A Knewton critique: Why computers shouldn’t teach calculus (or anything else)

Photo: Chegg Study

Isi, my teenager, is a college student taking calculus. The teacher is assigning homework from Knewton, an online learning system. I’ve been helping with the homework. Now I can see why computers can’t teach college courses — and why tools like Knewton are not just destructive, they’re evil.

The old way of teaching calculus worked pretty well

Once upon a time, nearly 40 years ago, I taught calculus. I taught recitations (interactive classes of a few dozen students who’d attended the professor’s large-scale lectures). In other semesters, I taught the whole class through lectures. One summer I taught it to a bunch of Navy grad students who were all older than me.

These experiences do not qualify me as an expert on teaching calculus, but they do inform the opinions I’m sharing here.

There are three things you need to do to teach calculus correctly. (Unsurprisingly, these are the same three things you need to teach just about anything else.)

You need to explain concepts, like rates of change, instantaneous slopes, and the idea of a derivative. To develop intuition, students need to know the meaning of what they are doing.

You also need to explain the rules. Students need to understand and memorize key formulas like the difference quotient and the chain rule and learn when and how to use them.

Finally, the students need practice. There’s a difference between knowing why and how to sing, and actually singing. So they need to actually work a bunch of problems. In calculus, the more complicated problems often involve a lot of algebra.

If you’re doing this kind of practice, you make mistakes. You can make conceptual mistakes, like applying the wrong rule. But you can also make many other kinds of mistakes: sign errors, failing to carry a number down to the next step, multiplying 12 x 7 and getting 86.

When I was grading calculus homework, you’d get no credit if you made a conceptual mistake. But you’d get partial credit if you understood the concept and applied it correctly, but then made arithmetic or simple algebra errors.

This type of grading encouraged the students. If the answer was cos(x) and you got -cos(x), you knew you were understanding the concepts, and you could work on getting the rest of the calculations error-free with practice.

Perfect students who were facile with math got great grades. Slower students who needed to practice got good grades, and got better with practice. Students who didn’t understand the concepts or didn’t put in the work got bad grades, which they deserved.

By broadly penalizing error, Knewton undermines learning

In Isi’s calculus class, the teacher appears to be explaining the concepts well. Isi is also quite capable of understanding the rules and how they apply. I was gratified to hear Isi say at one point, “I actually like this math stuff.” I think Isi’s logical mind and fastidious attention to detail is a good match for math and that is a good sign, since the kid wants to major in computer science.

But in Isi’s class, the homework is on Knewton. Here’s how Knewton works:

  • You get a bunch of assigned problems in each section.
  • You cannot skip a problem.
  • You don’t get the same problems as your fellow students, so you cannot compare your approach.
  • You read the problem and work it on a piece of paper. You use a sophisticated formula entry tool to type the answer into the web page.
  • If you get the answer exactly right, Knewton gives you a little dollop of encouragement (“Great job!”) and may even reduce the number of other problems you need to do.
  • If you get the answer wrong, Knewton gives you a hint and you can take another try.
  • When the answer is wrong for any reason, Knewton generates more similar problems, and you have to work those to finish the homework. You cannot skip any. You just sink deeper and deeper.

The Knewton company has big ambitions. It has raised $137 million in funding. Its former CEO, Jose Ferreira, had said that the company could replace teachers and human intuition. As he told NPR, “We can take the combined data power of millions of students — all the people who are just like you — [who] had to learn a particular concept before, that you have to learn today — to find the best pieces of content, proven most effective for people just like you, and give that to you every single time.”

In the same article, educational consultant Michael Feldstein said that Knewton was selling “snake oil.”

I can’t address what Knewton is doing in all those other courses, but I can judge what it is doing in calculus, which is to demoralize students and interfere with learning.

Imagine for a moment that you are doing a calculus problem that involves 20 steps, any one of which you might get wrong due to a simple transposition error that has nothing to do with conceptual knowledge or the rules of calculus. It has taken you 15 minutes to do the problem. Now it’s time to type that answer into Knewton.

If you get it wrong, you know Knewton is going to give you even more homework to do. That could be four more problems just like the one you made the mistake on. You just lost an hour or more. And heaven forbid you make a sign error on one of those problems.

You cannot get partial credit.

You cannot skip the problem and come back to it.

What you can do is go to Wolfram Alpha and type the problem in there and get the right answer without doing any work.

If you are extraordinarily moral, you will work the problem on your own, check it on Wolfram Alpha, and then type it in. But you won’t be getting the learning that came from making mistakes.

Making mistakes is a fundamental part of learning anything, whether it’s salsa dancing, tennis, writing, or calculus. We learn by doing things wrong. The teacher shows us what we did right (yay!) and where we went awry (ah, I see, I’ll do that better next time). This is normal, natural, and essential — it combines learning with the simplest and most obvious psychology of learning.

Despite the cheery mechanical pronouncements that Knewton makes when you get a problem right, students have a very strong incentive not to make mistakes, even if those mistakes are tiny errors. It’s as if you got an electric shock every time you used the backspace button when writing. Except this time, the penalty is having to do more of the same work that you just made a mistake on — without much in the way of diagnosis.

While Knewton’s public statements indicate that they can create a conceptual map of where students’ problems are, in the case of calculus, there are simply too many ways to go wrong. If you make a mistake on step three of a 20-step problem, you might go wrong in any number of different ways. And you’re not allowed to just give up and move on.

As so often happens in today’s automated society, Knewton replaces a manual system (teachers correcting papers) with an automated system that works poorly and chews up the student’s psyche in the process.

That’s no way to teach, and it’s no way to learn, either.

AI is practicing on us

Before AI can get good, it has to be bad. That is, it has to be worse than doing things the old people-intensive way.

AI gets better by making wrong guesses, just like people do.

But when it comes to learning, training AI by making guesses about what students are doing wrong is abuse. Students don’t pay college tuition to be fodder for AI experiments.

People need to make mistakes. Until computers can identify and act on all of those possible mistakes, teachers are going to be better than machines at teaching.

7 responses to “A Knewton critique: Why computers shouldn’t teach calculus (or anything else)

  1. There’s a special place in hell for the small minds who refuse to give partial credit.
    There’s a special place in hell for the small minds who punish all mistakes equally.

    1. That made me smile.

      Josh – you write:

      Despite the cheery mechanical pronouncements that Knewton makes when you get a problem right, students have a very strong incentive not to make mistakes, even if those mistakes are tiny errors.

      I’d love to see what Knewton’s data says about the matter.

  2. It sounds as if the Knewton approach is similar to speaking English louder and louder to someone who barely speaks the language, in hopes it will aid comprehension.

    In college I completed 5 semesters of calculus and a semester each of matrix math, differential equations, and statistics. I appreciated the professors and graders who took your grading approach and loathed those who did not.

    As you pointed out, Josh, there is a world of difference between a conceptual error and a calculation (algebraic or numeric) mistake. In the learning process, knowing why an answer is wrong is more important than it being wrong in the first place. When we had our homework graded line-by-line and discovered where we made the errors, it helped us in two ways: knowing where we tended to go awry in our calculations and increasing our confidence by ironing out any faulty concepts. An effective learning tool diagnoses and distinguishes such reasons.

    The line-by-line grading is tedious, and that’s why graders are employed in the university to handle those tasks. I understand that your daughter is a high school student, and funding is likely not available for such a position.

    Something using black-and-white reasoning like Knewton might be useful to the faculty in grading lots of tests quickly, once the students have acquired the skills, or it can be used for assessing a student’s knowledge level (albeit both inaccurately, as calculation errors beset the best of us). Without the discerning corrective diagnosis, Knewton fails as a teaching tool. And homework is a teaching tool.

    One way Knewton can be used as a teaching tool is to change its process. This is how I would reprogram Knewton: When the student chooses an answer, it is recorded to the teacher, who accumulates the aggregate results of the students. A student should be able to skip around to other problems and circle back, if they please. Another slot should be given at each problem: a comments section, where a stuck student types in the quick explanation (I got stuck at the _____ step) or enters their last step in symbols. The teacher can then see which problems are problematic (pun intended, you’re welcome), and then the next day the teacher can review these step-by-step on the board.

    Giving students more of the same problems they can’t complete correctly in the first place, WITHOUT telling them precisely where they went astray, is purely ridiculous, let alone frustrating. It may even ingrain bad habits or faulty thinking that you’re trying to fix in the first place.

    If you require perfect practice, you have failed as a teacher.

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