Today in precalculus we went over / reviewed / saw again / learned Long Division of Polynomials and Synthetic Division of Polynomials. I told the kids that I had mixed feelings about teaching the L.D. because I thought it was on its way out due to graphing calculators but I still thought it was worthwhile because it was cool and it allowed them to practice their algebra skills and solidify their knowledge of factoring.
Anyway, I mentioned that I had a friend that was about 10 years older than me (so she probably graduated high school in early 1970s), and she mentioned that she had to learn how to calculate square roots by hand, and these days we don't teach that anymore.
Another student asked, "didn't you all have to look up the sine and cosine and stuff in tables?" And that got me thinking. Yes we looked them up in tables every time we did trig, and so that visual memory was there, and we had constant reminders that you took sines of angles and the result was a number and all those numbers seemed to be between -1 and 1. (now I don't remember if it was between 0 and 1, and we had to think or not).
I seem to have too many instances now where in a situation, students don't intuitively know in sin x = y which is the angle and which is the number/ratio of sides. It seems like maybe we've/they've lost something from not having that table to help.
I agree. I was just saying that I miss the days when we used hte "arcsin(x)" notation. I think that little inverse button causes more confusion... does it mean reciprocal? Inverse?
ReplyDeleteAh... for the good old days.
I hate when I mis-type the really big words. Like "the".
ReplyDeleteNo fooling. Some (many?) students would really benefit from the comeback of arcsin x notation.
ReplyDeleteMs. Cookie
I think I still have a CRC book with busted spine hiding somewhere. I used to browse the thing...
ReplyDeleteJonathan
I have my dad's somewhere. I used it too. :)
ReplyDeleteYou may find it odd, but I lean towards teaching polynomial long division, but skipping synthetic division, because:
ReplyDeleteHalf of what you use synthetic division for is evaluating the function (to find the zeros), which is the thing which I find is closest to becoming obsolete with graphing calculators.
I find long division easier to remember (I have been taught two very similar, but slightly different versions of synthetic division, and I tend to get them mixed up.
Sometimes you need long division in a problem that can't be done with synthetic division (especially in Calc II), whereas the converse never happens.
And arcsin? definitely better!
I do personally like the long division process, and I agree with you about forgetting the synthetic division process (add down or subtract down??) every year until the next year I teach it.
ReplyDeleteI will probably still bite the bullet and teach it even though I have my doubts about its "real" purpose. I really REALLY like teaching anything, though, that forces my students to practice their algebra and manipulation skills, which seem to need to be consistently reinforced.
Ms. Cookie
Hi,
ReplyDeleteI'm not a teacher but a student... but this seems like a great place to go to get the attention of math teachers...
I am doing basic mathematics at university but it is very similar to high school level.
I have a question and was just wondering if any of you whizzes could give me a logical easy to understand answer:)
The question is as follows:
Choose any four consecutive even numbers. (For example 2,4,6,8). Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. What is the result each time?(it is always 8 using this method) Show algebraically that this is true for all sets of four consecutive even numbers.
I didn't realize it needed a comeback -- I use arcsin et. al. in my class partly to align with the Precalculus and Calculus books we have (which both use it) but also because it's less confusing when asking "what's the inverse of sine?" and having them say "the arcsine" rather than "the inverse sine".
ReplyDeleteThey seem to get just fine that the sin -1 thing on the calculator is equivalent.
Sapphire, since you want an algebraic proof, let x represent the first number. Now the next number is two higher than that, so it would be x+2. You can figure out what the other two numbers would be in the same way. Then, multiply the middle 2 algebraic expressions. And multiply the first and last algebraic expressions. And compare your answers. You will see why the answer has to be 8 when you do that.
ReplyDeleteAs far as tables go... I wonder if having students create a large sine, cosine, and tangent table in a spreadsheet could have the same affect of numerous repetitious problems in much less time. I think with the write prompts, students could reflect on the process and say "Oh... I always get answers from -1 to 1" or whatever else you wanted to elicit.
ReplyDeleteI believe firmly in the discipline of repetition, for the record, but at the same time, a little math history knowledge (of the use of tables) and the creation of one some, could have the same affect. I could even see some students preferring the table... at least for the conceptual understanding it brings.
That's an interesting idea. Then they could also scan their fingers down the columns and see that the numbers increase to one, then decrease to zero and less down to -1, etc.
ReplyDeleteMs. Cookie