How much of a class should be conceptual and how much should be procedural?

This question came up at a professional development session I led last Saturday. I tried to answer it by sharing my daily class routine and giving some examples of how I’ve taught for conceptual understanding. But as I left the session, I found myself wanting to consider the question in greater detail.

Ratio is a hot topic in pedagogy these days. Teachers think about the ratio of student talk to teacher talk. They think about the ratio of lower-order thinking questions to higher-order thinking questions. Ratio is important because it ensures differentiated instruction.

But as I tried to figure out an appropriate ratio of teaching for conceptual to teaching for procedural understanding, I realized that it couldn’t be done. Teaching for conceptual understanding is like being a vegetarian: Vegetarians don’t eat a high veggie to meat ratio; they stop eating meat. Not only that, most vegetarians stop eating meat for a reason, and that reason impacts many of their choices, not just what they choose to eat.

The same commitment applies to teaching for conceptual understanding. If you believe that math is something that can and should be understood by all students then you give up teaching rote procedure. Conceptual understanding becomes the foundation of everything you do and all the choices you make. Sure, just like vegetarians might have a little fish or egg now and then, the conceptual teacher might have to do a little “drill and kill.” But the basic belief about math making sense is always present.

This doesn’t mean that a teacher who is new to teaching for conceptual understanding can’t shift approach gradually. It means that teaching for conceptual understanding requires more than a five minute addition to the class routine. It requires a shift in mindset. To get yourself in that mindset, let me share a few things that I often see teachers who are committed to conceptual understanding do:

1. Introduce every topic in a way that relies on student sense-making.

“See how this looks like an array? That means it’s multiplication!”

2. Correct every misconception by reminding the students “why.”           

“You forgot the zero. Remember how this 3 is really 3 tens? When you multiply it, it’s still going to be tens.”

3. Encourage students to ask questions and make connections.

It’s almost the New Year. Lots of people make resolutions about their diet. Maybe this year it’s time to go on a different kind of diet.

Making Best Practices Automatic

School is on Monday. I have been going to professional development workshops, planning with my grade level team, discussing details with my principal, and producing: posters, seating charts, homework, activities… The list goes on forever.

Then last night I woke up in the middle of a deep sleep, suddenly realizing, “Oh my gosh, the students will be there.”

This might sound like a silly realization, but it takes a real mind-shift to start thinking about “the first day of school” as “the students’ first day of school.” If you do start thinking about it this way then you start to think about how intimidating it can be. And how exhausting. And how boring. Really, if we’re honest, about 99% of what we do on the first day is going to be blocked out by all the overwhelming emotion that comes with just showing up after a long summer break.

It is these concerns that make me so grateful for my lesson planning template (download sample here). I imagine that at first glance it would appear pretty complicated. It started out a simple place to list the objective, the agenda and the materials. But  as I have added new “best practices” to my repetoire, I have added to the template. For me, it is a constantly evolving document that reminds my present self what my past self found was important. When I make changes, it becomes a gift to my future self.

You will see that I have broken things down into 5 minute chunks. That’s because I used to be really bad at budgeting time and keeping the lesson flowing. You will see that I put a place to record “sparkle,” the place in the lesson where something magical will happen. Every lesson should have some sparkle. You will see that I broke my 5 minute chunks into what the teacher does and what the student does. I did that to remember to plan exactly how the students would move between activities.

The part that stood out to me today was the reminder to highlight my movement breaks. The students are going to need that.

Thank you, past self. All last week, I kept thinking about my first day back. Now it’s time to put myself in my students’ shoes. It is their first day back.

Coming Back For More

I am getting ready to teach place value for the seventh year in a row. But I am totally not bored. In fact, I am pumped that I’ve been able to collect better and better ideas for teaching the topic as time has gone on. It makes me proud. It makes me feel like a professional. And it makes me want to keep teaching.

For example, I used to give my students digit cards to do place value games. They were just cards with single digits. They worked. My students learned.

But…

…it is just as easy and MUCH more meaningful to use value cards.  They are cards with multiples of ones, tens and hundreds that can be stacked to make multi-digit numbers. With value cards, students have to think about a digit’s value as they are building numbers like 395 or 1,026. They are learning the purpose of place value while developing the understanding needed to write numbers in expanded form. 

I don’t remember who shared the idea of digit cards with me, but I am grateful. The idea didn’t just help my students. It got me excited to keep learning and growing as a teacher.

And now I’m attaching my digit card templates for someone else out there. Simply print them out so that the different place values are different colors, and go. I hope they get you excited to teach place value one more time.

Math on the Mall

It’s been a really long time since my last post. Did I really not take a single photograph of my students since Halloween? There is a dark period in teaching that lasts from November to April of every year. I still teach my heart out, but I can’t take pictures of it.

The bright period starts as the school year is wrapping up. Suddenly the exhausting days don’t seem so bad because a break is coming. I start to get excited about making changes for the next group of students. I try to work on things that I will need when I am tired. It’s like stocking up for the winter.  I HIGHLY RECOMMEND IT. Last year, I wrote my February decimal unit over the summer. I am convinced my students would not have learned decimals if I hadn’t.

I also start to take more pictures. Like these from our field trip to the National Mall…

If you’re like me then every time you hear about something like math on the mall you think vaguely about how learning is so much better in an authentic context. But it goes at the bottom of the To DoList because it is non-essential. I wish it didn’t.

Math on the mall gets students to talk about mathematics as if they actually care about it personally. I was amazed to hear my students eagerly discuss the angle of fountain jets.

Math on the mall is memorable, and it engages students with many different interests. There is plenty to see inside at the museums and outside at the National Sculpture garden. We explored lines of symmetry and tried to estimate the volume of the sculpture shown below.

One of my colleagues has three different versions of a sheet she’s written for various visits to the mall. They have a rich variety of problems for different locations (1, 2 and 3). There is also this math on the mall sheet for highschool students.  I adapted part of the MAA field guide because it was a bit difficult for my kids. I also incorporated ideas from an NCTM article about why things are shaped the way they are. This is the student sheet I compiled:

We did the sheet on our walk from the American History museum to the US Capitol. We did not get through it all, but we had a lot of fun on what we did. It is probably easier if the teacher shows the students a wrench and a piece of paper with different shapes cut out of it. I had to print paper wrenches and paper lug nuts. 

Washington, DC provides so many free opportunities for learning in many different subjects. It just takes that end of year burst of energy to take advantage of it.

G.E.T.S. Part Two

The essential, enduring understanding of algebra in middle school is that a graph, table and equation are different ways to describe the same function. That is why I have dedicated 3 posts to it, and that is why I’m going to explain the graph part of it here.

But first, let me entice you by saying that when you teach this lesson, you get to wear a costume:

This costume obviously took me about 2 minutes to put together. That’s so I could spend the rest of time making photocopies.

The Lesson

Before you get to graphing, students must already understand what a function is and how it can be represented using an equation and story (I discuss this in a previous post). They should also know how to graph coordinates. Then it is quite easy to take the points generated by the “FUNction machine” and graph those.

Students do not seem to struggle going from number pairs to a graph. They seem to struggle going from a graph to number pairs. This is where the acronym, G.E.T.S., is important.

I want to be able to say: “You see a graph of a function, what do you need to do?”

Students should say: “Find the equation by making a table.”

Of course, it’s really important for students to look at graphs on a bigger level before they do this or they won’t really understand why we bother with graphs at all. Here’s the plan I use:

Day 1: Students write functions as equations and tables. (Again, I wrote about this in a previous post.)

Day 2: Students make observations of graphs without numbers to see how graphs describe global behavior in a way that points on a table do not. This is a great way to informally introduce the idea of slope and y-intercept.

Day 3: Students use points to make a graph. They use a graph to find points. They say “G.E.T.S” over and over so many times it becomes drilled into their brains.

From here students can begin to use G.E.T.S in problem solving. I have created twelve pages of practice because we did one worksheet a week for the rest of the year after this unit. When students really understand functions, they can do this sweet enrichment activity that brings it all together through geometry. That is when your heart will really get racing.