Bad Idea: The coordinate plane is where we graph (points, lines, parabolas, etc.)
Welcome to math class! Here is the subject of our discussion today:
Now, it seems clear to anyone that upon first seeing this diagram, a good deal of explanation ought to accompany it. But, in the world of standardized assessment, and too-short a school year, nothing about how this object works is ever explained to students in an open, honest way. Instead, they are presented with how to "count boxes", as a way for finding locations. After some experimentation and practice, and dealing with enough numbered and labeled planes, students are supposed to just assimilate the notion that points in the plane and ordered pairs are "the same thing" (even though this is never explicitly expressed). They are still disconnected from what they are actually doing, and are thus usually bored by the monotony, to say nothing of the giant list of new vocabulary terms thrown at them pertaining to the plane as well.
Additionally, and this is perhaps a secondary issue, but one of significance nevertheless, students almost never see points graphed with fractional coordinates. As far as they're concerned, the only points that are "allowed" and are "actually points" are ones with integer coefficients; even though nothing is said on this topic.
Since time is short, students very quickly have to move on to how to graph lines. They are presented with the slope-intercept form, and in a good enough curriculum design, spend a unit on slope. Determining the slope of a line, or graphing a line with a given slope, will be something many math students will proceed to adulthood without ever actually understanding even remotely.
After slope comes dealing with linear equations and y-intercepts. Students have no clarity at all in why they never hear of x-intercepts, but that doesn't seem to matter to their teacher, so they deal with it, as they slip further away from the content.
Eventually, the end-all solution to the problems students have with the coordinate plane and graphing are "solved" once they get to the later years of high school, at which point they are given graphing calculators, taught how to type things in, and silently learn to remain disconnected from this bit of context forever.
The source for this disconnect, of course, comes from the context-free environment that the coordinate plane is to most students. Too much time and energy is wasted on what to do on the plane. Students are expected to be able to solve certain classes of problems on the plane (e.g. intersecting lines, graphing lines and finding intercepts, slopes, etc). Thus, they are taught only these procedures, and never taught what the plane really is.
Now, it seems clear to anyone that upon first seeing this diagram, a good deal of explanation ought to accompany it. But, in the world of standardized assessment, and too-short a school year, nothing about how this object works is ever explained to students in an open, honest way. Instead, they are presented with how to "count boxes", as a way for finding locations. After some experimentation and practice, and dealing with enough numbered and labeled planes, students are supposed to just assimilate the notion that points in the plane and ordered pairs are "the same thing" (even though this is never explicitly expressed). They are still disconnected from what they are actually doing, and are thus usually bored by the monotony, to say nothing of the giant list of new vocabulary terms thrown at them pertaining to the plane as well.
Additionally, and this is perhaps a secondary issue, but one of significance nevertheless, students almost never see points graphed with fractional coordinates. As far as they're concerned, the only points that are "allowed" and are "actually points" are ones with integer coefficients; even though nothing is said on this topic.
Since time is short, students very quickly have to move on to how to graph lines. They are presented with the slope-intercept form, and in a good enough curriculum design, spend a unit on slope. Determining the slope of a line, or graphing a line with a given slope, will be something many math students will proceed to adulthood without ever actually understanding even remotely.
After slope comes dealing with linear equations and y-intercepts. Students have no clarity at all in why they never hear of x-intercepts, but that doesn't seem to matter to their teacher, so they deal with it, as they slip further away from the content.
Eventually, the end-all solution to the problems students have with the coordinate plane and graphing are "solved" once they get to the later years of high school, at which point they are given graphing calculators, taught how to type things in, and silently learn to remain disconnected from this bit of context forever.
The source for this disconnect, of course, comes from the context-free environment that the coordinate plane is to most students. Too much time and energy is wasted on what to do on the plane. Students are expected to be able to solve certain classes of problems on the plane (e.g. intersecting lines, graphing lines and finding intercepts, slopes, etc). Thus, they are taught only these procedures, and never taught what the plane really is.
Good Idea: The coordinate plane is an extension of what students have seen before (the number line)
For those who have seen my earlier entry on adding and subtracting integers, now is when the proper approach to adding and subtracting comes into play in a very big way:
If students understand the number line, then the coordinate plane already has a legitimate, mathematical context!
This is why I don't understand modern mathematics educational practices. We've gone to such great lengths to make the tasks easier, and have tried to give students quick, easy, procedure-driven ways to solve problems, and the only tangible end-result is that future developments, (like the coordinate plane), are left incomprehensible.
Students should take a good look at the coordinate plane, perhaps one without grids, but with just the axes. They should be asked what they notice, and what perhaps looks familiar. In a pre-algebra class where the number line methods were taught, there should even be number lines posted on the walls. It is easy for students to see that each axis looks just like a number line. There's no trick here, its simply because they are number lines. The only thing about the coordinate plane that makes things different, is that now its 2-dimensional, and thus there are 2 number lines. Any point, therefore, is represented by 2 numbers; once which tells us how far on the horizontal number line we are (the abscissa), and the other which tells us how far on the vertical number line we are (the ordinate). Suddenly, these new words, among others, have a relate-able, tangible context for students.
Additionally, any time spent dealing with spaces between numbers on the number line (as a way of dealing with rational numbers other than integers) also makes it clear that there is such a thing as points on the plane outside of the grid intersections. Students should realize that the plane is a collection of points, infinitely many, in as small a space as they wish. While there is some very sophisticated mathematical content concerning some of these infinite relationships with respect to space, it is not necessary for students at this level. But they should still be able to appreciate "points between points".
Once students move on to discussions of slope, lines, and their equations, the number line approach comes in to save the day yet again. In my experience, students have never been able to understand equations that are not in what they call "standard form" (which is really slope-intercept form). In my opinion, when dealing with equations of lines, students can and should think of the equations themselves as rules (which is a great way to think of functions in general, even more context for their mathematical futures). The rule tells you what relationship points should have to satisfy it. For example, the equation:
y = 3x + 1
simply says that it concerns points where the y coordinate is three times the x coordinate, plus 1. It is readily easy to check points by their coordinates against this rule. Once some points are found, the line can easily be graphed.
Similar graphing exercises can be translated into problems of this form with ease. And, as a bonus, some previously difficult problems become far more simple. For example, how do students handle equations of this type?:
y = -1
x = 0
y = 5/3
Often these problems simply baffle students who don't have a context-rich understanding of the coordinate plane. But if lines and other functional equations are thought of as rules, as criteria on which points are chosen or not chosen, then the above lines are trivialities, as they should be. Horizontal and vertical lines, denoted as y = or x = a constant, should be the most trivial kind of lines to graph. The line y = -1 is clearly all points where y is -1. It couldn't be simpler than that. But the way we teach the plane, (or don't teach it, if you like) actually cheats students out of math's easier answers.
If students have a strong f
I agree with what you have said.
ReplyDeleteDo you have ideas on how we can implement the teaching of this in the ways you have described above?