps://www.blogger.com/profile/08234926780132269005noreply@blogger.com