Effort -> Learning

The little things

I think my most important goal as a teacher is for students to learn that effort leads to learning. This is one place where teachers need to show, not tell. I can tell students to put in their best effort as many times as I like. If students’ everyday experience is that when they put effort in, they don’t learn, they won’t believe me.

This poster isn’t going to cut it

An Example

Here’s something I’ve done over the last few weeks.

I’m just beginning to teach circumference and area of circles. The area of a circle is especially tricky for 7th graders.¹ One piece of notation students need to be fluent in is squaring numbers: knowing that 4² is 16, not 8. Students first learn this notation in 6th grade, but many forget, so it’s worth a review.

A few weeks ago I did a first round of review on squaring numbers. I modeled a few examples, had students try a few on mini whiteboards, gave feedback, and then handed out a worksheet for some practice.² This whole thing was fast: maybe 5 to 7 minutes total.

Two days later I carved out about the same amount of time. This round I began with a quick assessment on mini whiteboards to see what students remembered. Much of the class did remember, and the assessment helped me figure out which students needed a bit of extra scaffolding. Again, we did some mini whiteboard practice and paper-and-pencil practice, this time with more targeted support based on the assessment data.

The next week we did it again. This time almost every student got it right the first time. Again, some assessment, mini whiteboard practice, and then paper-and-pencil practice.

At this point I started putting questions about perfect squares on our daily Do Now. I gave some extra support to the final few students who needed it. Then we had a bunch of days to improve fluency before we began circles.

All that didn’t take tons of time! A few 5 to 7 minute rounds of modeling and practice, then a few questions on our daily Do Now. That’s it.

Effort → Learning

My impression is that what I’ve just described is unusual. More common is a quick review of exponents just before teaching the area of a circle. I find that multiple rounds of spaced-out practice and feedback are the best way to get every student fluently squaring numbers. This is important! Fluently squaring numbers frees up mental space for all the other parts of finding area of circles that are hard.

This structure helps students learn the area formula. The extra rounds of practice will also help students when they get to more complex exponent and root problems in 8th grade.

But more important to me is the idea I mentioned at the start of this post: teaching students that effort leads to learning. Here is a place where I can take something that’s fuzzy for a lot of students, and get every student confident in that skill. I can show them that their effort leads to learning.

A Different Approach

Here’s a different approach to squaring numbers, an approach I’ve used before.

The first day of finding area of circles arrives. I do a quick reminder of how to evaluate exponents. Some of my students remember right away and have no problem. Others have a faint idea that they’ve been taught this before, but they keep telling me that 4² is 8. I try to address it, but there isn’t much time. We need to move on to circles. We dive into the area formula, and some students keep getting questions wrong because they evaluate the exponent incorrectly. They feel frustrated. They’re trying, but they just can’t get it right. Even when they get the exponents right, they seem to mix up something else in the formula. These students are putting in effort, but the learning doesn’t stick.

I have seen this over and over again in my teaching career. I’m not describing all students. Much of my class will be successful with this type of teaching! But I’m describing a group who is consistently in the bottom 20%. They learn a clear lesson from school: even when they try their best to learn, that effort does not reliably lead to learning. Many stop trying altogether, or adopt an attitude of learned helplessness. Teachers put some nice posters on the wall and we repeat the message that perseverance is important. But it can all feel pretty useless. Those messages are outweighed by the sum of students’ everyday experience.

Don’t Leave Students Behind

Leaving a bunch of students behind is hardwired into many approaches to teaching. I often hear people say that teachers should aim for 80% mastery before moving on. If you’re happy with 80% mastery, great. Ignore this post. This post is about how to support that final 20%. If we accept 80% success, we teach that final 20% a very clear lesson: even when they put in the effort, that effort is unlikely to lead to learning.

Taking the time to teach squaring numbers well is just one little example of how I try to help students see that effort leads to learning. Show, don’t tell. Take the time to help every student learn. Don’t stop at 80%. Assess students. Give a bit more feedback. Provide more practice. Structure regular retrieval practice to remind students of what they’ve learned. I do stuff like this all the time, with all sorts of little micro-skills that matter in math.

There’s more to learning math than making sure every student knows how to square numbers. In a few weeks my students will take the state test, and they’ll see questions like this one:

That’s a hard question! I’ll do my best to teach my students how to solve questions like it. I can’t guarantee every student will be able to solve tough questions like this one.

What I can do is guarantee that every one of my students knows how to square numbers accurately, and lots of other similar micro-skills. Those micro-skills are the foundation. If students can’t square numbers, they don’t have much of a chance of getting that tougher question right. Lots of math teachers look down on skills like squaring numbers. It’s rote learning, it’s not relevant, it’s boring. I disagree. Math is worth learning.

But maybe more important, getting those micro-skills right shows students that they can learn math. Successful learning builds confidence, so students are willing to try harder problems like the circle cutout problem above. I’ve taught lots of students who simply don’t believe they can learn. They’ve tried in math, and years of experience have taught them that despite that effort, they just aren’t very likely to learn. My goal is to show students that they can learn, that I’m here to help them, that I’m not going to move on as soon as most of the class gets it.

I’m not a perfect teacher. I’m not successful with every student, every day. But I try, as often as I can, to pick out specific skills and provide a bit of extra practice and a bit of extra feedback, until every student learns. That extra time sends a message to the students who often feel unsuccessful in math class: they, too can learn, and I am here to make sure that happens.

1. I find that it’s often hard for adults to understand why finding the area of a circle is challenging for students. Here are a few reasons. First, area and circumference of a circle are some of the first formulas students learn. Sure, there are other formulas earlier on for the area of rectangles and triangles or the volume of rectangular prisms. The area formulas are different because this number pi comes out of nowhere. Finding the area of a rectangle makes sense: there are two numbers that delineate the rectangle, we multiply them, that’s it. But with circles, where does the pi come from? Do we use diameter or radius? Why? Why is something squared? There are reasons for all of these things, sure, but that’s a lot for students to keep track of. It all adds up to a much more abstract skill than the formulas students have seen before.

2. One note about this worksheet: my goal is for all students to complete the left side, which is focused on squaring numbers. The right side has some tougher problems with larger exponents, negative numbers, and equations to solve. Those problems are there to provide an additional challenge. Practice is good for everyone. For students who are already confident, they get to apply their knowledge in a few new ways to stay challenged.