tag:blogger.com,1999:blog-2036014053389751696.post8953122134315574732..comments2024-08-07T08:29:41.242+01:00Comments on Colin Foster's Mathematics Education Blog: Don't forget the units?Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-2036014053389751696.post-50221722958641733082023-02-16T15:44:08.578+00:002023-02-16T15:44:08.578+00:00This is my recent twitter thread. An engineer says...This is my recent twitter thread. An engineer says "the speed of a particle that travels d meters in t seconds is d/t meters per seconds". d and t are numbers, so you can put them into a calculator to work out d/t, and you don't have to tell the calculator what the units or dimensions are. In this usage, "d m" is a length, whilst "d" is a number.<br />OTOH, a mathematician, at least at University level, will talk about a distance d and a time t and a speed d/t. So here "d" is a distance, it has a dimension but no units. A distance can be divided by a time, but not on a calculator. The relationship can be illustrated by saying "this distance" (indicating two points) and "this time" (snapping fingers twice), and "this speed" (sweeping a finger from one point to the other in the indicated time). Understanding this equation does not require a system of units. So "d" can either mean a number (of meters) or a physical length. I detect some lack of clarity in school mathematics about which approach is being used, and I don't think that the distinction is taught either at school or university. The engineer will say the mathematician has forgotten to put in the units, but not so. The engineer's "d" is a completely different class of entity than the mathematician's.<br />Sio normally I'd say a numeric length needs a unit. But then we talk about unitless lengths and areas in the Cartesian (or Argand) plane, so here we are talking about the geometry of our abstract number system, and perhaps the words "length" and "area" are being misapplied in this contextJim Simonsnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-42703329352699180552023-02-16T13:21:06.682+00:002023-02-16T13:21:06.682+00:00I agree totally with all of this. I would add that...I agree totally with all of this. I would add that when I taught standard deviation I would always get the class to do one example (no more) - usually {4, 6, 7, 8, 10}, or {4, 6, 7, 7, 8, 10} if you insist on using (n - 1) - wholly without a calculator. I believe that "getting their hands dirty" (just the once) gave them an understanding of the formula and its meaning that would be lacking if all their calculations were simply "substituting into a formula" or using calculator software. <br /> What I would really like to know is how we can set public exams that test things like mean and SD realistically, i.e. requiring candidates to use computer software to analyse large and messy data. Surely we are at the stage when taking examinations online need not be restricted to candidates with learning difficulties?Owen Tollernoreply@blogger.com