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I remember being very discouraged (in the old traditional math days, no less) trying to understand mixture problems because the book we used approached it using tables and grids. When the problem changed a little bit, I couldn't figure out which numbers went into what boxes. I finally learned to approach the problems using governing equations and defining variables.
That understanding didn't come from solving one or two problems. I had to work at it. There were so many times when I thought I understood what I was doing only to feel completely lost when I tackled the homework set. That's when the real lightbulb goes on. Look at any proper math text book and you will see homework sets that give you all sorts of problem variations of the material in the section.
I also want to make a case for speed in helping understanding too. As you move along to more complex math, you need this speed or else you will be completely bogged down. In high school, I got really good at "seeing" right triangles in word problems, even if the triangles weren't explicitly drawn. I was very fast at finding any side or angle given "enough" information. I could state that a length was something like d*cos(theta) just by looking at it. I didn't have to draw a picture and stew over which leg is for sine and which leg is for cosine.
The mechanical monkey paradigm leads to all sorts of wrong conclusions. It also conveniently fits in with their predisposition to equate mastery with rote learning and drill and kill. When they talk of balance, they really don't mean it. They still think it's just for convenience rather than understanding.
This position might seem reasonable when it comes to the basic algorithms of arithmetic, but it falls completely apart as you head into algebra.
Reading this post makes me want to go do, right this minute, two things that cannot be done at the same time:
- fire up ALEKS and finish the geometry course I was taking before my mom fell last summer
- finally write my post on just exactly how much money Response to Intervention (pdf file) is going to cost us once RTI gets going in public schools with a) lousy curricula and b) no focus whatsoever on deliberate practice (pdf file) & mastery
2 comments:
One more comment.
There is always one right answer. There may be many feasible answers, but only one right one. Math gives you the tools to figure this out. Math doesn't select right or wrong for you.
A classic "no one right answer" problem they is something like the Christmas present problem where you can spend up to $2000 for 10 people. The feasible range is to spend no more than $2000, but what is the "right" answer? If you don't know what it is, figure it out. Define the equations. Decide on what percentage each person gets, plus a percentage for charity. (It better add up to 1.) Perhaps the right answer is to save a lot of time and just give the money to charity. Time can be a variable. Perhaps the right answer is to not start with a fixed budget.
To think that math does not account for no one right answer is the height of ignorance. It's mathematcal child abuse.
The Precision Teaching followers and researchers have the best information on the importance of fluency and how to measure it.
From the Binder, Haughton, Bateman paper at fluency.org:
If you carefully observe children in the learning process, it is easy to understand why behavioral fluency is an essential success factor in learning and performance of any kind. Both informal experience and scientific research (e.g., Binder, 1996; Wolf, 2001) suggest that fluency contributes directly to three types of critical learning outcomes:
- Retention and maintenance: the ability to perform a skill or recall knowledge long after formal learning programs have ended, without re-teaching in school year after year
- Endurance: the ability to maintain performance levels and attention to task for extended time periods while resisting distraction, and
- Application: the ability to combine and apply what is learned to perform more complex skills, creatively, and in new situations.
These are important outcomes that education is supposed to accomplish, but which are sadly lacking in the long-term results of many educational programs. Parents usually see the lack of these outcomes as symptoms, or problems that arise at homework time and when children try to apply what they’ve learned in school to life situations. Even in relatively successful students, who do not falter in obvious ways, a lack of fluency in essential skills and knowledge can seriously limit their ability to achieve the full learning potential of which they are capable.
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