To show or not to show work

I have a story on this. Our teacher in electronics was constantly heckled by us to do a multiple-choice test &quot;just once&quot; 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.<p>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.<p>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.

&gt; 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.<p>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 <i>and</i> asking them to do boring things.

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.<p>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:<p><pre><code> 1) Show that lim x-&gt;0 sin(2x)&#x2F;x = 2 </code></pre> That way a student can’t get away with just writing 2.

&gt; If a student can do it in their heads, then the work is too easy! ... Simply increase the complexity of the problem<p>This website is a resources for teachers of &quot;gifted and talented&quot; students - but, in my experience, this strategy absolutely will not work in a more general setting.<p>It&#x27;s often the case that students aren&#x27;t writing down the steps because they don&#x27;t understand what the steps are and don&#x27;t know how to formulate the steps as individual components (and may be answering the questions in unexpected ways!)<p>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.<p>I do use this as a strategy but only when I&#x27;m confident that the student has a very good understanding of what&#x27;s going on.<p>If they&#x27;re not writing down the steps, it&#x27;s more often that they are demonstrating that they don&#x27;t have a good enough level of understanding to do this. (This also applies to &quot;gifted and talented&quot; students)

There was a thread &quot;How to tell an over-confident student they still have a lot to learn?&quot; <a href="https:&#x2F;&#x2F;academia.stackexchange.com&#x2F;a&#x2F;17833&#x2F;49" rel="nofollow">https:&#x2F;&#x2F;academia.stackexchange.com&#x2F;a&#x2F;17833&#x2F;49</a>.<p>If a student is smart some problems may seem to them like &quot;I see you know that 2 + 3 is 5, but what&#x27;s the reasoning?&quot;. So indeed, making problems more complex is the only way to go.<p>And the &quot;reasoning&quot; 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&#x27;t work for machine learning in a similar way, vide:<p>&quot;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&quot;, as quoted in <a href="https:&#x2F;&#x2F;p.migdal.pl&#x2F;2019&#x2F;07&#x2F;15&#x2F;human-machine-learning-motivation.html" rel="nofollow">https:&#x2F;&#x2F;p.migdal.pl&#x2F;2019&#x2F;07&#x2F;15&#x2F;human-machine-learning-motiva...</a>.

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.

I taught &quot;College Algebra&quot; for one semester, long ago, at a big ten university. Naturally I told my students &quot;show your work&quot; 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&#x27;t do it.<p>Many of my students came from schools where they learned a method called &quot;guess and try,&quot; where you plug answers into the problem and see if one works. This is a speedy way to dispatch multiple choice tests.<p>To my students, &quot;show your work&quot; 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.<p>In my view, &quot;show your work&quot; 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.

What? This doesn&#x27;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&#x27;t know why that won&#x27;t work elsewhere.<p>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&#x27;d forgotten, we&#x27;d add an imaginary i•sin(bx), integrate the resulting exponential only and then separate real and imaginaries. But whether that&#x27;s legal is kinda not obvious. It&#x27;s just letter manipulation to do that.<p>In a proof, it&#x27;s for you, so you know you did a legal thing.

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&#x27;re slick.

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 &quot;show youve done it correctly&quot; 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<p>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<p>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)<p>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

It can save the student&#x27;s bacon.<p>I remember a physics test: Answer one of three problems. Problem 1: I thought not enough information had been given. Problem 2: didn&#x27;t remember discussing this topic. Problem 3: did not even understand the question.<p>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&#x27;t believe I deserved but I was happy to get anyway).

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).<p>Real life problems do not come with an answer key, and real life problems tend to be hotly contested.<p>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?<p>Numbers are just numbers. They are orthogonal to answers, orthogonal to arguments.

I can&#x27;t think of an argument that makes sense to me to &quot;not show&quot;. 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&#x27;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 &quot;grade and throw away&quot; manner. Consequently, easier grading has a very high priority.<p>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.

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.

I had a highschool calc teacher who told us &quot;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&quot;<p>(The problem with increase complexity is that a lot of work happens, but after a certain difficulty level it&#x27;s all on the calculator)<p>I think this is a good solution.

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.

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

Oh Come On!