I'd like to add to some of David's points and maybe disagree a little :) I teach chemistry, not one of the fields that David called technical. However anybody who wants to pursue an advanced degree in chemistry needs to get through a minimum of multivariable calculus by junior year of college (diff. eq, linear algebra, general algebra and significant computer programming experience is necessary too if you end up working at the physical chemistry/chemical physics edge as I do).
I second the comment about students being algebraically weaker than when I started teaching. I also agree that students should be required to write things out and show how it is done. This is the same as requiring students to be reasonably proficient at computation (+, -, x, etc..) before they are allowed to make extensive use of calculators. It is easy to reach the wrong conclusion and not realize it if you don't understand what the tool is doing. At the college level in chemistry an example is quantum mechanical simulations of molecules. All real computations of these types are done using computer programs that numerically approximate the solutions (there are few analytically soluble chemically relevant situations in quantum mechanics). However, we expect our students to understand the analytically soluble examples so that they have an understanding of how valid solutions behave before they use the computational packages, which give good answers most of the time, but can drift of into never- never land... So the students who are truly ready to pursue an undergraduate science program (we get few of these, so have to bring most students up to this level in college), will have the following mathematical capabilities upon graduation from high school. 1) Specific technical skills (training, got mostly by rote practice): a) be proficient at algebraic manipulations to isolate single variables in symbolic equations (no numbers). It is usually much more efficient to stick in the numbers for your particular physical situation at the end. Most of our students want to stick in all the numbers and then solve for the unknown. This leads to dropped digits, copying errors and usually erroneous significant digits. b) be proficient at solving simple systems of equations (2 variables). c) able to sketch what a function looks like without having to put it into a graphing calculator. Especially they need to know sine, cosine, hyperbolas, exponential growth, exponential decay, gaussians, circles and spheres. They should also be able to visualize what two of these functions might look like when multiplied together. d) able to look at a function of multiple variables and decide what happens to the value of the function if a variable is increased or decreased (yes, I know this is technically calculus, but it can be done without taking derivatives). e) (not required, but is what I would really like to see) understand basic 1-D calculus. Able to take derivatives of functions of one variable (including ln, exp, sin, cos, powers and can use the product rule). Can do simple integrals as inverse differentiation. Even more than that would be better, but I know I'm dreaming... f) (not required, but is what I would really like to see) some experience writing simple computer programs in any language. Even if they are just games or little more than "hello world" or flash card programs, it would be better than what we're getting now. 2) Higher level reasoning skills (what I consider education rather than training): a) able to read a story problem about a physical situation and combine that with known relations among the quantitative pieces of information in the problem to generate symbolic representations of these relationships. Then they should be able (and willing) to play (struggle) with manipulations of these relationships to extract the information requested. We see many, many students who can only solve story problems if they can plug them into an algorithm. These students are "trained" but not "educated". The difference is somewhat subtle. Trained students cannot solve any new quantitative problems without being trained in a new algorithm. Thus someone needs to train them to a each new situation. In college we try hard to educated them to develop general strategies (play) for finding solutions to unfamiliar problems. Although I would not expect most high school graduates to be proficient at this play they should understand that the struggle is what they are supposed to be doing. They don't have enough room in their brains to memorize every single kind of problem they will encounter. (((An aside: there are some students who are only trainable and should probably not go to college, but instead attend technical school and plan to get retrained when their skills are no longer useful.))) b) able to read for understanding and write their answers using correct grammar, spelling and punctuation. High school students should be required to do some proofs and they should get marked down in math class for grammar, spelling and punctuation errors. There are more things, but I think that's my fundamental list... Jonathan On Dec 19, 1:51 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? > > 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 > > 1. 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. > 2. 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. > 3. 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. > 4. 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. > 5. 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 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. > 6. 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 '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! > 7. 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? > 8. China is already uniting Computer Science and math classes at the high > school level. > > -- > "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. 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