...I resist it because, philosophically, I disagree that it's a good way to teach.
This is what I believe:
“Deeper conceptual understanding” in K-12 math comes with knowledge and practice to mastery, not with pointless struggle and reinventing of the wheel. Efficiency on paper is critical; the calculator tends to get in the way of learning. Each day, as I work through another chapter, I think, “Oh, yes. Right. I see that now.” Proper process is being reinforced for me; each time I cut a corner, I pay for it with an error. As I practice, I’m becoming faster, more efficient and more accurate. Recently I tweaked an algorithm to make it more efficient; this would not have come to me without skills and understanding.
And this:
“Creativity springs unsolicited from a well prepared mind.”
“Fundamental knowledge is the basis of creativity.”
– John Saxon, co-author of the Saxon Math textbook series
And this:
If proponents of fuzzy programs and constructivism had to use math in the “real world,” and were held accountable for the results, they would have to modify their views. In the “real world,” math is a tool, used to get a job done. What matters are clarity (understandable by others); efficiency (done relatively quickly); and accuracy (the result is correct). Math is a tool – like a hammer or drill. One doesn’t come to consensus on the philosophy of a drill; one learns to use the drill and then one uses it.
You'll want to read the whole thing.
I have reached the conclusion that grade schools should be run like high schools and that there should be dedicated math teachers and language teachers instead of one teacher in charge of all subjects. I think many of the problems could be nipped in the bud by simply providing quality instruction in the early years. I am horrified that some elementary teachers have to be taught how to work with fractions. Before being hired to tutor, I had to pass exams to demonstrate that I knew the material I was assigned to teach. I figured the same MUST apply to the hiring process for teachers. Doesn't it?
ReplyDeleteI think we need to separate math teachers in to two categories ... before algebra, and algebra and up. We also need to accept that most of the people teaching the before probably did not have significant math instruction ... which is sad, but true. The lower level math? Memorization and algorithms work pretty well --although I would argue that a greater understanding of what fractions are, along with a lowered desire to have approximate decimal answers would help. Algebra and after? I speak only for my self, but I became good at higher level math when my teachers started spending a period showing me WHY something worked ... and being correct about it. We would spend the entire period discussing the last night's homework, and the day's new topic... no doing HW in class -- ever. In the district where I most recently worked, my nightmare was having a student, whose teacher I could immediately identify, telling me that if you have an equation like X + 7 = 18, what you do is move the 7 to the other side and change the sign. The algorithm works ... but it also doesn't help for anything even moderately more complex.
ReplyDeleteWhen I was in 6th grade, our class was so large that our small rural school ended up hiring two teachers and splitting us between them. One teacher was so math phobic that the other assumed responsibility for that subject as well as science. The math hating teacher taught English and social studies, and the teachers switched classrooms halfway during the day. It worked very well, but I have not heard of such an arrangement being done elsewhere.
ReplyDeleteWhen I taught 6th grade several years ago, we did just that. I taught 4 classes of science because the other teachers feared it. We had another teacher with an aptitude in math teaching 4 classes of math and likewise for English and Social Studies. The other subjects were assumed by each teacher with their homeroom. It worked well. And after spending 9 years in that elementary school of 1200 students (k-6) I can state with certainty that most of the elementary teachers were there because they wanted to help kids feel good about themselves. I had to leave and move to a parochial school because I was so frustrated by the lack of high expectations and the lack of subject knowledge of my colleagues. They were wonderful, caring people but those students were far behind their parochial school counterparts because their teachers lacked either the skill, knowledge or desire to to teach math and science at the lower grade levels. Schools of education are not doing their job in preparing elementary teachers with subject matter knowledge. But those kids sure felt good about themselves. (Until they found out they were unprepared for high school). It's time to retire!!
ReplyDeleteI think the elephant in the room is that we expect elementary teachers to be good at everything ... and that's just impossible.It works out okay as long as you're just teaching basics ... but I would be awful trying to teach science or history, beyond what I got from the textbooks. At my kid's elementary school, they let the one teacher who actually knew math rotate around ... and she was really good. But hidden in this is also the issue of ... why do we need to press really important, fundamental math so far down the grade levels, and to place that burden on teachers who don't usually have that level of expertise in the subject? You can be a great teacher, but if you've got a multiple subject credential, chances are you weren't a math whiz. And that has an impact on the kids later...without a solid foundation in algebra, every thing else is harder.
ReplyDeleteAnyone who's graduated from college should be able to work with times tables, fractions, and long division. Anyone who cannot just doesn't deserve a college degree. Seriously.
ReplyDelete*Teaching* those--they should learn the basics of *that* in ed school.
Well, agreed. But we live in a world where that just isn't true. With the weight we put on the use of calculators, it's entirely possible to get through college without really understanding any of those ... or caring. And, I was sorely underwhelmed by my ed school ... lots of philosophy, very little practicality. I don't know how the multiple subject (elementary) program works, but for me, it was just expected that I knew stuff. You dropped down your expectations, though --the three areas you mentioned, should be easy. Algebra? as important as it is, it needs to be taught by someone who knows it. And honestly ... I never understood long division until I started teaching division of polynomials. I could do it, I could show it ... but I never knew why.
ReplyDeleteMax - I am not sure that it works out if you are "just teaching basics" because the basics are the fundamentals, and nowhere else is a thorough understanding more important.
ReplyDeleteI remember when our daughter was in the 5th grade, there was a change made in her school's approach to teaching. The school convened a parent meeting to explain the new methods. The emphasis was to shift away from "drill and kill" rote memorization without "substance" and toward teaching "higher order thinking skills" and "critical thinking" (phrases I have come to detest). There was even a chart with a hierarchy of mental processes; analysis and extrapolation topped the chart.
I remember the teacher who was delivering the presentation to us said this:"It doesn't matter if the student can remember that the formula for the area of a circle is pi times the diameter." Of course, I immediately thought that if the teacher herself had absorbed the idea that area is two dimensional, she would have never made that error! Critical thinking skills indeed! This is an example of the sort of basic understanding that gets taken for granted, but that could be so easily addressed by someone who really has an affinity for a subject, even at "low" levels.
It's no less important in teaching English. For example, how many times do you hear people use the word "I" in an inappropriate place because they are afraid of using the word "me"? Arne Duncan himself made a public gaffe involving those words. That sort of error doesn't happen if the person using the words understands their grammatical function in the sentence and knows the difference between a subject and an object. Again, a "basic" idea that would best be taught by someone who has a firm grounding in English.
BTW, I am a fan of diagramming sentences and doing complicated math problems that involve basic skills, because they require a good working knowledge of fundamentals. Once you have the main concepts internalized, multiple step problems can be used to improve "critical thinking". But, I haven't seen a lot of elementary school homework where, once the student has mastered a concept, he has to use it in anything more than a two-step problem. Is it any wonder, then, that students have difficulty transitioning to high school geometry or algebra, where finding a solution often requires more work?
PeggyU -- That presentation example was hilarious, in a sad way. Probably a better thing if students DON'T remember that the area of a circle is pi times diameter. I think elementary school is absolutely time for drill and kill, and rote memorization ... I'm not sure that we need to be doing multi-step algebra problems at that age. I know it looks good on paper but I think it doesn't work out well later. One of my degrees is in English and I share your frustration ... I have my own pet peeves: I feel badly; I impacted the students; If I was to ... not sure about diagramming sentences, though. I learned zero from doing that. All I learned about writing, I got from Warriner's and reading my essay aloud...
ReplyDeleteMax - Certainly, you have to memorize basic arithmetic facts. I do believe that problems which involve only arithmetic and some simple geometry concepts can be challenging and engaging. They can enhance math enjoyment in grade school. The owner of the mathcurmudgeon blog serves up examples frequently.
ReplyDeleteI enjoyed diagramming sentences - the harder the better! I took grammar alongside Latin,and maybe it was actually the Latin exercises which drove home the parts of speech. I am sure, though, that I make grammar errors from time to time. I have a friend who runs a tutoring service;every so often she politely corrects mistakes in our correspondence.