In an article for the *New York Times* Motherlode blog, MindShift contributor and author Annie Murphy Paul explains why the perception that U.S. students are bad at math might indicate schools aren’t challenging students enough.

“‘Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture, she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than ‘little manipulations of numbers,’ as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories.”

]]>For subjects like math and foreign language, which are traditionally taught in a linear and highly structured context, using more open-ended inquiry-based models can be challenging. Teachers of these subjects may find it hard to break out of linear teaching style because the assumption is that students can’t move to more complicated skills before mastering basic ones. But inquiry learning is based on the premise that, with a little bit of structure and guidance, teachers can support students to ask questions that lead them to learn those same important skills — in ways that are meaningful to them.

This model, however, can be especially hard to follow in public school classrooms tied to pre-set curricula. Class time, class size, assessments, resources, student buy-in, administrative pressures, and students’ learned helplessness are just a few of the reasons why it can be challenging to create learning experiences that are deep, authentic, and driven by inquiry, according to participants at EduCon 2.6 hosted by Science Leadership Academy (SLA), a public high school in Philadelphia recently.

Science Leadership Academy, which has an established track record as an inquiry-based school, has just opened a second campus in Philadelphia called the Beeber school, whose teachers are still adapting to the inquiry model. With one freshmen class and a new crop of teachers still adjusting to project-based and inquiry driven approaches to learning, the school is a good model for learning how these complex ideas flesh out.

“As much as we can say it’s okay for students to fail within the class, if they don’t pass the test at the end of the year, it’s suddenly not okay.”

“You have to spend a lot of time and a lot of energy supporting kids to unlearn how they’ve been taught to learn for the majority of their lives,” said Marina Isakowitz, a ninth-grade math teacher at Beeber. Students at both SLA campuses come from public middle schools across the city and enter with varying levels of proficiency and very little experience with inquiry learning.

Isakowitz starts the year by asking lots of low-stakes, but complex questions as a way of scaffolding a new kind of learning for her students. Gradually, she says, they realize that this math class isn’t going to be like others they’ve been in and they begin to understand and appreciate the freedom they’ve been given. It’s about providing just enough structure that the class holds together, but not so much that teachers are telling students what to do and how to do it.

Watching students struggle with how to ask good questions and discover answers can be hard. “I, as the teacher say, ‘I’m going to let you bruise yourself, and that’s going to be hard to watch, but I’m not going to step in and help,’” Isakowitz said. That can feel like she’s not doing her job, she said, but she knows it’s an important part of getting them to take ownership of their learning.

“It’s so terrifying as a teacher when you have this notion of what looks right, and they’re not doing it right and I’m failing as a teacher because of that,” Isakowitz said. She’s had to learn to hang back and watch what develops. She’s also had to challenge her own ideas about math education, including the notion that students must learn one skill in order to move onto the next one.

“How much of this hierarchy comes from the idea that they need to know A in order to do B,” Isakowitz said. She’s constantly asking herself, “How can I let them explore something and let them learn the skills along the way?” For example, Isakowitz designed a project based on students’ complaints over the unbearably high temperatures at the school in the summer. They are working to come up with a solution to a problem they are physically invested in by researching air conditioning systems, figuring out how many units their school would need based on its size and layout, and calculating costs. In the process, students are learning about things like systems of equations and slope.

“Sometimes that means letting them go down a dead end — that I know is a dead end — because they just need to figure it out,” Isakowitz said. Within the framework of research and discovery she is building in different ideas and units that the state requires ninth graders to learn. Even with the best of intentions, there are times when the class has to cover a topic that will be on the test but it doesn’t fit into a project. In those cases, Isakowitz tries to be honest about why the learning style has changed.

“As much as we can say it’s okay for students to fail within the class, if they don’t pass the test at the end of the year it’s suddenly not okay,” one teacher said in response to the discussion. That mixed message is a challenge to many teachers who understand that learning from mistakes is an important part of a good education.

Isakowitz faces that challenge every year when her students have to take a test that determines if they graduate. “I’m wagering my students’ ability to graduate high school on this teaching practice,” she said.

Using a project-based inquiry approach to learning math is not easier, but kids are learning the material in ways that are relevant to them. Consequently, there are some topics that Isakowitz knows she won’t be able to cover. She’s hoping that her students have learned the topics she covered deeply well enough to make up for any gaps in knowledge on the state tests.

“Parents are having a hard time watching their kids struggle,” another teacher participant said. “Especially kids who were successful learning the other way.”

Teaching a topic like math without the traditional sequencing can be hard for everyone in the community to understand and requires tolerance for failure. The payoff is when, for example, a student becomes a senior and chooses mechanical engineering as an elective because he loves solving problems and has been learning to do it all through high school.

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**By Sam Gliksman**

*The following is the second of a series of excerpts from Gliksman’s book iPad in Education for Dummies.*

We tend to split science and humanities as though they were separate branches of life. But no matter what profession we choose — artist, plumber, historian, or salesman — we all use some form of scientific inquiry in our daily lives. We learn about the world around us through the same vehicles of experimentation, trial, error, and experience. We use scientific inquiry to learn about the world around us.

Today’s interconnected world demands that the doctor, engineer, pharmacist, and scientist increasingly master skills that used to be classified within the domain of the arts. Skills such as communication, presentation, effective writing, among others, are now vital to all walks of life. In addition, scientific inquiry, critical thinking, exploration, and experimentation have never been more important skills than they are today. If we expect to produce independent learners who can thrive in a society that’s constantly changing, it’s vital that we educators search for opportunities to hone those skills in our students at every turn.

More than any other academic disciplines, science and math draw their meaning by relating to life in the “real” world outside the classroom. They seek to understand the world by inquiry and investigation. Giving children ample opportunities to develop sound investigative skills at an early age is essential to nurturing their ability to think critically and scientifically as they get older. Reading and learning about plants in a book is a far less meaningful experience than planting seeds and watching them grow. If you discuss the various needs of plants and then hypothesize and experiment with differing amounts of sunlight and water, you have a scientific experience that imbues children with a more intimate and fundamental understanding of what it takes for any living organism to grow.

Granting students the freedom to inquire and explore makes them the investigators of life’s mysteries. In the process, they are sharpening their all-important critical and creative thinking skills. Technology offers fantastic opportunities for the application of critical thinking skills toward an understanding of real-world questions and answers. It can be used to gather information about the world around us so that we can investigate real-world questions and test their answers. That’s the focus of this chapter. You find numerous apps that deliver content about botany or algebra, but I want to focus on how you can use technology to have students experience that knowledge from the inside out.

This chapter looks at tools that can be used with the iPad to help you investigate phenomena and collect data. After you collect that data, you look at tools that help analyze the information and present conclusions. You focus less on apps and instead discuss the “application” of mobile technology within a more inquiry-based approach to science and math education. It should be fun!

Whether it’s geometry, physics, or chemistry, scientific method starts with research, discussion, and the development of a hypothesis about the phenomenon being examined. A substantial amount of space in this book is devoted to research, communication, organization, and sharing of information — all important elements in the research and development stage. You start by jumping to the next step: using the iPad for investigating and gathering data for analysis.

**DEMONSTRATING MATHEMATICAL PROOFS**

**Submitted by:** Dr. Randy Yerrick, professor of science education, State University of New York at Buffalo

**Grade level:** 8th- to 12th-grade math

**Objectives:** Have students demonstrate knowledge of mathematical proofs.

**Apps/tools: **Keynote, Explain Everything app

Following the lesson where math teachers demonstrate the construction of an algebraic or geometric proof, students are given the opportunity to create their own. Using media-based presentations and audio recordings of their own voices, students create and share a narrated slide-show presentation.

Keynote is Apple’s presentation app that enables students to build a slidebased presentation with text and media, which they then present live in front of an audience. The Explain Everything app (detailed in Chapter 17) enables students to build and record a series of narrated slides with picture, text, and media content that explains and demonstrates a concept to an audience. Essentially, once students put the slides in the order they choose, they tap Record and step through their slides, explaining, highlighting with color, and accentuating with the laser pointer feature. The Record function also enables students to pause or to rerecord their presentation so that they can practice before presenting it to the class. Explain Everything also includes members’ libraries and upload links to such social media sharing sites as Dropbox and Facebook, making student sharing of work easy and manageable.

Students were given the option of using the preceding apps to explain some of the data that can be retrieved from scientific agencies such as the U.S. Geological Survey (USGS), National Aeronautics and Space Administration (NASA), and the National Oceanic and Atmospheric Administration (NOAA). Students used the latitude and longitude of approaching hurricanes to learn Cartesian coordinates and plot them. For example, one student provided an explanation of the calculated arrival time of a tsunami across the Pacific Ocean from an image acquired from the USGS. This is also a scenario in which students can use apps such as GarageBand or iMovie to collate specific photos they gather into a podcast or movie and explain what they know or how they apply the concepts to examples in the real world.

**EXPLORING CELESTIAL MYSTERIES**

**Submitted by:** Dr. Randy Yerrick, professor of science education, State University of New York at Buffalo

**Grade level:** 4th- to 12th-grade astronomy

**Objectives:** Navigate the sky for recognizable celestial objects; identify and name constellations, stars, galaxies, and planets.

**Apps/tools:** iPad 2 (or higher), iOS 5.1 (or higher), Star Walk app

Some of the best science apps created for the iPad are those geared toward the teaching of astronomy. One benefit of these apps is the integration with the built-in accelerometer and GPS information systems that enables users to simply point to the sky to obtain vast updated information about where they are looking. Celestial navigation is made simple through the interactive interfaces that allow the user to turn on and off the available layers of information when pointing at the night sky.

Teaching with the Star Walk app can be as easy as taking an independent evening stroll with the self-lit app in hand and noting the relative positions of the objects in the sky. At the same time, this app adds a robust environment to explore important questions such as:

**•** Does the sun really rise in the east? Does that change?

**•** Where do all the planets appear in the sky? How can you tell a planet

from a star or galaxy?

**•** Can people in the Southern Hemisphere see the same stars as people in

the Northern, Eastern, or Western hemispheres?

**•** Where do stars go during the day?

The power of this app can be seen in a very simple activity for children up through adulthood. Ask students, “How many planets are there? More important, how do you know?” Have students point out any objects in the sky that they think are planets, and before they use Sky Walk, ask them to point to any they can see. Ask them to draw where those objects would appear at night, and explain what they would look like.

*Excerpted with permission from the publisher, Wiley, from iPad in Education For Dummies by Sam Gliksman. Copyright © 2013.*

BEAUTY OF MATHEMATICS from PARACHUTES.TV on Vimeo.

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How can teachers help students find the beauty in math? There may be roadblocks already set up in math education — students’ disposition toward math anxiety, and pressure to cover material quickly. Or maybe it has something to do with the curse of knowledge — the gap between what experts know and non-experts don’t.

It’s easy for math professors to see the beauty in math, said New York University neuroscientist Pascal Wallisch, because they already have an obvious connection with it. “They perhaps had the luck to enjoy a positive math experience in school,” he said. “Or, frankly, from a cognitive neuroscience perspective, there is little doubt in my mind that they have quite a bit of a different brain than the average person who is trying to unlock the wonders of math, or just learn some math in order to get by.”

Wallisch said that as both a brain scientist and someone for whom math did not come easily, the key to mathematical beauty (and understanding) is visuals. “As primates, we are mostly visual creatures. A good amount of the cortex in primates (upwards of 30%) is dedicated to visual processing in one way or the other. Put differently, things that look interesting or appealing are bound to attract curiosity,” he said. Looking at a picture (or a movie or video) is the same thing as looking at an equation. But while it represents the same information, one method is inherently more appealing to our brains than the other.

According to Wallisch, mathematical imagery is what students are missing, and what causes confusion. He used the example of reading the words “Statue of Liberty,” and how it evokes an immediate image in the mind. But if a person couldn’t read, or had never heard of the Statue of Liberty, they would visualize only letters and words, not Lady Liberty holding her torch — and the same goes for math novices. Since they have no experience, the mix of mathematical symbols on the page don’t mean much. “Mathematicians see equations by imagery built by long-term practice manipulating them,” he said. “The trick is to use software to visualize the equations so that those who don’t have the practice (or the unusual brain) can see the same.”

“Mathematics is a way to read the world of nature and technology around us. If a teacher can convey this, the entire world becomes an exciting textbook.”

Wallisch began creating moving mathematical images for himself using technical computing environment Matlab, and said that, although he uses it for high-level research computations, high school students can just as easily build visual mathematical models with some guidance. By creating images of equations and playing with the variables, Wallisch now sees what all the fuss is about. He wrote in a blog post: “Personally, I’m betting on aesthetics, with Kant: ‘Beautiful is that which is appealing without interest.’ As we can’t presume interest, aesthetics can serve as an important bridge(head).”

California College of the Arts math professor Michael S. Schneider agreed that imagery is the best way to show students the beauty of math. He has been helping students connect mathematics to visual imagery for nearly forty years, and wrote the book *A Beginner’s Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art and Science* to show humans are surrounded with mathematical imagery — including right outside, in nature.

**[RELATED: How to Deal With Kids' Math Anxiety]**

“As a youngster I wasn’t particularly outstanding in math,” he said. “But at 16 I became earnestly interested in the variety of shapes nature produced and wanted to understand why regular shapes keep recurring in nature. I remember pondering the same hexagonal shape found in the beehive, quartz crystal and metal hexnuts. I could understand how a crystal grew mechanically in this precise geometry by accumulating atoms, but how did bees know how to produce the pattern which holds more weight of honey than, say, a checkerboard pattern? I wasn’t comfortable with the ‘trial and error’ explanation, and even if it was in their DNA, how did that knowledge of superior design get there?” Then, he said, he wanted to know more about logarithmic spirals – “in the bathtub, swirling leaves, tornadoes, hurricanes, solar systems, galaxies.” Schneider said he had good math teachers, but these topics were never covered in school; books he looked up on the subjects covered them one at a time, but never altogether in the same place.

Schneider said he became obsessed with understanding the language and shapes of “nature’s geometric alphabet”: circles, spheres, triangles, squares, and more — the shapes that surround us every day if we simply take the time to notice them. “A circle represents the number one,” he said. “Most people can feel why a circle represents unity, its wholeness, completeness. A circle holds more inside it than any other shape having the same perimeter. So it’s practical to know that round pizzas hold more toppings than squares or rectangles having the same length of crust.”

Armed with this set of nature’s images and symbols, Schneider found that numbers and shapes have personalities, each playing different roles in the cosmos. “The universe becomes a book and then a great play with great actors in great parts telling great stories,” he said. “Mathematics is a way to read the world of nature and technology around us. If a teacher can convey this, the entire world becomes an exciting textbook.”

Schneider admits that today’s math teachers are strapped for time and resources to really explore the beautiful part of math, partly due to the way textbooks are constructed, and the pressure to cover material quickly. He believes that for students to see the beauty in math, teachers need more time and freedom. “I think that math education gets too abstract too quickly without first providing a sense-based foundation,” he said.

But appreciating the beauty in mathematics could start by just having students look around them. “The universe may be a mystery, but it’s not a secret,” he said.

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