We have the technology. We can rebuild assessment. We can make it better than it was. Better, stronger, more accurate.

We all understand how assessment has served as a destructive force in our classrooms. And we’re all to blame. While the obvious perpetrators of destructive assessment are those foisted upon us by states and districts, let’s not forget to point the finger at our own shortsighted teacher-designed tests. And the latter culprit is one we have more control over.

I didn’t take a course in assessment in college. The only assessments I knew how to create were from a schema based on the problems at the end of the chapter or the standardized exams we all loathe.

I’d like to propose a model of assessment based on resources that we have largely at the ready. Now that we’ve got all these great tasks floating around out there, let’s put them to use. Let’s do this (Martin-Kniep, G. & Picone-Zocchia, J. 2009):

Let’s untether ourselves to the 50 question, multiple choice/short answer test and start assessing in ways that are less destructive: by posing rich (even engaging! yikes!) tasks throughout the school year, assessing in a way that honors the student thought process and allows for demonstration of growth, while at the same time compiling artifacts that would serve well as a thesis-style defense.

**Portfolio Problems**

(n) Rich problems that serve as potential assessments worthy of a student to demonstrate proficiency in a group of standards. At the culmination of a school year, the artifacts created through the solving of such problems could constitute a demonstration of learning, either along side or in lieu of a comprehensive exam.

These problems may take a two or three days of class time when you take in to account the posing of the problem, the facilitation of the problem, workshops that need be taught or retaught, and revision of student work to proficiency.

One could pose, facilitate, and assess such a problem, say, once a “chapter” (since we secondary teachers seem to be quite beholden to chapters), perhaps eight a year. Think of it: by the end of the year, you’d have eight rich artifacts that demonstrate student learning per student. These artifacts could tell quite a story.

Perhaps the format of the artifacts are well-organized solutions on poster paper. Or perhaps they’re formal texts with figures and tables embedded. Or even a recorded presentation.

A two day Portfolio Problem may look like this.

A Portfolio Problem should:

- Be accessible at some level to all students in the room
- Allow for multiple solutions or multiple solution paths
- Require a deep demonstration of content knowledge and adept application of content knowledge
- Align to standards you have taught, potentially synthesizing multiple content standards

Let me provide an example.

This task is from Illustrative Mathematics (simply, *the* go to spot for CCSS-aligned tasks).

*A Linear and Quadratic System*

I love this task. There’s so much going on in this problem. A student must understand

- how to find the equation of a linear function based on coordinates,
- how to find the roots of a quadratic,
- the actual
*meaning *of a solution of a system of equations.

If a student can demonstrate competency on this task, there’s no need for me to give a test.

**The Assessment: Rubrics with external standards**

Assessing such problems will require a bit more than a “8/10” slapped on the top of a paper. It’s going to require a more holistic approach to assessment, namely a rubric. Moreover, since we’re assessing eight-ish portfolio problems in the course of a year, it makes sense to have a generalized rubric set to external standards. That is, one could assess two different problems using the same rubric; the rubric is not task-specific (or at least, in addition to task-specific indicators, you include general ones as well).

For example, here are the rubrics we at New Tech Network co-designed with the Stanford Center for Assessment and Learning Equity (SCALE). Here is the 12^{th} grade math rubric. Note the generalized indicators for college-ready work.

But these particular rubrics aren’t the thing; any generalizable agreed-upon set of criteria for quality student work would do. Bryan has an excellent post *from four years ago* where he describes a portfolio system assessed on his agreed-upon Habits of a Mathematician. Be sure to check that post out ASAP both for the criteria he came up with and the system of portfolio/performance assessment he advocates.

**The portfolio and teacher learning**

I’ll leave the choice of platform for portfolio to the #edtech folks, but here are some nice recommendations from Tom Vander Ark, along with further motivation for a portfolio system. Manila folders also work well.

But now we have all these artifacts that demonstrate student learning and – provided the problems are rich enough – can expose gaps in student learning. Now we actually have work worthy of a session of learning from student work, which I’ve talked about before. The data provided by this work can be much more instructive than a spreadsheet telling you which students have answered 6-out-of-8 questions correct on HSA.APR.D.7.

**You’re making it sound pretty rosy, Geoff**

It’s true. I am. It’s a huge commitment. You have to take up, perhaps, twice as many days for assessment in your class than you currently offer. Perhaps three times if you want to give a more traditional assessment alongside a Portfolio Problem. It takes time to assess a Portfolio Problem against a rubric that you wouldn’t need in a scantron format. And if you’re going to use these artifacts as teacher learning opportunities (via LASW), that’s quite a lift for a math department there as well.

But, like I said, thankfully, many kindly bloggers and forward-thinking task designers have done some of the work for us: namely making quality tasks.

Which brings me to the “now what” portion of the post.

**Suggested Portfolio Problems (the rest of the school year)**

It’s late March (wait, really?) so we’re closer to the end of the year than the beginning. My imploration would be for teachers to undertake a year of Portfolio Problems for all the reasons mentioned above.

Still, the last few months of the school year might be a good time to get some baseline data for the 2015-16 school year with the facilitation of one Portfolio Problem. We wouldn’t be able to (or need to) get into the whole digital portfolio system now. But it would give you some reps in facilitating a rich problem, collecting the student work, and be able to compare next year to this one.

Here are some suggested Portfolio Problems that may match with late-in-the-year standards you may be tackling in class right now.

**Grade 4**

**Grade 5**

**Grade 6**

**Grade 7**

**Grade 8**

**Algebra 1 / Grade 9**

**Geometry / Grade 10**

**Algebra 2 / Grade 11**

These are problems that allow for significant demonstrations of student understanding of multiple content standards, and often the synthesis between multiple standards. With all the great stuff that our colleagues have created, we have the ability to do that UT’s Dana Center has been advocating for decades now.

**Suggested Portfolio Problems (going forward)**

I’ve added asterisks to (what I think are) exceptional potential Portfolio Problems in the ol’ PrBL-CCSS curriculum maps. I admit it’s only my intuition that’s marking them as such. In almost all sections of the curriculum maps I was able to identify at least one problem that I felt was rich enough to be a Portfolio Problem, or at least had a kernel of potential that could fully form it into a Portfolio Problem.

**Wait, so why are we doing this again?**

For the student, hopefully the implementation of 6-10 Portfolio Problems throughout a year will be a positive learning experience, even though it’s assessment. It could help blur the lines between assessment and instruction (see tweet above). Also, a student could be proud of and reflect on the work they’ve accumulated over the course of the year.

For the teacher, hopefully the implementation of 6-10 Portfolio Problems throughout a year will provide a much more instructive data set to determine whether students have

truly mastered the content. It could also help with our messaging around math. We can authentically say “this is math,” rather than saying “math is fun y’all, except we have to do this really boring math to see if you’re doing it right.”

One last note: I was chatting with some teachers and principals recently and was informed that their pro-active assessment system (similar to the one I describe here) found purchase in the district. They were able to provide documented evidence of student learning using a performance assessment-style system and now they are awarded the agency from the district to *not* give the common standardized benchmark assessments other local schools have to give. I’m not going to suggest your school, district, or state would be amenable to such a system of demonstrated student learning in lieu of standardized tests, but it’s possible. And I’d submit that you might be surprised how open district officials might be to an alternative assessment system if presented with a convincing case.

==

Martin-Kniep, G. & Picone-Zocchia, J. (2009) Changing the Way You Teach: Improving the Way Students Learn

See also: Raymond’s review of Shepard’s *The Role of Assessment in a Learning Culture*(2000)

## Is bad context worse than no context?

In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. We ask students to count and perform arithmetic on objects, even contrived ones. We ask students to group socks, slice pizzas, and describe snowballs. A critical person might suggest these are all examples of pseudo-context, and they’d be right! These are more-or-less contrived scenarios that don’t

reallyrequire the context to get at the math involved. Why do we provide such seemingly inessential context? I’m venturing a little far away from my area of expertise here, but I’d guess it’s because it helps kids understand the math concept to have a concrete model of that concept in their heads.My question is this: in secondary classrooms, is there inherent value in linking an abstract concept to an actualized context? Even if the context is contrived?

I mean, yeah that’s bad. Comically and tragically bad. It doesn’t do anything to enhance understanding. I’d say the context actually

hindersunderstanding.The thickness of an ice sculpture dragon’s wing?That’s about three bridges too far.But what about a slightly less convoluted, but also-contrived, example. Say:

This problem is still most certainly contrived: dimensions of tanks aren’t often given in terms of

x. I’m not even suggesting this problem will engender immediate, massive engagement, but it might help students create a mental model of what’s going on with a third degree polynomial. Or at least the context allows students to affix understanding of the x- and y-axes once they create the graph of the volume of this tank.We provide similarly pseudo-contextual in elementary classrooms in order to enhance understanding of arithmetic and geometry.

From Burns’ “About Teaching Mathematics”

Of course, we also compliment such problems with manipulatives, games, play, and discourse, which secondary math classrooms often lack. In the best elementary classrooms we don’t

justprovide students with that single task. We provide others, in addition to the pure abstract tasks such as puzzles or number talks.Perhaps the true sin of pseudo-context is that it can be the prevailing task model, rather than one tool in a teacher’s task toolbox. In secondary math classrooms pseudo-contextual problems are offered as

themotivation for the math, instead of exercises to create models and nothing more.(See also: Michael’s blog post on Context and Modeling)