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If you had asked them, most of my high school teachers would have called me an unmotivated student or said that I lacked discipline and didn't take learning seriously. And yet, that abandoned storage bin told another story: with the aid of my calculator, I'd crafted narratives, drawn storyboards, visualized foreign and familiar environments and coded them into existence. I'd learned two programming languages and developed an online network of support from experienced programmers. I'd honed heuristics for research and discovered workarounds when I ran into obstacles. I'd found outlets to share my creations and used feedback from others to revise and refine my work. The TI-83 Plus had helped me cultivate many of the overt and discrete habits of mind necessary for autonomous, self-directed learning. And even more, it did this without resorting to grades, rewards, or other extrinsic motivators that schools often use to coerce student engagement.

I've now begun to see Texas Instruments graphing calculators as unique among educational technologies in that they enable learning that is couched in discovery more than formal teaching.

take the notion of "correctness." School typically assumes that answers fall neatly into categories of "right" and "wrong." As a conventional tool for computing "right" answers, calculators often legitimize this idea; the calculator solves problems, gives answers. But once an endorsed, conventional calculator becomes a subversive, programmable computer it destabilizes this polarity. Programming undermines the distinction between "right" and "wrong" by emphasizing the fluidity between the two. In programming, there is no "right" answer. Sure, a program might not compile or run, but making it offers multiple pathways to success, many of which are only discovered through a series of generative failures. Programming does not reify "rightness;" instead, it orients the programmer toward intentional reading, debugging, and refining of language to ensure clarity. This is a form of learning that privileges the process of discovery over the interventions of formal teaching. It can fuel an intrinsic desire to pursue similar learning experiences, but even more, it gradually transforms the outlook of the student

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.”

It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar. Demanding algebra across the board actually skews a student body, not necessarily for the better.

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.”

But this progression actually “has nothing to do with how people think, how children grow and learn, or how mathematics is built,” says pioneering math educator and curriculum designer Maria Droujkova.

The current sequence is merely an entrenched historical accident that strips much of the fun out of what she describes as the “playful universe” of mathematics

“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.

As she headed into fifth grade, she assumed she was in for more of the same—lectures, memorization, and busy work. Sergio Juárez Correa was used to teaching that kind of class. For five years, he had stood in front of students and worked his way through the government-mandated curriculum. It was mind-numbingly boring for him and the students, and he’d come to the conclusion that it was a waste of time. Test scores were poor, and even the students who did well weren’t truly engaged.

Juárez Correa didn’t know it yet, but he had happened on an emerging educational philosophy, one that applies the logic of the digital age to the classroom. That logic is inexorable: Access to a world of infinite information has changed how we communicate, process information, and think. Decentralized systems have proven to be more productive and agile than rigid, top-down ones. Innovation, creativity, and independent thinking are increasingly crucial to the global economy. And yet the dominant model of public education is still fundamentally rooted in the industrial revolution that spawned it, when workplaces valued punctuality, regularity, attention, and silence above all else.

knowledge isn’t a commodity that’s delivered from teacher to student but something that emerges from the students’ own curiosity-fueled exploration. Teachers provide prompts, not answers, and then they step aside so students can teach themselves and one another. They are creating ways for children to discover their passion—and uncovering a generation of geniuses in the process.

She is also frustrated by the predictable nature of many of the “dirty dozen,” the teachers’ nickname for the basic lab exercises now recommended by the College Board. In one that her class did last fall, the students looked at pre-stained slides of onion root tips to identify the stages of cell division and calculate the duration of the phases. She and her students, who historically score 4’s and 5’s on the exam, were one of several schools asked by the College Board to road test one of the proposed new labs to see if it brought back the “Oh, wow!” factor. The basic question: What factors affect the rate of photosynthesis in living plants? The new twist: Instead of being guided through the process, groups of two or three students had to dream up their own hypotheses and figure out how to test them. Caroline Brown, a senior who stages the school’s plays, connected the lab to her passion for theater. She borrowed green, sky blue and “Broadway pink” filters from the playhouse to test how different shades of light affected photosynthesis in sunken spinach leaves.

Many years ago, people learned to weld in a high school shop class or in a family business or farm, and they came up through the ranks and capped out at a certain skill level. They did not know the science behind welding,” so could not meet the new standards of the U.S. military and aerospace industry. “They could make beautiful welds,” she said, “but they did not understand metallurgy, modern cleaning and brushing techniques” and how different metals and gases, pressures and temperatures had to be combined.

Welding “is a $20-an-hour job with health care, paid vacations and full benefits,” said Tapani, but “you have to have science and math. I can’t think of any job in my sheet metal fabrication company where math is not important. If you work in a manufacturing facility, you use math every day; you need to compute angles and understand what happens to a piece of metal when it’s bent to a certain angle.” Who knew? Welding is now a STEM job — that is, a job that requires knowledge of science, technology, engineering and math.

even before the Great Recession we had a mounting skills problem as a result of 25 years of U.S. education failing to keep up with rising skills demands, and it’s getting worse.

Taking a test is not just a passive mechanism for assessing how much people know, according to new research. It actually helps people learn, and it works better than a number of other studying techniques.

“We’re trying to take kids who haven’t been engaged in school and hook them to an expanded vision of what their future might be,” he said. When they return to their own schools, the hope is that they’ll be more interested in history, math and English — and have a sense of environmental stewardship as well.

“The most important thing really is teaching kids through hands on, experiential learning,” said Rees. “Our kids do this because they’re inspired to be there every week, to work with adults and do hands on things.”

used properly, testing as part of an educational routine provides an important tool not just to measure learning, but to promote it.

Various kinds of testing, though, when used appropriately, encourage students to practice the valuable skill of retrieving and using knowledge.

tests serve students best when they’re integrated into the regular business of learning and the stakes are not make-or-break, as in standardized testing.

11 Mathematics Resources to Try in 2011