We are in the midst of paradox in math education. As more states strive to improve math curricula and raise standardized test scores, more students show up to college unprepared for college-level math. The failure of pre-college math education has profound implications for the future of physics programs in the United States. A recent article in my local paper, the Baltimore Sun: “A Failing Grade for Maryland Math,” highlighted this problem that I believe is not unique to Maryland. It prompted me to reflect on the causes.
Not a bad essay, but I disagree with his tying the problem to "standards". There's nothing inherent in "standards" that makes them bad; in fact, I'll bet there were plenty of standards in math when you and I (and he) were in school! He's correct when he talks about confusing difficulty with rigor, and about inappropriate content (middle school kids' working with matrices? Please!). His discussion of mistaking process for understanding is also very elucidating, and also explains why I'm against using graphing calculators in K-12 education.
Again, he's mostly right, but we'll disagree on standards. The standards may not be good enough, or may be too stringent, or some people may try to teach them in superficial way just to get students to pass a standardized test, but those are not faults of having standards.
12 comments:
So all graphing will be done by hand? What other uses of graphing calculators do you not like? Standards are written so the student can do a task, not understand why, and there's too many of them. On a somewhat related note, how do you feel about rationalizing denominators?
Yes, all graphing is done by hand.
I understand your point about understanding vs. tasks, but especially when you get to "college prep math" (which is Algebra 2 and above, IMHO) I teach for more than just doing. Understanding "why" something works makes "doing" it even easier because then the processes make sense.
Rationalizing denominators was useful when things had to be looked up on tables, but now the main purpose it serves is to ensure the students can perform algebraic manipulation to get a desired result.
I'm going to slightly disagree with you on graphing calculators. I think students should know how to sketch a graph, know how to find key points (intercepts, vertices, etc.) But making them graph anything more than a simple quadratic is like making them use a table and interpolating sines or using a slide rule to approximate the square root of a number. I do think that there should be no calculators (and no problems written that need calculators) until high school. I must say though, I am impressed with my 7 year old's school. He is doing mental math that would baffle some of my high school students.
I think part of the reason students have so much trouble with logarithms is because they *don't* have to interpolate on tables anymore. The interpolation process and using tables combined to allow students in my day to grok the concept much more readily than do today's students who merely push a button on a magic box to get a log. And having read an Asimov book to teach myself how to use a slide rule, I wouldn't be against some use of those--but no, I'm not going to advocate for that!
Granted, I do not teach a core class, but it seems to me that every time a state supposedly makes exams more rigorous, the classes themselves become softer. Instead of encouraging learning, the harder tests seem to set up a situation where in order to get a good grade the teachers must spoonfeed information to the students. This in turn makes the students less willing to risk failure through asking questions or deviating from the rigid learning models. What happens in the end is you have a few students who succeed in truth, many who only succeed on paper and many more who end up hating learning on all accounts. It's a forced, systemic stagnation of learning as a process because it tries to ensure success by removing failure. The penalties for failure are so far reaching that cheating is often committed in desperation to give the appearance of success with students who could not care less.
It's weird because I have totally changed my way of teaching to be more standards based. Everyday in my Algebra 1 class, we do a graphing problem, system of equations, and factoring/solving a quadratic. When we go over them, I talk about how everything connects, like the roots of the quadratic are the solutions to the equations, or how the solution to a system is a point where the two lines intersect.
By the end of the year, most of the students get the problems right, but I wonder how many remember how everything is connected. I used to think most remembered, but now I am not too sure. I hope some can remember the what or why's and not just the how's.
BTW, I don't allow any calculators in my classes second semester except for one chapter in Algebra 2. Geometry and Algebra 1 go without...why? Because they aren't allowed on the CST's. I have changed my tests to reflect how the CST's answers are as well.
Isn't teaching to the test great?
And yes I do believe some sort of check should be in place, but not so weighted. Our school is at an 862 and instead of being happy we are being pushed to 900...good times.
Two finals left to give today, both Algebra 2. *No* calculators. Honestly, what do you need a calculator for in Algebra 2? There was some factoring, some graphing (lines, parabolas, cube roots, square roots, absolute values), some polynomial division (both long and synthetic), and the like.
First time I haven't allowed calculators on an Algebra 2 final and the average score's no different than before.
Darren: I have been reading your exchange with Skip with interest. I'm old enough to have had to use log tables and trig tables too, when I was in high school. I would agree with you that I think logs made more sense to our generation because of it. But Skip is right too that you can get bogged down in busy work.
A guy named Dan Umbarger came up with what I think is a good supplement for any students who are interested in a more detailed explanation of logs.
Don't for a minute think I support the concept of busy work.
I'll be sure to check out that link, especially since I'll be teaching logs this next semester.
Darren - I didn't think that you did support mindless projects! I was just referring to what some people call "drill and kill", although in my view in earlier grades I don't think drills are a bad thing.
Perhaps you mean "drill and skill" :)
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