onsider the following situation:
Take a room and draw a line down the middle of it. Fill the room with mathematics instructors. Open up the discussion with "should we use computer algebra systems (CAS) for teaching freshman math classes?". What do you get? Division.
This is a scenario that has begun to replay itself in schools not just all over the country, but literally all over the world. Some view the development of CAS as a great opportunity, claiming it will remove the drudgery of educationally contentless algebraic manipulation, and thus provide the chance to explore mathematical issues that were infeasible to address in the classroom before. Others see a great danger that students will become mathematically illiterate, will treat these software tools as though they are perfectly implemented black boxes, and will end up with little or no understanding of the mathematics that those black boxes purport to implement.
Which side of the debate is correct? As with all difficult social questions, the answer is both and neither. The optimistic viewpoint lets us see what we have to gain by the journey, and the pessimist viewpoint shows us the pit-falls that we must either avoid or bridge across if we are to come through the endevour safely. In other words, we need to find a position between the extremes were we get the most of the good and the least of the bad. For now we will defer discussion of the wonderful possibilities and instead examine some of the dangers in order to better understand and hopefully avoid them.
One of the most frequent issues that comes up in these discussions are concerns over the fabled "black boxes". Black boxes are bad, black boxes are evil, we cannot have black boxes or students won't learn anything... or will they? Let us take a closer look at this concept of a black box.
By a box we roughly mean a process that has inputs given to it and which then outputs some results or a behaviour of some sort. We call a box black when it is opaque and we can't see what is going on inside. We hope that the right thing is happening, but often we don't know for sure.
Programmers deal with black boxes all the time by using subroutines, libraries, and programs written by other people. As long those tools work properly and we know how to use them, we rarely concern ourselves with the internal workings. This leads to efficiency of work. This is a good thing.
In mathematics when we have a black box, then there are mathematical concepts hidden inside. If students can't see the internal workings then they may not learn those mathematical concepts intrinsic to the implementation of the black box. Students may work at assignments efficiently, but since learning the math is the true objective, the efficiently-performed work could be to little purpose. This is a bad thing.
Consider, however, an extreme alternative perhaps reflective of the current state of affairs. If we give students large volumes of the relatively simple problems suitable for manual algebraic manipulation, they may learn a few skills thoroughly by brute force repetition, but never have the time to tackle the more interesting and worthwhile mathematical issues. How many times have students cried "don't show me the proof, show me how to do the homework assignment"? Lots of work is done, some skills might be learned (at least until the final exam is over and done with), but the level of true mathematical understanding achieved can be minimal. This is also a bad thing.
Obviously we need some way to move forward. Consider for a moment a typical two-semester freshman calculus sequence, where students deal with the usual collection of topics in differentiation, limits, integration, and series. What gets taught in these classes? Maybe a little real and complex analysis, some topology, formal logic, Gödel-Bernays set theory... Hold on a second, those topics aren't taught in freshman calculus! They aren't taught, but they do underlay the simpler mathematical concepts that are being taught. In other words, we treat the more formal and rigorous mathematics as black boxes, if not explicitly, then by unmentioned assumption. We use the black boxes to hide confusing complexity from students when they aren't ready for it. This allows instructors to teach high-level concepts, just like black-box subroutines allow programmers to write high-level software code. Thus, black boxes are not inherently bad things, in fact they are a necessary fact of educational life.
The idea I am putting forward is this: mathematics instruction has always contained black boxes, and they are not inherently bad things. In fact, much of the process of designing any kind of course can be viewed as deciding which black boxes to present and which to ignore the existence of. If we now accept the idea that black boxes can be good things, namely when they provide us with useful abstractions and help us to maintain a focus on the topics that we really want to teach, when do black boxes become bad things? Here are a few possibilities:
When students are denied access to the inside of a black box.
Consider the Maple routine
int, which provides
symbolic definite and indefinite integration capabilities. This can
be a convenient way of quickly integrating functions, but it is the
ultimate in an undesireable black box with respect to the objective of
teaching freshman calculus. If this were the only functionality
available, then a CAS like Maple would be of no use in teaching such a
subject. What we really need is the ability to rip the cover off of
int and let students see the pieces of it in action.
With Maple we have such options: in this case we can use the
student package to perform smaller steps in problem
solving, for example by using the
intparts routine for doing
integration by parts. For other situations instructors are exploring
the Maple worksheet interface as a mechanism for presenting the steps
in an algorithm, instead of just providing pre-written subroutines as
a fait accompli. Students modify portions of the worksheet for a
given problem and step their way through the algorithm, and are thus
more involved with the CAS-based presentation of the material.
When students don't understand how the black box works even when it is opened.
This is perhaps one of the most important things that educators and administrators must realize when considering the use of a CAS for teaching. Just because you have a piece of software, even a really good piece of software with lots of online help available, students still need a teacher to help them understand a subject. The teacher knows from experience which black boxes are important to open, how to explain what the internal contents are, and explain why those contents were put there in the first place. A teacher provides context, relevance, and detailed as-need explanations. A piece of software just does whatever it is told to do. This may seem an obvious statement, but was been a source of problems in attempts to use the technologies lumped under the label Computer Aided Education. To put it in simple terms, if you expect your students to use Maple and learn something, then you need to be sure that you are prepared to help and guide them through that process.
When students never have reason to care about the internal workings of a box containing concepts critical to a subject.
Obviously if we give students simple questions, then they will use the most convenient Maple routines to quickly generate simple answers. Instructors will have the chance to become more creative in their design of assignments, replacing simple problems:
"integrate the following function"
with more challenging ones:
"integrate the following function both by parts and by partial fractions, showing the steps used and intermediate results. Discuss which method was more effective and why. If the results aren't the same fix the mistake or explain the difference."
While it will take effort to rethink the design of assignments in this way, the potential exists for giving fewer or at least less repetitive assignments that will require students to really think about what they are doing. If students aren't expected to spend their time in mechanical algebraic manipulations, they can shift their efforts towards learning more interesting content than ever was possible before.
This is a good thing.