# What is “fading out a worked example” and why would it work and also when?

The basic theory is that people often learn things via a two-step process: they watch what someone else does and try to do it on their own. This isn’t the only way that people learn, but in its broad strokes covers a surprisingly large number of cases, at many different scales of learning. Parent shows baby how to play with ball, baby tries to bounce ball. New kid sees cool kids wear trendy shoes, learns to imitate their interests and affectations. Elizabethan poet-playwright writes awesome murder play, contemporary screenwriter writes mediocre murder movie.

Sure, but what about when this is very hard? If it was easy to imitate excellence, everyone would do it. It helps if you can impose some sort of structure on the effort. Sometimes it’s useful to focus on imitating just one step at a time. Other times you need to better understand what that other person did. Hopefully this is starting to sound familiar, it is typically called “teaching.”

Specifically in the context of math or science and learning procedures, there is a method for structuring this kind of learning called “fading a worked example.” I don’t remember where I first heard of it, but it’s strongly associated with Alexander Renkl’s research. Here’s how it works:

• Show someone a task and perform all of it for them.
• Then, present a new task, and perform almost all of it for them, everything but the last step. They do they last step on their own.
• Present another new task, and let them handle a bit more of it on their own. Ask them to do the last two steps.
• Etc., until the person is performing the entire task on their own.

Last week I did this sort of activity with my calculus class. It started with a fully worked example, that I prepared before class and asked students to analyze:

Then, I started knocking out steps from the procedure, starting at the end. This lets kids practice just that step, but in a meaningful context. (That is, the context of the integration problem.)

And then I knock out some more steps.

And by the end, they should be able to try the problem all on their own.

Here is another fading activity I ran in class last week, this time with my 8th Graders. I was teaching writing expressions for linear situations. At first I gave them the expression and asked them to think about why the expression fits the scenario:

But soon after I was leaving out the expression and asking them to provide it:

As I describe in my book Teaching Math With Examples, these sorts of tasks do much more for students than I first expected them to:

“As my class got to work, I was surprised by how challenging the assignment was. Students were asking me good questions about mechanical issues I hadn’t even considered to be problematic for them. It turns out that lurking beneath their overall struggles had been a variety of smaller difficulties. The faded-out exercise gave them a chance to systematically work on all of them.”

I find myself using these faded worked examples the day after a rough class. One where every student seemed to do great right up to the moment when I asked them to try the problem on their own. It’s for those days when the class seemed nervous about even starting, not because the problem was opaque but because the solution seemed like just SO MUCH.

It’s worth pointing out that this is not rocket science. Which makes sense, because teaching is not rocket science. It’s not science? It might be fair to say that teaching is maybe cognitive science plus other educational research plus a lot of random practical knowledge multiplied by a ton of emotional intelligence. The point is that you don’t need a researcher to invent something like fading a worked example.

One the other hand, I learned about this from reading research. And then, after I read about it in research, I sat on that knowledge for a few years. I wasn’t sure if it would fit in with the rest of my random practical knowledge about teaching. I also was worried that it would be dull and repetitive to my students, throwing off the mood in the room. (I shouldn’t have worried, it’s great.) Now that I’ve tried it a lot in my teaching, I’m happy to share the idea with enthusiasm.

Research is great, but that will only ever be a part of what it takes for a good idea to make its way into practice. The people who use the idea need to chew on it for a while and make sure it fits in with everything else. It would be good if researchers understood this, because it implies different sort of studies than what is frequently done. (“We Did This One-On-One With a Bunch of Undergrads Which Reveals Universal Teaching Techniques and Now Go Ahead and Use It.”)

Anyway, if you’ve got a long-ish procedure and it’s a topic that is easy to create many similar-ish examples for, you might consider writing a worked example and fading out the steps, starting at the finish.

# My Billion Dollar Idea

I recently had a billion dollar idea. It’s for an education business, but there’s no way I could pull it off. I figured I’d write it up here, and hopefully someone else can create it, because it really should exist.

OK, so here’s the problem: there are young children who know a lot more math than their peers. The smaller the school, the bigger a problem this can be. Get five kids who know a lot of math together, you’ve got yourself a small math group. But if you just have one kid in your 1st Grade class who can confidently and fluently do 5th Grade whole number arithmetic? You’ve got yourself a problem.

Long-time readers know that we’ve discussed this before: What should a school do with an advanced 1st Grader?

But that was before the pandemic. And now I have a much better understanding of the pros and cons of online teaching. And I think that an online math class for young students — one that was integrated with their school day but didn’t focus on just advancing them further down the path of arithmetic — could really work.

The big downside of online teaching is that a demotivated student can slip through the cracks with the greatest ease. Kids who are excited by the material and eager to engage online, in my experience, have been able to handle the digital setting quite well. Even kindergarteners. (I’ve watched my son get quite good at using his favorite apps and websites.)

Going forward the most promising use of online teaching in K-12 is for increasing access to electives and advanced courses for motivated students. China has experimented, seemingly successfully, with beaming math courses with knowledgeable teachers to students in underserved rural areas:

New technology is helping close the gap. Since 2016, 248 under-served high schools in the poor areas have tuned into “live streaming classrooms” hosted by Chengdu No.7 High School, one of the top high schools in China. Over the livestream, the program puts together two groups of students that could not be more different — upper-middle class students and their peers from the country’s most educationally under-served families.

Similar experiments are happening in the United States:

That’s why there’s considerable excitement about the free program bringing AP physics to Mississippi this school year, courtesy of the Global Teaching Project, a Washington D.C.-based education company that is part of a nonprofit consortium in the state. A few years ago, the Holmes County school district offered a few college-level AP courses at only one of its three high schools. After the three schools consolidated during the 2014-15 school year, the newly formed Holmes County Central High School was able to offer five classes, including AP calculus, English language and English literature.

And last summer, I personally was involved with BEAM’s attempt to take their math enrichment camp for underserved students fully online. Of course, this move was a necessity of the pandemic. It was not easy. Read the director of BEAM, Dan Zaharopol, describe how difficult it was to provide reliable internet access and proper devices to every student admitted into the program. (They delivered and troubleshooted a lot of hotspots.)

On the teaching end, it wasn’t always easy either. Some kids didn’t really end up getting emotionally connected to the teaching. They sat through classes with their cameras off and didn’t truly engage. But by the end of camp it was clear that our teaching made a very real difference to a great number of students — students who would not have had this opportunity otherwise.

So here’s the money-making idea: a country-wide math class for elementary school students who know a lot of math. It would be online and (here is the crucial part) it would have nothing to do with the math they are learning in school.

Graph theory. Probability. Games. Puzzles. Logic. Incompleteness. Fractals. Anything! There are a million facts of mathematics and what we cover in school is just one, narrow slice of the mathematical world. If a student is ready for more, mathematically speaking, the solution is more mathematics. Yes, but what mathematics?

You can accelerate a student through many years of elementary math fairly quickly. Facility with arithmetic compounds; I’ve seen 3rd and 4th Graders that can do truly impressive things with numbers. But at a certain point the race forward itself becomes a problem. You can put a young student in a class full of teenagers, but do you really want to? The emotional needs and social environment is quite different. Anyway, kids like being with their friends. And all sorts of messy inequities emerge when some kids race forward and other kids are left to handle the grade level material.

But! What if you just teach kids who know a lot of math different math? In time, the gap between the knows and the know-nots will narrow, it will be less absurd for advanced students to share a 5th Grade math class than in 1st Grade.

And, with motivated students, you could totally do this online.

People would pay for it. Districts would pay for it, but individual parents might pay more. This is a way to keep their kids interested in math but without creating a schooling problem. It would happen during the day, a few times a week, with a device and headphones while the rest of the class is building up their basic number skills. The three ingredients are: young children; math that isn’t arithmetic; online learning. We can do this.

If you end up taking this idea and building your own business, it will take time. You’ll have to slowly develop a curriculum that is both accessible and advanced. You will need to learn how to help very young students do math over a platform like Zoom. You’ll need to gain a reputation and win the hearts of parents and the faith of a school district. This will be a steady, modest way of building a business, one that will enable you to slowly build up a wealth of capital.

At that point, you sell the business. You take the money, you put half in Bitcoin and the other half in Dogecoin. It’s too late to get in on the ground floor, but if you act quickly there is still a fortune to be made. Then you sit back and just wait.

That’s my billion dollar idea.