> 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.
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.
> Sometimes they’ll find a way to optimize the work away.
As would plenty of other adults, myself included. Whether that's evidence that children don't have much different self-motivation than adults, or that my self-motivation skills are still childish - that I'm not sure.
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.
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.
Sadly for you, Dave, the teachers from my youth are likely retired and/or dead. But, fear not, I didn’t care to “show my work” until late high school when it was mandated (as a comment upthread points out, for credit).
I ran into a situation that isn’t described: even if you’re right, they want you to prove that you didn’t cheat. Thanks to the absurdity of physics in imperial units, you can almost always show some bland dimensional conversions, but generally they love seeing an explicit change of basis for say straightforward calculus problems.
tl;dr: embrace your inner windbag, it won’t matter as long as they’re right (and one day they may appreciate it when they’re slow like us!)
When I tutor my siblings if the steps aren't satisfactory I make them repeat the answer from scratch. I immediately stop them when they're taking too many shortcuts. "Oh I just forgot this!" "Yes, now start over."
Each shortcut or omitted step is more mental load and for these beginners they will slip. Just missing a negative sign or adding a constant would cost the points on the test. Worse, if the tests are multiple choice questions (which are stupid imo) they get 0.
So to build their mechanical ability I don't even allow them to show how smart they are. I stop them when they're being too smart. I only reward complete steps.
It sounds rough and it kind of is but after using these methods for years they still come to me and ask me to tutor them, rather than our parents or a private tutor. They know that my diligent checking of their work cuts down on their study time and I'm still tutoring them into college now.
> they still come to me and ask me to tutor them, rather than our parents or a private tutor.
I wouldn't read too much into it... there's often times other factors involved. Just because you are a sibling and not a parent might be reason enough, at some age, to prefer you; and by college, it may already be a habit.
(not sure about the "private tutor' part, have they actually tried that, and was the tutor ok? because normally that's what a kid would prefer, since he/she doesn't feel the need to challenge the tutor's authority. Maybe a really poor experience/ personality mismatch with one private tutor? ).
I coach young people in other fields like public speaking so I have some skill in it. I use similar techniques.
The private tutors they've tried are actually math teachers. I think teaching a class is a very different skillset than tutoring.
According to my brother he came to the realization that an hour studying with me makes up for multiple hours with the tutors. I'm stubborn and authoritative when I coach, I challenge them and it takes me a lot of effort to butt heads with a teen, but I think they see the results.
Math is tough in that you don't get to enjoy it until mechanical concerns become an afterthought. So when my youngest sister, who initially didn't enjoy math, said "wow that felt good" about solving a problem I knew I was going somewhere.
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.
All through my life, I was a "plug and try" type of student when it came to things like algebra and quadratic equations etc. My rate (and accuracy) at solving those problems was roughly equivalent to my peers who did it the 'proper' way. Even in more fiendishly difficult maths like calculus, I found the 'shortcut' methods much easier to understand, digest and use than the longer, verbose first principles method.
As a corollary to that, I absolutely hated spelling bees at school because while I was excellent at spelling, I simply could not do it by visualising a series of letters in my head in order. I always had to write a word down, then correct the spelling (or not) depending on whether it 'looked' right to me. I had to see the full word before beginning the process of analysis and correction. I could not build the word (accurately) letter by letter in consecutive order.
I presume that my brain is just wired differently and processes information differently. Which is why nowadays I object to standardised methods of teaching and testing young people. Not everyone fits in the same thinking box IMO.
I would definitely say that if you want students to show their work, it behooves you to show what that means.
I had six years of Saxon math, and probably did upwards of 700 30-problem problem sets. After that kind of repetition and consistent grading, showing work was instinctive, I'd seen it lesson after lesson presented the same way, and had to do it thousands and thousands of times. Other math curricula I've seen require far, far less repetition and practice at doing the mechanics.
>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.
This doesn't just arise from students guessing. For any "difficult" problem, most students will end up doing useless steps, or possibly going down rabbit holes before finding the correct path. I don't think I have ever seen a proffesor that appreciated seeing all of this work (unless you couldn't find any path, in which case they often seemed to appreciate seeing what you did try).
a fictitious chain of reasoning and present it in a style learned from the textbook and classroom presentations
Communication is an important skill for scientists, it's not enough to discover some new science, you must also get the point across to your peers and the general public (and of course the examiner in an exam situation). There are accepted forms (although some textbooks are really lousy). I keep telling my students that and am known to take points off for poor writing or unclear reasoning. It's something I learned from my teachers.
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.
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
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.
Guess and try is completely valid way to show your work, as long as the question is "find a solution", not find all solutions. If you want to discourage that, draw problems from a larger solution space.
I wonder if our dialect differences change that perception. In NZ English (usually closer to British than American), it's "show your working". "Working" as a noun pretty much exclusively means mathematical steps, rather than the much more general "work", and I don't think I've run into the same mistaken student idea you did (I teach undergrads maths).
I totally disagree about the "fictitious" part though. I think without proper working most interesting problems are impossible to solve.
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.
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.
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.
I occasionally ran into exam questions that were "discuss the motivation behind making this maneuver instead of some other maneuver in this proof we discussed in class," which is a sort of meta-showing-of-work.
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.
if you're asking a proof question, it will be something of the form "prove that there are as many solutions as the degree for all polynomials" rather than a specific polynomial.
if you skip writing down each step in the proof, aka your work, you haven't completed the proof.
These types of questions are nice because they allow you (1) to check your work (2) work towards the answer. On the other hand, I feel like these questions make you think less and make you less cautious about your moves because "you know where to go." Furthermore, I had this multiple times in exams where the "supposed" answer was something like n^(-1) and I was so damned sure I did everything correct and my answer was n^(-1/2). I literally wrote on my paper:
"I know that the answer is n^(-1) but I did all the steps and checked it multiple times and did not get a different answer than n^(-1/2). Sorry."
Of course, I was right and it was a small typo. A good friend of mine /crossed out/ his answer because /he thought it was wrong/ (he had my answer).
Luckily the question was scrapped and I got bonus points, but my point stands. My friend wasted a lot of time staring at a wrong answer -- giving him less time to think about the other questions!
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.
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).
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
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).
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.
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.
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....
> 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)
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.
I realize that a student will occasionally get a particular question right by a method that she cannot write down, and also that, even more rarely, whatever way she got to that answer is actually a sound method rather than luck, but I question whether it is frequent enough to invalidate the advice you quote. In my experience, most of the time when someone does not know how to solve a problem, he cannot solve it in his head.
As you say in your last paragraph, "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", but that is not the situation addressed by the advice you object to. If they are answering the questions in their heads without understanding how to solve them, the questions would not seem to be a good match to the topic. If they do understand how to solve the problem but don't know how to put it in words, again that is a different problem, possibly in how it has been explained to them.
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.
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 :)
That is pobably why your teacher did not like multiple choice (correctly, IMHO): by seeing your work, he got a better understanding of how well you understood the subject. Understanding something is more than just getting the right answers on a test.
begging for a multiple choice exam in a math-heavy course can be a really bad idea.
I had one physics teacher who taught us this lesson. he designed a test where some of the answer choices would be the same number but with a different number of significant figures (0 pts if you choose the wrong one). in a different section of the exam, he made answers that were deliberately off from the calculator result, and you had to know your sig figs to know if it was correct (none of the above was also a choice).
after that, we were all very happy to show our work on the remaining tests.
Yes! The Seed of Doubt! If I ever have to create a multiple-choice test, this will always be one of the possible answers :-)
Just last month I had an exam where the examiner had already helped me correct a few basic errors by saying stuff like "well no you switched that around", then asked: "So what is the minimum distance here allowed by regulation?" Me: "Uuuuhh wait, it has to beeeee... 30!". Him: "So it's not 50 you sure?" Me: "wellllll Iii... (fuck you) no it's 30." Him: "Alright."
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.
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 "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)
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[ 0.21 ms ] story [ 115 ms ] threadThis 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.
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.
As would plenty of other adults, myself included. Whether that's evidence that children don't have much different self-motivation than adults, or that my self-motivation skills are still childish - that I'm not sure.
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.
I ran into a situation that isn’t described: even if you’re right, they want you to prove that you didn’t cheat. Thanks to the absurdity of physics in imperial units, you can almost always show some bland dimensional conversions, but generally they love seeing an explicit change of basis for say straightforward calculus problems.
tl;dr: embrace your inner windbag, it won’t matter as long as they’re right (and one day they may appreciate it when they’re slow like us!)
Each shortcut or omitted step is more mental load and for these beginners they will slip. Just missing a negative sign or adding a constant would cost the points on the test. Worse, if the tests are multiple choice questions (which are stupid imo) they get 0.
So to build their mechanical ability I don't even allow them to show how smart they are. I stop them when they're being too smart. I only reward complete steps.
It sounds rough and it kind of is but after using these methods for years they still come to me and ask me to tutor them, rather than our parents or a private tutor. They know that my diligent checking of their work cuts down on their study time and I'm still tutoring them into college now.
I wouldn't read too much into it... there's often times other factors involved. Just because you are a sibling and not a parent might be reason enough, at some age, to prefer you; and by college, it may already be a habit.
(not sure about the "private tutor' part, have they actually tried that, and was the tutor ok? because normally that's what a kid would prefer, since he/she doesn't feel the need to challenge the tutor's authority. Maybe a really poor experience/ personality mismatch with one private tutor? ).
The private tutors they've tried are actually math teachers. I think teaching a class is a very different skillset than tutoring.
According to my brother he came to the realization that an hour studying with me makes up for multiple hours with the tutors. I'm stubborn and authoritative when I coach, I challenge them and it takes me a lot of effort to butt heads with a teen, but I think they see the results.
Math is tough in that you don't get to enjoy it until mechanical concerns become an afterthought. So when my youngest sister, who initially didn't enjoy math, said "wow that felt good" about solving a problem I knew I was going somewhere.
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.
As a corollary to that, I absolutely hated spelling bees at school because while I was excellent at spelling, I simply could not do it by visualising a series of letters in my head in order. I always had to write a word down, then correct the spelling (or not) depending on whether it 'looked' right to me. I had to see the full word before beginning the process of analysis and correction. I could not build the word (accurately) letter by letter in consecutive order.
I presume that my brain is just wired differently and processes information differently. Which is why nowadays I object to standardised methods of teaching and testing young people. Not everyone fits in the same thinking box IMO.
I had six years of Saxon math, and probably did upwards of 700 30-problem problem sets. After that kind of repetition and consistent grading, showing work was instinctive, I'd seen it lesson after lesson presented the same way, and had to do it thousands and thousands of times. Other math curricula I've seen require far, far less repetition and practice at doing the mechanics.
This doesn't just arise from students guessing. For any "difficult" problem, most students will end up doing useless steps, or possibly going down rabbit holes before finding the correct path. I don't think I have ever seen a proffesor that appreciated seeing all of this work (unless you couldn't find any path, in which case they often seemed to appreciate seeing what you did try).
Communication is an important skill for scientists, it's not enough to discover some new science, you must also get the point across to your peers and the general public (and of course the examiner in an exam situation). There are accepted forms (although some textbooks are really lousy). I keep telling my students that and am known to take points off for poor writing or unclear reasoning. It's something I learned from my teachers.
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.
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
I totally disagree about the "fictitious" part though. I think without proper working most interesting problems are impossible to solve.
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:
That way a student can’t get away with just writing 2.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.
if you skip writing down each step in the proof, aka your work, you haven't completed the proof.
Proofs are sometimes a little unsatisfactory, the theorem is proved satisfactorily, but all the traces how you got to the proof are covered.
"I know that the answer is n^(-1) but I did all the steps and checked it multiple times and did not get a different answer than n^(-1/2). Sorry."
Of course, I was right and it was a small typo. A good friend of mine /crossed out/ his answer because /he thought it was wrong/ (he had my answer).
Luckily the question was scrapped and I got bonus points, but my point stands. My friend wasted a lot of time staring at a wrong answer -- giving him less time to think about the other questions!
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.
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).
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
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.
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....
Make them try to explain what they understood to their peers.
Give him a problem that's worth his time to solve.
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)
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.
As you say in your last paragraph, "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", but that is not the situation addressed by the advice you object to. If they are answering the questions in their heads without understanding how to solve them, the questions would not seem to be a good match to the topic. If they do understand how to solve the problem but don't know how to put it in words, again that is a different problem, possibly in how it has been explained to them.
I had this fight with my son years ago. Algebra is process and algorithms, what he was doing was not Algebra IMO.
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.
Eg. Given a polynomial with real coeficients, find the complex roots and tell me the sum of their imaginary parts.
I had one physics teacher who taught us this lesson. he designed a test where some of the answer choices would be the same number but with a different number of significant figures (0 pts if you choose the wrong one). in a different section of the exam, he made answers that were deliberately off from the calculator result, and you had to know your sig figs to know if it was correct (none of the above was also a choice).
after that, we were all very happy to show our work on the remaining tests.
Yes! The Seed of Doubt! If I ever have to create a multiple-choice test, this will always be one of the possible answers :-)
Just last month I had an exam where the examiner had already helped me correct a few basic errors by saying stuff like "well no you switched that around", then asked: "So what is the minimum distance here allowed by regulation?" Me: "Uuuuhh wait, it has to beeeee... 30!". Him: "So it's not 50 you sure?" Me: "wellllll Iii... (fuck you) no it's 30." Him: "Alright."
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.
(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.