
To show or not to show work - abrax3141
https://www.byrdseed.com/to-show-or-not-to-show-work/
======
lolc
I have a story on this. Our teacher in electronics was constantly heckled by
us to do a multiple-choice test "just once" for us. So he did! But with the
caveat that we had to show the work. Negating the whole reason why we wanted
that test. The bastard.

Comes the test and one of the questions required calculating total resistance
of a grid of resistors. The way the resistors were arranged made it impossible
for the result not to be an integer. Except there was one tiny half-ohm
resistor in series. And there was only one answer that was not an integer.

So I wrote two steps: The reasoning that the result can only be an integer
plus 0.5. Then the reasoning that only one answer was left. I likely spent
more time on that than I would have on the calculation. Still I got full
points and an extra smiley so it was worth it.

~~~
edflsafoiewq
In math you sometimes see multiple-choice questions that ask for the "hash" of
the answer (eg. find the complex root of this polynomial... and tell me the
sum of its real and imaginary parts) that are supposed to prevent you from
doing this kind of thing. They do take the fun out of having multiple choice
questions though :)

~~~
gizmo686
You also sometimes see questions that look superficially like thay, where the
"hash" is actually easier to computer.

Eg. Given a polynomial with real coeficients, find the complex roots and tell
me the sum of their imaginary parts.

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dahart
> If a student can do it in their heads, then the work is too easy! [...]
> Instead of battling over “showing work,” simply increase the complexity of
> the problem until the student must do the work out to get it right.

This is a good idea, and having tried it, I know it sorta-kinda works a
little. When I’ve tried it with my kids, what happens most often is one of two
things. Sometimes if it gets too intimidating they give up and won’t try
without hints. And sometimes it gets them to write some intermediate steps,
but they still skip over the easy pieces, try to do 2 or 3 simplification
steps at once and make mistakes. That should, you’d think, be convincing about
the importance of writing down more granular steps, but for whatever reason
they just hate making it mechanical and they keep resisting the idea of
writing incremental steps. I’ve tried too many times to point out how much it
help avoid mistakes, but they just think I’m a windbag _and_ asking them to do
boring things.

~~~
viklove
Kids don't have mature facilities for self-motivation; you can't expect them
just to do math problems because it's good for them. Tell them they can't do X
until they finish 20 difficult math problems, and I guarantee they'll find a
way to optimize their work habits.

~~~
swiley
Sometimes they’ll find a way to optimize the work away.

I still think it’s better to give them tests and show them that the only way
to pass the test is by practicing before hand.

~~~
viraptor
That may backfire. I remember my physics homework/tests and going to absurd
levels to satisfy the requirements and not do the boring work. I did things
like a freehand graph with important points marked and a curve labelled
"parabola", because that's the information the teacher was after and spending
time actually drawing it to scale and precisely was boring. (after 20th time
or so)

I actually lost interest in the subject around that time, even after scoring
high enough in regional competition previously to get a free choice of any
high school.

So watch out, you may kill someone's enthusiasm by making them practice simple
things the way you think is best to satisfy some silly rules. I ended up more
happy not getting full marks on something I could do well than showing the
work.

~~~
swiley
Yeah that's not really what I meant by "practice." I meant just give them the
text book and tell them to make sure they understand how to do the problems
without looking them up, and that the best way to do that is to do the
"practice problems" that look difficult.

------
chongli
As a 3rd year math student, I’ve seldom been required to show my work unless
it was a proof. On the other hand, a lot of the questions on exams gave me a
full page and awarded part marks for demonstrating any correct steps, even if
the final answer was incorrect or even missing. Some of these exams were so
hard you tended to pass on part marks alone and few people managed to get all
the right answers. If you did know the answer you could just write it down and
get full marks. That’s demonstrating a lot of self confidence (or hubris)
however.

One thing I think might be interesting to try is to give the student the
answer to the problem and ask them to show how to arrive at that answer. This
could give them the feeling that their steps actually matter, rather than
simply satisfying a teacher’s demand. For example, you could ask:

    
    
        1) Show that lim x->0 sin(2x)/x = 2
    

That way a student can’t get away with just writing 2.

~~~
lonelappde
That's tautological. Showing work is what a proof is.

~~~
im3w1l
Is it? Proof to me means I can say "here's the answer" without saying how I
found it, maybe I just tried a few random things, and just prove that it's
correct.

For instance, for a polynomial I might show that my answer solves the
equation, and that I have as many solutions as the degree.

Maybe that goes for "show your work" too? As I didn't study in the US, I don't
know what a teacher would expect when saying that particular phrase.

~~~
HarryHirsch
Yes, heuristics is important and underrated in math education, especially in
the US, where they place excessive faith in rote learning.

Proofs are sometimes a little unsatisfactory, the theorem is proved
satisfactorily, but all the traces how you got to the proof are covered.

------
learnstats2
> If a student can do it in their heads, then the work is too easy! ... Simply
> increase the complexity of the problem

This website is a resources for teachers of "gifted and talented" students -
but, in my experience, this strategy absolutely will not work in a more
general setting.

It's often the case that students aren't writing down the steps because they
don't understand what the steps are and don't know how to formulate the steps
as individual components (and may be answering the questions in unexpected
ways!)

Making the questions harder will often make things worse, on its own, because
the student will get stuck and demotivated and hard-questions-for-the-sake-of-
being-hard will seem pointless to them.

I do use this as a strategy but only when I'm confident that the student has a
very good understanding of what's going on.

If they're not writing down the steps, it's more often that they are
demonstrating that they don't have a good enough level of understanding to do
this. (This also applies to "gifted and talented" students)

~~~
travisoneill1
If they are answering math questions correctly in unexpected ways then they
are demonstrating skill at math. Math is not a procedure,

~~~
learnstats2
As randogogogo hints at, it's not necessarily the case that you are
demonstrating skill at the appropriate level.

If you take the question 3x + 1 = 10

And solve it (for example) by trialling x=1, x=2 and x=3, then you are able to
answer the question correctly. You may even be able to answer a set of
questions correctly.

But, you have not learned the relevant algebra skill to be able to generalise
this process.

------
stared
There was a thread "How to tell an over-confident student they still have a
lot to learn?"
[https://academia.stackexchange.com/a/17833/49](https://academia.stackexchange.com/a/17833/49).

If a student is smart some problems may seem to them like "I see you know that
2 + 3 is 5, but what's the reasoning?". So indeed, making problems more
complex is the only way to go.

And the "reasoning" part is difficult. We never know if something is a true
reasoning, something tangentially relevant, or rather something we were
trained to say. It works (or: doesn't work for machine learning in a similar
way, vide:

"Speaking as a psychologist, I’m flabbergasted by claims that the decisions of
algorithms are opaque while the decisions of people are transparent. I’ve
spent half my life at it and I still have limited success understanding human
decisions. - Jean-François Bonnefon", as quoted in
[https://p.migdal.pl/2019/07/15/human-machine-learning-
motiva...](https://p.migdal.pl/2019/07/15/human-machine-learning-
motivation.html).

~~~
travisoneill1
"How to tell an over-confident student they still have a lot to learn?"

Give him a problem that's worth his time to solve.

------
thomasedwards
I was always told by my teachers the main reason was for practicing taking
exams. If you can show the workings, but you made a simple mistake somewhere,
you could still be awarded points for getting the process correct, even if the
final answer is wrong. Say a trigonometry question that was worth three marks,
and you messed up a decimal, you could still get one or two marks for
correctly identifying the problem and solving it – albeit with the wrong
number.

------
analog31
I taught "College Algebra" for one semester, long ago, at a big ten
university. Naturally I told my students "show your work" without ever
wondering if anybody had ever explained what that meant. It began to bug me. I
asked some professors. None of them could explain it either, though they were
certainly indignant that the students couldn't do it.

Many of my students came from schools where they learned a method called
"guess and try," where you plug answers into the problem and see if one works.
This is a speedy way to dispatch multiple choice tests.

To my students, "show your work" meant showing some evidence that they had
solved the problem themselves. They thought I was policing them, when I really
would have liked to engage them at a bit higher level.

In my view, "show your work" means, loosely speaking, to create a fictitious
chain of reasoning and present it in a style learned from the textbook and
classroom presentations. I call it fictitious because they might have guessed
the answer and then worked backwards from it to obtain the steps. I could
handle saying that a bright student should be able to do this, but that it
should be taught.

~~~
diminoten
I'd push back on your characterization of "fictitious" here, since all you're
asking them to do is "document" their work.

From a software engineer's perspective, that's all I really want/need to do
for my colleagues. I don't need to "show my work" because they're making sure
I didn't copy/paste, they're trying to move quickly and don't have time to
work things out on their own about how I came up with the solution.

Maybe "show your work" _should_ be taught; less because it's a gateway into
the mind of the solution provider, and more because it's helpful to others to
see how someone arrived at a conclusion.

~~~
gaspoweredcat
i dont really think anyone cares if somethings been copy/pasted as long as it
works as it should, the most talented engineer i ever met told me theres no
need to be ashamed of it, theres no point reinventing the wheel if you dont
have to, and he was right if youve been able to adapt something else to your
task then evidently you understand what it does.

lets say youre building a car, maybe you take the engine from a ford and mount
it on the chassis of renault with nissan steering, just because you didnt
design those parts from the ground up doesnt make the end result any less
valid as a final product (obviously thats a very loose analogy but you get the
picture)

software is a bit different though, rather than showing that you followed a
rigorous formula its about letting people know how the formula you came up
with works

~~~
diminoten
I care _deeply_ if something has been copy/pasted, but not for attribution
reasons. I need to be certain the person who copy/pasted the code actually
understands what the code does, rather than blindly grabbing the first result
off of SO and exposing the codebase to vulnerabilities.

------
scarejunba
What? This doesn't make sense. Obviously you have to show work because you can
use the wrong techniques to get the right result otherwise and doing that is
bad because you won't know why that won't work elsewhere.

In high school, we had to know how to rapidly solve questions like integrate
e^(ax)•cos(bx) for constants a and b (this is one of those easy ones) and we
knew the closed form answer and if we'd forgotten, we'd add an imaginary
i•sin(bx), integrate the resulting exponential only and then separate real and
imaginaries. But whether that's legal is kinda not obvious. It's just letter
manipulation to do that.

In a proof, it's for you, so you know you did a legal thing.

------
spedru
I have no teaching experience but plenty of experience as a student. This is
right on the money; it would have made me a much less frustrated child. Kids
are sensitive to being patronised, and so many adults think they're slick.

------
gaspoweredcat
i have always questioned this, not only is it inefficient and even confusing
at times if you normally work perfectly fine in your head but often the
reasoning is to "show youve done it correctly" which i qualm simply because it
gives the idea that there is only one possible approach to something which
kind of stifles peoples ability to accept that things can be done a different
way

just because Pythagoras came up with a way to calculate the length of a
hypotenuse doesnt mean its the only way but thats what youd be led to believe
and as such no one considers other methods and will probably even dismiss any
alternatives without even bothering to check

thats not to say that we shouldnt check that people understand how to do
things but this can be achieved by posing a few questions, if the answers are
consistently right then we can generally assume the method used to get there
is valid (or they cheated which will be pretty evident when they attempt to
apply it to a real situation)

but i have a strange way of working with numbers, as long as i understand the
theory my brain works more abacus like than using arabic numerals so it
essentially creates an extra step in having to sort of convert the working out
back into something thats writable, in my school days i found that more
difficult than the problems themselves, thankfully its something i havent had
to do for a great many years

~~~
mkl
It's not about showing you used "the" correct method, but providing evidence
that your answer is right. Without working it's impossible to verify the
answer is correct without doing the whole thing again. For anything that
actually matters, it needs to be checkable by other people, and quite likely a
computer will handle the fiddly numerical details, so the working steps may
well be the _only_ part that matters (they give you a way to tell the computer
what to calculate).

------
gumby
It can save the student's bacon.

I remember a physics test: Answer one of three problems. Problem 1: I thought
not enough information had been given. Problem 2: didn't remember discussing
this topic. Problem 3: did not even understand the question.

I beavered away on problem 1, turned it in. Teacher found enough of a thread
to give me 20%. Came in 3rd -- and with the curve, I passed (which frankly I
don't believe I deserved but I was happy to get anyway).

------
NiloCK
The article presumes that the point of mathematics class is to produce numbers
that match the ones in the back of the book or an instructors answer key. This
is incorrect. The point of mathematics class is to build skills for deduction
- drawing valid conclusions from data (often numerical).

Real life problems do not come with an answer key, and real life problems tend
to be hotly contested.

My stock one-off conversation on this issue with my own students is to
consider (particular example varies based on background of the student) the
manager of an engineering firm who has tasked two teams for an calculated
minimum thickness of the main supporting cable on a suspension bridge. Team A
returns after a while and says 3 inches. Team B says 4 inches. Do we go with
the 3 inches, which one team believes is unsafe? Do we go with a bid based on
the 4 inches where we may be beat on cost?

Numbers are just numbers. They are orthogonal to answers, orthogonal to
arguments.

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kriro
I can't think of an argument that makes sense to me to "not show". Showing the
work captures the thought process which is really the only thing that matters
from an educational point of view (in my opinion). Mostly because that's where
one can step in and follow up after the exams and see which students have
problems and find the root causes of the problems. Of course most grading is
done in a "grade and throw away" manner. Consequently, easier grading has a
very high priority.

I really whish more educators would see exams as chances to evaluate how well
they taught the material and to find areas they could improve.

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abrax3141
One problem is that explanation is in-and-of-itself a separate skill, and is,
in many cases, harder than math (at least K-8 math). So asking for an
explanation in addition to doing the math is asking for 3x effort. Moreover,
explanation is highly ambiguous (whereas math ain’t!) So you’ll often get an
explanation that you don’t quite understand bcs the kid isn’t good at
explanation, even if they know perfectly well what they did.

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alexandercrohde
I had a highschool calc teacher who told us "Show your work. If you get the
answer correct you get full credit. If you get it wrong, and show your work, I
can give you partial credit for the steps you got right"

(The problem with increase complexity is that a lot of work happens, but after
a certain difficulty level it's all on the calculator)

I think this is a good solution.

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abrax3141
I’m told by multiple math teachers that being able to explain yourself has
mostly to do with being able to collaborate, which is an important skill for
any sort of very hard engineering activity.

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lonelappde
It makes me sad that this advice has to be written down.

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anonuser123456
Oh Come On!

