I have been following the discussion on fractions with great interest since I am one of those high school teachers who inherits many students who really don't understand any thing about them. In my school system our students start Algebra I as early as the sixth grade. Since I teach primarily Algebra I to ninth through twelfth graders, (and next year will teach an 11-12 class of Algebra I), most of my students are those who find math painful and have not truly mastered much beyond operations with whole numbers. I firmly believe that children must be able to manipulate fractions in order to progress beyond whole number arithmetic! Of course, it would be ideal if they also understood why the algorithms worked, but I'll settle for being able to add, subtract, multiply and divide fractions and decimals.
Now, about decimal numbers. It is very common for a student to not truly understand that a decimal is simply another way of expressing a fraction. To many of them, decimals and fractions are totally different. They like decimals because any claculator will work with them. However, with the exception of usual deciamals like ".5, .25, etc." they have no true understanding of the actual value of the number. (An interesting experiment for the classroom is to draw two circles on the board. Ask one student to color 7/19 of the circle. Ask another student to color .833333333... of his circle. Even though 7/19 is a strange amount, that student almost invarialbly will feel confident. The other student will usually just guess.) It is essential for a student to understand operations with fractions in order to work abstractly, as required in the study of algebra and beyond. Please, if you teach elementary and middle grades, teach fractions. And, tell your students the truth. Not everything they learn in school has a direct relation to "real life" but everything you learn, period, provides a foundations for future learning and success. Look at all those people who study the violin and never play for money. Are they sorry they learned to play?
Now, for my question about technology. This year I fortunate to be the "storage" area for our classroom set of TI-82 calculators. After teaching equation solving every possible way I could think of, I still had a large (50%?) number of students who were not proficient. Since some of them were on their third trip through Algebra I, I didn't think that further drill was going to help. So, I passed out the TI-82's and taught them to solve by graphing and using the 2nd calc- intersect feature. (I never showed them the solve feature.) The results were amazing! Those who had never mastered this but still wanted to were excited and interested. They were quite willing to sketch the graphs, worry with the window, and do anything else required to find the solution. We even solved higher degree equations and they understood why x^2 = 9 has two solutions and sqrt x = 9 has only one! So, is it acceptable for students to move on even if they are technology-dependent? Not all of the students were successful, but all of those willing to put in the time were. I really do want to hear comments on this, since there is an old-fashioned side to me that says if you canj't do it without a calculator, then you can't go on. But... (An interesting aside, even though the use of the calculators was sanctioned, I find that I will not be the "storage" area next year.)
Sorry this was so long and appologies for all typos!