This perfectly illustrates my complaint with the grading / instruction methods of much math in general (not just Common Core). The psychology of the student isn’t taken into account when teaching and grading of the techniques. Ideally, the student should only have to demonstrate the process for 1 or 2 examples, and then be allowed to skip over all of the “doing it in your head” steps instead of repeating the drudgery 20 more times. This teaching pattern follows long into college. Doing it the “easy way” should be encouraged, impatience can turn into a virtue where motivation can be encouraged with allowance of the “easy way” as the reward.
The student’s frustration at enforced demonstration of the full process long after they have internalized it into quick rote in-the-head capability is detrimental. IMO, a student that still requires writing down the explicit cookbook steps to solve each problem is still missing fundamental understanding.
I was in graduate school and still getting marked down in exams because I recognized a trig substitution and substituted in my head on the fly instead of wasting precious test time writing out all of the intermediate steps.
This 3x5 thing might be an example of bad teaching . . .
. . . or, it might not be. It amazes me that people are so eager to throw the teacher or system under the bus.
Taking for granted that there are practical reasons why 3x5 and 5x3 ought to be represented differently, then I see nothing wrong with that test.
Assuming that the lessons involved teaching specific methods, then failure to show those methods should obviously result in no/partial credit. I sometimes wonder if the real problem here is that adults looking at this stuff don’t remember what was expected of them in their math classes of long ago.
It’s perfectly reasonable and common for a teacher to grade on methods. I am fairly sure that I can remember answering basic calculus questions with guesses/brute force, and getting marked off for it, because I didn’t show mastery of the subjects discussed in class.
Just because the quiz/test doesn’t lay that out explicitly doesn’t mean the students weren’t told and didn’t understand. And, just because the teacher didn’t write an essay on the paper doesn’t mean that he/she didn’t explain after the fact why the student got only partial credit on those questions.
Either the methods are wrong or the teacher is. Any method (or teacher) that teaches a child that 5x3 is different to 3x5 is inherently wrong. Just as wrong as an English teaching method that taught you don’t need a capital letter at the start of a proper noun.
Never mind; I agree with you completely. I misread your first post on the subject as saying that there was a fundamental difference, and so thought, “well, doesn’t make sense to me, but it’s just likely he/she has more math experience than I.”
So yes, if there is no functional difference between 3x5 and 5x3 (and now I feel both gullible and illiterate for being willing to buy the idea that there is a difference), then those questions should have been marked right.
In response to The Tooth: baring context, 5x3 does not mean “five groups of three” any more or less than 3x5 does.
I disagree. The array method with one digit numbers is just a completely different problem than doing it with two digit numbers. I don’t think they are transferable except in the general process of using area for multiplication.
It’s really not useful until you are at least doing 2 x 1 digits. Then you can start to see that you’re putting different rectangles together. Then, when you get to 2 x 2, you can see you’re putting together more rectangles.
And then, if you get what the rectangles represent, you can basically derive the FOIL method by yourself. 2854 = 20 * 50 + 204 + 50 * 8 + 4 * 8 = 2500 + 80 + 400 + 32 = 3012.
In fact, I’m pretty sure that’s exactly how I learned using the area method at Montessori, where I maxed out their math curriculum by fourth grade and coasted in public until seventh grade when I finally got to learn something new–pre-algebra. And be fed up with a teacher that made us do 30 problems every night for no good reason.
And that, BTW, is my problem with Common Core. It still is adhering to the crappy “tons of homework” method of teaching. It’s busy work. At home you’ll either get it quickly and not need to practice a whole lot, or you won’t get it and will be stuck.
Arithmetic is the one subject that should be online now. And streamlined to each student and how they learn. We were doing that in fifth and sixth grade in the computer lab. What happened? Why did we revert to new math but old methods of giving the kids busywork?
BTW, i also learned multiplication using the times table square. That was the last step before you were to have your times tables memorized.
At any time when you were learning that, did they also demonstrate or explain what “Fundamental Law” of arithmetic you were using? Something I’m seeing (especially where the FOIL method, which I consider an abomination BTW, is taught by rote) is that kids aren’t even taught the basic laws of arithmetic, especially the one in question here because it’s complicated :rolleyes:
Hint:
The 1-digit x 2-digit case is the Distributive Law. The 2-digit x 2-digit case is just a repeated use of the Distributive Law.
But I’m not seeing children coming out of all those rectangle drawings knowing that. And I’ve looked at a few of those tutorial YouTube videos that taught and worked through the rectangle method. If I didn’t already know what they were thinking in that method, I’d have found them incomprehensible.
Be a dear, would you, and point me to the standard requiring “tons of homework”? Here are the third grade standards. I’ve studied them pretty carefully, but admittedly I don’t have them memorized. I teach common core, but get complaints from parents (and some kids! they’re weird) that I don’t give enough homework, since I tend not to think it’s the best use of a nine-year-old’s time. Kids that master material in my class have the option to complete some advanced work in Khan Academy as an alternate homework assignment.
But if you’re right about Common Core, I’m doing it wrong. So could you cite what you’re looking at?
Apparently the teacher thinks otherwise. I think this student was expected to come up with a six by four matrix, and he came up with a four by six matrix instead. If the kid was expected to apply what he was taught when it comes to putting numbers in groups and didn’t, the low mark is deserved. That’s the only reason I can think of for marking those questions wrong. I don’t know, I wasn’t there for the lesson. Maybe the fact that there’s a* representational *difference between 5x3 and 3x5 was never mentioned in the class before. But looking at the test, that’s what I would assume was taught.
It’s not a matrix. When the problem is that a teacher is wrongly implying that multiplication of numbers is not commutative, introducing the possibility of confusion with matrix multiplication is an absurdly terrible idea.
And that is precisely the problem. Not anything to do with Common Core: one teacher who does not understand the concepts (s)he is supposed to be teaching and mistakenly thinks that there is functional difference between 3x5 and 5x3.
Common Core Standard for Grade 3 is to understand and apply properties of operations as strategies to multiply and divide. The grading of that test is teaching the exact opposite of the correct information and in direct opposition to the Common Core Standard.
What goes on is that anything bad is being blamed on Common Core. Odd and different grading system? Common Core. A bad teacher doing the job poorly? Common Core. Too much homework? Common Core. Too little homework? Common Core …
It’s pretty clear that the teacher thought the kid arrived at the right answer the wrong way, and part marks for that sort of thing is nothing new, and I wasn’t there to see how this kid was taught to multiply. But I’m not going to argue further because whatever I think, it’s probably not relevant to the pros and cons of Common Core.
It actually does. If the teacher was NOT teaching that for multiplication 5 x 3 = 3 x 5, was not teaching the correct information about order of operations, then (s)he was IGNORING the standards of Common Core and doing it very wrong. Common Core Standard is to teach, explicitly teach, that those two are identical and not different in any way. The point that the Common Core wants emphasized and clearly understood is that 5 x 3 and 3 x 5 are the same thing, both represented additively as either 5+5+5 or 3+3+3+3+3 whichever one is written first. If this teacher was teaching to Common Core (s)he would not have taught it otherwise or scored it otherwise.
Be that as it may … a lot of “change for change’s sake” happened after states adopted Common Core standards. Ask your kids’ teachers most any “why it this the way it is” question, and the answer will invariably be “Common Core”. That response may be shorthand for “Dunno exactly, we got a lot of new materials and guidance after Common Core standards were adopted”. Why didn’t teachers and administrators know that existing methods, textbooks, etc. could be readily used to teach to Common Core standards? I mean, no one anywhere around here seems to have recognized that – why not?
In any event, it is common (no pun) for perceptions to trump reality in the marketplace of ideas. At the moment, Common Core desperately needs a PR makeover.
As a parent I lived through lots of changes for changes sake before and after Common Core. School administrators seem to be prone to do that. Multiple math programs and pendulums swinging from phonics to whole language to phonics while the best teachers always tried to keep doing both in balance. Hell, I am 56 and my parents were complaining about the “new math” I was being taught (big on learning different base systems as I recall it.)
No doubt however that the execution was flawed, again with an excessive embrace of lots of standardized testing as the evaluative method. That motivated many districts to try to quickly game their processes to test performance and many have done that badly.