Ask Marian

Transcript

An interesting thing happened to me. I was working with a teacher a couple of weeks ago, and we were working on children adding 3/8 to 2/8. And what we anticipated was that the kids would take a whole that’s 8 pieces and show 3 of them and 2 of them. But that isn’t what happened.

So, there was a child who showed 3 on top of 8 — he had 3 squares on top of 8 squares, then he had 2 squares on top of a different 8 squares — and he said, “\(\frac{3}{8}\) + \(\frac{2}{8}\) is \(\frac{5}{8}\).” So, the teacher turns to me and says “What should we do, because it looks like there’s 16 and not 8. But he did say the right thing, so what should we do?” And so, that was a real situation that happened recently. And it was interesting because I didn’t anticipate that the child would do that either, but he did.

So, if the child had looked at 3 compared to 8, and 2 compared to 8, and said 5 compared to 16, we would have had a problem because the answer isn’t \(\frac{5}{16}\). But that isn’t really what happened. The kid realized that if I have 3 of those and 2 of those, I have 5 of those. And that turned out not to be an issue. He had an unusual representation, but it was not an issue.

So, I think if students do things differently than you think they would or maybe you think they should, you can probably let it go if they can make sense of it. You obviously can’t let it go if they can’t make sense of it.

Transcript

A teacher was chatting with me and told me about something [that] happened in her classroom and she wanted an opinion. She was doing some work with patterns, and she had put up the pattern 4, 7, 10, 13 and asked kids to continue it. And one of the kids did something unusual. So, after the 13, he wrote 14, 18, 22, 26, and she was totally not expecting that, and she wondered just what kind of feedback would I give that kid if I were in that classroom teaching because it wasn’t what she expected.

So, I did respond to her, and one of the things that I suggested is that I kind of admired the student for his creativity. So, not only did he do something a little bit out of the box and a little interesting, but he actually showed understanding, as well, because essentially what he said to her was it used to be increasing by 3s, so now I’m going to make it increase by 4. And so he was really showing he really understands what patterns are about. They have to be sort of predictable, so first it was 3 and then it was 4 and presumably then it would be 5, and so on. And so, I talked about how we want to admire the fact that a kid does think like that.

But there are some worries that you have, as well, like what if there’s a standardized test; they won’t want you to think like that. So, one of the suggestions I made, and it’s something you might consider as well, is that, although we want students to understand that until you actually say out loud, “The pattern keeps going up by 3,” kids do have freedom to do all kinds of things. You might pose the question in one way and say, “Suppose you were considering this pattern: 4, 7, 10, 13,” whatever it was, “and [want to] continue it in an unusual way. What would you do?” and then another time say, “Suppose you were looking at the pattern 4, 7, 10, 13. What do you think most people would do?” So, you’re actually telling kids there is something that probably most people would do, but you still have room to do that creative stuff as well. So, I think you might want to play around with that on a fairly regular basis with patterns, where kids do understand you can go unusual and it’s okay, and it still shows understanding, but there’s also some conventional thinking too.

Almost by definition, no single approach to assessment is ever fair to everyone. Imagine that some kids understand the same ideas as other kids but have more trouble expressing themselves in writing than they do orally; is it fair that they all have to write? Or suppose some kids understand a particular idea, but the details of the situation (e.g., the complexity of the numbers) make them nervous and less able to express their understanding. Is that fair?

I believe that we have to be clear with ourselves about what it means to meet an outcome, standard, or expectation and allow for alternative ways for students to show us that they have achieved what is expected. My use of parallel assessments is sort of “in-between” allowing individual students their own unique way to express their learning (which should be our goal but might be hard to manage) and requiring the identical way from all students.

Sometimes students need space to figure things out. If another person (such as a teacher) is looking over their shoulder all the time, they may feel constrained in their exploration. I think this is as true for adults as it is for youngsters.

Of course, we must gather evidence to see how our students are doing, both to improve our instruction to them and to report to the system, but I’m not convinced we need to be doing it every single day for every single student. Taking a few moments to talk with a student might be more telling than the piece of paper you are collecting from them, since you’re not sure what led them to write down what they did.

Many teachers make a point of talking with a small group of students each day, rotating among students to ensure that data are collected about every student over the course of a week.

In this way, teachers aren’t hovering, and they’re gathering more reliable information.

Transcript

A teacher was telling me that she had been at a session, and they had told her that it was important not to write rates as fractions—that rates and fractions were different things. She was a little confused because, in her experience, you would write a rate, like if you wrote 90 kilometres per hour, you might put 90 over 1, and she didn’t know why it was a bad thing to do.

So, I was explaining to her that someone is being careful because the fraction 90 over 1 is actually a number. A rate is not a number. A rate is a comparison, but a fraction is a number. So, you don’t want to say a comparison is a number because it’s kind of a different thing.

But the trick is it kind of works the same way. So, just like we say that an equivalent fraction to one-half is two-fourths, we could say that an equivalent rate to doing one thing every two minutes is to do two things every four minutes. So, it’s really the same kind of mathematics that is used but someone is being very careful to help kids focus on the fact that a fraction is a number and a rate is not.

I think there is value in students learning not just to execute a lot of steps but also to show that they know what they are doing. Sometimes, a closing sentence accomplishes that. But maybe instead of the typical closing sentence (e.g., “The solution is …”), we might encourage students to write a sentence about the solution process. It could be about something they found tricky or about when they had an “aha” moment, or it could be about whether they thought it was like another problem they had already done. It could even be some advice they’d give someone who had a similar problem to solve.

I think it’s a great idea, but I probably would not tell students which is which level. My reasoning is that sometimes students have decided they are weak and won’t even try the hot and extra hot, but maybe they would be successful. I’d prefer just to offer the choices with no labels, making sure that the easier ones are sometimes first, sometimes last, and sometimes in the middle.