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  • Writer's pictureKatelyn Hayes

Her Mathematical Voice: Creating Authentic Math Assessments to Deepen Understanding

In this post by math teacher Katelyn Hayes, she continues the conversation about the intersections of math, writing, and student well-being begun by her co-teacher Claudia Mathison in an earlier post. In her article, Claudia highlights how reflective writing develops students’ growth mindsets, allowing them to process their struggles with particular skills following an assessment. Here, Katelyn focuses on the assessments themselves, discussing how creating innovative, authentic math assessments empowers students to more fully demonstrate deeper understandings of mathematical thinking. Together, their posts document their collaborative effort to reimagine the teaching of math as a space for students to move beyond recitation and into application, developing both the skills and the mindsets necessary to grapple with the inherent beauty of math and its real world applications.

I vividly remember getting a D on an Algebra II test in high school. Yes, a math teacher got a D on a math test. I even recall the exact classroom I was sitting in: our desks were in groups of four and I sat at the yellow pod in the back by the teacher’s desk. Because we couldn’t check our grades online, we had to wait to find out what we earned when the tests were returned in class. When the teacher put the test in my hand, I was immediately mortified. Math was supposed to be “my thing”… what happened?! I hid the test with the big red D under my binder as quickly as I could.

The teacher reviewed the test material that same day, but I cannot remember anything else from that class period aside from the big red D that haunted me from my binder. I couldn’t pay attention, was completely distracted thinking only about my grade, my GPA, and how I was going to recover from this failed assessment. What were my parents going to say? I hoped my friend who was sitting across from me didn't ask me what I got. I was sure my face was red with embarrassment.

At the end of class, I learned that the teacher was offering reassessment. Interestingly, this was the only opportunity I ever had to redeem a grade on an assessment in high school. A tremendous weight immediately lifted off my shoulders, yet I was so distracted with my poor performance that I was not paying attention… and so completely missed the teacher’s review of the material.

Now, I’m on the other side of this scenario as the high school math teacher. My own high school Algebra II experience has challenged me to be more intentional in writing assessments and to provide students multiple opportunities to demonstrate learning.

The Beginnings

In 2017, I participated in the Kettle Moraine School District TransformED Unconference on Learning without Boundaries in Wales, WI. While deep in discussion on the purpose and meaning behind assessments, all I could think about was my D on my Algebra II test so many years ago. What if I had gone through an assessment process that included an authentic assessment and an opportunity for reflection that allowed me to communicate to my teacher my understanding of the objectives? Would I have been able to learn from my mistakes instead of worrying about the consequences of one “bad” test grade?

My colleagues and I began brainstorming. How can we reimagine assessments in a meaningful way? How can we create an assessment process that facilitates engagement in deeper learning? We began with the following goals in mind:

1) Compose creative assessment questions that include critical thinking and reasoning combined with mathematical application, which allow students to demonstrate learning in their own way.

2) Incorporate a self-assessment protocol into the assessment process that offers opportunities for students to take ownership of their learning, learn from their mistakes, and deepen their understanding of the mathematical concepts being studied.

Together, Claudia Mathison and I developed the following system designed to engage students more deeply in their learning: do, evaluate, reflect, revise, and connect.

Goal #1: Develop Creative Assessments

Tammy Yung, our Mathematics Department Chair, led the way on writing creative assessment questions. She challenged the math teachers to reconsider their approach to writing assessments. What value or purpose does each question serve? How can we pose questions so that students can demonstrate mastery and apply their learning? How can we write purposeful questions that have multiple pathways to a solution or questions with multiple solutions?

A Sample Problem from Tammy Yung:

For the numbers 2 through 9 on a cellphone keypad, consider these two relations.

1) mapping numbers onto letters

2) mapping letters onto numbers

Are both relations considered to be functions? If so, what type of function? Explain your thinking.

Tammy’s work inspired me to create thought-provoking, real-life application assessments that challenge students to see the beauty of mathematics applied in everyday life. These questions often have multiple “right” answers and encourage students to present their understanding in a way that makes sense to them--as opposed to a traditional math test filled with “solve for x” type of questions. Moving away from “solve for x” questions and towards a more open-ended form of questioning has drastically improved the quality of the work that students do--and makes it more fun! While writing creative assessments is quite difficult, the process is ultimately rewarding.

Creative Assessment: Geometric Transformations Assessment on Desmos

Desmos is an interactive digital platform where students progress through a teacher-designed activity at their own pace. Activities are created with various math applications that enable students to develop visual representations of the mathematical concepts being studied.

The Geometric and MADI Art Museum in Dallas, TX serves as an ideal space for students to see the beautiful intersections of math and the real world. Here, students enjoy connecting the beauty of math and art by finding geometric transformations displayed in the remarkable art pieces. Without the opportunity to visit the museum this year, I brought the art to them through a Desmos assessment.

I provided students with several art pieces from the MADI on a sketch slide in Desmos. Students were then asked to identify and label the transformations that they see using proper mathematical notation.

Additionally, students engaged in video game-like questions. They were asked to transform an original object through an obstacle course maze using geometric transformations without hitting the barriers… or else their objects would be destroyed and they would need to begin again.

Students thoroughly enjoyed this type of assessment and asked for more assessments like it. They appreciated the immediate feedback and opportunity to try again to demonstrate mastery. By allowing students multiple attempts to demonstrate mastery, I could have a clearer understanding of their approach and thought process.

Creative Assessment: ADA Wheelchair Compliance

In this assessment, students were given an article to use as a reference document on the requirements for ADA compliant wheelchair ramps. Students used this resource to determine whether or not a specific wheelchair ramp was ADA compliant.

Scenario #1: Students were given a ramp that needed an additional support beam and were asked to identify the exact height of the beam using similar triangles.

Then, students read ADA Wheelchair Access for Buildings to determine the maximum height and slope of an ADA compliant wheelchair ramp. The assessment required students to use their critical thinking and algebra skills to determine if the ramp was compliant.

Scenario #2: Students constructed their own ramp that was ADA compliant out of similar triangles over a sand dune to a public beach.

Their initial reaction to the assessment was something like: what kind of math assessment is this? After completing the assessment and self-assessing, however, students typically have a change in heart.

Student Feedback on the ADA Wheelchair Access Assessment:

"Although it was a little stressful, Ms. Hayes’ assessments give a purpose to what we are learning and, in the end, helps the lesson stick with me."- Izzy S.

"This type of assessment enables my mind to think outside of the box, which I believe will be very beneficial in my future endeavors." - Maddie W.

Creative Assessment: UA Theater Design

Lastly, it is important to me to connect math to something meaningful in students’ lives.

The Geometry teachers at our school recently organized a virtual tour of the school’s new building. Claudia and I invited one of the architects for the new building--a young woman who also happens to be an alumna--to discuss her journey from a student at our school to an architect for her alma mater. She discussed the geometry involved in the building project on campus, highlighting the real-life application of many of the concepts that we are studying in class.

Building on her presentation, I created an assessment addressing viewing angles in the new theater. Students were challenged to apply their knowledge of right triangles and trigonometry to determine the angle a person in the top row must look down at to see the stage and to determine the length of the new curtain.

Student Feedback on the UA Theater Assignment:

"I really liked the Right Triangles and Trig test where I used measurements from the new UA stage being built right now. Problems like this prompt me to use all of the knowledge I've learned this year and challenge me in new ways. I love seeing how what we are learning in class applies to real life. When I successfully complete a test in Ms. Hayes' class, I feel accomplished, and like I fully understand the topic." - Sara D.

Critical thinking and written communications skills like analysis and problem solving, scientific and quantitative reasoning, critical reading and evaluation, critiquing and argument, as well as writing effectiveness and mechanics are skills students will use throughout their education and beyond. Interpreting and analyzing information, and then applying to a specific task, is what the "real world" is all about.

Goal #2: Create a Self-Assessment and Reflection Process

When teachers returned an assessment in school, I, like perhaps many of my readers, put it away and never looked at it again. I was concerned about the grade and just moved on. As a teacher, though, when I see students quickly put away their assessments, I think: why did I spend time reviewing their work and leave feedback for them to never look at the assessment again? I knew there had to be a way for students to engage with the feedback and continue learning.

Why was I finding their mistakes and not the students? Isn’t it important for them to know where there might be a misconception? These articles reinforce why students need to play a role in recognizing mistakes: Learning from Mistakes: How Does the Brain Handle Errors? and Why Mistakes Matter in Creating a Path for Learning.

Claudia and I created a self-assessment and reflection process to help students measure their own level of understanding and identify areas of growth to deepen their understanding. We constantly refer back to the process Do-Evaluate-Reflect-Revise-Connect when designing curriculum and assessments, creating time and space for our students to self-assess after completing an assessment.

So what does this self-assessment look like?

Students take a colored pen and an answer key, and start working through the assessment.

In supplying students with the key, I give them the opportunity to identify errors in their own work, at their own pace. The assessment key is used as a resource to determine where/why they made a mistake. Students then identify their error(s), write a sentence explaining their own thought process and where they went wrong, and explain what the correct answer is and why. No grades are discussed at this point.

The self-assessment process gives students the opportunity to take control of their own learning. Students determine their level of understanding, which gives them the chance to define their own goals and necessary action steps to reach mastery. When students engage in this process it fosters a deeper understanding, leads to a greater degree of student engagement, and challenges them to reflect on their overall progress. Students use these experiences to adjust their approach so that they can improve their performance on future assessments.

Student Feedback on Self-Assessment:

"I like the self-assessment process after we take tests because they allow us to personally see what mistakes we make and how to fix them. For most quizzes/tests in other classes, we don't completely go over assessments or we are just given the correct answers without any explanation. The self-assessment process is really beneficial and helps me continue growing my understanding of certain topics even after unit tests." - Sawyer B.

Reflections and Moving Forward

I am continuously striving to incorporate opportunities for students to demonstrate learning in a deeper manner rather than reciting information. The creative questions and assessments have challenged my students to draw on skills beyond the scope of the unit. Even though students are sometimes intimidated by the unknown of the assessments, they appreciate the opportunity to see math in real-life examples and to self-assess. They review their assessments at their own pace and reflect on their understanding, improving their life skills.

After self-assessing, students spend time reflecting on their understanding, considering their journey from the beginning of the unit through the self-assessment process and identifying which activities, lessons, and assignments helped them gain mastery. Students identify what they are proud of and areas of growth to deepen their understanding:

"Ms. Hayes’ assessments require more than just math, but also application to the real world. Math tests that require this type of application help put into perspective what the math we are learning is used for in our everyday life." - Charlotte R.

"I really enjoyed this class this year and I liked doing reflections after we completed an assessment because it gave us the chance to see where we went wrong and to fix it and to learn from our mistakes which I thought was helpful." - Anonymous

"I will never forget freshman year math tests." - Macy M.

"The assessments provided are the epitome of what is expected in an honors class. Not only are we challenged, but we are pushed to apply the knowledge we've learned in real situations." - Anonymous

To learn more about this reflective process, read my colleague’s Claudia’s post: Writing for Math: Developing Growth Mindsets in Young Mathematicians.


Overbye, Knut, et. al. "Learning from Mistakes: How Does the Brain Handle Errors?" Frontiers for Young Minds, 20 June 2020. Learning from Mistakes: How Does the Brain Handle Errors?

Wallis, Claudia. "Why Mistakes Matter in Creating a Path for Learning." KQED, 26 July 2017. Why Mistakes Matter in Creating a Path for Learning.

Katelyn Hayes has taught high school mathematics since 2014. Through her years studying math education, Katie has discovered a passion for developing a student-led learning environment. The focus of her work centers on building student agency in mathematics learning. With the do-evaluate-reflect-revise-connect model, she provides opportunities for students to take ownership of their learning by teaching them skills they can use in their education and beyond. Katie has presented her work at local and national conferences, such as the National Council of Teachers of Mathematics 100 Days of Professional Learning. Katie has an unique appreciation for finding palindromes in unexpected places, such as her car mileage and even her address... living at 17671!

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