Why is it nuts?
And how would you teach it, if your goal was to help her understand what it means to divide by a fraction?
Me too (except Applied Math instead of CS).
A lot of these complaints come from the fact that as adults, we understand basic arithmetic so completely that we’ve forgotten that it’s not obvious and someone who doesn’t understand division conceptually could benefit from some time drawing the pictures explicitly to get a handle on it. It’s like marathon runners criticizing aqua-aerobics.
These kinds of exercises were common before Common Core, too, but no instead of saying, “this is stupid!” people say “Common Core is stupid!”
I was never taught to “understand what it means to divide by a fraction”. I was taught a simple thing - dividing by a fraction is the same as multiplying by the inverse of a fraction. The lack of “understanding what it means to divide by a fraction” has not hampered me in either my academic studies or professional career.
Mean is not defined for infinite sets. What is the mean if you had infinitely many 4s and infinitely many 12s? And don’t tell me it is 8. It’s going to depend on the order in which they are presented.
The first seven questions were based on common notions and quite easy. The 8th is based on a word I had never heard before, not in that context (the range of a function is the set of values it takes on). Still, I was able to guess it.
Wouldn’t you rather your child have a better understanding of what they were doing, rather than just “do it that way because it works that way.”
You gotta love it. One guy says something like “Just teach them to invert and multiply!”, and the other guy says “Math just doesn’t relate to the real world!” and a third says “My teachers never explained to me what the numbers and symbols really represented.”
One thing is for sure: Everyone’s got an opinion.
How do you know? No, seriously, how do you know that? It’s akin to saying “Not owning a metal detector has not once hampered me in finding buried treasure.” You can’t say that with any certainty.
I would rather he didn’t have to do the equivalent of counting on his fingers in fifth grade.
I don’t think they taught the concept of range when we were in school, I’ve never heard it either. But it is a fairly simple concept once someone gives a definition.
Based on my kids, both of whom were very good in math, and me, (ditto) having a fraction in the denominator is the least intuitive part of division, so I think models would help. In any case this is just how you teach it - I assume we all agree that dividing by a fraction is a reasonable thing to expect fifth graders to know.
Sure, but why is fiddling around with concrete models necessarily a better way of understanding what’s going on rather than understanding it more algebraically? Certainly if you need to visualize boxes as in the linked pages in order to divide 5 by 1/8, then you can’t claim to have a very solid grasp of the material.
Because people learn in different ways. Terr doesn’t need the image of a shape with units of area marked in order to understand division by fractions, but that doesn’t mean someone else doesn’t. If students are exposed both to the abstract concept of division and a visualization of division, it increases the chance that more of the students will understand division.
The test is not that students must have an image in front of them to understand it, but students who understand it in the abstract will also understand what the image means because they already understand division. Students who understand the image might not get division in the abstract, but they will also understand division.
I wish I had learned with all the models that are hip these days. My teacher purely taught the algorithm, and so I never understood the reason why they worked.
Math, to me, seemed like my teachers had picked an arbitrary number, and then demanded that I go through some arbitrary steps to get some other arbitrary number, basically for no gold reason. It seemed almost tortuously pointless and meaningless, and I never learned much beyond what I needed to squeak past the next rest.
If I had learned a bit more about why the algorithms functioned, my life might have a pretty different path.
All math, up to and including elementary calculus, makes perfect sense to me. I really didn’t study math in school because it was intuitive.
Advanced calculus? Fuuuuuuccckkkkk. That’s when the wheels came off the cart for me. Maybe I had a bad teacher, but pretty much none of it made intuitive sense to me. I got a D.
ETA: I think they gave me a complimentary D just so I could graduate.
I taught math in fifth and seventh grades, and the test is perfectly reasonable for a fifth grader.
I retired the year before they started Common Core, thank goodness. It’s not so much what they want the kids to know, it’s how the teachers are required to teach it. I would be rolling my eyes the whole time. I understand that you want them to know why hey are doing what hey are doing, but sometimes that comes later, and all that writing about why you d each step in solving a problem isn’t always useful.
Of course I’m old, and was perfectly happy to learn “Mine is not to reason why, just invert and multiply”. And one day, yeah, I knew why.
Reminds me of the way I learned English in the beginning (was not in the US at the time). It was called a “ball-box method”. Each period I was given a sheet with something like a dozen very primitive pictures on it - stick figures and geometrical objects. Each one had a caption in English - something like “The boy is picking up the ball” or “The boy is giving the box to the girl” or even something complex like “The boy is making a mistake in leaving the box behind”.
We had to memorize those captions, and were tested on that. Just rote memorization. The theory (and it worked) was that when those phrases stuck in your head, your brain would figure out the grammatical underpinnings of the language automatically. The vocabulary was taught separately, also mostly by memorization. Worked wonders.
In any higher level class that involves math, most concepts are taught by showing derivations that students will never use ever again. They’re important for teaching concepts and helping to show why something works. It would have to be a pretty poor calculus class that did not include a derivation of the derivative by starting with limits and taking the limit of h->0 for [f(x+h) - f(x)] / h. After that, the focus is generally on all the nice algorithmic rules, but at least the full derivation is shown.
When it comes to things like using number lines for subtraction or models for division by fractions, what is wrong with a “this is why it works” derivation before teaching the simpler method? As long as the students learn how to do the simpler rule, it certainly can’t hurt to present what amounts to a derivation.
Yes, but this ambiguity won’t arise for the example Saffer had in mind. No matter in what order you present infinitely many 12s and one 4, you will get an asymptotic average of 12, which seems fair enough to phrase the way Saffer did in certain contexts.
(That having been said, there are of course other, perhaps more familiar contexts, in which it is perfectly natural to speak the way the Common Core question did instead)
[This ambiguity being in just the same way as that, for example, the answer to the question “The dataset E consists of each item in D, along with one extra item. Can D and E have the same number of items?” depends a great deal on the implicit context within which it is being interpreted…]
Nothing at all.
In fact it helps teach the idea of "think about what it means and think about your answer - is it a reasonable answer ? "
The response to jack’s numberline shows its stupidity by suggesting that the employee who did that would be terminated… A bit what the hell ? So he is actually marking doing it on paper in a way that shows the process… what if he has to actually do this process, eg using lengths of rope. 3 X 100 metres of rope, then a ten metre rope, and then six one meter ropes… its not so daft.