Your not going to learn math without rote memorization , repetition.
The building blocks of understanding are memorization and repetition
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Re: The building blocks of understanding are memorization and repetition
#42TL;DR: Study foundational blocks by repeating then until you understand. Keep repeating as you build knowledge and move to "harder" things. I like the message, but not the messenger. Articles like this, in my point of view, fail to delivery their messages in a better way because they are simply too long and with too much noise for my taste. The point he is trying to get across could be explained in a simpler way with…
I remember first learning Quantum Field Theory. One week, my head would ache from what I was reading but by the next week, that bit would be incorporated into new parts, so the "confusing stuff" become foundation for the next week's confusing stuff. Eventually, it became intuitive.
This was my exact experience learning how to program at 24.
Re: The building blocks of understanding are memorization and repetition
#43The author is arguing that 'the latest wave of educational reform in mathematics' (in the U.S.) is overemphasizing conceptual understanding, and that doing so will be damaging to students. The author advocates instead using a memorization and repetition based approach. I can see where she's coming from, but the argument she is making is flawed. She describes a process for developing fluency both in foreign language a…
> You'd be wasting your time trying to get a deep understanding of the Russian phrase, but there's a reason to do it with the mathematical phrase. I think you're simply wrong here. You're not talking about the amount of conceptual depth underlying it, you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning. On the whole, the Russian phra…
> you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning
I don't think so. In both of my examples, the one from mathematics and the one from Russian, language is used as a way of representing some concepts—now, the language itself also has concepts underlying it, but these are about grammatical structures, lexicon, etc.—not about the subject the language is used to talk about.
To be clear, I'll call the first category of concepts 'subject concepts' and the second 'language concepts.'
When you learn a new foreign language you must learn new language concepts, but not new subject concepts. You aren't re-learning what it means for one to visit a store, you are only learning a new scheme for representing that concept, which isn't nearly as deep as the initial concept itself.
In learning mathematics you're having to pick up the language concepts and the subject concepts, and the subject concepts (as in the example I gave) can be quite deep.
Re: The building blocks of understanding are memorization and repetition
#44In maths you learn to solve problem by expanding the abstractions by adding new elements. There is a crucial difference between problems easy to solve by repetition and problems that require some insight.
Exactly, otherwise all those IOI and IMO contestants would be pure geniuses, probably they are, but they practiced a lot to be good at a single task - problem solving in programming or mathematics for competitions. That's a good foundation built on insane amounts of repetition and memorization, but still, not many turn out to be Terence Tao or Peter Shor.
Although https://en.wikipedia.org/wiki/List_of_International_Mathemat...
seems many did "pivot" from competitive memorization and repetition to some lovely mathmakers.
Re: The building blocks of understanding are memorization and repetition
#45Earlier quoted context omitted.
> You'd be wasting your time trying to get a deep understanding of the Russian phrase, but there's a reason to do it with the mathematical phrase. I think you're simply wrong here. You're not talking about the amount of conceptual depth underlying it, you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning. On the whole, the Russian phra…
> you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning I don't think so. In both of my examples, the one from mathematics and the one from Russian, language is used as a way of representing some concepts—now, the language itself also has concepts underlying it, but these are about grammatical structures, lexicon, etc.—not about the su…
You've outlined my point exactly: you're comparing a case of learning language concepts to a case of learning language and subject concepts. Of course one of those is easier, but it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point.
Your comment is no more insightful than to say that it's easier to learn set theory knowing category theory than it is to learn English while knowing no languages, since it's purely acquisition of language rather than subject concepts. (This is actually untrue -- which is why it's easier to say, switch to romance languages than going to an Asian one from English; there's a bit of subject conceptualization in the nature of the language concepts.)
It's not comparing apples to apples, which sort of reduces the point about the relative complexity of statements and the way that you learn the underlying concepts -- both language and subject. You're comparing the complexity differential of two encodings on the one hand and the total complexity involved in the other. Nonsense comparison.
If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory?
Because the language differential between spoken languages (eg, you do learn new subject concepts if you learn Japanese versus English) is comparable to the difference in mathematical underpinnings, eg, the switch from set theory to category theory.
Re: The building blocks of understanding are memorization and repetition
#46I talk to a lot of young adults about their challenges in school. Many are interested in computer science but don't have a strong foundation to ace a C++ course. I wish I could convince them that failing the course is not as big a deal as it seems. Repeating the class is the right choice. Giving up is not the answer. Sometimes repetition means taking a class 3 times.
To help younger kids with repetitive learning, consider smartmadre.com (beta). Once configured, it will deactivate your child's access to time wasting websites. In order to get internet access back, the child has to spend a few minutes earning points on readtheory.org, quizlet.com, khanacademy.org or typingclub.com etc..
Re: The building blocks of understanding are memorization and repetition
#47TL;DR: Study foundational blocks by repeating then until you understand. Keep repeating as you build knowledge and move to "harder" things. I like the message, but not the messenger. Articles like this, in my point of view, fail to delivery their messages in a better way because they are simply too long and with too much noise for my taste. The point he is trying to get across could be explained in a simpler way with…
Yeah, advice I heard from parents and teachers countless times growing up.
"Memorize this." "Don't quit." "Keep trying."
Then I had to repeat verbatim whatever it was I was supposed to memorize, whether a reading selection, scales on an instrument, or multiplication tables. I didn't get to move forward till I got it. And I didn't get to choose what I studied.
Seems like we need more of this today.
Re: The building blocks of understanding are memorization and repetition
#48TLDR: This seems to be a strawman, at least in how it presents common core. However, I have to give a concession to the criticism that superficial understanding is often what common core implementations look like. Still, in my opinion fluency comes from understanding, not the other way around.
>The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t. By championing the importance of understanding, teachers can inadvertently set their students up for failure as those students blunder in illusions of competence. As one (failing) engineering student recently told me: “I just don’t see how I could have done so poorly. I understood it when you taught it in class.” My student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it. He had not developed any kind of procedural fluency or ability to apply what he thought he understood.
Teaching for understanding means that teachers are responsible for ensuring that students are understanding. If a student is mistaken about understanding something, but the teacher doesn't probe their understanding to expose their misconceptions, that's not "teaching for understanding".
Common core encourages repetition through its focus on multiple representations. One might study linear growth as repeated adding, as a table, as a graph, and in applications to various real-life phenomena. Common core places emphasis on the student being fluent (as the author states, common core has fluency as one of its three major focal points) with all of these representations, and also in seeing the connections between them. This repeated exposure brings out misconceptions, builds understanding, and (over time) results in fluency.
I really don't see why the author has a bone to pick with common core since the sort of practice she describes would fit perfectly into a common core curriculum:
>I memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side?
Common core (and contemporary education movements) are against "rote" or "procedural" learning. They would be against making up a song to memorize f=ma, and merely using that song to plug-and-chug through a small collection of problem types.
One recent example I saw (a colleague works on coaching teachers in common core) was a class of elementary students who could correctly multiply 4/7 * 5/9, but couldn't shade in 1/4 of a square. They memorized and rehearsed the procedure for multiplication, but never built understanding of what they were doing.
The unfortunate thing is that they are able to demonstrate fluency in this skill - and they will likely score well on standardized tests as a consequence of this fluency. This skill, however, is shallow - and will be easily forgotten without continued practice. Furthermore, when the time comes to learn proportional reasoning, or rates of growth, or any other thing that has to do with fractions, they will have nothing to build their understanding on.
I have to make a concession to the author, however. It is easy to get this impression of common core from the sidelines. Most teachers, departments, and schools were dumped into the core (which is merely a set of standards) without much support or training. Implementing the core requires a major shift in how one approaches teaching, and whether it is due to a lack of understanding, a lack of will, or most likely - a lack of resources, many classrooms are merely cargo-culting the sorts of things that common core demands.
My favorite introductory book to the subject is https://amzn.com/0325052875 happy to chat!
Re: The building blocks of understanding are memorization and repetition
#49Earlier quoted context omitted.
> you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning I don't think so. In both of my examples, the one from mathematics and the one from Russian, language is used as a way of representing some concepts—now, the language itself also has concepts underlying it, but these are about grammatical structures, lexicon, etc.—not about the su…
You've outlined my point exactly: you're comparing a case of learning language concepts to a case of learning language and subject concepts. Of course one of those is easier, but it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point. Your comment is no more insightful than to say that it's easier to learn set theory knowing category theory than it is to learn English while…
> ... it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point.
That was not my initial point. You've read the 'fundamentally' part into it.
> If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory?
That is what I was doing. I'm not saying learning math versus foreign language is fundamentally different, I'm talking about practical differences in actually learning one or the other: who in the audience here doesn't already speak one natural language? So, with any foreign language they will be in the position of somebody knowing category theory and attempting to learn set theory. That is by definition not, however, the case for someone trying to break into mathematics for the first time. So you always have language concepts only for a foreign language and language concepts + subject concepts for mathematics—practically speaking.
Re: The building blocks of understanding are memorization and repetition
#50Earlier quoted context omitted.
Thanks for the TL;DR the topic was interesting but I was tired for read more after all day. My grandma used to say "brain is a muscle". And I used to see it as an over-simplification, because I was studying, etc... With the time, I've come to a similar conclusion, a different muscle, and with a micro-services architecture, but at the end of the day, it inherits the attributes, methods and behavior, of the "muscle" cl…
> at the end of the day, it inherits the attributes, methods and behavior, of the "muscle" class Really, it inherits from the general "biological system" class, of which "muscle" is the common subclass referenced, since it's the one most people have familiarity in utilizing. The differentiation of "muscle" from other "biological system" classes is entirely unrelated to the "use-it-or-lose-it" feature, however, and yo…
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