Monday, 7 August 2017

Beyond Tests in HS Math (Part 1)

I’ve been a high school math teacher for 17 years and I’m still working on and intrigued by the classroom assessment piece. In this space, I am taking assessment very broadly to include all of the assessment for/as/of learning, formative/summative assessment, evaluation etc.

I am still thinking about something that came up in a Twitter chat from a few months ago (January 2017).  It was a #caflnchat (Canadian Assessment for Learning Network Chat) and the exchange involved Jimmy Pai (@PaiMath) who is a Teacher in the same Ottawa school district as me, and Peter Liljedahl (@pgliljedahl) who is a Professor in the Faculty of Education at Simon Fraser University in Vancouver. 

A question in that chat asked: "How can/do we make use of information we collect every moment in the classroom?" In a response to a thoughtful comment Jimmy posted, Peter suggested thinking about both process and product, as well as "in the moment" and "after the moment".
I chimed in with a sketch of what I thought Peter meant by the 2 by 2 matrix (hastily drawn on the whiteboard in my kitchen) which Peter later concurred matched his description. At the time, I wrote "feedback" at the top of that grid, but I have since been thinking more broadly about this grid in terms of "assessment".

Peter asked, "What does assessment look like in each quadrant?". I arbitrarily threw the numbers 1-4 in so I could refer to the specific boxes later - I won't address them in order here.

Here, I offer my thoughts on this question: What does assesssment look like in each quadrant?

Quadrant 4: Assessment of a product /after the moment

In my experience, this is where most evaluation (and much formative assessment) occurs in high school math. The most common example of this is math tests. Students finish a section/unit of learning and write a test. This is usually done individually (more on this and group tests in an upcoming post). The teacher then marks these tests later in the school day or at home; away from the students. These hopefully get returned in a prompt manner, but can take a few days or longer sometimes as I can personally attest to. Most of the feedback is written and might involve short phrases, check marks and circles or other notations. Research has show that students have a difficult time interpreting this sort of feedback (e.g. Weimer, 2013's review of Sadler, 2010). In addition, the delay between the "performance" and the feedback or judgment reduces the power of the assessment to serve LEARNING.

While tests are most common, formative quizzes and exit tickets are often also largely assessment of a product "after the moment" when the teacher responds the next class. For example, I routinely use "not-for-grade" quizzes. These quizzes are very short (usually 1-3 questions) and I give comment-only feedback. No grades, no levels, just written feedback. I also post solutions for these quizzes electronically so students can fully review solutions. Where multiple solutions are possible, I often post two interesting solutions and discuss in class. In addition to providing feedback to students, formative quizzes and exit tickets can also inform the teacher about next steps in instruction.

Quadrant 3: Assessment of a product /in the moment

I can think of a few examples where I have assessed a product "in the moment". In a grade 12 Data Management class I teach (combines statistics & number theory), students do an oral presentation in front of the class based on a project they have spent most of the semester working on. I have a rubric ready before each presentation begins (which students receive in the early stages of the project) which lists the standards (or "expectations" as we say in Ontario) along with four levels of achievement. I re-write these in student-friendly language and demonstrate an oral presentation from a previous year. During the presentation, I make observations and write notes and questions so in pencil so I can revise "in the moment" as I hear more of the presentation. Towards the end of the presentation I make tentative "tick marks" about the level of achievement which I solidify in the 2-3 minutes after the presentation as the next student is getting set. Students can receive the feedback that same class period. I admit here that I am thinking about assessment "in the moment" from the teacher's perspective rather than the student's perspective.

Last year in my grade 9 and 10 math classes, I did exit interviews at the end of the course. Students chose parts of the course they wanted to give more evidence for (certain standards/expectations). The interviews occurred during regular class time. I labelled whiteboard stations with course topics and had several problems posted at each station for students to work on. I circulated and students talked to me and to their groups about math concepts as they solved problems. I observed student work and asked questions. Classmates also chimed in with productive comments and questions. It was very helpful to see how the student I was focusing on responded to those prompts from peers. At times I posed other problems than the ones posted at a particular station. I carried around a clipboard and made notes about what each student knew and could do and then converted these comments into levels to incorporate into the student's grade. Again, I note that I am thinking about assessment "in the moment" from the teacher's perspective rather than the student's perspective.

Quadrant 1: Assessment of a process/in the moment

The most frequent example from my class of assessing a process "in the moment" is the conversations and interactions I have with during regular classwork. This often occurs in #VNPS format (see Peter's work on Thinking Classrooms). I pose a problem or ask a question and students begin working on whiteboards. The vertical nature of the work allows me to constantly scan to look for students who are stuck, or interesting ideas. It also allows students to see the work of other groups and access each others ideas. The majority of my prompts in these learning settings is to ask questions like "are you sure?" or "how is your solution the same/different than group X's?". My prompts usually focus on process rather than product during the learning (although I do often summarize content/product ideas after we have worked for a while).

I feel that the work for me in this setting is to try and understand students' thinking about mathematics - I am still surprised by methods or representations I hadn't considered, especially the first time I use a problem in this way. The assessment "in the moment" occurs in a variety of ways and from a variety of sources as students have conversations within their groups, between groups and with me. I have had an enthusiastic response from students about this type of classroom environment, and I think part of it is that they do get more feedback "in the moment" than they would sitting down at their desk working on textbook problems.

A similar example is the conversations I have with students individually during class or at lunch. Observing a student work through a problem and then responding to questions they have is also assessment "in the moment".

What is different *for me* between the examples of #VNPS and exit interviews is that in the former I am more focussed on the learning and the mathematical processes (NCTMOntario). The exit interviews occur in the last few days of class and while the learning is obviously still important, I am focussed on assessing the level of achievement of the learner at that point. The product that we discuss allows me to make decisions about grading and reporting.

Quadrant 2: Assessment of a process/after the moment

An example of assessment of a process "after the moment" that I have used is journal writing. I have done this both formally and informally at the grades 9/10 levels. A few years ago, I did formal journal writing in grade 10 where students kept a notebook through the semester of weekly prompts that I responded to (feedback). The prompts I provided to students were largely about their thinking PROCESS. I would often respond to a question that troubled the class by allowing them more time to work on it, followed by a journal entry that asked about their process. As I write this, I am recognizing that it has been a while since I have done this.

Another example is a "process portfolio" that I did with my grade 9 classes a few years ago. In my district we have been required to have two pieces of summative (course-end) evaluation. For a few years, I asked students to provide two samples for each "process expectation" and a piece of reflective writing. The evaluation piece was the reflective writing.

As I write this, it is becoming evident to me that I haven't focused on assessing process "after the moment" in a few years. Any suggestions?

On Feedback:

This week I read an article by Alice Keeler (@alicekeeler) which included this beautiful quote on feedback that I think really connects to this idea of "in the moment"/"after the moment".

"The impact of your feedback is highest when students are not mentally done with a task"
- Alice Keeler

Caveats and Cautions:

I don't claim to fully represent what assessment looks like in each quadrant, just offered some thoughts and examples. I'm very interested in what others (maybe you?) have to say about this.

In addition, since assessment can be "partitioned" in so many ways, this 2x2 matrix is certainly not the only way to think about assessment but has been very helpful for me in thinking about assessing products vs observations and conversations. As Peter noted:
Jimmy asked some thoughtful questions that I am still thinking about:

And Jimmy also introduced the idea of "premoment" which I take to mean "before the moment" adding to Peter's "in the moment"/"after the moment".

What are your thoughts? I'd love to hear from you.


  1. Thanks so much for sharing your thoughts! Definitely reminded me of this wonderful conversation that was waiting to be had. Maybe now is the time to continue it?

    I typed up some thoughts and wonderings here:

  2. Very thought provoking. I have been thinking about the quote:

    "The impact of your feedback is highest when students are not mentally done with a task"
    - Alice Keeler

    I am trying to think of a better way to take up tests that students are mentally done with. One possibility is getting them to engage with common mistakes in groups before giving the tests back.

  3. I am very intrigued by this idea and way to break down assessment. I am teaching a preservice teacher class this semester, and am trying to change a project I have them do to really examine assessment (formative / summative) and figure out the differences. This helps a lot.

  4. I am a student teacher and we just talked about this in class today, thank you for sharing your experience!

    On giving students feedback on tests, one of my university profs would have us do a test individually, and then immediately have us redo the same test in groups of 3 after we were done. Our mark would then be the better of our individual mark or an average of our individual mark and our group mark.

    After tests in other classes, we would all tend to gather in the halls and discuss problems and possible solutions, i.e. we were not done with the problems. Through this strategy, we got peer feedback on our efforts immediately.

    I thought this was a neat example of providing feedback before students are done with a task, and I am hoping to try it out in a high school in a few months. What are your thoughts?


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