
Many elementary teachers don’t understand math, and it makes them anxious - denzil_correa
https://www.latimes.com/opinion/story/2019-11-21/math-anxiety-elementary-teacher
======
nopinsight
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!)

~~~
nopinsight
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...](https://gato-
docs.its.txstate.edu/jcr:450cce10-3b6a-4ddd-a195-5c26efe39bed/2012_PWMC_individual.pdf)
"

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

~~~
chongli
_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.

~~~
username90
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.

~~~
chongli
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.

~~~
username90
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.

~~~
chongli
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.

~~~
username90
You could do it without a calculator like this:

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.

~~~
chongli
_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.

~~~
vworldv
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:

17, ..., 65, 66 => 17, ..., 65

~~~
chongli
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.

------
_bxg1
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.

~~~
geomark
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.

~~~
chrisseaton
Because the algorithm is so astronomically complicated that people can’t see
any pattern or reason in it anymore and it becomes an opaque mystery.

I have a PhD in computer science but I cannot understand long division.

~~~
_bxg1
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.

~~~
chrisseaton
No I have genuinely tried to learn it and I have no understanding why I am
doing any of the steps in the algorithm.

~~~
SamReidHughes
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.”

~~~
chrisseaton
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.

~~~
seppel
> Why are we doing things in decimal?

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:

    
    
      (3*10^3 + 3*10^2 + 7*10^1 + 6*10^0) / 4 ==
    
      3*10^3/4 + 3*10^2/4 + 7*10^1/4 + 6*10^0 / 4 ==
    

(3/4 does not work, so let merge the first two again)

    
    
      (3*10^3 + 3*10^2)/4 + 7*10^1/4 + 6*10^0 / 4 ==
    
      (33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
    

(now we have progress)

    
    
      (33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
    
      (32+1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
    
      (32)/4*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
    
      8*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==
    

(now merge the 1 and the 7 group)

    
    
      8*10^2 + (17)/4*10^1 + 6*10^0 / 4 ==
    
      8*10^2 + (16+1)/4*10^1 + 6*10^0 / 4 ==
    
      8*10^2 + 4*10^1 + (1)/4*10^1 + 6*10^0 / 4 ==
    
      8*10^2 + 4*10^1 + (16)/4*10^0 / 4 ==
    
      844
    

The 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.

~~~
dclowd9901
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!

------
renaudg
_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.

------
Gunax
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.

------
dnprock
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.

~~~
barry-cotter
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

~~~
buckminster
And the best way to test you really understand all the details is to teach a
computer the algorithm.

------
Gunax
> 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.

------
alfiedotwtf
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!

~~~
Gunax
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.

~~~
matwood
> 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.

~~~
barry-cotter
> 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.

------
rahuldottech
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".

~~~
kkarakk
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

~~~
iteratorloopmap
I can pretty much agree with you. Especially in state boards, it is just
knowing how to solve problems given at end of textbook..

------
projektir
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.

------
GordonS
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.

~~~
orbifold
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.

~~~
Filligree
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?

~~~
eropple
Because you did not have a * button?

~~~
Filligree
You don't have a calculator, so you're calculating logarithms by hand?

~~~
eropple
You understand that slide rules existed before calculators, yeah?

------
optimiz3
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.

~~~
xiphias2
Why does it even matter?

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.

~~~
optimiz3
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.

------
spodek
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.

~~~
dannypgh
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.

~~~
galangalalgol
Also consider FAANG pay 2-3 times more than other software development jobs.
When the next bust comes some of these problems will be alleviated.

------
olooney
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.

~~~
koonsolo
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.

~~~
pharke
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.

~~~
koonsolo
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.

------
BrandoElFollito
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.

~~~
Tharkun
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.

~~~
pharke
I believe they are called Cuisenaire Rods
[https://en.wikipedia.org/wiki/Cuisenaire_rods](https://en.wikipedia.org/wiki/Cuisenaire_rods)

------
jimhefferon
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.

------
symplee
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.

~~~
hanoz
_> > To [students], [the equals sign] doesn’t signify equality, but instead
means “put the answer here.” Imagine their confusion when, in algebra, they
first encounter problems with numbers on both sides of the equal sign._

Is this really true? I'm sure I and my children (in the UK) were presented
with fill in the box questions like:

7 + [ ] = 10

at the very earliest stages of arithmetic.

------
LegitGandalf
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.

~~~
_bxg1
That seems like a stretch.

~~~
LegitGandalf
University of Chicago study for you to peruse

[https://news.uchicago.edu/story/female-teachers-can-
transfer...](https://news.uchicago.edu/story/female-teachers-can-transfer-
fear-math-and-undermine-girls-math-performance)

~~~
wnoise
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.

~~~
LegitGandalf
Yeah it could be that the boys noticed and ignored the fear. The impact on the
girls is troubling though.

------
joker3
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...](https://www.amazon.com/Why-Dont-
Students-Like-School/dp/047059196X).

------
Falkon1313
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.

------
dclowd9901
> 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?

------
6gvONxR4sf7o
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."

------
dboreham
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!

------
roenxi
> 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.

~~~
purple-again
What mistake is that? The left hand variable equals the right hand value
assigned to it.

Foo = bar and 1 = 1 are equivalent in my mind.

~~~
ken
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.

~~~
purple-again
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!

------
ausjke
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.

------
nyxtom
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.

------
elchief
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

~~~
barry-cotter
Your ex was correct.

[https://spaceplace.nasa.gov/seasons/en/](https://spaceplace.nasa.gov/seasons/en/)

What Causes the Seasons? The Short Answer: Earth’s tilted axis causes the
seasons. Throughout the year, different parts of Earth receive the Sun’s most
direct rays. So, when the North Pole tilts toward the Sun, it’s summer in the
Northern Hemisphere. And when the South Pole tilts toward the Sun, it’s winter
in the Northern Hemisphere. It's all about Earth's tilt!

Many people believe that Earth is closer to the sun in the summer and that is
why it is hotter. And, likewise, they think Earth is farthest from the sun in
the winter.

Although this idea makes sense, it is incorrect.

It is true that Earth’s orbit is not a perfect circle. It is a bit lop-sided.
During part of the year, Earth is closer to the sun than at other times.
However, in the Northern Hemisphere, we are having winter when Earth is
closest to the sun and summer when it is farthest away! Compared with how far
away the sun is, this change in Earth's distance throughout the year does not
make much difference to our weather.

~~~
username90
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.

~~~
barry-cotter
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.

~~~
username90
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.

------
esmi
I suppose this is what happens in a teacher shortage.

[https://www.epi.org/publication/the-teacher-shortage-is-
real...](https://www.epi.org/publication/the-teacher-shortage-is-real-large-
and-growing-and-worse-than-we-thought-the-first-report-in-the-perfect-storm-
in-the-teacher-labor-market-series/)

------
epx
Was evicted once from the class by a teacher that said 5^0=0 and I proved on
board it was =1.

------
blt
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.

~~~
zamfi
> 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.

~~~
rocqua
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.

~~~
zamfi
> 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...

~~~
rocqua
It should be spread to the state / country level rather than being based on
districts is my point.

~~~
zamfi
At least in California, it is indeed apportioned at the state level.

------
acollins1331
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.

~~~
analog31
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.

------
gok
This could be written about essentially any school subject. Many elementary
school teachers don't understand history, science, or writing either.

------
m3kw9
Or they think there is actually more to elementary math than meets the eye

------
scscsc
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).

~~~
adriand
(2 + 3) and 5 are not “the same”. If we’re using ordinary English language to
discuss this issue I think “equivalent” is a sensible choice.

~~~
scscsc
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.

