Researchers Test Out Primary Sources for Math Class
Writings of Euclid become class texts
Over the next five years, as many as 1,000 college students will be asked to grapple with the original writings of mathematicians like Euclid and Archimedes in a project that many experts say has good utility in high school math classrooms as well.
Under a $1.5 million grant from the National Science Foundation, seven university math professors are writing lesson plans around primary sources—common practice in history and English classes, but rare in the math world—which they'll then pilot with students around the country.
The idea, based on earlier grant-funded work, is that understanding the origins of important mathematical concepts will help students fully grasp and remember them later, and that exploring mathematicians' motivations will be inspiring for students.
With primary sources, "you see why people want to study math, what problems it was designed to solve," said Jerry Lodder, a mathematical sciences professor at New Mexico State University, who is working on the grant project. "You don't see that if you just look at the algorithmic model."
Typically, math textbooks in both high school and college include a short blurb about the mathematician behind a particular concept, but then quickly move into a clean, modernized presentation of the math. "What we miss when that's all we see is all the thought, work, and understanding that goes into creating that presentation in the first place," said Dominic Klyve, an associate professor of mathematics at Central Washington University and a principal investigator on the project. "We forget that mathematics is a human endeavor."
"Can you imagine a literature class where you just read about literature and never study what people were writing 200 years ago?" he added.
Previous NSF-funded work found some evidence that college math students were motivated by the use of primary sources and showed some performance gains in subsequent math classes—but the data were mostly anecdotal.
As part of a project being underwritten by the National Science Foundation, students will see original renderings and writings from early mathematicians, such as the presentation on the left of binomial coefficients created by Blaise Pascal in about 1654. The drawing, known as Pascal’s triangle, is more likely to be shown in modern textbooks as it is on the right. The modern rendering is intended to help students see that each number is the sum of the two directly above it.
Lodder, who also worked on a prior NSF grant on which this study is based, said he believes students will gain "more enduring knowledge" through the use of primary sources. "When you read the formula or algorithm in a textbook, sure, you memorize it for the exam, but a few days later you forget it because there was no context," he said. "When students see the original problem or solution it sticks with them longer."
The professors will be devising both long primary source projects that could take up to several weeks, and shorter projects that are meant to be completed in a single class. About 50 math professors around the country have already agreed to test the tasks with their own students. Researchers will track students' learning growth and compare results with those for students in nonparticipating math classes.
For Kathy Clark, an associate professor of math education at Florida State University who is also a principal investigator, the project is about enhancing students' ability to build their own mathematical arguments. Students who are learning Pascal's triangle and "have gone through the words of the actual authority from 1654, when they go to work on problems based on that mathematical concept, we hope their articulation will be more nuanced because they've had this rich experience," she said.
The experts involved in the project say it translates well to high school—in fact, some of the primary source projects they're designing will cover precalculus and trigonometry concepts. "I don't see any way in which high school students in their junior or senior years will be less mentally prepared to tackle these than first- and second-year [college] math students who are in precalculus," said Klyve.
Clark, who taught high school for 12 years, periodically used primary sources with her students and found them beneficial. "That's a missed resource in high school teaching today," she said. "I hardly see math students in high school open a book for anything other than math exercises. It perpetuates this notion that math is just a bunch of exercises you do—you crunch numbers and solve for x." Through primary sources, students learn to "tear apart the mathematics and the meaning and put it together at the end."
The Common Core State Standards, which are being implemented in K-12 public schools in more than 40 states, require the use of primary sources in teaching literacy. According to Clark, the NSF project dovetails with the common standards in math as well, by encouraging behaviors such as perseverance and critiquing the reasoning of others. Those behaviors are part of the Standards for Mathematical Practice, the overarching standards in the common core that describe the behaviors of proficient math students and apply to all grades.
Diane Briars, the president of the National Council of Teachers of Mathematics, agreed that there is overlap with the practice standards, especially one that says students should construct viable arguments.
"When you're reading these original writings, you're directly building on that [standard]," she said.
But incorporating these primary sources into classrooms isn't easy work, the experts said, and there are some caveats.
"You read one of these for the first time and you have no idea what it says," said Klyve. "The mathematics is explained in vocabulary you haven't seen before or notation you haven't seen before." And that can be hard for all students—but especially English-learners and those with language or reading deficits.
"What you don't want to have happen is students are so distracted by the writing style or it's so hard to make sense of the language used that you're investing a lot of time in that rather than the mathematical ideas," said Briars of NCTM.
For those reasons, the writings must be carefully selected and excerpted to ensure they're digestible. "It's very hard to pull an original source off the shelf and drop it in class and hope students read it and do something useful with it," said Klyve. "We're putting all these projects through peer review, because teaching the math while being true to the original source is just hard."
That said, Klyve would "love high school teachers to try this." The projects will all be open source and adaptable, so "I'd easily imagine somebody could modify them to [explicitly] match the common-core standards," he said. "I'd do that."
Vol. 35, Issue 03, Pages 10-11