tag:blogger.com,1999:blog-8231784566931768362.post7666739868187001954..comments2021-06-16T05:56:38.112-04:00Comments on MathNotations: Instructional Strategy Series for Middle School and Beyond: Developing Direct & Inverse Ratio ConceptsDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-8231784566931768362.post-71162294676691963922008-12-14T15:12:00.000-05:002008-12-14T15:12:00.000-05:00I wanted to expand this idea into arbitrary number...I wanted to expand this idea into arbitrary numbers:<BR/><BR/><A HREF="http://homeschoolmath.blogspot.com/2008/12/solving-direct-and-inverse-variations.html" REL="nofollow">Solving direct and inverse variations in chart form</A>.Maria Millerhttps://www.blogger.com/profile/00230743954246449727noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-25966851599007653922008-12-13T06:49:00.000-05:002008-12-13T06:49:00.000-05:00Burt,Thanks...If these critical concepts are devel...Burt,<BR/>Thanks...<BR/>If these critical concepts are developed early on, high school and college instructors should have an easier time. For years I had to endure comments from my friends in the Science dept like: "Why do we end up having to teach them the math they should have learned from you!" My reply is usually: "Students often do not transfer knowledge well from discipline to discipline. They often do not relate what they learned in science to what I'm teaching them in math! We need to jointly help them to make these connections. Further, our roles are somewhat different. The scientist is generally more interested in math as a means to an end -- a tool. The mathematician is very concerned about using that tool properly: One needs to understand the restrictions on how that tool is to be used or the results may be invalid -- the theory, not just the application. <BR/>Both views are necessary for knowledge." Sorry for the rant on science vs. math instruction!<BR/><BR/>Overall, I'm hoping to identify some of the major conceptual blocks in the foundations of mathematics learning and suggest ways to overcome these. I am fully aware that the problem I posed and the methods offered are just one avenue to direct and inverse ratio. But I do know that it seemed to help many of my students to make sense of these ideas. I didn't begin on Day I of my teaching career knowing this. I had an idea of how to teach based on my understanding of math but I never learned this in a methods course. It took years of experimentation and communication with other educators to refine my techniques. To the end I was still learning how to make it better. Perhaps, most importantly, I learned to stop talking and listen to my students. What good was that perfect lesson that no one understood! This blog is all about networking and sharing ideas that might help the teacher who is going through that same process...<BR/><BR/>Agai, thanks for the kind words!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-13466043825060177502008-12-13T03:36:00.000-05:002008-12-13T03:36:00.000-05:00Good post, Dave. I agree that developing an intuit...Good post, Dave. <BR/>I agree that developing an intuitive understanding of proportion should come before doing the algebraic formulas. Changing one variable at a time makes a lot of sense. Here is some evidence that focusing on the formulas alone generally doesnâ€™t work. It is a post from what seems to be a physics college instructor explaining what he went through to get his students to understand proportions, at http://www.usca.edu/math/~mathdept/hsg/<BR/>ProportionPaperV03.htmlAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-45554731522092281562008-12-12T12:10:00.000-05:002008-12-12T12:10:00.000-05:00Since we're into puns, tc, just remember Darwin's ...Since we're into puns, tc, just remember Darwin's 2nd Voyage on the Beagle and we all know what 'naturally evolved' from that!<BR/><BR/>On a more serious note, do you look at the digging problem more as a puzzle or more as a vehicle to develop important concepts? I see it as both. I think there are some fundamental teaching precepts here. The idea of analyzing 3 variables by keeping one fixed is important. What generally happens is that students get to see these later on and they're presented as formulas:<BR/>D = kH/B, where D = # of days, H = # of holes and B = # of beagles. In my day this was termed joint variation and the problem would be expressed something like: The number of days varies directly as the number of holes and inversely as the number of digging dogs, etc...<BR/><BR/>My intent for this post was to demonstrate how to informally introduce to middle schoolers important ideas via engaging problems/puzzles.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-69867868859157454512008-12-12T09:32:00.000-05:002008-12-12T09:32:00.000-05:00Interestingly, this exercise can also be used to i...Interestingly, this exercise can also be used to introduce the students to the concept of 'dog days' :-)<BR/><BR/>TCUnknownhttps://www.blogger.com/profile/06449079338919787252noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-30296414477932169732008-12-11T17:00:00.000-05:002008-12-11T17:00:00.000-05:00Paul,I agree with both of your observations. The i...Paul,<BR/>I agree with both of your observations. The independence of the dogs is often overlooked. The "linear" relationships are even more important. <BR/><BR/>I was attempting to build on youngsters' intuitive understanding of these kinds of relationships and I had to make some decisions regarding simplifying the problem for the grade levels involved. In most problems of this type that appears on assessments, the direct or inverse variation is often taken for granted. On a more advanced level, I would have indicated how the variables are related. <BR/>Thank you!<BR/>Dave MarainDave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-3022562683648588682008-12-11T15:16:00.000-05:002008-12-11T15:16:00.000-05:00I'd say there are other natural assumptions being ...I'd say there are other natural assumptions being made, all amounting to an assumption that beagle quantity and hole digging speed have a linear relation.<BR/>It might seems petty at the level this is aimed at but why not also point out the assumption that the beagles are working independently?Paul Carpenterhttps://www.blogger.com/profile/10054030430724849356noreply@blogger.com