I like to imagine a fictional but possible world in which computer programming for everybody (CP4E) means smoother integration of math and computer science in the lower grades. In the real world, there's a push to make the teaching of algebra more widespread at the 8th grade level. What I imagine is that this is the grade level where we might formally begin using some programming language (e.g. Python, Scheme) in the classroom -- in the sense of having students look at and write source code. In 7th grade, they might watch the teacher project some command line stuff, but getting under the hood wouldn't happen quite as much. A link between programming and algebra is in this concept of variables. A polynomial in the 2nd degree is typically written as Ax^2 + Bx + C, which in Python is more like A*x**2 + B*x + C. The capital letters are called constant coefficients and they "fix" a polynomial, give it its characteristic "call letters" (like in radio: the is KQPD...). Then x is what varies -- you might stipulate over some domain, e.g. from in [-10,10] (square brackets means inclusive). At the command line, 8th graders would have one kind of function called a "polynomial factory" that turned out polynomials with a specific set of coefficients. These would then be floated as functions in their own right, ready to take in x domain values and spit out f(x) range values. There may be a better way to write the factory function than I've shown below. I'd like to see other solutions:
def makepoly(A,B,C): """ Build a polynomial function from coefficients """ return eval("lambda x: %s*x**2 + %s*x + %s" % (A,B,C))
f = makepoly(2,3,4) # pass coefficients as arguments f(10) # f is now a function of x 234 2*10**2 + 3*10 + 4 # check 234 f(-10) 174 [f(x) for x in range(-10,11)] # remember, 2nd arg is non-inclusive [174, 139, 108, 81, 58, 39, 24, 13, 6, 3, 4, 9, 18, 31, 48, 69, 94, 123, 156, 193, 234] g = makepoly(1,-2,-7) # make a new polynomial g(5) 8 g(f(5)) # composition of functions 4616 f(g(5)) # f(g(x)) is not equal to g(f(x)) 156
The same technique might be applied to a sinewave function. Here the coefficients may appear is follows: f(x) = A sin(Bx + C) + D You can imagine doing the factory function, based on the above example, or using some (better?) strategy. Below is a web resource that implements some of these same ideas, also using a computer language, but to my eye it all looks less intuitive in the J language (what the text says is featured). Is it just me? Is this just because I've spent more time with Python? I'm sure that's partly it. http://www.jsoftware.com/pubs/mftl/mftl.htm Moving beyond 8th grade, we want students to understand what's meant be D(f(x)) at point x, i.e. dy/dx at x -- the derivative. Again, Python makes this easy in that we can write a generic derivative taker:
def deriv(f,x): """ Return approximate value of dy/dx at f(x) """ h = .0001 return (f(x+h)-f(x))/h
deriv(f,2) # f(x) = 2*x**2 + 3*x + 4 11.000200000026439 deriv(g,2) # g(x) = x**2 - 2*x - 7 2.0001000000036129
If you remember how to take the derivative of a polynomial, you'll know that f'(x) = 4*x + 3 and g'(x) = 2*x - 2 -- so these are pretty good approximations. Somebody challenged us on math-teach to think of how we might teach crypto and number theory to 7th graders. My suggestions are in this thread (Number Theory question, reply to Wayne Bishop, 25 Apr 2001 http://www.mathforum.com/epigone/math-teach/zilbangkan ). I use the IDLE CLI quite a bit, along with my ciphers.py Another post re Python + Intro Calculus, as per the above is at: http://www.mathforum.com/epigone/math-learn/wonbloryeh I think any of us who know some Python, and some math, can write this kind of stuff. It's not esoteric. I'd just like to see more of it, is all, coming from a lot of different corners. That's because I think a CP4E world would have some major advantages over the current one, including more students motivated to comprehend technology at deeper levels (I think adding more computer stuff to math will make it less of a turn off for many students, if done in a cool and interesting way (which doesn't have to mean super glitzy at every turn -- a bare bones CLI ala IDLE can be part of it)). Kirby