You stare at the stack of Problem Solvers on your desk. You flip through them. One paper has well-labeled work and shows clear thinking, but the student has the wrong answer. Another paper has the correct answer but the evidence of conceptual understanding is unclear or is scattered throughout the paper.
How do you mark the papers?
Before delving into the procedures and rubrics, some assumptions must be stated:
- Assessment is different than grading.
- No assessment system is perfect, but some are better than others.
- Students and teachers should both have an idea of what the ‘grade’ will be before students hand in a paper.
The assessment process should have started earlier in the week – before students received the papers that are now in your pile.
Backing up
The papers on your desk should not represent students’ first attempt at a type of problem – especially at the beginning of the year.
If a concept or strategy is new, it is not unusual to spend a full class period allowing students to construct concepts related to one or more strategies. When students receive similar (but slightly more complex) problems on subsequent days, they will solve the problems more quickly.
Those first couple days, you assess student work, but you don’t collect it to be graded. Instead, you assess progress using checklists and anecdotal notes.
Formative Assessment #1
Checklists and anecdotal notes are invaluable forms of assessment. Remember that, on the first day, students have individual work time, pair share time, more individual work time, and class consensus time.
During the initial individual work time, look for the students who demonstrate understanding right away. Indicate such on your notes. Visit those students. Increase the difficulty of the question if necessary. What if this pattern continued to…?
Look at the students’ pictures and diagrams. Are there any that have misunderstood the language of the problem? Make a note and help them understand the context and the question.
Note those who are experimenting – and what kinds of experiments they are trying. Are they doing random operations all over the page? Are they making charts or tables? Finding patterns? Looking at their neighbors’ papers?
During the pair share time, watch for students who are carefully explaining their processes, those who are just stating an answer, those who are passionately defending their processes, and those who are shaving their pencils with scissors.
During the second individual work time, note who changes strategies, who is now able to get started, and who regularly asks to go to the bathroom at this point in the lesson.
You should now have identified three groups of students for differentiated instruction: (1) students who need to see a significantly more complex problem or different type of problem because they’ve already nailed the concept, (2) students who are getting it and just need a bit more practice, and (3) students who will need some more intensive coaching.
While students are coming to consensus, note the way students explain their reasoning to the class. Note who asks clarifying questions and who uses the vocabulary of mathematics.
Continue with the checklists and anecdotal notes during subsequent lessons. Especially note improvements and further misconceptions.
Formative Assessment #2
Students begin the self-evaluation process by reflecting on their work, making corrections, and writing notes to themselves.
Once your anecdotal notes indicate that students are capable of independently completing a homework problem solver, send one home.
Rather than collecting the problem solver the next day, have students share strategies and answers. You might go through the consensus process once again. While sharing or comparing, students can use a different color pen or pencil to make changes and/or write reminder notes to themselves.
The homework with notes can be glued into their math journals for future reference.
Formative Assessment #3
Students should know how their work will be assessed. Hand out the rubric on the first few days of school. Talk through it column by column. Students can mark on it, highlight it, and glue it into their math journals for future reference.
When you hand out the “test” papers, or the ones that will end up on your desk, require students to self-assess their work on a rubric before handing it in. What descriptors match their work?
Some colleagues and I have spent the last month piloting the rubric below – with a great deal of success:
EE: Exceeds expectations, ME: Meets exp, MEA: Meets with Assistance, DME: Does not meet
The rubric above is based on the NCTM process standards. Remember that Problem Solving is at least as much about the process as it is the final answer. Students see the importance of process more clearly when they see that the answer is only a small part of the final ‘grade’.
Summative Assessment
By the time students turn in a “final” problem-solver to be graded, you should have a high degree of certainty that all students will at least meet expectations. The students should be confident too.
Representation involves the pictures and diagrams students use to make sense of the problem. Students have probably used tables or charts to play with numbers and number patterns.
Connections is about connecting the problem to other areas of math or to the real world. Where have you seen patterns or ideas like this before?
Communication. Notice how nothing in the descriptors include sentences like “First I…Then I…Next I…”. A student should not be required to write a tome about their thinking IF the work already includes representations, the numbers/numbers/charts/tables are labeled, and you can follow the student’s train of thought. I’m passionate about this for a few reasons:
- Students’ grades should reflect mathematical thinking, not writing ability.
- Writing requirements turn reluctant writers into reluctant problem solvers.
- It’s a pain to read through prose when the thinking is already clear by looking at the work (yeah, this one is selfish).
- The time it takes prolific writers to write out their processes in complete sentences could be used to teach more mathematical concepts.
Accuracy. How do you mark the papers when work is done well but the answer is wrong? Dock accuracy, but give the student credit for what he or she did well in other areas.
Reasoning and Proof. I usually don’t mark this column but I keep it there because i want to report to students and parents when my anecdotal notes indicate students did an outstanding job of defending their answers or asking clarifying questions during the consensus-building time. I can also indicate whether students know more than one way to solve the problem.
You don’t have to mark all the columns all the time.
Rubrics help clarify expectations for both students and parents. Students can self-assess and you can confirm or discuss their self-assessments. Parents understand why you are praising their child’s mathematics even though the answer may be incorrect.
So here is my question for you: Which of the formative and summative assessment procedures are assessment for learning? Which is/are assessment as learning, and which is/are assessment of learning?
Final Thoughts: Back to the Stack of Papers
This final paper that is handed to you should be the only thing represented in the grade. Why?
If we ‘grade’ students according to their early work, we end up grading the speed of their learning. If, in the end, two students can solve a problem equally well, should one be penalized for learning the concepts more slowly?
When giving students grades for problem solving, the final grades should reflect math. Final grades should not reflect reading ability, writing ability, or speed of learning.
The stack of papers should now be demystified. Look at the papers, refer to your anecdotal notes, and refer to students’ self assessments. Students are often harder on themselves than I would be. Expect smiles of pride when you mark a paper higher than a student expected. If there is a huge discrepancy between the student self-assessment and your assessment, have a conversation. In my experience, those discrepancies are rare.
If this series has helped you, please consider doing one or more of the things in the storyboard below…
photo credit: DaveCrosby via photopin cc
Expat Educator 
these concepts are so important. They sound time-consuming–and are at first, but quickly become standard, the Way to Grade. Thanks for the reminders.