Two-Legged, Four-Legged, Winged, Finned: Patterns from Indigenous Art

Back when we were all together, I’d often stop on my way in or out of DEC to play with the 3-D printed First Nation shapes on display. These manipulatives were a collaboration between Nadine McSpadden (Aboriginal Helping Teacher), Eric Bankes (ADST Helping Teacher), and the Bothwell Elementary community (Bea Sayson, Principal). Like others who passed by, I just had to rearrange them to create repeating patterns or symmetric designs.

Photo: Nadine McSpadden

Before having students explore mathematics using these materials, it’s important to first teach the cultural significance of Indigenous works of art. In Surrey, we work and learn on the unceded shared territories of the Coast Salish. We acknowledge the Katzie, Semiahmoo, and Kwantlen First Nations who have been stewards of this land since time immemorial. 

Students should understand that, although there are similarities, not all First Nation art is the same. Both Coast Salish and Northwest Coast art reflect a worldview of connection to the land and environment. There are differences in design: Coast Salish artists use three geometric elements — the circle (or oval), crescent, and trigon — whereas Northwest Coast artists use formline — the ovoid and U-shape. The use of circles, crescents, and trigons is unique to the Coast Salish! These elements suggest movement and make use of positive and negative space. In his video covering Coast Salish design, Shaun Peterson invites viewers to “imagine a calm body of water enclosed by two borders and dropping a pebble in to create ripples that carry the elements away from the centre.” Just as there is diversity within both Coast Salish and Northwest Coast peoples, there is diversity within both Coast Salish and Northwest Coast art (e.g., compare the Northwest Coast styles of the Haida and the Tsimshian).

Patterns play an important role in aboriginal art and technology. Coast Salish art could provide opportunities for students’ across the grades (and into Pre-calculus 12!) to expand their ideas about “what repeats.” Dylan Thomas is a Coast Salish artist from the Lyackson First Nation whose work in silkscreen prints, paintings, and gold and silver jewelry is influenced by Buddhist imagery and M.C. Escher’s tessellations (see Mandala or The Union of Night and Day or Salmon Spirits or Ripples or Swans or…). Share this video in which Dylan Thomas talks about connections between geometry, nature, and art as well as the importance of noticing and wondering (4:00-4:40) with your students. In Mandala, Pythagoras — or a ruler — tells us that the ratios of successive diameters of circles or side lengths of squares is √2:1. Have your students investigate this relationship. This illustrates that sometimes it’s the repetition of a rule that makes a pattern a pattern. To learn more about the artist’s interest in mathematics, I recommend reading his essay on the topic. Now is a perfect time to remind students of protocols: students should not replicate a specific piece but can instead create their own piece that is “inspired by…” or “in the style of…”; if displayed, an information card acknowledging the artist, their Nation, and their story should be included.

I’m really interested in geometry and the reason I think I am is geometry is nature’s way of producing really intricate and beautiful things. I hope that when someone sees one of my pieces they see the correlation between what I designed and what you see in nature, these sacred geometries that have shown up in nature since life evolved. And I’m hoping that when they can look at my piece they can take that wonder into their everyday life and start noticing the things that I notice and the things that inspire me.

Dylan Thomas

My numeracy colleague, Jess Kyle, recently created a lesson around the 3-D shapes above to teach students about Coast Salish culture and repeating patterns with multiple attributes (shape, colour, orientation). I wanted to expand on this lesson and zoom out from these shapes to the animal forms seen in Coast Salish art. These animals — two-legged, four-legged, winged, and finned — are connected to the land. I’m imagining these math investigations within a classroom where learners understand that animals were and continue to be an important part of the lives (and art) of First Peoples. For example, see Maynard Johnny Jr., Coast Salish, Kwakwaka’wakw, talk about his work Ate Salmon, its past-tense play-on-words title displaying humour while addressing the effects of overfishing and stock depletion on Indigenous communities (3:00-4:00). In many First Nations, certain animals are significant or sacred to the teachings, histories, and beliefs of that Nation. Each will have their own protocols around ways in which these animals are portrayed. In some parts of British Columbia animals appear on crests and regalia while in other parts of Canada animals are sacred gifts from the ancestors.

The City of Surrey has commissioned several public Indigenous works of art. Four Seasons, by Brandon Gabriel and Melinda Bige, Kwantlen First Nation, is located in the Chuck Bailey Recreation Centre. 

Photo: City of Surrey

I have some mathematical noticings and wonderings but, again, it’s important to first teach the cultural context and meaning.

Throughout this cancellation of in-class learning due to COVID-19, Surrey’s cultural facilitators have been creating and sharing videos to show and discuss with your students. Chandra Antone, Squamish First Nation, shares her teachings about drumming with us in the videos “Honour Song” and “Animal Hides.” As well, Surrey’s Aboriginal Learning Helping Teachers have generated sets of questions to ask your students about each of these videos.

Display images (below) of the four drums and ask “What do you notice? What do you wonder?”

Students might notice the blues, greens, yellows/whites, and reds/oranges; they might wonder if these colours represent winter, spring, summer, and fall. They might notice the moons (“Why just two?”), two wolves, four salmon, and trees/leaves and wonder how they tell the story of the four seasons. They might also wonder “How big are they?” (30”), “What are the drums made of?” (buffalo hide) or “Who is the artist?” Introduce your students to Mr. Gabriel through this video:

We wanted to make sure that we captured the essence of the space that we were in, that Surrey didn’t begin as Surrey, that its beginnings are much more ancient and go back many more years than the current incarnation of it. This place is very special for Indigenous people — it was also home to multiple Indigenous communities that were established here for thousands of years — so we wanted to make sure that we were honouring those people in a way that was respectful and dignifying to them. We thought, what can we use as part of the narrative that we’re going to tell with these drums that not only speaks to the Indigenous community that’s always been here but to the people who now call this place home?

Brandon Gabriel

Students may also make many mathematical observations. For example:

  • in the winter drum, there is line symmetry
  • in the summer and fall drums, there is rotational symmetry
  • in the spring drum, there is line symmetry in (just) the moon and rotational symmetry in (just) the surrounding running water design
  • in the summer drum, there are two repeating yellow-white patterns (salmon and border)
Line Symmetry
Rotational Symmetry

Again, students should not replicate Four Seasons but can instead draw their own symmetric piece that is “inspired by/in the style of Brandon Gabriel and Melinda Bige, Kwantlen First Nation.” Challenge students to use pattern blocks to build designs that satisfy mathematical constraints such as:

  • has more than three lines of symmetry
  • has rotational but not line symmetry
  • has oblique — not horizontal or vertical — lines of symmetry
  • order of rotation is three/angle of rotation is 120°
inspired by Four Seasons, Brandon Gabriel and Melinda Bige, Kwantlen First Nation

For more symmetry in Surrey Public Indigenous Art, seek out and visit:

Like night following day (or moon following sun), the cyclical changing of the seasons is something that young children can connect to when introduced to the concept of patterns. With changes in the seasons comes changes in their own lives. This is an opportunity for students to learn how seasonal and environmental changes impacted the village of qəyqə́yt (now known as Bridgeview) and continue to impact the lives of First Nations peoples today.

We Are All Connected to This Land by Phyllis Atkins, Kwantlen First Nation, is installed on a small bridge on King George Highway spanning Bear Creek. The design features three salmon (one male, one female, one two-spirited), a sun, an eagle, a moon, and a wolf, cut from powder-coated red aluminum and mirrored on both sides of the bridge.

Phyllis Atkins at blessing ceremony for We Are All Connected to This Land
Photo: Surrey Now-Leader
Photo: City of Surrey

The animals are described on the artwork’s page on the City of Surrey website:

“Salmon are resilient creatures that make an arduous journey to return to their freshwater spawning grounds, such as Bear Creek, to give new life and sustain eagles, bears, wolves, and people. The wolf represents the teacher and guide of the Kwantlen People while the eagle flying closest to the sun is carrying prayers to the Creator. The inclusion of Grandfather Sun and Grandmother Moon contrast day and night and indicate the passage of time.”

Teachers should avoid giving “meaning” to each animal as it often leads to appropriating spirit animals. Instead, ask “Can you think of characteristics of each animal that might be important?”

What if these figures were the core of a pattern? What if, like Nadine’s 3-D shapes at the top of this post, we could pick up and play with these figures? We could create repeating patterns like salmon-eagle-wolf or finned-winged-4legged. We’re not limited to left-to-right patterns arranged in a line. Different displays of patterns will bring to light different patterns. For example:

AB

Maybe this example better illustrates this idea:

ABBC three ways

In the second and third arrangements I interrupted the black-red-red-white pattern core in the first row to offset the pattern in subsequent rows. What (new) patterns can you find? What would the fourth arrangement look like? What’s the pattern in the patterns? Like the idea of patterns as “ripples that carry the elements away from the centre” above this structure provides us with new ways of thinking about the core of a pattern: we can think in terms of repeating vertical columns just as we would if we were bead looming. (To learn more about bead looming, please register for Nelson’s Culturally Responsive Math webinar series. It’s free!)

Teachers can use First Nation rubber stamps — available from Strong Nations — to explore repeating patterns of animal images. While we strive to embed local content, this is not always possible so we may blend Coast Salish and Northwest Coast art.

AAB

A playful approach is to begin a pattern — say wolf, raven, … — and ask “What comes next?” Some students will suspect an AB pattern and predict wolf. Others will suspect that you’re trying to trick them by not revealing the entire pattern core; they might predict raven (ABB) or orca (ABC). Ask “How confident are you?” Repeat this a few times. Suppose that you’ve revealed wolf, raven, wolf, raven, wolf, raven. By now, students will be very confident that wolf will come next. Mess with them: add bear instead. Ask students “What’s my pattern rule? Would you like to revise your thinking?” and have them share their conjectures. Next, add eagle. Can students identify the pattern as 4legged-winged? And what if we throw colour or orientation into the mix? Multiple attributes can add ambiguity to pattern tasks. Invite students to use these stamps to create their own repeating patterns.

Beginning in Grade 2 (and continuing into Grade 10), students learn about increasing patterns. In Grade 2, it is expected that students describe the salmon pattern below as “start at 3 and add 1 each time”; in the upper intermediate grades, students describe the pattern as n + 2; and in Foundations of Math and Pre-Calculus 10, this is formalized as slope (or rate of change) and y-intercept (or constant).

3, 4, 5, …

Presenting only the first and second terms of a pattern is another way to add ambiguity. (For example, “Extend the pattern 5, 10, … in as many ways as you can.️”) I’ve been playing with this approach to visual patterns. Take a moment to consider the pattern below. What comes next? What else might come next?

You might have noticed that three tiles were added and imagined a linear pattern — 3n as either n groups of three or three groups of n:

You might have saw this as doubling and visualized an exponential pattern — 3(2)ⁿ ⁻ ¹:

Or you might have spotted squares and pictured a quadratic pattern — n² + 2:

In later grades, these more complex patterns (quadratic, exponential, triangular numbers, Fibonacci) can be introduced. Again, there’s a chance to spotlight First Nations art. Here’s a different arrangement of 3, 6, …

What comes next? What else might come next?

(If there’s a way to see a quadratic pattern in this arrangement, I can’t make it out.)

I’m more than a bit apprehensive about sharing these last two examples. They feel inauthentic: swap in dots for the images of animals above and the task remains the same. However, in using these images and first teaching their cultural significance, I’m hopeful that this communicates my respect for First Nations culture, especially to Indigenous learners (and outweighs my concerns about curriculum design).

Huy ch q’u Nadine McSpadden and Heidi Wood for continuing to help me make connections between the cultural practices and perspectives of First Peoples and the teaching and learning of mathematics.

Egg, Head, …

Take a moment to think about the following image:

#whatrepeats? #patternchat

What comes next? What comes before? How do you know?

You might have sensed (the start of) a repeating pattern. Whether you considered the materials that make up the egg cups (glass, porcelain, …) or the position of the eggs (down, up, …), it’s a simple AB pattern. Or rather, like 🍀💎🍀💎🍀💎…, two synchronous AB patterns. If you were to extend the pattern, you’d get this:

AB; AB

Not so fast. Check out the video in the following tweet:

An AB pattern is maintained in the materials: still glass-porcelain. But the video hints at a new possible pattern–an ABAA pattern–with respect to the elliptical “dome”: egg-head-egg-egg.

AB; ABAA

Again, not so fast. These first four elements may not be what repeats; they may not be the pattern core. What if the pattern core were instead egg-head-egg-egg-egg-head (all the while still maintaining glass-porcelain)?

AB; ABAAAB

Patterns repeat. Repetition is what makes a pattern a pattern. Sometimes items repeat, sometimes a rule (e.g., add 3 each time) repeats. How would you describe what repeats in the following pattern?

All of these possibilities illustrate that without knowing what repeats, you can’t know for certain what comes next. For example, consider the following open question: Extend the pattern 5, 10, … in as many ways as you can. Common classroom responses include: 5, 10, 5, 10, 5, 10, …; 5, 10, 25, 5, 10, 25, …; 5, 10, 15, 20, 25, …; 5, 10, 15, 25, 40, …; 5, 10, 20, 40, 80, …; etc. (Variation: Extend the pattern ▲◾️… in as many ways as you can.)

The two attributes in the egghead examples–container and “contents”–made the task more interesting. In the classroom, this plays out by looking at repeating patterns with multiple attributes (i.e., colour, shape, size, orientation). Consider the pattern below:

What’s missing?

What’s missing? If you focus on colour, it’s an ABC pattern; it must be teal. If you focus on shape, it’s an AABB pattern; it must be a triangle. If you focus on orientation, it’s an ABBA pattern; it must “sit” on a vertex. If you hold all three asynchronous patterns in your mind, it must be a teal square resting on a vertex (a/k/a “diamond”). But I’m not looking for one right answer. In the classroom, I’d happily accept a teal triangle (or circle) from a student who sees a teal-orange-green pattern; an orange (or purple) square from a student who spots a triangle-triangle-square-square pattern; etc. If the claim is true, the answer is correct.

Pattern Fix-Its present another opportunity for students to examine patterns involving multiple attributes. Here, a pattern is messed with by adding or removing an element, changing one or more attributes of an element, or swapping the order of two adjacent elements. The math picture book Beep Beep, Vroom Vroom by Stuart J. Murphy provides a context: Molly plays with her big brother’s toy cars and must put them back in the right order before he returns. Using this context, I swapped the last two cars in a big-small and yellow-blue-green pattern:

Can you fix it?

Press Here by Hervé Tullet also includes some mixed-up pattern pages. That probably inspired my shaking effect here:

Like Which One Doesn’t Belong?, these questions allow all students to confidently contribute to and benefit from the discussion, whether they notice one or many patterns, whether they attend to simple (colour and shape) or more challenging (orientation) attributes, or whether they examine single or multiple attributes at a time.

* * ** *** ***** ********

I’d be remiss not to include Marc’s tweet somewhere in this post:

[TMWYKS] Rainbow Loom

Christopher Danielson brought you #tmwyk, or talking math with your kids. I bring you #tmwyks, or talking math with your kid sister.

It happens to every parent, I think: the kid says something and nobody has to ask “Where’d she hear that?” Maybe it’s the kid’s choice of words. Or maybe it’s the tone, pitch, or rhythm that gives you away. Rare is it for me that these are proud parenting moments.

A recent exception:

Gwyneth (9 years old): What patterns do you see?

Rainbow Loom

Keira (6 years old): Red, white, yellow, red, white, yellow, red, white, yellow.

Gwyneth: Great! Can you find another pattern?

Ann, Brad, Carol, …

One of my favourite open questions we present to teachers:

Extend the pattern Ann, Brad, Carol, … , in as many ways as you can.

That’s it. Simple, but brings out some big ideas.

So what’s next? Daniel gets a lot of early votes: starts with D, male, six letters. At some point, the increasing pattern–start at three letters and add one each time–becomes challenging. Take Elizabeth. Starts with E? Check. Female? Check. Seven letters? Crap. Extending the pattern in this way eventually means hyphenated names.

Ted
Wait; was it any of those names with a “Lynn” after it?

After exhausting Ann, Brad, Carol, … as an increasing pattern–Eleanor!–teachers get creative with repeating patterns.

For example, looking at one attribute:

  • Aaron, Blake, Caleb (ABC)
  • Olivia, Jackson, Isabella (female-male-female)
  • Max, Liam, Jacob (3-4-5)

Looking at two or more attributes:

  • Andrew, Brooklyn, Christopher (ABC & female-male)
  • Ava, Bono, Chloe (ABC & female-male-female & 3-4-5)

What if Ann-Brad-Carol wasn’t the core of the pattern?

  • Ann, Brad, Carol, Connor, Amy, Bryn, Caden, Carter (ABCC & 3-4-5-6)

A different attribute:

  • Ann, Brad, Carol, Elijah, Genevieve (1-1-2-3-5 vowels)

Not mathy enough for you? Remember, not all teachers will have a positive attitude towards mathematics. This is a safe icebreaker. You can always follow it up with the mathier “Extend the pattern 5, 10, 15, … in as many ways as you can.”

The big idea? Patterns involve something that repeats. Sometimes items repeat, sometimes its the rule that repeats.

Ann, Brad, Carol, … can focus teachers/students on another big idea: the way you show information can make patterns easier to see. Moving from names to SET, spot the pattern in the photos below:

SET1

SET2

When I last posed the Ann, Brad, Carol, … problem, I encouraged teachers to rearrange the names to highlight patterns. One teacher connected this to 100 charts–an aha moment for her.

Big ideas above paraphrased from Marian Small’s Big Ideas.

This is part of this.

iPad as Mathematical Communication Tool – Part Deux

I have been learning about educational uses of the iPad. My daughter has been learning about repeating patterns at school (Grade 1). Also, she has been asking me to show her how to use iMotion. A win-win situation.

She built four patterns, taking a photo each time she added a piece. Then, she created a video which I dragged into iMovie. Finally, I recorded her as she talked about her patterns. The movie would be better if the audio were synced to the video, but I wanted to see what we could create in ten minutes. Here it is:

In primary classrooms, students could share their videos and have classmates describe or translate the patterns. Similarly, in high school mathematics classrooms, students could build functions and have classmates determine equations. See an interview of UC Berkeley Math Education Professor Dor Abrahamson for the inspiration behind this idea.

These student-created movies could be used by classroom teachers to assess what students are able to do. There are nine mathematics learning outcomes in the BC Kindergarten IRP. One addresses Patterns:

B1 demonstrate an understanding of repeating patterns (two or three elements) by
– identifying
– reproducing
– extending
– creating
patterns, using manipulatives, sounds, and actions [C, CN, PS, V]

In Grade 1, two small changes are made to B1 and a second PLO is added:

B2 translate repeating patterns from one representation to another [C, R, V]

What judgements could you make about my daughter’s performance in relation to the prescribed learning outcomes? A rhetorical question – I’m not expecting or even wanting a reply.

My daughter also told me that sometimes shape and size can be used to describe patterns (e.g., “circle, circle, square, circle, circle, square” or “small, big, small, big”). Our movie doesn’t demonstrate this knowledge. This speaks to the importance of having conversations with our students – from Kintergarten to Calculus.