You can press "Enter," but think twice before pressing "="

I just had a epiphany tonight while reading an article by Alibali et al. (2007) about students' understanding of the equal sign. While some students see it properly as a relational symbol, the most common misunderstanding is that equals is operational -- a sign that indicates "get the answer" or "add them up." It is this operational conception that leads some students to believe x = 10 in a problem like 5 + 5 = x + 3. (Some students also incorrectly believe x = 13, figuring the three has to be added with the two fives somehow.)

So here's my surprise: I had never considered that students might be using a tool every day that is reinforcing that operational conception -- their calculator. Go ahead and search Google Images for calculators. Doesn't every one use an equals button to perform the "get the answer" function? Should that button be labeled with something else? Some say "Enter" but still have an "=" sign on the button.

This is what's fun about being a researcher -- I suddenly want to do an experiment with two sets of classrooms, one that gets traditional calculators and one that get modified calculators without "=" signs for the "Enter" button. Let them go about their business for a year without any other attention paid to the issue, and measure students' understandings of the equal sign at the end of the year and see if the treatment group has better understanding than the control. I know it sounds trivial, but it's often in these small steps where we make new knowledge.



Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9(3), 221-247.

3rd International Realistic Mathematics Education Conference (#RME11)

Starting tomorrow I'll be attending the 3rd International Realistic Mathematics Education Conference (#RME11), hosted by the Freudenthal Institute USA (FIUS) here at the University of Colorado at Boulder. The three-day conference features four keynotes, three plenaries, and only 18 breakout sessions, one of which I have the privilege of leading. I attended the previous RME Conference in 2009 before I really had a chance to become familiar with the theory and those who develop and promote it. RME is a theory of mathematics education worth knowing, but for this post I'd rather focus on some of the people who will be presenting. They're well-known in the field of math education research, even if they might not be in the math education blogosphere. I'm hoping this post helps change that.

David C. Webb is the Executive Director for the Freudenthal Institute USA and an assistant professor of mathematics education at the University of Colorado at Boulder. (He's also my advisor.) His involvement with the Freudenthal Institute goes back to his graduate school days at the University of Wisconsin, where we was advised by Tom Romberg and worked on the Mathematics in Context project, an NSF-funded curriculum that combined the goals of the NCTM Standards with the philosophies and design theory of RME. Following Romberg's retirement and David's move to the CU, FIUS came to Boulder in 2005. For RME11, David will lead Friday's plenary titled, "Informed Classroom Practice: Progress and Challenges" and will co-lead Sunday's closing plenary titled, "Design, Research, and Practice: Building a Community of Designers and Practitioners."

Henk van der Kooij (pronounced "koy") is a senior staff member at the Fruedenthal Institute, University of Utrecht, the Netherlands, where he conducts research and trains mathematics teachers. I had the pleasure last summer of taking a class co-taught by David and Henk, and even after ten days of going 8am to 3pm, it was not unusual to catch Henk sitting to the side of the room, molding a not-so-great mathematical task into a much better one. For this year's conference, Henk will co-lead the closing plenary with David Webb, participate in a Q & A Friday afternoon with Keono Gravemeijer and Mieke Abels, and conduct one session called, "What Mathematics is Important for (Future) Work?"

Koeno Gravemeijer gets the honor of the opening keynote at this year's conference, titled, "Helping Students Construct More Formal Mathematics." I've read a number of his articles and book chapters (see here for a sample), and seen many more referenced, so I'm quite excited to see and hear him in person. I'm not sure what areas of math ed research Gravemeijer hasn't tackled, from design research to statistics education, and the list of articles returned in Google Scholar makes me want to just stop what I'm doing and read for about a month.

Doug Clements will deliver Saturday morning's keynote, "Learning Trajectories -- The Core of Standards, Teaching, and Learning." My introduction to Doug Clements came last spring when my advisor asked me to read Clements's chapter in the Second Handbook of Research on Mathematics Teaching and Learning. The chapter, "Early Childhood Mathematics Learning," written with Julie Sarama, is perhaps the most thorough, dense, yet well-organized and enlightening (in a near-overwhelming kind of way) reading I've done yet as a graduate student. Some suggest we know (or we're close to knowing) all there is to know about early childhood mathematics, so summarizing that knowledge is no easy task. If you ever cross this chapter, take the advice my advisor gave me: "Take your time."

There are so many more excellent people presenting at this conference. Mieke Abels. Debra Johanning. Meg Meyer. The point of this post wasn't so much to drop names, or to think you'll be star-struck by this lineup (in a nerdy math ed researcher way), but to let you put a few names and faces together of people who share a common interest -- they can't stop thinking about how we can better teach and learn mathematics. And if you can think of them that way, then the walls of the ivory tower seem to crumble.

I plan on blogging, tweeting (with hashtag #RME11), and posting to Google+ throughout the weekend, although I still have to find enough spare moments to keep up with my other classwork due next week. (And finish putting together my own presentation!) So far I don't know of any other bloggers or members of the math ed Twitter community who will be attending, but be sure to make your presence felt if you're lucky enough to attend the conference.