
Why Math Word Problems Fail - CarolineW
https://www.noodle.com/articles/why-math-word-problems-fail-and-how-we-can-get-them-right238
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sgentle
I think this whole thing is missing the point. People, even young people, are
extremely savvy when it comes to figuring out what they need to do to get what
they want.

What they want is to do well in school. How they do well is to pass their
tests. How they pass their tests is by learning the answers to the questions.
Making indirect questions that "appeal to real-life problems" is obviously
worse, because the main real-life problem students have is passing their
tests.

Don't get me wrong, I'd love for this to not be the case. If students could be
assessed on their ability to actually understand and use the material to
generate new insights and solve interesting problems, I think mathematics
education would be a lot better. But as long as we're assessing students on
equation-memorising, let's not pretend otherwise in the way we teach them.

~~~
gohrt
The standardized tests have word problems, so your objection doesn't make
sense.

~~~
scotty79
I guess his point was that world problems don't describe situations that are
problems for the students. So they can't relate them to their lives and it
makes things harder because they add to the complexity of the task.

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analog31
I taught freshman algebra for a semester at a big ten university. Therefore
it's also a safe bet that I did a shit load of word problems myself when in
school. Here's how you solve word problems:

1\. From the text of the word problem, remember what section of the textbook
it belongs to, and the standard form of equation to be solved in that section.

2\. Match the parameters of the problem to the parameters of the equation.

3\. Solve the equation.

For instance, "find the maximum" means creating an expression that has the
form of a parabola (the only function they learned with an extremum, up to
that point in the course), and solve for the vertex of the parabola.

~~~
HonestQ_902F
From your description and context, it sounds like you're describing elementary
algebra and not abstract algebra.

Honest question that sounds mean: isn't elementary algebra required for
graduation from high school and admission to university? My university didn't
offer elementary algebra, and I thought students without elementary algebra
skills needed to at least start out in the community college system. In my
public school system, the advanced math kids took elementary algebra in 8th
grade, and the less mathematically inclined took it in 9th grade.

~~~
analog31
Yes, elementary algebra.

Where I taught, they gave the incoming freshmen a math test, and the kids were
put into Calculus, Algebra, or a "remedial" math course. Algebra was the
lowest level course that could be offered by an accredited college, and the
"remedial" kids eventually had to pass Algebra.

This was a large state university, so the kids came in with a wide range of
skills, from a state where there is a lot of variation in how math is taught
in high school. They were bright, affluent kids. Many of them had taken
algebra and even calculus, had gotten good grades, but had not really learned
those subjects well enough to skip them in college.

I had pretty much the same thoughts as you express, but as the semester
progressed, I developed the opinion that these kids had been screwed by the
system and still deserved a chance. They were not going to be engineers. Most
of my students were planning to major in psychology or business.

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unabst
Their value would be so much more heartfelt if students actually encountered
math word problems outside class. But they don't. And part of it is because
they don't exist. They are hostile games. By hostile I mean they are designed
with gotchas and purposely ambiguous, and they have unnatural rules like the
inability to ask for more information or to get help. In real life, problem
solving does not present itself like this.

Those that do well are the students that understand the game, and play along.
They are not learning math, but do well "against" math word problems.

The question as educators is simply, "is this the definition of good?"

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EliRivers
Almost every situation for which I have needed mathematics outside the
classroom has been a "word problem". Nobody has ever simply come to me and
asked me to multiply two numbers together and then add the result to something
else.

Instead, I get word problems like "how long is this task going to take, if we
can give you this much resource?" and "how long is it going to take to drive
from A to B at this time, given this knowledge of traffic conditions?" and
"what's the best tradeoff of time spent travelling against airfare, to travel
from here to there, given these limiting conditions?" and so on and so on.

Word problems don't exist? Nonsense.

~~~
unabst
Where are the gotchas and purposefully ambiguous terms? And none of those are
difficult math problems either. Regarding traffic, a smart human (including
students) would just refer to an app on their phone.

There is a clear line between problems described with words, and these "word
problems". The topic was about math, but physics is worse. My high school
physics teacher had a hobby of writing to text book publishers about how
"silly" some problems were. They're designed to be difficult and relevant to
specific material, but in the process reality goes out the window.

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force_reboot
From what I've seen of the common core, 90% is fine and just plain old math
stripped of any historical baggage. The 10% that is bad is the ideology that
math is more ambiguous or less certain than it has been portrayed in the past.
This is partly based on politics, and partly on the misuse of mathematical
philosophy.

I remember being given problems that were plain wrong, e.g. Q: How many miles
would a person walking 3miles/h walk in 2 hours? A: 6 miles. I always asserted
that the answer is 6, not 6 miles, since the unit is given in the question.
Small errors like this in teaching add up to math that looks like a collection
of arbitrary commandments that must be memorized. It's understandable people
want to move away from this. But this solution is to fix all aspects of
teaching so that everything is completely correct, not to pretend that math
allows multiple correct answers.

We should be teaching that you can express a well formed question in words, to
which there should be a unique correct answer. Word problems shouldn't be an
added layer of obfuscation, but rather a map from the semantics of math to the
semantics of the real world, to test that the student understands this
mapping. It's a bit like talking to a backend programmer and saying "if a user
did X, then they did Y, what would your function return, and is that correct"?
It forces the programmer to consider not the internal logic of their own code,
but how that ultimately maps to what the user sees.

~~~
lotharbot
from what I understand of common core, the correct answer to your example
would be "They would walk 6 miles total." There's an emphasis on clear
communication and complete sentences. Many of the textbooks would even
encourage the student to write out the sentence stem "They would walk _____
miles total" prior to performing any calculations, and then at the end of the
problem to check that the number was actually measured in miles and that the
sentence made sense and was a reasonable way to answer the initial question.

This seems to be what you're encouraging in your final paragraph. This brings
up the question: why is the common perception of common core so different from
what common core is designed to do?

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nevdka
Common core is a specification that is implemented many, many times. It's not
just each textbook or school system - each teacher will implement the spec in
a unique way. With so many implementations, some of them are going to be
terrible.

But the common perception isn't built off an average or large-scale aggregate,
it's built on what people see and hear. A single comically bad implementation
can be seen by millions of people thanks to Reddit, Facebook, and the like. A
single good implementation will be seen by the students, maybe their parents,
and maybe a few more teachers. Most examples people see are comically bad, so
the common perception is that common core is comically bad.

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Sniffnoy
I have to disagree with this pretty strongly. The best thing I've seen on the
subject of word problems is the following piece by Andrei Toom (warning, this
is 98 pages long): [http://toomandre.com/travel/sweden05/WP-SWEDEN-
NEW.pdf](http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf) The claims
made in this article seem like the sort of thing Toom is decrying; see section
IV in particular.

To attempt a summary -- the way math is taught in America is broken, and word
problems are horribly misused. Because neither the students nor the teachers
really understand math, they attribute their distaste for these useless word
problems to the inherent nature of word problems, or to them not being "real-
world" enough. Word problems are not the problem; Russian word problems
accomplish their goal quite well and are not held in such distaste by Russian
students. And no, it's not because those ones are more "real-world".
Basically, none of this is the real problem, these sorts of attempts to
address it will not help, and attempts to address it by teachers and
curriculum-writers who don't actually understand math can actually make the
problem worse.

But really, I recommend reading the whole thing.

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delish
Here are two math problems. The first is from an article linked-to in the
post. The second is from Challenging Problems in Linear Algebra.

 _Two planes, which are 2400 miles apart, fly toward each other. Their speeds
differ by 60 miles per hour. They pass each other after 5 hours. Find their
speeds._

 _Show that a Matrix A is of rank 1 if and only if A can be written as A =
xy^t for some column vectors x and y._

I'm more excited by the second problem. I'm going to use concepts I mostly
understand to _prove_ ("if and only if") a concept. In so doing, I'll
understand all those concepts better.

In the first problem, it looks too much like I'm trying to tell the teacher
something he/she wants to hear: the speeds. It's hard for the reader to see
how he/she learns something about arithmetic.

When doing the math that mathematicians do, what you learn is often explicit:
"prove that for all..." It's more motivating.

Of course I imagine the makers of Common Core have more expertise and
experience in teaching students; I have no plan to reform math education.

~~~
saretired
Why are you comparing a middle school problem with a college level problem? I
believe the purpose of the first problem is not to teach something about
arithmetic, but to test (or reinforce or expand) the child's understanding of
d = rt, a pretty important thing to understand.

~~~
delish
I'm pointing out what makes--in my experience--a problem interesting.

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nitwit005
I see the whole word problem approach as a bit dubious. Usually the aim is to
help students learn to solve "real world" problems instead of artificial math
problems.

And indeed, I would say that I've learned math much more successfully when
I've had a real world math problem. The trouble is, real world problems tend
to be horribly complex.

You're in an awkward position where you don't want to give artificial
problems, but you also don't want to give real problems. She recommends this
expii site, but the first problem I clicked on (the most recent one) seems
ridiculous:
[https://www.expii.com/solve/11/1](https://www.expii.com/solve/11/1)

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mjevans
Maybe a math book should start with a few problem questions, built around
every day scenarios.

Then it should break down the skills that are needed to solve each of these
problems, and cover those skills and the smaller steps of those problems.
Finally building to a cohesive solution to the initial problem.

~~~
lotharbot
> _" start with a few problem questions, built around every day scenarios"_

One of the difficulties here is termed "cultural competency". Textbook writers
sometimes try to write in "every day scenarios" that end up being much more
familiar to middle-class suburban white kids than to lower-class inner city
latin-american immigrant kids. Math problems that are supposed to reference
familiar everyday scenarios sometimes end up turning into lessons about
language or culture.

~~~
vitd
In my opinion it's way worse than that. I was a middle class suburban white
kid and it was not an everyday scenario to figure out when 2 trains were going
to pass. While I do recall hearing a train somewhere in my hometown, I don't
recall ever seeing it, let alone seeing more than 1 at a time.

~~~
lotharbot
in this case, I'm specifically referencing circumstances where they've tried
to write problems based on "everyday experience" \-- not trains passing
halfway through Kansas, but perhaps things like going to a shopping mall or a
museum. Things that you think "that's just an ordinary part of childhood"
until you spend some time in an area where a lot of the kids have _never been
to the mall_ , and the only museum they've ever seen was on a school field
trip. Perhaps they don't know the term "supermarket" but they know what a
"mercado" is, and they might expect very different products to be on sale from
what the textbook author expected.

The idea of "cultural competence" is about figuring out how to eliminate
unnecessary cultural barriers -- to make sure that a kid trying to solve this
type of word problem is working through the problem, and not getting
sidetracked by the fact that they don't have the context to make sense of what
it's even about.

~~~
taejo
I once volunteered to help kids in a rural school in my home country with
their math homework. There was a word problem in their textbook that seemed to
hit every single possible cultural barrier.

It was about budgeting for a couple. The couple had a difficult-to-read name
in a language not spoken in that province. Most of the kids couldn't even read
past the first sentence without stumbling, because of this. The couple in the
problem probably earned about as much as all of the students' parents
combined. Both of the people had their own car - while in this village there
were about four cars total. The couple had no children, and didn't look after
any family, while many of the students had children of their own, and anybody
in that village who earned an income had a large family to support on it.

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Animats
The Gary Larson cartoon version: [1]

[1] [https://s-media-cache-
ak0.pinimg.com/736x/be/55/08/be5508f4c...](https://s-media-cache-
ak0.pinimg.com/736x/be/55/08/be5508f4cf2c4c9d79730e2c88fbcc2f.jpg)

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scotty79
Word problems shaped the way how I take in information. In any given uttering
there's usually just few actual pieces of information relevant to what
interests me. Word problems taught me how to filter out the gibberish about
how Johny felt and whom he bought potatoes for.

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bsder
Math word problems fail because _math_ word problems are useless.

Good word problems are science or engineering. Once you abstract abstract
those away, there is just not a lot of anything actually useful or interesting
left.

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namelezz
Does anyone know how the second expression in the picture is evaluated?

"12 - 3 = 15"

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blahedo
That's meant to be "12 + 3 = 15" (or maybe it's an intentional error to make
the thing look more confusing?). The idea behind this particular subtraction
algorithm is to reinforce two number-sense ideas: 1) a subtraction problem
like "32 − 12 = ?" is equivalent to an addition problem "12 + ? = 32", and 2)
we can group additions in ways that are convenient to use numbers that are
easy to add or to use convenient intermediate results that are multiples of 5
or 10 or 100.

EDIT: to fix my own math typo. :P

~~~
namelezz
LOL, "New math" was misleading. Thanks for your explanation.

