Putting Sums back into Summer

by Scott Taylor

Colby College

Waterville, ME

 Every math teacher hears the “What’s it good for?” complaint. Even elementary students want to know what math is good for. But children, especially those who are at risk of not succeeding academically, have little prior math knowledge for us to draw on and rarely care about such adult concerns as being a good employee or making wise financial decisions, much less applications to physics or data science. But nearly every child cares deeply about the quality of their artwork, their ability to enter into a story, their ability to make beautiful music, and having fun. Through my experiences developing a summer camp, I believe that tapping into these desires for beauty, movement, and explanation can support children who are in danger of future academic failure.

The Sum Camp Story

 I live in Waterville, Maine a city of about 17,000 people in central Maine. Like many rural cities across the U.S., it is a wonderful place to live and work, but it has its challenges. A large percentage of families live in poverty and the vast majority of elementary school children test below grade-level in math. Our public schools have many dedicated, excellent teachers and administrators, but resources are stretched thin. As a professional mathematician watching my own sons move through the public school system, I was confronted by the question: “How can I help?”

An answer came during another contentious school board meeting. During a lull, a volunteer in the schools, Sara Taddeo, shared with me an idea she’d had for a summer math camp that would use the visual and performing arts to help children develop basic math skills. She believed that trauma, and its effect on executive functioning, was at the root of why many children have difficulty in math. She was interested in the ways that the arts can help people of all ages process trauma, and believed they could also help children mathematically. Her thoughts dovetailed with some of my own reading and thinking about ways to expand students’ experience of math.

We looked to local resources to get the camp running. With a one-year grant from the Davis Family Foundation (Davis Family Foundation  https://www.davisfoundations.org/) and additional assistance from the local public schools and Colby College, we were able to pull together a small team of incredible art and math teachers from area public and private schools. Sum Camp was born!

We recruited 20 children to participate in Sum Camp. Most of them were identified by a mathematics specialist as currently mathematically weak. But all of them did math for 5 hours every weekday for an entire month during the summer of 2019 — and they loved it!

The vision

 At Sum Camp, a month-long summer day camp for rising fourth and fifth grade public school children who are behind grade-level in math, we use music, art, theatre, and math games to immerse children in mathematics. We design each activity to strengthen grade level mathematics and/or to help children develop a growth mindset.

In his famous essay On Proof and Progress in Mathematics  (https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf), William Thurston (1994) discusses the plethora of ways that mathematicians think about mathematics. “People have very powerful facilities for taking in information visually or kinesthetically, and thinking with their spatial sense.” Children who grow up playing board games are constantly confronted with physical realizations of number lines (for instance, as scoring tracks) and simple probability. Part of the goal of Sum Camp was to give children who may not have had those opportunities the experience of working with abstract mathematical concepts in fun, spatially embodied ways.

A large number of children in Waterville come from traumatic backgrounds. One effect of trauma is to suppress reasoning abilities and executive functioning. The arts are a demonstrated method of helping people of all ages process trauma. They provide an approach to learning and internalizing math that has the potential to partially circumvent the effects of trauma. We saw this at work in 2019: children who were prone to outbursts could focus on mathematics embedded in art, music, and theatre.

We have learned a lot from the “growth-mindset” approach promoted by Carol Dweck and Jo Boaler. Prior to the start of the 2019 camp, we interviewed all of our campers. Despite the fact that most of the campers were currently behind grade level in math, they uniformly said they loved math. In fact – division was a favorite! They loved how math helped them make sense of the mysterious world of numbers. But when we probed a bit deeper, we could tell that almost all of these children were at risk of not succeeding mathematically in the long-term. The simple question, “How do you feel when you get a math problem wrong?” instantaneously elicited tears. As all mathematicians know, being wrong and learning from mistakes are necessary components of mathematical success. Over the course of the camp, we helped children to learn to embrace mistakes, analyze them, and learn from them. As Boaler writes in What’s Math Got to do with It?, “We need to change students’ mind-sets so that they know it is productive to struggle and make mistakes and that they should feel comfortable doing so.” (p. xx)

How it works

 The morning starts with music. Kids sing, dance, and make music: all the while doing mathematics. Their singing is explicitly connected to moving back and forth on the number line, often skip counting forwards and backwards by 2s, 5s, 10s, and sometimes 7s! They also learn the basic units of rhythm and used them to practice using fractions with small denominators. Musical notes are represented using both musical notation and with fractional notation. We begin with whole notes (representing a unit) and work our way to using half notes, quarter notes, eighth notes, and triplets, corresponding to fractions with denominators of 2, 4, 8, and 3. Children initially learn the rhythms by echoing the music teacher through speaking, singing, clapping and drumming and eventually design their own rhythms using both musical notation and mathematical notation. One child was so inspired, he wrote his own song about learning math and all the kids sang it at the final performance.

Children learn scales and rhythms easily and learning to notate them is not difficult (likely because of prior exposure in school music classes). The corresponding mathematical ideas then seem natural and relevant. Certainly care is needed in how lessons are designed. Musical and mathematical notations do not align perfectly: if one note is a major third above another note it is only two whole steps above it and how one represents note lengths mathematically should depend on the time signature. Our music educator (Christine Little) is adept at navigating these subtleties and keeping the focus on helping the children develop an inner sense for the relationships between the notes and the relationships between the numbers; e.g. a half note is twice as long as a quarter note and 1/2 is twice as large as 1/4.

After music class and a snack break, we spend 90 minutes playing math games. The games are designed for the camp by Thom Klepach, a biology PhD with a deep love for math and games. The games incorporated a mix of running games and sitting games and all are explicitly connected to Common Core standards for children in grades K – 3. I’ll share two of my favorites with you.

We play a version of the classic “night watchman/statue” game, which Thom calls “Statue Times.” The children are divided into four teams, each positioned in the corner of the tiled gym floor. Thom, dressed in academic robes and with a wand, stands in the center of the rectangular playing area. When his back is turned, one child from each team creeps towards him. If Thom sees them move, they become frozen. The child’s score is the number of tiles enclosed in the rectangle determined by their starting position and their ending position. To figure out their score, the child has to multiply the number of tiles along each side of the enclosed rectangle and compare that to the number determined by directly counting the number of tiles. Frequently, the children obtain different answers, as the multiplication usually involves two digit numbers. When they get the wrong answer, it provides an opportunity for a growth mindset in figuring out what went wrong and gives an opportunity for number talks with camp staff.

One of the most popular outdoor games is called “Super Mega Kwa-kwa Ball!” The exact rules evolve over the summer in response to suggestions from the children. Roughly speaking, it is like kick-ball but with number lines and a gallon of craziness thrown in. We built two giant number lines out of rope and numbered cattle ear tags. These are laid out along the 1st and 3rd base lines of a kickball field. The children are split into two teams, with any child that doesn’t want to run designated as a “statistician.” The pitcher for the team in the field rolls a kickball to the “batter,” who kicks it. The batter then grabs a frisbee, a football, and a 20-sided foam die and throws them into the field. Pairs of fielders are assigned to each object. The fielders run orthogonally from their thrown object to the number lines, read off the numbers they arrive at, and report them to the statisticians. While they are doing this, the runner proceeds around the bases. Between each pair of bases they need to do some wild activity, such as run with a ball between their knees or balance a baseball bat on their hand. The batter’s score is determined by how many runners they get in and the number from the die that had been thrown. We put all the numbers the statisticians record into a spreadsheet and plot the results. In subsequent days, we discuss the data with the children. The children find, for instance, that most of them are right handed, based on the football’s coordinates. This activity connects to the grade one and two standards for using a number line and emphasizes the way that abstract representations of physical situations can help us see patterns. Of course, it also looks forward to the Cartesian plane that students will encounter in later grades.

After math is lunch and recess, and then the children are split into two groups. One group goes to theatre class and then art class, while the other group attends classes in the other order. Theatre is devoted to team-building and role-playing responses to failure. These activities are explicitly connected to building good math habits and skills. One activity involves using a goniometer (an angle-measuring device used by physical therapists) to measure the angles formed by elbows, knees, and necks during theatre sports. Over the weeks, the children work with the theatre teacher to write a mathematical play. They perform the play on the final day of camp to an audience of about 100 family and community members. In 2019, the play involved two fairy tale realms that had to work together to solve a math problem that would save the world.

Art projects are designed to emphasize the mathematical aspects of design, measuring tools, and working with different units. The kids’ favorite is tie-dyeing camp T-shirts. Before applying dye, they sketch what they think their shirt will look like and what proportion of the shirt will be each color. After the shirts were done, they overlay a grid of squares to estimate what proportion of the shirt each color actually occupied. The children discover how far off their predictions are and that there is a method for actually determining the (approximate) proportion each color occupies. The individual nature of their designs gives them ownership over both the problem and its solution. This project reinforces grade 1 and 2 standards for counting, adding, and multiplying and grade 3 standards for area.

One of the threads that held the different aspects of camp together were the camp journals. Each child had a journal for working math problems and recording different aspects of camp life. Several pages were devoted to keeping track of their camp points. Camp points could be earned by analyzing a mistake in a math problem, helping another child with a math problem, or helping out a teacher. Campers had to add all their camp points by hand and have camp staff sign off on the result. By the end of the four weeks, all the campers were adding numbers with 3 or more digits: the biggest numbers they had ever worked with. Many kids identified working with numbers that big as one of their favorite aspects of camp.

In Conclusion

 I believe that the principles behind Sum Camp — of making mathematics a full-spirited, full-bodied experience — are critical to its success. Parents describe it as “life-changing” and students say that it transformed their motivation to learn mathematics. Common comments are that they now understood why math is worth learning. Most children say that Sum Camp increases their ability and desire to do math.

Robert was a rising 4th grader who often became very frustrated with his propensity to do math incorrectly. He likely has a traumatic home situation and often blew up at himself, camp staff, and other campers. He said that Sum Camp “tested [me] to the limits. It was fun to solve all the problems.” He also pointed to one particular problem he solved in the course of the Statue Times game and said, “First I got it wrong, but I kept on trying until I got it right.” Robert linked the game to an experience of perseverance.

Charlotte, a rising 4th grader who had trouble sitting still during a math lesson, said “I used to think math was boring. What’s the point? [But now] I have realized math is an important part of life.” Her favorite activity was using fractions while measuring ingredients to make slime during art. Her kinesthetic experience with fractions may have helped her understand them more deeply than what is possible using just pencil and paper on a page.

Emma was a rising 4th grader whose body language often indicated disengagement. She said, “I used to not really care about math. But now when I go home, I get my multiplication flash cards out of my closet and do them. I want to do math.” Perhaps Emma’s experience with numbers in games like Statue Times and camp points helped her develop a disposition to seek out mathematical activity.

Amelia’s mother said, “Amelia now realizes math can be found all around her.” She also said that she herself was made more aware of the different daily applications of math. These comments suggest that the camp’s emphasis on connecting playground games and everyday experiences, like cooking, to mathematics can help students and their parents see mathematics around them.

Combining ideas from William Thurston and Jo Boaler may seem surprising, as they come from very different kinds of mathematical careers. However, there are interesting parallels between their views of mathematics, particularly with regard to expanding the channels that we use to communicate mathematics in our teaching. Both believe that mathematics can draw on observations from lived experience, and that human language is a powerful tool for understanding mathematics. Thurston emphasized different ways of thinking about mathematics, for instance, infinitesimally, logically, symbolically, or geometrically. He also described some of the many faculties we draw on when doing mathematics: natural language, metaphor, kinesthetic sense, vision, and logic. Boaler leans on Carol Dweck’s notion of growth mindset, and on developing mathematical tasks that are highly accessible and that afford multiple solution strategies. Sum Camp combines these ideas by developing mathematical experiences that tap into different senses — motion, touch, hearing, rhythm — and that lead to accessible mathematical problems that children can solve in a variety of ways. The activities are constructed so that children know it is okay to make mistakes, try again, and learn from others.

Sum Camp will return to Waterville starting this summer, funded by an NSF grant (NSF grant https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104022&HistoricalAwards=false), Colby College, and the Waterville Public Schools. We’re excited to see how creative experiences of mathematics empower children to engage with mathematics in new, life-changing, ways.

Further Reading

 One of the main pedagogical tools, Boaler promotes (and which can be used in any setting) is the “Number Talk.” These are useful in everyday conversation with any child or non-mathematical adult. The basic idea is easily absorbed and implemented by any mathematician. I’ve found the book Making Number Talks Matter by Cathy Humphreys and Ruth Parker, the original creators of Number Talks, to be helpful. Sherry Parrish has a series of books published by Math Solutions, implementing them for various mathematical content and audiences. Videos of how they work are easily found online. Many people, besides Thurston, have observed the value of an embodied approach to mathematics. The Algebra Project famously helps students understand the number line by riding the subway. Anyone working with children from distressed communities will have to confront the effects of trauma. The classic The Body Keeps Score (Penguin 2015) by Bessel van der Kolk gives a helpful overview of how the experience of trauma can affect an individual’s behavior, brain, and body. Anyone working with children who may have experienced trauma should seek out at least minimal training, so as to not exacerbate the effects. In his book, van der Kolk describes both the ways that trauma can shut down both creative and rational thinking and the positive ways the arts can help individuals process and move through trauma.

Scott Taylor is an associate professor of mathematics at Colby College in Waterville, Maine. He does research in geometric topology and teaches across the undergraduate curriculum. He works closely with the Waterville Public Schools to promote and support mathematics and was awarded the 2019 Waterville Board of Education Community Award for his efforts.

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