For most, mathematics education begins in elementary school. Students learn the basics from teachers: addition, subtraction, multiplication, and division. The fundamentals, right?
Professor Eva Thanheiser of the Fariborz Maseeh Department of Mathematics & Statistics knows there is something even more fundamental than basic mathematics operations: how people conceive numbers. Much of Thanheiser's research focuses on content knowledge—how well preservice elementary teachers (PTs) know the mathematics they will teach to children, PT's conceptions about math, how those conceptions develop, and what motivates PTs to learn math.
According to Thanheiser, there is a connection between how PTs conceive multidigit numbers and their ability to explain algorithms conceptually. Understanding the value of a number and the meaning of digit placement is critical when conveying why algorithms are mathematically valid. In the classroom setting, teaching elementary school children why basic algorithms operate the way they do strengthens the foundation of their mathematics education and better prepare them to engage more complex mathematical processes.
"If I write 128 on a piece of paper, most preservice teachers see a one, two, and eight grouped," said Thanheiser. "If that is how you see the number, it can be difficult to think about the fact that the one represents 100, which is also ten tens, five twenties, and 100 ones, and that the two represent 20, two tens, and so on. The focus is on the digits and not the value their placement assigns them. This kind of thinking makes it hard to explain why algorithms work."
Consider the example of subtracting 75 from 328 using the standard algorithm. The first step is easy: taking five from eight leaves three. The next step complicates things; seven can't be subtracted from two without introducing negative numbers. The operation requires a regrouping of the numbers. Thus one hundred is regrouped from the hundreds place to the tens place, leaving two hundred in the hundreds place and adding ten tens to the two tens in the tens place; hence we have 12 tens in the tens place; seven tens from 12 tens leaves five tens. The solution then is 328-75=253.
"I can regroup the numbers like this," Thanheiser explained, "because I know the one I am regrouping from the hundreds place is 100 ones or ten tens and I'm combining the ten tens with two tens making 12 tens or 120 from which five tens, 50, can be subtracted. I've found that most people who enter preservice teacher content courses can't explain this in general. They've never had to think about it. They see a three, two, and eight, and they know the rules work, but they can't explain why they work."
In her research and classroom instruction, Thanheiser motivates PTs to understand numbers' value and meaning better. Along with colleagues and students, she studies and establishes tasks intended to help children and PTs alike grasp concepts essential to the foundations of mathematics education.
Thanheiser has authored and co-authored numerous peer-reviewed papers detailing her studies' findings on how PTs conceive of multidigit whole numbers and the role children's mathematical thinking plays in preparing PTs for the classroom and designing tasks in mathematics content courses. She has presented her work nationally and internationally. She has contributed chapters to books on mathematics teacher education. Thanheiser has also guest-edited a special volume of the Montana Mathematics Enthusiast.
A recently published paper, "Developing prospective teachers' conceptions with well-designed tasks: explaining success and analyzing conceptual difficulties," addresses a gap in the knowledge of how PTs develop conceptions of number. In the study, PTs completed mathematical operations using digit cards in one case and the Mayan number system in another. The tasks were designed and augmented by Thanheiser to help PTs connect digits in a number to their actual value and to understand the relationships between digits. The study's findings showed that PTs have limited conception of the meaning and value of numbers and that well-designed tasks can help PTs develop a better understanding of content, amongst other findings.
Thanheiser also employs innovative in- and out-of-the-classroom methods to motivate her students to develop a deeper understanding of numbers and the operations used to manipulate them. At the beginning of the first class in the Foundations of Elementary Mathematics cycle she teaches, she individually meets with each student for a recorded interview. They solve a math problem and explain how they came to their answer. According to Thanheiser, many of her students are stumped when asked to explicate the algorithm's function.
"Often, the students have never been asked to solve a problem and reflect on how they came to an answer," said Thanheiser. "Many don't realize the problems can be explained. It is a revelatory moment for them. It motivates them to learn, to take advantage of the resources available in the class. At the end of the course, I show each student their video; it's a way for them to see what they have learned."
Thanheiser has also partnered with public schools to create a family math night. She noted that these events add a sense of authenticity to the PTs' learning and work throughout the Foundations courses.
"During the family math nights, the preservice teachers get a chance to work with children. They see how children think. It's a way to connect the university classroom experience to the elementary classroom. They design activities to help the kids understand the material and later analyze and improve the tasks they've created. It brings together all of what they're learning."
For Thanheiser, the classroom where she instructs PTs is also her laboratory. What does it take to motivate PTs who may not have strong math skills or be frightened by math to engage the content and get excited about learning? How do you change the conceptions PTs have of numbers to help them better understand the relationship between digit placement and the value of numbers? How do you help them know how to design tasks that children can perform, learn from, and gain a better understanding of the principles of mathematics by having the experience? Thanheiser's research explores the nuances of questions like these.
If our schools are to begin graduating students who meet or outperform their peers in mathematics, if we are to provide an education in which math is more than problems in a book, but a tool with countless practical applications, then having educators in elementary classrooms capable of teaching students the most fundamental of concepts from the very beginning is essential. By educating PTs and adding to the literature informing mathematics teacher educators, Thanheiser contributes to Oregon's considerable efforts to reform public education from cradle to career.