M. J. McDermott talks about math education in Washington state.
She talks about and demonstrates some “reformed” methods for solving arithmetic problems that have been commonly taught in the 4th and 5th grades: Cluster problems (taught in the “Terc” books), and partial products and partial quotients, and the “lattice method” of multiplication (taught in books called “Everyday Math”). She says that these “reform” math curricula discourage the teaching of what’s called the “standard algorithm” for multiplication and division (called “long division”).
In my own life I’ve used a method similar to the “cluster problems” to do multiplication and division in my head if I don’t have pencil and paper handy, or a calculator. The best thing I have when I just do it in my head is my own internal multiplication table. I find it difficult to do the standard algorithms in my head, though I think they’re fine if I have pencil and paper. If I don’t have to do it in my head I prefer the standard algorithms. I wouldn’t want to write out all the different sub-answers I’d have to write with the other methods. I remember when I did math drills in elementary and jr. high school. It would’ve taken me forever to finish math problems using the “reform” methods she describes. By the way, she’s for teaching the standard algorithm.
McDermott illustrates as well how problems are posed. They don’t do math drills. Instead they plan vacations, with geographic data. Group work is emphasized, as opposed to students working alone.
She describes how when she returned to college about 10 years ago to get a degree in atmospheric science, she encountered traditional aged students, just out of high school, in her classes who couldn’t do the math. She said they had the inability to work alone. They always felt like they had to check their answers against others’. They had a lack of basic math fluency in the symbolic language of math, and they were lacking in the ability to think logically. They lacked basic math skills such as arithmetic (which I assume means things like order of operations), algebra, and trigonometry. I’m really curious about the reason behind the problems with algebra and trig, but she doesn’t explain. I wonder how they got into college, especially the atmospheric science program without these skills. She said they had a complete dependence on calculators.
She paints a rather scary picture. It illustrates something that Justin James, a tech blogger, talked about a while back (unfortunately I can no longer find the blog entry where he talked about this), that math is important for developing higher order brain functions. Even if you don’t end up using the type of math you’ve learned, you gain mental abilities through the math work that you can use in other areas of life.
I think what this presentation illustrates is it’s easy to take traditional math skills for granted, because we don’t realize what we’ve gained from them unless we use those math skills directly. If we don’t use the math directly we tend to think that we had the cognitive skills we developed as a result all along. It looks like that’s a misconception we should be wary of.
One of the areas she mentioned where the skill of thinking logically is critical is computer programming. I agree. I’ve written previously about the lack of interest in enrolling in computer science in college, and about people who are in non-computer science courses actively trying to get out of them when computer programming is made part of them. Would this explain part of the aversion?
Edit 2/14/07: As I investigated this video further I found that some people were calling the demonstrated “reform methods” constructivist education. In some ways they may be right, just as people picked up on this when “whole math” and “whole language” were the “reform du jour” of the late 1990s. However I am suspicious that what’s been tried so far is not entirely representative of this education method. Sadly, I think that constructivism has been misunderstood by its practitioners in this country.
The idea is supposed to be that the teacher is an active participant in the education of their students, and that the students are an active part of their own education as well. What I used to hear about the older attempts at constructivist reforms is they put teachers in a passive role, leaving the learning totally up to the students. Further, the teacher was not to correct the students. The curriculum shunned rote methods. Anything that smacked of memorization or repetition was excluded. Students were supposed to come up with their own answers, and the teachers were not supposed to judge their work for fear of damaging their self-esteem.
Why they insisted on excluding repetition is beyond me. When children play they do things repeatedly. Repetition is part of their natural learning process. I don’t think what the educational establishment has come up with is what the creators of constructivist theory had in mind.
My sense of this teaching style is that overall the goal is to teach students how to learn, by using the students’ innate process of experimentation and play, while at the same time teaching them about real subjects they need to learn. The teacher should be a guide to the students. Yes, the students are to explore and experiment to arrive at their own answers to problems, however the problems are posed by the teachers, giving the students a goal to achieve. If the students are coming up with answers that are off, the teacher can introduce further problems prodding the students to explore the subject more, until they arrive at a correct answer. The goal is to build understanding of the subject. Yes, students build their own model of the world through this process, but it does a disservice to the student if that constructed model is not compatible with the real world. Further, the teacher provides some concrete examples of the subject matter, giving the students a grounding in it, before they are allowed to go off and explore it.
My impression is that the goal is to help students arrive at correct answers, but process is emphasized so that it’s the students who obtain the answers, through their own process of discovery, without the teacher imposing a single way to arrive at those answers. This creates a sense of accomplishment in the students, and self-esteem, encouraging them to continue the process. I may be wrong on some points.
I think there are times when rote methods are best for gaining a grounding in a subject. One of the things I thought was insane about “whole language” was they banned phonics from the reading curriculum. Phonics has consistently worked well as a way to teach children about reading. I think the reason it’s essential is that language in large part is standardized in our modern civilization. It provides a way of conveying information that is logically consistent from the disseminator, to the receiver. It’s essential that people’s understanding of language be consistent. It’s how we make connections with each other, and conduct transactions. Students who don’t get this are going to be handicapped.
I think the goal of reform efforts has been noble. The problem has been in their execution. The goal was to get the schools out of the old Industrial Age model of assuming that the students were blank slates upon which the teacher would impress their knowledge. The teacher was pretty much the only one who was active in the process. The students would just passively receive the information (hopefully) and then either regurgitate it later or synthesize it into something new.
The goal with constructivist methods was to create active learners and explorers. That sounds good to me. The problem seems to be that the ways in which this method is being used leads to students who do not fully understand the subject matter. It seems they end up behind other students who were taught using the older methods. I think the old methodology provides a useful benchmark. If using a new methodology results in students who are behind grade level I think that indicates that the methodology needs to be revised and some assumptions about it need to be questioned.