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?
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!