Toward an Education Science (I)
Time seems to move faster as we get older. But every time an experience reminds us of this, we're surprised (and probably a little disappointed) to find that it is still true.
For me, last night was a case in point: I sat down to write something about this post from Michael, imagining at the time that I was coming back to it after a month or two-month hiatus. Yet, with an admixture of surprise and disappointment, I found that nearly six months had passed since I had read it.
Of course, age isn't the only factor here. Another reason Michael's post seems much younger than six months to me is that one of the points of disagreement in the discussion was that of the value of student errors--a "theme," if you will, that can be found in one way or another almost daily in discussions about education.
Still, even though student error became a sticking point for myself and others, the main idea of the post was not the value of student error, but the value of student creativity in the mathematics classroom.
In order to see why this approach is both novel and interesting, let's briefly review the traditional, or textbook, way students are taught to find the slope of a line given two points.
Now let's look at horizontal distance. Point C is located at 6 on the horizontal axis, or x-axis, and point B is at 4 on the horizontal axis. So, we can describe the horizontal distance between points C and B as 6 &minus 4, or 2. There is a horizontal distance of 2 between points B and C.
The slope of line f can be described using a ratio, as shown below. The first term (top number) of the ratio is the vertical distance between any two points on the line, and the second term (bottom number) of the ratio is the horizontal distance between those points. Between points B and C, which are points on the line, there is a vertical distance of 1 and a horizontal distance of 2. So, line f has a slope of 1/2.
As I mentioned, though, the slope of a line can be found using any two points on the line. If we use points A and C instead of B and C, we get the same ratio. Look at the graph and table below.
The vertical distance between points A and C is 2 (3 &minus 1), and the horizontal distance between the points is 4 (6 &minus 2). Using points A and C, then, we would write the slope ratio as 2/4 (vertical distance over horizontal distance). And, of course, this ratio simplifies to 1/2.
To write the formula for the slope of a line given two points, we actually write two expressions--one for the vertical distance between the two points and one for the horizontal distance between them. Then we combine these two expressions together into one ratio.
To find the vertical distance between the points in our examples above, we subtracted the vertical, or y-axis, location of the first point from the vertical, or y-axis, location of the second point. So, we can write y2 &minus y1 to describe the vertical distance between points.
To find the horizontal distance between the points in our examples, we subtracted the horizontal, or x-axis, location of the first point from the horizontal, or x-axis, location of the second point. So we can write x2 &minus x1 to describe the horizontal distance between points.
When we combine these expressions together into one ratio, recalling that vertical distance (y2 &minus y1) is the first term and horizontal distance (x2 &minus x1) is the second term, we get the following:
Now, it should be noted that the order of the terms in each expression is not important, so long as they are consistent with each other. That is, the vertical distance could be described using the expression y1 &minus y2 instead of y2 &minus y1. So long as the second term of the ratio uses the same order, the ratio will accurately describe the slope of a line.
Typically, we teach students to find the slope of a line given two points in two different ways--using a graph and using ordered pairs. The difference between these two approaches has to do with how the locations of the points are presented, either as points on a line or as ordered pairs of numbers. In our example above, we found the slope of a line using points graphed on that line, but typically students would also be expected to find the slope of the line when given the ordered pairs describing the locations of any two of the points on the line. The ordered pairs for the points in the example above are as follows: A (2, 1), B (4, 2), C (6, 3).
First, students are taught (correctly) that, in ordered pairs--those pairs of numbers in the form (x, y) that can locate points on a coordinate grid--the x-coordinate comes first and the y-coordinate second. For slope it is, in some way, the opposite. The y-coordinates are listed on top (first), and the x-coordinates are listed on bottom (second). Second, the slope formula does not keep the ordered pairs intact, though students work with them as such. The formula strips out the y's and places them on top while it does the same for the x's on the bottom. The ordered pairs no longer exist as independent entities in the formula.
Another issue worth considering--an issue that is independent of prior teaching--is that simple tracking errors might occur when students (or anyone) attempt to substitute information written in one form (e.g., horizontally as ordered pairs) into a very different form (e.g., vertically as a ratio).
As you can see, in "Lisa's Method," for each horizontal line, the x's always come first and the y's second, and the ordered pairs remain intact: the top line shows the ordered pair of the first point, and the second line shows the ordered pair of the second point.
The last line in "Lisa's Method" is, in fact, the complete slope formula shown above on the right. Each triangle symbol can be read as "change in," and, in this case, "change in" simply refers to the distance between the points. So, the last line in "Lisa's Method" shows the distance between the y-values divided by the distance between the x-values. The fraction bar in the traditional formula is also a division symbol, so the meaning is (pretty much) the same.
Let's take that up next time.
For me, last night was a case in point: I sat down to write something about this post from Michael, imagining at the time that I was coming back to it after a month or two-month hiatus. Yet, with an admixture of surprise and disappointment, I found that nearly six months had passed since I had read it.
Of course, age isn't the only factor here. Another reason Michael's post seems much younger than six months to me is that one of the points of disagreement in the discussion was that of the value of student errors--a "theme," if you will, that can be found in one way or another almost daily in discussions about education.
Still, even though student error became a sticking point for myself and others, the main idea of the post was not the value of student error, but the value of student creativity in the mathematics classroom.
Recently, I had one of those much-desired opportunities to see a student spontaneously come up with what was, to me at least, an original approach to something that is easy for and familiar to many, but distressingly hard for a significant number of students: calculating the slope of a straight line given two points.
In order to see why this approach is both novel and interesting, let's briefly review the traditional, or textbook, way students are taught to find the slope of a line given two points.
The Basics
Look at the figure below. Line f is a straight line that passes through points A, B, and C. Point C is located at 3 on the vertical axis, or y-axis. Point B is at 2 on the vertical axis. So, we can describe the vertical distance between points C and B as 3 &minus 2, or 1. There is a vertical distance of 1 between points B and C.Now let's look at horizontal distance. Point C is located at 6 on the horizontal axis, or x-axis, and point B is at 4 on the horizontal axis. So, we can describe the horizontal distance between points C and B as 6 &minus 4, or 2. There is a horizontal distance of 2 between points B and C.
The slope of line f can be described using a ratio, as shown below. The first term (top number) of the ratio is the vertical distance between any two points on the line, and the second term (bottom number) of the ratio is the horizontal distance between those points. Between points B and C, which are points on the line, there is a vertical distance of 1 and a horizontal distance of 2. So, line f has a slope of 1/2.
As I mentioned, though, the slope of a line can be found using any two points on the line. If we use points A and C instead of B and C, we get the same ratio. Look at the graph and table below.
The vertical distance between points A and C is 2 (3 &minus 1), and the horizontal distance between the points is 4 (6 &minus 2). Using points A and C, then, we would write the slope ratio as 2/4 (vertical distance over horizontal distance). And, of course, this ratio simplifies to 1/2.
To write the formula for the slope of a line given two points, we actually write two expressions--one for the vertical distance between the two points and one for the horizontal distance between them. Then we combine these two expressions together into one ratio.
To find the vertical distance between the points in our examples above, we subtracted the vertical, or y-axis, location of the first point from the vertical, or y-axis, location of the second point. So, we can write y2 &minus y1 to describe the vertical distance between points.
To find the horizontal distance between the points in our examples, we subtracted the horizontal, or x-axis, location of the first point from the horizontal, or x-axis, location of the second point. So we can write x2 &minus x1 to describe the horizontal distance between points.
When we combine these expressions together into one ratio, recalling that vertical distance (y2 &minus y1) is the first term and horizontal distance (x2 &minus x1) is the second term, we get the following:
Now, it should be noted that the order of the terms in each expression is not important, so long as they are consistent with each other. That is, the vertical distance could be described using the expression y1 &minus y2 instead of y2 &minus y1. So long as the second term of the ratio uses the same order, the ratio will accurately describe the slope of a line.
Typically, we teach students to find the slope of a line given two points in two different ways--using a graph and using ordered pairs. The difference between these two approaches has to do with how the locations of the points are presented, either as points on a line or as ordered pairs of numbers. In our example above, we found the slope of a line using points graphed on that line, but typically students would also be expected to find the slope of the line when given the ordered pairs describing the locations of any two of the points on the line. The ordered pairs for the points in the example above are as follows: A (2, 1), B (4, 2), C (6, 3).
Potential Problems
There are at least a few issues that immediately present themselves when considering the typical slope formula and the teaching that has (again, typically) occurred prior to the introduction of the formula.First, students are taught (correctly) that, in ordered pairs--those pairs of numbers in the form (x, y) that can locate points on a coordinate grid--the x-coordinate comes first and the y-coordinate second. For slope it is, in some way, the opposite. The y-coordinates are listed on top (first), and the x-coordinates are listed on bottom (second). Second, the slope formula does not keep the ordered pairs intact, though students work with them as such. The formula strips out the y's and places them on top while it does the same for the x's on the bottom. The ordered pairs no longer exist as independent entities in the formula.
Another issue worth considering--an issue that is independent of prior teaching--is that simple tracking errors might occur when students (or anyone) attempt to substitute information written in one form (e.g., horizontally as ordered pairs) into a very different form (e.g., vertically as a ratio).
Lisa's Method
The approach that Michael dubbed "Lisa's Method" eliminates at least the first two problems listed above. (Though, as I will point out in a future post, this is not a satisfactory criterion on which we can base an evaluation of Lisa's method.) Let's briefly compare the two "approaches":As you can see, in "Lisa's Method," for each horizontal line, the x's always come first and the y's second, and the ordered pairs remain intact: the top line shows the ordered pair of the first point, and the second line shows the ordered pair of the second point.
The last line in "Lisa's Method" is, in fact, the complete slope formula shown above on the right. Each triangle symbol can be read as "change in," and, in this case, "change in" simply refers to the distance between the points. So, the last line in "Lisa's Method" shows the distance between the y-values divided by the distance between the x-values. The fraction bar in the traditional formula is also a division symbol, so the meaning is (pretty much) the same.
Scientific Thinking
Given that background, there are an extraordinary number of questions, arguments, hypotheses, etc., that could be generated, all starting with the simple comparison between two approaches used to find the slope of a line given two points. Each of these could, in turn, lead to other questions, other hypotheses, and other areas of inquiry. For example, one might question the conceptual equivalence between the long division symbol used in Lisa's method and the fraction bar used in the formula. Do they really mean the same thing?Let's take that up next time.
Labels: education, mathematics, textbooks