1. What is IBL?
A pedagogical strategy in which students are led to develop mathematical concepts and discover mathematical connections for themselves.
In an IBL class, we collectively develop the cognitive tools and structures necessary for individually making sense of the mathematical content.

2. Student Inquiry is Supported by:
(Carefully crafted) materials
Substantial class time is spent in discussing the mathematical insights of the students.
carefully crafted materials that engage “students in sense-making activities, tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate” mathematical ideas.
Substantial class time is spent in discussing the mathematical insights of the students.

3. Learning is more important than teaching
Axiom
Learning is more important than teaching
It is not what I do, but what happens to my students that is important.
I have found that when I can devise a good way to substitute something that the students do for something that I have previously tried to do for them, it is always a good idea.

4. Teacher as Amateur Cognitive Scientist
How do we get our students to think and behave like mathematicians?
Getting into our students’ heads:
How do they learn mathematics?
And, thinking cognitively, what do they need to learn mathematics?

5. Teacher as Amateur Cognitive Scientist
How do we operate mathematicians?
Getting into our own heads: This is perhaps the crucial question, but in some weird and subtle ways, it is harder for us than figuring out how our students think and learn.

6. Let’s try an experiment . . .
What does the B in Benoît B. Mandelbrot stand for?
Benoît B. Mandelbrot.
Nothing screams culture like inside jokes that only those immersed in the culture will understand.
Theorem
Mathematics is a Culture

7. Cultural Elements Presuppositions and assumptions Skills and practices
We know where to focus our attention and what can be safely ignored.
We hold presuppositions and assumptions that may not be shared by someone new to mathematical culture.
We have skills and practices that make it easier to function in our mathematical culture.

8. The Purpose of Proof
Our students (and most of the rest of the world!) think that the sole purpose of proof is to establish the truth of something. But sometimes proofs help us understand connections between mathematical ideas.

9. We know where to focus our attention and what can be safely ignored
We know where to focus our attention and what can be safely ignored. And it is sometimes difficult for us to really comprehend how difficult this is.

10. “The best way to learn is to do; the worst way to teach is to talk”
—Paul Halmos
David Bressoud writes an MAA column called Launchings. Two of his columns were titled “The Worst Way to Teach” and “The Best Way to Learn.”
It is said that “lecturing is that it is an efficient means of conveying a large amount of information, and it can be inspiring. Most instructors expect that students will do the mathematics back in their rooms as they go over the lecture notes. The unfortunate fact is that few students know how to engage mathematics on their own. As Uri Triesman famously observed and documented back in the 1970s, most mathematics students do not know what they do not know, nor how to begin to overcome this deficit. The most effective way for them to discover the gaps in their own knowledge is to discuss the mathematics with their peers who also are struggling to put the pieces together.” Research shows that “class time is better spent helping students learn how to engage the subject by making them active participants than in simply communicating information.”

11. It’s Very Effective! Sandra Laursen et. al (Univ. of Colorado) :
Large comparative study of IBL vs non-IBL courses
Several (large) institutions.
Courses at different levels.
---Courses at different levels from math for elementary teachers to upper level courses for math majors.

12. It’s Very Effective!
IBL courses produced better results than parallel non-IBL courses in
Both cognitive and affective areas.
Both within the semester of the IBL course and in later courses, taught by IBL or not.
The strongest gains were observed among women and students with prior weak achievement.
The percentage of time that the instructor spent on student centered activities was the best predictor of student gains.
IBL courses produced higher cognitive gains, including understanding of mathematical concepts and improved thinking and problem-solving skills; higher affective gains, including increased confidence, improved attitude, and greater persistence; and higher social gains, including ability to collaborate and explain mathematical ideas to others.
Laursen also found that the percentage of time that the instructor spent on student-centered activities was the single best predictor of student gains.
The strongest gains were observed among women and students with weak prior achievement. These gains appeared not just in the IBL class but also continued through subsequent required mathematics courses, whether or not they were taught using IBL. This happened without decreasing the achievement levels of men and students with strong prior achievement.