MathemAntics: a database of strategies for teachers
Numbers Games (1/31/11)
New software based on research by TC’s Herb Ginsburg harnesses the everyday math abilities of very young children
One day this past fall in the basement cafeteria of P.S. 76 in Harlem, Steven, a preschooler, was testing out a software program called MathemAntics.
The screen displayed a farm scene, with goats scattered across a barnyard.
“How many goats are there?” the program’s computerized voice asked.
“One, two, three, four, five,” Steven counted, pointing his finger. To verify his answer, he clicked the number five on the number line at the top of the screen. The voice told him he was correct.
Now nine yellow chicks appeared on the screen. This time Steven’s answer – 11 – was wrong. “Nice try, but that’s too many,” the computer told him.
Esther Yoon, a first-year TC doctoral student in Cognitive Studies, pointed to another button. “Try the ten box,” she said.
Steven moved the cursor and clicked on the 10 box button, and the scattered chicks lined up in a row, each in a box, with one empty box left over. The visual prompt seemed to help; this time, Steven gave the right answer.
“How do you know?” Yoon asked. “How did you get that number?”
“I counted,” Steven said.
MathemAntics, which is being developed by Educational Network Services (ENS), a Massachusetts-based educational software developer, is b ased on the research of Herbert Ginsburg, TC’s Jacob H. Schiff Foundations Professor of Psychology and Education. Backed by a $1.4 million grant from the Institute of Educational Studies, Ginsburg and seven of his doctoral students (including Yoon) are evaluating the program at P.S. 76 and two other Harlem schools. Based on their feedback, ENS, over the next two years, will develop software for children ages three and up through third grade.
Building on the work of Jean Piaget and other cognitive psychologists, Ginsburg has devoted his career to debunking the common American misconception that – notwithstanding 25 years of research and the demonstrated capability of children in Singapore, Finland and elsewhere – very young children are unable to think mathematically. He has spent hundreds of hours interviewing and videotaping toddlers and preschoolers in the classroom, capturing examples of their “everyday mathematical thinking” – awareness of shape, number and other concepts employed in play, eating, and other informal situations.
“What baby would not know the difference between a lot of food and a little food?” Ginsburg asked an audience at TC’s Academic Festival in April 2010. “Or a lot of attention versus a little attention?”
Technology holds enormous potential to harness such abilities, though Ginsburg says that many flashy educational games fail to make use of what’s known about how children of different ages think. In contrast, MathemAntics, whose programmers received the first grant to develop the precursor to the current software, “gives us an opportunity to see what’s in kids’ minds as they work on both informal and formal mathematical problems,” says Ginsburg. That’s something that a score on an annual achievement test --- which is based almost entirely on whether an answer is right or wrong --- can’t hope to touch. And by gathering information on what young children know and how they know it, MathemAntics c ould become the foundation for an ever-expanding library of knowledge that could help generations of future learners learn math.
Once the grant is completed, MathemAntics will feature six different “environments,” all incorporating educational goals and standards for early math learning established by the National Governors Association Center for Best Practices: “how many,”—the one Steven was working with—which covers counting, cardinality and subitizing (the ability to instantly recognize a small number of items, without counting); addition and subtraction; equivalence; symbolism, using a base-10 system to understand column addition; multiplication; and negative numbers. Each area is loosely geared to a different age-level, yet allows for a wide range in comprehension and proficiency.
Each area also is the specific province of one of Ginsburg’s seven doctoral students – Yoon, for example is in charge of the negative number environment -- but the team’s work on MathemAntics is ultimately highly collaborative. The group— the “Mac-nificent Seven,” as Ginsburg called them at a recent team meeting— meets once a week to exchange notes and discuss what they’ve witnessed during data collection. The process of revising the software is continuous, with video and audio recordings of the sessions sent weekly to the developers at ENS.
“It’s a fantastic back-and-forth,” says Ginsburg of the partnership. “If we send something to be fixed, it’s often done by the following Monday.”
The process differs from the way most education software is designed, he says, because the software and guiding theory are under constant revision as a result of observed practice. “This program is built on children’s needs and abilities as we have observed them in real world settings,” he says. “The designers have an extraordinary understanding of their end users.”
Ginsburg and his students constantly exchange ideas online with ENS. This past December, the developers made the trip down from Massachusetts for a meeting at TC in which all issues about the project were put on the table.
TC doctoral student Kara Carpenter reported that after pressing the 10 box, two of her second graders debated whether or not the animals would fit in the boxes. She thought that the slow speed of the application allowed them the necessary time to think and discuss.
After another graduate student, Samantha Creighan, suggested revising the program to display up to 100 animals, a software developer countered that, because this would mean changing the scale of the animals, children would learn that the size of an item has no relevance when it comes to counting quantity.
The exchange typified the discussion. Even the most mundane logistical concerns were parsed and refracted through though the prism of the larger question: Are students learning? And if so, why?
“We could make a kid learn that seven plus three equals ten by having them repeat it a million times, but that wouldn’t necessarily mean that they understood the underlying concept,” says Ken Schroder, ENS’ computer science programmer.
In addition to recording a child’s strategies for problem-solving, MathemAntics is designed to create reports that give teachers and parents a fuller understanding of the child’s thought process. It can “speak” to users in multiple languages. And by also requiring children to read questions, it lays the ground for reading skills, as well.
The program also has the touch of magic that computer activities and games bring to any subject, performing functions that could never be replicated in real life. As Ginsburg explains, “When you’re placing six large elephants into six small boxes, the elephants have to shrink. That’s a lot of fun.”
The preliminary results—which should please teachers—show that MathemAntics can help younger children learn a lot more math. Back in the cafeteria of P.S. 76, after a handful of trials with the 10 Box, Steven was no longer counting to determine the number of animals on the screen. Instead, he simply pressed the 10 box and declared his answers based on how many boxes weren’t filled.
“After understanding and comprehending the meaning of the 10 box, within five or six minutes, he started using a completely different strategy,” Yoon said afterwards.
In fact, by the end of his session, Steven was working easily with the base-10 concept, using the “10 box” to evaluate numbers in the teens. He was comfortable subitizing small numbers of animals on the screen and counting from “10.”
“Why do you like using the boxes?” Yoon asked.
“Because it’s more better,” he said.