Quote:
Originally posted by Milz:
...A monkey, could be taught to push the buttons of a calculator and communicate it's output, but that doesn't let him understand what multiplication is. The same thing happens when you teach a student to hit the integrate key on his calculator and write down the answer in a test blank. He needs to focus on the concept of what integration is, we agree on that, but I think he also needs to do it by hand at least a few times, so that he knows where his calculator got the number.
With current technology, examinations that would only require a student to "punch and write" would certainly be poor examinations indeed. Don't you know and agree that very good examination questions can be written which measure the degree of a student's concept comprehension, and are far removed from the style you describe above? Writing good examinations is considered an art-form in itself by educators, and the current technology forces us to rethink the "whats and hows" of good examination questions and formats. Focusing on this new style of examination writing is still a bit new to the math ed community, but is highly effective when it is done properly and with good professional insight.
Also, BTW: You certainly don't believe that a calculator uses the same process to find derivatives, integrals, or even logarithms, etc. that the human being does... do you? Few people are ever really aware of exactly "how their calculator got the number". For instance, do you personally know the specific algorithms your calculator implements to calculate such things as trig function values? These are numerical algorithms, which don't generally resemble the computational steps that the human being uses to calculate - and this example actually further illustrates and reiterates my above points about concept focus.
Quote:
Originally posted by Milz:
...I suppose that depends on what the student's future career is... Many people can obviously get by without theese skills... I believe, however, that there is more to math than reaching an answer. Learning different ways to approach a problem...
These are some of the specific points that math ed reformers themselves make, but in support of the opposite point from yours. The usual goal of a college education is not merely to satisfy somebody's personal curiosity... in fact, most people in college today are there for a very different reason, viz. to get a good resume item and to hopefully land a 'good job' afterward. It's quite an understatement to say that most people can 'get by' without knowing how to, say, integrate by parts - in fact they never need to know how to do a technique like that, even if their profession requires calculus skills (which is itself quite rare).
And also, why is it so hard to conceive of technology being effectively utilized in academia as a main way to approach a problem? The technological approach is by far the most common one taken in professional practice, in fact it is even now by working mathematicians!
Quote:
Originally posted by Milz:
...My whole point is that calculators don't teach concepts, they give answers...I have seen many students learn which button to push so that they can pass the test, and then they completely ignore the concepts. How do you design a test that allows a TI89 and that will accuratly guage a student's knowledge of concepts? The only questions like that I've seen have been ones where the calculator isn't necesary anyway. Where calculators are important is when numbers don't come out nice and neat, like all real world data, but when you give a student a real world problem he can push buttons and give you a real world answer without the teacher knowing whether he really understands what's going on...
The point here is not to ask the calculator to teach. The point is utilizing the calculator as a powerful tool to aid the teaching and learning process, by providing students with better access to the important concepts themselves. Calculators don't anymore just 'give answers'. And if you haven't seen good tests that measure a students concept comprehension when a modern calculator is available, you've probably either been out of school for a while or have witnessed the fault of teachers that still need lots of work bringing their curriculum and teaching up to speed to effectively incorporate technology. It is a changing need of the times, and makes for a better and more relevant course when this is done. Good exam questions require the student to demonstrate recognition of the important concepts, and also to demonstrate their skill in properly setting up a problem prior to attacking it with technology (real world data or not). This has always been the reality in the workplace, so why shouldn't it be a reality in the classroom as well?
Quote:
Originally posted by Milz:
...Is 'Mathematics Education Reformer' a political party or something? It sounds like you're saying anyone who wants to reform mathematics has the same opinions you do. I'm certainly not a member of a conservative old guard or anything. I just don't believe a student should give a teacher an answer he obtained by pushing a key, when he doesn't know what the key really means... BTW, Trigonometric substitution may be one thing that can be skipped over. If the student understands basic trigonometric functions and algebraic substitution there's really no new concepts. Their only purpose is to make things fit cookie-cutter formulas in a book.
LOL - Sometimes it does feel alot like a political party! Neither do I think that simply "pushing a key" is an appropriate demonstration to a teacher that learning has taken place. This was never the point being made, don't you see? The point is about concept comprehension, and this can be measured effectively simply by designing examinations which focus more on the 'recognition and setup' stages of solution rather than merely on the 'final execution' stage of computation. To take any of this in a different way would be to miss the important points being made here entirely. Please don't do that!
And your point about eliminating trig substitution could equally be made about things like 'integration by parts' for instance too, as well as a host of other things in lower-division math curriculum that are totally irrelevant to the students involved and their future career experiences.
These topics, i.e. about math reform of both curriculum and pedagogy, currently form a lively and heated debate involving math educators from all levels and from all around the world. I have personally participated in professional discussions on a national level regarding these issues. A general concensus among educators is emerging, but not everyone agrees with one another of course - and it is an extremely time-consuming and tedious process to overcome old traditions and boilerplate teaching practices, even when they are no longer relevant or meaningful to the learners involved. We need to assist our college students to be better prepared for the actual realities of the society (and world) they will face outside of academia, and help them to become more effective and productive citizens without also wasting time in school on ideas which don't pertain to these important goals. Now if one is going to be a mathematician, such things as we're discussing above will in fact be much more important, and will better fit into the upper-division curriculum - although along with a healthy dose of tech as well! https://www.sharkyforums.com/images/.../2002/02/3.gif
