Spiro Inquiry


The Spiro Inquiry is a project designed to explore ideas in art, math, and digital fabrication. Learners move between the physical and digital worlds of design as they play with and create “spirograph” patterns with both physical gears and pens and with the free Spirogator app. By enabling learners to determine their own creative ends, this inquiry can structure not only academic math content, but also the potential for new ideas of what math looks like and what it is used for. (Learn the math below the surface of spirographs)

Mathematical phenomena are the materials with which learners explore the aesthetic possibilities of their designs. The drawings and digital files can then be fabricated into stickers, prints, stencils or new custom gears, made to their specifications.


Materials & Tools


Varying Approaches to Get Started

Below we share the varying approaches of three educators who have introduced Spiro in their contexts:

Paula Hooper begins a workshop with everyone in circle. She passes the physical gears around so that students can hold them in their hands. She asks if it reminds them of anything? Students have related the gears to things like bicycle gears or the launching mechanisms for spinning tops. Connecting students with what they already know can be especially important for children who experience gendered or racialized stereotypes about who is good at math in math classrooms. (See theory corner at the end of this guide.)

Walter Kitundu introduces the physical gears to elementary age youth by talking through the process of learning to use the gears. He chose to start with a demonstration because he had seen that young children are sometimes overwhelmed by the frustration of learning the dexterity of keeping the gears from slipping: “The internal gears are small and they all have little teeth on the outside and those teeth will lock into the teeth on the outside gear. Now I’ll try to get this to run along the edge so the teeth all stay together. But I’m gonna try and do that with my pen. I’m not trying to draw a shape, I’m just trying to keep the gears rolling smoothly together. And because of the way the gears are, it ends up being a really nice image.”

Fan Kong facilitates lifelong learning programs in Chinatown, NYC for youth and senior citizens. She likes to begin by reminding students to slow down and to let the pen follow the movement of the gear, as if we are “listening to the objects speak.” When we let the objects themselves do the work, their regular pattern and rhythm feels like a form of meditation. This is especially helpful for young children and elders alike, who are learning (or re-learning, like with elders with arthritis) the dexterity of their hands, while coordinating their hands and eyes. When they find their rhythm with this process, they will see the designs emerge, revealing themselves as they move their hands with the gears.





We like to start with pen and paper and then move on to the app so that students build on the familiarity of the physical experience.

Working with physical spiro gears involves an engagement with dexterity and coordination with our hands, the table, and different moving parts. It is important first to find a way to fix the outer gear on the paper so that only the inner gear moves freely. Clipboards are the easiest solution we’ve tried but limits where the gear can be placed on the paper. You can also use a small amount of mounting putty so that the gear can be placed anywhere on the paper.

Finding the rhythm and ease of the amount of outward pressure needs for the gears to stay interlocked is usually a bit of a challenge at first. Treating this as important intellectual work and thinking of the first drawings as drafts or experiments can ease the frustration students might feel when their pen slips. This experience can contribute to an appreciation for working in the digital app for some.


What looks good to me? What do the numbers mean?

Initial explorations with physical gears is a time for students to notice how different combinations of gears (they are all inscribed with the number of teeth), and different pen holes make different designs. We have seen their engagement fall anywhere along a spectrum between aesthetic output to the desire to “figure out” what the relationships of the teeth numbers and pen positions mean for what kind of graphs are produced. These are some of the questions we’ve used to deepen inquiry and exploration with Spiro gears:

Questions to spark inquiry and exploration with Spiro gears:

  • Start making designs and let's see what we notice about them.
  • Which designs do you like?
  • What do you see in these designs?
  • What new designs do you want to make? How do you think we can do that?
  • Look at someone else’s drawings. How are they different from yours? How could you make a similar one?
  • What do you notice about how the numbers of the gears change the drawings?
  • What if your friend wanted to make a design like yours? What would you tell them so they know how to do it?



We find that a brief introduction to the app and how to use it to make spirographs is helpful to ensure that everyone starts with a sense of familiarity. We generally do not explain every aspect of the app. This allows students to find the use and significance of those features on their own, or together with peers and facilitators. Students often use those features in ways we wouldn’t have anticipated or instructed them to.

Features of Spirogator:

At the top left of the screen there are two tabs for modes of use: “make gears” and “make drawings.” The make gears tab will produce a pdf file for gears with your custom specifications. This file can be used with a laser cutter to create custom gears. (see “Take it Further” in this guide for more on digital fabrication)

In “Make Drawings” Mode:




Both the digital and physical modes allow for parallel explorations in the creation of a range of designs. Participants may find themselves investigating the same question in both modes; sometimes an insight from one mode can travel and build upon in questions initially generated while in another mode. For example, experience with the physical gears can help ground learners’ understanding of what is happening in the app. Because these 2 modes are independent from one another, it is the movement between the worlds that encourages an iterative process and a refinement of questions.



After students have some time to play with the gears or the app, we like to encourage them to voice or write their questions about what they are noticing.

Why are some of the patterns more curved and others sharp?

When educators focus on question generation as the learning goal, these questions will serve as guides to help learners deepen their understanding or investigation. Some educators choose to do this more explicitly by asking students to draft their questions on paper to put on the walls and then ask them to focus a second wave of Spiro investigation on answering one of those questions. Others may choose to informally support question-generation in one-on-one conversations while working with children.

What happens if I increase the outer and inner gear by the same number?

When educators support kids in collectively answering questions about Spiro, we have the opportunity to identify math within students’ desires and imaginations. In other words, the process of creating a meaningful design/object engages learners in the possibilities between art and math, and between school learning and everyday wondering.

What affects the rotations a gear must make to complete this pattern?

After generating questions, ask students to write their favorite question on a sentence strip or piece of paper and put them up on the walls. Invite students to choose a question that is not their own and investigate that question with a second round of Spiro exploration. This is an approach that works well in more structured settings like classrooms.



Where is the Math?

In our own teaching practice, we don’t start with presenting the math theory when introducing the Spiro Inquiry activity to students. However, having these explanations available when students or educators become curious is always helpful. In this page we offer three explanations that help us understand the math below the surface of spirographs.


Good Problems

The gears slip and mess up the drawing

It’s easy for the gear teeth to slip and cause the pen to make errant marks which can be a cause for frustration. Facilitators can take this as an opportunity to draw our attention to the teeth of the gears and what kind of pressure and coordination they need from our hands in order to move smoothly together. A focus on coordination allows us to become more aware of how we position our bodies while we work. This allows us to slow down and pay attention to our breathing as we steady our hands. Once youth gain fluency with how the gears physically move together, they might use those skills in novel ways. For example, we’ve seen students stop a drawing mid-design to switch pen colors. In order to continue the drawing seamlessly, they carefully lined up the inner gear where the drawing last stopped.


They’re Interested in the Designs but not the Numbers

Even when a person is primarily driven by an aesthetic interest in Spiro, the methods and process they are working with is made up of math “material”. Likewise, others who are more driven by engaging specific math questions, will find that their investigations result in artful patterns that are intrinsically tied to the math. As a facilitator, we keep this in mind when observing what students are interested in, knowing that authentic questions that drive learning are not always ones that sound explicitly mathy. For example, one student was observed trying to make a rainbow spirograph. To achieve this, the student had to figure out how to change the parameters so the design got smaller and smaller without overlapping colors. For each color to follow without interruption, they lined up the teeth of the gears to where the last drawing ended before starting the drawing. In this example, like so many others, an aesthetical goal inspired technical expertise.


Is it okay to use the App this way?

Sometimes students become interested in pushing the boundaries of the App. “Glitching” the app can be a wonderful exploration of the possibilities of the digital realm. Some of the explorations we have seen students explore involve making the number of teeth on the inside gear larger than the number of teeth on the outside gear; or students will try moving the sliders while the app is drawing. This can open up “what if” conversations about what is possible when we are not constrained by the laws of physics like “what if we could draw through plastic?”


They don’t want to use the computers

It might seem like kids are exclusively focused on either the physical or the digital and not willing to move between them. This presents an opportunity to observe and talk with the student to understand what unique abilities this mode is offering them. For example, some children prefer to work in paper because they are able to use metallic colored pens on colorful paper- an aesthetic that isn’t possible with the digital app. Although moving from physical to digital is one possible pathway for this activity, it doesn’t have to be. Physical gears are a kind of technology just like a computer is and the potentials for learning and exploration are not superior in the digital space, just different.




Make Color Prints:

Some of the more intricate and colorful designs young people make on the app make for beautiful prints. We have often brought small, portable inkjet printers to afterschool workshops so that children can share their creations with friends, family and each other. Moving work off individual computer screens and into the physical workshop space gives it new, communal life to be learned from, appreciated and built upon. We once constructed a string of prints and hung them in the afterschool center like papel picado or paper flags. Here are some examples made by our students that make great prints:



Cut Stickers, Iron-on Patches, Stencils, etc:

Using the craft cutter to cut Spiro designs is another great way to bring the digital designs into physical form. Simpler designs with thick lines make great stickers or stencils. Designing with this purpose in mind can be an excellent opportunity for students to design intentionally using the knowledge they have developed around what kind of parameters would be easy for the craft cutter to cut. Below are examples of designs our student have made into stickers:



TIPS for cutting spiro stickers or stencils in the craft cutter:
  • Use dark colors like black or blue which will be easily traced in the craft cutter software.
  • Export your design and save the file where you’ll be able to find it
  • Open the file in your craft cutter’s software and use the trace function to create a cut line.


Designing for a Laser Cutter:

If you have access to a laser cutter, your gears can be cut onto 1/8” or 1/4” plexi or wood. The desktop app has a feature to design your own cut files. We also have premade cut files available on the resources page.



If you want to learn more about how spiro is designed and why it can be a good tool for learning and making… we invite you to think with these ideas.


Constructionism as a theory and a design principle: Spiro as a good tool for learning, play, exploration, & making

One way to understand why Spiro can be used to support learning in informal making spaces is through a lens of constructionist theory (Papert, 1993). Spiro embodies the constructionist theory of learning that explains that learning through the creation of projects is a particularly good way for learners to form new ideas (Hooper & Freed, 2013). Seymour Papert’s path to forming this theory of learning grew from his lifelong fascination with gears and their ratios (Papert, 1980). Constructionism is both a theory of learning and a principle for designing learning environments. As a theory of learning, constructionism explains that through play with making spiro shapes, learners form their own mathematical ideas by making sense of the patterns within their designs. As a design principle, constructionism uses computational ideas as tools for learning math through making and play. For example, in Spiro the process of working with physical and digital tools helps learners to create their own mathematical ideas as they make their designs (See section on Where’s the Math? for more on hypotrochoid curves). From a constructionist perspective, it is important for teachers to support learners in forming their own questions and using the tools to further investigate designs that are interesting to them.


Eisenberg’s thinking about the power of tools that relate the physical and digital

Another important aspect of the Spiro exercises is what Eisenberg called “bi-directional digital to physical fluency” where young people can learn how digital activities relate to physical representations of a practice and vis-a-versa. Important to this practice is how educators choose tools and materials to support young peoples’ inquiries into STEM and computing practices (Eisenberg & Eisenberg, 1999; Blikstein, Kafai & Pea, 2019). In particular, Spiro has complementary physical and digital tools that are related by offering different ways to create similar representations. Several lines of research have formed along these lines and found that creating spaces for reflection, communication among the group, and time for iteration allows young people to blur the lines between art and math and between individual and group understanding and pursuits (Hooper & Freed, 2013, Sherry Hsi, Eisenberg, Buechley and Jie Qi). Through fluid play and construction, the mathematical and artistic inquiries become intertwined often informing one another. In settings where young people have engaged in Spiro, explorations across physical and digital spaces while also sharing in a collective activity allows for artistic and mathematical insights to emerge (Vossoughi, Hooper, & Escudé, 2016).


Support for making environments nurture equity and inclusion

The collaboration and communication that can happen when learners are engaged in Spiro is a particularly good way to support equity and inclusion within maker spaces. Specifically, support for equity and inclusion can happen through countering racialized discourses about who can develop complex mathematical ideas, how mathematical inquiries can begin, and highlighting the diverse forms that mathematical thinking can take. Spiro activities in maker spaces can become an example of nurturing cultural ways of knowing intertwined with constructionism (Hooper, 1998).

Learners who create Spiro inquiries are often working through complex mathematical problems regardless of their levels of achievement or mathematical identities in school. Through engaged practices outside of school, young people are learning math and developing mathematical practices without also confronting individualized and racialized discourses that often circulate in math classrooms (Nasir et al., 2008; Shah & Leonardo, 2017). Spiro is one such engaged practice that can be used to shape mathematical experiences connected to personal, artistic, and creative endeavours, while subverting racialized discourses about mathematical ability. For example, Nasir et al. (2002) examined the domino play of African American young people where they developed complex mathematical strategy through joint problem solving and traditions of game play rooted in cultural histories of the African diaspora. Like dominoes, Spiro offers opportunities for mathematical inquiry alongside collective joy and creative expression. The way that young people can engage with mathematical practice outside of school can change the paradigm of what it means for young people to be mathematical.



Paula and Natalie created the Spiro activities through years of working together at the Exploratorium and in many other settings. Their work with young people doing Spiro Inquiry allowed them to see the possibilities of teaching and learning with these exciting concepts and tools.


We designed the Spiro Inquiry as a way to help schools and other learning environments to become places where people are excited about figuring things out and building on their interests. We wanted to support learners in developing fluency with technologies that are accessible to them for building new ideas and to encourage ways of working together. If people really are attuned to listening to each other’s ideas, then moments can emerge where the group thinks together. We also have a commitment to equity that is embedded in how we envision Spiro will be used. Equity is not a hook. Equity is not a thing. It is a stance. It is a characteristic of an experience. Equity is created within learning environments that are open to cultural connection and through providing multiple modes of access to ideas. By creating a culture of learning in your classroom that says ‘we’re going to develop very sophisticated ideas together.’Dr. Paula Hooper


We get questions about what math is this? And how do you get them to the math? We were asking that at different points, too. It just turns out the more we have explored it, there is a lot of different math in it. And then there is algorithmic thinking and there is parametric thinking. I think if you speed to quickly to “this was the connection you were supposed to make” or “this is the formalized version of the math that is present in this” you are going to miss a lot of the discoveries that are possible. It turns out there really isn’t just one answer.Natalie Freed


HOW IS IT High Tech Low Cost?

We use the term High Tech Low Cost for projects that use “high-tech, low-cost” digital tools while engaging in high-complexity thinking and creating. We believe that cost shouldn’t be a barrier for meaningfully including technology in creative learning.

The Spiro Inquiry is a low-barrier entry point for youth to create and explore with both digital platforms and physical objects in tandem. Working with physical gears, pens and paper invites attention to the coordination between hands and eyes, keeping the teeth of the gears engaged, and noticing how the shapes change depending on which gears are used. In the digital Spirogator app, students can investigate the same questions and designs using an expanded set of parameters to work with that can be more precisely adjusted. The app can also be used with fabrication tools such as a craft cutter, printer, sticker maker, or laser cutter. Our students have turned their designs into stickers, stencils, gift cards and wall hangings. The Spriogator app is free and is available here.

Thanks to the generous funding of the National Science Foundation.