Essay on MAA’s MathValues.org

It’s October, and I have no idea what I’m doing.

So begins my essay “Let’s Remember the Year Everyone Wants to Forget”, which appears on the Mathematical Association of America‘s website MathValues.org. It’s a reflection on our shared year of pandemic teaching and learning, and it offers something to think about as we look to return to normal.

The essay draws heavily from my weekly reflections on remote and hybrid learning, and I’m happy to have to opportunity to share it with the MAA. You can read the piece here.

Related Posts

Regents Recap, January 2020: Isn’t the Quadratic Formula Completing the Square?

Completing the square is one of those mathematical techniques that should be taught but probably shouldn’t be assessed on state exams.

First, when you insist that a student use a specific technique to solve a problem, you penalize flexible and creative thinking. Second, completing the square probably isn’t important enough a problem solving technique to warrant its yearly appearance on the New York State Algebra 1 exam.

Of course, the one situation in which completing the square is absolutely indispensable is in deriving the quadratic formula. Which makes this sample student response from the exam’s official scoring materials a bit puzzling.

The student lost a point because, instead of completing the square, they used the quadratic formula to solve this equation. But the whole point of the quadratic formula is that it completes the square for every trinomial. The quadratic formula is completing the square.

As I clarified in comments on Twitter, I find this more amusing than objectionable. But these little windows into the testing process often tell us more about what is valued and understood than test scores themselves.

Related Posts

Regents Recap, January 2020: What is an Irrational Number?

This type of problem frequently appears on New York State Algebra 1 Regents exam.

There’s really no way for an Algebra 1 student to properly “explain” their answer to this question. Proving that a number is irrational is a concept from elementary number theory and is not part of the Algebra 1 course. What the test makers expect is for the student to simply state that a rational number times an irrational number is irrational. Not only is this not an explanation, but such a question reinforces the idea, for both students and teachers, that mathematics is a collection of facts to be memorized and regurgitated.

I’ve written about this issue before, but I didn’t think this kind of problem could get much worse. I was wrong. Here’s an example of a full credit response from the official scoring materials for the 2020 Algebra 1 Regents exam.

Image

In this exemplar full credit response, the student erroneously represents 3\sqrt3 as a number with a terminating decimal expansion, i.e. a rational number. Then the student incorrectly claims that the number 5.19615243 can not be expressed as a fraction and thus must be irrational.

The student has demonstrated some understanding of the situation, but doesn’t grasp the fundamental issue of what an irrational number is. This response shouldn’t get full credit. More importantly, the official scoring guidelines should not communicate erroneous mathematics to those who use them. How many teachers will walk away thinking this is valid? And then teach their students that to show a number is irrational, all you have to do is plug it into your calculator and see if it has eight digits past the decimal point?

This is kind of work that makes this other exemplar full credit response seem not so bad by comparison.

Image

A lively Twitter thread on this problem can be found here.

Related Posts

Ends and Beginnings

In a past life I worked in office buildings. One of the reasons I quit that life was the inescapable perpetuity of it all. There was no end, no beginning. Just work.

The well-defined beginnings and ends of the school year are important to me. Anticipation creates excitement, closure creates opportunities to reflect. Working in perpetuity made me realize how important this cycle was in my life.

I always have my students write end-of-year reflections, in part to bring their attention to this cycle, but also because there are questions I don’t know the answers to. And I want to know. Although I could have guessed many of those answers.

What worked for you? Breakout rooms, because I got to learn from others and build relationships with my classmates.

What didn’t work for you? Breakout rooms, because they were awkward and no one talked.

What did you learn about yourself? I learned that I need social interaction more than I thought.

How do feel about next year? I’m nervous that I won’t recognize any of my classmates because I don’t know how tall anyone is. (Ok, I never would have guessed that.)

What would have improved your experience this year? In-person classes. (No surprise there.)

My own end-of-year reflection begins with “I made it”. I think this is where many teachers would start their self-evaluations. But as I look deeper, I realize how far I’ve come, how much I’ve learned, and how many new skills I’ve developed. I now feel comfortable teaching a live remote class. I have a sense of how to structure asynchronous learning, and knowledge of tools that can make it meaningful. I can build relationships with students in a virtual environment. I can run a workshop for teachers in zoom. I can facilitate teacher teams remotely.

I couldn’t say any of these things twelve months ago. It feels good to be able to say them now at the end, and at the start of a new beginning.

Related Posts

Follow

Get every new post delivered to your Inbox

Join other followers: