Worked Examples in Math Ed: Theoretical and Empirical Support

WrongAnswer-isoIn preparing our Small Business Innovation Research (SBIR) proposal on the topic of developmental math games aimed at community college students, we’ve partnered with, and sought feedback from, several experts in math education, notably Dr. Michelle Chamberlin of the University of Wyoming. Dr. Chamberlin volunteered to write a short briefing on the theoretical and empirical support for worked example problems (click to see our first post on the topic).

What follows is Dr. Chamberlin’s sketch on the background of worked example problems, which she has kindly agreed to let us publish here.

Worked Example Problems: Theoretical and Empirical Support

by Michelle Chamberlin, Ph.D., Associate Professor, Department of Mathematics, University of Wyoming

Worked examples have a strong theoretical foundation in the field of cognitive science Continue reading

Working Memory and Lost Keys

Not my family’s fabled key ring but about the right size and variety.
Photo by Jim Pennucci via Flickr
[License: Creative Commons Attribution 2.0]

Keys lost

I share a personality trait with at least two of my closest blood relatives. When I get in a hurry or under stress, I lose my car keys.

It took me a long time to isolate this pattern as something predictable and, indeed, diagnostic.

Lost key episodes were a dramatic feature of my childhood. My parents owned a large and rambling old motel and, when I was very young, they shared custody of a master key ring of epic proportions. Once when my dad was out of town for a weekend, my mom had to cope with an unexpected spring snow storm and with all the plumbing and heating problems that were usually dad’s to deal with. (For the most part, my dad took care of the physical systems; my mom dealt with the people.) Mom did a masterful job of getting us through the crisis. But by the time my dad got back to town, her triumph was overshadowed by disaster. She had lost the big key ring and none of us could find it, no matter how hard or long we searched. Continue reading

Wrong Answers: Mistakes or Misconceptions?

Even the equals sign can be the target of misconceptions

The first use of an equals sign, equivalent to 14x+15=71 in modern notation. From The Whetstone of Witte by Robert Recorde. [Public domain] Courtesy

I’ve been reading a lot lately about learning and teaching algebra. It’s not a topic I thought much about when I was learning algebra myself. It was just another math class for me. Not as much fun a geometry proofs, admittedly, but a reasonably good time nonetheless. (Yep. I was one of those strange kids that actually liked geometry proofs. Some people like to run 25+ miles at a time. Some folks are willing to listen to jazz. No accounting for taste, is there?)

Turns out, though, that learning algebra is actually pretty hard and teaching it is very tricky indeed. It’s full of little “threshold concepts” — those hard-to-teach, hard-to-learn, see-the-world-differently-after-you-know-them ideas that frustrate learners and teachers alike. And oddly enough, it is also a topic where the math the student already knows can turn out to be full of deep-seated misconceptions, lying in wait to trip her up and make new learning painful.

Take the humble, ubiquitous equals sign for example. Continue reading

Wrong Answers in Developmental Math

This is the second of two posts on the role that wrong answers can play in the algebra classroom. The student in a developmental math class generally doesn’t react in either of the two ways described in my previous post.
Developmental Math is not formulaic

Wrong Answers — Student Perspectives II

It is easy to picture the developmental classroom as being filled with undirected young slackers who just didn’t try hard enough when they were taught algebra the first time around in public school. Clearly this is the theory of the many state legislators around the country who are restricting funds for developmental education in their post-secondary education systems.

Ask a developmental education instructor, though, and what you will consistently hear is, “Our students have complicated lives.” Whatever the back story of an adult learner sitting in a remedial pre-algebra class, you can be pretty certain that some aspects of their current situation are not that pretty. Most of them are making an extraordinary effort to pull themselves up by their bootstraps.

So, while the instructor is up there at the board, working through that hard one, a thousand things unrelated to the correct math process are probably running through the students’ minds. Continue reading

Wrong Answers in Algebra Class

Teaching algebra is not an easy formula to parse

It’s been my privilege, recently, to sit in on several algebra and pre-algebra classes being taught by master instructors.

Wrong Answers — Teacher perspective

Think about an experienced algebra teacher going over a set of practice problems with her class. It’s yesterday’s homework assignment that she wants to correct at the start of class today, so she and the students can get quick feedback on how they are doing before she moves on to new material.

She asks for the students’ attention. She reads each answer out loud, repeating it if requested, while each student marks up his or her own paper. At the end, she asks, “Do you want me to work through any of the problems on the board?”

Because she’s experienced, she really doesn’t need the students to tell her which problems to go over, but she would rather they do so. It’s one way she knows she’s developing rapport with the class, when students feel comfortable enough to raise a hand and ask out loud, “Can you please do #11?”. Continue reading