When a person writes a computer program, she has to run small parts of the program at various points along her work to see if the code does what she intended it to do. Frequently the programmer finds a “bug,” that is an error of some sort. The programmer then has to “debug” her program by correcting the error. Usually this involves evaluating what the program actually did and comparing that with what the programmer intended the code to do, and then analyzing the discrepancy to identify what changes he needs to make. Unless a programmer is working on a trivially simple program, she expects that the process of writing a computer program will involve debugging. In other words, identifying and correcting “bugs” or errors is understood as a normal part of computer programming.
This attitude toward errors is a very productive and useful way to think about mistakes. In many math classes, however, there is a very different attitude: Mistakes are treated as failures and students are penalized for them in one way or another. If you ask students what the difference is between a good math student and a bad one, the chief criterion is usually whether or not the student makes errors.
One of the main reasons that errors are often treated in math class this way, is that, as I have discussed in earlier blog posts, the dominant method by which teachers present new material to students is to demonstrate a new procedure and have the students parrot it.
When a teacher exposes students to new material primarily by means of demonstration, she is circumscribed in how she can respond to a student error. She must focus on helping students memorize decontextualized steps that often don’t make much sense to the students. Thus in many classrooms students learn various mnemonics such as DMSB (for, divide, multiply, subtract, bring down, i.e., the steps involved in standard long division) or "keep-change-flip" a set of symbol manipulation procedures for dividing a number by a fraction (i.e., keep the first number unchanged, change the operation from division to multiplication, and flip, or use the reciprocal of the second number). Techniques for solving words problems are also frequently taught in a similarly mechanical way, with students taught to translate certain words into certain operations.
If a lesson has been presented in this manner and a student makes a mistake, when the teacher works with a student to correct the error, it is rarely about whether the student’s answer makes sense based on an understanding of basic number patterns, but whether the student has implemented the series of steps correctly or not. The consequence of this approach is that in the minds of many students, doing math means the memorization of steps that make little sense, and many students, even many who have been generally successful in school, graduate with weak math skills and great trepidation when they are required to do any math.
There is an alternative. It turns out that students can have experiences exploring fundamental patterns and relationships that lead them to understand how to use new techniques and perform new tasks. An essential part of such experiences and investigations is an analytical stage in which students try to explain new patterns and relationships that they are observing. As they analyze their new experiences and try to come up with general observations, they will inevitably make mistakes. These mistakes, it is important to stress, aren’t because of some failure or defect of the person’s mind. They are an essential part of how humans learn. That is, learning is a process that involves repeated doing followed by evaluation and modification. Learning most things, one might say, involves lots of little bites instead of one big gulp.
Thinking about student errors in this light gives teachers a powerful framework for understanding how to respond to them. The teacher should understand that her role in many cases isn’t simply to give the student the correct answer in response to an error, but to show her why her answer is incomplete or inaccurate so that she can figure out what the “bug” in her thinking was. This sort of response keeps the responsibility for “fixing the bug” with the student and ensures that she is always trying to make sense of the material she is investigating.
This approach also helps students develop strong self-correction skills because they are constantly in an environment in which they are testing the adequacy of their responses. This in turn helps build confidence because the student amasses countless experiences in which she had to continue to analyze some task until she herself comes to recognize and understand a new pattern.
Student engagement in math is also improved with this approach. One of the joys of doing mental work is, as it were, putting the pieces together oneself. The student has legitimate pride of ownership about her new understanding because it was the result of her own initiative and effort. Knowledge acquired in this way is much more satisfying than knowledge that is simply “given” to one by someone else.
I owe much of the previous thought to the ideas of Seymour Pappert in his classic book, Mindstorms. I highly recommend that book. I owe a debt as well to my training twenty years ago in Lindamood-Bell reading programs. Their mantra, “respond to the response” helped me understand the importance of precious error correction and to see it as an essential part of good curriculum design.
I know it is difficult to grasp the abstract points I'm making without examining specific interactions between teachers and students. I do have videos on the ABeCeDarian YouTube channel with regard to correcting oral reading errors, but I don’t yet have any for math. The general principles, however, are the same: the goal is to provide the student enough information that he can figure out what he has to adjust to perform a given task correctly. Over the next months, I’ll try to make some videos of good error correction in math.
Until next time,