I have set a personal goal for myself to work on in my pedagogy this year. I want to more fully explore and implement the use of assessment as learning into my teaching practice. This is not to say that I feel I've "mastered" the use of assessment of learning (what we call summative assessment in Canada) or assessment for learning (formative assessment).

I see tremendous value in assessment for learning as it helps to move my students forward in their learning and provides me with feedback to help them learn.

Summative assessment I think is more controversial -- it is very difficult to arrange for common standards that all teachers everywhere adhere to such that several different teachers assessing the same student's work all come up with the same mark or grade. (I once participated in a workshop where the facilitator had us -- a bunch of experienced math teachers -- assess the same piece of student work. No grade scheme was provided. We were told to decide on how many marks each of the three items should be worth and assign a percent grade to the paper. The results: grades varied from marks in the 30s to the 60s.)

Here are some of the resources I am reading/using to guide my exploration:

• »Assessment Is For Learning web site from Scotland.
• »Assessment as Learning in Mathematics, a research paper by Corinne Angier and Hilary Povey from the University of Wolverhamton in the UK.
• »Lessons Learned from Research, articles from NCTM's Journal for Research in Mathematics Education rewritten to be more accessible for teachers as opposed to researchers.
• »Rethinking Classroom Assessment with Purpose in Mind published by Western and Northern Canadian Protocol for Collaboration in Education, 2006
• »Formative Assessment in the Secondary Classroom by Shirley Clarke
• »Inside the Black Box: Raising Standards Through Classroom Assessment Black and William, 1998 (pdf)
• »Knowing Knowledge by George Siemens, 2006
• »Mathematics Assessment: A Practical Handbook for Grades 9-12 another NCTM publication ... on my wishlist (don't have it yet).

According to the Assessment Is For Learning site from Scotland, three of the key features of assessment as learning are:

• »Self/peer assessment
• »Goal-setting
• »Reflection

The assignments I've given my students to date highlight the first and third of these key features; self/peer assessment and reflection. I have to work more on including goal-setting activities into my practice. (A colleague of mine, Erin Armstrong, does this very well.)

The wiki solution manuals I started with my classes last year also exemplify assessment as learning in line with peer assessment and reflection. I'm planning to evolve these assignments this year to incorporate goal setting by having this year's students do some work on last year's wikis before they begin the creation of their own wiki solution manuals. More about this in a future post.

I began this year by calling these assessments, in my head at least, Learning is a Conversation assignments. I think learning conversations naturally facilitate self/peer assessment and reflection. Blogs are ideal tools for having these sorts of conversations. Students participate without feeling self-conscious or shy because the computer screen creates an arm's length distance between them and I. I also believe they feel less "judged" and so are more forthcoming in sharing their thinking. There's something about typing your thoughts on a blog. The keyboard leads to more writing (not sure why that is) and the fact of publishing their thinking for an authentic audience makes the task more meaningful ... maybe that's why they write more.

What follows are synopses and links to the first four assignments I've given in two of my three classes along the lines outlined above. I will be exploring this further over the course of this school year.


First Assignment
I've blogged about this previously.
AP Calculus Exponential Functions Lab
Danny, Manny, Mark, Linger
In Class Lab
Exponential Functions Lab


Second Assignment
The structure of this assignment was exactly the same as the previous one. There were five different True or False statements about logarithms. The groups had to decide whether or not the statement they were working with was true or false. If true they had to explain, justify, why they thought the statement was true. If false, they had to explain the nature of the error and illustrate the correct application of the misconstrued concept.
TF Log assignments in AP Calculus
Group 1
Group 2
Group 3
Group 4
Group 5


Third Assignment
This was the first assignment of this sort with my grade 12 Pre-Cal class. They were given a trigonometric modeling problem. (A real life application of trigonometric equations ... well, maybe not "real life," but the principles are the same in all problems of this sort.) Whoever left the first comment had to solve the problem. All subsequent students had to leave a meaningful comment. A "meaningful" comment was one that (a) if the student thought the solution was correct offered a rationale for why they thought that or (b) if the student thought the solution was incorrect they had to explain why they thought that and correct it. Students received full marks for making a reasonable attempt and no marks for comments that were not "meaningful;" comments that lacked any evidence of thought. This is what Shirley Clarke (see above) would call "attempt marking." The type of problem assigned has historically been exceptionally poorly done by students in this course. Jho-ahn was scribe that day so she was excused from participating in the problem. She had the responsibility of posting it as anindependant post, for which she received full marks.
Jho-ahn's Bicycle


Fourth Assignment
Another assignment for my grade 12 Pre-Cal class. This wasn't planned in advance. In class we were discussing the nature of the domain of logarithmic functions. I asked the class" "Can the argument of a logarithm function be zero? Explain your reasoning." We had discussed this several times in class before. Their oral answers were confused, lacked confidence and illustrated a host of misconceptions about an idea I had thought was very clear. We ended up having a 15 minute class discussion around this topic and still failed to have one student state the correct answer with confidence. It is not my habit to answer questions directly if I have covered the material before in class. I prefer to ask a series of questions to encourage students to recall what they've learned and draw out the connections between concepts that have been previously acquired ... I want to help them figure it out for themselves; if I just tell them the answer they don't learn it and come to rely on me over much for answers in general. Time was running short and I had new material to cover that day. On the spur of the moment I told them we would do with this question as we had done with Jho-ahn's bicycle.
Why Not Zero?

Overall I've been pleased with the depth of thinking the students are doing on their blogs given these sorts of assignments. Just the other day the domain of a logarithm function came up again in class. Everyone immediately understood what was being discussed and seemed to have a good grasp of the problem and solution. In the coming weeks I plan to do more of this and include this kind of assessment in my grade 11 Pre-Cal class. At least two other teachers have also picked up the idea and are using it with their students. Vicki Davis (see her post Wikis and the Power of Internet peer review!) and Chris Harbeck (see his posts Oh no I have done something wrong and Oh No I goofed Again). Chris is taking a different approach. He posts a problem and solution but the solution is wrong. In the comments to the post students must identify the error and correct it.

It'll be interesting to see how this evolves in our classes over the rest of the school year. ;-)