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What is in a Number?

For most, mathematics education begins in elementary school. Students learn the basic from teachers: addition, subtraction, multiplication, and division. The fundamentals, right?

Dr. Eva Thanheiser of the Fariborz Maseeh Department of Mathematics & Statistics knows there is something even more fundamental than the operations of basic mathematics: the way people conceive of numbers. Much of Dr. Thanheiser’s research focuses on content knowledge—how well preservice elementary teachers (PTs) know the mathematics they will teach to children, PTs conceptions about math, how those conceptions develop, and what motivates PTs to learn math.

According to Dr. Thanheiser, there is a connection between how PTs conceive of multidigit numbers and their ability to explain algorithms conceptually. An understanding of the value of a number and the meaning of digit placement are 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 prepares 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 together,” said Dr. 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 10 tens, five twenties, and 100 ones, and that the two represents 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 the introduction of 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 hundreds in the hundreds place and adding 10 tens to the 2 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,” Dr. Thanheiser explained, “because I know the one I am regrouping from the hundreds place is 100 ones, or 10 tens and I’m combining the ten tens with two tens making 12 tens, or 120 from which five tens, 50, can be subtracted. What I’ve found is that in general, most people who enter preservice teacher content courses can’t explain this. 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, Dr. Thanheiser focuses on ways to motivate PTs to develop a deeper understanding of the value and meaning of numbers. She, along with colleagues and students study and develop tasks meant to help children and PTs alike grasp concepts essential to the foundations of a mathematics education.

Dr. Thanheiser has authored and coauthored numerous peer-reviewed papers detailing the findings of her studies on topics such as how PTs conceive of multidigit whole numbers, the role children’s mathematical thinking plays in preparing PTs for the classroom, and designing tasks in mathematics content courses. Her work has been presented nationally and internationally. She has contributed chapters to books on mathematics teacher education. Dr. 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 Dr. Thanheiser to help PTs connect digits in a number to their actual value and to understand the relationships between digits. The findings of the study 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.

Dr. 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 in which they solve a math problem and explain how they came to their answer. According to Dr. Thanheiser, when asked to explicate the function of the algorithm, many of her students are stumped.

“Often times 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.”

Dr. 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 learning and work the PTs do 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 really brings together all of what they’re learning.”

For Dr. 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 who may 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 understand how to design tasks that children can perform, learn from, and gain a better understanding of the principles of mathematics by having the experience? Dr. Thanheiser’s research explores the nuances of questions like these.

If our schools are to begin graduating students who meet or out preform 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. Through educating PTs and adding to the literature informing mathematics teacher educators, Dr. Thanheiser is contributing to the considerable efforts underway in Oregon to reform public education from cradle to career.

Authored by Shaun McGillis
Posted April 23, 2014