Positive and negative criticism

I’m preparing to teach a class about mathematical proofs. I want to teach my students to write proofs, but I also want to teach them to think critically about their own work.

This is really important. If you can understand and check mathematics on your own, you’ve earned a kind of intellectual self-reliance. You have the ability to determine the value of your own work, without having to rely on a teacher or textbook. It’s a particularly beautiful kind of freedom.

So how can my students learn to criticize themselves? Firstly, how should they criticize themselves? I want them to reflect on their errors, understand them, and learn from them. The criticism shouldn’t be scary. I don’t want them to be so afraid of making mistakes that they aren’t willing to take risks and experiment with different writing styles.

Here’s a simplistic way to think about this. There are two kinds of criticism, positive and negative. It might be better to call them “growth criticism” and “set criticism,” following the idea of “growth and set mindsets” from education science (see the work of Carol Dweck). Positive criticism is helpful, important, essential to your development. It gives you strength and energy and propels you into the future. Negative criticism is hurtful and destructive. At best, it makes you too afraid of making mistakes, slowing you down and making you less productive. At worst, it saps your energy and will, leaving you paralyzed, disengaged, and unable to produce anything at all.

Both kinds of criticism start by identifying the mistakes and areas for improvement. They have that in common. What distinguishes them is the interpretation, the attitude about how to think about errors and shortcomings, and what to do next.

To do negative (set) criticism, you turn each mistake into a weapon and make it hurt, the way you might train a dog. A bad grade means that you are a failure. So you work towards a good grade in order to avoid that pain. When surrounded by criticism of this kind, you might shut down and give up, or try very hard and punish yourself for every little mistake. Negative criticism doesn’t guarantee you’ll fail, but it certainly makes it harder to succeed.

To do positive (growth) criticism, you welcome mistakes because you can improve by learning from them. Your mistakes become signposts that tell you what to work on next. A bad grade means that you aren’t done yet. So you work towards a good grade because that’s the eventual finish line. When surrounded by criticism of this kind, there is a rising tide that pushes you to achieve the best version of yourself. It is easier to take risks and try new things, because every screw-up brings you one step closer to success. Positive criticism doesn’t guarantee you’ll succeed, but it gives you the best possible shot at success.

In short, criticism and critical thinking is an immensely powerful tool, and that tool can be used for evil, or it can be used for good. And in reality, most criticism falls somewhere between these two extremes.

The other important point is that the critic and the person being criticized both play a role in making the criticism positive or negative. So it’s not enough to think about my own tone. I also have to encourage my students to see their errors as opportunities for growth.

How can I structure this class so that we lean as much as possible toward positive criticism? One idea is to assign very few problems, with the expectation that they will be turned in multiple times. So instead of the grade being “correct” or “wrong,” the grade is either “done” or “not done yet.” Another idea is to assign homework that asks the student to be the critic for a proof. Then I could base the grade on how constructive their criticism is.

By the way, this dichotomy doesn’t just apply to math education. It’s important to think critically about relationships too. You need to give your partner positive criticism, engaging honestly with their issues, forgiving mistakes, and using the mistakes as an opportunity to improve. Punishing your partner for mistakes sometimes seems like a good way to help them improve, but in practice, it only slows or reverses their personal growth.

The gist of positive criticism is that mistakes are not completely bad. We strive to avoid them, of course. But every time we mess up, it is a springboard that propels us forward, teaching us more about ourselves and other people. As we go on, we make fewer mistakes over time, but we never attain perfection, and we don’t really want to be perfect, anyway. Every mistake is a tool that helps us build a better life.


One thought on “Positive and negative criticism

  1. Stan Kolpakoff

    I am very thankful to Professor Malkiewich for the foundations of mathematical proofs which he taught me in my 3rd year of my undergraduate studies at UIUC. I am now pursuing a PhD in Economics and feeling well prepared for mathematical rigor which is a must in this program.


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