I wonder if part of the problem that northTOmom’s twins encounter in their math classes (http://northtomom.blogspot.com/2010/10/this-math-depresses-me.html?spref=tw) is that we don’t teach students how to learn and study math?
Here’s what I’m thinking about: to really understand and make use of something in math (like long division), you have to understand the CONCEPT, have METHODS for solving problems, and have a PROCEDURE that lets you effectively and efficiently solve any problem where division is useful. I wouldn’t expect students to discover the efficient long division algorithm. But I do expect that they can and will construct an understanding of long division concepts (like what division means — how many of these are in that, place value, etc.). I also expect that they are perfectly capable of finding ways to do meaningful division tasks (like figuring out how many 1 1/4 yard-long banners they can cut from 10 2/3 yards of cloth). Then I want them, once they have a concept and meaningful methods, to learn a procedure. That can be by teacher demonstration, by building up from student methods, by rote, whatever. Then, to do the exact task that northTOmom described in which students break down the procedure and connect it to their conceptual understanding and personal methods.
For some students, those concepts and methods are easy to come up with and the mathematical symbols are as sensible as the yards of material. For those kids, I’d be happy to show them a procedure and ask them to pick it apart. But for all kids I want to be sure that the have the trifecta of CONCEPT, METHOD, and PROCEDURE.
The various reform curricula are, I think, designed to be more explicit about focusing on concepts and methods. But too often in their implementation, students are expected to get to procedures very quickly without instruction, rather than being taught how to understand math problems, strategies to solve them, and learning strategies to help them move from solving one problem to developing a procedure. I’ve worked with too many teachers who see that the topic is “long division” and try to get students to do and understand long division procedures without actually teaching them. In good spiraling curricula, there is something to master at every level, but it might not always be the efficient algorithm. It might be getting good at reading and making sense of problem contexts that involve long division. Or getting good at recording the methods you used to solve long division problems. If those learning steps are noticed and celebrated, then the frustration element decreases, and the students’ understanding of what they are learning, and why, is increased.
I could go on, but I think I’ll wait to see what others might say in the comments…
I understand what you’re saying about the “trifecta” of concept, method, and procedure, but does the order have to be set in stone? Current reform curricula and pedagogy favor that order, but isn’t it possible that some children learn the other way around, by practicing a procedure and, by means of thoughtful practice, seeing thorough the procedure to the underlying concept? I ask this because I believe I learned that way. I also was a kid who only began to enjoy math when it became more abstract, from about 8th grade on. Sure, I liked real-world problems, but the fun of math problems for me was in reducing them to formulas and numbers on the page. I think reform curricula does not encourage that kind of purely symbolic enjoyment of math–in fact it actively discourages it in the early grades. One of my daughters showed an interest in Algebra, so we taught her the basics, and she enjoys doing simple Algebra problems a lot more than anything she does in math at school. They are like puzzles to her, puzzles she can solve with the tools we’ve given her. Does she fully understand what she’s doing? I don’t know, but I don’t think doing this little bit of Algebra is hurting her mathematical understanding. (To keep up her interest in math, my husband is also planning to introduce her to very basic computer programming.)
One other thing: you say that “In good spiraling curricula, there is something to master at every level, but it might not always be the efficient algorithm.” I do understand this, but I think the proponents of these curricula don’t appreciate the extent to which it can frustrate a child not to be given the full picture. My girls, for instance, are always sensing that they don’t fully understand something in math–or don’t have the full set of applicable tools–but they don’t realize that they’re not expected to fully understand it at their current (6th grade) level. For instance, this year, in the unit about perimeter, area, and volume, they were given a question that required them to divide a whole number by a decimal. Because dividing by decimals had not been covered, the question had a calculator icon beside, meaning that kids can use a calculator; however, the teacher had told them not to use calculators. My husband and I had to teach the girls to divide by a decimal–in a rushed half hour before dinner–in order for them to be able to answer the question. Unfortunately, elementary-level reform curricula (taught most often by non-specialists in math) often necessitates this kind of “after-schooling”–which is one of the reasons parents find it annoying. It’s not that we’re lazy, but most of us are not trained to teach math and, in our household at least, math homework causes more tears than any other type of homework.
More and more I think the best teaching has got to be differentiated (but not because we’re all doing different worksheets). Your children, who clearly love the puzzles and elegance of formal math, need instruction that builds on that (and makes sure they don’t get lost in manipulations they can do but don’t understand). And the other kids in their class might need very different paths to get to the same understanding.
I tend to find more opportunities to differentiate when I’m working with really good problems, and when my students are good at learning to learn, and learning to learn from each other. I like to get students solving problems, with multiple resources available, identifying individually what they don’t know and want to find out, helping them learn it, and facilitating their teaching it to others. In that case, running into a problem where you need to divide by a decimal isn’t a cause for tears, it’s time to make a decision: what’s the best tool to solve this problem, and do I want to learn how to do it a different way because it’s worth it? And then I share back what I figured out with a class that’s eager to hear about my thinking and learning.
Of course, that’s a long way from reality in most math classrooms, but I don’t think it has to be…
I would like to echo one thing that northTOmom says. My experience is that these curricula (our district uses Everyday Math) freak kids out because they feel that they cannot do what is being asked of them — which is practically the dictionary definition of stress. The response from math teachers always seems to be, Well, they’re not really being expected to do that yet. As you note, in good spiraling curricula, there is something to master at every level, but it might not always be the efficient algorithm.
I could be fine with all of that, but that message is clearly not getting through to a lot of kids, who, as a result, completely stress out and develop what may be a lasting math anxiety. At some point, if that happens frequently enough with enough students, it’s insufficient to say that it is a flaw in the teaching; it’s a flaw in the curriculum.
I wonder if part of the problem is a failure to recognize the context in which this math teaching occurs. The teacher may think that he or she is letting the kids know what they are and aren’t expected to master at any given time. But that message is drowned out by the larger school environment, where the emphasis on right and wrong answers and on achievement is pervasive, and where the sorting out of kids based on “smartness” is happening constantly. A child in that environment is naturally going to feel bad if she feels like she is unable to figure out the “answer” to a math problem that the teacher has given her. At the very least, curriculum designers need to take those facts into account when they assess how clear they have made their expectations on any given problem. At worst, it may be that a spiraling approach to math teaching is just unrealistic in the environments we’ve created at our schools.
Hi Chris,
Thanks for the comment. I was excited to see your blog, since my family is Iowan and both my mom and aunt have taught in the Iowa City schools (my aunt is now a sub in Washington, IA, which has the best Italian restaurant in Eastern Iowa).
I agree with you that it’s really important to look at the context and culture of school overall — can we effectively teach math through exploration and problem solving when students are facing tremendous pressure to be the best, fastest, most correct? Can we make the culture in one classroom strong enough to override the predominant culture of schooling?
Max
Hey,
I was wondering if you have any solid information to share on the site on why they teach other strategies. It would be very helpful as I am currently doing an assignment on prefered strategies, however to find this out I need backround information on why they teach different methods instead of old fashioned algorithms.
-Thanks
I’d be happy to help if it’s not too late but I’m wondering who you’re thinking about? Why does *who* teach different methods? The reason I do it because I think it’s really hard to learn *why* “old-fashioned” algorithms work without learning some of the methods that they came from. I think people are really smart and can figure out quick ways to do things like multiply big numbers, divide fractions, etc. Then they can relate the methods they came up with to the old-fashioned algorithms. Relating two ideas makes them easier to understand, remember, or come up with an alternative if you forget what to do. More methods = more remembering & fewer mistakes!
Max