Wednesday, September 18, 2013

Conceptual Understanding and Procedural Fluency

I know I am supposed to share my first few years of PBL. I have it all in my head. I just got distracted with school. Things are starting to be "normal" for me so I am going to try and write weekly. Key word is try :). I want to share something that happened in class this week. It is not PBL but a key aspect of helping all kids learn math deeply which is the heart of PBL.

In my Algebra II classes, we started the systems of equations unit on Monday. I always start a unit with a small pre-assessment. It is never more than four multiple-choice or written response questions and it is basically skills students need to be successful during the unit. See this link if you want to know the pre-assessment questions. Well, the pre-assessment let me know that most kids didn't have the necessary background and I would have to do some reteaching as we learn how to solve a system of equations. 

We started with the simplest way to solve--graphing. We reviewed how to convert an equation into slope-intercept form so that they could graph the equations. The example I used was 2x + 4y = 36 and 10y - 5x = 0. I knew from the pre-test that some students knew how to convert from standard form to slope-intercept form so I just asked for volunteers on how to convert the two equations. This gave me the opportunity to see how students approached converting the equation. For the first equation, some students wanted to divide by four for each term. Although this is a perfectly acceptable first step, I asked for another way just to make sure I didn't throw off students by putting fractions in early. 

In each class, various students helped me follow the typical process of subtracting/adding the term with the x variable, then dividing by the number with the y variable at the end. With each step, I reminded students of the desired result (isolating the variable of y) and why we are doing each step. We then completed the last step of placing the equations in the graphing calculator to determine the solution. I send them off to work time like normal and this is where the class becomes interesting. 

I completely expected to help students with using the graphing calculator. We have only had them a week. However, I was shocked by the number of students who were still lost with converting the equation. At the end of the day, I discovered the root of the problem was students lacked the fundamental understanding of equations. They had practiced such a specific algorithm they didn't know what to do if it came in a different form. For instance, the first problem in the book was 2y - 3x = 7 and 5x = 4y - 12. A few didn't know where to begin while others were doing the typical process. As I helped student after student, everyone wanted to subtract 5x as for he second equation.   

Just like other teachers, I thought about what to do the next day. The plan calls for me to move on to substitution. They have a test on Thursday that includes them solving by graphing and substitution. The students don't have a conceptual or even a solid procedural fluency of equations. 

What would you do? How do you help students who have misinformation while staying on track with learning new information?


earlsamuelson said...

I always incorporate graphing along with the algebraic solutions as well. When introducing systems, I make sure solutions are integer values. In terms of distinguishing between "substitution" and "elimination" methods, I connect those as well. The two procedures are essentially doing the same thing (relating two scenarios with only one variable); all that differs in the "order" in which the manipulation occurs. I start with slope y-intercept form for both and draw comparisons. Systems of non-linear equations are all together similar.

Systems of linear equations can also be solved from a "vector" perspective......linear combinations of vectors. This is a much different perspective but has definite value later on.

Max Ray said...

One thing I think, in general, is that students don't really get that solving equations is all about equivalence. Some of them might even have that lovely misconception that = means "the answer" so if you ask them to solve 3 + 6 = __ + 8 they will write 9 instead of 1.

But I'm not sure that students realize that they are writing lots of equivalent expressions and that some small subset of those expressions help them get closer to a target goal.

I think of solving equations as like solving a maze, rather than following a recipe. There are lots and lots of possible paths, and some are more efficient than others, but as long as you don't cross any walls and you persevere, you'll get there eventually.

I wants someone to invent a Scattegories game for equations in which students get a base equation or expression and come up with as many equivalent forms as they can and count who has the most unique forms.

Then, I want them to go about simplifying and solving equations by writing as many equivalent statements as they can. Like, dozens and dozens! And only then choosing -- which of these are closer to my goal of getting y alone? Have them focus first on maintaining equivalence, and only later on the kinds of steps that lead efficiently to an equivalent equation with y on one side.'s online graphing calculator is great for making sure your expression/equation really is equivalent because you can enter expressions that aren't in y= form and see if their graphs exactly coincide.