Many elementary teachers don’t understand math, and it makes them anxious
latimes.com
latimes.com
We usually employ teachers with degrees in math or science, or sometimes engineering. Occasionally for early elementary levels (grades 1-3), some teachers have a degree in another discipline. Regardless of major, we always test them for math aptitude. The test includes competitive exam questions focused on mathematical understanding and problem solving skills, rather than advanced math knowledge. (The other part of teacher selection is interviewing for teaching skills and trial teaching with student and expert evaluation.)
Somewhat surprisingly, even some PhD candidates in engineering may fail these tests, while a few liberal arts degree grads passed the lower levels of these tests. So math aptitude is not limited to those who majored in math, science, or engineering. (Note that many of our elementary school students who have studied with us for a couple of years also pass them at a high level.)
It seems that countries with successful math programs, like Singapore, also utilize a rigorous teacher selection process that favors subject matter specialists even for teaching at an elementary level.
This is crucial for math, which is particularly hard to catch up later on with shaky foundation.
(In fact, kids are great at absorbing their teacher’s attitude toward the subject. It is no surprise many kids may sense “math fear” from their teacher. Having a teacher with positive attitude toward the subject they teach is essential!)
Back when I thought I was going to be an engineer, I went through a brief phase of doing everything by hand. Indeed, my brain is capable of "rapid" arithmetic, but it requires practice and discipline to achieve and maintain what amounts to a billion-time slowdown.
In my experience, many kids are much more enthusiastic about this kind of "challenging" problems than the drills in many standard textbooks, as long as the problems are chosen to match their level. They definitely learn a lot more as well.
Note that although they do require a little arithmetic to solve, the challenging part is not arithmetic.
Some problems focus on geometry, logic, patterns, or other kinds of puzzles. More examples can be found at the source below.
"- The edge of a cube is 8 cm. All the faces are painted orange. It is then cut into small cubes of edge 1 cm. How many small cubes have exactly two faces painted?
- What is the greatest possible number one can get by discarding 100 digits, in any order, from the number 1234567891011121314151617…57585960?
- Eleven consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2012, what number did Maria erase?
- You must color each square in the figure below in red, green or blue. Any two squares with adjacent sides must be of a different color. In how many different ways can this coloring be done? Figure at question 11 here: https://gato-docs.its.txstate.edu/jcr:450cce10-3b6a-4ddd-a19... "
Source: the Primary Math World Contest (usually held in Hong Kong) https://www.txstate.edu/mathworks/PMWC/previous-pmwc-tests.h...
What grade level is question 3 considered?
It would be suitable for students between grades 4 and 6 (the highest level we currently teach), depending on how talented the kid is.
To be clear to other readers, these are hard questions that most people, including older kids and adults, probably can't answer without proper learning and practice.
https://en.m.wikipedia.org/wiki/Scientific_notation#E-notati...
I misread the question though and now realize it’s not needed.
So for those who have let their math skills lapse or need help to get back in the right frame of mind to learn, where online is the best resource? Khan Academy?
I have not checked them out in detail, but it appears that https://artofproblemsolving.com/ is a good resource. They also have interesting online and physical book series: https://artofproblemsolving.com/store (I have no affiliation.)
99999785960?
199?
1. 12 edges * 6 non-corner pieces per edge.
2. There are 111 digits there. You can't keep six nines and have enough digits left. So you keep five (from 9, 19, 29, 39, 49), and now have six digits to keep from 5051525354555657585960. It can't start with 8 (or you still won't have enough digits left), so you choose 785960.
3. First notice that 0+...+10 = 55. To make it end in 2, you have to discard the 3. In general if you have n+...+(n+10) you discard n+3. Now n+...+(n+10) = 5.5*(n+(n+10)) by Gauss' trick, and subtracting (n+3) you have 10n + 52 = 2012.
Very much doubt I could have solved them in elementary school(!) though.
This is a very challenging problem. Without a calculator (or computer), it took me a system of an equation with an inequality in 2 variables, using the fact that the solutions are integers, to solve it.
I would be very surprised (and excited) to meet an 8th grader (let alone a 6th) who could solve this problem without any external help the way I did. I would not expect an elementary school student to have this kind of algebra knowledge. Most elementary school students around here don’t even know what algebra is.
I solved the problem for n, the largest number in the sequence of eleven, and a, the number Maria erases, by the following:
n - 11 < a < n + 1
11(n + n - 10)/2 = 2012 + a
I’m interested to know what other methods may be used to solve it (apart from brute force).
Forcing yourself to only rely on the solution strategies taught by school makes you very bad at problem solving.
This sort of puzzling through play is good for solving math contest problems that have been specifically designed for the purpose, and little else.
Example: The Josephus Problem - Numberphile https://youtube.com/watch?v=uCsD3ZGzMgE&t=86
Also ny solution wasn't brute force or trial and error, there are very few possible sequences of 11 numbers around 201, so you are guaranteed to find the solution quickly.
Perhaps the elementary school math contest permits the use of a calculator? That would make it very quick for sixth graders to solve it using your method.
Assume the middle of the sequence is 201 since 2012 / 10 ~= 201. Then the sum of the 11 numbers between 195 and 206 would be 11 * 201 = 2211. Subtract the sum of the values we didn't remove to get 2211 - 2012 = 199. 199 is between 195 and 206 proving that it is a valid answer.
Edit: Also I did my masters in pure math, doesn't mean that I cannot use common logic without the formalism.
Try the same trick with this modified problem:
Fifty consecutive positive integers are written on a board. Maria erases one of the numbers. If the sum of the remaining numbers is 2009, what number did Maria erase?
You'll find that it does not work. The trick you did relies on the specific problem as stated. It is not true in general.
2009 / 49 = 41. 41 * 50 = 2050.
2050 is not the sum of the numbers 17..66, 2075 is, and subtracting 2009 from 2050 just gets you 41 again, which is not the correct answer.
I think a school student can do this math manually. The hardest part is the sum of the first N(in this case 10) numbers and then finding the divisor. Yes there might be a better solution.
But I think a kid would think up a brute force algorithm like this and it is easier for small N. The computation is only in O(N) complexity haha. And if an answer exists it will always be found - so it is a real solution. Might be the first algorithm you write if you are tasked to do this with software.
In this case, let's consider the sequence of 50 consecutive integers 16, 17, ..., 40, 41, ..., 65 where 40.5 is in the middle and the sum is 40.5 * 50 = 2025. Subtracting 2009 we get a supposed erased value of 2025 - 2009 = 16 and this indeed correspond to a valid solution:
16, 17, ..., 65 => 17, ..., 65
We also could've considered the sequence 17, ..., 41, 42, ..., 65, 66 centered at 41.5 and this would've led to an erased value of 41.5 * 50 - 2009 = 66 which is another solution:
17, ..., 65, 66 => 17, ..., 65
For instance, if you want to find the global minimum of some function on the integers, you might realize you can find the roots of the derivative and test only the neighboring integers. If you want to divide 144359305 by 241, you're going to start by guessing how many times 241 goes into 1443, and then multiply to check whether that's right. If you want to compute floor(29489238428^(1/3)) by hand, you're better off smartly guessing and checking than trying to figure out a digit-by-digit cube root algorithm on the spot.
The eleven numbers are 196,197,..., 206, out of which 199 was removed.
I began with n being the largest of the sum so that 11n - 55 = 2012 + k. I solved for k in terms of n and then used an inequality to find n which gave me k.
Now I see that the method I used relies on math beyond elementary school, which makes the problem a bit more complicated than it ought to be. On the other hand, the elementary school only method relies on the fact that you have 10 numbers remaining and so you can easily see the remainder. It wouldn’t generalize as well, especially in cases where there may be multiple solutions and you’re asked to find all of them.
10*n+sum_conseq=2012
therefore erased number is n+3 because (2012-55-3) divisible by 10 -> n=196.
I was also pretty poor at this kind of thing in high school and college, but fortunately the skills involved are almost but not quite disjoint from what is needed to do research on math.
One is doing math PhD at Harvard; another received a PhD in theoretical CS from Berkeley with stellar track record (over 20 papers when he graduated after just 4 years in PhD--many/most in top venues, as well as perhaps the most prestigious best paper award for CS Theory PhD student when he was in 2nd year). Not all went on the research path though; some are working in applied software engineering. A few who did math undergrad degrees at places like MIT but I'm not sure where they are going next.
Another conspicuous example: Terrence Tao was also the youngest ever International Math Olympiad gold medalist (he won it when he was 12-13!).
There appears to be a significant correlation between subsets of people who are good at certain types of math and theoretical CS research and at solving this kind of problems when younger.
I read that there are mathematicians who focus on solving problems and those who build frameworks. Perhaps the skills for solving these problems are closer to the former?
Seems like it might be close to a subset of the sorts of questions one encounters on an IQ test.
I've always had a very intuitive grasp of math, which meant I excelled at solving things in my head but struggled with (and resented) having to show my work in the arbitrary algorithms we were taught. I understand division despite my education.
They really should try to come up with algorithms that are more intuitively linked to a spatial comprehension of the numbers. That underlying grasp is really all that sticks with you into the real world (and it's also what makes math engaging). If students aren't getting that, they're wasting their time in my opinion.
Multiplication = repeated addition
Addition = a form of counting
Maybe they need to explain it in those terms. Or use pictures.
I don't think you're wrong, just seems an odd way to think about it, and fails to account for any remainder.
Oh :) I just worked it out - I was taught to fold the subtraction and carry right in one step. I'm an idiot!
I can distinctly remember learning long division as a kid, and I wonder why this isn't immediately taught afterwards as the logical progression
(All these methods had the same asymptotic complexity. Mine was slower by a small constant factor; on the other hand it actually made sense. It was just repeated subtraction!)
I could breeze through higher-level logic courses, but was essentially remedial for higher-level math.
And it's because you need long division to do division in algebra
After looking at the English Wikipedia I found that
a) what I learned in school is long division
b) the notation with the dividend and the divisor separated by a parenthesis and the result on top of a horizontal bar is totally alien to me
c) I've never seen short division before
d) the German Wikipedia doesn't even have an article about short division
Edit: although if someone else can explain it better I'd really like to know.
From what I have searched,(quickly mind you), it seems to be a mind set.
I went to school in Sweden where we do teach all this things, but in a strange order.
the "short" notation relies on the ability to squeeze the remainder in between the digits of the dividend. if you have a divisor greater than ten, it's possible to have a remainder with more than one digit. two or more remainder digits is a lot harder to write legibly between the digits of the dividend. you also end up subtracting larger numbers in your head, which is error prone. finally, most people only memorize the multiplication table up to 12x12 or so. once they can't simply do a lookup into their memorized table, they switch to more of a "guess and check" approach, where they will inevitably have to cross out or erase their previous work.
in the end, it introduces a lot of unnecessary opportunities to make mistakes and hides stuff that people can't do reliably in their heads.
For anyone interested in going down the rabbit hole of those infinite levels.
I have a PhD in computer science but I cannot understand long division.
I think you're trolling us a little: you could sit down with pencil and paper and think about how it must work and 20 minutes later you'd understand it, surely.
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.
https://web.stanford.edu/class/ee486/doc/chap5.pdf
Edit: standard deviation is just a fancy name for root mean squared over the sample offsets. The square is there to deal with the fact that there are positive and negative offsets. The mean is the whole point of the exercise, and the root is to negate the fact that we squared earlier.
This is how I always find maths and particularly statistics - I always just want to answer ‘but why’ and then people lose interest in explaining after a while.
Standard deviation is mathematically easier to work with though and has nice properties e.g. you can compute the standard deviation of combining a bunch of independent things by combining their individual standard deviations.
Also, consider that the correlation coefficient consists of similar formulas. They all look like the theorem of pythagoras. This ties in nicely with the fact that observables can be seen as having angles in between them where the correlation coefficient is the cosine of these angles.
You can read up on the central limit theorem if you want to understand more.
This is also where I get frustrated.
I'm taught about the CLT but every place I think I might use it, the samples aren't really IDD when you look hard enough, so it doesn't apply.
Computer benchmarking (which is where I'm usually trying to apply statistics) is the classic example. One iteration changes the state of the computer for the next iteration. They aren't remotely IDD, but people still try to claim the CLT. Seems like all their stats beyond that point is broken.
I'd assume that computer benchmarks have very low correlation between runs, unless you leak ram or your components don't keep a stable temperature. So in that case the CLT seems to apply just fine.
And they are not in a computer program. The state of, to just give one example, the processor cache depends on what happened in the previous run.
We have all this rigour in how statistical techniques are applied, and we make black or white decisions based on the numbers that come out of them, but at their very foundation we're saying 'whatever I'll just pretend this rule applies it's probably fine'. How can we build on that flawed foundation? I don't get it!
"All models are wrong, but some models are useful." Just remember that there's some imprecision in the answer a model gives you, and keep that fact in mind when applying that answer to make real-world decisions.
e.g. https://socratic.org/questions/what-is-the-difference-betwee...
Put simply, standard deviation is arbitrary for most people.
Speaking of language, have you ever thought about how all names and grammatical structures are arbitrary as well? There are parallels.
My larger point is that, something can be arbitrary and still be meaningful. If people agree on a common meaning, it formalizes a communication protocol that allows for much better information sharing.
This means that any distribution with standard deviation of X will behave as a normal distribution with standard deviation X when you have enough of them. Or in other words, any distribution with standard deviation of X will behave like any other distribution with standard deviation of X when you have enough of them. There is no other measure like this as there is no other equivalent to the central limit theorem. Therefore it makes sense to have this as a universal measure of random processes.
Now that we have a normal distribution, the mean is an obvious metric to pick because it captures the notion of 'middle' in a useful sense. Then we have proofs that we can characterise the normal with the mean and 1 other parameter (normal normal) or a matrix (multivariate normal). We call that the standard deviation^.
The exact formula wasn't a coincidence. The normal can obviously be characterised by the mean and a statistic from another formula. There were a bunch of experiments tried (eg, using |x| instead of sqrt[x^2]) but it turned out that sqrt[x^2] had some other nice property that minimised some sort of error so they went with it as a standard. I forget what one, might be error of estimating the true parameters from a sample or similar.
We could characterise the the normal as an infinite sum or something quirky, but when people say 'easy to work with' they mean instead of a function or something quirky we can simply pick a number.
Standard Deviation isn't as important when working with non-normal distributions, although I think it still turns out to be useful. But its importance is that it characterises a normal apart from the information captured in the mean. I'm not a mathematician, YMMV, could be wrong, standard disclaimers.
[0] https://en.wikipedia.org/wiki/Central_limit_theorem
^ I'm not going to edit this but it occurs to me that we call it the Variance. Same thing as std. dev in my opinion.
' There is a fairly subtle observation to make - Normal is characterised by mean and std. dev, but the most efficient ( https://en.wikipedia.org/wiki/Efficiency_(statistics) ) unbiased estimator of the std. dev of the population is the adjusted std. dev of the sample. Therefore, accounting for mean, you can't get a more efficient characterisation of the normal than mean & std. dev. Ie, if you picked a formula other than std. dev then the most efficient estimators to characterise it would still be mean and std. dev. Don't recall if there are equally efficient choices but I think that proves there are none better.
That might have been the logic for why std. dev was chosen. Just a guess.
There are many "right" metrics - this is one of them. The big picture is we want a measure/idea of how spread out the data is. One can come up with many ways to do this, and absolute value and root mean square are two of the most common. They correspond to L0 and L2 norms. There is also the L-infinity norm (just use the farthest point).
The L2 norm is the most complicated of the 3, but as people pointed out, it is one of the easiest to use analytically because of its nice mathematical properties.
Have a look at https://en.wikipedia.org/wiki/Statistical_dispersion for other measures.
To illustrate, perhaps an example closer to home:
If someone asked you to tell them "how many lines of code in this project?" you'd start wondering things like "do I count comments?", "do I count dependent libraries?", "perhaps some metric that is equivalent of a statement count but not strictly counting lines?" and on and on.
Statistical measures are like that. Someone with a decent knowledge of the problem space came up with the best stab at how to characterize the data.
You can. It's called the mean absolute deviation. Which one you use depends on your application. If you just want to give a summary of how spread out the data is then either would work. Lots of people say that we should use the mean absolute deviation as the default rather than the standard deviation.
I too have a problem with math tools passed down without any surrounding context - without telling why are we using this formula, instead of any other variant from the family of formulas that would satisfy the same goals. I'd have much easier time dealing with statistics in school if someone told me that a) deviation with ABS instead of root-square is also a thing, and b) we use the root-square one because it amplifies offsets from the mean.
Also one of my peeves. However over the years I've come to realize that in addition to teachers who omit the context, there is also a class of student that actively doesn't want to hear the context. I'm not sure exactly why this is, but I see it in my immediate family quite a bit. A sort of <cover ears, lalalala...too much detail> kind of thing perhaps due to difficulty taking on too much information?
I feel like the standard deviation is a black box to basically everyone without statistics degree. It also bothers me that it is so widespread, because almost nobody know how to interpret this value. It seems to me like most scientists treat the standard deviation as a magical number that allows to compare spread between datasets and otherwise doesn't mean anything on it's own. I feel like a lot more insight could be gained from the mean absolute deviation.
I'm also curious how are scientific insights affected by defaulting to something that is extra-sensitive to outliers. The effect can't be big per single research, but in collective?
Unless you try to solve more problems yourself, you won't understand the advantages or disadvantages of some methods in solving the problems. So the real answer is: construct the examples, compute and compare, don't approach it "philosophically."
All the methods used today survived because they were the solutions to some problems. It doesn't mean that everybody applies them properly: for that you have to get some experience yourself.
Learning about the historical development of the methods is also for me satisfying experience: e.g. logarithms aren't a concept devised for philosophical purposes, Napier developed them to save astronomers time and limit "slippery errors" of (their) calculations:
https://www.thocp.net/reference/sciences/mathematics/logarit...
"I have a PhD in computer science but I cannot understand long division"
For that, personal experience in devising the examples, evaluating them and comparing the results is the only way to get the understanding. One can't complain that one doesn't understand Greek texts if one personally never tried to learn the Greek letters. One could have learned "about" Greek but at the end one hasn't done the inevitably necessary preconditions to actually read Greek texts.
The error in thinking is believing that because one already learned about something else somewhere else he should somehow "understand" something without doing the necessary work on that something. The solution is "make your own homework" to fill the gaps.
The topic of what, how and why children learn in schools is much more complex.
Edit: Also, now reading the Wikipedia article: https://en.wikipedia.org/wiki/Long_division I do understand why people don't understand it: the taught notation for the process, used in many countries, is utterly confusing to me, so my reason for confusion when looking at various notations is not that I don't understand the nature of the method but that I haven't spent time analyzing what was built on top of the basically simple idea -- I can also "understand" all the notations but I'd also need more time for that. Still, having enough experience, I can claim that I understand the idea and the process even if I never learn all the notations and conventions in different countries (which part of the process you write where, what you write and what you don't when etc). The idea is simple, the conventions enforced don't have to be. Those who only learned the conventions maybe never invested any time to figure out what's behind the conventions: what is actually being done.
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 / standard deviation very nicely describe a Gaussian distribution (bell curve). This is a very important distribution because of the law of large numbers. Because we see bell curves so often, it is nice to have tools (std.dev) that work well with these curves.
It should be noted that, in optimization problems, there can be reasons to try and minimize something other than variance. We tend to pick the variance / std.dev because it is familiar, easy to work with, and very efficient. Notably, the derivative of variance tends to be linear, which makes it pretty efficient to use in gradient descent.
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.
Take a course in ring theory if you want to see how all of this machinery gets built up.
also maybe they are specific about they way they do long division in the US? it was a surprise to me in college too and i have a french background. it's not how the french system does long division...
6240 / 5
= divide(6240, 5)
= divide(1240, 5) + 1000
= divide(240, 5) + 1000 + 200
= divide(40, 5) + 1000 + 200 + 40
= 1000 + 200 + 40 + 8
= 1248
----------
2. Regarding standard deviation, one of these bullets might help:
- The normal distribution has exactly one shape, centered at x=0. But it's useful to apply two transformations to it: translation and horizontal stretch. To translate a distribution left/right, change its mean. To horizontally stretch a normal distribution, change its standard deviation.
- Mean has units of length. Standard deviation has units of length. They tell you where the normal distribution is offset, and how wide it is. Mean and standard deviation are just measuring sticks/rulers for normal distributions.
- When people talk about the standard deviation with any arbitrary data, they're usually assuming the data is normally distributed. If the data is not normally distributed, standard deviation no longer refers to the width of the normal distribution, so we lose that visualization.
- With non-normally distributed data, the standard deviation is still useful as an analytical tool because taking the (sqrt of the) summed squares still gives us a number that grows as the data spreads further apart or if the distribution grows wider. There's a center point for the data (the mean), so to ensure you're measuring the overall spread of the data, you subtract the center point off each data point before squaring them. In other words, you're squaring deviations from the mean. And unlike the sum of absolute differences, the sum of squared differences (variance) is differentiable. Differentiability is a great property, so this is the standard way to compute a sum of deviations from the center.
func Longdiv(A, B) {
// Assume A > 0, B > 0.
N = floor(log10(A))
Q = 0
R = A
while (N >= 0) {
// Invariant: A = B * Q + R
// Invariant: B * 10^(N+1) > R
C = 0
while (B * (C + 1) * (10 ^ N) <= R) {
C += 1
}
// C is the largest C such that B * C * 10 ^ N <= R
R -= B * C * (10 ^ N)
Q += C * (10 ^ N)
// R is still positive.
// Invariant A = B * Q + R is maintained.
// We know B * 10^N > R because otherwise we
// would have picked a larger C.
N -= 1
// Invariant B * 10^(N+1) > R is maintained.
}
// By loop invariants, we have
// A = B * Q + R and B * 10^0 > R.
return (Q, R)
}
I mean your very first line has a logarithmic operation that was not mentioned anywhere in your text or your comment! Just pops in there out of nowhere. I guess it's about the number of decimal digits? But why? Why are we doing things in decimal?
A dense ten-line algorithm like this seems far more complicated than other algorithms we try to get school children to memorise.
We’re subtracting off multiples of 10^N because that’s easy when you write your numbers in base 10. Try subtracting off multiples of 9^N, or multiples of N!, or some other choice, and you’ll see why.
His method for polynomials/derivatives is also interesting if you watch until the end, dead simple calculus.
That could be meant shallow or deep. The shallow answer is because that's all you need to do to function in society (and conversely, while arguing with a police officer about the presumed base on speed limit signs may be fun, it is also pointless).
The other answer probably needs to explore the question a bit further. Perhaps a good starting point is the fact that we almost universally share the physical characteristics of having ten fingers.
I think this is maybe the crucial part that you might be missing. All this long division (and long multiplication) stuff works on base-n represenations of numbers. I.e. if you have an number like 3376, it is actually a short hand for
3*10^3 + 3*10^2 + 7*10^1 + 6*10^0.
(3*10^3 + 3*10^2 + 7*10^1 + 6*10^0) / 4 ==
3*10^3/4 + 3*10^2/4 + 7*10^1/4 + 6*10^0 / 4 ==
(3*10^3 + 3*10^2)/4 + 7*10^1/4 + 6*10^0 / 4 ==
(33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
(33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
(32+1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
(32)/4*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
8*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
8*10^2 + (17)/4*10^1 + 6*10^0 / 4 ==
8*10^2 + (16+1)/4*10^1 + 6*10^0 / 4 ==
8*10^2 + 4*10^1 + (1)/4*10^1 + 6*10^0 / 4 ==
8*10^2 + 4*10^1 + (16)/4*10^0 / 4 ==
844
The idea of long division is just that to solve A/B, we can find another problem C/B we know the answer to and break it into the smaller problem A/B = (A-C)/B + C/B.
For example, to solve 742/13, the normal way is to observe that 13×5=65 is as close as we can get to 74 without going over, so we can reduce this to
742/13 = (742 - 13×50)/13 + 13×50/13
= (742 - 650)/13 + 50
= 92/13 + 50
But you could also use 13×4=52 and work out
742/13 = (742 - 520)/13 + 40 = 222/13 + 40
222/13 = (222 - 130)/13 + 10 = 92/13 + 10
92/13 = ( 92 - 91)/13 + 7 = 1/13 + 7
35/350 (to simplify) Once is 325, twice is 300, X, X, X, we end up at 10 times is 350 divisible by 35.
I'm wondering if I'm missing something (and if I am I'll own up to it lol)
https://commons.wikimedia.org/wiki/File:LongDivisionAnimated...
In essence, repeated subtractions. The blank areas to the right of each number are really just zeros.
Try it with 1/7?
The same process still works if you want to calculate the decimals though. Just pretend you're doing (1000000/7) * (1/1000000) or however many digits you want.
Afaikr it's just division, how man X are in y.
083.28
2.26...
___
7 |583.00
Long division:
083.28
2.26...
___
7 |583.00 [<-dividend]
56 [=8x7]
--
23 [2 from subtracting answer to 8x7 {8 is put in answer line}, 3 from the dividend]
21 [=3x7]
--
2.0 [2 is remainder, 0 from dividend]
1.4 [=2x7 is remainder, 0 from dividend]
---
.60
.56 [=8x7]
--
4 [is the remainder in 100ths]
I'd do an algebra example but it's a pain in the arse just using ASCII.
https://www.solipsys.co.uk/new/SquareRootByLongDivision.html
NN = (10d + e)(10d + e) = 100dd + 20de + ee
Group the digits of the square by two, and add extra values to the "divisor" so that the extraction of the square root in effect undoes the squaring above.
See, for example, https://sciencing.com/calculate-square-root-hand-5081134.htm...
Just show long division for “proofs” while introducing the concept of division, and let kids use a calculator for division after that (supplemented with calculator-free drills). That way there will be more lesson time for advanced topics. Same thing with long multiplication. Show the proofs using long multiplication, then have kids do progressively harder multiplication drills where all the math is done in their head. In fact, kids might grok first principles better that way.
https://apps.apple.com/us/app/dragonbox-algebra-12/id6344441...
https://news.uchicago.edu/story/female-teachers-can-transfer...
He got stuck on a level where he had to solve a simple equation, and tried to do it in the “wrong” order. He eliminated an unknown from one side first, and then the system wouldn’t allow him to do the type of operation necessary to complete the other side, maybe because that particular technique hadn’t been taught yet. It would have been a perfectly natural way to solve the equation on a blackboard. I could see they probably wanted him to do the other side first.
My nephew tried the same basic approach 3 times in a row, and then gave up and threw the tablet down and went off to play with Lego or something.
https://en.wikipedia.org/wiki/Long_division
Children are mostly idiots. Education is the process of painfully breaking them out of their idiocy.
Bonus points if you implement a long division algorithm that works on strings in your favorite language.
You're right that it could be a fun programming exercise, though
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?
I have impaired executive functioning and this is basically my experience in tying shoelaces, I struggled to accommodate the mental model of the process myself so I sought out others so manage it to try and learn from them.
What instead happened was exasperation on their part and frustration on mine, so after repeating that a few times a year until I was 22 I just stopped and bought shoes without laces.
When you go at it for long enough without succeeding it can make the experimentation not very fun or interesting.
I don't think I'd call it a red flag for a PhD to have fatigued themselves out of interest in a problem, it's something that happens to everyone at some point.
37>1, 37>12 but 37*3<=123.
Write down 3 (meant for 3000) and start again with 123456-37x3000 as the new dividend.
I wish the algorithm was taught by writing the full 3000 instead of only the digit 3; this appears to be a source of major confusion and to hide the underlying additions/multiplications.
Have you heard of "chunking"?
This video shows old style long division (and you're right, the method is confusing). It also shows an easier method that is based on understanding numbers rather than just applying an algorithm by rote. https://www.youtube.com/watch?v=K2rBaSj2L74
Here's another explanation: https://www.theschoolrun.com/what-is-chunking
$SIMPLE_CONCEPT that you’re trying your best to understand is easy if you think of it as a specific example of $MORE_COMPLICATED_CONCEPT that you surely can’t understand without understanding $SIMPLE_CONCEPT
E.g.: I distinctly remember that many of the original articles about the Amplituhedron spent like a page explaining what a triangle was, then jumped straight into functions over higher-dimensional analogues without explaining any of the intermediate steps. Something-something-logarithm-of-a-simplex.
I can grok what a simplex is, but they literally just hand-waved over the bit where they take a logarithm of it in some way.
The page of explanation of what a triangle is was of zero assistance.
My 17 year old was struggling with the division algorithm they were teaching in Elementary school at the time (late 2000s). I can't remember the details but it was something I had never seen before that was a step up from throwing poop at the wall.
I showed her how to do long division and she still uses it to this day. It's simple and there is a history of work to easily identify errors.
The homework scene from The Incredibles 2 really hit home for me. I feel like book publishers development new methods or ways to teach the methods as a means to sell new editions of books, not because the new ways are better. This causes a rift in society because people no longer share a common understanding of a subject.
It’s completely different than the way I was taught so I have to invest in figuring it out but it’s led them extremely quickly through multiplication & division and into algebra.
I’d characterize it as “multi-algorithmic”. They come at it from a variety of algorithms, assumedly so that each student can find one that works for them.
As someone who had to teach themselves a different algorithm than I was taught many painful years later this resonates.
Perhaps the rift was always there and you ended up in the lucky side of it?
But in reality the common core way explained the actual algorithm and the reason it worked so much better, and further, the only reason it appeared more complicated was because the people who were targeted by the meme already knew long division. If you took someone completely new to both methods, they’d find the more “complicated” algorithm far easier, because a large part of its complexity was explaining why the division worked, whereas long division was about memorizing a process.
https://en.wikipedia.org/wiki/Common_Core_State_Standards_In...
If you actually understand the math picking up the common core methods is minutes of work at worst.
Just another anecdotal data point but my experience was completely opposite yours.
I was taught long division in the 4th grade (age 9) but wasn't taught a proof of why it works until a college algebra course (age 19).
For whatever it's worth a different type of long division is taught now under common core. The common core math stuff that I've seen is impressive - there is more focus on teaching things in a way where comprehension of deeper concepts should be possible. It's a shame there's so much fear of it from parents who are upset kids are learning differently than they did.
Eg, say you are dividing 76543 by 5. The normal procedure is to say something like:
- 5 goes into 7 once. The first quotient digit is 1.
- 1 times 5 is 5, write that under the 7, and draw a short line under the five
- 7 minus 5 is 2; write that under the line you just drew
- the next digit is 6, so write that next to the 2 you just drew, ie, 26
- 5 goes into 26 five times; put that up in the quotient, etc
That minimizing the writing, but it hides what is really happening.
Really what is going on is this. How many times can you subtract 5 from 76543? One procedure would be to subtract five over and over until the remainder is less than five. The number of times you subtracted is the quotient. But in this example you'd have to subtract more than 15000 times. Instead of subtracting 5 at a time and incrementing a count, it would be faster to subtract 50 at a time and increment your quotient by 10 until the result is less than 50, and then switch to subtracting by 5s and incrementing by 1 as before. If you can see why that works, then you realize it would be even faster to subtract by 500 at a time and increment by 100, then switch to subtracting 50 at a time and increment by 10, then subtract 5 at a time and increment by 1. Etc.
So rather than minimizing the amount of digit copying, it would be better at first to write everything out. In this particular case:
- write the dividend: 76543
- remove groups of 50000. we can see we can subtract 50000 from this once; the leading digit of the quotient is 1
- write 50000 (150000) under the 76543
- subtract, writing 26543 (this is the current remainder)
- remove groups of 5000. now we see we can subtract 25000 from this, so the next digit of the quotient is 5.
- write 25000 (55000) under the 26543
- subtract, writing 1543 (current remainder)
- remove groups of 500. 500 goes into that 3 times, so the next digit of the quotient is 3
- write 1500 (3500) under 1543
- subtract, writing 43 (current remainder)
- remove groups of 50. it is larger than the current remainder, so the next digit of the quotient is 0.
- remove groups of 5. 5 goes into 43 eight times, so the next quotient digit is 8
- write 40 (85) under the 43
- subtract. the final remainder is 3.
The sequence of quotient digits was 15308.
The standard way long division is taught, the child skips writing down all the digits of the remainder and it is confusing as to why this all works.
Reading this paragraph gives me a strong hunch that it could be gender related. I'm wondering what the gender skew is like in elementary school teaching, similar to nursing ?
While research hasn't shown any difference in STEM ability between genders, it most definitely has when it comes to intrinsic interest (and I know this is controversial, but contrary to what is commonly said it's not because of some evil patriarchal plan to keep girls out : gender-related "things vs people" affinity has been found in baby monkeys too, and seems correlated to pre-natal testosterone exposure !)
One can easily imagine a bunch of people-orientated young female students going into this field for the kids and people interaction not being too thrilled with the topic.
My high school had an elaborate system of ways for people to avoid doing math. Even though Algebra was technically required to graduate, there were enough loopholes and alternative classes that students could effectively skip learning actual math. Fail an exam? The teacher would give extra credit: presenting about a famous mathematician--certainly interesting work, but does not really show that the student actually understands the core math material.
This is how you get incoming first-year undergrads who don't know how to reduce fractions even though their curriculum indicates they should have learned it multiple times over. They have managed to avoid jumping through the hoop so many times because we gave them a rope to skip instead.
I come from a skill-focused math education. I was pretty good with skills. But I didn't understand fundamental concepts. I still remember my struggle with prime numbers. Those problems were the hardest for me. I didn't understand why prime numbers are popular. In college, I discovered the meaning of prime numbers and their applications. Then, I was reading a lot about prime numbers just for fun.
My kids are now learning in the American education system. I feel like there is a shift to skill-focused math education. I suppose American educators feel like we need to "catch up" with other countries in terms of test scores. I find those objectives silly. Comparing test score is very easy. Gauging concept understanding is much harder. I myself will encourage my kids to understand the fundamental concepts.
> Young man, in mathematics you don't understand things. You just get used to them. - John von Neumann
That's easy to say, but hard to do. I find in my math experiences that I often don't truly master a concept until I am a level or two beyond it. I learned fractions when I was in grade 5 or 6 and could demonstrate adding, multiplying, reducing, etc. I think if you asked me 'why can you multiply across when multiplying fractions' I may have said I knew, but I don't think I actually mastered that until high school--when I had the skills to derive the fraction operation rules. Ditto with things like FOIL, completing the square, etc which I don't think I mastered until I was out of high school.
And of course, everything seems easy and obvious in hindsight.
I suppose what I mean is: I don't think there is anything new about what the author lauds as a goal. It's what everyone has been trying to do for 50 years. It just isn't easy, and so far no one has found a magic bullet aside for lots of hardwork and practice.
Edit: Oh :) I just worked it out - I was taught to fold the subtraction and carry right in one step. I'm an idiot!
Also in the states, engineering students ask the same question about Laplace transforms. ;-)
Because I want you to build things that don't fall apart because of resonance?
Many engineering students go through their education, having heard from someone, that they won't need their math after they graduate. And things do fall apart.
Don't get me started on engineers who think floating point numbers can be treated like real numbers for equality testing.
Laplace transforms are wonderful because they protect you from the differential equations that would otherwise await you if were foolish enough to take a step outside the S-domain.
It makes perfect sense that prospective educators would specialize in different areas of math than those with a pure math (or engineering, or physics, or statistics, etc) degree.
What university did you attend that Laplace transforms were part of the standard calculus curriculum, let alone a requirement for primary school teachers? From my experience in the US, universities typically offer different sets of calculus classes based on the major. Undergrads in business school might have to take a simplified calculus class or two, but the content is not as in depth as the calculus classes required for STEM majors. I learned about them in a signal processing class, not from the math department. I can't think of a single reason why a primary school teacher should have to study them.
I find it odd that laplace transforms would be part of the general calculus curriculum, but even more odd that non-STEM students are required to take those classes.
I think it is a mistake for our education system to treat math as something superfluous and abstract from daily life. While neither Laplace transforms nor Shakespearen analysis are necessary to live, the former is probably more useful than the latter.
Without context, such an overbroad generalization is at best meaningless.
If someone is choosing between potential spouses, or addressing a dishonest business colleague, or deciding how best to recognize a selfless samaritan, few branches of mathematics would be of any use. Whereas, one of Shakespeare's hundred and an half sonnets might be of direct "use".
Usefulness and utility depend on context. Contextless generalizations about the value of different branches of human knowledge is generally meaningless.
I don't know in your school, but in our people would complain and complain and complain about even theoretically having humanistic course compulsory. And those courses were designed to be free grade and any expectation on student to actually put effort into it caused complains. I did not wanted them either, it is not like I would just complain about other people.
Nobody expected tech students to read Shakespear much less analyse it.
This is the underpinning of every conversation where engineers say college is a waste of money. If anything, I see engineers complaining more loudly about taking any course that isn't directly related to engineering. This not only includes the humanities, but also things like business.
> While neither Laplace transforms nor Shakespearen analysis are necessary to live, the former is probably more useful than the latter.
Learning Shakespeare is typically tied to learning how to communicate through reading, writing, and comprehension. IMO, that is much more generally useful than learning anything about a Laplace transform.
Learning to do a poor quality imitation of early twentieth century literary criticism is at best a really inefficient way to teach communication through reading, writing and comprehension. Vocational pseudo reasons to teach the humanities are and always have been horse shit, detracting from the pleasures of those who are actually interested, whether as teachers or students.
I suppose I agree with you, but I think math is not given enough credit.
Edit: the textbooks are freely available on the web, if anyone wishes to take a look. Look up "NCERT class 10 maths textbook".
Information is laser focused on getting into an "Engineering college" and not actually on "knowing" anything. It's like teaching someone about spoons and forks but never giving them any food to actually interact with but vaguely mention "food will be relevant when you get a job, not right now"
Better than american kids sure but i'm not sure anything more of value is earned by making these kids essentially memorize theorems and algorithms
You should also think about if it's more important (to the people you're telling that message) to get a good wage or if they're fine taking a few years to catch up.
"Understanding" concepts is great and all, but unless they're doing that additionally to training the simple application, they'll most likely start having issues once the tests are up.
Having good grades (and being able to prove their ability on the spot) is definitely correlated with good starting wages. Not so much later on, but it's a massive difference at the start.
Moved to the US, math was easier and it kinda fell off. Was back to Ukraine for some time, but it was never the same afterwards, and now I have the same math anxiety many other people do.
I really think the abstract nature of the way maths is taught (at least in the UK) is a big problem that holds people back - kids don't understand the point of the more complex stuff.
I recall asking my secondary school maths teacher what the point of learning about some concept was (logarithmic equations, I think), what practical applications it had - he couldn't answer that.
If I'd understood how such concepts could be used in the real world for interesting things, I'm certain I and others in the class would have been better able to "get it", and would certainly been more enthused.
Then I tutored people doing "business math" - calculus but using finance for all its applications. Most people struggled not because of the math but because of the difficulty in understanding the finance part. And it was hard for me to give help because I had to learn the financial aspects to help them with their HW.
If the application doesn't click with you, then you now have two problems.
I was done with the problem before my peers even hit the right keys in their pocket computers (TI-83 is not a calculator, but a computer pretending to be a calculator).
I still think the non-calculator way of teaching math gave me better skills down the road.
Perhaps the problem is that there is no difference between 2x + 1 = 0 and 3x + 3 = 0, they are the same kind of question but in my school days we have to do all these similar tasks for ... whatever reasons.
Then again, practice is a really good way to solidify one's understanding of a concept so its a hard thing to balance.
I’ve taught at a number of “nerd camps”, generally in the fields of Gamedev and 3D modeling. I found that, more than anything else, showing young students what they can use a tool directly for triggers the imagination and inspires them to engage further with the field. These students were able to produce Google Cardboard VR content, which at the time was impressively close to the state of the art.
Anecdote: I’ve personally seen a 10-12 year old student tinker with and research the shadow settings in Unity3D [1], which is quite complicated! And she was willing and excited to put in the work because she could see with her eyes how the options affected the shadows directly. She wanted to match the shadows with a drawing she brought in.
Too bad Logarithms are not as engaging to look at as video games...
The example you give is an interesting one. Logarithms are quite hard to motivate[+]. Once they could be introduced as analogous to a slide rule as something to speed up multiplication (indeed that was their actual original motivation). But that isn’t convincing these days. Perhaps a motivation could be found in science (something about gradients and log-log plots). The problem is that the real reason you learn about logs is because they are related to exponential and exponential are important. Consider the differential equation:
dy/dx = k y
k dx/dy = 1/y
k x = log(y) + C
log(y) = k x - C
y = exp(k x - C)
y = A exp(k x) (where A = exp(-C) is chosen to fit initial conditions)
[+] By motivate I mean “put into context of some practical application”.
In the “real world” such mathematical things aren’t necessarily well motivated either. It may be that one has to calculate certain weird things because that is what clients/customers/taxes/regulations/circumstances dictate.
In applied mathematics things aren’t so well motivated. Either “we see these equations in physics sometimes” or for fluids you mumble something about boats or planes. In pure mathematics the motivation basically always boils down to “we need this crazy thing to solve a certain problem to tell if two objects are the same/different/somehow interesting because we are interested in this sort of object and in particular how to classify the different kinds of it.
https://www.mentalfloss.com/article/76144/why-no-one-wanted-...
Sub-Satoshis are used only as transaction fees in lightning network, and even there 1 milli Satoshi is the smallest amount that's allowed for now AFAIK.
Once someone learns math, what incentives do they have to choose teaching elementary school over alternatives?
I could imagine that the skills to learn math don't overlap with the skills to teach elementary school and that time spent one detracts from the other. If mastery in each takes a decade or so and few people dedicate themselves to double time in school for a lifetime in communities that misunderstand them, I could see teaching k-12 math as a career and life disaster.
I guess there's more money in advertising than educating the next generation.
It's a vicious cycle. I'm a data scientist in SF and I'd make 5-10x less if I taught grade school.
Arithmetic in base 2 is so easy, you can literally teach your pet rock to do it (assuming your pet rock is a reasonably pure silicon wafer with scattered boron and phosphorous impurities shot through with thin veins of copper.)
And no, I'm not advocating teaching kids to convert between different bases, which they almost universally find confusing - I'm saying we should teach kids arithmetic purely in base 2, and only expose them to base 10 once they understand the fundamental concepts.
Am I being serious? Well, it's certainly a lot easier to dismiss this approach out of hand then to pinpoint what - if anything - is actually wrong with it.
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 opinion the most practical thing that kids need to learn first. It's more about remembering than logic.
When the get older, math comes into the picture. They can leave calculations up to calculators or computers, and learn the language of math.
So in my opinion, there needs to be a more clear distinction between calculations and math, because they are definitely not the same.
Both have their applicability, but knowing one doesn't imply knowing the other.
Each one also has a proper age group to start teaching it. Little kids have great memory but limited reasoning skills. That changes when they are older, so at that point you should switch from calculations to math. Dropping calculations for math is just as stupid as dropping math because you think calculations are enough.
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 just these freaking fractions and their convoluted addition I do not know how to properly convey.
It is their addition which is not, as it does not have any obvious counterpart in simple nature.
I had a math teacher at the university who told us once "there is a neat trick which is normally taught next year but is useless by then and super useful today. Just use it without too much thinking."
This is my reasoning too, I showed them the always working way to add fractions (cross multiplication) and hope that someday they will appreciate the beauty of the thing.
Many kids never "get" fractions, but do remember cross multiplication! And so they use it for anything with a fraction. 1/3 + 4/5=? CrossMiltipy! 1/3 x 4/5=? CrossMiltipy! 1/3 - 4/5=? CrossMiltipy! 1/3 ÷ 4/5=? CrossMiltipy! 1/3 + ? = 4/5 ... CrossMiltipy?
Lots of kids complain about math because there is so much to remember. That's a bad sign because there is so little to remember if you understand the concepts as you can always re-discover forgotten tricks.
In similar fashion, as a little kid, I was not allowed Velcro shoes when learning to tie my shoes. After I showed I could master tying them, then I had the option to go back to the easier Velcro option.
1. Cut the first sheet into 3 equal parts. Take 2 of them (labelled A and B) and place them side by side on the third sheet.
2. Cut the second sheet into 4 equal parts. Take 1 of them (labelled C) and place it on the third sheet next to the other two pieces from part 1.
3. Now, figure out how much area the three pieces cover of the third sheet of paper. If you do it right, you can cut the third sheet of paper into 12 pieces, such that the 11 of these pieces will cover A, B and C exactly.
Sometimes you will need a fourth sheet of paper in case the sum is greater than 1.
There are many variations of this. You can pour 2/3 of the water from one 100 ml graduated cylinder and 1/4 of water from another 100 ml graduated cylinder into a third one. Or use sand and weight it. If all numbers are divisors of 12, you can use the clock.
Which isn't so bad for a recipe because most things aren't that precise in cooking - but not so easy for all students to work out, especially without having a tablespoon to use. This is even so when you have tablespoons: Those 1/3 cups don't divide neatly into tablespoons.
That said, it isn't like you cannot illustrate it, of course, but just that this method isn't the best. I've just memorised the cooking stuff since I only have a 2-person household.
But they do into teaspoons.
This allowed for a great visual (and tactile) illustration of how fractions work. 1/2 + 1/4? No worries, just break off some pieces and add them together. Then you can compare the length to various other configurations, and you can viscerally experience that 3/4 really does equal 6/8.
I don't know whether this is a standard way of teaching fractions, or whether this teacher was particularly motivated or whatever. But it made a lasting impression on me. Thanks, miss Annie.
The benefit of this approach is you can easily go to 1/8 and 1/16 and do addition and subtraction. Even multiplication and division. Get two pizzas to show 8/4 and 16/8 etc.
And you can eat the whole thing afterwards. Fractions are delicious.
The best way I found to learn fractions was through carpentry. Measuring and cutting over and over quickly builds the intuitive sense of how fractions work.
One thing it helps is that they are quite employable since their resume stands out.
>> A colleague told me of vainly trying to persuade a college student that .015 was less than .05; the student insisted “but 15 is more than five.”
>> To [students], [the equals sign] doesn’t signify equality, but instead means “put the answer here.” Imagine their confusion when, in algebra, they first encounter problems with numbers on both sides of the equal sign.
Is this really true? I'm sure I and my children (in the UK) were presented with fill in the box questions like:
7 + [ ] = 10
at the very earliest stages of arithmetic.
7 + x = 12
"The 'x' represents the answer, so we have to get it on the 'put the answer here side'; lets write 12 = x + 7 ; now we just need to take 7 from the 'put the answer here side' and we're done. But whatever we do on one side we have to do on the other. Take 7 off. 12 - 7 = x + 7 - 7 => 5 = x ... bingo".
https://news.uchicago.edu/story/female-teachers-can-transfer...
They could have taught us about arithmetic, but instead it was "Just stare at this table until your attention wanders off, then daydream until the bell rings and hope by chance some bit of it sticks in your memory."
No phonetics or linguistics to explain how things were spelled, instead just "Write this word over and over until your hand cramps, then do the same with the next one." As if by getting enough hand cramps you would magically learn how language works.
Just recite "In 1492 Columbus sailed the ocean blue." Never any question into why the Portuguese, with their lead on exploration and colonization, did not end up with a bigger overseas empire than the British. As if that should be obvious as long as you can recite that rhyme and get the date right.
If the teachers are around my age, most of them were probably taught the same way, which explains why they wouldn't understand math.
This is, of course, a well known and reliable way of solving fractional division problems, but it seems to be the _only_ way I’ve seen people solve fractional division problems (by using the reciprocal rule). What would be the approach that doesn’t involve using multiplication?
Or maybe require more stringent testing to become an elementary school teacher in the first place. If this means too few eligible candidates, pay more. It seems like a no brainer use of tax dollars.
I'm appalled that it's so acceptable to let people who don't understand a thing teach it. Maybe it just comes down to "pay teachers more."
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.
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/dictionary/d001.htm
(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.
Foo = bar and 1 = 1 are equivalent in my mind.
How do you square that circle, "fine" for assignment; wrong for equality.
Depending on which programming language you're using, "foo = bar" can mean either "make foo now equal to what bar is" or "is foo equal to bar now?". Neither one is the same as in mathematics, which is "foo is categorically equal to bar".
Consider:
foo = 1
bar = 2
foo = bar
In programming, this is perfectly valid, and means "ok, ignore what I wrote just 2 lines ago, and now make foo equal to 2, also".
You say you think it means "the LH var equals the RH value assigned to it", but you don't say what you think it means to assign to the literal 1.
The fix is to have specific teacher undergraduate or graduate degree, or education departments or state-funded full-blown normal universities, pay them well, make teacher profession a honorable career, only then the best will seek a position in k-12 schools and our next generation can keep getting better.
“X... and that should scare you”
I would really wish the emotional appeal journalism would make its way to the door but it doesn’t look like that will ever go away.
TLDR: the tilt does in fact affect the season
Yes, the earth is tilted on its axis. Yes, this tilt affects how much energy is absorbed (seasons). No, the seasons do not change because of the tilt of the earth.
The tilt of the earth does not change. The seasons change because the earth is rotating around the sun. That is, on one side of the sun the northern hemisphere experiences summer and on the other side, winter. It _is_ our orbit that causes seasons to _change_. Said another way, if the earth stopped orbiting seasons would stop changing.
https://spaceplace.nasa.gov/seasons/en/
What Causes the Seasons? The Short Answer: Earth’s tilted axis causes the seasons. Throughout the year, different parts of Earth receive the Sun’s most direct rays. So, when the North Pole tilts toward the Sun, it’s summer in the Northern Hemisphere. And when the South Pole tilts toward the Sun, it’s winter in the Northern Hemisphere. It's all about Earth's tilt!
Many people believe that Earth is closer to the sun in the summer and that is why it is hotter. And, likewise, they think Earth is farthest from the sun in the winter.
Although this idea makes sense, it is incorrect.
It is true that Earth’s orbit is not a perfect circle. It is a bit lop-sided. During part of the year, Earth is closer to the sun than at other times. However, in the Northern Hemisphere, we are having winter when Earth is closest to the sun and summer when it is farthest away! Compared with how far away the sun is, this change in Earth's distance throughout the year does not make much difference to our weather.
The changing tilt of the earth relative to the sun is what the seasons are.
it's tilted. and it moves to the other side of the sun. so its tilt relative the sun changes. but it doesn't rock back and forth on its own
https://www.epi.org/publication/the-teacher-shortage-is-real...
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 profit. Society as a whole would obviously benefit from exceptional math teachers at every school, but we would need to get creative to pay for it. Perhaps we should be spending less on wars.
This isn’t true.
From the CPC Popular Front funded Brookings Institution.
https://www.brookings.edu/research/how-progressive-is-school...
> School funding systems vary by state, but in the typical state, the financing of K-12 education is shared roughly equally by the state and local governments (with the federal government covering less than 10 percent of the cost). Most states have provisions designed to target additional funding to districts attended by low-income students, with the goal of making up for these districts’ limited capacity to raise revenue through property taxes and other local sources.
> Nationwide, per-student K-12 education funding from all sources (local, state, and federal) is similar, on average, at the districts attended by poor students ($12,961) and non-poor students ($12,640), a difference of 2.5 percent in favor of poor students.
I think this is exactly the intent — get the whole county / district to pay, not just the parents, right? And these payments are usually compulsory regardless of whether you have children or send them to private school...
TEACHER QUALITY
https://faculty.smu.edu/millimet/classes/eco7321/papers/hanu...
> Perhaps most remarkable is the finding that a master’s degree has no systematic relationship to teacher quality as measured by student outcomes. This immediately raises a number of issues for policy, because advanced degrees invariably lead to higher teacher salaries and because advanced degrees are required for full certification in a number of states. Indeed, over half of current teachers in the US have a master’s degree or more. Teacher experience has a more positive relationship with student achievement, but still the overall picture is not that strong. While a majority of the studies finds a positive effect, only a minority of all estimates provides statistically significant results. Even the subset of studies that use a value added approach and information from a single state produce a highly variable set of results (see bottom panel in Table 1). If anything, the value-added estimates within individual states suggest more strongly that experience has an impact, although still only 41% of the estimates are statistically significant. It is quite likely that a number of these studies lack the statistical power necessary to identify precisely the experience effects.
And yet, I cannot find anything on the subject. That is very suspicious. It’s like no one wants to look into it. Can anyone tell me what’s going on here? Did I miss something, or is it just common sense that the intelligence of your teacher doesn’t matter?
Are they going to spend their extra cash on mathematics lessons? In their copious free time, I presume. Just like these rich software developers spend all their extra cash on more school. I see a lot of Porsches and BMWs in my neighborhood but curiously none parked at the community college.
Are you going to hire different people, who chose to go into computers just because it paid better? I don't see many software developers who would make good elementary teachers. Or would want to try.
I saw this, teaching college math for one semester long ago, and also in talking with adults about their experience in high school and college math.
As a consequence, a person may be completely functional at basic math, such as balancing their checkbook, while simultaneously believing that they don't understand math.
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).
When talking about arithmetic expressions themselves, you would use a meta-language (e.g., English). The arithmetic expressions would then be called the object-language (the language being studied). All of this should be pretty clear after taking a serious course in Logic.
I am not saying the choice of words is not sensible, I'm saying it it factually wrong.