Making Connections, One Concept at a Time

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. She said “Yes!” excitedly as if I had understood the holy grail of education. But then I added. “There’s so much going on, that I’m not sure what they’re learning or if they’re learning anything at all.” My teacher’s smile 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 observed, 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. Having said this, I add that it’s entirely possible to do horrendous things extremely well. Her sessions were a mixture of letting the students “struggle” with a problem and then afterward, after determining that students had met her quote of struggle, she would provide some explanation through explicit instruction and questioning; More questioning than instruction.

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 then made to struggle with some problems to eventually see what the y-intercept is. This was followed up with “working with patterns” from a table of values to find the y-intercept. They were to find and describe the pattern, and ultimately the y-intercept using logic. First, they had to calculate the rate of change. If the zero value was not listed for x, they had to use other values to lead them to it. For example, if x = 2 and y =5 and x = 4 and y = 9, students were to see the pattern of 4/2, that as x increases by 2, y increases by 4 and thus conclude that the rate of change is 4/2. Going the other way, when x decreases by 2, then y decreases by 4. Going backwards from the point (2, 5), the pattern tells us that at x = 0, y = 5 - 4 = 1.

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. They had been working on this unit for about three 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:

CCSS.MATH.CONTENT.8.F.B.4
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.

CCSS.MATH.CONTENT.8.F.B.5
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.