If that is the lesson then the method for showing the result is flawed. It could make more sense to the child to group the ones by vertical lines instead of horizontal. If you are testing the understanding of 5 groups of 3 vs. 3 groups of five then you should require the child to circle the groupings on the grid or use bunches instead of a grid. This method would remove any ambiguity and help to reinforce the lesson.
Ted Cruz explained his opposition to Common Core when he declared his candidacy at Liberty University. After promising to repeal Obamacare when he became President, he went on:
This article explains that Common Core is not a Federal program. But being wrong has never been a problem for Cruz & his fans…
The right’s sudden hatred of it is odd. My guess is it is just related to the general hatred of public ed that the right has. It is an easy (and newish) target to point to and say “hey, look how crappy public schools are! They actually teach this stuff!”. When in fact, common core was driven by policies pushed by the right (standards, NCLB, etc.).
Testing aside, the biggest complaints about the way it teaches math stem from the fact that it teaches math differently than we learned it, and people don’t like that.
The way I was taught to do 31-19 doesn’t do me any good when I’m doing math in my head. It involves putting a slash through a digit and “borrowing” and creating a whole 'nother subtraction problem on top of the one I was asked to solve.
On the other hand, when subtracting 19 from 31 in my head, I add 2 to get to 21, which is 10 less than 31. Ten plus two is 12. This is what Common Core math teaches kids-- a different way of looking at and manipulating numbers to get the answer.
This comes in handy when giving change. If a bill is $6.70, and someone gives you a $20. You’re not going to have time to whip out a piece of paper and write this out:
$20.00
-06.70
But that’s what many of us were taught in school.
Instead, you’re going to mentally add 30 cents to get you up to the nearest dollar (7), 3 to get you up to the nearest 10 (10), and 10 to get you up to the the amount you were given (20). 10+3+.30=the answer.
People are all up in a tizzy because Common Core math methods are having kids use manipulatives and use addition to solve a subtraction problem, etc. But that’s real-world shit. Slashes and borrowing aren’t. Sure, good old-fashioned memorization and times tables are still necessary, but so is finding new effective ways to teach when old ways aren’t ideal.
The student did grasp the lesson. It’s the teacher who is too stupid to see the forest for the trees, and who marked the test wrong in violation of the fact that multiplication of numbers is commutative.
I suspect it’s because math is such a tremendous source of stress and bad memories for many people who had terrible experiences learning it in school. I teach high school math, and when I tell people…competent, intelligent adults… my profession I’m used to seeing a pall of existential terror pass over their faces. I suspect if I taught English I wouldn’t get that reaction (as often).
People struggled through shitty math instruction and hate it as a result. And they think that their kids should have to struggle through shitty math instruction. Combine that with the fact that math problems are easy to take pictures of and share on Facebook.
I think far too much of Ender talking to his squadmates and telling them, “The enemy’s gate is down.” His point: forget what you think is the right way to approach the problem. As long as you solve the problem, the way you used way the right way to solve it.
This requires a couple of caveats. First, there are times you can use a blatantly incorrect method to solve a problem, through sheer coincidence. 11x 11 = 121, which I know because I put the first 1 in the hundreds place, added the next two ones together for the tens place, and put the last one in the ones place. Cool, but 12 x 12 does not equal 132.
Second, there are times you must jump through a very specific hoop. The EOG requires students to be able to analyze a number line, so I must teach them that method. I once took a typing test for a temp job, and the backspace key was disabled. It was inane, but for that test I had to slow way down because I couldn’t afford to make the errors I’d normally fix through an almost subconscious use of the backspace key.
Third, there are times you must jump through a much more general hoop. The best way to solve 343 x 596 is to use a spreadsheet or a calculator, but I might give you the option of solving it using any pencil-and-paper method that doesn’t involve another person’s aid. That’s a legit hoop.
But the enemy’s gate is down. If your method reliably gets the right answer, makes sense to you, and doesn’t take too long, it’s a good method.
As much as I hate discussing particular Internet memes, I gotta agree in this case it looks like the teacher was grading poorly. It’s the teacher’s job to interpret a test, figure out what students are thinking, and grade their work based on sound mathematical reasoning, not on whether a student is showing the work in precisely the format the teacher is expecting.
A friend of mine once taught a remedial math course for teachers, and he was dejected when I asked him about his day. “This teacher believed that 1/2 plus 1/2 equaled 2/4, and nothing I told her persuaded her otherwise,” he told me. “She said, ‘I’ve always taught it that way, and I’ll always teach it that way!’” There are dummies out there.
I’m pretty torn. I have kids going through elementary level math now and for the most part it seems fine. The only times I have issues with it is when pretty simple questions are overly complicated to force a particular method. 3x5 doesn’t need an array to solve.
I tell the kids that they need to learn the method that is being taught, and understand that in math there are many ways to solve the same problem and get the correct answer. Part of the answer they are trying to get includes the method to solve it, so think of it that way. If they know an easier way, use it to check their work.
This is a fair point. However, if you learn the array method with a problem like 3 x 5, there’s two advantages. First, it’s helpful to learn a tricky technique with a simple problem. Learning arrays now means that when you’re learning an array model for solving 28 x 54, you already know the baseline strategy. It’s analogous to learning how to draw a straight line with a rule before you learn to draw a 60 degree angle.
Second, common core includes multiplication and area in the same year. If you learn multiplication via arrays, when it comes time to learn area, it’s a very easy fit.
That said, I believe a good teacher will not force a student to draw an array every time she’s solving 3 x 5, once she shows she understands the underlying relationship between multiplication and arrays. My three criteria for a good method are that it
- Gets the right answer.
- Makes sense.
- Doesn’t take too long.
Forcing a kid to draw an array every time violates #3.
I kind of disagree. Kind of. Learning a complex technique using arrays for simple problems also teaches something else unintended. It taught my kid that the teacher is forcing them to do something complicated and made them frustrated because it’s pointless. If a teacher wants to convey the importance of a technique, they should use an example that requires the technique, not force the technique in an example where using it would be inefficient.
I distinctly remember thinking like this all through school. The sentiment can be summed up with the question, why is this useful to me? Using an array to solve 3x5 is not useful. Using an array to solve 28x54 may be useful (I still don’t get the technique actually).
Here’s what I did - recently actually. I printed a times table from 1-12 in a grid. Then I asked the kids what they saw. They saw the patterns. I told them times is just like adding, over and over. That clicked instantly. Then I told them to use the array to satisfy the homework.
For those just joining us: the thread the OP linked to was originally started in September, but starting at Post #94 of that thread is a discussion of (and link to an article about) this particular example of “very wrong and bad maths.”
I get what you’re saying, but the problem is that by the time the technique becomes important, the cognitive load to learn both the technique and its extension is so high that many kids can’t do it.
A major component of teaching is taking complex processes and breaking them down into smaller steps that can be mastered bit by bit. Another major component is teaching kids how to complete an entire process. These two components are often at odds.
The array model of multiplication is pretty cool, actually, and if you want to learn, follow this link. You might be able to understand the method just by looking at the initial picture on the embedded Youtube video: just find the area of each of the shaded arrays within the larger array. This array method is very effective at getting kids to understand why you can solve 13 x 14 by finding (10 x 10)+ (3 x 10) + (4 x 10) + (3 x 4)–which is essentially what you’re doing with the traditional algorithm. It’s also very effective at getting kids to come up with mental math solutions, such as (14 x 10) + (10 x 3) + (4 x 3), or even (13 x 13) + (1 x 13). In other words, it beautifully models the distributive property of multiplication in a way that allows kids to exploit this property for mental math.
But if you learn the array model while simultaneously trying to apply it to problems with multiple place values, again, that’s too much for a lot of kids to handle. So you break the process down: first teach the array model, then teach how it’s applied to two-digit problems.
That said, if your kid’s teacher isn’t explicitly explaining this pedagogical technique, that’s a problem. I often explain to kids that the method they’re practicing on easy problems now will become very useful when they attack more complex problems. This is sometimes a hard sell for impatient students, I know, but when we get to those harder problems and they’ve already mastered the basic skill, I remind them of my promise, and they appreciate the method to my madness.
Usually :).
(TLDR: it sounds like the teacher isn’t explaining why the teaching method isn’t “pointless,” or else your kid’s not listening to that part. That, I think, is the problem.)
I don’t fault the teacher. I expect the teacher is explaining it but the capacity to digest these things isn’t quite there yet - and given answers are easy to arrive at via simpler methods I’m sure there is a bit of tuning out.
Thanks for the link - that actually explains it pretty well, especially the interactive nature of moving the grid. I still think that it is faster in nearly every case to do multiplication the traditional way (i don’t know what it’s called, but you add a zero for each additional place for the bottom number and add the sum of all the products). And that method works for multiplication of any size, including decimals, faster than you can do with an array.
My kids will always try to do things the easiest way possible. They are impatient, as I was impatient, as many kids are impatient. I distinctly remember a math test where I could determine the answer in my head so I did so, and got marked down for not showing my work. I complained that the test instructions said “Solve…” which I did. It also said “show your work” and my complaint was, there is no work to show. I did not prevail but I was pissed off at the teacher.
I’m not really a fan of Ted Cruz. But I am even less of a fan of this kind of absolutist suggestion of yours, when there are nuances about the issue – specifically, the federal government’s Race to the Top grant program, in which states that adopted Common Core standards earned bonus points on their ability to receive a share of the $3.4 billion available.
While Cruz’s statement was not a model of clarity either, he’s not a poster here and thus the light of my sweet reason cannot shine upon him. You, however, are reachable and should understand that although Common Core is itself not a federal standard, it is incentivized by federal grant money, over which Cruz would presumably have some control, so the gravamen of his claim is not entirely without basis.
But teaching him that 5x3 is ONLY five bags of three and NOT three bags of five is WRONG. Its actually teaching something that is mathematically incorrect.
With the caveat that people should have cursory understanding of other methods, and when they’re more appropriate. In pretty much all of my classes from English to Mathematics, tests usually require you to use a certain method (or essay format or rhetorical device in English) once or twice, but also have a “freeform” section where you can solve the problem whichever way makes sense to you the most. (Usually the problems where you’re meant to use a certain method are significantly more difficult if you try to use a different one).
This is less of an issue with arithmetic or other 3rd grade math, but once you start getting into, say, linear systems having a small breadth of knowledge of the different methods and being competent at all of them even if you usually default to one can make things a lot easier.
This.
I was watching this closely, in a professional context, from the beginning. You can pinpoint the first blog posts, the first local politicians and school board, and finally the first Tea Party candidates to latch on the the “Obama Core” rhetoric. It was very deliberately moved from a bipartisan effort to a wedge issue.
Everyone feels strongly about schools, but few people know much about state education policy. So the Common Core was a chance to tell people whatever wacky scare stories you can come up with (The UN and Bill Gates are watching your kids with facial recognition technology!), and have it be taken at face value.
The student could merely have stated “By the commutative property, 5x3=3x5 = three groups of five, so suck it”
That may have been the case at the outset, but is that still relevant now? People can identify issues they have with the Common Core efforts and related curriculum independent of any anti-Obama or Tea Party or other political angles.