Changing Assessment within a Changing CurriculumMichael Pruner

When planning my assessment strategies for the 2018-2019 school year, there were a few issues that I wanted to address. I wanted to develop a plan that recognized the curricular competencies and not just the content. I wanted to move away from a points system where everything is recorded as a score and tallied in weighted bins. And I wanted to create a system that was efficient and effective in informing students where they are and where they are going in their learning. Working with a colleague at my school, we discussed possibilities around how to keep some of what we were already doing with assessment and how to create some new instruments that suited these goals. Before I share exactly where we landed, I would like to first describe why I felt that my assessment practices needed to change.

Years ago, assessment in my mathematics classes was comprised almost entirely of test writing. My students would write quizzes within chapters, tests at the end of the chapters and a final exam at the end of the year. Quizzes were the formative assessments for students to see how well they were understanding the new content and tests (including the final exam) were the summative assessments for student learning. Points were awarded throughout the year and collected in weighted bins. Here is an example of the weighted bins from my 2013-2014 Mathematics 10 course outline:

Figure 1: Final Mark calculation from 2013 Mathematics 10 course outline.

I have many concerns with respect to this dated system of assessment (not the least of which being the comic sans font chosen for this outline). Quizzes here were intended to be a type of formative assessment that informed the teacher and the student on how well students were learning the material. This was very ineffective. Students would focus on the mark they received and not the mistakes that they made, and so students rarely learned from their quizzes. Moreover, students who performed poorly on their quizzes and then showed excellent understanding on their chapter tests would end up with lower than deserved grades in the course, because the lower quiz points were still affecting their overall grade. I was also concerned with the perceived objectivity of these assessments. I was not so certain that a student scoring 26 out of 30 actually knew more than a student scoring 24.5, and yet these students were separated by letter grades on a report card (Romagnano, 2000). The biggest concern that I have with this system is that everything that was assigned value in this mathematics class was related to test writing. As a mathematics educator, I value so much more in the development of my students.

Thankfully, we are now working under a new curriculum from the Ministry that also places value on other aspects of a student’s development within a mathematics classroom. This new curriculum

motivated me to re-think how and what my students were learning in my classroom. I always thought highly of the seven “mathematical processes” that were emphasized in the old WNCP curriculum, but these processes were mere supporting features to the high and mighty content. In the new curriculum, the mathematical processes have evolved into curricular competencies, and more importantly, the curricular competencies are alongside the content as part of the Learning Standards for the curriculum. From the ministry’s web page on assessment: “Curriculum sets the learning standards that give focus to classroom instruction and assessment” (https://curriculum.gov.bc.ca/assessment); consequently, curricular competencies are now front and centre with the content as my focus for instruction and assessment.

What I will share with you now is the assessment system that my colleague and I have implemented for this 2018-2019 school year. It is not perfect–in fact it has a lot of room for improvement; it does however, approach my initial stated goals of placing value on the curricular competencies, moving away from a points system, and being an effective and efficient means for assessing student learning.

Within the new curriculum, the learning standards are made up of the curricular competencies and the content. Our assessment model has a similar breakdown; it contains tools for assessing

development within the curricular competencies and the content. Some of these tools are new and developed with purpose and others are old assessment tools that have been adjusted to suit our needs. Although each of these tools could probably be used to assess a variety of different aspects, we decided to assign each tool to a specific learning standard for simplicity. Below is the list of assessment tools that we are currently using in our classrooms matched with their corresponding learning standard. I have also included a detailed description of each assessment tool and student exemplars where appropriate. For context, I should point out that I teach in a linear timetable with 80-minute classes meeting on alternating days.

Figure 2: Mathematical Processes from the 2008 WNCP curriculum.

Figure 3: Pre-Calculus 11 revised curriculum

Assessment Tool Learning StandardProblem Solving Term Assignments (New) Reasoning and ModellingProblem Solving as part of test (New) Understanding and SolvingUnit Test (Adjusted) ContentReflection Journals and Practice (New) Reflecting and ConnectingGroup Quizzes (Adjusted) Communicating and Representing

Problem Solving Term Assignments

My 80-minute classes always begin with students solving a non-routine task like this one (below):

I have shared these tasks in the past through twitter (#weeklymathtasks) and the BCAMT listserv. I like to begin classes this way because it places a focus on collaborative problem solving (something that I truly value as a math educator), students tend to enjoy these types of problems and these types of tasks help in developing and maintaining a thinking culture in the class. Within each of our three terms, I ask students to submit a problem-solving

assignment based on one of these tasks. This is a take-home assignment where students are asked to demonstrate their mathematical reasoning, estimation, analysis and modelling, and curiosity. It is marked by me, using a rubric with letter grades A (extending), B (proficient) and C (developing). A copy of the rubric and criteria can be found here: https://bit.ly/2EZovKJ . This assessment tool is being used to measure progress under the curricular competency, Reasoning and modelling (Analyzing). Each row in the rubric corresponds to a specific learning standard from the curricular competencies and is written in

language that students and teachers can understand. Over the course of a year, each student will have three opportunities (one for each term) to demonstrate their progress in their Reasoning and Modelling within mathematics.

Unit tests and Problem Solving as part of test

Figure 4: Reasoning and modelling curricular competencies.

To assess the Understanding and Solving category in the curricular competencies, we thought it would be a good idea to have this as part of the student’s normal unit test. Unit tests are the more traditional forms of assessment in our classrooms that most resemble our assessment from earlier days. We are now using this existing tool to serve two motivations in assessment. Unit tests are a good tool for assessing student knowledge of the content. We have four unit tests over the course of a year (each test covering about two chapters of study). Unit tests are made up of multiple choice and free response questions usually scored out of 20 points. These tests are re-writable if students show corrections and complete a small practice assignment. To keep re-writes manageable from a teacher’s perspective, they take place for all classes during a single lunch hour once per month. We have also adjusted this tool slightly to include an assessment of the Understanding and Solving curricular competencies. We do this by including an open-ended non-routine task at the end of each test. Below are two examples of these tasks, one is from our Mathematics 9 test covering rational numbers and scale factors, and the other is from a Pre-Calculus 12 class covering exponential and logarithmic functions.

Figure 6: Sample task from Math 9 Unit test

Figure 5: Understanding and solving curricular competencies.

Figure 7: Sample task from Foundations 12 Unit test

We include the rubric for marking this final task in the test, so students can see what we are focussing on for this assessment. After grading the unit test, students receive the test back with a content grade out of 20 and an understanding and solving letter grade (A, B, C or Incomplete). A copy of the rubric can be found here: https://bit.ly/2snX9pp . The description for this curricular competency includes: “Engage in problem-solving experiences connected with place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures.” To address this competency, we decided to design tasks that were connected to place or included cultural perspectives relevant to local First Peoples communities. This is a challenge to accomplish and a work in progress, as you can see from the two examples above. Over the course of a year, each student will have four opportunities (one for each test) to demonstrate their progress in their Understanding and Solving within mathematics.

Reflection Journals and Practice

Reflecting on learning and practice is a daily activity in our classrooms. At the end of each class, students are provided with a reflecting question that captures the main concept being taught and a set of questions to practice at home. During their home practice, students are expected to take time (20–30 minutes in total) to think and respond to the reflection question and then spend the remainder of their time practicing the mathematics. Having the advantage of digital technologies (we use Scholantis in our district), students upload a photograph of their practice and reflection response to their class portal. I do not have the time to check this daily; but, once every five or six classes, I view one of their uploads and make comments on the quality of their reflection and practice. Every two weeks, students complete a self-assessment of their Reflecting and connecting by completing this rubric: https://bit.ly/2AITKGa .

Figure 8: Connecting and reflecting curricular competencies.

Figure 9: Self-Assessment for Reflecting and Connecting

When completing these self-assessments, I ask that students provide evidence if they think they are at a “B” or “A” level in any of the rows. This evidence can either be a detailed description or a photograph of their work. I quickly scan these self-assessments before awarding students their self-assigned grade. Most of the time, I agree with their assessment; however, sometimes there are a few that I need to adjust down due to lack of evidence or some that I need to adjust upward due to students being too hard on themselves. Each term, I have three to four of these assessments to look at before deciding on a term grade for this competency.

Group Quizzes

I started giving group quizzes to my classes a few years ago, and I was struck by the level and quality of engagement by my students. This is one of the few times where I would hear students arguing with one another about mathematics, and I found it to be very effective for student learning. These quizzes were typically shorter than a regular quiz with only one or two questions, and the questions were more open and complex. For example, instead of asking students to factor x2-x-6, in a group quiz, I would ask students to show all possible factorizations for x2+bx-6 with corresponding values for b. I eventually moved to a point where all quizzes are now written in group form. These quizzes are either written in groups at whiteboards or at their tables, each having its own unique benefits. The highly collaborative nature of these quizzes aligned very nicely with making this a tool to measure student’s Communicating and representing curricular competencies. I still mark, make comments and record each quiz using a 4-point scale: A, B, C or Incomplete, but these marks are used strictly to inform students on their progress and understanding of the concepts, they are not used in determining a reporting grade. At the end of each quiz, students complete a self-assessment that attends to how well they represented their mathematical ideas and how well they communicated within their groups: https://bit.ly/ 2 Mb40LW . Each term, I have two to three of these assessments to review and decide on a term grade for the curricular competency heading: Communicating and Representing.

Figure 10: Communicating and representing curricular competencies.

Figure 11: Self-assessment for Communicating and Representing

Reporting

After each term and at the end of the year, I am required to produce a summative letter grade for my grades 8 and 9 classes, and a summative percentage for my grades 10–12 classes. If you have been with me up until now, you will have noticed that I have a collection of letter grades (from the competency assessments) and percentages (from the content assessments on the unit tests). To meld all of these into a single result, we use an Excel spreadsheet for organizing all of this data. Here is an example of what a term report looks like for one of my Mathematics 9 students:

Figure 12: Student term report

Now that the data is organized, I can use my professional judgement to determine a suitable letter grade to represent this student’s progress. This student has demonstrated B’s and A’s through her curricular competency assessments (note: the 0’s are a result of the spreadsheet not having a value to show, they do not represent “no achievement” and do not factor into my overall assessment). She also demonstrated a C+ level of achievement on her Unit 1 test, so when I put these two assessments together, I decided on a B for her Term 1 report. For Grade 8 and 9, this is sufficient; however, for grades 10 – 12, I need to turn this into a percentage. My colleague and I decided on using cut scores to help with this conversion:

High A A Low A High B B C+ C Low C C-98% 90% 86% 80% 75% 70% 65% 60% 55%

For the student above, I would have given her a 75% if I was required to give a percentage. I see that her curricular competencies showing B’s and A’s melded with her 71.5% content work justifies a B for her term. I should also add that because this letter grade is a result of my professional and subjective judgement, I also consider my experience of working with this student. This includes observations and conversations throughout the course of the term. It is freeing to not be bound by an accumulated percentage based on points earned when deciding on a student’s level of achievement in a class.

Summary

I feel that these assessment tools go a long way towards meeting my goals from the beginning of the year. They recognize the significance of the curricular competencies within the new curriculum, and they have transitioned away from a points system to a system that recognizes growth and requires professional judgment in reporting progress. Efficiency was my third goal, and I do believe that it has the potential to be an efficient assessment system. The two self-assessments were intended not only to provide opportunities for autonomy in student learning, but also to make the entire assessment system more efficient. Right now, I still find that I am spending too much time reviewing and adjusting my student’s self-assessments; and therefore, the efficiency is lower than intended. In reflection, perhaps this is something that I need to let go of. I think that I can be satisfied with the fact that my students are spending time thinking and writing through these assessments and this demonstrates to them that it is a valued part of my mathematics classes. Perhaps this is good enough, and I do not need to review them so carefully.

Thinking forward to next school year, I would like to further refine the competency assessments, and I would like to begin the transition to standards-based grading (SBG) in course content. I do believe that SBG is a better tool for providing information to students on where they are and where they are going with respect to the content, but it was too much for us to take on for this year. I have always known the mathematics classroom to be a complex space of content intertwined with actions and social interactions to support learning. In this new system of assessment, I am seeing students also understanding the classroom as this same complex space.

The conversation in all of my classes has changed. Students are not focused on half points earned on quizzes and tests, nor are they asking about percentage bumps to improve overall grades. There is now more conversation around actual content learning and genuine efforts to improve in specific competencies. I have seen students go out of their way to provide multiple solutions or explain their thinking on group quizzes in order to provide an improved outcome on their communicating and representing self-assessment. By placing value on other aspects of learning in a mathematics classroom, students are now seeing the importance of the curricular competencies in their own learning of mathematics. Students are working to develop and improve in the curricular competencies and they are seeing first hand how this is helping them with learning the content.

References

Romagnano, L. (2001). The myth of objectivity in mathematics assessment. Mathematics Teacher, 94(1), 31-37.