Mr. Clement, For many years I have followed with admiration your carefully crafted and knowledgeable replies to the naysayers of interactive engagement reform in teaching of math and physics. I am now an emer. prof. of math. but still interested in problems of teaching elementary math/science in the schools. I would appreciate if you could point me to a few articles, books or websites which present detailed explanations of some ideas mentioned below, in what you consider the best of the reformed PER curricula. (I can still get almost anything by interlibrary loan at my local univ. library.)
Here are some things I'm not clear about in your note below: sequencing the learning cycle what exactly are you looking for on the subject of variables, in math & science what is a reasonable way to discuss measurement of all types what kind of treatment of units is appropriate and useful what do you mean by the grammar of functions?
By the way, may I ask where you teach? All physics?
Ladnor Geissinger 1105 Old Lystra Rd, Chapel Hill, NC 27517 ======================================================== On 7/18/2011 10:28 AM, John Clement wrote: > > > The real difficulty that students have is that they are thinking at a low > level. So they do not understand variables, do not have proportional > reasoning, and have difficult with things like sequencing. Most students > who go to MIT or Rensselaer are probably not in this class. > > Previous posts have used proportional reasoning to do some proofs. But > that > is generally unavailable to the vast majority of middle school > students, and > is only used by maybe 25% of graduating HS seniors. The problem with > lecture courses is that students do not develop proportional > reasoning, but > the learning cycle has been shown by Karplus, Renner, and Lawson to > promote > development of these various skills. The reformed PER curricula should > also > do this, but I now have evidence that some deliberate bridging > stragies are > also necessary. > > As to understanding variables, this is definitely a problem in math. > But it > doesn't come up because it is usually avoided by the math curriculum. A > simple example is the phenomenon of frictional forces. The equation F_f = > mu F_n is an empirically determined formula which is true of most surfaces > encountered in the simple labs. Of course there are other factors, but > basically the surface area and the sliding speed is unimportant. So one > asks questions like if the surface area doubles is the frictional force > multiplied by a 4, b 2, c 1/2, d 1/4, e it doesn't change? If you just > look at the formula e is the only answer. But when students think at a > level below the formal operational they have difficulty rejecting a > preconcieved idea and using the evidence that they have. So they will > reject answer e and give one of the others. In other words they don't > understand that if a variable does not appear in the equation, that it has > no effect. This is just as much math as physics. Of course if math used > the "proper" variables in problems students might begin to understand > variables. Generic X and Y are very unuseful when you have to deal > with any > number of variables, and they conceal the meaning. (proper in this case > refers to the meaning particular and not the meaning correct) > > The idea of having multiple variables some of which have no effect and > others which are important is a difficult idea for most of the students. > Then when asked a question such as "how much" they automatically > reject the > null answer, even when it is one of the multiple choices. > > Then there is the perfectly awful use of units as if they varied. So > students will write and equation such as T=2 pi sqrt (kg/ N/m) instead of > T=2 pi squrt(m/mu). They will use units as if they were variables or just > as bad they will write T(s) = 2 pi squrt (m(kg)/mu(N/m)). For those who do > not recognize this formula it is for the period of a mass spring > system with > a mass of m and a linear density of mu. They have absorbed the look of the > grammar of functions without understanding it. This is a common problem > even with some higher level students. It actually stems from the early > grades where they write things like apples=5 rather than a correct > number=5 > apples. Number is the variable, and apples is the unit. This even appears > in conventional textbooks and I can remember it appearing in my elementary > texbooks which were so heavy to lug home. Those stone tablets were much > heavier than the modern paper books. > > John M. Clement > Houston, TX > > > >
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