Math Difficulty / Math Teaching / Pedagogy

Showing Work in Mathematics

Last week, Karen Young stopped by this blog and made a comment that led to a great discussion that has taught me a lot, so I decided to pull it out and capture it in this post. First, she said in the first comment on this post:

As a kid I could do math in my head, doing basic mathematics without using a calculator or pencil (when I do laps I still do fractions in my head). Every test, there were marks off for not showing my work. If I answered a question on the board I had to show my work. Well sometimes I couldn’t do that because my brain “knew” the answer.

I replied to that point (and yes, I edited a typo in my original comment):

I really think your point about showing work is interesting — I know that the curricula used in my area are all about showing work, often showing multiple methods. I can get that we want students to be able to communicate their thinking, but can we do that without making it drudgery?

To which Karen made a response that really blew my mind:

Angela, when did we decide, as educators, that we had to show work in math? My grandfather was an accountant and brilliant with figures but he did them in his head. That is how I did math when I was young. Having to show it actually made me have to rethink my answer, leading me to doubt my accuracy, which lead to me “showing my work” and making more mistakes. Math was intuitive, almost instinctive prior to that point. Now it is something I fumble over, except when I am swimming laps and almost in a trance. Why is intuition in learning a bad thing?

She further expands this in a later comment:

We always remember how to tie our shoe because of motor memory, so if we’ve created a motor memory (or song memory) through the physical teaching of math, doesn’t that link remain? Especially if we exercise it everyday? At some point, if we approach a subject in the wrong way, I think we can break the ” old link” in the brain, by creating the new “show your work” path. And do the two conflict? In my case, yes. I am intuitive by nature and am used to my brain making what appears to be sudden connections but I am in fact just allowing it free rein to make associations that lead to “aha’ moments.

So, why do we ask student to show their work in math? Here are some reasons I came up with (if you have more reasons  or more interpretation of these reasons, please lay them out in the comments):

  • We believe that explaining the mathematics is an integral part of understanding the mathematics. That is, just as Karen says, we mistrust intuition. After this conversation with Karen, I think that this mistrust may be wrong-headed. Ive known plenty of students that could see answers that they couldnt fully explain, and I think it is patronizing of me to not believe in their understanding simply because they cant walk me through a solution step-by-step in a way that I expect their mathematical learning should have trained them to do.I see both my own 6-year-old and a fourth grader I tutor working with the TERC Investigations curriculum which asks them to draw out solutions to problems (for the first grader) and to show two different methods for a problem (for the fourth grader). These both seem like they might artificially interfere with a students process. You should of course draw out solutions to a problem, but only if thats the way you solve the problem if you count on your fingers or see the answer in your head, the drawing step is artificial, meaningless, and can possibly get in the way of your own method. And you should explore different methods for doing, say, three-digit addition so that you can understand the process and settle on a method for yourself that is actually meaningful, but why would you need to show two different methods for the same problem simultaneously? (And see Karens comment below this if you were comparing answers with a peer you would get exposure to multiple methods for the same problem in a more authentic way.)
  • Some problems are complex enough that they require record-keeping. This might be careful recording of useful data, recording results and methods so that the problem can be tackled again after a break without losing momentum, or writing an explanation to be shared with others. Asking students to show work even on less complex problems may help to train them to do this kind of recording so they have it available as they problems they are going to tackle get more intense. I think this is actually a good reason to show work, and the question for me becomes whether we can do this kind of training in an authentic way that still honors intuition (which is really just deep, non-verbal understanding), but helps students gain facility with communicating.
  • We dont trust that the students are really doing the work we set out for them, or we dont trust that they are using the methods we want them to use. This could be because we think the students are cheating or because we think they are using techniques or technology that we dont want them to use on the problem. I have been noticing lately when I make choices as a teacher because I dont trust students. It is more often than I would have thought, and I want to find a way to stop and increase my trust of students.
  • As instructors, we want a window on students thinking in order to help them. If we only have an answer, and that answer is wrong, then we dont know anything about where the student went wrong it could be as simple as an error in what numbers were used, but it could be a misunderstanding about the mathematical concepts. This is, I think, another good reason for having students show work, but it is really only needed if students are struggling.

I laid out these reasons for Karen, more or less, as part of the comment thread then asked:

How do we support student intuition and still help them to develop record-keeping and communication skills that will serve them well as problems get more complex? And if we are having them “show work” to help develop those kind of skills can we make sure that it is authentic, not made up after the real work is done to satisfy requirements?

Karen answered:

When we look at teaching math we are, in a sense, trying to develop two skills within math once the basic math foundations are in place. The ability to edit numerically (see your mistakes) and to be able to reason mathematically (knowledge, logic and intuition). This is predicated on the foundation being strong (but we both know sometimes it isn’t.) In English, I have had many students who have a wonderful writing style, a true “voice”, but their grammatical skills are terrible. I have always counseled them to keep writing and find themselves a good editor. Not every student can see their mistakes, which is why we have groups share papers to help proof at all grade levels. So why can you not have math conversations and math proofing shared between students? If sharing is how we build knowledge and understanding this would help support student math learning and math intuition.

I think this is a great suggestion, and a profound one in mathematics. If students are in conversation with each other, then communicating mathematics and showing a record of your work become authentic tasks that allow you exchange ideas with other people.

Thanks, Karen, for the great conversation! If you have any thoughts about your own experiences with showing work or asking students to show work, or if you have thoughts about why students should (or shouldnt) be asked to show work, chime in!

15 thoughts on “Showing Work in Mathematics

  1. Here are two more reasons I want students to show work:

    1. Communication skills are important and should be practiced. An engineer who can intuitively build a bridge but cannot explain why it will stay up will not have that bridge built (this is waaaay oversimplified as an example).
    2. I want to engage my students metacognitive skills. These skills are associated with better learning, and intuiting an answer engages in zero metacognition.

    And I also want to second your idea that student work helps us to help them. It is easy to think that showing work is unimportant when the answer is correct; showing work is most useful when the answer is wrong.

    • In high school, I was told to show my work and I didnt know how to. I would invariably turn in a homework paper that was completely correct and get a 0 on it for not showing any work. This discouraged me from turning in any homework (I still did it actually, because I enjoyed it, but I didnt want my teacher to know that).

      Yet, I make my differential equations students show work. I believe that at that level, you dont really understand what you are doing if you cant explain why it works. Its too easy to use a recipe to solve an ODE without having any idea why that recipe works. So I ask my students to explain why their methods work.

      Once you get to a proof class, showing work is all there is. Students who already have experience explaining solutions will be better served by the time they get to a proof class.

      When my daughter was young, she had strong intuition about mathematics, but making her show her work made her feel like there was a right way and a wrong way to do a problem and she really wanted to please her teachers so she was always worried about doing problems the right way. In the end, this made her stop relying on intuition and start relying on cookbook methods, without understanding where they came from. Her intuition was railroaded and she stopped doing as well in mathematics.

      So Im torn on the showing work issue. Intuition has its place. But so does forcing a student to explain why his or her method works. Maybe thats the issue- maybe we should have students explain why their method works rather than have them show work. The two are slightly different.

      • Hi Wendy,

        I agree that explain why their method works is preferable. I also think that my comment below about grading less might go a long way. I think that your daughter might have kept her creative ways more if she had not been graded on it. If the teacher just gave her feedback on the method (after thinking hard about whether the reasoning was correct, and not just the same as the teachers method), then it seems like we get the best of all worlds.

      • I think the thing Im struggling with is the forcing. If it really is valuable, we shouldnt have to force it. And if we believe it is valuable and we are forcing it, maybe we need to approach things another way. Bret, I think you are right that we have to let go of some of our grading and need to judge!

  2. Intution will only take most students so far, and in K12 education (I cant speak to higher ed personally) its not easy to tell which students can go all the way in intuition alone. I know, because intution failed me, literally. I was a facile math student through ninth grade and relied on intuition and getting it in advanced courses from third grade on. In high school, the lack of deep understanding began to be an issue, and by the time I began calculus, I was screwed. I understood the higher level concepts, but had no foundation to which I could apply them, because my glib intake of precalc had not created deep pathways. I failed Calc not once, not twice, but three times. Could I have applied myself and passed? Sure. But Id have been better off and passed sooner, I believe, if my earlier education had put more emphasis in the process and not simply the answer.

  3. Great comments, thanks! It makes me think that we really have to walk a line. Both intuition and explanations are important, so how do we balance them for each student.

    @Bret, Im not convinced that intuition means no metacognition. I think we can reflect on a problem we solved through a flash of insight without showing our work. I think thats what we need to be looking for, opportunities for that reflection that are more meaningful than show your work or solve the problem with two methods.

    @ThatJen I think you are right, and I think that not being able to tell encourages us to develop one-size-fits-all solutions to develop that step-by-step thinking. You are right, we need to make sure people can do that kind of careful thinking, but Im not sure we go about it in the right way.

  4. Hi Angela,

    Hmmm. . .to me, intuition seems incompatible with metacognition (I am a fan of intuition, too, so I am not saying it is bad). To me, intuition means I came up with this answer, and it seems right, but I cannot explain why. Any sort of metacognitive thought would be trivial: What was I thinking? I dont knowit just came to me.

    Perhaps I have an odd definition of intuition? What do you think?

    • Ah, but cant we start with intuition and then reflect and get into metacognition. If we can do it in a way that doesnt destroy or belittle intuition, then we may be able to grow the intuition in to something bigger.

      Heres my working defs wrt math problems:
      * Intuition: Feeling an answer in your bones, feeling rather than laying out an argument that an answer is correct, being able to skip step to arrive at an aswer
      * Metacognition: Reflecting on problems, answers, and solution methods, asking not just why about an answer but why about where the answer came from

      Great question, would be curious about what others think

  5. And just to clarify: I think that a student can engaged in metacognition without showing work (they can silently think about their thinking, for instance). But there would have to be a mental process that the student is aware of in order for that to happen. To me, this is not intuitionthis is a method.

  6. My previous response was done simultaneously with yours, Angela, so it was not a response.

    Here is my response: I absolutely agree with everything you said. I think that many students start with intuition, and that is good (even if it isnt, it is probably reality). But to advance, we need to coach them into thinking about what is really going on in their brains. This can be done well, and it can be done poorly.

    Perhaps it is because I lack imagination, but I cannot think of how to coach students through this without having them show their work. This does not necessarily mean writing down every step, but I think that the student needs to communicate what they were thinking in some fashion (perhaps verbally). This is what I mean when I think show your work.

    So I think that we agree?

  7. And not to get off on a rant here, but this is where we need to be very careful about grading. It seems like one way to do things wrong is to say: How were you thinking about that problem? Really? You were thinking about it wrong: you get a D. That does not seem helpful. It seems like there should be A LOT of formative assessment before we even think about assigning a grade to the thinking.

  8. Wow! What an intense back and forth conversation. I have had bad and good experiences with having to show how I arrived at a conclusion of a math problem. I recall a teacher in grade school who was very big on showing the work of our problems. The problem for me in that class was that I always got the correct answer, but I was never able to arrive at that answer with the formula that the teacher wanted us to use. I could not grasp it and he was unable to accept my answer with my formula. I did not do well with that particular math. I always felt that I was punished for not being able to complete the math the way my teacher wanted even though I was getting the correct answer. But then there are times when showing my work has helped me understand the problem better. I certainly can understand how difficult it is to make students show thier work when you may be hindering a student who just operates differently. Tough call.-J. Morin

  9. I like that sort of balance you give, J. Morin it seems like we should be punishing students because of their methods (or their difficulty explaining/showing), like Bret says above, we dont want to just leap to grading students harshly for wrong thinking.

    Bret, I do think that we agree. I end up getting myself all tied up in knots around the compulsory qualities of what we do, but I think you wouldnt make them quite so compulsory! And then I also get worried about the degree to which we shut down alternate methods of knowing or understanding in math, compelling people to do things our way. But that may just be me going to far down the feminist/critical/queer pedagogy rabbit hole.

  10. I think that part where karens speaks of math conversations and sharing between students is a great idea. In my experience I didnt have much of that and when we did thats when I learned that most. When I was stuck with problem and we were in a group I was able to learn other ways that others were doing or figured out what step I was missing. Ive always been the student that has to work the whole problem out and write each step because one step will throw me off. Being in groups helped me a lot.

  11. Sometimes the answers just come to you and youre not sure how or why! The thing that used to really make me mad was partial credit because you either didnt do your work for the answer the way that the instructor asked you to or because you only gave the answer. If its right, its right! Right?

    Bret, I like what you said about formative assessment. Maybe there should be more of a conversation around the problems involved/ processes in a test/quiz before grades are given.

What do you think?

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Connecting to %s