As the three or four readers of my missives know, one of my big concerns is “What sort of questions can we ask our students in the internet era?”

So today, let’s talk about throwing axes:

Now you might wonder what this has to do with math. The answer should be obvious: Some people are born with “axe arms.” They instinctively know how to throw axes. It’s why people say “I’m just bad at throwing axes,” and when you try to teach them, they’ll say “When am I ever going to use this?” (and they’re not convinced when you say “Suppose there’s a zombie apocalypse and you need to throw an axe…”).

No, wait, wrong script.

I got good because…wait for it…I practiced.

OK, that should be obvious. And it’s something we always tell our students: You won’t get good unless you practice, so here’s 25 problems to practice on.

This works up to a point, but sooner or later runs into a fundamental problem: human beings are intelligent, and intelligent beings get bored doing the same thing over and over again.

Say I give a student 25 linear equations to solve. They may struggle with the first five. But by the second five, they’ve started to figure things out. And the third set of five they’re in the groove.

At that point, boredom kicks in, because they realize they’re doing the same thing, over and over again. So they start to make mistakes, not because they don’t know what they’re doing, but because they’re not paying attention. Those last ten problems will be increasingly riddled with simple errors.

And now for the really bad news: what they’ve practiced in those last ten problems, and what they’ll most likely remember from the assignment, is how to solve the problem incorrectly.

How can we avoid this? Let’s get back to throwing axes. I practice, but I don’t just stand at the throwing line and try to hit the target. Once I’ve established that I can hit the target reliably, I take a step back and try to hit the target, which is now a little further away. And once I can reliably hit the farther target, I take another step back and repeat the process.

In other words: You acquire skill by tackling increasingly difficult tasks.

I call this “shifting the lines.” To begin with, we can change the starting line. This is more comfortable for students: they know what the goal is, and the idea that they’re starting a little farther away isn’t too daunting.

So let’s consider those linear equations again: “Solve 3x + 5 = 25.” It’s easy enough to come up with five or so problems like this, and students can practice. However, it’s just as easy for students to run the problem through an online calculator and get the answer, with all steps shown. Consequently, this type of question is meaningless for an exam: it fails to distinguish between the students who “know how” to solve the problem, and the students who can use an online calculator.

But move the starting line. Where might such a problem come from? Typically, these come from word problems. So wrap these linear equations with some words: “The width of a rectangle is 5 feet longer than three times the length. If the width is 25, what’s the length?” This type of problem will produce nonsense results when run through an online calculator, so it’s a far better question to ask.

Notice that this question places a premium on the ability to translate a written statement into a mathematical one. While this is a worthy goal, once the student realizes the problem is “Solve 3x + 5 = 25”, they can run that through an online calculator. Arguably, that’s not a problem: it’s a proper division of labor, with the human doing human things (reading and interpreting) and the machine doing machine things (mindless application of an algorithm). But maybe we want more.

The problem here is that the question gives us an equation to solve: “The width is 25.” Since solving equations is the epitome of an algorithmic process, it’s exactly the type of thing computers are good at. So if we want to internet proof our questions, the answer cannot be the solution to an equation.

So let’s move the finish line. Solving our equation gives us the length. If the length isn’t the answer, why are we finding it? Maybe we need to find the area. So change the problem: “The width of a rectangle is five feet longer than three times the length. If the width is twenty-five feet, what’s the area?”

Now we have a problem that’s internet resistant. Providing a correct answer requires identifying an equation to be solved; solving it; then using the solution to answer the question. Of these, only the middle step can be done by machine; the others require human intervention.

So here’s a general framework for writing questions in the internet era: If you want to ask a question but it’s something that can be run through a problem solver, shift the lines. Change the starting point, so a problem solver can’t solve it. And change the ending point, so that even with a problem solver, there’s much that a human must do.

It’s nearly certain, at this point, that we’ll be able to resume “business as usual” for Fall and act like nothing happened.

Whups, wrong script.

It’s nearly certain, at this point, that even if we resume “business as usual”, it won’t be “business as usual.” Like it or not, online learning and online assessment are here to stay, and COVID-19 only accelerated an existing trend towards distance education. Institutions and faculty who don’t adapt to the new reality are going to be at a competitive disadvantage.

Math is particularly challenging. The existence of “freemium” online problem solvers that show all steps needed to find a solution means that any question that can be answered by following a sequence of steps is meaningless as an assessment instrument. If I ask a student to find

they can use an online problem solver that will also show all the steps; at that point, they just need to cut-and-past the answer. The difficulty is we can’t distinguish a textbook perfect answer generated by a computer algorithm; a textbook perfect answer because they modeled their answer after the textbook perfect examples; and a textbook perfect answer because the student actually listened to us in class. So how can you write problems for the internet age?

We need to constantly ask ourselves: Why did we ask this question?

In this case, the answer might be “To evaluate student understanding of the chain rule.” Leaving aside what we mean by “understanding”, the chain rule is:

and so maybe we want to see if a student understands this concept.

Let’s break this concept down a little further. To correctly apply the chain rule, a student needs to identify an “inside” function and an “outside” function; find the derivatives of both; then use the “inside” function as the argument of the derivative of the “outside” function. The problem is that if we provide a formula for the two functions, an online problem solver can grind through this purely mechanical process.

This means that we can’t provide the formulas.

So how do you ask a math question without providing formulas? Here are some options.

First, if you want a “plug-and-chug” type question (and remember, even good students like these type of questions, because they know what to do with them), you can give partial formulas. For example:

A well-organized student could solve this by following the chain rule, step-by-step. But a problem solver can’t—at least, not yet. (I make no promises for what will happen six months from now)

Online problem solvers are also very bad when it comes to dealing with values from a table, so another possibility is to give a table of values:

which requires some differentation “by hand”, or

which does not.

If you’re one of the three people who’ve read my posts, you’ll know that I believe (a) cheating is going to happen, (b) there’s nothing we can do about it. You should also get the impression that (c) I’m not too worried about it. Don’t we always say “Cheaters only hurt themselves”? If we say it, shouldn’t we stand by it?

But academic dishonesty is an issue: we want the “A” to really mean “good work” and not “knows how to defeat webcam monitoring.” The way forward is through:

We have to scrap the notion that exams in ANY form are meaningful evaluations of a student’s knowledge.

If your department has common finals, try this exercise: take the most recent final you can find. Then type the questions, exactly as they’re presented, into Symbolab (or Cymath, or Wolfram Alpha, or Mathpapa). See what grade a free internet app would get on your final exam. If the free internet app can get a passing grade on the final, what does this say about the student who can do the same?

I don’t know what it says to you. But what it says to the student is that your class has the value of a free internet app. And the question we hope they don’t then ask is, “So why did I pay $4000 for the course?”

So what can be done? The obvious solution is making sure that no internet app can solve a question you’ve asked. That’s easy: just make every question require the student to do a Ph.D.-level dissertation. Of course, most students would end up failing such exams, and an exam that fails everyone is a useless exam. (An analogy that’s helpful: If you want to evaluate the physical fitness of a group of people, you could have them run a 100 mile marathon. Those who cross the finish line are in great shape! But the test tells you nothing about those who fail to cross the finish line)

While you couldn’t write Ph.D.-dissertation questions for a final exam, it does at least suggest a way forward. The thing about a Ph.D. dissertation is that it relies on putting together a lot of little things, over a long period of time. It’s a project, in other words.

In the age of the internet, we must move towards project-based evaluation.

The problem is that doing so is very difficult, especially if you haven’t trained your students to do projects. Telling them, three weeks before the semester ends, that their final evaluation will be a project will cause riots, especially this semester when everything else has changed. Be kind to your students: live with the fact that your final exam is effectively going to be an open book, open note, open internet exam, and accept that some students will get 100s even though they should be getting 30s.

But next semester…now is the time to start thinking about what those project-based questions are going to be. What sort of problem requires putting together everything from the entire semester, and can’t be answered by typing the problem into Wolfram Alpha? (Modeling questions are good: maybe they have to gather some data, construct a mathematical model, cross-check it against reality, and then come to a conclusion about its predictions. In a form suitable for a memo to their boss, whose last math class was “Introduction to ENRON Accounting”)

Wait, you won’t be teaching online next semester because by then the stay-at-home orders will be rescinded?

Sure…but the same issues still apply. Again: If your final exam can be passed by someone using a a free internet app, then what is the value of your course? The “silver lining” of the crisis may be that we scrap the idea of an exam once and forever, and find valid ways to assess student learning.

Last time I talked about the impossibility of monitoring students in the online world: if they’re going to cheat, they’re going to cheat, and there’s not a lot we can do about it. Just the other day, I had a student who got an “11” on the first (in-person exam), based on indecipherable scribbled work, get a “90” on the second (online exam), based on letter hand perfect work.

Can I prove academic dishonesty? No.

Should I worry? One of two things is going to happen. Either the student is going to go to the next class, and their inability to do the work without help is going to hurt them…or they’re never going to use the skills the test was evaluating them on.

Still, there’s the hope that we can still do something resembling exams online with some level of integrity. In the longterm, there’s only one good solution: scrap exams entirely, and shift to project-based learning: You pass an engineering course by building a bridge that doesn’t collapse. Of course, that’s easier said than done, and what it really takes is one-on-one interaction with the students. Fortunately, that is the longterm solution, and we don’t have to implement it by the end of the semester.

The short term solution: Lots and lots of words.

Ideally, every problem should be a word problem. But once upon a time, I got into a debate with a colleague over why we didn’t make every problem a word problem. Their response? While they though it was a good idea, the weaker students needed the problems they could do by following an algorithm. In other words, the non-word problems are there so the weaker students have a chance of passing the exam. The student who won’t flinch when you say “Solve 3x + 5 = 35” will be unable to solve “Three times a number, and five, is thirty-five. Find the number.”

However, in the internet world, this is our (short term) solution to apps like WolframAlpha, Mathpapa, Sybmolab, Cymath, and others. They don’t (yet) know how to handle word problems, even simple ones like the one above. WolframAlpha thought we were asking about newspapers:

Symbolab responded in what looks like Old English (Beowulf, anyone?)

MathPapa’s response:

Of course, a student could figure out that “Three times a number, and five, is thirty-five. Find the number” is the same as the problem 3x + 5 = 35, and then run that problem through their favorite problem solving software.

But maybe that’s not such a bad thing. It’s using a human brain to do what a human brain is good at (understanding questions asked by another human brain), while using technology to do what technology is good at (applying an algorithm that requires no logic, thought, or insight).

The big problem confronting many of us right now is academic dishonesty: How do we make sure our students don’t cheat?

Here’s the easy answer: You can’t.

More specifically: Unless you intend to “Big Brother” your students and watch every move they make via webcam, it is impossible to prevent students from using unauthorized resources.

For example, there’s been some buzz about “lockdown browser” software: once a student starts the exam in a lockdown browser, they can’t open a new window and look up the answer. That works great…except that there are several smartphone apps (Symbolab and Socratic among others) that will allow students to take pictures of questions; the app will then produce a solution complete with steps. Students can then just copy the steps down onto paper, and they’ve “shown their work” (a traditional remedy against cheating).

So what can we do? The reality is there is nothing we can do to prevent this type of cheating. Add to that an important idea: Teachers are not policemen. None of us entered this profession because we got joy and satisfaction catching miscreants. None of us like the idea of having to monitor students to make sure they don’t cheat, and quite a few of us have used the line “If you cheat, you’re only hurting yourself.”

Still, the academic dishonesty issue is important: the student who sails through their classes with As may well become your doctor, and you’d like the reassurance that their academic performance reflects their competence. So what can we do?

I’ll pass on something I learned from my first years of teaching:

Always ask yourself “Why am I asking this question?”

In this case, let’s ask ourselves: Why do we care if students “look things up?”

Anyone remember the birth and death dates of James Monroe? How about Article 5 of the US Constitution? The speed of light (in m/s)? The 1735th decimal digit of e? In the real world, in our professional lives, we look things up all the time. If “education” is supposed to be training students for the “real world,” we should encourage looking things up.

“But they’ll need it for the next class.”

Ah, that’s a good point. Students will need to know how to solve quadratic equations for their next class.

But the number one criticism of math by everyone who’s not a mathematician? “I don’t see how this applies.” Remember the vast majority of math students are not going to become mathematicians. So while it’s true that a calculus teacher might want a student to solve a quadratic equation “by hand,” the engineering teacher just wants the student to design bridges that don’t collapse. The nursing teacher just wants student who correctly calculate drug doses. The marketing teacher just wants students who can find the correct price point for a product.

And when they get to the real world, their supervisor doesn’t care if they know the quadratic formula. They want the bridge that doesn’t collapse, the patient that doesn’t die, the marketing campaign that makes money. The ability to solve a quadratic equation “by hand” becomes a skill like knitting or brewing: it’s really neat when you can do it, but you still buy your clothes off-rack and buy beer at the packy store.

So stop worrying about academic honesty and focus on the real task: helping students learn. I’ll talk about that next time.

CUNY is going to be shut for another week or so as part of a “recalibration”.

The problem is that the sudden shift to online teaching has placed a burden on the collective infrastructure of society. Internet traffic has exploded so much, with everyone now video lecturing or putting materials online or checking their emails, and while we joke that every student has a smartphone that’s better than my laptop, the reality is that it takes more than a smartphone.

What this plague is going to reveal are all the little cracks in our society that have been masked by those near the top.

K-12 schools closed? Good idea. Except, for a “civilized country,” we have an embarrassingly large number of kids whose only meals of the day come from school lunches.

Online classes? Good idea. Except, for a “civilized country,” we have an embarrassingly bad broadband infrastructure in rural areas.

Perhaps the fundamental problem comes from the sentiment, begun by Texas Lieutenant Governor Dan Patrick, that seniors are willing to die for their country:

https://www.snopes.com/ap/2020/03/25/texas-lieutenant-governor-says-us-should-get-back-to-work/

But that misses the point. It’s not whether you are willing to die for the country.

It’s whether the country is willing to kill you to make money.

Some people and some companies get it: they understand that human life is more valuable than cash flow.

Comast is offering a variety of free services: https://corporate.comcast.com/covid-19

Zoom has expanded the functionality of its free accounts: https://www.forbes.com/sites/alexkonrad/2020/03/13/zoom-video-coronavirus-eric-yuan-schools/#618bd2594e71

Kahoot! has done something similar: https://kahoot.com/blog/2020/02/27/kahoot-free-access-schools-higher-education-coronavirus/

These things help, of course (and after the crisis, they’ll have a whole batch of new customers who’ve learned to rely on them, so what they’re doing is both altruistic and good business). But the fundamental problem remains our digital infrastructure. Free service from Comcast, with videoconferencing by Zoom and polling with Kahoot! won’t help the student trying to access the internet on cables put down in the 1990s.

Digital infrastructure is (in the 21st century) as fundamental to living as water and electricity. It’s as important as roads. It’s a critical as fire and police departments.

So the question you have to ask yourself is this: How much should the government support this critical part of society?

A tip from my long past days as an HTML coder: Things look different from the outside.

‘way, ‘way back in the day, when everyone who didn’t use Netscape used Mosaic (except for those oddballs who spoke ftp), the rule was: see how your web page looks on every browser available, because even though HTML was a standard, it appeared slightly differently on different browsers.

Many videoconferencing platforms provide you a window of how you appear to others. The problem is that this window is “local”, meaning that it’s using information that hasn’t passed through the internet. The problem is that if your internet connection is spotty or dies midcall, you won’t know it.

So here’s a quick tip: If it’s at all possible, join your own meeting from another account. That way you can see yourself exactly as others see you.

(Another quick tip: If you do this, make sure you mute the sound on the secondary device. Otherwise you’ll get some horrific feedback.)

CUNY has suspended all on-campus classes for the semester. The big question on everybody’s mind is: So how are we going to do exams?

Let me start with a basic assumption: Unless you are physically present, it is effectively impossible to prevent students from using disallowed resources on an exam. I don’t care what online activity monitoring software you want to use, students who want to cheat will cheat.

So what can be done? Some suggestions include having students do their work in front of a webcam, which might work as long as they don’t have “technical difficulties” (and let’s face it, we’ve all been on a videoconference where someone’s screen froze). And if a student accidentally knocks their camera over a and doesn’t fix it immediately, are we going to assume they’re cheating?

Thinking this over, it occurred to me that we’re trying to solve the wrong problem: trying to prevent students from cheating on an exam is not the problem we should be solving. Rather, the problem is making sure our exams reflect student understanding. If the exams are structured so they reflect student understanding, then it is (by definition) impossible to cheat.

There’s an old saying in math (and, I’m sure, many other disciplines): the best way to learn the subject is to teach it. To that end, here’s one possibility. Imagine that part of the assessment is having the student create a short video where they solve the problem while explaining every step they take.

Why does this work? I don’t know; I haven’t tried it. But why do I think it will work? There are many places on the web you can type in a question like “What is the derivative of f(x) = x^3 sin (ln x)”, and a handy little CAS (computer algebra system) will spit out the answer. In some cases, you can pay for a step-by-step solution. If the exam has a question like “What is the derivative of f(x) = x^3 sin(ln x)”, it’s easy enough for a student to cut-and-paste the solution.

But now make them explain the steps of the solution. At this point, the student who has cut-and-pasted the answer won’t be able to do much more than say “That’s the rule that you follow to get the next step.” If they can’t provide an explanation of how they got from A to B, it’s a safe bet they don’t know what they’re doing.

Will this work? I don’t know. But at this point, anything is better than nothing…

If you’re planning on videoconferencing, one of the things you might want to do is construct a screen so that students won’t be able to see the piles of laundry or undone grading behind you, and if you live with somebody (a spouse, kids, feral dingos), they won’t distract students by walking behind you. So here are step-by-step instructions for making an improvised greenscreen for about $25 plus the cost of a bedsheet. The only tool you need is a saw.

First, you’ll need to get some supplies. I used 1″ PVC, available at any hardware store. You’ll need four 10 foot segments. You’ll also need some connectors: two elbows and two T-junctions. Finally (and optionally) you can get some end caps:

Now cut 3 foot segments off the end of four of the PVC pipes.

If you got the end caps, cap one end.

Now attach two of these to the opposite ends of the “T” junctions.

Remember you cut 3 feet off of a 10 foot pipe? That leaves 7 feet. Take one of the 7-foot pipes and put elbows on both ends. (The picture shows two of the 7-foot pipes, but don’t join them yet)

Now take the (unelbowed) 7-foot pipes and join them to the T junctions. These will give you the uprights:

Now for the only difficult part: while you can probably do this on your own, with a chair and some choice expletives, it’s probably easier if you get someone to help. If you’re living with somebody, you can draft them (the feral dingos aren’t so helpful; I recommend the high school age kid who’s taller than you…).

Take the bar with the two elbows, and attach it to the two uprights. Then hang a curtain over the top:

And voila (random professor not included)! You now have a greenscreen and can webcast like the best of us.

If you’re looking for time-honored wisdom and erudite insights regarding how to proceed in this plague semester…you might want to look elsewhere.

This is mostly about how I managed (or failed to manage) the transition to an online course to finish out Spring 2020. A little background:  I’m teaching Math 1006 (college algebra) to a group of about 35 students.  The students did not sign up for an online class, nor did I sign up to teach one, so when CUNY suspended on-campus classes in response to COVID-19, these students (and myself) were plunged into a totally unexpected situation.

We (my students and myself) started with a slight advantage: we’d been using an online homework platform (MyOpenMath) and polling platform (Kahoot), so they had some familiarity with remote instruction.  And I have my own set of lecture videos, which cover subjects from prealgebra to cryptography and differential equations, so I’m no stranger to putting material online.  And during Winter 2020, I had the distinction of teaching the first ever online math class at Brooklyn College (as well as being one of the first ever intersession math teachers).

But remember:  the students of Spring 2020 didn’t sign up for an online class.  So even though the Winter 2020 online class was, by any reasonable standard, successful, that success couldn’t translate into “best practices” for the Spring 2020 online class. So what does?  We’ll find out…