That seems subjective. I don’t think they are easier.
The first step is listed as 12 -3 , but it should of course be 12 +3 . Whether that was changed on purpose or accidentally, is unknown.
It is an example of how someone might use their knowledge of the multiples of 5 to use addition to step through the problem.
Any normal person might try to do it that way.
They might think, well what do I need to add to 12 ? . If they forget was 20 - 12 is, then they might just see 12+3 = 15. So add 3. then the next step, add 5, and and remember that 3+5 is 8… so then add a few 10’s …
Its actually how lots of people might solve the problem, rather than doing it as direct subtraction.
But of course, the smarter way is to get close with the 10’s column (hey, add 20 !) and THEN look to see a smart way to solve the ones column… thats 0.
There’s nothing wrong with getting kids to practice doing maths this way, because its just an excercise in using the numbers and maintaining accuracy in mathematics… thinking of multiple things at once.
While it might appear to be teachig how to do what is obviously to most a simple question a long way, the idea is that you might do it for "76 - 29 " ? 29 is one short of 30… add 40 to get to 70, add 6… so 1 + 40 + 6 = 47.
So the picture is not wrong, its a valid way for any intelligent person to do subtraction. Apart from the error with the + being written as a - at the first step.
The current trend is for teachers to advise to get toward 10’s and 5’s. But you can do it as “toward 10” only.
toward 5’s is the first step because its easier to mentally count a group , and subsets of, five. (eg 2 +3 = 5.)
But one would think that rote learning could easily get one to know all the sums to get to 10…
2+3, 3+7, 4+6… Why bother with 5’s ? Thats all the argument is about… whether to rote learn the "add to make 10’s , or do it by counting toward 5’s.
And then whether to rote learn the times tables, or just pick it up by usage.
And you think that is easier than just “Subtract the ones column, then subtract the tens column”?
When giving a problem to a student in a math class, the goal of the teacher is not to get the answer. The teacher already knows that 32-12 is 20, and they have no need of using the students as an efficient calculating device. The goal is to teach the students a variety of methods and concepts that they may reapply later when they are needed. Therefore, your method being easier isn’t important in the moment. They’ll be taught your way too.
For a metaphor, imagine if the only roads in the US were the interstates.
See, I think the goal is to teach the easiest method to subtract numbers.
For a metaphor, I use Google maps to find the easiest way to a destination, not to explore all the alternate routes.
No, because the box is nowhere near the -. What’s missing is another - in front of the 3: 12 - [-3]=15 (brackets represent the box)
“Easy” is relative. For this particular problem, yes, I think the usual method is easiest. But once you get to stuff like 35-17, all bets are off for me. I like doing it the way I was taught, which was with borrowing tens and all that, but the “counting up” method must be easier for the majority of people, given how almost everybody who works in retail and handles money seems to “count up” when giving change. Honestly, I never understood it myself, since when I was handling money, I would just subtract the “normal” way and not bother with the counting up stuff, but a lot of people seemed to find that easier. That’s perfectly fine with me, as long as they get to the right answer.
New Math is a bit older than that. Tom Lehrer was satirizing it in 1964, and Peanuts did a strip on it in 1965.
So, if you were born after 1960 or so it’s likely what you saw in school, so the age border is more like 57 or so.
I haven’t seen anybody in retail not use what the register says to count change. Have you seen many people using their brain to calculate change?
That’s kinda what I was thinking in the OP, the complements method. Or the way my old Commodore 64 treated 256 as -254 in some instances.
Born 1961, about all I remember is base 8 and some Venn diagrams.
Yes.
Define “easiest”. It may be easiest for you but what if someone else finds another method easier? For a metaphor, I use Google Maps to find a route to a destination, then I use my own knowledge and experience to modify it so that it suits me.
Personally, I would almost always subtract using the old fashioned method, but I recognise that other methods are valid and may be easier to “grok” for some kids.
No, in the original the box is obscuring the +.
My wife is an elementary school teacher, and she has learned from experience what people in this thread have been saying: What seems obviously the easiest method for you is not always the easiest method for every kid.
ETA: How do you think Google maps works? I used to work on mapping software very similar to Google maps, and I can tell you it did explore alternate routes.
Next we’ll be arguing what method to teach the kids to use for extracting square and cube roots…
(Cf. Feynman vs The Abacus: )
To illustrate the way that different people solve problems; When I looked at that, I immediately saw that 35 is twice 17 plus one, so the answer must be 18.
When I was at school (in ye olden days) we learned tables (up to 12 times) by rote and I still remember them. We also had to learn the squares up to 12 squared and memorise the prime numbers up to 100. If you are wondering why “12”; it was because at that time there were 12 pennies to a shilling and 20 shillings to a Pound.
Forty years later, my daughter never learned tables. She calculated them in her head by simple addition - not quite as fast as remembering, but nearly. Of course by that time we had 100 new pennies to the pound.
Fair enough I guess. Hard to answer MY kids though when they ask me “Why do we have to do this stupid way of subtracting?”
Some form of magic?
This way is really good for some problems, such as what is 111-98. And learning both this and subtracting in columns with borrowing increases the number of kids who walk out of class understanding how and why 111-98 is 13, which is important.
At every stage of the way you run the risk of some kids walking out with only the ability to do the ascribed math problems with a rote learned algorithm, and those kids are on a path to innumeracy.