Dylan Wiliam and Ruth Sutton

I’m working my way up to strategies for embedding formative assessment in high school math, honest. Just before I do that, let me remind you that formative assessment isn’t always a “thing”. Formative assessment is about feedback. Ruth Sutton tells me she wishes she had called it “feedback for learning” instead. The word “assessment” has too many connotations that cloud our understanding of formative assessment.

The strategies I will share are based on the work of Dylan William, and his book Embedded Formative Assessment, in particular. His book, which contains many examples from math classes, outlines 5 key strategies for embedding formative assessment. Those strategies, in no particular order, are:

• Clarifying, sharing, and understanding learning intentions
• Providing feedback that moves learners forward
• Activating students as instructional resources for one another
• Engineering effective discussions, questions, and activities that elicit evidence of learning
• Activating students as owners of their own learning

In subsequent posts, I will outline specific classroom strategies for high school math (and very likely applicable to other subject and grade levels) that align with Dylan’s key strategies. All I have to figure out before I start doing that is whether I should share my best one first so that you keep checking back, but gradually see less and less exciting ideas. Or, should I start with a less interesting one and build to the best one I have?

Defining Terms

Before I get into what I think formative assessment looks like in secondary math, let me define my terms. Since I work for these guys, I will base my discussion around their definitions.

AAC Definitions

Summative Assessment

Assessment of Learning – assessment experiences designed to collect information about learning to make judgments about student performance and achievement at the end of a period of instruction to be shared with those outside classrooms (also called summative assessment; refers to performance data compiled as a grade)

Formative Assessment

Assessment for Learning – assessment experiences that result in an ongoing exchange of information between students and teachers about student progress toward clearly specified learner outcomes (also called diagnostic and formative assessment; refers to information not used for grading purposes)

For me, what makes something summative is that it is a judgment that gets shared outside the classroom. Schools that have bins weighted zero in an electronic grade book labelled “formative”, have crossed into a grey area here, in my opinion, partly because those zero weighted grades get reported out to parents, and partly because a grade is not an effective form of feedback (Butler, 1988)

Formative assessment is all about the feedback. It is feedback for the student. What does he/she still need to learn? It is feedback for the teacher. What can I do to help my class and my individual students master the curriculum?

Remember, formative assessment isn’t always a thing. I am going to spend the next few posts elaborating on classroom practices, rather than photocopyable things. Certainly that quiz you use mid-unit to see how your kids are doing but don’t count for marks is formative assessment. We all know that. What I will look at is other ways of getting feedback to and from students.

Formative Assessment

I haven’t posted anything here in a long time. It’s not because I’ve been working too hard (even though I have been working hard). It’s not because I haven’t had thoughts to share (because I have lots). It’s mostly because my work has shifted from math pedagogy to math assessment. For a long time, I wondered if anybody wanted to read more from me about assessment.

Here’s the thing, though. My understanding of formative assessment has grown dramatically since I started my new job just over a year ago. I now know that math pedagogy and formative assessment in math class go hand in hand.

When I left the classroom four years ago, my rudimentary understanding of formative assessment was that it was quizzes that used to count for marks that no longer counted for marks. Here’s a slide I’ve been using in presentations recently.

This slide pretty much describes where I’ve been for the past year or so. I have shifted from thinking formative assessment is a thing that I photocopy and hand out, but don’t count for marks. I now think of formative assessment as an embedded classroom process.

I’m going to use this space to describe some examples of what I think embedded formative assessment looks like in high school math classes. I’ll try to post a lesson or two very soon.

DragonBox

Browsing blogs last week, I came across this post about math apps. I had not heard of DragonBox before and decided to give it a whirl. At $5.99, it was pricey, but came highly rated.

DragonBox teaches algebra without calling it algebra. The object is to manipulate the two sides of the screen so that the box is isolated on one side. As you progress through the levels, the box eventually becomes an x.

I think the concept and execution are nothing short of brilliant. I worry that the questions I’m going to pose below may highlight a deep pedagogical deficiency on my part, but after working through all the levels, and introducing my 8-year-old daughter to the game, I do have some questions.

Me

At first, I had a really hard time understanding what was going on. I wanted to see the box and cards as variables, coefficients, and constants. They didn’t work exactly the way my algorithmic training wanted them to. I had a hard time separating my prior and totally engrained training from the game. Does the fact that I was taught equation solving in an algorithmic way, and the fact that I practiced those methods over and over again, actually hinder my ability to understand this concept at a deeper level?

My 8 Year Old

She loved this game. She progressed through the first two worlds and enjoyed it, despite the fact that she’s probably not developmentally ready for algebra yet. Her comments were that she didn’t see how this was math, and that she didn’t see what it was that she was supposed to be learning. I took one of the levels she had completed, and translated it into a concrete mathematical equation, and tried to walk her through it. It didn’t work. What’s the bridge between this brilliant app, and the concrete? Is the next step apparent to older children and to better teachers than me? I’m embarrassed to admit that I’m not entirely sure what my next step is, after my students play with this app.

Hello Goodbye

You may not know it, but the Amazing Race is big in Canada. It is so big, in fact, that they are planning on running a Canadian version. I’ve already started training and I am currently accepting applications from people who would like to join me on the winning team. But I digress…

Last week’s season finale (season 21, I think) included a challenge that asked contestants to put banners containing the words “hello” and “goodbye” beneath country flags, in that order, and in the language from that country. The contestants struggled and the challenge took over 2 hours, but one contestant tackled it systematically by trying all possible combinations. It was made for a math classroom. In the WNCP, this fits Permutations and Combinations from Pre-Calculus 12 (Math 30-1) in Alberta. It also fits Math 30-2 in Alberta. Here it is, in 3 Acts.

Act I

Play the video by clicking the photograph of one of the contestants working on the challenge.

With any kind of luck, the students will wonder how many combinations of the “hello” and “goodbye” banners are possible. They will require more information.

Act II

This video is longer than the Act I video, and by watching it closely, they should be able to determine that they are working with 9 country flags, and 20 banners with words on them. There are 2 extra banners.

Act III

I don’t have a video with the answer. It is fun playing with this problem, though. Initially, there are 1,216,451,004,088,320,000 combinations (20 x 19 x 18 x…x 3). By getting France and Spain correct immediately, the contestants reduced that number by a factor of 116 280, and now only have 10,461,394,944,000 possible combinations to try. If they had truly had to guess them all, they’d still be at it.

Enjoy. Fix my math.

Domain and Range Lesson

A few weeks ago, I agreed to teach a lesson on domain and range in a Math 10C class. I told the teacher I’d make it interesting and engaging. Shortly after making that promise, I realized I had no idea at all how to make it interesting and engaging. I did what I always do in those situations. I begged for help on Twitter.

It turns out I wasn’t alone in looking for ways to teach Domain and Range. Marshall Thompson admitted he was also interested in finding something. Don’t worry, Marshall, I’m about to hook you up.

Dan Meyer jumped in and was, as advertised, not all that helpful. And he spelled cheques wrong.

I had given up hope. I was about to plan a typical boring lesson. Then Peter Vandermeulen came through.

Peter’s link was to this file. It’s really all you need. It’s a nice, fun, compelling and engaging way to get at Domain and Range. Peter tells me he got the idea from a workshop in his district. I made some additions and modifications, and I’ll explain the lesson below. I’ll present it how I would do it if I ever did it again. I learned a few things.

Domain and Range Lesson (2 Classes)

Introduction and Hook – Pictionary

Run off the documents below. They contain some blank grids and lots of different types of graphs. Cut them out. The idea is that one student will be given a graph and have to describe it to a partner, who will draw it without looking at it. You make it tougher if you don’t let the describor see what the describee is drawing until the graph is done.

There are two ways you can go from here. Peter’s lesson plan suggests pairing students off and giving them each one graph and one blank. Partner 1 describes his graph, while partner 2 draws. Then they switch. I’d put a time limit of 1 minute on each drawing. When the time is up, they can look at the original and the drawing and see how accurately the drawer was able to replicate the graph based only on the verbal description of the partner.

I tried to make this competitive, like pictionary. I put them in teams of 2 and had them compete against another pair. I photocopied  the completed graphs on card stock and gave each group of 4 the whole set, shuffled and face down. Each student was given the sheet with the blank graphs on it. Then the students took turns pulling the top card, and describing it to their partner. The pair sitting out in a round had to judge and decide if the pair doing the drawing did well enough to earn a point. I tried my best to make sure that the person describing the graph couldn’t see what his partner was drawing. It’s much more challenging that way. We played 12 rounds of 1 minute each, so that each student got to describe 3 times and draw 3 times. The competition was fun, but the noise level got pretty high in the room. Peter’s way might be simpler, quieter,  faster, and every bit as engaging.

Graph Templates

• Word document so you can modify my graphs if you want.
• PDF file in case my graphs look terrible when you open them with your version of Word.

The Lesson

After the game, have a class discussion about what kind of words they were using to describe the graphs to their partners. Students will throw out words like arrow, axis, quadrant, stops, keeps going, points, curves, straight, ends, begins, lowest, highest, farthest right, farthest left and more. Their language leads nicely into domain and range.

Give every student two different coloured pencil crayons for the domain and range lesson. Walk them through several graphs from the game, and show the set notation appropriate to the various types (set of points, between two values, going on forever in one or both directions). All I did was make a quick notebook file with screen shots of some of the graphs from the game. What I tried that was new to me, was using the coloured pencil crayons. I asked students to identify the farthest left and right points, mark them, and then colour the x-axis in that same colour. Then I had them switch colours, find the highest and lowest points, and colour the y-axis in that colour. It really made the domain and range pop out for them.

Formative Assessment

After the brief lesson, give them a short sheet with 3 questions. In a 60 minute class, this will pretty much be an exit slip, which is what I called it. In an 90 minute class, you’ll have time for the next part. Use the exit slip to see who understood the lesson, and who needs more help. As students hand them in, you can sort them pretty quickly. I sort them into three piles – “Got It”, “Mostly Got It”, and “Didn’t Get It”.

Day 2

Practice Time (I use this instead of assigning homework) Group the kids according to how they did on the exit slip. Those in the “Got It” pile are given some higher level questions to practice. Normally, I just pull these right out of the student resource. Those in the “Mostly Got It” pile are given some basic practice questions, as well as some higher level practice questions. Those in the “Didn’t Get It” pile work with me. We will go over some more examples together before I turn them loose.

Closing Activity

Peter’s materials contained a set of cards that I used for a closing activity. Half the cards have domains and ranges on them, and the other half have corresponding graphs on them. I didn’t modify these at all, and used them as-is. Peter’s lesson plan suggests giving each student one card, and having them match up with the person who has the corresponding card.

I went a slightly different way. I copied these cards on coloured card stock and separated the kids into groups of 4-5. I gave them an entire set of cards and had them pair them all off, working as a group.

Closing Activity Materials

• Word Document of the cards
• PDF File of the cards

My Thoughts

Thanks to Twitter, I think I delivered a way better lesson than I used to do on this topic way back when I was in the classroom. I’d like to try this lesson once more, exactly as described here. It is ready to go. If you try it, let me know how it goes for you.

Book List

I haven’t read this book, but I intend to, based solely on the quote below. I’m intrigued.

There are some kids growing up with way too much adversity in their lives and what they need more than anything is protection from that adversity. And then we have other kids, especially kids who grow up in affluence, who just don’t have enough adversity in their lives. I think that is a hard message for parents to hear… In trying to protect them from adversity, we can sometimes be doing more harm than good.