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When the first author started teaching, she used to brush the chalk dust off her clothes before going home. Now, she checks that all changes to the Web sites have been uploaded. Her average student was right out of high school. Now, the average student is an adult learner, returning to be educated for a second or third career. Many of these adults have not had a mathematics course in many years and find college mathematics daunting. The methods behind mathematics teaching and learning need to change to accommodate this changing learner population. One change with which the authors have had success, especially for adult learners, is moving mathematics online.

How people learn and teach has changed dramatically in the last twenty years. Online adult learners have become a significant population. However, mathematics learning and teaching has remained more static than other subject areas. What has occurred to date has largely been successful; in fact, some colleges have put entire mathematics programs online for the purpose of recruiting adult learners.

The authors have taught mathematics at both the secondary and post-secondary level. The first author taught secondary mathematics and college-level mathematics for seven years before earning a doctorate, and is now in her fourth year of teaching undergraduate mathematics. The second author recently received her license to teach mathematics at a secondary level, and, while working on a graduate degree, is teaching college-level mathematics courses. We have seen mathematics teaching go through many different phases, but recently it has become clear that the teaching of mathematics is in crisis for all types of learners.

Today's K-12 students are less and less interested in mathematics, and standardized test scores are among the lowest in the world. The United States is facing (and this will only get worse in the immediate years to come) a severe shortage of qualified mathematics teachers. Adults are finding their career paths blocked by the inability to master mathematics.

Most people agree that technology (and in particular online learning) is changing how teachers teach and students learn. Yet many people believe that how one teaches or learns mathematics cannot change because mathematics itself is so unchangeable that the traditional methods must be used. However, we have found that learning mathematics online can also be efficient, effective, and fun.

In our teaching, we decided to see if the quality of mathematics learning and teaching can improve through the use of technology. While teaching secondary geometry supplemented with online material and college-level algebra completely online, we have tried to be teacher-scholars and reflect on our experiences. We have become convinced that learning and teaching online mathematics works.

We have gained insight into what our students think of learning mathematics online. Out of our 56 secondary students in one course, 75% found the experience very positiveâ€”and it is difficult to get 75% of secondary students to think anything is positive, let alone a mathematical experience. Although some described online mathematics as "just pushing buttons" and thus lacking in "explanations," others thought online mathematics had advantages that traditional methods do not. For example, online mathematics offers a perspective that a textbook or even a teacher's words and drawings on a chalkboard cannot. Online mathematics is dynamic (e.g., figures can change size and shape in time and variables can truly vary before a student's eyes). In addition, online mathematics strikes our students as simply more engaging, more fun, and more current than traditional mathematics methods. The following are quotes from our students:

- "Textbooks can only show things at one angle, [online] you can move to fit your needs."
- "You could see the numbers changing really fast so it was easy to see how all the volumes and surface areas are related."
- "I liked to be able to see all of the prism instead of just a 2D image in a book."

If attitudes about learning mathematics improve, without demonstrable results, we would hesitate to support online learning (even though it is nice to have positive attitudes). However, we found, in another experiment, that skills improve. Secondary students and adults using online learning performed significantly better (p-value = 0.0195) on harder problems than a control group in the classroom. There was no difference between the groups on easier problems.

We believe that there are two primary reasons why learning mathematics online improves attitudes and skills. The first is that online mathematics can provide seemingly infinite examples for students to experience. As the students said themselves, they can see relationships among theorems and visual evidence as to why things are true. Although we feel we must caution readers that merely providing a large number of examples does nothing, it certainly can build understanding. We are also convinced that having that many examples available dynamically (the examples can change under the student's control) increases the students likelihood of making conjectures and their level of curiosity and interest in mathematics.

The second facet of learning mathematics online that is superior to traditional learning is the hands-on nature, in which students receive instant, individual feedback as they learn. This is simply not possible in a traditional setting.

Besides the benefits we have mentioned, we are convinced that the teaching methods of teaching of twenty years ago, which may have been suitable for students in the 1980s, are inappropriate for current students, who are used to information presented online. Failure to use this powerful resource in the classroom may be robbing some students from truly understanding mathematics.

Of course, technology is fallible and students need to have some practice developing mathematical reasoning with and without technology. This can still be accomplished online. The technological tools (such as a graphing calculator) may be present or not. Finally, we feel that just because a technology does not work with a particular curriculum doesn't mean that it will not work with any curriculum or student. And when technology does work with a particular curriculum, it won't necessarily work with all curricula or all students.

Online mathematics learning, especially for adults, can be a response, even an answer, to the crisis in mathematics teaching and learning. However, teaching and learning will continue to change, and twenty years from now, we may shake our heads with a smile at what we did today. Although we are convinced that learning mathematics online works, there are many future directions that need exploring. We have given the two main reasons that online mathematics learning provides benefits (seemingly infinite number of examples that change dynamically, and the one-on-one attention). Are there other reasons that we have not noticed? How can these reasons and any others be explored and exploited so as to maximize the benefit of online mathematics learning, especially for adults?

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