On Sat, Dec 19, 2009 at 2:51 PM, michel paul <mpaul...@gmail.com> wrote: > Since most on this list probably work at the college level, as a high school > teacher I'd be interested in the math expectations you'd have for incoming > high school graduates today? In an age of ubiquitous computational > technology, what should they know? What background skills should they > have? Both in a traditional math sort of way, but also in a computational > sense?
These are great questions. Thanks for opening this thread. I'll reply with my thoughts, even though they may not be entirely Sage-related. First, I think teaching programming using Sage is an excellent goal and I hope you do and are very successful. Students like the interactive aspects of Sage and if it helps them learn more, the more power to it. I think, very generaly speaking, that there are two main components to student learning: (1) repetition and working problems, (2) an emotional connection to the material. If students like Sage then they will learn mathematics better for item (2). (I could go off on a wild tangent and rant about the amount of money speat on mathematics education which I think is merely a expensive way to implement item (2) but I will not:-) *However*, the symbolic "langauge" of calculus (what my student call "super high-school algebra":-) is a language which must be learned by any student who wishes to seriously pursue a technical major. It should be drilled into their brain that if they neglect learning the "language" and "grammar" of symbolic manipulation they are making a decision as to how good or bad they want to be in a technical career. By "technical", I mostly mean engineering (electrical or systems) or physics or mathematics, though there are some exceptions. I think there is no question that, at least where I teach, the algebra skills are getting worst. One can blame brain rot caused by the over-use of symbolic calculators. The advantage is that students come in much more computer-savvy, which in my view (being a Sage fan) is a plus. > > I may have an opportunity after winter break to discuss why creating a > computational math course would be a really good thing to do, and I'd like > to be able to back up what I say. I don't want to just make stuff up. > > These are some points I've come up with. Please correct me if I'm off, and > please add anything else you consider essential. > > Thank you very much, > > - Michel Paul > > Our secondary math curriculum arose in the age of handwriting and > handcomputing (handcomputing includes the use of calculators), and most of > what we teach has to do with the needs to express thoughts precisely and > succinctly in order to minimize the number of hand calculations needed when > evaluating expressions. I'd guess that's not the entire reason for our > traditional syntax, but I bet a lot of it does have to do with those needs. > Our culture is shifting very rapidly because of technology, and literacy > regarding it is important for general education. This need can be answered > efficiently and quite elegantly via math classes. Computer Science classes > are usually electives, but everyone has to take some math. > We often pay lip service to the assertion "Math is a language", but we > really don't teach it that way. We teach it as a set of techniques to use > for solving certain kinds of equations we might run into. We might 'use > technology' to help us in that process, but we are still not thinking of > math as a language when we do so. Exactly. > In a computational age, it is more important to grasp relations between > concepts than to memorize particular formulas. Better to learn how to > analyze a concept as a set of inter-related concepts. Example - the > quadratic formula. The traditional schoolish expression minimizes the > number of hand calculations necessary. However, a more conceptually > valuable expression might be to express it as h +/- r, where h =axis of > symmetry, and r = distance to the roots. The traditional formula already > does contain that relationship, but the structure of the related parabola is > hidden for most students. I think it would be a good exercise for kids to > think about it in this slightly more analytical way, spell it out, code it, > and test it that way. Using Sage, it would be very easy to unite the > articulation of the various components and the visual representation. > Especially with @interact! Per the recent thread, even the ones who might > not be able to code it could still interact with it and perhaps learn to > understand the code that way. This is a good idea for a class exercise, agreed, but my personal feeling is that it would be valuable in direct proportion to how it helps them embed that formula into their brain. See points (1) and (2) above. > With Sage, students could be creating their own mathematical papers. You > want writing in the curriculum? Well, there you go! It's very easy to open > up a text cell in Sage, so kids at many levels could create math reports > that actually DID things. I don't even think it's that far fetched to have > the more advanced kids learn some TeX. I just recently discovered the I could not agree more. > insert equation feature in Google docs. It's cool. Even if you don't know > TeX, you can learn it just by using the editor. With this kind of stuff in > the environment, I think this might be good for kids to experience. > There is always a tension between the use of calculators and 'showing ones > work'. Kids hate having to write it all out if the calculator has already > done it. All kinds of discussions go on about how 'much' work needs to be Sorry. Zero sympathy here. Of course, I *like* writing, so ... > 'shown'. All of this becomes irrelevant if we instead focus on the 'work' > being a functional decomposition of a problem or a concept. If one does > ones 'work' correctly, the 'work' will then work for you! You can use it! > Instead of spending so much time teaching kids how to isolate variables in > equations, perhaps it would be better for them to learn how to construct > sutes of simple interacting functions? > China is already uniting Computer Science and math classes at the high > school level. This raises an important issue in my opinion. There are some very technical jobs where US citizenship is *required*. We cannot expect to outsource Chinese technical proficiency for *everything*. This technical training must start at the US high-school level. Thanks again for raising these questions. > > -- > "Computer science is the new mathematics." > > -- Dr. Christos Papadimitrious > > -- > > You received this message because you are subscribed to the Google Groups > "sage-edu" group. > To post to this group, send email to sage-...@googlegroups.com. > To unsubscribe from this group, send email to > sage-edu+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/sage-edu?hl=en. > -- You received this message because you are subscribed to the Google Groups "sage-edu" group. To post to this group, send email to sage-...@googlegroups.com. To unsubscribe from this group, send email to sage-edu+unsubscr...@googlegroups.com. 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