Explicit Success Criteria

In the video I linked yesterday, John Hattie identifies a few of the most important factors for student success in school. I will take a light-hearted look at a few of them. In particular, I’ll be talking about assessment for the next few posts. But before I get there, I’d like to visit the notion of explicit success criteria.

Can you state the Fundament Theorem of Arithmetic? Can you prove it? It’s a fair guess that only a very small number of readers can answer either of these questions. I’ll state the theorem for you:

Any positive whole number greater than 1 can be uniquely written as a product of prime numbers.

That’s not so hard, provided you remember what a prime number is, and if you are prepared to accept the convention that a prime number is a product of a single prime—itself. For the numbers that aren’t prime, it’s easy to come up with examples: 6=2×3; 45=3x3x5. Oh yeah, and we have to agree that order doesn’t matter: 2×3 is the same factorization as 3×2.

But notice what just happened. Just stating this very simple result of arithmetic, we needed to set some rules straight. We needed to define a few key terms. We needed to make sure that we were all talking about the same thing. It isn’t too much of a stretch to see that the same kind of “stage setting” is important in the classroom.

Now prove the theorem.

Back already? Did you prove it? Why not? I am being serious here: why did you not prove the theorem?

For most people the answer is either you didn’t care to prove the theorem, or you had no idea how to begin. Not only that, you probably wouldn’t know how to tell if you’d done it correctly. And that’s what we’re on about today.

Think back to when you were a student (or if you are any kind of teacher, think to some of your own teaching experiences). How often was the instruction given to “write a story”? Or “research” something? Or “solve a problem”? And how often did you find that some students knew exactly what to do, while others sat around uncomfortably, waiting for help? Or worse yet, put their heads down and didn’t even try? And how often is “not trying” interpreted as a problem with the child’s character/work ethic/ambition etc.?

Allow me to make the following speculation. Most of the waiting for help and not trying is not because of character weakness, or even fear of failure. It’s a result of not knowing the rules of the situation. Students don’t do something because they don’t know what to do, and because they wouldn’t recognize a promising strategy unless you told them that it was promising. These students are in a sense “incompetent”. But it’s a treatable incompetence. And not by punishment, lines, detentions, ridicule or phone calls home. None of that will come close to the real issue.

Any positive whole number greater than 1 can be uniquely written as a product of prime numbers.

Did you notice, by the way, that to prove the theorem you have to prove two things? Take a second and see if you can spot the two propositions to be proved. Don’t worry if you didn’t notice; why would you know? You’re clearly not stupid and you are curious enough intellect to be reading a blog like this. You don’t know because it’s not part of your educational background and experience. If the point of this blog post were to teach you how to prove the theorem, it would be a complete bust, mainly because I have not adequately set the stage for you to meaningfully tackle the task. I have provided no indication of what successful proof might look like, and I have provided no “scaffolding” for you to move from your present knowledge to the proof. In fact, I have made no attempt to understand what you currently can do with regard to the relevant mathematical knowledge for pursuing the proof (more on this another day). Thankfully, my goal today is merely to get you to think about academic stage setting in general.

It is a rare student who wants to fail. It is a rarer student still who wants to be perceived as unintelligent. And it is almost equally rare for a student not to desire approval from her/his teacher. Yet we are often lulled into believing these things about our students. Students want to be successful; many are simply not very good at being successful. And I agree with Hattie that one of the easiest and most powerful ways to have students be successful is to help them to understand what counts as success before they begin their tasks, not after.

Any positive whole number greater than 1 can be uniquely written as a product of prime numbers.

The two things to be proved are:

  1. every positive whole number can be written as a product of prime numbers; and furthermore
  2. the product is unique

You are now much closer to being able to begin. But before you can begin, you need strategies. This (sadly) isn’t the place to teach strategies, but I’ll give you a name: proof by contradiction. In each case, you assume that what you are trying to prove is false, and then show that the assumption leads to a logical contradiction. You then conclude that your assumption is false, i.e. that what you are trying to prove is true.

Let’s bring this back to the classroom. Any teacher would be thrilled if every student willingly and joyfully participated in every activity. They way into participation includes clear instructions, clear goals, a grasp of possible strategies, and the means to evaluate success. I will flesh this out in a few more posts, but if I have convinces you of this much, we’re well on our way.


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