During a course in math teaching methods I took in ed school, I watched a video of a teacher leading his students to do a variety of tasks, ostensibly to teach them about factoring trinomials, such as x2 + 5x + 6. But rather than teaching factoring techniques, as is done in traditionally taught classes, the session was a mélange of algebra tiles (plastic squares and rectangles used to represent algebraic expressions) and a graph of the equation being factored (a parabola).
The teacher “facilitated” the class into making connections between the factored equation and where the graphed parabola crossed the x-axis. The class had not done factoring nor solved quadratic equations before, nor a host of other things that would have been important for understanding the lesson.
After the video, our teacher asked for our reactions to the video. I said that rather than teach students factoring first and having them practice it, they were doing things that generally came after such mastery. “There’s so much going on, that I’m not sure what they’re learning or if they’re learning anything at all,” I said. My teacher’s face went into a frown, and she called on another student.
What I Saw During a Recent Classroom Visit
I recount the above because of an eighth-grade math course I’ve been observing, which aligns with Common Core math standards. The teacher does a good job teaching and I do not hesitate to say she is excellent at what she does. Her sessions are a mixture of letting the students “struggle” with a problem and then providing some explanation through direct instruction and questioning. During one such observation, the class was learning about linear equations, graphing and functional form.
I watched as the teacher explained that an upcoming test would require students to write two sentences describing how to find the slope and y-intercept from a) a graph, b) a table of values, c) an equation, and d) a word problem.
“You’ve learned about slope before, in seventh grade,” she told the class. “You were told, ‘Here’s the procedure, now let’s do the procedure, now you do it alone’: Wash, rinse, repeat—and repeat and repeat and repeat. Lots of practice, practice, and more practice. Now you are being asked to analyze, not just ‘plug and chug.’ You will have to explain what you’re doing, not just perform the procedure. Common Core is about thinking and understanding, not just doing. The goal of this school is to work on improving your writing skills, so you have to be able to explain what you do and why. Don’t just say ‘The slope is 4.’ Tell me how you got the slope, how did you find the intercept. Don’t just tell me ‘because when x is 0, y is 3.’ Tell me why that’s the intercept.”
In finding equations from a table of values, students were instructed how to find the change in x and y values (delta x and delta y), then to “make a fraction” of delta y/delta x, and that was the “rate of change,” or slope. They were made to “struggle” with some problems to eventually see what the y-intercept is, and then to be able to “work with patterns” from a table of values to find the y-intercept. They had to do this using logic, the teacher told them. First, they had to calculate the rate of change. If the zero value was not listed for x, the other values could lead them to it. For example, if x = 2 and y =4, and x = 4 and y = 8, students were to see that the rate of change is 4/2; that is, the “pattern” is that as x increases by 2, y increases by 4. Going the other way, when x decreases by 2, then y decreases by 4. Going backwards from the point (2, 4), the pattern tells us that at x = 0, y = 0.
As was the case with the video I had seen in ed school, these students were being given multiple concepts all at the same time, and expected to make and “explain” the connections between them. Unlike the students in the video, they had learned some of the material previously, and had been working on this unit for about five weeks. Nevertheless, based on the questions the students asked, it was clear they were confused: “How do you explain this in writing?” “How do I find the equation from the table?” “How do I find the y-intercept from a word problem?”
What Do the Standards Say?
For reference purposes, here are Common Core standards for functions in eighth-grade math:
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Nothing about these standards is different from what would be in any conventional algebra 1 course. A look at the algebra books I used in high school (see Figures 1 and 2) illustrate how to obtain an equation from a table of values. They also illustrate the same four ways of expressing a function that the CC standards require.
Figure 1 (Aiken, et al., 1957)
Figure 2 (Aiken, et al., 1960)
Unlike what I saw in the classroom, the traditional approach presents each concept in order—not all at once—and presents exercises for students to master each successive skill. Mathematics is relentlessly hierarchical; one concept serves as the foundation for the next to build upon. In the book I used in high school, the topic started with deriving an equation from a table of values. Then came how to graph a table of values on an x-y Cartesian coordinate plane, how graphs represent relationships that words can also express, the slope of a line and how to determine it from a table of values, the slope intercept form of the linear equation (y = mx + b), what is the y-intercept and how to determine it, and, finally, how to construct a graph from an equation without using a table of values. This was accomplished in about three or four weeks, with relative ease.
Those who view the traditional approach as strictly procedural believe it does not result in a “deep understanding.” They believe students do not learn the “why” of a procedure, just how to do it through “meaningless” drills. Yet the teacher I observed was still giving the students instruction after allowing them some time to “struggle.” She also gave them opportunities to practice. The difference was that her exercises included having to explain (orally and in writing) various connections—i.e., how the table of values related to the equation of the line, how the line related to equation, and so forth. The writing was also instructed: she gave them examples of sentences that she would say aloud but with blanks where key words would go.
Drilling for Understanding
This teacher’s method is consistent with what I’ve described in my previous articles in this series. That is, students are being made to do what amounts to “rote understanding.” Instead of adding 9 + 6 in any manner students want, students must do a number of problems by “making tens.” Instead of adding multi-digit numbers using the standard algorithm, students must do problems using inefficient strategies that purport to show what is happening when doing such operations. And in the case of functions, instead of being taught concepts and skills sequentially, they are required to “show the connections” between concepts–something that would likely happen on its own. They are taught to reproduce explanations that make it appear they possess understanding—and more importantly, to make such demonstrations on the standardized tests that require them to do so. And while “drill and kill” has been held in disdain by math reforms, students are essentially “drilling understanding.”
Some people, like Elizabeth Green, whose article I discussed in the first of this series, may view the teacher I described here as doing reform math poorly. Green might say this teacher was putting traditional methods in the “clothing” of reform math. Such arguments are red herrings. The key questions are whether one method is better when well-taught, which ones can be well-taught by properly prepared and resourced teachers, and which ones provide worthwhile educational outcomes. Once we have established this, we need to work on ensuring that teacher use such a system, and teach it well.
This article is the last in my series on common-sense approaches to the Common Core math standards. I have taken the view that the standards can be met with traditional approaches in math education. My purpose in highlighting alternative interpretations of the standards has been to show that the Common Core standards can be met sensibly. I have purposely taken examples from old textbooks to show that meaning and understanding were always integral to accepted methods of instruction that critics of traditional methods have denigrated.
It is both intriguing and disturbing to see that in the name of Common Core not a few school districts have made it much harder for a student to qualify for a traditional algebra 1 course in eighth grade. Because of this, some students who would have qualified are being forced into courses conducted in the manner I have described here. The take-away message from such policies is: “You are not bright enough to benefit from traditionally taught math; you would just be taking part in rote memorization with no understanding.” I hope such policies are challenged and reversed, and that common sense will once more prevail in math instruction—Common Core standards or not.
Aiken, Daymond J., K.B. Henderson, R. E. Pingry; “Algebra: Its Big Ideas and Basic Skills, Book 1”; McGraw-Hill, New York. 1957.
Aiken, Daymond J., K.B. Henderson, R. E. Pingry; “Algebra: Its Big Ideas and Basic Skills, Book 2”; McGraw-Hill, New York. 1960.
Image by Alexander Farley.