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Posts Tagged ‘assessment’

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)

Slide1

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.

Wiliam

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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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.

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