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鈥攃ommon practice in history and English classes, but rare in the math world鈥攚hich they鈥檒l 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鈥� motivations will be inspiring for students.
With primary sources, 鈥測ou see why people want to study math, what problems it was designed to solve,鈥� said Jerry Lodder, a mathematical sciences professor at New Mexico State University, who is working on the grant project. 鈥淵ou don鈥檛 see that if you just look at the algorithmic model.鈥�
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. 鈥淲hat we miss when that鈥檚 all we see is all the thought, work, and understanding that goes into creating that presentation in the first place,鈥� said Dominic Klyve, an associate professor of mathematics at Central Washington University and a principal investigator on the project. 鈥淲e forget that mathematics is a human endeavor.鈥�
鈥淐an you imagine a literature class where you just read about literature and never study what people were writing 200 years ago?鈥� he added.
鈥業t Sticks鈥�
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鈥攂ut 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鈥檚 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.
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Lodder, who also worked on a prior NSF grant on which this study is based, said he believes students will gain 鈥渕ore enduring knowledge鈥� through the use of primary sources. 鈥淲hen 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,鈥� he said. 鈥淲hen students see the original problem or solution it sticks with them longer.鈥�
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鈥� 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鈥� ability to build their own mathematical arguments. Students who are learning Pascal鈥檚 triangle and 鈥渉ave 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鈥檝e had this rich experience,鈥� she said.
The experts involved in the project say it translates well to high school鈥攊n fact, some of the primary source projects they鈥檙e designing will cover precalculus and trigonometry concepts. 鈥淚 don鈥檛 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,鈥� said Klyve.
Clark, who taught high school for 12 years, periodically used primary sources with her students and found them beneficial. 鈥淭hat鈥檚 a missed resource in high school teaching today,鈥� she said. 鈥淚 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鈥攜ou crunch numbers and solve for x.鈥� Through primary sources, students learn to 鈥渢ear apart the mathematics and the meaning and put it together at the end.鈥�
Common-Core Connections
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.
鈥淲hen you鈥檙e reading these original writings, you鈥檙e directly building on that [standard],鈥� she said.
But incorporating these primary sources into classrooms isn鈥檛 easy work, the experts said, and there are some caveats.
鈥淵ou read one of these for the first time and you have no idea what it says,鈥� said Klyve. 鈥淭he mathematics is explained in vocabulary you haven鈥檛 seen before or notation you haven鈥檛 seen before.鈥� And that can be hard for all students鈥攂ut especially English-learners and those with language or reading deficits.
鈥淲hat you don鈥檛 want to have happen is students are so distracted by the writing style or it鈥檚 so hard to make sense of the language used that you鈥檙e investing a lot of time in that rather than the mathematical ideas,鈥� said Briars of NCTM.
For those reasons, the writings must be carefully selected and excerpted to ensure they鈥檙e digestible. 鈥淚t鈥檚 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,鈥� said Klyve. 鈥淲e鈥檙e putting all these projects through peer review, because teaching the math while being true to the original source is just hard.鈥�
That said, Klyve would 鈥渓ove high school teachers to try this.鈥� The projects will all be open source and adaptable, so 鈥淚鈥檇 easily imagine somebody could modify them to [explicitly] match the common-core standards,鈥� he said. 鈥淚鈥檇 do that.鈥�
