This is part 2 of a 4 part series on Khan Academy and math education, specifically American math education. Part 1- Khan Academy and the mythical math cure set up the context, my point of view, and a bit of learning theory. It also discussed one prevalent myth of American math instruction, that math is a discrete set of sequential skills. It wrapped up with some research on effective multimedia in math and science instruction. And of course what all these things have to do with Khan Academy.
I’ll continue with another American math myth — that math is best taught by having students practice step-by-step procedures that lead to the right answer.
The prevailing theory goes — experts figure out the best set of steps to solve any problem, we show students these steps, then they practice the steps until they can easily solve problems. If you believe this myth, it follows that if students don’t learn math: 1) it’s the teacher’s fault for not being clear enough, or 2) it’s the student’s fault for not practicing enough.
Khan Academy fits this myth perfectly. Here’s a quick (and even better, free) way to help with both of these. Replacing or supplementing a teacher with a video solves #1. It solves #2 by saying that these videos should be watched outside of class, thus freeing up time for more student practice in class.
But here’s a question… What if it’s not the teacher’s fault or the student’s fault? What if the assumption that people learn math by watching and practicing the pre-determined steps is wrong?
Alfie Kohn has famously said that you can’t practice understanding. The confusion is that we think math is similar to tennis or other skills that demand muscle memory and reptilian-brain reaction.
“By contrast, when students are simply told the most efficient way of getting the answer, they get in the habit of looking to the adult, or the book, instead of thinking things through. They become less autonomous, more dependent. Stuck in the middle of a problem, they’re less likely to try to figure out what makes sense to do next and more likely to try to remember what they’re supposed to do next – that is, what behavioral response they’ve been taught to produce. Lots of practice can help some students get better at remembering the correct response, but not to get better at – or even accustomed to — thinking.” – Alfie Kohn Do Students Really Need Practice Homework?
And worse, assuming that practice creates proficiency backfires in the worst way with students who are furthest behind. Students who understand the material and made to complete a lot of practice will be, at worst, bored. But students who do not understand are being drilled into desperately guessing, never quite sure why they get some answers right and some wrong. It develops into a feeling of dread, of never being sure that they are doing anything right, but mostly that they just aren’t cutting it, and never will. Students who develop a deeply-held belief that they are not “good at math” may never overcome this.
The trouble with algorithms We double-down on the assumption about “learning by practicing” by breaking problem-solving into bite-sized chunks. We teach children specific ways to solve types of problems – tricks, mnemonics, and step-by-step processes (algorithms) like borrowing, carrying, or FOIL. We prompt students to look for clues in word problems, like if you see the word “more” it means to add. At the end of the day, this trains kids not to think, but to quickly try to guess the hidden rule and move on. All the help is well-intentioned, but reinforces a guessing game approach to math.
“Algorithms are harmful to most young children for two reasons: (1) They encourage children to give up their own thinking, and (2) they “unteach” what children know about place value, thereby preventing them from developing number sense.” Constance Kamii and Ann Dominick – The Harmful Effects of “Carrying” and “Borrowing” in Grades 1-4 (also in PDF, sometimes the Google doc doesn’t seem to work..)
This quote is from a research study that found that teaching carrying and borrowing to children significantly damaged their ability to solve addition problems. This is a must-read from 1998 that points out that despite numerous research studies that confirm the damaging effects of training children in carrying and borrowing algorithms, we continue to do so in most U.S. classrooms.
“The Harmful Effects of Algorithms in Grades 1-4” was published in 1998, four years after Kamii (1994) had published even more data. But 15 years later, most curricula still include the teaching of “carrying” and “borrowing.” When educators use research to inform practice and teach mathematics as a sense-making discipline, we will have a much better chance of helping all children be successful in mathematics.”
Our beliefs, even when refuted by research, allow us to continue to hope for magic wand solutions that make our beliefs real. Math myths keep us on the lookout for an easy answer that isn’t there. When something doesn’t work, myths allow us to ignore evidence and keep doing the same things because we “believe” in them. (If practicing isn’t working, practice more!) It makes us less willing to do the hard work of actually dealing with students individually and grappling with deep and difficult questions about how best to teach math. It’s all too easy to say, let’s push play on the video! Hurray, all our problems are solved.
The problem with “problems”
Additionally, we confuse solving problems with answering test questions and textbook exercises. Khan Academy deals with the later – specific steps for finding the right answer to “problems” that students are mostly likely to find on tests and in textbooks. (Some good examples of the differences between the two can be found in this blog post: Khan Academy is an indictment of education by Frank Noschese, a physics teacher and blogger.) This tricky word swap is confusing, because we DO want kids to have good problem-solving skills, but we certainly mean more than just answering textbook exercises. If we break a student’s confidence by imposing someone else’s problem-solving algorithm, when they encounter a real problem, one that isn’t made up for a test, they lack the confidence to explore their own solutions once they’ve gone through the list of algorithms we’ve had them practice with such fury.
The curse of the right answer
Math is viewed by many people as being logical, somewhat cold, and very rigid. Math is seen as the one subject where there are cut-and-dried right answers. But we forget that there are many ways to the right answer, and exploring these different paths helps strengthen existing mathematical understanding. Instead, we give kids lots of problems to work on so they can show us that they can get right answers quickly. It becomes about the product, not the process.
What ends up happening is that we spend a lot of time telling kids they are wrong, hoping that they will “get it” and start being right. Constance Kamii, an eminent math educator and a protégé of Piaget, says that this is completely the wrong approach – that if you destroy a child’s sense of autonomy and self-confidence, they will never recover that. She says that you should allow children to solve problems and LISTEN as they do, preferably in a group setting, as they discuss their answers. Let them convince each other based on their own observations and problem-solving ability. Let them defend their answers – even when they are wrong. Because it is destructive to tell a child they are wrong, but constructive to let them move from their first answer to an answer they come to like better.
I wrote a post about seeing Dr. Kamii do professional development with math teachers using this model – Questioning assumptions with Constance Kamii. Constantly telling children they are wrong creates a sense that right answers are simply mysteries that appear out of nowhere, and some people can guess them and some people can’t. And if you fall into the “can’t” pile, you are doomed forever.
This is not a made-up, one-off fantasy
You may be thinking, well, I’m sure a few teachers here and there do this, but not at the scale we need in this country! (You can also read my thoughts about the scaling question – Big problems require small solutions.) Take a look at this one national example (there are others). The New Zealand Numeracy Project encourages flexible strategies for solving numerical problems, and discourages reliance on standard computational algorithms. The project supports teachers with professional development, resources, and coaching. It gives parents information so they understand why their children aren’t being taught the same problem-solving rules they were taught.
Here’s just one evaluation done on it – The Algebraic Nature of Students’ Numerical Manipulation in the New Zealand Numeracy Project showed that, “…that students who participated in the Numeracy Project solved numerical problems that required manipulation with more success than did students who had not participated in the project.” Why are they doing this? Because New Zealand decided to pay attention to research about how children learn, not myths.
Americans have a bi-polar view of youth autonomy. We want them to be empowered AND to do what we (adults) tell them to do. We want them to find their voice AND sit still and listen. We want them to think outside the box AND bubble in the right answers. We are fooling ourselves if we believe that we can tell children that math is fun and creative, but only if you do it MY way. We must be able to answer “why do I have to learn this” with something better than “because you’ll need it in grade n+1” (where n = the grade they are in now.) This just reinforces the message, “… shut up and do what I say.”
When we think about how students learn math, it’s all too easy to discount how they feel about themselves as math learners and users. We want them to just do the work, pass the test and move on. There are students who are compliant and do just that. But there are many many students who get caught in extended power struggles with teachers, parents, and the school system. Some of these power struggles are overt, some quiet, but it’s a waste of potential all the same. (I can’t think of a better book about this than Herb Kohl’s I Won’t Learn from You and Other Thoughts on Creative Maladjustment.)
Constance Kamii has this quote front and center on her website, “A classroom cannot foster the development of autonomy in the intellectual realm while suppressing it in the social and moral realms.” Why would a math educator care so much about autonomy? I would encourage anyone exploring this question to take a look at some of the videos on her website that show classrooms where great care is given to this question.
In other words
Teaching math is not like Teach Me How to Dougie.