For over a decade, I have run a successful tutoring center focused on math and science for elementary school students in a developing country. Our students won numerous awards both at the national and international levels.
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!)
For those curious, below are some kinds of primary math questions we model after to test teacher applicants.
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... "
I'm a working mathematician and would need a non-trivial amount of thought to get any of these problems.
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.
I know some students who excelled at this kind of problems when they were in elementary school and now possibly on path to be good researchers in math and theoretical CS.
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?
and here I sit totally flummoxed by the questions yet later on you state these were suitable to grades up to sixth?
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?
To be clear, these are hard contest questions that most people, including older kids and adults, probably can't answer them without proper learning and practice.
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
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.
You don't need algebra. You know that 10 numbers are 2012, so on average 201, so the middle of the numbers needs to be around there. After that you just try a few sequences around 201 and you'll find it pretty quickly, most can add 11 numbers pretty fast in 6'th grade.
Forcing yourself to only rely on the solution strategies taught by school makes you very bad at problem solving.
I specifically set out to find a solution that doesn’t involve any brute force (trial and error). Trial and error can waste all of your time and get you nowhere on nontrivial problems.
You wondered how a sixth grader could solve it, I showed you how. Being able to do it using formal methods is good, but my point is that sixth graders are still pretty smart and can use more creative methods which still works very well as I showed.
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.
I'm a pure math student and when I first found the problem I sent it to my friend who is a stats student. She solved it very quickly, but admitted to using a calculator. None of my exams permit calculators, so I've gotten used to never using them.
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.
199 is between 195 and 206 proving that it is a valid answer.
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 solved it like this:
The numbers are x, x+1..., x+10 = 11x + 55. One is erased therefore 10x + 55 - y = 2012. 10x - y = 1957. We know X is whole so we need a number between 1957 and 1967 that divides 10 and that is X. 1960 so X = 196 and therefore y = 3. Answer is X+Y = 199.
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.
Due to a parity issue (50 here is even) we need to adjust username90's solution. In particular, a sequence of consecutive integers of even length has a half-integer in its middle.
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:
Right, but we are talking about sixth-graders here. If a sixth-grader saw username90's solution and tried to apply it to the adjusted problem, they would not know that the result is incorrect without actually summing up all the numbers to check, an undesirable chore on a math contest with other problems to worry about.
The act of figuring out how to solve the problem is itself an act of trial and error. And in many problems, the clever part is figuring out how to target your trial and error.
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.
OTOH, puzzling something out through trial and error fosters curiosity and teaches a "fail fast" mindset. For a sixth grader, this "play" may be a more beneficial lesson than only being able to tackle problems for which you already know the optimal strategy (in this case, algebra)
I disagree. The hard problems in math, science, and engineering are not amenable to a "fail fast" mindset. They take years, decades, or sometimes centuries of study to solve.
This sort of puzzling through play is good for solving math contest problems that have been specifically designed for the purpose, and little else.
The eleven numbers are n,n+1,..., n+10. Their sum is 11n + 55. Let the number removed be n + k, then the sum of the remaining is just 10n + 55 -k = 2012. Or 10n-k = 1957. Then it is obvious that k=3 so that n is an integer. Hence, n=196
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.
What do you mean by 'engineering form'? Using variables in the answer?
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.
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.
You’re allowed to delete whichever digits you like. But you aren’t allowed to reorder the digits which are there. I.e. if you had to delete one digit you could go from 189 to 89 but not to 98.
I always found it amusing that most of my math and cs professors in college seemed to have terrible arithmetic skills. I'm sure part of it is just that it's hard to concentrate on the work when you're also narrating the process to the class, but I suspect it's also that basic arithmetic is a relatively unimportant skill in most advanced research. hiring this sort of person to teach/tutor grade school math might be like hiring a porsche engineer to repair your car. they'll get the job done, but maybe not as efficiently as your local mechanic.
I'm a mathematician, not a calculator. Mathematicians spawned an entire field, computer science, to alleviate the tedium of computation!
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.
I still don't know why long-division works. I don't think I even remember how to do it as an adult.
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.
I don't understand why long division would be a mystery. It's just repeated subtraction - you are finding how many times you can subtract the divisor from the dividend, in a systematic way.
It's almost never explained with such lucidity. If you find a hundred people who can perform long division, I would be surprised if twenty of them have any idea how it works.
Multiplication tables are the most lucid way to explain long division. In order to get from 5 to 40, you need to keep adding 5s until you can't. So to divide 40 by 5, keep subtracting 5s until you can't any longer.
Exactly, I find it far easier to explain LD as just fast subtraction combined with multiplication to my kid than common core methods which I had to stare at for a while. The most critical method here is how to break up components of a number into addition "parts" as well as decimal positions.
Sure you can! Just look at a typical fifth- or sixth-grade classroom. Most of them can perform long division, briefly, before the test. They may even remember it for several days after the test.
Thanks! What you said, except my own point d) that Swedish Wikipedia tells that apparently short division is also used in Swedish elementary schools now.
Interesting, I'll start using that short version now. I always hated the long version because it took up a ton of vertical space and I didn't need the extra verbosity. In fact, I'll probably teach me kid that method when he gets to division in school.
Holy shit, I didn’t realize the shorter way of writing long division that I invented in early elementary school because I got sick of taking up so much space and introducing copying error potential had a name!
I've looked at both Wikipedia pages and still can't really understand the difference. Is it anything more than notational? Either way, you're calculating divisors for each component of the problem, and it only seems to matter whether you calculate it on paper or in your head.
the way I learned it, it's just a notational difference. either way, you are dividing the divisor into each place of the dividend (starting with the most significant digit) and carrying over the divisor to the next place. short division is just a more implicit way of doing this that saves space. the shorthand version doesn't really work for divisors with more than one digit though.
it's not that the algorithm "breaks"; it's just that it stops being a convenient notation for humans.
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.
It’s notation only. I “invented” it as a kid to save space on division problems and reduce the chance of copying errors. It’s the same method as long division (and TIL it already existed!)
It definitely looks nice, and I'll probably start using it. If something is too inconvenient to use short division for (e.g. large divisors), I'll probably just use a calculator.
Went to school in Sweden too in the 90-00s. I wasn't taught long division until I had to divide polynomials in my last year of high school, and that course was an elective.
Because knowing both helps realise how it works. Yes it's faster to use short division, but long helps you se the step by step process, giving a better understanding.
>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 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 / 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.
This is the right answer but maybe not to the right question. For most applications the linearity of variance does not really matter (e.g. when taking descriptives of a sample), at least in the way they are used/taught. Not that many appliers of statistics do computations where the mathematical properties of variance have any significance, and even when it does (e.g. ANOVA), most appliers don't really understand where it comes from.
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.
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.
But why not use the abs function or operator rather than squaring and rooting?
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.
> But why not use the abs function or operator rather than squaring and rooting?
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.
And suddenly another small subarea of maths starts to make sense for me. Thank you, and thanks to everyone commenting in this subthread.
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.
By the way, statistics as its own discipline is taught rather differently from the "here is a toolkit of magic tables and formulas and how to use it" method used when teaching it to non-statisticians. The magical bits actually make sense.
>I too have a problem with math tools passed down without any surrounding context
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?
The mean absolute deviation should be always introduced to people before introducing standard deviation. When trying to figure out what the heck is the standard deviation I also stumbled upon the mean absolute deviation. Now, to me, the standard deviation is the mean absolute deviation. It's just a version that is more sensitive to outliers. Otherwise, I still don't feel like I intuitively (visually) understand the standard deviation.
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?
Okay, we have deviations from the mean. They are (0.1, 0.1, -0.1, -0.1). Or we have deviations from the mean (0.2, 0, 0, -0.2). I think is is pretty intuitive that in the second case the deviation is larger. In the first case there are only two differing values that are 0.2 apart. In the second case there are three differing values and some are 0.2 apart and others are 0.4 apart. Also, in the first case there are two pairs of equal value and in the second case there is only one pair that is of equal value. The squaring and rooting does this for you. Of course, other functions could do a similar thing for you. The squaring and rooting has nice mathematical properties especially if you consider the variance, i.e., omitting the square root. That is much more differentiable than an absolute value.
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.
The standard deviation is not arbitrary, it tells us what the final normal distribution would look like if you averaged a lot of independent distributions.
You can read up on the central limit theorem if you want to understand more.
Normal distribution can be parameterized with almost any measures of location and dispersion. The reason why standard deviation (which is just variance in hiding) is used is because it makes the equations (that non-statisticians practically never need to look at) a bit nicer given the other (mostly arbitrary) conventions used in the maths.
Put simply, standard deviation is arbitrary for most people.
I don't want to keep going on because it sounds like I'm trolling, but when you tell me 'it tells us what the final normal distribution would look like if you averaged a lot of independent distributions' I just want to say 'why do I care about that? where did that requirement come from?'
Economics of stocks is a good example, when you put a lot of stocks in one portfolio you can calculate its risk (standard deviation) and returns (average value), and see how much each piece of stock contributed to that. This is because the results of many variables looks like a normal distribution with the combined average and standard deviation. Now it gets a bit more complicated in the real world since stocks are not independent from each other, but it shouldn't be hard to understand why knowing the standard deviation of something is useful.
> Now it gets a bit more complicated in the real world since stocks are not independent from each other
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.
Right, that is how statistics usually is done. Note however that you don't need IDD, independence is enough. So you assume independence, from that CLT follows which means that standard deviation is all that matters. Then the real world deviates from this based on how much your distributions fails to be independent.
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.
I can't understand this. Surely either samples are independent, or they are not?
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!
Statistical techniques are a model of reality: Reality is not rigorous, but when our models are rigorous, imprecision enters the computation at one place only. If our models aren't rigorous, imprecision occurs at more places, blurring the answer even more.
"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.
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.
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.
I think you're right that there is an arbitrariness to this algorithm, and there are other ways of normalizing your data. And you could probably build entire new branches of statistics off of them (like non Euclidean geometry), and they'd probably even be better in some way. But, you'd be speaking a different language from everybody else.
Speaking of language, have you ever thought about how all names and grammatical structures are arbitrary as well? There are parallels.
No, standard deviation is special. If you have a large enough amount of independent distributions which gets averaged or added in some way, then the only thing which matters are their standard deviations and average values for what the final distribution looks like.
I'm well out of my wheelhouse, so anyone else chiming in would be appreciated. I'm not sure what you mean by "looks like", but I would imagine other "nonstandard" measures of deviation would also result in a consistent model.
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.
"Looks like" as in the difference approaches zero as the number of times approaches infinity.
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.
The mathematical representations and operations people see at school were picked because they are easy to work on most common problems. There is not this "correct math" thing, you pick what works best.
Start with the Normal or Multivariate normal. We start with this because it is the distribution that emerges naturally when we have lots of little errors adding up around a basically consistent effect (see also [0]). So basically any situation where something ought to happen but 101 things could also happen to cause small changes will turn out to be normally distributed. An exceedingly common occurrence + one that almost inevitably turns up when dealing with a practical error prone process.
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.
^ 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.
> 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.
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:
I think you just described the common core method. The main problem is teachers don't understand all the methods they are teaching and kids pick up on this.
Then the parents don't understand all the methods and the few intellectually curios help their kids figure it out; but he vast majority of people just tell their kids it's stupid.
I didn't address the topic of what and how children learn, I've given some ideas to the one who wrote:
"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.
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.
Taylor series, calculating maxima/minima, etc. also standard deviations compose in a very nice way. All throughout math, we use squared terms instead of absolute value all the time because it’s convenient.
1. Long division uses a similar idea to Euclid's algorithm for GCD. Are you familiar with that? Regardless, both algorithms are divide and conquer algorithms.
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.
GP must be talking about the shortcut procedure to do it that is taught in schools.
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...
Now you’re talking about something different. The proof of the division algorithm is built up from a bunch of other machinery. It relies on the fact that the ring of integers is a Euclidean domain.
Take a course in ring theory if you want to see how all of this machinery gets built up.
Bud, I really hope you can grasp more complicated algorithms than that if you have a PhD in computer science. It's one thing to call it unintuitive and not worth your time, but let's not be hyperbolic.
Try again? Long division, dividing A by B, is a for loop that maintains the invariant A = QB + R, where Q is the number written on top (initially 0) and R is the number written on the bottom (initially A), by subtracting values of the form B(C)(10^N) from R and adding values (C)(10^N) to Q to compensate.
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)
}
This is ten lines, not “astronomically complicated.”
This is just re-stating the algorithm for the umpteenth time. I can read the algorithm. I just can't intuit why it works or why it has been designed this way.
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.
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.
And if want to divide it 4 and, suppose, you cannot do it in your head, you do it step by step, by clever regrouping with the distribute law:
The same works for other bases. If you were to implement some bignum library, you would also choose some base n representation for you numbers. Base 10 is not so optimal for computers, so maybe you chose base 2^32. If you then were to implement a division function, you would use similar algorithms.
Hell yes! You finally made this clear for me why we can go through the dividend number by number (or power of 10 by power of 10) to apply the divisor. Thank you so much!
You don't have to do things in decimal, it's just convenient since you get the answer digit-by-digit.
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
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.
Is the question really why we train children on basic arithmetics in base 10?
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.
It sounds like you've memorized the very specific notational steps. But if you would have to come up with an algorithm for doing division using only integers with multiplication, subtraction and comparisons, you would come up with some equivalent to long division.
Lol I can't understand any of that. 'Once is', 'twice is' what does that mean? Once what is what? Twice what is what? You aren't even writing coherent English phrases how am I supposed to follow that?
They're referring to the number of times to subtract 35 from 350. Do it 10 times and you end up with 0, there is nothing more to divide and so 10 is the answer.
Well if long division is unintuitive, the procedure to compute square-roots by hand is even more mysterious, as if it has transcended dealing with numbers and instead become a symbolic manipulation. Thankfully it is something I never had to apply, even once in adult life till now.
The calculation of the square root, relies on this result from squaring a number N = 10d + e (that is, single digits d and e, but d is in the 10s place):
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.
It wasn't until I learned about for-loops that I started thinking of multiplication as repeated addition and division as repeated subtraction. Before then I thought purely in terms of geometry.
I think it depends on how you were taught the concepts in elementary school. I think a lot of it is based on things like bar charts[0] and visual representations at first, so it's possible you were never taught how it was repeated subtraction and addition, how powers are repeated multiplication and so on.
At my kid's school they start teaching division as "sharing" - a kid has 10 pieces of candy he wants to share with 5 friends; how many pieces can he share with each friend. Later they describe it as repeated subtraction - how many times can his 5 friends take a piece of candy from the pile of 10 pieces.
Yep. I remember in college when I made that connection and realized you could create infinite "levels" of operation, with each repeating the previous one N times.
Not that story in particular, it's a perennial story on HN - I first learnt about that notation through the "largest number you can write on a postit note" challenge.
Long division calculates the multiple at each step; short division just finds the remainder. That doesn't seem that different but when you come to do it algebraically you need to do long division. I did both at primary school (age 10), but it wasn't until 6th form (age 16) that we used long division again, those who hadn't done it at primary struggled a little.
083.28
2.26...
___
7 |583.00
That's short division (sometimes called the "bus stop method" in UK), top line is answer [quotient], next line is "remainders". You say "7 in to 5 won't go; 7 in to 58 is 56 [just from knowledge of times tables, it's 56 tens your dividing], with remainder 2 [write remainder down, usually as a superscript to the dividend]; 7 in to 23 goes 3, remainder 2 [write remainder down]". Now you have the answer 583/7 is 83 remainder 2; but you can continue and divide the 20 tenths by 7, and so on.
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]
What we're doing is taking five-hundred and saying can we divide that by seven-hundred, we can't. So then we say well how about fifty-eight tens, can we divide that by seven-tens. The tens cancel each other out ( 10/10==1 ), so 58/7 = 8r2 but this is really saying 580/70 is "80 lots of 7" and 20/7 left over. So now we add that 20 units to the 3 units we have already from the dividend we started with, so now we need to do 23/7. And so on ...
I'd do an algebra example but it's a pain in the arse just using ASCII.
School usually start with remainders instead of trying to teach repeating decimals at the same time. So here the answer is just 0 remain 1.
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.
Shift the decimal point of the operand right by N places (where N is how many decimal places of precision you want), do the long division, then shift the decimal point of the result left by N places, and you have your answer.
Then you never knew it? Perhaps what you were taught wasn't long division. My kids were not taught the same math that I was taught in the 80s.
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.
Conversely I have been immediately impressed with the way my first grader is being taught math.
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?
I remember when the common core debates were taking place, one of the memes running in parenting circles was a comparison of long division vs how common core taught division, and how much more “complicated” the common core version was.
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.
My little brother was always better at Math than me. Whenever he tried to show me how to do something it was different than how our teachers showed us. He was really good at figuring out a method that made more sense for him, and it certainly worked for him.
Anyone who legitimately thinks common core is worse than how math was previously taught does not understand math at all. It is at worst just different with the dream of being better. There is no conspiracy theory about wanting to sell more books or to "separate the masses".
If you actually understand the math picking up the common core methods is minutes of work at worst.
It isn't a federal mandate, which can be seen because not all states adopted Common Core. There were federal incentives for states adopting a standard which has been recognized as effective, including Common Core.
That’s a good point. You know what, it’s way easier to teach how to solve for x when y + x = 3x + 3 and y = 4 than it is to teach long division. As far as I could tell, that basic algebra is also much more important than long division, so why not teach it at an earlier age? The symbology of algebra is probably great for developing minds.
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.
Yeah,it's amazing, I've hooked multiple friends on this app. It turned my friend's girls math fear around, which is really important given that girls are learning to fear math from teachers who are themselves afraid of math.
I saw my nephew play this. It’s a great idea but I found the implementation lacking.
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.
Generally a lot of the formalism of Algebra requires specific brain development milestones dealing with abstraction to be reached. That’s why it is taught at the end of middle school. Now could some concepts be introduced earlier? Yes a common exercise in elementary school is to solve questions in the form of 7 + _ = 15. In this case x is replaced with a _ because variables are hard to grasp.
Children arnt idiots, they are people seeking knowledge. What they need is patience, understanding and slow explanations as they grasp both the concept of what you are saying, the language you use and the minor concepts that you take for granted.
Thing is, even now I find it useless. I can do nearly any division in my head whose digits themselves fit in my head (Miller's Law). For anything bigger, I have a pocket supercomputer with me at all times.
You're right that it could be a fun programming exercise, though
If you do code in any capacity not being able to understand long division is a huge red flag. In terms of difficulty it's somewhere between regex and being able to write a recursive function.
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.
From the other comments that user's posted I can kind of understand their sentiment, they've tried to form an understanding from others and then found the same roadblocks repeatedly.
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.
Sometimes even if you drag your brain kicking and screaming through a mathematical topic, it just doesn’t give a shit. I felt the same thing about polynomial long division back in high school — it was just a bunch of gobbledeygook — so I’d forget how to do it, and then divide polynomials by ad-hoc trial multiplication/subtraction, until I figured out the most convenient way to do that was to first cancel away the leading term. And today, I’m uncomfortable with multivariable calculus. I’ve made a few abortive attempts to acclimate to it, and I’m sure I could force myself to get there if I needed to, but right now that’s just not important.
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.
It's weird because I was taught long division in primary school that was different to everyone else in class (different maths class). Everyone seems to do some weird subtraction for each digit, whereas I was taught to carry right (i.e almost like how carries work with addition). Way faster, and it works the same all the way to decimal places, no weird remainders. Not sure why that wasn't the standard approach.
This is the problem with rewarding conscientious kids, they just follow instructions without ever asking themselves why. If you were more rebellious and refused to follow steps you didn't understand then you'd get worse grades but you'd learn more.
> They really should try to come up with algorithms that are more intuitively linked to a spatial comprehension of the numbers.
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
My parents invented a simplified division algorithm so they could teach me the four basic arithmetic operations by third grade, and it was so much easier than long division that I could never remember how to do things the conventional way. I just pretended to follow the teachers' instructions for the next few years.
(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 repeatedly got in trouble for not showing my work in the format they wanted, whether because of intuitive leaps or because of tricks I came up with myself.
This worked for me up until Algebra 2. Up through Algebra 1 I'd just do everything in my head, never show my work, and thus solved a lot of problems using non-standard/creative methods on the fly. Ended up hurting myself a lot. I didn't have any muscle memory for solving the trivial parts of problems as I progressed from Algebra 2 through BC Calc. This led to me choosing a non-mathy degree in undergrad.
I could breeze through higher-level logic courses, but was essentially remedial for higher-level math.
Teachers are a very mixed bag. Some of them are delighted to see a student learning, in whatever way that learning may happen -- and some are just annoyed that the classroom's reality fails to match up perfectly with their lesson plans. If you're a nail that sticks up, some will try to uplift you further and others will angrily try to nail you down. (And most are somewhere in between.)
I prefer to think of subtraction as the inverse of addition, and division as the inverse of multiplication. You only need addition and multiplication if you're dealing with rationals.
I've never thought of division as repeated subtraction: it's the inverse of multiplication, or literal division as in I have a pile of beans that I divide equally between N piles.
I don't think you're wrong, just seems an odd way to think about it, and fails to account for any remainder.
The remainder is what's left after you've repeatedly subtracted, and can't subtract any more before going negative. (I wish that one of my teachers had said this to me, but they were too busy getting everybody to memorize the steps.)
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.
Yes I love the Facebook memes by parents talking about how math is so complicated now. Most American adults struggle to solve a basic algebraic equation, because it’s not used in day to day life. If a new teaching style gives us a better way to teach concepts, seems worth a try..
I think one reason why it doesn't make sense to a lot of people is that when doing long division, it is taught with notational shortcuts that obscure what the student is really doing.
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.
This comment is hilariously on the nose for how a lot of math texts read (at least in my experience)
$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
Wait till you get into Theoretical Physics, where the same thing is being explained by people who don't even have a firm grasp of $MORE_COMPLICATED_CONCEPT, but know just enough to be dangerous with it.
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.
That was quite an eye opener though when we were taught polynomial division in high school. Not necessarily for how long division works because I already understood that, but because it was an example where short division could not be used.
Research also shows that, compared with other college students, future elementary teachers are especially prone to math anxiety —apprehension about doing math that’s so severe it interferes with actually doing it. That anxiety remains once they are in classrooms, and studies show that students learn less math from a math-anxious teacher.
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.
This is a consequence of refusing to acknowledge math as a required course. Teachers and schools loathe to fail a student--especially one who is actually displaying effort and is otherwise a good student.
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.
The author of this article does not understand math :). Learning math is difficult. There're two things you need to learn: fundamental concepts and skills. These two things are related and commingled. Mastering both requires extraordinary effort.
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.
Being able to mechanically perform an algorithm is the first step on the road to in some sense actually understanding it. If you think you understand the fundamental concept but you can’t solve a problem using the algorithm you probably don’t.
> Young man, in mathematics you don't understand things. You just get used to them. - John von Neumann
> If math knowledge is limited to “when I see this sort of problem, I do that” it’s inflexible; if problems are phrased just a little differently, the student is often stumped. And once students hit algebra, math increasingly demands thinking through a sequence of steps and picking the right mathematical tool for each. That’s hard if you don’t understand what the tools actually do.
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.
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 one step. I'm an idiot!
There seems to be a culture of proud ignorance around math more-so than other subjects. While I am sure there is some engineering student out there griping about having to take a psychology course, they don't seem to complain as much as when non-mathy people have to study math.
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.
> 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.
If you are using sonnets to decide that, everybody would be better off if you did study Laplace transformations instead. It is harder to be confused with real life.
> While I am sure there is some engineering student out there griping about having to take a psychology course, they don't seem to complain as much as when non-mathy people have to study math.
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 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.
If learning about Shakespearean sonnets is actually about building writing and reading skills, then I would say the same for Laplace transforms: they are about building logic and math skills and not about actual laplace transforms per se.
I suppose I agree with you, but I think math is not given enough credit.
> While I am sure there is some engineering student out there griping about having to take a psychology course, they don't seem to complain as much as when non-mathy people have to study math.
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.
Maybe we ought to meet people where they are. What if math class was a series of practical problems that the students had to figure out together and make proofs for and the teacher has the classes compete with one another for progress through the lessons? You could make sure no one gets left behind by having a test with a minimum threshold of passing scores. Gets those less motivated, yet competitive kids behind it, even if they couldn’t give a hoot about math. Plus it’ll make nerds heroes!
>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"
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.
Nah, not an intro to Calc... it was second year Electrical Engineering calculus, but students were from all different courses (I was doing Compsci at the time).
Maybe I wasn't clear enough, but I wasn't referring to intro classes either. I had to take several calculus classes that were for STEM undergrad students and I still didn't encounter laplace transforms until a signal processing class. Students in other majors like business or education had their own separate set of less in depth calculus classes, if they had to take any at all.
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.
My response to primary school teachers is that they need some mastery of math so they are not anxious about the subject in front of students. Because if the girls in her class lookup to her and she is anxious, nervous, or dismissive of math then those girls will not go into any STEM field. If you want more females in STEM careers then it starts with elementary school teachers loving STEM.
Primary school maths is not really maths, is arithmetics, asking a primary school teacher to go into some crazy higher-maths details in order to teach arithmetics is just a huge waste of resources.
I would argue that calculus and Laplace transforms aren't math if there are no proofs involved; it's just multi-step arithmetic and symbol manipulation.
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.
Your school had a special fake Math degree for aspirant teachers? What was it called, Math for Educators? Was there any indication on the transcript or diploma that it wasn’t a real Math degree?
Damn, what a toxic and misguided comment. I have a MS in Applied Mathematics and there are many areas of math (especially “pure” math) that I’ve never touched. Same goes for my professors. The field is so vast and so deep that it’s impossible to learn even a negligible portion of it (unless you’re immortal). Are our degrees fake?
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.
Yes my college had a math for educators degree. https://www.bsu.edu/academics/collegesanddepartments/math/ac...
I was a math education minor and I had to take 9 classes (3 credit hours each) including 2 on teaching math. It didn't seem very much like a "fake math degree" to me.
Seems unlikely, though it may vary from one country to another. Here in the states, you would probably need to take a year of math, but there are levels of math "below" calculus that you can take to get a K-12 general education certificate.
Also in the states, engineering students ask the same question about Laplace transforms. ;-)
> Also in the states, engineering students ask the same question about Laplace transforms. ;-)
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.
I'm playing devils advocate here, but the argument you'll get is that there's software for that. Also, in a typical department, there may be one or two engineers who handle the quantitative problems for the team.
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.
>I won't need math, so I'll blindly trust this 20 year old 32 bit half precision floating point thing someone coded up in an afternoon before their holiday break.
Don't get me started on engineers who think floating point numbers can be treated like real numbers for equality testing.
Huh. I'm from India, and from what I can tell, maths is taught much better here than in the US, at least till 10th grade, after which it's no longer compulsory. It's actually taught in a very intuitive manner, and most students "get" it quite well.
Edit: the textbooks are freely available on the web, if anyone wishes to take a look. Look up "NCERT class 10 maths textbook".
Im my experience there is so much variation between countries, between different regions within the same country, between schools within the same region, between teachers within the same school and even students' subjective experiences within the same classroom that I am generally skeptical of broad conclusions drawn between countries like that.
My experience has been the opposite. Indian kids can solve math problems in the "curriculum" quite well but ask them to do anything WITH the math and they're lost. Ask them how the theorem relates to the natural world and they don't know. Ask them what applications it has and they don't know. Ask them to extrapolate their current knowledge base to a theoretical future and they are completely lost on what a "next step" would look like.
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
Try asking the average western student the same question. Most people -be they young or old- aren't able to do that. Nor is that a skill you actually need in real life.
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.
Wonder why math is such a problem. Growing up in the Ukrainian school system I was doing very well in math for 3 years and was doing double exams in the span of one.
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 struggled with maths as a kid, but gained a much better grasp as an adult, when I had to use it for practical applications.
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.
It's a double edged sword. I loved math because I liked physics, and calculus text books had plenty of physics examples.
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.
Exactly - the applicability has to mean something for the person. It could require several regressions before you get to a core interest of any specific student.
I never understood people that questioned the practical applicability of math. Surely they must realize that these things are so elementary that there is no way there was no practical use for them. Logarithms specifically allow you to multiply values quickly by looking up their logarithm in a table, add the values and do a reverse lookup in the same table. This is the foundation of slide rules, which is the way engineers did calculations before computers were common.
Of course we all know this, so you’re preaching to the choir. However I think it would be quite demoralizing to tell a young student who is struggling with logarithms (to follow your example) that “it’s so elementary that there must be a practical use!” Indeed, middle and high school math is, to many, the most intellectually challenging thing they’ve ever had to deal with at the time these classes are taken. Considering they haven’t even taken higher order maths yet, students likely don’t have the context to realize that “elementary ≠ easy to understand”
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...
I have never once in my life used logarithms in that fashion. I've used them in other ways, but why would you do it that way instead of just hitting the "*" button?
I this sums up my first math experience in the US high school. Learning before moving to the US, we were not allowed any calculators. My first class the teach wrote something basic like quadratic equation to solve. To my shock, out went the TI-83 graphing calculators.
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.
They are practical only to the one using them. I don't care about fluid dynamics when I stir my coffee with my well-designed spoon.
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 only learned about this application of logarithms years after learning the definition. Usually the applications are not obvious to students and this goes on even at the university level.
I sympathise with you. It certainly seems that those people who do best at mathematics in school are just those who are best at abstract manipulation of symbols.
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
This kind of equation comes up a lot in modelling things in science and other parts of the real world (eg compound interest). We can solve it:
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)
But this motivation isn’t much use when one doesn’t have differential equations or calculus. When I was in school I think differential equations where only in A-level further maths (age 16-18). I’m not sure whether we had logarithms at GCSE (14-16) but I think they were in A level maths which included calculus up to integration by parts and integrals/derivatives of various trig functions.
[+] 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.
Routinely we have to explain to users in a crypto related business that 0.00835 Satoshis is the same as 0.008350 Satoshis. Adult humans don't handle decimals.
That reminds me of A&W killing off the 1/3 lb burger from their menu because people didn't realize it's more than a 1/4 pound and thought they were getting ripped off.
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.
It's a hypothetical example based on actual customer cases. Change the units to BTC/LTC/whatever if you'd like to make it more precisely simulate the user interaction.
Or: why do so few people who understand math teach elementary school?
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 would love to teach math to elementary students but I make a base-10 order of magnitude more as a software engineer for a FAANG. I'd take a pay cut of a factor of 4 or 5 to do it in a heartbeat... but having to go back to school to get a teacher's certificate (and likely a master's degree) to get paid 1/10th as much is challenging. I may end up doing it in a couple of years when I'm even more solidly financially independent, but we as a society should probably just be paying teachers a lot more across the board.
I guess there's more money in advertising than educating the next generation.
The fewer people who are capable at something, the more lucrative it is to know that thing. The more lucrative it is to know something, the more of a pay cut it takes to teach it, assuming teacher pay at this level isn't subject dependent. The more of a pay cut it takes to teach, the fewer the skilled people willing to take a pay cut, and the fewer teachers there are with those skills. The fewer teachers there are with this skill, the worse the students turn out at it, leading to fewer people who are capable at it.
It's a vicious cycle. I'm a data scientist in SF and I'd make 5-10x less if I taught grade school.
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) instead of using the "borrowing" algorithm. You never have to "guess" when doing long division in base 2. All of these operations can be justified in terms of the Peano axioms with very short arguments: prove the associative and distributive properties, and the multiplication algorithm is obvious, etc.
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.
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 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.
Wasn't the premise of the article that the reason U.S. students perform so poorly in math in higher grades is because they are only being taught to memorize arithmetical facts and algorithms without properly learning the reasoning behind them? It seems you're prescribing the cause of the problem.
No, I'm saying that they are 2 different things, which seems to both end up in the "math" bucket.
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.
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 just these freaking fractions and their convoluted addition I do not know how to properly convey.
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 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.
To me what is utterly crucial is to use pie pictures to
convince children (and adults...) that 1/2 + 1/3 cannot possibly equal 2/5. If they actually buy this that’s a pretty good inflection point in their education on fractions.
After going though the education of two children who like math very much I realize that fractions on their own are very natural.
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.
As a former high school math teacher, chances are you did a disservice to them.
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.
If you want to add 2/3 + 1/4 (=11/12), take three square sheets of paper. Call the area of each one.
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.
Pizza slices. Different sized slices. Fractions of a pizza.
Half a pizza plus a quarter plus a quarter = one pizza.
A pizza is 4/4 of a pizza = a pizza sliced into 4 but still together.
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.
> I have a hard time explaining to my children how addition of fractions work.
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.
Just get a few measuring cups and some water. Half a cup and a quarter cup makes three quarter cups, because really, half a cup is two quarters of a cup.
Which is all well and good... Until you need a half of a quarter (which I'd use tablespoons for as there are 16 tablespoons in a cup), a half of a third of a cup, or need to add 1/3 and 3/4 together.
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.
I am a math professor. I teach in a SLAC, St Michael's College in VT. In our state education majors must also major in another subject (on an approved list). Every year we graduate one or two future Elementary Educators who are math majors. They take the same courses as any other math majors, and do just as well.
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.
>> 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:
Sounds like you can just say (as happens) we have to get the answer we're looking for on the "put the answer here" side.
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".
The worst part is that female students are more likely to notice the teacher's fearful body language, so you get this never ending cycle of girls developing fear of math, going into teaching and passing the fear on to the next generation.
Even assuming the study holds up, that doesn't mean that only the girls picked up the fear. It means only the girls were affected by the fear, by overly generalizing from this one example.
The author of this article is a researcher in cognitive psychology specializing in education. He wrote a pop-sci book detailing how cognitive science explains what's wrong with school, and it's very much worth a read if you're at all interested in the topic. Link is https://www.amazon.com/Why-Dont-Students-Like-School/dp/0470....
When I was in school, the problem was the way it was taught. There was no real explanation of anything or questioning that would lead to learning the fundamentals until the higher grades, when kids were already expected to understand things that they'd never been taught.
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.
> If you know that to divide one fraction into another you should “flip the numerator and denominator of one, then multiply them,” is it important to be able to explain why it works?
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?
This is such a vicious cycle. The worse we teach math, the worse our math labor force, including teachers. The worse our labor force, the more of a pay cut it takes to teach math for the rarer and rarer people who really get it. To cut this cycle, I'm tempted to say we just need to throw money at it for a while. Pay people who really get math to teach it. Pay them enough to draw them away from whatever industry would pay them to do. Pay for them to be trained to teach well.
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."
In my kids elementary school time. I had the interesting situation that while the teachers weren't strong in math, their boss the principal was an actual mathematician (or at least he had a real degree in Mathematics). Generally principals seem to have a psych background but not this one. Good times!
> 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.
So do Pascal and Ada, but using "the = character with a tiny dot next to it" doesn't sound like it will avoid the fundamental issue here. That's just extra syntax to help remind the programmer whether they're assigning or comparing. It still basically looks like "=". That's why it was chosen.
(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.
The mistake is assuming "=" is a statement of fact (as in mathematics), rather than a request for assignment (as in programming).
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 mathematics, this would be a logical contradiction, and either you'd realize you made a mistake, or you'd jump back to the step where you said "now assume ..." and say "now we've proven this can't be true".
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.
Thank you for your well thought out explanation! It is clear now and I was committing the cardinal sin of speaking generally while assuming context that was explicit only to my understanding of the situation. Have a great day!
Absolutely correct, but it's worse, since they don't truly understand math, they taught in such a way that confused/misled students, no wonder USA had to import so many STEM engineers(and graduate students to fill college classrooms) from other countries, and it's getting worse.
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.
My ex taught grade 6 geography. Is a principal now. Thought the seasons were due to earth tilting back and forth. Didn't get that the tilt was the same, but the position around the sun was different
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.
Yes the seasons are due to earths tilt, but the tilt still doesn't change between seasons as his wife thought, only earths position compared to the sun does.
The tilt of the earth relative to the sun does in fact change. That’s what the seasons are. If the earth was tilted but the tilt didn’t change over the course of the year the areas that had longer days due to being tilted towards the sun would be permanently hotter than the areas tilted away. There would be a permanent summer and winter, rather as at the equator it’s always summer.
The changing tilt of the earth relative to the sun is what the seasons are.
I'm not sure what you are trying to say, you probably misunderstood what elcheif said. Earth is tilted towards the sun in the summer and away from the sun in the winter due to its tilt changing, not due to its position changing. That is what you are saying, what I am saying and what elcheif was saying. What elchaif's ex thought was that earths tilt actually changed over the course of the year instead of earth just moving to the other side of the sun. There is no energy which could alter earths tilt like that, instead the tilt was set in motion a very long time ago and has been basically fixed ever since.
Um, isn't it primarily about tilt? Australia and the UK are both at aphelion (furthest distance from the sun, averaged over the day [if you want to be pedantic, and why not!]) at the same time but aren't both in Winter at the same time.
While I am also appalled, I imagine everyone has some fundamental misunderstanding about something that is 'obvious' to everyone else. Something akin to 'today's 10,000"
thats not incorrect though. The orbit of the earth is eliptical and during summer (for the northern hemisphere) the earth's tilts north end towards the sun. Fun fact, it is actually farther away from the sun as well during that time.
The particular non-fact is unremarkable, but the lack of inquisitiveness is off-putting. It falls down after not so many "why"s. You'd like to imagine a greater love of discovery in the people who are (hopefully) shepherding dozens of kids through their own discoveries, day in and day out.
I think what they are saying is that framing the answer to the question, “What causes the seasons the change?” in terms of “tilt” conveys a misunderstanding of how the seasons work.
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.
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 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.
Its weird that education is payed for completely by property taxes in the US.
The benefits of education go to more than just the (parents of) the recipients of that education. You also get weird incentives for people to reduce these taxes if they send their children to private schools.
> 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 believe the idea of schools being funded by property taxes is that you then spread the cost load out over people of all age groups, in a way to acknowledge that having educated children is a benefit to the community.
> The benefits of education go to more than just the (parents of) the recipients of that education.
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...
Teachers should be paid more, we all agree, but how will that solve this problem?
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.
> 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.
Shouldn’t that study focus on the distribution of types of master’s degrees, and a sub-population analysis showing which ones correlated most with teaching ability? I bet someone with a master’s in math would be better at teaching math than someone with a master’s in teaching theory. Ditto at the undergrad level.
Undergrad/grad major subject correlates with IQ. The population of teachers that majored in math should have a higher average IQ than the population of teachers who majored in other subjects, controlling for university. I would be shocked if teaching ability didn’t correlate with IQ. I’m sure it’s not a perfect correlation, but it should be there.
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?
I feel like anyone should be anxious if they don't understand math. How does one get through life without understanding math? Probably by getting had a lot.
Math is taught so badly, that it may seem more complicated than it really is. For instance, many students come away from their math classes, convinced that math is subjective. They believe that, like in English class, the correct answer is the one that the teacher wants to see, and that there is a "trick" for producing this answer.
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.
Ha,I see why you flipped an exception on my comment now. Let me assure you that I'm not attributing the decline of Western Civilization to women teaching math!
Did you read that Chicago study? I'm interested to see if your view shifted any.
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 equivalent, we would use another sign, like ≡ (unicode U+2261).
As expressions, sure, they are not the same. But "2 + 3 = 5" is a statement about numbers, not about expressions. When you talk about numbers (the object of study), you do not (usually) talk about arithmetic expressions. Arithmetic expressions are part of the language used to talk about numbers.
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.
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!)