Archive for the ‘Assessment’ Category

In the spring, I was working on a series of posts about formative assessment in math class. I got sidetracked by starting a new blog, and kind of let it drop. This morning, however, I read this great post from Max Ray about questioning, and it brought me back to formative assessment.

One of Dylan Wiliam‘s 5 Key Strategies is “engineering effective discussions, questions, and activities that elicit evidence of learning.” From Dylan William’s book, Embedded Formative Assessment:

There are two good reasons to ask questions in classrooms: to cause thinking and to provide the teacher with information that assists instructional decision making.

Max is right. Good questions that cause thinking in math are tricky. Most of us lean towards asking recall and simple process questions. With practice, we can learn to throw out deeper questions as easily as we ask recall questions.

Max’s post contains a number (26 to be precise) of great questions that prompt discussion. My two favourites are:

  • What do you notice?
  • What do you wonder about?

Questions like the two above feel safe to students. They don’t have to worry about being wrong. They can think and respond without fear.

Sometimes, questions can be improved by turning your lesson around. I spoke to a teacher last year who was working on 3-D shapes with his class. He had the nets all copied and ready to have the students cut out, fold, and tape. It seemed more like a lesson on cutting, folding and taping, so he scrapped it. Instead, he brought out models of the 3-D shapes, and asked the students to create the nets that could be folded up to make the shapes. It ended up being an incredibly rich discussion.

One of my favourite conversation-extenders comes from Cathy Fosnot. When a student responds to a traditional question, extend the conversation by simply stating, “convince me.”

The more we can engage students in conversation with each other through effective questioning and planned activities, the more likely they are to come to their own understanding of the topics.

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I’ve been writing here about getting students to practice in class. Many of our students don’t (and probably shouldn’t) be doing practice math questions at home. We need to build opportunities into our lesson for them to do some questions. Kate Nowak has provided us with two great ways to get students to practice some questions in class in more compelling ways. This practice is formative assessment. I would classify both of her activities as activating students as instructional resources, using the language of Dylan Wiliam.

I have nothing to add to Kate’s work, other than to tell you I’ve used this stuff and it works. I’m just pointing you in its direction and making a link between what she has shared and the formative assessment I’ve been writing about in my recent posts.

Check out Kate’s Row Game and Speed Dating. She has several already prepared that fit our WNCP curriculum. It looks like she has been collecting other people’s Row Games here.

Both activities are easily differentiable, and allow kids to practice their math. They have built in accountability because students are responsible to each other. They are both fine examples of embedded formative assessment.

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An earlier post discussed how to use exit slips as practice. A great way to activate students as owners of their own learning (another of Dylan Wiliam’s 5 key strategies) is to use exit slips to have students self-assess.

I was in a classroom last week where the teacher had prepared a review activity for his Math 20-1 (Pre-Calculus 11) students on radicals. He prepared 5 stations and had the students set up in groups. He chose to group them so that each group had a blend of abilities. The three groups at the front of the room completed station 1 (converting from entire radicals to mixed radicals) while the three groups at the back of the room completed station 2 (converting from mixed radicals to entire radicals). Each station contained an envelope with 6 questions of varying difficulty. After a few minutes, when students were done, the groups got up and switched stations.

Once they had done stations 1 and 2, the entire class did station 3 (adding and subtracting radicals) simultaneously. Groups that finished were instructed to get up and circulate and help those that hadn’t finished. Once the class was done station 3, they split up again. The front of the room did station 4 (multiplying radicals) while the back did station 5 (dividing radicals). Once completed, they all got up and rotated to the last station they had left. The 4’s moved to 5 and vice versa.

This is a nice, non-worksheety way to have students complete some practice before a summative assessment. It allows students to converse and help each other (feedback and activating students as instructional resources as defined by Mr. Wiliam).

The teacher greatly enhanced this activity by making students owners of their own learning with an exit slip.  He prepared an exit slip for them to track their progress through the 5 stations. As they completed each station, students self-assessed as Excellent, Satisfactory, or Limited. Based on their self-assessments, they left the class knowing exactly what, if anything, they still had to work on before the summative test.

This use of exit slips is an effective way to activate students as owners of their own learning. It allows them to articulate precisely where they are still struggling. The resources used in this lesson can be accessed below. Video of the lesson in action is posted on on the AAC website.

5-Station Cards

Exit Slip

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I take no credit for this one and I’ve never tried it. It’s all Rick Wormeli. He shared this strategy in a session I attended several years ago. It would certainly qualify as activating students as instructional resources for one another.

Rick works with one student a day ahead of the next lesson. He makes sure that student understands what is going to be taught the next day. That student becomes the “expert” on the material during the next day’s lesson. Students who have questions are expected to go to the expert first.

Simple. Shortest blog post I ever wrote. This strategy would be effective in showing students that support and help doesn’t always need to come from the teacher. Remember, formative assessment is about feedback. That feedback doesn’t always need to come from the teacher.

Rick has a great story about what he does when it is a weak student’s turn to be the daily expert. I won’t ruin it for those of you who haven’t seen him yet. And if you haven’t, I highly recommend him. He’s an engaging and compelling speaker.

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Providing feedback that moves learners forward is another of Dylan Wiliam’s 5 key strategies. Research has shown that feedback in the form of comments only, motivates students to learn more and ultimately improves their grades. Feedback in the form of a grade actually de-motivates students and has no effect on their performance. (Butler, 1988)


In math class, the only things my students get back with grades on them are summative assessments. I have significantly reduced the number of summative assessments I use in high school math classes. Most courses are adequately covered with 5 to 10 well constructed summative assessments.  Everything else goes back to the students with comments only.

The feedback I provide instead of a grade varies by student needs. Some students simply need me to circle the place in a problem where they started to go wrong. Those students can take it from there and correct their work with little direction from me. Others need some comment on what the next step might be. I try to provide as little scaffolding as I can get away with, while still letting them have enough to move forward. It’s a fine line. I want them to take ownership without me giving them everything. I don’t spend a lot of time on written comments. Most of the time I look at things quickly and arrange my class so I can talk to the students personally, as I described in the previous post. Those kind of groupings also allow students to get feedback from each other instead of just from me.

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There are several ways to use exit slips as formative assessment tools. One way is to simply have the students complete 2 or 3 questions based on the lesson that was done in class. I use exit slips in this manner to avoid giving homework. I believe that some practice in math class is necessary. There are certain things I need my students to be able to do, and some students need to practice these things. I do not, however, believe that students should be practicing these things at home. Home is for family, community soccer, dance class, piano lessons, and all the other important things that our schools are eliminating.

I’m going to tell you a secret now. The students who don’t need to practice math will go home and do every single question you assign. It’s a waste of their time. The students who need to practice math will go home and do none of the questions you assign. Then you will argue with them, call their parents, and devise elaborate schemes to collect and grade homework. It’s a waste of your time. If I am not going to assign homework, I need to build places into my lessons for students to practice a little.

I do not grade these exit slips. I do not put any marks on them. I look at them and get feedback about how my students did with today’s material. I sort them quickly into three piles: Students that got it, students that partially got it, and students that didn’t get it at all. Based on Dylan Wiliam’s 5 key strategies, I would classify this use of exit slips as providing feedback that moves learners forward. Based on how the students do on their exit slips, I can adjust my instruction as necessary. I start the next day’s class with activities that allow the students also receive feedback.

Here’s how the old John’s math classes usually looked (based on 80 minute block schedules).

  • 20 Minutes – Go over homework questions on the board that a few students had tried. Some students listened and copied down the solutions.
  • 40 Minutes – Teach new material.
  • 20 Minutes – Students had time to work on questions. Those that didn’t finish were expected to take their math home and complete the questions.
  • Wash, rinse, and repeat 80 times per semester.

The old John typically assigned 10-15 homework questions. Very few students ever did more than a couple of them.

Here’s how exit slips as practice can really activate students, involve far more students in the practice component, and frankly, be a much more efficient use of class time.

  • 20 Minutes – Students are grouped based on the previous day’s exit slips. Those that got it are sitting in small groups working on a few extension and/or application questions. Those that partially got it are in small groups correcting the errors on their exit slips and then working on a few practice questions that build to the extension and/or application questions. Those that didn’t get it are in small groups working with me. We do some re-teaching as necessary, and some practice. I don’t make up those questions. I just assign them from the textbook like I would have before.
  • 40 Minutes – Students learn something new. (Notice that the old John “taught” something new, but the new John gets students to “learn” something new.)
  • 20 Minutes – Students complete an exit slip with 2 or 3 questions based on what they were supposed to have learned. These slips are sorted quickly and used to begin the next day’s class.

In a method like this, every student does between 3 and 8 practice questions. That’s far more practice than I used to get them to do when I assigned homework regularly.

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Another of Dylan Wiliam’s 5 key strategies is clarifying, sharing and understanding learning intentions. Students can’t hit a moving target, so we need to make sure our outcomes (standards, for my US friends) are clearly defined for them. This process is sound formative assessment. Remember, formative assessment isn’t always a quiz that doesn’t count for marks. It can also be a classroom process. Are you tired of me saying that yet?

I used to give my students outcome checklists at the start of every unit. These were pulled verbatim from the Alberta Program of Studies and the Assessment Standards and Exemplars. I have created one for Math 30-1 (Pre-Calculus 12) Exponents and Logarithms as an example below.

As we work through the unit, I classify everything. Every single question I do and every single question the students do is labeled by outcome number and either acceptable standard or standard of excellence. This labeling occurs during class, on quizzes, in the textbook, on exams, and anywhere else we encounter questions.

I was doing this before I heard of standards based grading, and my use of these outcome checklists is similar, though not nearly as in-depth. And although I still reported holistically (different than in SBG), these outcome checklists did allow my students to articulate exactly where they were struggling. Saying, “I’m having trouble with outcome 8 at the standard of excellence” is a way better way for students to articulate their difficulties than the more traditional and vague, “I don’t get it.”

Students were expected to fill in the check boxes when they felt they had mastered the outcome at the various levels. By doing so, I was clarifying the learning intentions for them, and they were activated as owners of their own learning (another of Dylan’s 5 key strategies). It was a great way for us (student and teacher) to give each other feedback about progress to the clearly defined learning outcomes.

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The first strategy I’d like to share for embedding formative assessment (Remember, formative assessment isn’t always a thing. It’s feedback for learning) is referred to by the consultanty-types as “two by four”. While generally working in the field of consulting, I should point out that I’m not much of a consultanty-type. I’m not a fan of fancy processes and the equally silly names we give them. So, for classification purposes only, I present to you the two by four.

One of Dylan Wiliam’s 5 key strategies is to activate students as instructional resources for one another. This strategy is my go-to one, mostly because I’m trying hard to embrace the philosophy of the revised program of studies, which suggests I should orchestrate experiences from which my students can extract their own meaning. When I active them as instructional resources for one another, they learn from each other, rather than from me writing notes on the board.

This strategy is really simple, and can be used to teach just about anything I can think of in our curriculum. Two students learn something through some kind of structured activity. They check in with two more students who learned the same thing and compare strategies and results. When they agree, they move on.

Let me tell you how this worked in a class I tried recently. I was invited to model formative assessment in a Math 20-1 class (Pre-Calculus 11 elsewhere in the WNCP). The topic of the day was multiplying radicals. In my classroom days, I would have written some rules on the board, done some examples, and then given an assignment for practice. The students would not have been activated much at all.

For this lesson, I built a structured activity that I hoped the students would work through and learn what I needed them to learn about multiplying radicals. That activity is posted below.

Some notes about how it went:

  • First of all, I should point out that this does not need to be photocopied and handed out. It would work just fine as a set of questions that the teacher poses to the class. I ran it off, because I was going into a stranger’s classroom and I wanted to make it as simple as possible for the students.
  • As the students worked, I circulated and listened in. I helped when needed and as little as I could. I regrouped them at the appropriate times and with the appropriate people. By the end of class, almost all of the students taught themselves how to multiply radicals. I gave exit slips at the end (another formative assessment strategy) and the exit slips showed me that most students had met or exceeded my expectations. Two students didn’t manage to learn the material. This next part is huge. Wait for it…
  • It wasn’t my class. I didn’t know many of the student names. Despite that fact, when I saw the two exit slips that showed little or no understanding, I described to the classroom teacher exactly who I thought those two students were in terms of where they were sitting. I nailed it. Because I was circulating and working closely with the class, I knew exactly who was getting it and who wasn’t. That’s feedback, folks. That’s formative assessment, folks. The old me (remember, the one who stood at the board writing example after example) would have had no idea who was getting it and who wasn’t. Unit tests were frequently sources of great surprise for me.
  • I was fascinated by the strategies they were coming up with. On the first page of the activity, all students concluded that \sqrt{a}\times\sqrt{b} was equal to \sqrt{ab}. What they did when I introduced coefficients on the next page was really interesting. The room was split into two camps when they came across 3\sqrt{2}\times6\sqrt{3}. Many of the pairs of students got right to it and multiplied the coefficients, then multiplied the radicals, and came up with 18 \sqrt{6}. They confirmed their answers using their calculators, and moved on. The other camp wanted to rely on the only rule they knew at that point (\sqrt{a}\times\sqrt{b}= \sqrt{ab}). Their solutions looked like this: 3\sqrt{2}\times6\sqrt{3}= \sqrt{18}\times\sqrt{108}=\sqrt{1944}=18\sqrt{6}. They were converting to entire radicals, multiplying, and then simplifying. It worked, but it was highly inefficient. So when it became time to let the pairs meet with other pairs to see what they had learned, I made sure that each group of four contained one pair with the efficient strategy and one pair with the inefficient strategy. Students were taught the more efficient strategy by other students, rather than by me. That’s formative assessment.

I did something similar in a Math 30-1 (Pre-calculus 12) class earlier this year. The description and resources are available here.

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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?

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District Assessment Plan

In an earlier post, I suggested a school assessment plan, even though I have never been a school principal. In this one, I intend to outline a district assessment plan, even though I have never been a superintendent or trustee.

This plan is aimed at secondary (7-12) assessment. The elementary school world is too foreign to me to comment on its assessment practices.

I need to begin by saying that I am glad that the board of trustees in Edmonton Public Schools has voted to review their assessment procedures. Never before has assessment been such a hot topic, and we need to make sure we do this right.

The review must accomplish one thing, and one thing only. We need to emerge from the review with a clear understanding of what grades represent. By “we”, I mean teachers, students, parents, the general public, and the media. Let’s clearly define what we are assessing, and do a better job of communicating it.

I have confidence that our board and superintendent will do this right. Just in case, here’s what I think needs to be included in a district assessment plan. I’m sure theirs will be quite a bit longer.

The Plan


A grade represents demonstrated level of achievement compared to curricular outcomes.

Notes about definition

  • Notice the use of “demonstrated”. Kids have to do it so we can assess it.
  • Kids can demonstrate 0 ability compared to the curriculum. We can enter 0 if we have evidence that shows they know nothing.
  • If the student doesn’t demonstrate (ie misses the assessment) we do not assign any grade. We put in one of two codes. (See below)


In curricula that include behaviour, we will assess behaviour. In curricula that do not include behaviour, we will not assess behaviour. We will, however, report on behaviour. We will do this in the form of comments, or a grade in categories like employability, effort, ability to meet deadlines, post-secondary preparedness, or whatever other rather subjective measures will make people happy. We are limited here by our current grade book software. I won’t name them, but it is written by a giant textbook company.

Grade Book Software

We need to take over how our grade books work. They need to adapt to our needs, not the other way around. If our current provider won’t adapt, we need to find a new one. Here is what we need our grade books to do.

The grade book will allow only one of three things to be entered into it.

  • A Mark – The mark entered must be supported by evidence. Lack of evidence is reported differently.
  • An “Omit” – The omit code would not factor into student grades. This code is the option for teachers to leave out small assignments, or to excuse students who have extenuating circumstances. It should not be used for major assessments.
  • An “Incomplete” – Incomplete indicates just that. The student didn’t do the assessment, and the teacher needs him to. We need to get our grade book to function such that as soon as an “incomplete” is entered, it no longer calculates the cumulative grade. Instead, it reports the cumulative grade as, wait for it…           “Incomplete”

Notes on Incomplete

The entry of the incomplete would also automatically generate a comment which would appear on all reports, and be sent home to parents via School Zone. The comment would say something like this:

Due to missing assessments, I am unable to determine Kevin’s grade at this time. Once the missed assessments are completed, a cumulative grade will be calculated. Please contact me to make arrangements for Kevin to complete any missing assessments. Failure to complete the assessments may result in Kevin having to repeat this course.

I used a similar process in my last teaching position, and it worked wonderfully. Parents really took an active role in making sure those assessments came in to me.

How This Plan Affects Various Kinds of Students

Top End Kids

Our top end kids will react well to this plan. They tend to react well to everything we do to them. They will learn that we value performance compared to the curriculum. They will learn that the way to get high grades is to master the course material. They will learn that there are certain assessments that them must do. They will be well prepared for university life, where most courses use only a handful of assessments to determine a grade.

Our 26% Who Drop Out or Fail to Complete High School

26% of Alberta students fail to complete high school. That’s thousands of dropouts lacking the skills to be productive members of society. There are all kinds of reasons that students fail to complete high school. We need to fix the ones that are our fault. This plan will help most of them.

  • We won’t end up failing lazy students who could actually do the work. This plan ensures that lazy students are forced to do the work so we can assess them.
  • We will teach them that we expect them to do the work, which will serve them well in employment even if they drop out.
  • We will be able to extend the course for the kids who are truly struggling. A final grade of “Incomplete” is more supportive than 32%. It says, “Come back to school, and we’ll try to get you through this course so you can graduate from high school and have a better life.”

The Middle Kids

We mistakenly assume that our middle kids all want to be top kids. Some do. Many are happy where they are. Kids that are happy where they are “game” the system that allows zeros. They do the minimum amount that they can to get by with a 50 or 60 or whatever they are happy with. We have taught these kids, by allowing them to take zeros, that they can pick and choose what to do, and still get by. No wonder employers think they are lazy. We’ve taught them to be like that. By allowing them to take a zero, we may have let them believe that their employers will let them pick and choose which tasks they are willing to do in their jobs.

The plan above, whereby the kids have to do the essential assessments, teaches them the opposite. It teaches them that they have to do all aspects of their job, even the ones they may not like as much. That’s responsibility and accountability.

Who We Have to Educate, and How

To make this work, we need to get the following messages to the following people.


They need to know that they can’t skate by and do nothing. We have a set of non-negotiable assessments, and they are expected to do them.


We need to help teachers understand that many of us give kids too many little things “for marks”. We need to get deeply into formative assessment and how to embed it in our classrooms. Most of what I used to chase kids for should have been formative.

We also need to get into the notion of “professional judgement”. Proponents of zeros argue that they want professional judgment to allow them to give zeros. Professional judgement, in an assessment context, dictates that the teacher use other evidence (other assessments, observations, re-dos, etc…) to put a mark in a grade book that reflects what the student knows. See above: The mark that goes into the grade book must be supported by evidence.

If you believe the student truly knows zero, then put in zero. When you do that, you’ll realize you’ve really been harsh on your own teaching skills, though. If I had a kid who had been in my class for any length of time, and I thought he truly knew nothing, what does that say about my teaching ability?

School Administrators

For this to work, you have to be the ones chasing kids. Teachers are busy and stressed.


You need to be taught that our grades will now reflect how your child is performing compared to the curricular standards.

Right now, unfortunately, you can’t be sure. If my daughter comes home with 40% in Math 10, I don’t know what that means. It might mean she is bad at math. It might also mean she is good at math, but didn’t do all the things her teacher asked her to do. We need to clear this up, because the interventions I need to take as a parent are very different. In one case, I need to get her some math help. In the other case, I need to kick her butt.


We are not giving kids free passes. We are not giving kids free passes. We are not giving kids free passes. The grades they get will be earned. They can earn a zero. But we define an earned zero as demonstrated lack of knowledge. Undemonstrated knowledge is not assessable.

We need to assure the public that the grade assigned represents performance compared to the curriculum. We didn’t artificially lower it by inserting zeros that weren’t earned, and we didn’t artificially inflate it by giving bonus marks or other rewards. The grade represents our best guess as to how well the student mastered the material in the course.

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