To Win This Numbers Game, Learn to Avoid Making Math Patterns — Quanta Magazine

My latest column for Quanta Magazine starts with a simple game of numbers and ends with some unsolved problems in mathematics.

For example, let’s change the rules to make the loser the first person to complete three in a row of any step size. This means you lose if you make 2-3-4, as in the original game, but also if you make 1-3-5 (three in a row of step size 2) or 1-4-7 (step size 3). These patterns are “arithmetic progressions”: sequences of numbers with a common step size, called the common difference.
Let’s return to our first game board and use the new rules. It’s still your turn. And you’ve lost.

This simple game, where each player tries to avoid completing an arithmetic progression, leads to some complicated math, involving open questions about Salem-Spencer sets and a new result about polynomial sequences.

The full article is available here and includes several exercises to test your game play!

Remote Learning — Week 7

One thing I’ll miss from Remote Learning is rolling submissions of student work. As I discussed last week, students have 4-5 days to submit their assessments, and work that comes in before the deadline is evaluated and returned with feedback. Students can then choose to revise and resubmit their work before the deadline for full credit.

The approach has successfully created a consistent space to interact with students around their mathematical work. But it has also brought a welcome change in my grading workload. From the moment it’s assigned, student work trickles in slowly and steadily. This lets me evaluate a few at a time and quickly return them. Of course there are still plenty of students who wait until the deadline to turn in their work, but the incentive to submit early means there’s nothing like 100+ tests waiting for me all at once.

The ability to spread out my work has been invaluable during Remote Learning. The chaos of everyone working and schooling at home means uninterrupted blocks of time to focus are even rarer now. Knowing I can always turn an unexpected free 15 minutes into productive work helps keep all the balls rolling.

This process also has me approaching feedback differently, in a way that’s more like an ongoing conversation between teacher and student. I think we’ve both benefited, and it’s something I’ll continue to think about after Remote Learning has ended.

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2020 AP Calculus BC Practice Exams

The 2020 AP Calculus BC exam will be very different in scope and structure than previous years, as a consequence of logistical issues brought on by the COVID-19 pandemic. The College Board has indicated the test items will be similar to those on past exams, but there aren’t many existing practice materials designed with the 2020 format in mind.

I’ve created two sample practice exams for my BC Calculus students and will share them here for teachers and students looking for additional resources. All are welcome to use them, but keep in mind that these are merely guesses about what the exam might be like in terms of scope and difficulty. I will say that I intentionally tried to make these more challenging than the College Board’s sample exam, which is just a re-combination of existing 2019 items. (I referred to this as Practice Exam #1, which is why you see #2 and #3 below.)

Please use as you see fit. And let me know if they are helpful!

Remote Learning — Week 6

Assessment of learning is one of the great challenges of teaching. It’s a complex endeavor under normal circumstances, made even more difficult in emergency Remote Learning. To math teachers, assessment often means testing, and some teachers are trying hard to replicate the experience of classroom tests. They’re experimenting with different delivery platforms, administering tests synchronously under time constraints, and even having students test in front of their webcams.

Other teachers have entirely abandoned tests in the traditional sense. I’m firmly in this group, but I do empathize with the former. I enjoy writing and giving a good test, and a steady stream of supervised student work is fuel for my instruction. But I just don’t see how I can give a traditional timed test in a fair and reasonable way in the current situation.

One approach I’ve taken is inspired by the principles of Mastery Grading. I’m assigning assessment items similar to what students would see on a test, but giving them several days to complete the work and allowing them multiple resubmissions after feedback. In this model, students can revise and resubmit their work until it is as complete and correct as they wish. It’s not an approach I’ve used much in the past, but it’s working for us: I like the interactions around mathematics I’m having with students as I provide them feedback, and I’m getting that stream of work I need to tune my teaching.

Some teachers are very concerned about cheating. I’m not one of those teachers. I’ve been teaching long enough to know that the vast majority of students will do the right thing when they, themselves, are treated right. There may be a few students looking for a shortcut, but those cases are isolated incidents to be handled individually.

While the system in place may be vulnerable to some abuse, I feel it treats students right. Everyone has a chance to master the assignment to their satisfaction, and the weekly assessments are frequent enough to dilute the stakes of any individual task. No student will feel penalized for doing the right thing, and the incentives to do the wrong thing have mostly been removed.

And all the evidence points to students doing the right thing. I grade the assignments on a rolling basis, and the work that comes in late shows the same kinds of correctable errors as the work that comes in early. Often students will submit something that’s mostly complete and say “I’m not sure what to do from here”. Usually a single hint or a suggestion is enough for them to finish up. The personal exchanges, in comments and resubmissions, serve as an audit of the student’s process.

I always assess in many different ways, some of which are less impacted by the circumstances of Remote Learning: individualized assignments, technology projects, and oral exams all work mostly as well now as before. But this new approach is working for us, and seems particularly well-suited to the circumstances. It requires trust, but then, all good teaching does.

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STEM for All Video Showcase

Next week I’ll be a facilitator and judge for the 2020 STEM for All Video Showcase. This event highlights hundreds of innovative projects designed to improve Science, Mathematics, Engineering, and Computer Science education in both formal and informal settings.

Thousands of educators, researchers, policy makers, parents, and students actively participate in the event, and the videos are watched by hundreds of thousands of people worldwide. The showcase runs from May 5th to May 12th, and everyone is invited to watch the videos, participate in the conversations, and vote for their favorites!

Find out more at the STEM for All Video Showcase website.

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