Desmos wants to help every student learn math and love learning math. To accomplish that goal, we build math activities for students, and we build them to the specifications in this document.

Our design code folds in our collective understanding of mathematics, identity, culture, curriculum, cognition, and pedagogy. Together, these ideas can increase the likelihood that a student will come to identify themselves as a “math person.”^{1}

We intend this document to be *descriptive* of choices we have already made and *prescriptive* for choices we will make in the future. When we have doubts about our design decisions, we use this document to settle them. We share these principles publicly in case they’re of use to you and, especially, so you’ll hold us accountable if we fail to meet them.

### Create opportunities for students to be right and wrong in different, interesting ways.

We often ask students questions that have a single, correct answer. Those questions have value. More emblematic of our work are questions that lead to *many* interesting ways to be correct, and where even *wrong* answers offer a class something interesting to think about. In many of our activities, students create representations of their thinking that are so creative and interesting that labels like “right” and “wrong” feel inadequate.

In our curriculum, students create and share mosaics, turtles, robots, polygons, kaleidoscopes, transformations, water slides, card sorts, stories, problems for their classmates, and many other mathematical creations that point toward their brilliance.

When we anticipate and design for mathematical creativity, we make room for more kinds of mathematical brilliance and, by extension, more kinds of mathematical people.

*Questions We Ask Ourselves*

- How can we provide opportunities for students to be creative?
- How do we create space for answers that are wrong and brilliant?
- What are different ways that people come to understand a particular mathematical idea?

### Give feedback that attaches meaning to student thinking.

Whenever possible, we use computers to *interpret* rather than *evaluate* student thinking. Where *evaluative* feedback says, “You’re right” or “You’re wrong,” *interpretive* feedback says, “Here’s how we’re understanding your answer. What do you think now?” ^{2, 3} For example:

- We don’t tell you whether or not your sketch is correct; we use it to control the path of the parachutist.
- We don’t just tell you if your parabola is right or wrong; we show you whether or not it passes through the slaloms.
- We don’t just tell you your paint ratio is right or wrong; we mix the colors for you.

We show students what their ideas mean in context, without judgment, giving them the freedom to revise their thinking and the incentive to build on their earlier ideas.

*Questions We Ask Ourselves*

- How can we help students learn
*about*their answer, beyond telling them it’s right or wrong? - What concrete, visual contexts can we build to help students make sense of this abstract idea?
- What new ideas might students have as a result of our feedback?

### Create a need.

We want students to experience math as power, rather than punishment; as purposeful, rather than pointless. We want students to experience mathematics as though it were aspirin for a headache. We want students to understand the power of the intellectual tools they bring from home, like estimation, their imagination, and their visual senses^{4}. And we want them to experience a need for *new* tools. When we connect new tools to existing ones, we help students strengthen their understanding of both.

For example, in our lesson *Make It Scale*, we invite students to sketch a scaled figure without a grid before then inviting them to experience the precision that a grid offers them.

*Questions We Ask Ourselves*

- Why did different groups of people
*invent*this idea? What was its*need*?^{5} - How can we help students
*experience*that need?

### Use a variety of resources.

Helping students learn math and love learning math is a huge, demanding task. It wants every resource we can offer it, so we try to offer as many as we can. We’re best known for connecting students to *digital* resources, but we also connect students to resources like paper, pencils, and other physical manipulatives. Jere Confrey has said that “students are the most underutilized resource in a classroom,” so we try to connect students to *each other*, creating opportunities for conversation and making visible every student’s experiences, strengths, and knowledge for the benefit of the entire class.

*Questions We Ask Ourselves*

- Should students engage with this lesson on paper or on a computer?
- Should we ask students to type a text response into the computer or ask them to discuss their thinking with a partner?
- What early experiences, personal intuitions, or cultural knowledge will students make use of in this lesson
^{6}? - How can we get students out of their desks and moving?

### Interrupt our biases.

When students use our activities, they’re experiencing the sum of the values and experiences of everyone involved in their creation. So we invest significant time in developing those values, especially in interrogating our own beliefs and interrupting our biases. We hone our values in internal reading and discussion groups, in our interactions with educators in the Desmos Fellowship, in our collaboration with external organizations, and by using our equity principles^{7} to guide our decisions, all so we can help every student experience the kind of mathematics education they deserve.

*Questions We Ask Ourselves*

- What stereotypes might this lesson perpetuate?
- What messages about math are we communicating through the questions we ask and the ways we give feedback to student responses?
^{8} - How might we ensure that every student will have access and opportunity to engage with grade level content?
^{9} - How can we help teachers invite, celebrate, and develop the brilliance in early student thinking?
- How will a student’s identities affect their engagement with this activity?
^{10}

### References

- Goffney, Imani, and Gutiérrez Rochelle, eds. Annual Perspectives in Mathematics Education 2018: Rehumanizing Mathematics for Black, Indigenous, and Latinx Students. National Council of Teachers of Mathematics, 2018./)
- Butler, Ruth. “Task-Involving and Ego-Involving Properties of Evaluation: Effects of Different Feedback Conditions on Motivational Perceptions, Interest, and Performance.” Journal of Educational Psychology 79, no. 4 (1987): 474-482.
- Wiliam, Dylan. Embedded Formative Assessment. Bloomington: Solution Tree Press, 2017.
- Gutierrez, Rochelle. “Why We Need to Rehumanize Mathematics.” Annual Perspectives in Mathematics Education 2018: Rehumanizing Mathematics for Black, Indigenous, and Latinx Students. National Council of Teachers of Mathematics, 2018.
- Fuller, Evan, Jeffrey M. Rabin, and Guershon Harel. “Intellectual Need and Problem-free Activity in the Mathematics Classroom.” (2015).
- Moschkovich, Judit. “Using Two Languages When Learning Mathematics.” Education Studies in Mathematics 64, (2007): 121–144.
- Desmos Equity Principles
- Hammond, Zaretta. Culturally Responsive Teaching and the Brain: Promoting Authentic Engagement and Rigor among Culturally and Linguistically Diverse Students. Thousand Oaks: Corwin, 2015.
- “The Opportunity Myth,” 2018 TNTP report.
- Aguirre, Julia, Karen Mayfield-Ingram, and Danny Bernard Martin. The Impact of Identity in K-8 Mathematics Learning and Teaching: Rethinking Equity-based Practices. Reston: National Council of Teachers of Mathematics, 2013.