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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... "

Source: the Primary Math World Contest (usually held in Hong Kong) https://www.txstate.edu/mathworks/PMWC/previous-pmwc-tests.h...

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.
> The test includes competitive exam questions focused on mathematical understanding and problem solving skills, rather than advanced math knowledge.

Seems like it might be close to a subset of the sorts of questions one encounters on an IQ test.

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.
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.

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.

You should probably read about it as an adult:

https://en.wikipedia.org/wiki/Long_division

Children are mostly idiots. Education is the process of painfully breaking them out of their idiocy.

Bonus points if you implement a long division algorithm that works on strings in your favorite language.

Dividend is 123456 and divisor is 37.

37>1, 37>12 but 37*3<=123.

Write down 3 (meant for 3000) and start again with 123456-37x3000 as the new dividend.

I wish the algorithm was taught by writing the full 3000 instead of only the digit 3; this appears to be a source of major confusion and to hide the underlying additions/multiplications.

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

Here's another explanation: https://www.theschoolrun.com/what-is-chunking

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!)

Division is a repeated subtraction operation, and the long division algorithm is just a nice way to lay it out.

Multiplication = repeated addition

Addition = a form of counting

I was taught math via spatial properties and it wasn’t until I came back to it from a logical perspective that I was able to understand it.

Just another anecdotal data point but my experience was completely opposite yours.

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.

> 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!

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".

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.

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.
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.

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.

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.

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 couple examples from the article:

>> 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.

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.
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.

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.

Leave to journalists to constantly dictate this so often used phrase:

“X... and that should scare you”

I would really wish the emotional appeal journalism would make its way to the door but it doesn’t look like that will ever go away.

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
I suppose this is what happens in a teacher shortage.

https://www.epi.org/publication/the-teacher-shortage-is-real...

Was evicted once from the class by a teacher that said 5^0=0 and I proved on board it was =1.
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.
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.
This could be written about essentially any school subject. Many elementary school teachers don't understand history, science, or writing either.
Or they think there is actually more to elementary math than meets the eye
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).