Calculus Is the Peak of High School Math. Maybe It’s Time to Change That

EdWeek

May 22, 2018
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For more than 30 years, calculus has been seen as the pinnacle of high school math—essential for careers in the hard sciences, and an explicit or unspoken prerequisite for top-tier colleges.

But now, math and science professionals are beginning to question how helpful current high school calculus courses really are for advanced science fields. The ubiquitous use of data in everything from physics and finance to politics and education is helping to build momentum for a new path in high school math—one emphasizing statistics and data literacy over calculus.

“We increasingly understand the world around us through data: gene expression, identifying new planets in distant solar systems, and everything in between,” said Randy Kochevar, a senior research scientist at the Education Development Center, an international nonprofit that works with education officials. Statistics and data analysis, he said, “is fundamental to many of the things we do routinely, not just as scientists but as professionals.”

He and other experts are still debating the best way to integrate a new approach in an already crowded high school curriculum. One of the most difficult philosophical challenges: how to prevent a statistics path from replicating the severe tracking and equity problems that have long existed in classical mathematics.

“There’s a sense that calculus is up here and statistics is a step below,” said Dan Chase, a secondary mathematics teacher at Carolina Day School in North Carolina, adding that he often struggles to suggest to students that, “if you are interested in engineering, that might be a good reason to go to calculus, but if you are interested in business or the humanities or social sciences, there are different paths you might go, even if you are a top-achieving math student.”

On face value, new expectations for students already seem to be moving toward statistics. Both the Common Core State Standards, on which many states’ math requirements are based, and the Next Generation Science Standards call for teaching data analysis and statistics, both on their own and in the process of learning other concepts.

But Kochevar warned: “There’s a huge disconnect; if you look closely at the science standards, they are expecting students to have tremendous faculty with using data by middle school, but if you look at the courses, it’s really not clear where those skills are supposed to be filled.”

Both sets of standards need more integration of data and statistics, he and others argue, because they were developed in the early years of the big data boom. Studies tracking data worldwide through the years have found people produced 1.5 exabytes of new data in 1999—or roughly 250 megabytes of data for every person alive—but by 2011, when states were adopting and implementing the math standards, people produced more than 14 exabytes a year. Today, people worldwide produce 2.5 exabytes of data every day, and the total data have doubled every two years.

Ironically, the rapid expansion of big data and statistics use in the broader society and economy comes at the same time American students seem to be struggling with those concepts. From 2007 to 2017, 4th and 8th students’ scores on the National Assessment of Educational Progress in mathematics fell significantly on problems related to data analysis, statistics, and probability—a decline that helped drive overall dips on the math test in 2017.

In part, experts say, that’s because statistics and data analysis have traditionally taken a back seat to calculus in high school math, and most students already have difficulty completing the classical path.

“The idea that statistics is hard is grounded in that fact that if you took statistics 10 years ago, you had to take calculus first, and the statistics used formal probability … with theorems that built on calculus,” said Uri Treisman, a mathematics professor and the executive director of the Charles A. Dana Center at the University of Texas at Austin. He’s been working with K-12 and university systems to develop a statistics pathway as an alternative to classical calculus.

It’s an idea that others have pushed back on, by situating a high school statistics pathway as either advanced material only suitable for students who have already passed calculus—or a less-rigorous path for students who can’t hack it in classical math.

“Any time you have multiple pathways, the advantaged will capitalize on one and that will become the ‘real’ one,” Treisman said. “If we are going to create data science pathways, they had better be anchored in things that lead to upward social mobility and have a rigor to them. We have to make sure new pathways have at least equal status as the traditional one—and ensure everyone has access to them. If we allow [statistics and data] to be the easy or weaker path, we relinquish the commitment to equity we started with.”

Mixed Signals in Calculus

For a picture of how severe that inequity can get, one only has to look at calculus.

Until about 1980, calculus was seen as a higher education course, primarily for those interested in mathematics, physics, or other hard sciences, and only about 30,000 high school students took the course. That began to change when school reformers glommed onto calculus as an early example of a rigorous, college-preparatory course, said David Bressoud, a mathematics professor at Macalester College and a former president of the Mathematical Association of America, who has examined the evolution of calculus studies.

“The more schools did this, the greater the expectation that they would do it” from parents, and district leaders—and in particular from colleges and universities, Bressoud said. “It’s not just math majors or engineering majors; this has become an accepted requirement for admission to top universities. You are not going to get into Duke if you haven’t taken calculus, even if you plan to major in French literature.”

Today, some 800,000 students nationwide take calculus in high school, about 15 percent of all high schoolers, and nearly 150,000 take the course before 11th grade. Calculus classes have been and remain disproportionately white and Asian, with other student groups less likely to attend schools that offer calculus or the early prerequisites (like middle school algebra) needed to gain access to the course.

For example, in 2015-16, black students were 9 percentage points less likely than their white peers to attend a high school that offered calculus and half as likely to take the class if they attended a school that offered it. And if black students did get into a class, their teachers were also less likely to be certified to teach calculus than those of white students, according to an Education Week Research Center analysis of federal civil rights data.

And despite the rapid growth of calculus as a gold standard, university calculus experts argue it is a much weaker sign that a student is actually prepared for postsecondary math in the science fields than it appears.

In fact, a new report by the Mathematics Association of America and the National Council of Teachers of Mathematics found many students who took Advanced Placement Calculus AB still ended up retaking calculus in college—and 250,000 students end up needing to take even lower-level courses, like precalculus or algebra.

In the end, the report found taking calculus in high school was associated with only a 5 percentage point increase on average in calculus scores in college—from 75 percent to 80 percent. Rather, the best predictor of earning a B or better in college calculus was a student earning no less than As in high school Algebra 1 and 2 and geometry.

So if high school calculus isn’t the best indicator of a student prepared for college-level math, what does it signify in college admissions? In a word: Money.

More than half of students who take calculus in high school come from families with a household income above $100,000 a year, according to a study this month in the Journal for Research in Mathematics Education. By contrast, only 15 percent of middle-income students and 7 percent of those in the poorest 25 percent of families take the course.

“Math is even more important to upward mobility now than it was 20 or 30 years ago, because … it’s seen as related to your general ability to solve problems quickly,” Treisman said, adding that as a result, “there’s general anxiety and panic about equity issues for anything new, even though the current [calculus] pathway is a burial ground for students of color.”

Forging a New Path

Statistics and data literacy advocates hope diversifying the field of interesting and rigorous math courses could broaden students’ path to STEM and other careers. As of 2017, the U.S. Bureau of Labor Statistics estimations showed that jobs that require data literacy and statistics are among the 10 fastest-growing occupations in the country.

“We have two paths forward,” said William Finzer, a senior scientist at the Concord Consortium, which works with school districts to improve their math curricula. “The easier one—like the path computer science took—is to develop a course or a subject area and get schools to give it time. … The problem of that is, it doesn’t spread the opportunity very widely. It becomes concentrated in the small group of kids who elect to take the course—and it’s just one more subject to take.”

Progression for Statistics and Data

EDC’s Oceans of Data Institute is building learning progressions for statistics and data literacy at different grades. Randy Kochevar, who directs the institute, said they are based on the acronym CLIP, meaning students learn how to use:

Complex, multi-variable data (“We’re not just looking at hours of sunlight and heights of bean plants,” he said);

Larger data sets than students need to answer any one question, so they are forced to sort and understand relevance;

Interactively accessed data, rather than sample graphs just written out on paper; and

Professionally collected data that forces students to think about how and why it was collected—and what biases may exist in the samples.

Finzer instead envisions a more holistic approach in which at least one class a year—be it math, biology, or even civics or history—asks students to grapple with making sense of large data sets. Such an approach, he said, “would make a huge difference, because it would mean when you came out of high school, data would not be foreign to you.”

EDC’s Oceans of Data Institute is building learning progressions for statistics and data literacy at different grades. The progression would include concepts in statistics and data literacy, but also computer science—to be able to use common programming and tools used by data professionals—and more philosophical concepts, such as the ethical use of statistics and privacy protections.

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Finding the Beauty of Math Outside of Class

Edutopia

Math trails help students explore, discover, enjoy, and celebrate math concepts and problems in real-world contexts.

©Shutterstock.com/Jon Bilous

A math trail is an activity that gets students out of the classroom so they can (re)discover the math all around us. Whether out on a field trip or on school grounds, students on a math trail are asked to solve or create problems about objects and landmarks they see; name shapes and composite solids; calculate areas and volumes; recognize properties, similarity, congruence, and symmetry; use number sense and estimation to evaluate large quantities and assess assumptions; and so on.

This is one of those creative, yet authentic activities that stimulate engagement and foster enthusiasm for mathematics—and so it can be particularly useful for students in middle and high school, when classroom math becomes more abstract.

A math trail can be tailored to engage students of any age and of all levels of ability and learning styles. Its scope and goals can be varied, and it can include specific topics or more general content. And best of all, it can make use of any locale—from shopping malls to neighborhood streets, from parks, museums, and zoos to city centers, to name a few. Any space that can be walked around safely can work.

A DAY OF EXPLORATION

My school has used a ready-made math trail designed by the Mathematics Association of America. Although it’s designed for Washington, DC, its general ideas can be applied in any city or town. It could be particularly appealing for teachers because it’s open-ended and can be tailored to the curricular and educational needs of the students. In addition, as schools from all over the country visit the nation’s capital, it provides a math activity that can be easily added to the many history, art, and civics lessons elicited by such a field trip.

Our math trail is loosely structured on purpose: As the whole Grade 7 takes part in this trail and as many chaperones are not math teachers, we make it clear that the purpose of the day is for the students to explore, discover, enjoy, and celebrate the beauty of math and its presence all around us. Using the MAA’s Field Guide and a map of the National Mall, each group spends the first hour of the day planning their route. How they spend their time is up to each group—this freedom is what the students like the best about the day.

Some students want to visit the newly renovated East Building of the National Gallery of Art—a treasure trove of 2D and 3D geometric ideas, patterns, and artifacts. Can we calculate or estimate the volume of its octagonal elevators or triangle-base stairwells? Even without measuring tape? Others can’t wait to ride the Mall carousel while thinking of the trigonometric function the ride’s motion describes.

Other students look at a more pressing problem: Considering the scale of the map, what is the shortest route to Shake Shack from the Sculpture Garden? Is that path unique? Is the distance the same in Euclidean geometry? Walking at a fast pace—say 4 mph—how long will it take to get there? Can we get our food and make it back to the bus on time?

Another group might estimate how many people visit the Air and Space Museum in a day. And how do we go about solving this problem?

©Shutterstock.com/Lissandra Melo

In Washington, DC, students can use their math skills to discuss architectural elements like the curves at the Hirshhorn Museum.

Another group will examine the shape of the Hirshhorn Museum complex. Why does it look so appealing?

Where do we see symmetry in the World War II Memorial? Where do Fibonacci numbers and/or fractals appear in the National Garden? What’s the scale of the Voyage model of the solar system along Jefferson Drive? Given that scale, can we estimate the distance between Mars and Saturn? Turning our thoughts back to Earth, how is the map of Washington, DC, structured? If transferred to Cartesian coordinates, what is the origin? The possibilities—in DC and in your town—are probably endless: Teachers can tweak all of these questions to fit other contexts.

BACK AT SCHOOL

Each group visits various sites and takes photos, and after we’re back at school the students research the mathematical significance of the symbols or objects they’ve chosen, write and/or solve the problems they posed, annotate their photos and post them on an electronic bulletin board or map of the Mall, and express what they’ve learned and enjoyed in other creative ways such as movies, kahoots, songs, game shows, etc. These projects are shared at a later time in assembly and affirm the fun mathematical times.

There are many excellent resources about math trails, including already created trails and virtual trails as well as clear directions on how to create your own. Math trails are cooperative—not competitive, as mathematics learning is often seen—and they offer the opportunity of doing and talking about math. Making and using connections among mathematical ideas, recognizing and applying mathematics in contexts outside of math class, communicating mathematical thinking to others clearly, and analyzing and evaluating the mathematical thinking and strategies of others are all basic tenets of the NCTM process standards.

The collaborative nature of a math trail makes it a great opportunity for bonding. After our first math trail, we understood better the potential of this activity and realized that it could be integrated in our seventh grade Advisory program, particularly at the beginning of the school year, when it can help facilitate the transition into middle school.

The Way We Teach Math Is Holding Women Back

Time

March 29, 2017

A Stanford math professor encourages a different teaching approach

First Daughter Ivanka Trump and Education Secretary Betsy DeVos toured the National Air and Space Museum with a group of middle school students Tuesday, encouraging girls to pursue careers in science, technology, engineering and mathematics — even while President Donald Trump’s administration put forth a budget proposal that suggests cutting funding for education and research. There is nothing more important than advancing the STEM fields — and those groups who are underrepresented within them.

One area in desperate need of examination is the way we teach mathematics. Many Americans suffer from misconceptions about math. They think people are either born with a “math brain” or not — an idea that has been disproven — and that mathematics is all numbers, procedures and speedy thinking. In reality, mathematicians spend most of their working lives thinking slowly and deeply, investigating complex patterns in multiple dimensions. We sacrifice many people — women and students of color, in particular — at the altar of these myths about math.

Math is a prerequisite for most STEM fields, and the reason many students abandon STEM careers. In higher levels of mathematics, gender imbalances persist: In 2015, about 76% of math doctorates were awarded to men. This figure should prompt alarm in mathematics departments across the country — and encourage focus on an area that is shockingly neglected in discussions of equity: teaching methods in classrooms.

At Stanford University, I teach some of the country’s highest achievers. But when they enter fast-paced lecture halls, even those who were successful in high school mathematics start to think they’re not good enough. One of my undergraduates described the panic she felt when trying to keep pace with a professor: “The material felt like it was flying over my head,” she wrote. “It was like I was watching a lecture at 2x or 3x speed and there was no way to pause or replay it.” She described her fear of failure as “crippling.” This student questioned her intelligence and started to rethink whether she belonged in the field of math at all.

Research tells us that lecturers typically speak at between 100 and 125 words a minute, but students can take note of only about 20 words a minute, often leaving them feeling frustrated and defeated. “I’ve essentially given up in my math class right now,” another student of mine wrote. “In such a fast-paced environment where information is constantly coming at you, there just isn’t time to think deeply about what you are learning.”

The irony of the widespread emphasis on speed in math classrooms, with damaging timed tests given to students from an early age, is that some of the world’s most successful mathematicians describe themselves as slow thinkers. In his autobiography, Laurent Schwartz, winner of the world’s highest award in mathematics, described feeling “stupid” in school because he was a slow thinker. “I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent,” he wrote. “And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully.”

When students struggle in speed-driven math classes, they often believe the problem lies within themselves, not realizing that fast-paced lecturing is a faulty teaching method. The students most likely to internalize the problem are women and students of color. This is one of the main reasons that these students choose not to go forward in mathematics and other STEM subjects, and likely why a study found that in 2011, 74% of the STEM workforce was male and 71% was white.

Women are just as capable as men of working at high speed, of course, but I’ve found in my own research that they are more likely to reject subjects that do not give access to deep understanding. The deep understanding that women seek, and are often denied, is exactly what we need to encourage in students of mathematics. I have taught many deep, slow thinkers in mathematics classes over the years. Often, but not always, they are women, and many decide they cannot succeed in mathematics. But when the message about mathematics has changed to emphasize slower, deeper processing, I’ve seen many of these women go on to excel in STEM careers.

When mathematics classes become places where students explore ideas, more often than they watch procedures being rapidly demonstrated by a teacher or professor, we will start to liberate students from feelings of inadequacy. In a recent summer camp with 81 middle school students, we taught mathematics through open, creative lessons to demonstrate how mathematics is about thinking deeply, rather than calculating quickly. After 18 lessons, the students improved their mathematics achievement on standardized tests by an average of 50%, the equivalent of 1.6 years of school. If classrooms across the country would dispel the myths about math and teach differently, we would improve the lives of many students and enable the creation of a more diverse STEM workforce. It will take a generation of young, creative, adaptable and quantitative thinkers to tackle our society’s problems — thinkers that we are currently turning away from mathematics classrooms and lecture halls in droves.

Jo Boaler is a Stanford professor, co-founder of youcubed.org and author of best-selling book, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching.

How Kids Benefit From Learning To Explain Their Math Thinking

MindShift

same-different-jh

Math teachers of older students sometimes struggle to get students to explain their thinking with evidence. It’s hard to get kids in the habit of talking about how they are thinking about a problem when they’ve had many years of instruction that focused on getting the “right answer.” That’s why educators are now trying to get students in the habit of explaining their thinking at a young age. The Teaching Channel captured kindergarten and first grade teachers pushing students to give evidence for their answers in situations where there are several ways to think about a problem.

Pattern recognition is a fundamental part of mathematics and kindergarteners are not too young to notice, compare and describe simple patterns. In this video, kindergarten teacher Donella Oleston describes how she had to back up and explain to these young learners what it means to “explain your thinking,” because at first students would only answer, “My brain told me so.” With practice, she says students have gotten to deeper levels of noticing, moving past the obvious and picking out more abstract similarities and differences between two pattern sets.

 

Can Teaching Spatial Skills Help Bridge the STEM Gender Gap?

For all the emphasis placed on science, technology, engineering and math instruction, not much attention is given to a skill set that’s closely related with success in STEM: spatial skills.

The ability to mentally manipulate objects is key to success in many fields, including physics and engineering. Spatial skills are an early indicator of later achievement in mathematics, they “strongly predict” who will pursue STEM careers, and they are more predictive of future creativity and innovation than math scores. In fact, a review of 50 years of research shows that spatial skills have a “robust influence” on STEM domains.

However, women generally score lower than men on tests of spatial reasoning — particularly measures of spatial visualization and mental rotation. Some researchers point to evolution as the culprit, while others have tied the discrepancies to hormone levels or brain structure.  As one researcher put it, “Sex differences in spatial ability are well documented, but poorly understood.”

Sheryl Sorby said she’s not interested in arguing about why the gap exists because training and practice can close it.

“A lot of people believe that spatial intelligence is a fixed quantity — that you either have good spatial skills or you don’t — but that’s simply not true,” said Sorby, an engineering professor. This misperception is particularly harmful to girls who may not be encouraged to engage in spatially rich activities that would set them up for later STEM success.

“We may start with this small biological difference, but it grows because of our environment,” said Sorby.  For example, starting at an early age, boys are more likely to engage in activities that boost spatial reasoning. Research shows that boys play with spatial toys more than girls do — and spatial toys are often marketed explicitly to boys. In addition,studies find that parents are “less likely to restrain the exploratory behavior of boys,” such as allowing them to roam further from home than girls their same age.

The Ripple Effects of Spatial Reasoning

Boosting girls’ spatial skills can have a positive effect on other domains. Sorby believes that the small but persistent gender gap in standardized math scores can be largely explained by differences in spatial reasoning: Girls tend to do worse than boys on test items that have a spatial component.

A 2014 review of middle school physical science exam scores found that the gender difference boiled down to a few specific questions that required mental rotation. According to one report, “after students’ scores on the mental rotation assessment were taken into account, there was no longer a gender difference in physical science scores.”

Early in her career, Sorby wondered if spatial skills training could help colleges retain female students in engineering, a field with an acute gender disparity. As of 2011, 19 percent of all undergraduate degrees in engineering were awarded to women, and 3 percent were awarded to women of color. Sorby said that at many colleges, the first engineering courses for beginning students cover design graphics, which is highly spatial. 

When Sorby taught at Michigan Technical University, she noticed that some female students — who otherwise excelled in math and science — would struggle with the class and choose to switch majors. “They assumed they didn’t have what it took to be an engineer,” said Sorby, “when the real issue was a weakness in spatial skills.”

Spatial-test.png

From “Educational Research in Developing 3-D Spatial Skills for Engineering
Students” by Sheryl A. Sorby. 

To help her incoming engineering students, Sorby developed a “short introduction to spatial visualization” class. The course is 15 hours of instructional time —  “a miniscule amount of time” in the scheme of things — but the payoff has been worthwhile. Sorby taught students how to sketch figures from multiple perspectives, look at cross sections of objects and create 3-D objects through paper folding exercises. Students who took the class not only improved their spatial skills, but also their grades in all STEM classes improved, and they were more likely to graduate with an engineering degree.

In ninth grade at the Columbus School for Girls, students can take a version of Sorby’s spatial visualization course as a spring elective. The course is nine lessons and is taught by Linda Swarlis, director of information services. Swarlis says she often hears from graduates about how this course helped them in their college STEM classes. One young woman described how she found herself the only female enrolled in an inorganic chemistry class at a competitive college.

“The professor introduced the concept of chirality, and she recognized the concept as the right hand rule in engineering, something that she learned in her spatial visualization course,” said Swarlis.

Given that spatial skills can be learned, what can parents and teachers do? Sorby offers these suggestions:

Encourage Block Play: Playing with blocks and puzzles correlates with spatial development. Lego kits are particularly good for strengthening spatial visualization because kids have to examine a 2-D diagram and turn it into a 3-D model, said Sorby. She also recommends trying out some of the new engineering toys that have hit the market, such asGoldiblox.

Involve Girls in Practical Spatial Tasks: When planning a road trip, hand a map to your daughters and ask them to plan the route, said Sorby. When putting together a piece of IKEA furniture, involve girls in reading the instructions and screwing it together. These types of activities build skills and confidence.

Hold, Build and Sketch 3-D Objects: Sketching 3-D objects improves students’ mental visualization and rotation skills. Have children build an object out of blocks and then sketch it. Then have them rotate the object and sketch it again. Recent research also suggests that “holding an object in your hand seems to help you visualize it,” says Sorby. For example, showing students a 2-D model of a molecule does not help them nearly as much as handing them a model that they can hold, turn and examine from different angles.

Play 3-D Video Games: One study found that a mere 10 hours of “playing an action video game can virtually eliminate this gender difference in spatial attention and simultaneously decrease the gender disparity in mental rotation ability.” The authors speculate that more exposure to 3-D video games “could play a significant role as part of a larger strategy designed to interest women in science and engineering careers.”

Remember the power of expectation:  “If we have a child with poor math skills, we don’t say, ‘That’s too bad — you’ll have poor math skills for the rest of your life.’ But with spatial skills we tend to do that,” said Sorby. “Instead we need to tell kids, ‘You can develop these skills just like you develop any skill.’ ”

Want Your Kids To Do Better In Math? Have Them Trace Math Problems With A Finger

By Angela Laguipo, Tech Times | January 31, 2016

Not all kids are fond of math, and for them, solving math problems can be a tedious task. A new study suggests that students who trace certain math problems using their fingers are able to solve them more quickly and easily.

Researchers from the University of Sydney found that students who used a technique called finger tracing were able to solve math problems with more ease than others.

The researchers said that students who used their fingers to trace over examples while at the same time reading arithmetic and geometry material were able to perform better by completing tasks more easily and quickly than those who did not apply the technique.

Tracing involves using the index finger to physically trace and touch the angles of a triangle in geometry, for example. The research team believes this may help reduce the load on working memory and enhance the ability to retain complex information.

“Our findings have a range of implications for teachers and students alike. They show math learning by young students may be enhanced substantially with the simple addition of instructions to finger-trace elements of math problems,”says corresponding author Dr. Paul Ginns.

In the study published in the journal Learning and Instruction and Applied Cognitive Psychology, the researchers recruited 275 children from ages 9 to 13 years old. They discovered that tracing over math elements while reading them enhanced the children’s understanding of problems in algebra and geometry. Previous studies have also confirmed that finger tracing helps kids recognize shapes and letters.

Dr. Ginns says this simple and zero-cost teaching technique can help teachers assist students by giving them specific instructions to “trace over” important elements in mathematical textbooks.

The researchers are now looking for more ways to use the technique on more complex and harder math problems that require higher levels of cognitive ability.

They add that they are confident that the new technique can be used in the classroom setting and even in subjects other than math. Further research is needed to explore the technique.

Teaching Math With Modular Origami

Scholastic.com

By Alycia Zimmerman on January 22, 2016

  • Grades: 1–2, 3–5, 6–8, 9–12

Several years ago, I had the good fortune to attend a workshop by Rachel McAnallen (aka Ms. Math) about teaching geometry with a fun and tactile method: origami! Since then, introducing my students to modular geometric origami is one of my favorite teaching moments each year. Origami math gives my tactile and spatially gifted students a chance to shine, it helps students with sequencing and direction following, and it’s a fun way to introduce a wide range of geometry terms and concepts.

I had NEVER created origami before the abovementioned two-hour workshop. You absolutely do not need to be a talented origami artist to pull off these lessons with your students. With the straight-forward tips below and a few minutes of practice, you’ll be ready to guide your students through an origami math experience that will have them clamoring for more. (That’s when you can hand them an origami book and challenge them to figure it out!)

 

I was bursting with pride upon making my first “skeletal octahedron.” Students feel a similar sense of accomplishment when completing their origami structures.

 

What is Modular Origami?

Modular origami is the fancy name for geometric origami that is made up of many repeating “units” that are then assembled to create a more complex geometric form. Unlike traditional origami that uses a single sheet of paper to fold a figure, modular origami uses many sheets of paper that are folded into basic modules or units. Once you learn how to make the basic unit for a design, you repeat the process to make enough copies of the unit to assemble your final form. (For a look at some modular origami projects, check out my Pinterest board.)

Although the process of making the units is repetitive, I find that many students enjoy it as a calming, almost meditative process. I often introduce this activity right before standardized tests, because the repetitive folding soothes some students and gives them a purposeful active for jittery hands. I always have a few students who find folding the units to be a chore. I team these students up to divide and conquer the unit folding work and then assemble a joint final product. I’ve had so many students become nearly obsessed with folding units — they bring origami paper to lunch and recess (especially on rainy days) to get in extra folding time.

A student shows off his first modular origami creations: sonobe cubes. (See the video tutorial below to make these simple cubes.)

 

What Supplies Will We Need?

I buy very inexpensive origami paper for my students since we go through a fair amount of it, like this 500-sheet pack of 6”x6” paper. I keep a pack or two of fancier paper on hand for special projects that individual students tackle. Colored copier paper cut into squares also works well.

The only other supplies you’ll need are a Popsicle stick and a Ziploc bag for each student. The students use the Popsicle sticks to press “crispy creases” into the paper, and the bag to hold all of their units before they assemble the modules into the final design.

 

How Do I Incorporate Math Into Origami?

The math comes entirely through the discussion as you guide the students through making a module/unit. I sit all of my students down and VERY slowly go through the stepwise process of making the first module for a design. I model the process using the document camera, and I have a couple of student experts circulate to help other students who get stuck. (I pre-teach the folding process to these student experts so they are available to be my assistants.)

Before, during and after each fold, we discuss the shapes that we are pressing into the paper, we classify the angles, and I invite the students to name each step to help them remember the sequence of paper folding. This way, students can remember that “the large trapezoid comes after the double horizontal rectangle step.” By folding while discussing geometry, students are also more likely to memorize vocabulary-heavy geometry; they create kinesthetic associations to go along with the geometry terms.

Two of my student experts show off their icosahedrons. Empowering these guys to assist their peers not only helps to build their confidence, it also means that struggling students get timely hands-on support.

 

How Do I Get Started With Modular Origami?

The sonobe cube uses a very simple modular origami unit: the aptly named Sonobe unit. As a cube with six faces, this design requires six units. That makes for a pretty short project. Students can get the feel for modular origami without having to create dozens of units for a single project. And Sonobe cubes are so much fun to assemble! You can find plenty of online tutorials about the Sonobe cube, or follow along with my video below. Plus, once your students have mastered the Sonobe cube, they can use the same units to make octahedrons and icosahedrons.

 

What Do We Do After Our First Project?

After you teach your students how to make the Sonobe cube (and possibly the other Sonobe shapes), you might like to help them through one other project. An octagon-star is another favorite because it is a transforming shape — the final design transforms from a star to an octagon and back. For the second project, I provide written directions, but I still walk them through the process step by step. I have the students refer to the written directions (and diagrams) so they can learn to follow origami directions independently.

Students who caught the modular origami bug will be so motivated after learning the first two projects, that they will likely want to figure out other origami designs. I provide a basket of modular origami books and printouts that they can peruse to choose other projects. At that point I step back and let my students become the expert origami crafters — their skills soon surpass my basic ones, and I am very happy to take on the role of appreciative spectator.