i was referring to the guy who wrote the comment in the picture, i understand you have a different complaint.
i suppose it is not enough to simply know how to use an algorithm, and the 2nd grade is as good a time as any to challenge the assumptions behind them. what else would they spend the time on in 2nd grade maths anyway? i am assuming the standard algorithms are used as a matter of course with questions like the Facebook post used as open-ended, non-standard questions as a matter of exercise.
The two methods are manifestly equivalent; it’s just a matter of whether you want to subtract 1 from the top digit or add 1 to the bottom digit.
On the other hand, insisting that addition and subtraction be done with reference to the number line or by crossing out circles, or performing mutliplication by assembling grids of n by m blocks (I think ‘manipulatives’ was the buzzword the last time I checked this sort of thing) is a significantly different trend in math education. I’ve never understood why people think that abstraction is inherently scary or needs to be reduced to something concrete and tangible in order to be comprehensible. Abstraction is what math <i>is</i>. It’s fine to think of addition as, say, tying ropes of length n and m together to form a rope of length n + m when one is just learning what addition is, but that’s missing the point entirely and makes for an awful algorithm to actually compute anything.
Right. If you’re going to teach something, teach it fully and properly. I don’t care whether students are taught using either method in Tom Lehrer’s song above; they’re simple and equivalent algortihms. I do care whether children are taught subtraction by that method or by drawing a bunch of circles, crossing out some, and counting the number left. It’s just subtraction; you don’t need to waste so much time on intermediate steps that will ultimately be made unnnecessary and obsolete by the full method you’re planning to teach anyway.
Why is that perfectly acceptable? The article suggests the US is average, and average is perfectly fine: education doesn’t necessarily correlate with things we care about anyway (we care more about a strong economy, and who knows what standardized tests actually measure?), and we should be happy other nations are doing well and try to work with them rather than compete with them. For that matter, what are the educational objectives of the United States, and how do they differ from other countries’?
It is unnecessary and obsolete if all you care about is teaching kids to apply an algorithm. If you want a human calculator. That’s not math though.
Understanding what subtraction IS, is much more useful in life than memorising how to subtract. I don’t see what all this fuss is about. I’d rather they teach what subtraction IS, and then teach how to subtract, rather than just teach how to subtract and trust that the kid somehow “gets” what subtraction is later.
I wish I had been taught more of the theory behind math.
I am a big picture person, and not a great details person, and likely have a touch of dyscalculia. So when I would just run the algorithm by rote, I’d often make small errors. I’d get the wrong answer, have no concept of why the algorithm worked and how it fit in to the large picture, and end up wrong and frustrated. Math always felt like a bunch of arbitrary steps I needed to do in order to get the right arbitrary number.
The point of manipulative a is not to get rid of abstraction. It’s to show how that abstraction works. If I had understood why you carry the one (other than that the teacher said you do) that would have helped me.
I suppose “trending steadily downward” could be interpreted in light of a comparison to other countries, but it seems to me the more likely measure would be in relation to what they previously were, as measured without regard to other countries’ scores.
That was my intended meaning. I will admit that I don’t have statistics at hand to demonstrate a downward trend — just an intuition that average competency in math was higher when I was a child than it is today. (There is no question in my mind that this is true with regard to English language and writing skills, so this makes a similar trend in math more likely.)
I will concede that, as the article points out, the way these skills are measured matters. As noted, perhaps more is being demanded of a 4th grader than was in my day (though the article also states that this is not necessarily a good or productive thing).
My overall point was to question whether all of the newer and supposedly “superior” ways of teaching math have left graduating high school seniors (all of them, not just those with a career that will depend on advanced math skills) in a better place than they were a couple of generations ago, when the “inferior” methods were used.
Before the term “minus” was introduced to us, we used to say “9 take away 4 = ?”
I’m curious to know what you think an expanded definition of subtraction might be that goes beyond this and would enable children to more fully grasp what it “is.”
Put 9 beans on the table, take away 4 and count the number remaining.
You can’t do that with 427 take away 316. You have to use an algorithm, which stacks the numbers, has you do specific instances of “take aways” and carries numbers back to the top. It’s very different, if you don’t internalize the idea that you’re subtracting 10’s and 100’s it’s just a black box.
For everyone knocking the “number line shit” as being useless crap, remember how folks used to make change before computers did all the thinking for us. You have a bill for $3.16 and need to make change for $5.25. It’s number line stuff.
$3.16, you gave me $5.25
Pennies - 3.17, 3.18, 3.19, 3.20
Nickel - 3.25
Dollar - 4.25, 5.25
Here’s your change. Number line arithmetic, fast and useful.
So your contention is that, while as second graders we could grasp “9 take away 4,” once the numbers got larger we were somehow unaware that it was the same concept with just…well…larger numbers?
The poster I was responding to wanted to talk about what subtraction “is.” I’m saying that back in the day, we got that just fine. I’m extremely doubtful that, if the question had been posed in 1967 to both me and one of my younger brothers (who came up learning “new math”), he would have demonstrated a firmer grasp of the concept and articulated it better than I.
I don’t know if we’re talking about my example or not but a few points I should make about mine:
1)It’s kinda moot since I thought mine was ‘common core math’ which I guess it’s not so it’s sort of out of the realm of this thread which is now way off topic. I just glanced at the title, something about PETA?
My issue with the way my daughter is being taught math isn’t that she had to draw a few circles, then take some away, it’s that she did that all last year and now this year they’re still doing it. If last year she was doing 8-5 by drawing pictures, then this year she should be doing 28-15 the ‘normal’ way. I mean, she can add and subtract ‘easy’ numbers in her head.
2b) It’s not that they’re drawing some circles and rearranging them for arithmetic, it’s the overly complex way they’re doing it that, IMHO, isn’t conducive, to learning basic math skills. As a math major I fully believe in learning math theory, but some of this is just making life harder, taking them longer to get to where they need to be and leaving the struggling kids even more confused.
And this is in fact one of the problems that Common Core was created to address. Without the standards, individual teachers (or principals, or districts, or states) decide what to teach when, and if different teachers (etc.) make different decisions, then a student going from one to the next might be presented with the same material multiple times, or never be presented with some material at all. By contrast, if everyone is working to the same standards, then it doesn’t matter what teacher you have when: You’ll be doing the same things as other third graders in third grade, and the same thing as other fourth graders in fourth grade, and the things you learn in fourth grade will build and progress from what you learned in third grade in a natural way.
Subtraction is used to:
-Break a large quantity into two smaller quantities (9 take away 4).
-Find the missing value in an addition problem (Johnny had 4 rhinos, now he has 9, how many more did he gain?)
-Determine the distance to a goal (Johnny collects dead squirrels. He has four, but he wants to have 9, how many more does he need to collect?)
-Compare two quantities (Johnny has four dead squirrels, his best friend Sue has 9, how many more does Sue have than Johnny?)
Expanded enough for you?
The algorithm is awesome for solving a problem when you know the problem. When you don’t? Not so much. And you would not BELIEVE the things that are intuitive to you that aren’t intuitive to a kid–especially to a kid whose parents don’t expose him to math regularly. All the stuff I describe above with subtraction is totally non-intuitive to a lot of my students.
Edit: here, by the way, is the second grade common core standard dealing with addition and subtraction:
Well, perhaps there was a moment in my grade school career when these concepts weren’t “intuitive” to me, either. And yet, once my fellow students and I absorbed the “old,” “outmoded” methods of arithmetic instruction, I can’t imagine any of us having a problem with any of the four scenarios you outline.
For example, we very quickly got that there was a relationship between “9 - 4 = 5” and 5 + 4 = 9. Seems to me that covers all four of them.
It was long ago, but I’m not sure how regularly my mom “exposed” me to math regularly. Perhaps she did, but it seems my elementary school teachers did a pretty good job of it too.
Actually, I retain to this day a very fond memory of my second grade teacher running tic-tac-toe and bingo games in the class, using arithmetic problems as their basis. We got tic-tac-toe grids with numbers in the squares, or individual tiles with numbers on them. She would write an equation on the board, and if we had a square or a number tile with the correct answer, we would cover it/turn it over. The first to complete a row or turn over all of his/her squares would shout out, and after verifying that all was correct, the teacher would permit the winner to get a piece of candy from a bowl on the windowsill.
I reckon this would strike horror into the hearts of educators on several levels today…but it worked for us!
I think this is likely untrue as well, based on all the usual suspects of faulty evidence-gathering, such as selective memory, limited experience, reliance on anecdotes, confirmation bias, etc.
The fact is that 50 years ago, a lot smaller proportion of the population was educated in language, and for the most part they didn’t need to be. Now we’re in a world in which language skills are necessary for a much larger proportion of jobs, so that we are more likely to run into weaknesses in that area. Also, it’s likely that when you were younger, you were simply less likely to encounter weak language skills.
Contemporary studies that I’ve heard about show that the current generation of American youth is the most literate in history and quite skilled in language use. It’s just that sitting in our positions, we don’t have a good point of view to understand the broad trend.
“Any of us?” Really? You don’t remember that one dull kid? And you don’t know any adults today who say, “I’m no good at math”?
That may seem nitpicky, but that’s actually the point. Mediocre math instruction will work just fine for folks like you and me for whom math comes naturally–indeed as a teacher my quick ability to pick up new math concepts works against me, as I tend to assume kids will realize things that they won’t realize. But the old style of math instruction left a lot of kids behind, kids who grew up into adults who say they’re no good at math. Part of the idea behind teaching math conceptually rather than algorithmically is that the kids who struggle in math will get what they need.
I’m not convinced it works, mind you, but I’m also far from convinced that old-style teaching worked for that many kids, either.
You’re right that I shouldn’t generalize my experience and imagine it applies to everyone.
On the other hand, I came up in a time when kids were not routinely passed along to the next grade regardless of performance. I’m assuming that even the “dull” kids had to demonstrate at least a basic competence in math (we really should be calling this arithmetic to be accurate), or they would be “flunked,” as we used to politically incorrectly say back then.
I guess our difference of opinion comes in how “mediocre” math instruction was back in the day. I can only say that it worked fine for me, and most of the kids I knew. And my additional point was to question how superior subsequent methods have proven to be. As I noted from my own experience, I saw no evidence that my younger brothers by virtue of their “new math” curriculum had skills or a general understanding of math any greater than my own (and math was never my favorite subject by any means…I was more of an English guy).
My understanding that the specific new math regimen my brothers came up was deemed a failure and abandoned after a few years. Perhaps something superior to both it and my more standard instruction took its place?
To me this is the question. What evidence has been amassed demonstrating that, on the whole, the “non-algorithmic” approach has proven to turn out students with better math skills?
I’m not sure that you’re right–do you have some evidence to back up this claim? As in, stats on how many kids in your day were retained/flunked/whatever, compared to today?
And do you have any evidence that these retentions did any educational good? Be aware that the efficacy of retentions is a hotly-disputed topic in education; I’ve looked hard for answers, and to the best of my knowledge, there’s no gold-standard research that definitively settles the question.
Math instruction was wonderful back in the day. It was also terrible. I’m absolutely convinced that a good teacher using a bad curriculum will far surpass a bad teacher using a good curriculum. An individual’s experiences as a child are nearly worthless when trying to determine whether a curriculum is overall a good one.
Start here, focusing on the section about conceptual understanding.
