Vignette: it's not about getting the right answer

At the beginning of the first class I visited at my placement school, Ms C (a pseudonym for my placement teacher's name) gave the students a few do now problems on the smart board, and a Keystone practice worksheet. After letting the students work on these problems individually for a while, she got all of the students’ attention and they begin going over the problems together. She asked students to volunteer and explain their answers, and told them that she will give them a point on their next exam if they answer correctly today.

On the first problem, a girl eagerly volunteered to explain her answer. The problem asked students to simply the following monomial: 5^2/5. The girl explained that she found the solution by evaluating 5^2,then dividing that result by 5. She arrived at the correct answer, but Ms C told her that she didn’t solve the problem the right way (but still gave her a point).  She told the class that while the method the first student used will work on problems that use smaller numbers and smaller exponents, it wouldn’t work if the problem had asked them to simplify (5^23)/(5^20) or a similar problem.

Ms C explained that she wanted the students to learn how to solve the problems in a way that they would be able to apply to other similar problems, and problems that have variables instead of specific numbers. She asked if another student solved the problem in a different way, and one student raised his hand and explained that since 5 = 5^1, he used the laws of exponents in fractions, and found that the answer was 5^(2-1)=5^1=5. Ms C said that this was what she was looking for, and gave him a point also. She then went over the exponent rules that they had learned, and how they could apply those rules to the types of problems that are on their worksheets and will be on the Keystone exam.

I see this incident as an example of my placement teacher trying to break away from the rote-memorization, teach-directly-to-the-test curriculum that some urban high schools have adopted as a last effort to increase the number of their students who pass state exams such as the PSSA or the Keystone. Although I haven’t seen Ms C teach very much (there have been substitutes a few times that I visited), this small part of class showed me that she wants her students to be able to think about how to solve a problem, and see patterns in how certain types of problems are solved. This will make it easier for the students to remember what they learned after they leave this class or after they take the Keystone, and it will help them to solve problems that may be different from the examples they used in class.

We’ve talked a lot in class about how problematic standardized testing, and the pressure it puts on teachers, can be, and I see this example of conceptual teaching as a small step in minimizing the negative effect that these exams might have in the classroom. I wonder if it would be possible for my teacher to eventually not use worksheets made specifically for Keystone practice, or to maybe adopt a curriculum which focuses even more on making sure students understand WHY certain rules are in place in math. I don’t know if my placement school has adopted the common core standards, but it seems like Ms C was using teaching concepts in a way similar to what the common core standards outlines.

This theme of conceptual learning has come up other times in my placement. A worksheet the students had to complete the last time that I visited asked them to solved for a variable in a polynomial equation, but the students were instructed to factor the polynomial first. They have been learning why some methods of solving equations are easier to apply to specific equations, and the work they’ve done has focused on the intermediate steps of problem solving, and not just the final answer.