You mean (7 X 8) / 2…
My mind, it’s starting to go. I can feel it, Dave. I can feel it. Please stop, Dave. Please stop.
Daisy, Daisy, give me your answer true . . .
Matrices have not been de-emphasized.
High school mathematics teacher. 
Better is 70-14. 
Repacking the box was the best part. And licking the rods and sticking them together. Except then the colours all started to blur, destroying the whole point of the exercise.
We got a little bit of New Maths, but it was tempered by a lot of the Old Maths too. Looking back I’d say there was quite lot of “ticking the boxes” going on…“The Education Department says we have to teach you this stuff, so here it is. And now that we’ve got that rubbish out of the way, here’s the real proper maths”.
By “sound it out” I meant phonics style–putting the sound of the word together out of the individual phonemes roughly denoted by the letters in the spelling of the word.
With “frylock” you know how it’s pronounced because you know how “fry” and “lock” are pronounced. My bet is that you don’t know how “fry” is pronounced because you know how “f” “r” and “y” can be pronounced. Rather, (my bet is that) you know how “fry” is pronounced rather more “automatically” than that.
As an adult, a word I hadn’t seen before I’d figure out by looking at roots. I doubt individual letter pronunciations would cross my mind at all.
As a kid, I can imagine “sounding it out” might have had its uses. But I’m not certain. I’d have to see the research. Do kids really gain vocab through sounding it out? Or is vocab gained through habit gained by the repition of the act of seeing the word in context? Or some of both? This is an empirical question. I don’t remember finding phonics to be very useful when I was a kid–but I have a bad memory.
I think I know where **PoorYorick ** is coming from.
6x8, and I immediately know it’s 48. But with 7x8 and 6x9, I know they’re both either 54 and 56, but I have to think a bit to make sure I don’t get them mixed up.
I’ve tried lots of short cuts like the ones suggested in this thread, but none have really stuck.
Oh, Dear Lord, Help Us.
I suffered through the “whole language” debacle. I finally got a symposium speaker to admit that it was based on flawed reserach.
The fact is that written English is a (mostly) phonetic code to reproduce aural communication. If you abandon that, you’ve lost the thread.
Curse you, explotive educational self-promoters!!
As a student in the 70’s, is it possible that my school district never taught “New Math”? I don’t have a clue what most of you are talking about…and I got good grades in math. I remember hearing the phrase “New Math” but modulo arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra and abstract algebra? 
And to add to the confusion of “new math” we had one teacher that tried something called “chisenbopping.” Chisanbop - Wikipedia
Needless to say, it went over real well, just like trying to teach us kids the metric system. No wonder math and me do not get along to this day!
Hmm, I just tried it, doesn’t seem too bad? It’s certainly easier than my old method of counting on your fingers in base 2. Though I’m not certain why the method recommends using a table, it seemed easier to do it with your hands floating and using the usual finger closed/open way we usually count.
What was wrong with it?
Since (x-y)(x+y)=x^2-y^2 (difference of squares), I envision the product (x-y)(x+y) as an approximation for x^2 for y small. Thus, 78 would “approximate” 7.5^2, as would 69. However, 6*9 would correspond to a larger y value, so it would have a larger offset. IOW, given that the sum of two numbers is the same, the product of the ones with a higher difference is smaller.
“Frylock” was a fortuitously decomposable example, but suppose you’d never seen the word “harem” or “blimp” or “glee” or “Congo” or “zig-zag” or, hell, “fry” or “lock” before. What would you do? You’d appeal to various semi-regular orthography-sound correspondences to try to figure it out (i.e, “sounding it out”). Surely, were I to write down the word “morchivicate”, which almost no one has ever written before and which draws from few recognizable “roots”, you would still have a decent idea of how to pronounce it, and there would be considerable, if not exact, agreement with others upon this.
But, that having been said, discussion of what processes underlie reading in the experiencedly literate is somewhat besides the point. As far as reading instruction goes, the research is overwhelmingly, nearly unanimously, against “whole language” instruction, to the preference of phonics-based instruction instead. There’s some discussion and even more useful links in this Language Log post.
Aaaand my brain just faded to gray.
I vaguely remember a Schoolhouse Rock song called “Little Twelve Toes” that was about something called “base 12”, and everyone’s talking about base 2, base 10, etc, in this thread; what the hell is it?
Well, it looks different, but you can see how it’s pointing out the reasons for everything you’re doing, rather than giving you an algorithm and telling you to accept it by faith. Of course, that’s kinda how I was taught in school, so I might be biased. (Though not as much as the video; they write down all the scratchwork for the second way, and absolutely none for the first. If I cluttered up the screen for long division and made the second way look sparse, then the YouTubers would’ve come to the opposite conclusion.)
It applied to ‘Principia Mathematica und verwandter Systeme’, so any formal system that contained enough arithmetic to be worthwhile. Anyhoo, that’s not relevant either, since I think Wendell meant ‘rigorously proved’ as shown to be consequences of the axioms of ZFC, and ‘all mathematics’ not as ‘all possible statements that can be said’, but rather ‘whatever stuff André Weil happens to like’.
It’s the grouping system that numbers use (roughly speaking). Computer nerds tend to be the ones preoccupied with it nowadays, but it was a big concept in new math.
By “grouping system” I mean the digits. Our system is “base 10” or “decimal” it is in groups of 10:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
The presence of each digit to the left indicates another “group” of 10 of the previous digit (meaning 100 is a group of 10 10s, or 10[sup]2[/sup]). Similarly Base 2 (commonly called binary) is in groups of 2:
0, 1
so
0 (0 in base 10)
1 (1 in base 1)
10 (2 in base 10, but written as 10, because it’s one group of 2)
11 (3 in base 10)
You can go higher than 10 using letters (because they didn’t want to invent new symbols)
The most common is hexadecimal (again, used by computer scientists) -
1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
With A - F being what would normally be 10-15 in base 10, hence 10 in base 16 would be (you guessed it) 16, 11 would be 17 and so on.
If you’re still having trouble visualizing it, think of why we may count to 10 and not, say, 12. We have 10 fingers so we go to the next digit when we run out of, well, digits to count on. If we had 12 fingers (or toes, which I’m guessing is the example that schoolhouse rock used) we probably would roll over when we ran out of all 12 of our fingers*.
- Honest question, do people who are born with physical abnormalities like extra fingers ever have slight delays in learning their counting? It doesn’t seem like it would be too difficult either way, but I do honestly wonder.
To be honest, I have no idea what I’m doing when I see a word like “morchivicate.” I do immediately know how to pronounce it, but it’s not clear to me that this requires that I’m doing phonics. For all I know, I’m just drawing unconsciously on patterns between syllables and pronunciations established by the countless other words I do know. (Consciously, that’s what seems to me to be happening. Individual letter pronunciations don’t really “cross my mind” so to speak when reading the new word–just syllable pronunciations. But that’s consciously–I’m sure a lot more could be going on under the hood.)
As you said, what’s going on in the head of an experiencedly literate person when he reads is beside the point in the phonics/whole-word debate. I was discussing it because someone else said it’s just because of phonics that they know how to pronounce new words–by “sounding it out”. I’m not so sure I “sound it out” when I see a new word. But as I said before, maybe I did when I was a kid and was newer to this whole “reading” thing.
I also don’t think the purpose of phonics is really supposed to be just to let you know how to pronounce new words–it seems likely to me that the benefit from phonics is supposed to be that it lets you gain vocabulary when you’re learning how to read. The answer to whether Phonics is helpful with this requires empirical study, which leads us to:
Thanks, and I must remember, “always search Language Log.”
-Kris
Now, there really is a nice trick for keeping those two straight if they’re a pair that’s mixing you up. The digits in 54 add up to nine, so it’s a multiple of nine. So it can’t be 8x7. So 8x7 must be the other one.
You forgot ‘0’.
It’s important to understand why computer science people use base-16 (and, in the past, used base 8 fairly often as well): It’s a convenient way to represent bit (binary digit) patterns, which is how everything is represented when it comes to computer data.
Everything your computer does or can do can be represented as patterns of ones and zeros, which (usually) correspond to the presence or absence of electrical charge. It’s not convenient to constantly have to use numbers that look like 10010101 (base 2) all the time, so instead we switch to base 16 and write 95.
Notice that each hexadecimal (base 16) digit represents four bits (base 2 digits). This is very convenient on most modern hardware, which likes to move data around in multiples of four bits (8 bits, 16 bits, 32 bits, 64 bits): Every piece of data can be represented by some whole number of hexadecimal digits with no waste.
Octal (base 8) was used when computers were designed to move data around in multiples of three bits: 6-, 12-, 18-, and 36-bit groupings were commonly used as the fundamental building block of data. As you might expect, a single octal digit represents three binary digits with no excess. Since those computers are mostly obsolete now, octal isn’t used nearly as much as it once was.
Both hexadecimal and octal numbers could be confused with base ten numbers. Various communities have different ways to disambiguate which base a number is in. For example 0x10 is 16 (the ‘0x’ means base 16) in a lot of programming languages influenced by the common C programming language.