# I teach math, and I didn't give a single test last year. Here's what I learned. (9 min read)

### Not wanting to police students on zoom, I assessed with problem sets and projects. Students still learned—and found new confidence along the way.

Imagine this: you’re 16 years old. You’re sitting in your bedroom trying to take a math test. The camera watches you the entire time. You haven’t seen your friends in months. You glance over at the picture you took at the school dance last year. It seems a lifetime away.

Then, you get a chat from your teacher: *Where are you looking? Please keep your eyes on your test. No phones allowed.*

Sounds brutal, doesn’t it? Yet, this year, teachers across the country—especially math teachers—held online examinations in this manner. As we prepared for the school year, conversations about online learning and pedagogy quickly turned into discussions of exam security. Instead of asking, *How do we make math meaningful at a distance?,* we asked* How can we stop students from cheating when they take tests at home? *(the cheating happened, anyway).

So I opted out. My instincts told me that policing students would erode the relationships we were building, and these online relationships were tenuous at best. Instead, I hoped to create and implement assessment strategies that develop positive mathematical identities and a strong sense of agency, as these are crucial for mathematical success across a wide range of learners.1

Keep scrolling for the long version, but here's the TL;DR: I’m not convinced that students’ **recall of math facts or procedures** was any better than they would have been with traditional tests; however, I found tremendous evidence indicating that most students **understood** the concepts and their connections because the assessments **gave students agency** while requiring them to **justify their thinking in connected, authentic contexts**. Furthermore, the vast majority of students improved their relationship with the subject because the assessments were **more meaningful and less stressful** than traditional tests.

#### 🏫** Curriculum & Context:**

It’s worth mentioning that I’m lucky. I lead the precalculus team at a private school, so I had both agency and authority. The only person I had to convince was myself. My department chair was supportive, and the other teacher on my team went right along with me.

I implemented my no-tests curriculum in my three precalculus sections (and my team-teacher implemented the same assessments in his one section): a total of **51 students**. My school has three tracks of precalculus—*Honors Precalculus, Advanced Precalculus, *and* Precalculus*. My class is the lowest track.

I based the curriculum largely off of the precalculus curriculum developed by my brilliant colleague, Rachel Chou. There is no textbook for the course—I created original handouts and problems or re-used materials created by Rachel.

**Assessments:**

Coming into the year, I prepared four different kinds of assessments I wanted to try out:

**Problem sets:**I gave students 5 problems of which they would choose 4 to solve. These problems blended multiple standards and usually required transfer (read: they’re hard!). Students collaborated in small groups, but each submitted their own write-ups for evaluation. [assessment example]**Student-written quizzes:**students created four problems, then submitted them with answer keys. [assessment example]**“Teaching math” videos:**similar to problem sets, but students recorded themselves explaining and justifying their work. These videos were submitted via Flipgrid. [assessment example]**Modeling projects:**these usually required students utilize mathematical tools to analyze data to infer conclusions. [assessment example]

It’s worth noting that these assessments are designed to be *interesting*. I blend concepts from multiple units when I write them. I incorporate current events and inside jokes. I require students to use technology like Sheets and Desmos. I tie the math to its applications.

**Grading and Retakes:**

The students’ course grades depended entirely on these assessments. Neither homework nor participation were graded (though the students did receive feedback and encouragement to do both).

My students and I co-created a retake policy. Most importantly, a C+ or below led to a *mandatory* retake, but any student could retake an assessment, regardless of their grade. Joe Feldman’s text highly influenced the construction of this document.

The assessments were always evaluated using the same rubric. To give a student a grade, I selected a box in each row that best fit the student’s understanding. Each column, from left to right, translates to A, B, C, D. I would average the three together to get a student’s grade. *For example: if a student received 3, 4, 4, I would average these together to 3.66, which is an A-.*

**Data Collection**

Teaching being what it is, the most meaningful data collection came through direct observation of my students: watching them in class, evaluating their assessments, chatting with them one-on-one. To supplement this, I surveyed my students twice (at the end of each semester), repeating many questions to gauge any changes.

#### 📖 **The story of the year:**

From August-October, my school was fully online. In the four weeks leading up to Thanksgiving break, we had 25% of our students on campus to test-drive our hybrid model. January-April, we were at 50% on-campus, which increased to 100% for the last 6 weeks of the year.

The course ended up having 9 units, each with one major assessment. I quickly realized in unit 2 that the **student-written quizzes did not work.** They are an excellent activity to create learning, but a terrible assessment. Grading one took nearly 15 minutes. I remember looking at the other 37 waiting to be graded. I believe the sound that left my lips was *hnnnnnnnn*. I wouldn’t recommend it. Anyways.

As the first semester came to a close, it became clear to me that most students responded well to the teaching math Flipgrid videos. This was especially clear among a group of students who struggled to express themselves in written form. I believe the flipgrids allowed students to more easily incorporate their own experiences and language. This, in turn, helped them better learn the math.2 3 4

So, going into the second semester, **I modified the assessments to combine problem sets and Flipgrids**. Most units (unless they involved a modeling project) culminated in a problem set as the assessment: I gave students ~6 challenging problems. However, students had the *choice* to either write their solutions or record them via Flipgrid. Here’s an example from our unit on complex numbers. I still graded them with the same rubric and gave feedback in the same forms.

#### 🔎** Major Findings:**

**I’m not convinced that students’ recall of math facts or procedures was any better than they would have been with traditional tests.**In fact, in the short term, it may have been worse. I believe this is because my students did not experience the rush to memorize and replicate that traditional tests require. This was apparent to me on in-class formative quizzing, which happened nearly daily. I also suspect fewer students did the homework given the pandemic’s challenges. I did not count homework toward the grade, nor did I provide effective incentives and interventions for students who were not doing the homework.

*Next steps: I’d like to try new strategies to monitor and encourage meaningful practice of procedures and facts next year to supplement this assessment model.***While the retake policy increased motivation and decreased anxiety, the feedback system was vague and untimely.**It is important to note that retakes work. They reduce anxiety, and they don’t demotivate students from trying the first time around. But by restricting myself to a single rubric, I could only give feedback to students along three crosscutting skills. This wasn’t sufficient to help a student feel in control of their grade and understanding. Plus, I didn’t have any experience evaluating the students’ videos or modeling projects, so these took a lot of time to grade. As a result, the students received feedback long after it would be most effective (in some cases, a month later). This is neither sustainable nor effective.

*Next steps: I’ll try giving students feedback along different categories. I’m considering collecting data in three categories next year: content clusters, analysis clusters, and mathematical practices. I hope this will give students more control over the learning process. I also hope to implement a “double-dipping” model to prevent “snowballing” from my students who need more time to learn.***Most students understood the concepts and their connections because the assessments required students to explain and justify their thinking in challenging, authentic contexts.**I’m proud of this, but I’m not surprised. When math classes give students agency while helping them see connections between different concepts—that math

*makes sense*—student learning is more powerful.5 6 7 8 9 By designing my assessments around these principles, my students rose to the occasion. High expectations were fulfilled—and this is the*lowest track*at my school.*Next steps: I want to bring authentic problems and contexts into the units both at the beginning (to inspire the learning) and at the end (to assess the learning). I also plan to experiment with unit-ending “puzzle days” on which we all review the assessment problems together (to give students feedback more rapidly and to allow them to self-assess).***Furthermore, the vast majority of students improved their relationship with the subject because the assessments were more meaningful and less stressful than traditional tests.**This was abundantly clear to me as their teacher and observer. Still, I think it’s important to let the data speak for itself. Here’s what my students said after semester 1:

And here’s what my students said after semester 2:

Now, I always assume student data is skewed toward whatever they think the teacher

*wants*to see. Still, I take this as a very clear message from my students:*we prefer this to high-stakes testing because we no longer dread the assessments. We have more time to fully understand the material, so we feel more control and confidence in our understanding of the math.*While there are ways I intend to use traditional tests next year—formative quizzing of specific facts and procedures—I find this data extremely convincing. Here are a few highlights from their explanations:

*I have always done well in math whether that be math tests or projects. However, over the past year of attending precalculus I have come to appreciate that completing problem sets via flipgrids and modeling projects is much more interesting than the traditional test.**I don’t feel stupid when I make silly mistakes or can't do timed math problems. I am able to see math in a different way which improves my confidence.**Testing in math classes has always given me a great amount of stress it generally makes up a large portion of one's grade in math classes, and they also just create this environment of anxiety unique to math classes. With the pressures of testing removed I enjoyed learning a lot more and felt that I could learn for the sake of learning rather than for a test.**I always used to dread having to take traditional tests because I feel like they don't truly grasp what someone has learned over the course of a unit. I actually enjoy math now and I feel more comfortable with math in general.*All said, I am convinced that removing the anxiety of traditional test-taking

*vastly*improves the emotional experience for many students (and doesn’t make a difference for the rest).I imagine a skeptic would would respond, “Yes, they’re less stressed, but what about the satisfaction of overcoming a challenging test? Don’t students develop meaningful confidence when they work through genuine intellectual challenges?”

To this, I say: yes. But do the challenges need to create an environment of anxiety through a largely inauthentic task? I don’t think so. Think of it like a sport: let’s say an athlete practiced dribbling for a week, then took a dribbling exam. Would they learn to dribble. Yes, I think they would. But would they want to become a basketball player after that? No, I don’t think so.

It seems absurd, but this is what we’ve been doing in mathematics. Why not invite them to play a game instead?

#### 🧐 Questions on my mind

Who uses a

*double-dipping*retake model? How did you do it? Did it help students continue the learning process? Was it motivating?What do traditional tests do well — and more effectively than problem sets or projects? If testing doesn’t

*have*to be the backbone of assessment in a math course, what role*should*it play?

Aguirre, J., Mayfield-Ingram, K., & Martin, D. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. The National Council of Teachers of Mathematics.

González, N., Moll, L. C., & Amanti, C. (Eds.). (2006). *Funds of knowledge: Theorizing practices in households, communities, and classrooms*. Routledge.

Turner, E. E., & Celedón-Pattichis, S. (2011). Mathematical problem solving among Latina/o kindergartners: An analysis of opportunities to learn. *Journal of Latinos and Education*, *10*(2), 146–169.

Moschkovich, J. (2013). Principles and guidelines for equitable mathematics teaching practices and materials for English language learners. *Journal of Urban Mathematics Education*, *6*(1), 45–57.

Boaler (2019). *Limitless Mind. Learn, Lead and Live without Barriers*. Harper Collins.

Boaler, J. (2019). Prove it to Me! Mathematics Teaching in the Middle School, 422–429.

Boaler, J., Cordero, M., & Dieckmann, J. (2019). Pursuing Gender Equity in Mathematics Competitions. A Case of Mathematical Freedom. Mathematics Association of America, *FOCUS*, Feb/March 2019.http://digitaleditions.walsworthprintgroup.com/publication/?m=7656&l=1#{%22issue_id%22:566588,%22page%22:18}.

Anderson, R; Boaler, J.; Dieckmann, J. (2018) Achieving Elusive Teacher Change through Challenging Myths about Learning: A Blended Approach. Education Sciences, 8 (3), 98.

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Hi! Thank you for sharing your experience so thoughtfully. I teach physics, and last year I tried only take home, open resource tests. We made up the problems ourselves, so students could not look up the answers on the internet, although internet research was allowed (as were all other resources). The one rule was that students couldn't collaborate and I trusted that they did their own work. My colleague and I tried it for the entire year. We were in a hybrid model through May, then fully in person. We did not want to waste one of our two precious class periods per week on taking tests, which was our original motivation. Once we returned to fully in person, and two to three days per week, we still felt the same way, so we did not switch back to timed, in person tests. At the end of the year, I received this note from a student:

"I was thinking about our conversation yesterday about how physics causes me the least stress of all my classes. I definitely like physics a lot more than previous science classes, but I think that is a result of the less stress and not the cause of it. I think the real reason it's not as stressful as other classes is because of the take-home assessments!

Which got me thinking more, because I can see how it might be worrying as a teacher to give students take-home assessments that they could cheat on. But, the take-home assessments (in my opinion) have made me actually learn the material more. It feels less like I'm learning the info for a test, which makes me actually remember the content after we take the assessment. In other science classes, I always forget about the material after we take a test and I think that's because when I learn it, I'm learning it so that I can remember it on the test. With the take-home assessments, I feel like I'm learning the material so that I can actually learn the material, and the assessment is just part of the learning process. I don't know how much sense that made, but I was thinking about it and thought I'd tell you!"

We are still talking about what to do this year, but if it were entirely up to me, I would continue this practice indefinitely, along with timed, ungraded, formative short quizzes from time to time to develop fluency and recall, but without the pressure of that skill being included in a grade.

I really did feel, anecdotally, that my students had a deep, thorough, meaningful understanding of physics this year. I also figured out that, giving about 12+ tests all year, I saved myself almost two weeks of contact time with students that I could use for interesting discussions, activities, group practice and community building, instead of sitting in a room listening to kids' hearts beat out of their chests as they try to remember three week's worth of physics in 55 minutes.