Totally. Memes about Common Core are, in my experience, too stupid to address directly, full of fail on the part of most folks involved. My comments are general, not specific.
That said, I want to comment on the idea that 5 x 3 = 3 x 5 :).
I do teach students to say that 5 x 3 is five groups of three, and that if they want to talk about three groups of five, they should say 3 x 5. But I do it in a very bounded, specific way: when students are first learning what multiplication is, it’s far easier for them to have one specific way to understand it, and understanding that a x b means a groups with b in each group is much easier than explaining that it can mean a groups with b in each group, or b groups with a in each group.
If a student reverses the equation in discussion (Six dogs have 4 x 6 legs), I point out that we normally describe it the other way–but I also point out that you get the same answer using the equation they gave, and that if it’s easier for them to solve it that way, it’s fine. I would never in a bajillion years count a student wrong for representing the legs on six dogs as 4 x 6. That drives me bonkers. Instead, the reversed equation is just a chance to introduce the commutative property.
But rote memorization isn’t the only way to do that. I’ve memorized thousands of things in my life- without making any special effort to do so. Sell enough Big Mac meals and you’ll memorize the price with tax, without even trying. Multiply 2x5 enough times using rows of dots, and you’ll memorize that as well. Memorization doesn’t have to involve the sort of effort that leads me to mentally sing part of the alphabet song every time I catch up on my filing.
And in my statistics class, over 30 years ago, the professor was more concerned about teaching us to understand which formula to use when. We weren’t expected to memorize them and were free to bring our textbooks to the tests- because as she said , no one did these calculations themselves anymore, not even with a calculator.
Statistics, of course, is largely a practical applications class with lots of heavy-duty arithmetic involved for doing the actual problems. In that way, it’s somewhat similar to a chemistry or physics class, with lots of arithmetic, calculators allowed.
Yes, my stat teacher also didn’t require us to memorize a lot of formulas. The most basic ones we had to know (formula for average, standard deviation and a few like that). But from the second test on through the final, we were allowed to bring our own cheat sheets. Part of the challenge for students was making those cheat sheets and deciding what to put on them.
My experience is that if you don’t give students the time and opportunity to practice fluency with these facts, they don’t learn them. Indeed it’s a recurring complaint among third grade teachers that students come to us not knowing addition/subtraction facts fluently, and fourth grade teachers make the same complaint about multiplication facts.
That said, this practice should not be the bulk of mathematical practice. This is what I have students do after they turn in homework and before morning announcements come on. During our math time, we focus on concepts: what’s the relationship between addition and subtraction, and how can I represent this difference using a number line or decomposed numbers? Are squares rectangles, and why? How can I program this robot to roll around the perimeter of this shape? The automaticity work is a separate job, just as I teach phonics separately from teaching reading and writing.
bordelond, you said that the proper way to solve “10 - 6” was to just instantly recognize the answer as “4”. But if the method is “just instantly recognize the answer as 4”, then what about the problem 23 - 17? OK, you might say that there the correct approach is instead “just instantly recognize the answer as 6”. But why? Suppose you had a kid who did instantly recognize 23 - 17 as 4… How would you demonstrate to him that he’s wrong? Is it just “Because I’m the teacher and I say so”? If that’s your approach, then you’re going to handicap him once he starts taking math and science classes far more than he ever would be by mere slow calculation speed, because in math and science classes, “because the teacher says so” never works.
Mind you, rote memorization has a place, and I’m not denying that. But its place is after students know what addition (or subtraction or multiplication or whatever) is, and that’s not something that can be learned by rote (but can be learned via number lines or counting fingers or arrays of dots).
CONCRETE: Students learn what a mathematical process is by playing around with it. They discover that if you have ten blocks and cover six of them up and count the uncovered blocks, there are always four.
SEMI-ABSTRACT: They learn different ways to show that on paper. Draw ten lines, cross six of them out, you’ve got four lines left.
ABSTRACT: Students learn to represent this process with traditional mathematical symbols. They learn to write 10-6=4. They may still count on their fingers, or even just count backwards, at this stage.
FLUENCY: Given enough exposure to this fact in isolation (that is, they’re not trying to find its answer while also trying to remember how many crates of oranges Ms. Smith bought at the market), they remember, “Oh yeah, 10-6=4!”
If you start by saying, “Trust me kid, 10-6=4,” a lot of kids are going to be lost. Start with convincing them it’s true, and THEN get them to memorize it.
What do you mean by addition and subtraction “facts”?
Whatever they are, 60 per class sounds like a pretty heavy plateful, as in thousands over the course of two months, and whatever they are I wonder how they can be ?solved ?assimilated without heavy expenditure of brainpower. Not that there is anything wrong with exercising brain power, of course.
Addition problems with sums, and subtraction problems with minuends, up to 18.
It’s not that high at all: when you’re fluent in the facts, a set of sixty problems takes about a minute to solve. Even kids who are slow can do all of them in about ten minutes.
At first when you do them, you’re expending brain power to solve them. But eventually they become sight problems: you as an adult can probably recall the answer to 8 + 5 without having to count up from 8. That’s the goal. When students can recall them as easily as they can recall the number of eyes a cat has, it takes very little brain power, and they can dedicate their brain power on problems like 432 + 598 to more complex mathematical questions.
What a great example, btw, to illustrate how so-called “common core” methods that people get mad about can be more efficient and intuitive than the traditional algorithmic method.
Once one has noticed the second number is just two less than 600, this problem very nearly almost becomes a mere sight problem!
A “fact” is not the same as a “problem”. Why not stick with “problem”? Too threatening, or something?
I got as far as college algebra 101 (grade: B), I majored in English Lit, I have been a lifelong reader, and I had never heard of the word “minuend” until today. I see it means the number from which another number is subtracted, IOW in A-B A is the minuend (and B is the “subtrahend”- why did you leave it out of the explanation). I guess no harm in knowing those words, but the equivalent of “sum” in a subtraction problem is “difference”, isn’t it?
One problem a second??? That is not a serious reply, no matter how easy the problems are, if there are 60 of them. Sight problem or not I would want to slow down to significantly less than one solution per second just to avoid careless error. And when are the students’ answers going to get reviewed by the teacher or an aide to make sure all are ready to move on to 432+598?
That’s all very good to teach students, at some point, to be able to see shortcuts like that in arithmetic problems. I do that myself a lot. I remember being taught some of those kinds of things in sixth grade.
But tricks like that are entirely ad hoc, only pertaining to any particular problem where they just happen to pertain. You can’t have any systematic mechanism for doing arithmetic that is based on quick tricks like that. You still need to teach standard arithmetic practices – and teach them first, I think – with or without (preferably with, I agree) these kinds of embellishments.
The arguments I see about Common Core mostly seem to be that they rely too heavily on tricks and visual aids like rectangles as “crutches” instead of just, you know, teaching how to do arithmetic. It just lends ammunition to this sort of complaint when students are required to solve problems in tricky or strange ways that their parents never learned, and worse still, aren’t learning the basic arithmetic techniques, and even getting their answers marked wrong when they don’t follow some teacher’s pet methods.
So you don’t teach the students the tricks themselves. You teach them to be able to find the tricks, so that no matter what problem they encounter, they can always find the right trick to solve it.
:dubious: Did you not notice the next part where I used the word “problem”? Because you quoted it. If you really want to delve into why “fact” is a useful term in this context, I suppose we can, but that’s kind of far afield, and I get the impression you’re more interested in snarking at me than in an honest answer.
I left out “subtrahend” because it wasn’t relevant to my comment. Yes, “difference” is similar to “sum” in some ways, but in a subtraction problem, if you’re capping one value, the minuend is what you cap. Capping the difference at 18 can still allow questions such as 19,384,239 - 19,384,211, which isn’t something I want students to acquire as a sight fact :). Capping the minuend (and, of course, working with natural numbers) limits you to the facts analogous to 0+0 through 9+9 – in other words, the subtraction facts you’d want kids to hold in their memory.
Just to make sure that I’m not talking out my ass, I tested myself. You can too. I completed 40 problems in 32 seconds with zero errors; that counts the time it took me to find and click the “finished” button.
Yes, one fact per second is an ambitious goal (remember, I got one per 0.8 seconds). I’m okay if a student needs two seconds to get a fact. But the idea is that they should be retrieving answers from memory, not solving the problem each time. If you see 8 + 7, you shouldn’t have to say, “I know 7 + 7 is 14, so add one more and you’re at 15.” You should know it automatically.
This is what I do. I’m not talking out my butt here. You certainly might disagree with me–there are people on both sides of the issue who do so, thinking that I put too much or too little emphasis on memorization–but I don’t get the impression that you’ve gathered many facts before deciding you disagree with me. I gently suggest treating this as an opportunity to educate yourself about the fascinating topic of the acquisition of mathematical skills by children.
You have lost your own train of thought, and I am not going to lead you back through it.
I would like nothing more than an honest answer as to why “fact” should be given a brand new 21st century meaning as a synonym for “problem”. I suspect your retreat behind accusations of impending snark only means you do not have an honest answer which puts your profession’s use of such jargon in a favorable light.
Such seemingly little things are the opposite of “far afield” because they needlessly mystify and subsequently do much to embitter students, parents and the general public. One thing the teaching profession absolutely must do if it is to avoid antagonizing its constituencies is embrace the familiar unless there is a damn good reason not to.
Let me try again here:
Sum = the solution of an addition problem
Difference = the solution of a subtraction problem
Product = the solution of a multiplication problem
Quotient = the solution of a division problem
If you don’t cap the subtrahend then the possible solutions will include negative numbers.
Took me 1:32. Maybe cause I type 20-40wpm, but also because it was drummed into me, as it should be to all students, not to rush.
You said before, without disapproving (in reply #128), that some students took 10 minutes. Is that not OK after all? I do not see how justice can be done to all students where such a large gap exists in their attainment levels.
And I notice the word “fact” has reappeared in new guise as a synonym for the word “answer”, and maybe the word ”solution” too.
And I suggest, none to gently, that you educate yourself about US students’ national and international performance level trends:
I learned the words like “minuend” and “subtrahend”, “multiplier” and “multiplicand” and “addend”, and “divisor” and “dividend” (But I never learned “augend” until I saw it, years later, in an even older book.) AFAIK, children these days aren’t even learning the vocabulary of arithmetic. How can they even think about it if they don’t have the words for it? (This was Orwell’s argument for Newspeak.)
And I learned – memorized! – addition and multiplication “facts”. That’s what we called the addition tables and the times tables. (The ones I had to learn, circa 1960, went from 0 to 10. We didn’t have to learn the 11’s and 12’s. But in sixth grade, our teacher showed us the trick for multiplying numbers by 11 quickly.)
The concept of “the addition facts” and “the multiplication facts” is so commonplace, I have assumed it stemmed from my parents’ or perhaps even their parents’ generation or further. I was certainly taught them under the term “facts” in the 80s when I was in elementary school.
I think it may be you who is unfamiliar with what is… familiar…
In the original, ten of the words are replaced with symbols. It was fun.
My third graders have a section of their math journal where they record vocabulary, for pretty much the reason you suggest. I don’t use minuend and subtrahend with them, to be fair–“difference” and “sum” suffice for third grade. But I hope they learn these terms later in their academic careers, since they’re useful words to know.
By that logic you could mark them wrong even if they did write “3+3+3+3+3”. If a teacher does not know whether a student that showed the correct reasoning and reached the correct answer really knew what they were doing or only did so by accident, they need to give the student the benefit of the doubt or the entire concept of testing collapses.
I’m kind of surprised people use minuend and subtrahend, all the math people I know (and I’m talking grad students and professors in math-heavy fields) just say “the part on the [left | right]” (with maybe “of the minus sign” if context doesn’t make it obvious). I think the only time I’ve ever used minuend and subtrahend is when I was fishing for unique names for the arguments of a subtraction function in a program I was writing.