Teaching

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.

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.

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

Teaching

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.

Teaching

Questions and Answers

“Never ask a question you don’t know the answer to.” This is an old saw among trial lawyers that I learned growing up watching courtroom dramas on TV. For some reason it’s stuck with me my whole life, but it’s not advice I follow. As a teacher I ask these kinds of questions all the time.

I often ask math questions in class I don’t know the answer to. Sometimes it’s because a student’s curiosity has sparked my own, and I want to share that excitement with others. Sometimes it’s to model not knowing. It’s important for students to become comfortable operating in the space of not knowing, because that’s where mathematicians live.

And every assessment, every quiz, every test I give asks a question I don’t know the answer to: “Do you understand this?” Or, perhaps more to the point, “Did I teach this successfully?” I usually have a feeling about the answer, but I never really know. That’s why I have to ask.

In my end-of-year surveys I asked students a lot of questions I didn’t know the answer to. Did you feel productive in math this year? Do you feel like you learned as much as you would have in a normal math class? Do you feel prepared for next year? I wasn’t sure what I would hear. But I had to ask. That’s what teachers do.

Teaching

The Gift of Hope

I met my calculus class for the final time this week. We commiserated about their awful experience with the AP exam (“I spent so much time typing parentheses!”), did some math (“Don’t you ever take a day off?”), and spent plenty of time in breakout rooms so students could socialize and say goodbye.

Before I said goodbye myself, I thanked them. This group of advanced students were ready to succeed in September. More ready than I was. As I stumbled my way through those first weeks of turning myself into an online math teacher, they did more than their part. They engaged in class when I didn’t give them much to engage with. They took initiative and tried to make breakout rooms productive. They invested themselves in their work and responded to my feedback.

In those first few weeks, they did their jobs better than I did mine. They were learning, despite everything, and that gave me hope. Hope that maybe I just had to figure a few things out and I would be ok. Hope that maybe my 9th graders were learning, too, even though it was harder to tell. Hope that maybe I just might make it to the end of the year and feel like I accomplished something.

Students give so much to their teachers: their enthusiasm, their open-mindedness, their creativity, their skepticism. These gifts are part of what makes teaching so fulfilling. This year my students gave me hope. And I wouldn’t be here without it.

Challenge Teaching

Who Needs Trig Sub? Part 2

I recently wrote about an ingenious integration performed by two of my students. But that wasn’t the whole story. Here’s the rest.

It begins with this integral.

Usually when challenged to evaluate this integral students will try a few substitutions, fail to find an antiderivative, and realize something new is needed. This sets the stage for trigonometric substitution, which we then use to untangle the algebraic obstacles in the integrand.

But this was an unusual year. Not only did two students surprise me with a purely geometric approach, a third student found another completely different solution I had never seen before. She used integration by parts!

This application of integration by parts leads to the following equation.

Which she simplified to this.

This is the point where I would probably abandon integration by parts, because it doesn’t look the situation has improved. But my student did something incredibly inspired. She rewrote the integral

like this

And this is definitely an improvement. The integral on the right is known: it’s just $arcsin(x)$ . And the other integral is the one we were trying to find in the first place. When this is substituted back into the integration by parts equation you get this

The integral we want to evaluate is now on both sides of the equation, so just collect like terms and solve

Before this year I had never seen any alternate derivation of this formula. This year students produced two completely new ones!

I can understand why students had never attempted this technique before. Integration by parts typically comes after trig substitution in the course, so it wouldn’t usually be an option for them. But this year, because of the way I rearranged the curriculum, integration by parts came first. And I’m thrilled that my student thought to try it. And persevered!

It also made me feel good about my integration by parts lesson. Usually the last problem I present in that lesson is the integral

But this year I almost didn’t. This problem requires repeated integration by parts and quite a bit of perseverance to solve. At the end of a long lesson I considered passing on it, but we pushed through. And it was worth it! Seeing this technique set up my student to find her own ingenious integral. And to teach me something new.

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

Ends and Beginnings

Questions and Answers

The Gift of Hope

Who Needs Trig Sub? Part 2

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