# Performance Standards in Mathematics (brief)

In mathematics, we have traditionally taught skills, and then taught applications or “word problems” that go along with those specific skills and use those skills in specific ways. For example, after learning subtraction, pupils read and answer “take away” problems. Later, after learning subtraction with decimals, students work on change problems. After learning percents, students will answer discount questions, or calculate tax or tips. At the level I teach, algebra, after learning to factor, students might solve area problems. After solving equations with fractions, my students apply the skill to “work” or “mixture” problems. Even knowing the skill, these problems are hard.

But performance standards encourage something else. They give the student an unfamiliar situation, and ask the student to identify and apply the correct mathematical skill. Now, at the “Integrated Algebra” level the range of mathematics is not that great, so this is not impossible. But it is far more difficult than the standards writers understand. It is what colleges do with engineers (come to us with a large mathematics skill set, and we will teach you how to choose the appropriate equations). Using performance standards means the mathematics assessments are littered with science, technology, and artificial and contrived context.

New York State introduced performance standards in mathematics with Math A and B ten years ago. They reduced their role with the switch to Integrated Algebra, Geometry and Algebra 2/Trigonometry, but performance standards must be eliminated.

You indiciated that one of the problems with the exam is that it didn’t stretch strong kids enough. It seems to me that “performance standards” is *exactly* what you need, in some questions, to do this. Of course you also need more basically-stated questions for the weaker kids too.

A couple of comment…

First, at the risk of lapsing into “back in my day” (the early 70’s) rose-colored glasses, I remember one of the strengths of the Regents exams being that they covered the range of abilities pretty well. They were also wonderfully predictable — you got the Baron’s guides, and worked your way through old exams, and you were prepared. In essence, the collection of old exams set the content standards for the course.

Second, I have mixed feeling about your dismissal of performance standards, for two reasons. One, mathematical skills aren’t very useful unless you have some sense of what they can be used for, and I think students do need to get some practice in figuring out how to dissect a problem and turn it into equations. The challenge is to find problems where they can do this without needing other content knowledge that isn’t part of the course. Two, careful thought about what students will use their math skills for will help shape a better curriculum. I spent a summer teaching summer school Algebra II, and focusing primarily on skills so that my students could at least pass a “local” exam, and what was frustrating was that the skills seemed pretty meaningless without any context (I drew the line at teaching them to interpolate log tables). I think the kids would have been better served by a course than didn’t try to cover quite so many skills, but took the time to provide the context for them.

Dr. Rick –

performance standards would stretch strong students – perhaps appropriate for an algebra exam – but harder algebra problems and canning the 50% stats and probability would be better.

But performance standards tend to crush weak or even average students – inappropriate for a high school exit exam.

“The challenge is to find problems where they can do this without needing other content knowledge that isn’t part of the course.”

I think traditional word problems do this, without being any more artificial than the A and B style questions.

Within that, I do agree that more depth and fewer topics would make sense.

Why not just change the way math has normally been taught. Instead of just computing what the right answer is, have the student come up with a series of questions that can fit a give number. Or have that student explain why they feel an answer is correct and what lead them to that belief?

Unless the teachers start asking and creating deeper level questions, the students will fully understand the depth of what the question is asking.

Rachel,

I think most mathematical skills take time to ‘set.’ I would expect that skills taught a few years back are solid when kids reach me, and that the previous year’s needs refreshing and practice.

The application comes. But this is not social studies, where it comes the same day. Or where almost everything is application and it is the rarer occasion where the class steps back and notes more abstract phenomenon at play behind the details being studied.

Ultimate – I ask questions backwards from time to time. I ask students to explain answers. I like the ideas you mention, and use them. But I use them

as well as, notinstead ofthe old fashioned regular stuff. They complement each other.“Unless the teachers start asking and creating deeper level questions, the students will fully understand the depth of what the question is asking.”

priceless.

Vlorbik,

that comment is from last summer. I entirely missed it. Priceless indeed.