181–190 of 356 posts
Quoting him, "The equal sign is another mathematical concept that’s often misunderstood. It means, of course, that whatever is on either side of the equal sign is equivalent." This is actually wrong.
The equals sign (=) is a shorthand for stating not that the two sides are equivalent, but that they are the same (i.e., they are equal).
If they were just equivalent, we would use another sign, like ≡ (unicode U+2261).
On the flipside, I was in a Calculus lecture at uni and turned to the girl next to me who was crying. I asked her if she was ok, and she said she didn't understand why she had to sit in the lecture learning Laplace transforms when she "only wanted to be a primary school teacher". From the sounds of it, it was a prerequisite? Edit: Oh :) I just worked it out - I was taught to fold the subtraction and carry right in on…
This comment surprised me, as I have a BS in math and Laplace transforms were not something I was required to learn for the degree (although I had the option to take courses as electives that covered them). My university offers a math degree intended for those who want to go into teaching that doesn't get any hairier than Number Theory for required courses.
> But many elementary students don’t understand the meaning of the equal sign. Some people might not have made the connection, but by and large programming languages make this mistake too. All the major ones are using the = glyph for assignment.
Foo = bar and 1 = 1 are equivalent in my mind.
The author unfortunately has a misunderstanding of equality. Quoting him, "The equal sign is another mathematical concept that’s often misunderstood. It means, of course, that whatever is on either side of the equal sign is equivalent." This is actually wrong. The equals sign (=) is a shorthand for stating not that the two sides are equivalent, but that they are the same (i.e., they are equal). If they were just equi…
Perhaps base 10 is the problem. In base 2, you can learn the "multiplication table" in 10 seconds (0 X n=0, 1 X n=n, we're done.) The long multiplication algorithm for n and m is just multiply by n by the least significant bit of m, add to the accumulator, left shift, repeat. You can do subtraction by adding the two's complement (just flip every 1 to a zero and every zero to a one - trivial for a child to learn) inst…
The real world depends on base 10, and so it's way more practical to learn calculations in base 10. I said calculations, not math. I think there is a big confusion between math and calculations. Kids learn calculations, not math. Learning the multiplication table is just a way of how to do quick calculations inside your head. I need to pay 35.15 euros, so how much do I get back? That is calculation not math. In my op…
Earlier quoted context omitted.
I didn't understand why it worked (a comment here pointed out that it's just division-by-repeated-subtraction which does make sense now). But I was still able to perform it, during the time when it was relevant to my life.
What's a red flag is being frustrated by long division and at the same time not being curious or resourceful enough to resolve your frustration. You blame your lack of understanding on other people. I can understand if most people shared your frustration, but not a PhD holder. Your degree basically says that you're capable of doing long, intensive research on arcane topics. What am I missing here?
If we are serious about trying to "break the cycle", one obvious solution is to pay teachers more. Imagine if teaching paid as well as software development.
> Imagine if teaching paid as well as software development. I’d love this personally. But the organizations that employ teachers do not make the same amount of money, per employee, as organizations that employ software engineers. Depending on your state, your property taxes would need to double or triple to make the math work out. To be clear: I’m not making a comparison of the value provided, just the accounted-for…
> But many elementary students don’t understand the meaning of the equal sign. Some people might not have made the connection, but by and large programming languages make this mistake too. All the major ones are using the = glyph for assignment.
Languages like J make this distinction: `=` is equals in the mathematical sense [1], while `=.` and `=:` are the assignment operators (the former being local and the latter global) [2]. Although the assignment operator and the equals operator are similar there is a preference to have the equals sign retain its original meaning. 1. https://www.jsoftware.com/help/dictionary/d000.htm 2. https://www.jsoftware.com/help/di…
(I wouldn't call J's = "the mathematical sense", either, as I don't know any mathematicians who would say that = is an operator that evaluates to 1 when the two sides are within 2⁻⁴⁴. The mathematical sense is merely a statement of fact, e.g., "x+10=y+10", when x is 5 and y is unknown, should mathematically result in knowing that y is also 5. Wolfram is the only language I know offhand that does anything like this.)
Forth, Lisp, Scheme, APL, R, Logo, Cobol, and assembly all have a variable assignment syntax that doesn't use the "=" character at all. Only 2 of them even use the form "varname SYMBOL expression".
I probably wouldn't recommend assembly language as a teaching language, but you have to admit no assembly programmer ever confused LDA and CMP.
I have a hard time explaining to my children how addition of fractions work. Not how to do that, but how to feel the intuitive way. And percentages, to some extend. The sad part is that I have a PhD in physics and an engineering degree in CS. I used to teach physics at the uni and loved it. My children like me to explain them physics because I love it so much and they appreciate the analogies, their limits etc. It is…
When I was first taught fractions in primary school, the teacher had a bunch of rods of equal length. Each rod was divided into segments of different colours. Each segment could be "fractured" off the rod because they were attached by magnets. I think they went from 1 to 10 or so segments. This allowed for a great visual (and tactile) illustration of how fractions work. 1/2 + 1/4? No worries, just break off some piec…
Earlier quoted context omitted.
I have no intuitive understanding or mental model of the algorithm. I could follow the instructions if they’re written in front of me but I have never managed to understand why they work or to memorise them. I have a similar mental block for standard deviation - I always seem to have another ‘but why?’ question that eventually people answer with ‘because!’ and then we’ve both given up.
Standard deviation is a form of variance that has the same units as the mean. Whereas the variance has square units. This means the quantity of mean/variance depends on your choice of units, whereas the quantity of mean/standard deviation. Is constant no matter the units. Why do we care about variance? Because it is a nice and linear property. This means it is easy to calculate and manipulate. Moreover, the variance…
I think the OP's frustration in how computation of standard deviation as a quantity is just taught by rote in a cookbook manner is quite understandable. If one's not thought more in depth about distributions and estimation of their parameters, it really makes little sense.
Whether the statistics should be taught more in depth is another question. The current state of the mathematics of statistics and probability is such that it needs quite heavy tools to do rigorously. And countless of hours of getting to know the quirky inconsistent notation.
But until it is taught more rigorously, more intuitive measures of dispersion like MAD make a lot more sense for most applications. And I don't think it would be an unreasonable ask for the statisticians to lay out the "porcelain" for working with the more intuitive, if somewhat more analytically inconvenient measures. A bit like we don't require programmers to understand how transistors work.