By Chris Robinson *@Isomorphic2CRob* & Jonathan Osters *@callmejosters*, The Blake School

*What does a grade in your course represent? How does the way you calculate your students’ grades describe what you value?*

Most of us think about two types of assessment on a daily basis. ** Formative Assessment **should give students feedback on how close they are to meeting class expectations of knowledge and skill.

**is a measure of what they know at the end of the learning process.**

*Summative Assessment*Another two pieces of the assessment picture that we often think about is the balance between a student knowing the ** skills of mathematics** and students being able to

**. In the past, Mathematics education movements have swung far in each direction, at times over emphasizing skills and at times sacrificing skills practice.**

*solve novel problems*As the Blake Mathematics Department renewed its discussion on teaching problem solving in 2009, we began thinking about how we could assess more effectively. We wanted to utilize frequent formative assessment but were unsure how to walk the line of making them worth enough that students took them seriously, but not worth an unfair amount as students are still in the learning process. We also wanted to assess authentic problem solving. The last post discussed our Skills Quiz System, while this post will discuss our Problem Solving Assessments.

**Building a Culture of Problem Solving BEFORE Assessment**

Being able to assess problem solving effectively requires that students are “practicing” solving novel problems on a daily basis. There is a great deal that can be said about buliding a culture in the classroom focused on problem solving and collaboration and every teacher’s experience will be a little different: see our blog *5 Things Teachers Can Do to Establish a Cooperative Classroom Environment *and Carmel Schettino’s blog is a great resource as well. Carmel is a teacher at Deerfield and major proponent of problem based instruction. Suffice it to say that building a culture is more than selecting good problems but as we are trying to talk about assessment here, we shall move on.

My colleagues and I each take a different approach here but in all cases a majority of our lessons are centered around problems that explore new topics and are either scaffolded by printed leading questions or are scaffolded by interjected questions in small group discussions. Some problems are taken from the Phillips Exeter Academy Curriculum, some from Carmel Schettino, some from our current hodgepodge of curricular resources and many written by us. The goal in general is for students to encounter something approachable but new and for them to conjecture and test out approaches to solving a problem. Then students share their approaches in small group and then large group verbally or visually through doc cams or writing on the boards. There is a summary process of the approach and sometimes an urging of “if you want to remember the steps you just went through, you may want to take notes now.”

**Problem Solving Under Pressure**

The threat of assessment is real for all students, and although our skills quiz systems alleviates some pressure, solving novel or near-novel problems can be a hairy experience for most students and teachers. The constant practice of problem solving in the classroom should help prepare students. Our Problem Solving Assessments (PSA) come in a few varieties: Exploration Labs, Reflection Journals, In-Class PSAs and Take Home PSAs. Not all classes use all types but it is safe to say that all classes use at least two of the four.

*Component 1: Exploration Labs*

Exploration Labs come in many forms. A majority of them are guided questions utilizing a technology like Geometer’s Sketchpad, Fathom, or Desmos. In Geometry, students may be tasked with creating a diagram, manipulating the diagram and observing how things change. In algebra, students may use Desmos sliders to see how adjusting one part of an equation affects the graph of the equation. In statistics, we may create some statistic “from the ground up” that measures some particular facet of a distribution. Each of these activities puts students in new situations, where building something from scratch is required of them, but the stakes are low, so they can try different things until a “good,” “best,” or “most efficient” way to approach the problem comes up.

Pros: +Students have opportunities to conjecture in low risk situations (often using dynamic software).

+ Students often work together and develop math communication skills.

Con: – They can take a while to grade, depending on depth of expectation.

*Component 2: Reflection Journals*

Inspired by the work of Carmel Schettino and her frequent metacognitive journal assignments, we have experimented with a variety of writing assignments. My most recent Honors Algebra II written assignment was the following:

*Which is your favorite representation for a line? (standard form, slope-intercept form, point-slope form) Be sure to compare and contrast your favorite to each other type regarding*

*a) ease of graphing.**b) ease of writing an equation given two points.**c) ease in finding the intersection of two lines.*

Students get a few nights to compose their first response. A grade and a written commentary is given to each student, and they can then revise and resubmit to regain half the points they lost.

Pro: + Reflection and metacognition are powerful learning tools.

Con: – They can take a while to grade.

*Component 3: In-Class Problem Solving Assessments*

A traditional test has a mixture of skills and problem solving. Often coming in the form of 15 skill problems and 2-4 “word problems.” Those word problems were often the *same as problems previously encountered with numbers changed*; this is by necessity of students having so little time to approach them on a test filled with skills. And because there is so much to do, it was relatively common for students to skip or provide only a minimal attempt of the “word problems.”

It is important to have summative assessment on skills, but now that our skills quiz system is accomplishing both formative and summative assessment , we don’t need to do that on our tests. And thus was born the **Problem Solving Assessment**, or PSA. A PSA is essentially a traditional test with all the skill portions removed. It is a set of 2-6 problems that require synthesis of a number of skills.

Our PSAs have novel problems in the sense that they may have the same theme as a previously seen problems, but there is a major twist. In an Algebra 2 class, for example, we may have a two-variable systems word problem, but instead of giving them the problem and they find the solution, we might give them a solution and a framework like a paint mixing problem, and ask them to write the question. In order to write the problem they will have to create a system of equations with the correct solution, and then create the sentences in the word problem. Another example might be that of a race between several racers, where speeds, head starts, and starting points all vary, leading to different equations of position for the different racers. We may ask them a straightforward problem like “who won the race?”, but we also could ask them more thought-provoking questions like “ which racers were in 2nd place at any point in the race? How long were they in 2nd place? Use the graph to justify.”

Like any exam, these in-class PSAs have a time element to them. Students must complete the problems during the period. At times we don’t predict properly how long students will take to “solve a problem,” and so we at times let them take them home to revise and at times allow them to revise in the class after teacher comments. This has worked well for us so far, since the novelty of the problems make them such that students can’t find a similar problem in their book or online.

Pros: + Students have the extra time to problem solve, compared to a traditional test.

+ They are faster to grade than a journal, students gain comforter in a timed situation.

Con: – It’s a timed situation, and problem solving is tough with limited time.

*Component 4: Take Home Problem Solving Assessments*

Take-home PSAs are essentially the same as in-class PSAs, the only difference being the location. The advantage is students have more time to work through the problem if needed. The disadvantage is an increased risk of academic dishonesty. We might give students only one in depth problem rather than a couple of shorter problems as in the in-class PSAs.

Pro: + Students have more time to complete it.

Con: – Managing academic honesty becomes much trickier.

A teacher can use all or some of these components for a successful assessment of problem-solving. But what happens if a student has trouble even getting started? Or what if their work is haphazard and difficult to follow? Are these PSA’s graded differently than traditional exams? We will discuss those questions in next week’s post!