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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. I’ve known plenty of students that could see answers that they couldn’t fully explain, and I think it is patronizing of me to not believe in their understanding simply because they can’t 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 student’s process. You should of course draw out solutions to a problem, but only if that’s 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 Karen’s 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 don’t trust that the students are really doing the work we set out for them, or we don’t 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 don’t want them to use on the problem. I have been noticing lately when I make choices as a teacher because I don’t 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 don’t 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 shouldn’t) be asked to show work, chime in!

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19 thoughts on “Showing Work in Mathematics

  1. bretbenesh says:

    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.
    Bret

    • Wendy Hines says:

      In high school, I was told to show my work and I didn’t 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 didn’t want my teacher to know that).

      Yet, I make my differential equations students show work. I believe that at that level, you don’t really understand what you are doing if you can’t explain why it works. It’s 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 I’m 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 that’s the issue- maybe we should have students explain why their method works rather than have them show work. The two are slightly different.

      • bretbenesh says:

        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 teacher’s method), then it seems like we get the best of all worlds.
        Bret

      • Angela Vierling-Claassen says:

        I think the thing I’m struggling with is the “forcing.” If it really is valuable, we shouldn’t 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. thatjen says:

    Intution will only take most students so far, and in K12 education (I can’t speak to higher ed personally) it’s 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 I’d 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. Angela Vierling-Claassen says:

    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, I’m 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 that’s 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 I’m not sure we go about it in the right way.

  4. bretbenesh says:

    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 don’t know—it just came to me.”

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

    • Angela Vierling-Claassen says:

      Ah, but can’t we start with intuition and then reflect and get into metacognition. If we can do it in a way that doesn’t destroy or belittle intuition, then we may be able to grow the intuition in to something bigger.

      Here’s 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. bretbenesh says:

    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 intuition—this is a method.

  6. bretbenesh says:

    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 isn’t, 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. bretbenesh says:

    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. clenora says:

    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. Angela Vierling-Claassen says:

    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 don’t 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 wouldn’t 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. guevarayazmin says:

    I think that part where karen’s speaks of math conversations and sharing between students is a great idea. In my experience I didn’t have much of that and when we did that’s 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. I’ve 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. kjohns28 says:

    Sometimes the answers just come to you and you’re not sure how or why! The thing that used to really make me mad was “partial credit” because you either didn’t do your work for the answer the way that the instructor asked you to or because you only gave the answer. If it’s right, it’s 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.

  12. brettrubicristian says:

    I jus finished writing a blog about this and was looking for others.
    http://wp.me/p2woZN-2r

    I enjoyed this blog and the comments. I think an important issue is with level of ability of the student. Another important issue is level of difficulty of the math.The lower the level of math, the less work should be need to be shown. And the most interesting element is every student with different abilities has a different point at which the math becomes so difficult that work must be shown. One last idea is that I find it is easier to get students to show more steps if I make the problems so difficult they can’t do it without showing steps.

    • Avery D says:

      Using longer strands of numbers and numbers with more intricate number values sounds like a great idea, although eventually you get down to the small easy problems within those, and whose to make the call to weather its right to skip the small problems inside of the larger one, its an inevitable cycle at that point

  13. Armaxx says:

    I used to know things without fully understanding the reason, I just knew it to be correct. This way of thinking went beyond mathematics, I was good at music, reciting lyrics and conversing with people beyond my demographic and age group from a very early age. Things just came naturally to me but in school I had to explain my reasoning and to do that I had to break down what I knew and reassemble it in a way that suited other people and not myself, which I was unable to do effectively without losing what it was that allowed me to come up with the answer. The end result of this was the feeling that my way of thinking was unworthy or arrogant. I struggled to break down my thought’s into what was considered acceptable in school and ended up feeling stupid despite having answers that most kids my age didn’t. If you have a child that has a natural ability to solve problems, why force them to break that structure and rebuild it in a way that suits the majority. This seems counter-intuitive to development. For every step forward (evolutionarily speaking) our communal nature as a culture forces us to take two steps backward. Why not trust that, left to their devices and encouraged along the way, this child might grow up to find an environment and like-minded people that nourishes their thought process and amplifies their ability to invent and discover from a completely subjective viewpoint rather than being straddled by patterns developed by others and shaped to adhere to the consistency of pre-established thought, developed by previous, less informed social constructs. In my case, things used to work naturally for me and I was well above the average of my peers. As I progressed through school I found it increasingly difficult to change my patterns to accommodate the curriculum’s requirements and as a result I began to fail and fall behind. While I could still work things out in my head and come up with correct answers faster than most, I was unable to do it to the standards of the curriculum I.E. showing your work. By high school I stopped thinking I could cut it in school, I didn’t feel like I was stupid but I didn’t know how to communicate my thinking to the standards expected. By Year 10 I had given up on education but continued on to year 12 covering the most basic subjects, falling further and further behind the peers I considered intellectual equals (They never needed me to show my work they just accepted that I was intelligent). I missed out on learning the hard stuff that’s drilled into kids in high school, like algebra, economics, chemistry etc… because of this one institutional inadequacy. When my friends moved forward, I moved backwards. I won’t go into the vices I used to dull my over-thinking and feelings of social ineptitude that were associated with feeling alienated by the education system. All I will say is that I am now 27, painfully alone, painfully unaccomplished and desperately longing for a world where education systems were able to better identify and accommodate individual requirements for cognitive learning. I guess things are better now than they were but due to a mixture of circumstance and the resulting bad choices I will never feel like I reached my potential, far from it. That is a horrible feeling and I don’t wish it on anyone. Thanks for listening.

  14. Avery D says:

    As a student in the 8th grade this is a very annoying issue of mine, my teacher is asking for us to show “all” work from the simplest 4+2-3, now for most of you reading this you have solved that already, you don’t know exactly how but you did, it was an instinct, now if you try writing such a problem out you’ll realize that its harder to write out the entire problem with out trying to skip to the answer, longer problems like finding M.A.D
    have lots of little things that can be done in your head but need to be shown although they still have complex parts to them, it makes it difficult to show work, i wish there was a balance for students, enevidibly I feel like I’m on the same path as ARMAXX

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